[ { "id": "Physics_496", "problem": "Relativistic particles obey the mass energy relation\n\n$$\nE^{2}=(p c)^{2}+\\left(m c^{2}\\right)^{2}\n$$\n\nwhere $E$ is the relativistic energy of the particle, $p$ is the relativistic momentum, $m$ is the mass, and $c$ is the speed of light.\n\nA proton with mass $m_{p}$ and energy $E_{p}$ collides head on with a photon which is massless and has energy $E_{b}$. The two combine and form a new particle with mass $m_{\\Delta}$ called $\\Delta$, or \"delta\". It is a one dimensional collision that conserves both relativistic energy and relativistic momentum.\n\nIn this case, the photon energy $E_{b}$ is that of the cosmic background radiation, which is an EM wave with wavelength $1.06 \\mathrm{~mm}$. Determine the energy of the photons, writing your answer in electron volts.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nRelativistic particles obey the mass energy relation\n\n$$\nE^{2}=(p c)^{2}+\\left(m c^{2}\\right)^{2}\n$$\n\nwhere $E$ is the relativistic energy of the particle, $p$ is the relativistic momentum, $m$ is the mass, and $c$ is the speed of light.\n\nA proton with mass $m_{p}$ and energy $E_{p}$ collides head on with a photon which is massless and has energy $E_{b}$. The two combine and form a new particle with mass $m_{\\Delta}$ called $\\Delta$, or \"delta\". It is a one dimensional collision that conserves both relativistic energy and relativistic momentum.\n\nIn this case, the photon energy $E_{b}$ is that of the cosmic background radiation, which is an EM wave with wavelength $1.06 \\mathrm{~mm}$. Determine the energy of the photons, writing your answer in electron volts.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{eV}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{eV}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1303", "problem": "如图 15a, 间距为 $L$ 的两根平行光滑金属导轨 $\\mathrm{MN} 、 \\mathrm{PQ}$ 放置于同一水平面内, 导轨左端连接一阻值为 $R$ 的定值电阻, 导体棒 $a$ 垂直于导轨放置在导轨上, 在 $a$棒左侧和导轨间存在坚直向下的匀强磁场, 磁感应强度大小为 $B$, 在 $a$ 棒右侧有一绝缘棒 $b, b$ 棒与 $a$ 棒平行, 且与固定在墙上的轻弹簧接触但不相连, 弹簧处于压缩状态且被锁定。现解除锁定, $b$ 棒在弹簧的作用下向左移动, 脱离弹簧后以速度 $v_{0}$ 与 $a$ 棒碰撞并粘在一起。已知 $a 、 b$ 棒的质量分别为 $m 、 M$, 碰撞前后, 两棒始终垂直于导轨, $a$ 棒在两导轨之间的部分的电阻为 $r$, 导轨电阻、接触电阻以及空气阻力均忽略不计, $a 、 b$ 棒总是保持与导轨接触良好。不计电路中感应电流的磁场, 求\n\n[图1]\n\n图 15a$a$ 棒从开始运动到停止的过程中产生的焦耳热 $Q$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图 15a, 间距为 $L$ 的两根平行光滑金属导轨 $\\mathrm{MN} 、 \\mathrm{PQ}$ 放置于同一水平面内, 导轨左端连接一阻值为 $R$ 的定值电阻, 导体棒 $a$ 垂直于导轨放置在导轨上, 在 $a$棒左侧和导轨间存在坚直向下的匀强磁场, 磁感应强度大小为 $B$, 在 $a$ 棒右侧有一绝缘棒 $b, b$ 棒与 $a$ 棒平行, 且与固定在墙上的轻弹簧接触但不相连, 弹簧处于压缩状态且被锁定。现解除锁定, $b$ 棒在弹簧的作用下向左移动, 脱离弹簧后以速度 $v_{0}$ 与 $a$ 棒碰撞并粘在一起。已知 $a 、 b$ 棒的质量分别为 $m 、 M$, 碰撞前后, 两棒始终垂直于导轨, $a$ 棒在两导轨之间的部分的电阻为 $r$, 导轨电阻、接触电阻以及空气阻力均忽略不计, $a 、 b$ 棒总是保持与导轨接触良好。不计电路中感应电流的磁场, 求\n\n[图1]\n\n图 15a\n\n问题:\n$a$ 棒从开始运动到停止的过程中产生的焦耳热 $Q$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-06.jpg?height=362&width=739&top_left_y=1161&top_left_x=1184", "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-15.jpg?height=60&width=506&top_left_y=484&top_left_x=244" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1535", "problem": "在固体材料中, 考虑相互作用后, 可以利用“准粒子”的概念研究材料的物理性质。准粒子的动能与动量之间的关系可能与真实粒子的不同。当外加电场或磁场时, 准粒子的运动往往可以用经典力学的方法来处理。在某种二维界面结构中, 存在电量为 $q$ 、有效质量为 $m$ 的准粒子, 它只能在 $x$ - $y$ 平面内运动, 其动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}+\\alpha p$,其中 $\\alpha$ 为正的常量。\n\n(1) 对于质量为 $m$ 的真实的自由粒子, 动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}$,试从动能定理出发，推导该粒子运动的速度 $v$ 与动量 $\\boldsymbol{p}$ 之间的关系式;仿照（1）的方法, 推导准粒子运动的速度 $v$ 与动量 $\\boldsymbol{p}$ 之间的关系式;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n在固体材料中, 考虑相互作用后, 可以利用“准粒子”的概念研究材料的物理性质。准粒子的动能与动量之间的关系可能与真实粒子的不同。当外加电场或磁场时, 准粒子的运动往往可以用经典力学的方法来处理。在某种二维界面结构中, 存在电量为 $q$ 、有效质量为 $m$ 的准粒子, 它只能在 $x$ - $y$ 平面内运动, 其动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}+\\alpha p$,其中 $\\alpha$ 为正的常量。\n\n(1) 对于质量为 $m$ 的真实的自由粒子, 动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}$,试从动能定理出发，推导该粒子运动的速度 $v$ 与动量 $\\boldsymbol{p}$ 之间的关系式;\n\n问题:\n仿照（1）的方法, 推导准粒子运动的速度 $v$ 与动量 $\\boldsymbol{p}$ 之间的关系式;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_714", "problem": "A person is moving away from two speakers $S$ and $S$ ' along the y direction as shown in the figure below. The speakers are in phase with each other and emit a sound wave with wavelength $6 \\mathrm{~cm}$. As the person moves away from speaker S', they mark the points on the y axis where there is no sound. Which of the plots below shows the position of the marked points correctly?\n\n[figure1]\n\na)\n\n[figure2]\n\nb)\n\n[figure3]\n\nc)\n\n[figure4]\n\nd)\n\n[figure5]\nA: A\nB: B\nC: C\nD: D\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA person is moving away from two speakers $S$ and $S$ ' along the y direction as shown in the figure below. The speakers are in phase with each other and emit a sound wave with wavelength $6 \\mathrm{~cm}$. As the person moves away from speaker S', they mark the points on the y axis where there is no sound. Which of the plots below shows the position of the marked points correctly?\n\n[figure1]\n\na)\n\n[figure2]\n\nb)\n\n[figure3]\n\nc)\n\n[figure4]\n\nd)\n\n[figure5]\n\nA: A\nB: B\nC: C\nD: D\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-07.jpg?height=301&width=468&top_left_y=457&top_left_x=1273", "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-07.jpg?height=94&width=591&top_left_y=842&top_left_x=1165", "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-07.jpg?height=92&width=634&top_left_y=968&top_left_x=1163", "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-07.jpg?height=92&width=623&top_left_y=1079&top_left_x=1160", "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-07.jpg?height=106&width=614&top_left_y=1188&top_left_x=1162" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_882", "problem": "A single electron transistor (SET) consists of a quantum dot, which is a small isolated conductor where electrons can be localised, and of several electrodes in its vicinity. The gate electrode couples capacitatively to the quantum dot, while the two other electrodes --- the source and the drain --- are connected via tunnel junctions, through which electrons can tunnel due to quantum mechanics. A simplified circuit diagram for an SET is shown in the figure.\n[figure1]\n\nCircuit diagram representation of an SET. QD is the quantum dot, $\\mathrm{S}$ is the source, $\\mathrm{D}$ is the drain and $\\mathrm{G}$ is the gate.\n\nThe capacitance of the gate is $C_{g}$ and the capacitance of the tunnel junctions is $C_{t} \\ll C_{g}$. Consider $C_{g}$ to be the total capacitance of the quantum dot. In this part of the problem, the source and the drain are held at zero potential, and the voltage on the gate electrode is fixed at $V_{g}$.\n\nConsider a state of the SET in which the quantum dot contains $n$ electrons.\n\nFind the electrical potential $\\varphi_{n}$ on the QD.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA single electron transistor (SET) consists of a quantum dot, which is a small isolated conductor where electrons can be localised, and of several electrodes in its vicinity. The gate electrode couples capacitatively to the quantum dot, while the two other electrodes --- the source and the drain --- are connected via tunnel junctions, through which electrons can tunnel due to quantum mechanics. A simplified circuit diagram for an SET is shown in the figure.\n[figure1]\n\nCircuit diagram representation of an SET. QD is the quantum dot, $\\mathrm{S}$ is the source, $\\mathrm{D}$ is the drain and $\\mathrm{G}$ is the gate.\n\nThe capacitance of the gate is $C_{g}$ and the capacitance of the tunnel junctions is $C_{t} \\ll C_{g}$. Consider $C_{g}$ to be the total capacitance of the quantum dot. In this part of the problem, the source and the drain are held at zero potential, and the voltage on the gate electrode is fixed at $V_{g}$.\n\nConsider a state of the SET in which the quantum dot contains $n$ electrons.\n\nFind the electrical potential $\\varphi_{n}$ on the QD.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-3.jpg?height=582&width=868&top_left_y=2076&top_left_x=594" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1484", "problem": "质量为 $M$ 的绝热薄壁容器处于远离其他星体的太空（可视为真空）中。在某惯性系中观察, 该容器的初始速度为零。容器的容积为 $V$, 容器中充有某种单原子分子理想气体, 气体的初始分子数、分子质量分别为 $N_{0} 、 m$, 气体的初始温度为 $T_{0} 。 t=0$ 时容器壁上出现面积为 $S$ 的一个小孔, 由于小孔漏气导致容器开始运动, 但容器没有转动。假设小孔较小, 容器中的气体在泄漏过程中始终处于平衡态。已知气体分子速度沿 $x$ 方向的分量 $v_{x}$ 的麦克斯韦分布函数为 $f\\left(v_{x}\\right)=\\sqrt{\\frac{m}{2 \\pi k T}} \\exp \\left(-\\frac{m v_{x}^{2}}{2 k T}\\right)$ ( $k$ 为玻尔兹曼常量)。在泄漏过程中, 求:\n\n已知积分公式: $\\int_{0}^{\\infty} x \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{2 A}, \\int_{0}^{\\infty} x^{2} \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{4} \\sqrt{\\frac{\\pi}{A^{3}}}, \\int_{0}^{\\infty} x^{3} \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{2 A^{2}}$$t$ 时刻容器运动速度的大小（假设 $M \\gg N_{0} m$ ）。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n质量为 $M$ 的绝热薄壁容器处于远离其他星体的太空（可视为真空）中。在某惯性系中观察, 该容器的初始速度为零。容器的容积为 $V$, 容器中充有某种单原子分子理想气体, 气体的初始分子数、分子质量分别为 $N_{0} 、 m$, 气体的初始温度为 $T_{0} 。 t=0$ 时容器壁上出现面积为 $S$ 的一个小孔, 由于小孔漏气导致容器开始运动, 但容器没有转动。假设小孔较小, 容器中的气体在泄漏过程中始终处于平衡态。已知气体分子速度沿 $x$ 方向的分量 $v_{x}$ 的麦克斯韦分布函数为 $f\\left(v_{x}\\right)=\\sqrt{\\frac{m}{2 \\pi k T}} \\exp \\left(-\\frac{m v_{x}^{2}}{2 k T}\\right)$ ( $k$ 为玻尔兹曼常量)。在泄漏过程中, 求:\n\n已知积分公式: $\\int_{0}^{\\infty} x \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{2 A}, \\int_{0}^{\\infty} x^{2} \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{4} \\sqrt{\\frac{\\pi}{A^{3}}}, \\int_{0}^{\\infty} x^{3} \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{2 A^{2}}$\n\n问题:\n$t$ 时刻容器运动速度的大小（假设 $M \\gg N_{0} m$ ）。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1371", "problem": "光电倍增管是用来将光信号转化为电信号并加以放大的装置，其结构如图 a 所示: 它主要由一个光阴极、 $n$ 个倍增级和一个阳极构成; 光阴极与第 1 倍增级、各相邻倍增级及第 $n$ 倍增级与阳极之间均有电势差 $V$; 从光阴极逸出的电子称为光电子，其中大部分（百分比 $\\eta$ ) 被收集到\n\n[图1]\n第 1 倍增级上, 余下的被直接收集到阳极上; 每个被收集到第 $i$ 倍增级 $(i=1, \\cdots, n)$ 的电子在该电极上又使得 $\\delta$ 个电子 $(\\delta>1)$ 逸出; 第 $i$倍增级上逸出的电子有大部分 (百分比 $\\sigma$ ) 被第 $i+1$ 倍增级收集, 其他被阳极收集; 直至所有电子被阳极收集, 实现信号放大。已知电子电荷量绝对值为 $e$ 。求光电倍增管放大一个光电子的平均能耗, 已知 $\\delta \\sigma>1, n>>1$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n光电倍增管是用来将光信号转化为电信号并加以放大的装置，其结构如图 a 所示: 它主要由一个光阴极、 $n$ 个倍增级和一个阳极构成; 光阴极与第 1 倍增级、各相邻倍增级及第 $n$ 倍增级与阳极之间均有电势差 $V$; 从光阴极逸出的电子称为光电子，其中大部分（百分比 $\\eta$ ) 被收集到\n\n[图1]\n第 1 倍增级上, 余下的被直接收集到阳极上; 每个被收集到第 $i$ 倍增级 $(i=1, \\cdots, n)$ 的电子在该电极上又使得 $\\delta$ 个电子 $(\\delta>1)$ 逸出; 第 $i$倍增级上逸出的电子有大部分 (百分比 $\\sigma$ ) 被第 $i+1$ 倍增级收集, 其他被阳极收集; 直至所有电子被阳极收集, 实现信号放大。已知电子电荷量绝对值为 $e$ 。\n\n问题:\n求光电倍增管放大一个光电子的平均能耗, 已知 $\\delta \\sigma>1, n>>1$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$e V$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不含任何单位和等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_49158ed36459a540f197g-03.jpg?height=440&width=845&top_left_y=1002&top_left_x=1028" ], "answer": null, "solution": null, "answer_type": "EX", "unit": [ "$e V$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_955", "problem": "Estimate the mass of the Earth.\nA: $10^{20} \\mathrm{~kg}$\nB: $10^{22} \\mathrm{~kg}$\nC: $10^{24} \\mathrm{~kg}$\nD: $10^{26} \\mathrm{~kg}$\nE: $10^{28} \\mathrm{~kg}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nEstimate the mass of the Earth.\n\nA: $10^{20} \\mathrm{~kg}$\nB: $10^{22} \\mathrm{~kg}$\nC: $10^{24} \\mathrm{~kg}$\nD: $10^{26} \\mathrm{~kg}$\nE: $10^{28} \\mathrm{~kg}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_423", "problem": "The figure shows a load of mass, $m$, supported by a simple pulley system with a tension $T$ in the cord. \n\n[figure1]\nFigure: Two light pulleys and a light cord.\n\nHow much work is done by the external agent pulling cord towards $\\mathbf{X}$?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe figure shows a load of mass, $m$, supported by a simple pulley system with a tension $T$ in the cord. \n\n[figure1]\nFigure: Two light pulleys and a light cord.\n\nHow much work is done by the external agent pulling cord towards $\\mathbf{X}$?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-4.jpg?height=497&width=391&top_left_y=742&top_left_x=838" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_328", "problem": "Consider points $A$ and $B$ separated by height $H$ in the vertical direction and distance $L$ in the horizontal direction, placed in a gravitational field $g$ as shown in the figure below. A point mass can slide along a rail of fixed shape frictionlessly (including taking $90^{\\circ}$-turns) from $A$ to $B$. The brachistochrone curve is the curve minimizing the total travel time.[figure1]\n\nThe obtained equation can explain mirages, which occur when the index of refraction increases with height. Consider light ray coming from the sky that graze the surface of the Earth $(y=0)$ and hits the eye of an observer at height $h$ (for this task choose the y-axis in the opposite direction bottom of the page to top). If the refractive index varies as $n(y)=n_{0}(1+\\alpha y)$ with $n_{0}$ and $\\alpha$ constant, find an expression for the apparent distance that the ray of light is emanating from $d$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider points $A$ and $B$ separated by height $H$ in the vertical direction and distance $L$ in the horizontal direction, placed in a gravitational field $g$ as shown in the figure below. A point mass can slide along a rail of fixed shape frictionlessly (including taking $90^{\\circ}$-turns) from $A$ to $B$. The brachistochrone curve is the curve minimizing the total travel time.[figure1]\n\nThe obtained equation can explain mirages, which occur when the index of refraction increases with height. Consider light ray coming from the sky that graze the surface of the Earth $(y=0)$ and hits the eye of an observer at height $h$ (for this task choose the y-axis in the opposite direction bottom of the page to top). If the refractive index varies as $n(y)=n_{0}(1+\\alpha y)$ with $n_{0}$ and $\\alpha$ constant, find an expression for the apparent distance that the ray of light is emanating from $d$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_706aca6df357b4c9a255g-2.jpg?height=336&width=664&top_left_y=436&top_left_x=56" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1110", "problem": "## To Commemorate the Centenary of Rutherford's Atomic Nucleus: the Scattering of an Ion by a Neutral Atom\n\n[figure1]\n\nAn ion of mass $m$, charge $Q$, is moving with an initial non-relativistic speed $v_{0}$ from a great distance towards the vicinity of a neutral atom of mass $M>>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nWhat is the electric potential energy of the ion-atom interaction in terms of $\\alpha, Q$ and $r$?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n## To Commemorate the Centenary of Rutherford's Atomic Nucleus: the Scattering of an Ion by a Neutral Atom\n\n[figure1]\n\nAn ion of mass $m$, charge $Q$, is moving with an initial non-relativistic speed $v_{0}$ from a great distance towards the vicinity of a neutral atom of mass $M>>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nWhat is the electric potential energy of the ion-atom interaction in terms of $\\alpha, Q$ and $r$?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_51c4dc0e7c52a1226310g-1.jpg?height=462&width=1495&top_left_y=701&top_left_x=315" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_214", "problem": "A man standing at $30^{\\circ}$ latitude fires a bullet northward at a speed of $200 \\mathrm{~m} / \\mathrm{s}$. The radius of the Earth is $6371 \\mathrm{~km}$. What is the sideway deflection of the bullet after traveling $100 \\mathrm{~m}$ ?\nA: $3.1 \\mathrm{~mm}$ west\nB: $1.8 \\mathrm{~mm}$ west\nC: $0 \\mathrm{~mm}$\nD: $1.8 \\mathrm{~mm}$ east \nE: $3.1 \\mathrm{~mm}$ east\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA man standing at $30^{\\circ}$ latitude fires a bullet northward at a speed of $200 \\mathrm{~m} / \\mathrm{s}$. The radius of the Earth is $6371 \\mathrm{~km}$. What is the sideway deflection of the bullet after traveling $100 \\mathrm{~m}$ ?\n\nA: $3.1 \\mathrm{~mm}$ west\nB: $1.8 \\mathrm{~mm}$ west\nC: $0 \\mathrm{~mm}$\nD: $1.8 \\mathrm{~mm}$ east \nE: $3.1 \\mathrm{~mm}$ east\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1638", "problem": "电子偶素原子 (Ps ) 是由电子 $\\mathrm{e}^{-}$与正电子 $\\mathrm{e}^{+}$(电子的反粒子, 其质量与电子的相同,电荷与电子的大小相等、符号相反) 组成的量子束缚体系, 其能级可类比氢原子能级得出。根据玻尔氢原子理论, 电子绕质子的圆周运动轨道角动量的取值是量子化的, 即为 $\\hbar$ 的整数倍。考虑到质子质量是有限的, 氢原子量子化条件应修正为: 电子与质子质心系中相对其质心的总轨道角动量取值为 $\\hbar$ 的整数倍。这一量子化条件可直接推广到其它两体束缚体系, 如电子偶素等。以下计算结果均保留四位有效数字。\n\n已知: 氢原子基态能量 $E_{n=1}^{\\mathrm{H}}=-13.60 \\mathrm{eV}$, 电子质量 $m_{\\mathrm{e}}=0.5110 \\mathrm{MeV} / c^{2}$, 质子与电子的质量之比为 1836 。写出电子偶素基态能量。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n电子偶素原子 (Ps ) 是由电子 $\\mathrm{e}^{-}$与正电子 $\\mathrm{e}^{+}$(电子的反粒子, 其质量与电子的相同,电荷与电子的大小相等、符号相反) 组成的量子束缚体系, 其能级可类比氢原子能级得出。根据玻尔氢原子理论, 电子绕质子的圆周运动轨道角动量的取值是量子化的, 即为 $\\hbar$ 的整数倍。考虑到质子质量是有限的, 氢原子量子化条件应修正为: 电子与质子质心系中相对其质心的总轨道角动量取值为 $\\hbar$ 的整数倍。这一量子化条件可直接推广到其它两体束缚体系, 如电子偶素等。以下计算结果均保留四位有效数字。\n\n已知: 氢原子基态能量 $E_{n=1}^{\\mathrm{H}}=-13.60 \\mathrm{eV}$, 电子质量 $m_{\\mathrm{e}}=0.5110 \\mathrm{MeV} / c^{2}$, 质子与电子的质量之比为 1836 。\n\n问题:\n写出电子偶素基态能量。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{eV}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{eV}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_147", "problem": "An introductory physics student, elated by a first semester grade, celebrates by dropping a textbook from a balcony into a deep layer of soft snow which is $3.00 \\mathrm{~m}$ below. Upon hitting the snow the book sinks a further $1.00 \\mathrm{~m}$ into it before coming to a stop. The mass of the book is $5.0 \\mathrm{~kg}$. Assuming a constant retarding force, what is the force from the snow on the book?\nA: $85 \\mathrm{~N}$\nB: $100 \\mathrm{~N}$\nC: $120 \\mathrm{~N}$\nD: $150 \\mathrm{~N}$\nE: $200 \\mathrm{~N} $ \n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn introductory physics student, elated by a first semester grade, celebrates by dropping a textbook from a balcony into a deep layer of soft snow which is $3.00 \\mathrm{~m}$ below. Upon hitting the snow the book sinks a further $1.00 \\mathrm{~m}$ into it before coming to a stop. The mass of the book is $5.0 \\mathrm{~kg}$. Assuming a constant retarding force, what is the force from the snow on the book?\n\nA: $85 \\mathrm{~N}$\nB: $100 \\mathrm{~N}$\nC: $120 \\mathrm{~N}$\nD: $150 \\mathrm{~N}$\nE: $200 \\mathrm{~N} $ \n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1677", "problem": "一很长、很细的圆柱形的电子束由速度为 $\\mathrm{v}$ 的匀速运动的低速电子组成, 电子在电子束中均匀分布, 沿电子束轴线每单位长度包含 $\\mathrm{n}$ 个电子, 每个电子的电荷量为 $-e(e>0)$, 质量为 $\\mathrm{m}$ 。该电子束从远处沿垂直于平行板电容器极板的方向射向电容器, 其前端 (即图中的右端) 于 $\\mathrm{t}=0$ 时刻刚好到达电容器的左极板。电容器的两个极板上各开一个小孔, 使电子束可以不受阻碍地穿过电容器。两极板 $\\mathrm{A} 、 \\mathrm{~B}$ 之间加上了如图所示的周期性变化的电压 $V_{A B}\\left(V_{A B}=V_{A}-V_{B}\\right.$, 图中只画出了一个周期的图线), 电压的最大值和最小值分别\n\n[图1]\n最小值的时间间隔为 $\\mathrm{T}-\\tau$ 。已知 $\\tau$ 的值恰好使在 $\\mathrm{V}_{\\mathrm{AB}}$ 变化的第一个周期内通过电容器到达电容器右边的所有的电子, 能在某一时刻 $\\mathrm{t}_{\\mathrm{b}}$ 形成均匀分布的一段电子束。设电容器两极板间\n[图2]\n的距离很小, 电子穿过电容器所需要的时间可以忽略, 且 $m v^{2}=6 e V_{0}$, 不计电子之间的相互作用及重力作用。满足题给条件的 $\\tau$ 和 $\\mathrm{t}_{\\mathrm{b}}$ 的值分别为 $\\tau=$ $T$, $\\mathrm{t}_{\\mathrm{b}}=$ T。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一很长、很细的圆柱形的电子束由速度为 $\\mathrm{v}$ 的匀速运动的低速电子组成, 电子在电子束中均匀分布, 沿电子束轴线每单位长度包含 $\\mathrm{n}$ 个电子, 每个电子的电荷量为 $-e(e>0)$, 质量为 $\\mathrm{m}$ 。该电子束从远处沿垂直于平行板电容器极板的方向射向电容器, 其前端 (即图中的右端) 于 $\\mathrm{t}=0$ 时刻刚好到达电容器的左极板。电容器的两个极板上各开一个小孔, 使电子束可以不受阻碍地穿过电容器。两极板 $\\mathrm{A} 、 \\mathrm{~B}$ 之间加上了如图所示的周期性变化的电压 $V_{A B}\\left(V_{A B}=V_{A}-V_{B}\\right.$, 图中只画出了一个周期的图线), 电压的最大值和最小值分别\n\n[图1]\n最小值的时间间隔为 $\\mathrm{T}-\\tau$ 。已知 $\\tau$ 的值恰好使在 $\\mathrm{V}_{\\mathrm{AB}}$ 变化的第一个周期内通过电容器到达电容器右边的所有的电子, 能在某一时刻 $\\mathrm{t}_{\\mathrm{b}}$ 形成均匀分布的一段电子束。设电容器两极板间\n[图2]\n的距离很小, 电子穿过电容器所需要的时间可以忽略, 且 $m v^{2}=6 e V_{0}$, 不计电子之间的相互作用及重力作用。\n\n问题:\n满足题给条件的 $\\tau$ 和 $\\mathrm{t}_{\\mathrm{b}}$ 的值分别为 $\\tau=$ $T$, $\\mathrm{t}_{\\mathrm{b}}=$ T。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[the value of $\\tau$, the value of $t_0$]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_40a5e1e69014b22d267bg-04.jpg?height=54&width=1345&top_left_y=761&top_left_x=201", "https://cdn.mathpix.com/cropped/2024_03_31_40a5e1e69014b22d267bg-04.jpg?height=286&width=950&top_left_y=899&top_left_x=583" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "the value of $\\tau$", "the value of $t_0$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1658", "problem": "如图, $1 \\mathrm{~mol}$ 单原子理想气体构成的系统分别经历循环过程 $a b c d a$ 和 $a b c^{\\prime} a$ 。已知理想气体在任一缓慢变化过程中, 压强 $p$ 和体积 $V$ 满足函数关系 $p=f(V)$ 。\n\n[图1]定量比较系统在两种循环过程的循环效率。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, $1 \\mathrm{~mol}$ 单原子理想气体构成的系统分别经历循环过程 $a b c d a$ 和 $a b c^{\\prime} a$ 。已知理想气体在任一缓慢变化过程中, 压强 $p$ 和体积 $V$ 满足函数关系 $p=f(V)$ 。\n\n[图1]\n\n问题:\n定量比较系统在两种循环过程的循环效率。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[循环过程的循环效率, 循环过程的循环效率]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e3a9fdbbef225ad3aefbg-03.jpg?height=982&width=785&top_left_y=1682&top_left_x=1092" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "循环过程的循环效率", "循环过程的循环效率" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1286", "problem": "如图, 一张紧的弦沿 $x$ 轴水平放置, 长度为 $L$ 。弦的左端位于坐标原点。弦可通过其左、右端与振源连接, 使弦产生沿 $y$ 方向的横向受迫振动, 振动传播的速度为 $u$ 。\n\n[图1]\n\n固定弦的右端 $P_{2}$, 将其左端 $P_{1}$ 与振源连接, 稳定时, 左端 $P_{1}$ 的振动表达式为 $y(x=0, t)=A_{0} \\cos (\\omega t)$, 其中 $A_{0}$ 为振幅, $\\omega$ 为圆频率。已知弦上横波的振幅在传播方向上有衰减, 衰减常量为 $\\gamma(\\gamma>0)$, 求弦上各处振动的振幅; (已知: 在无限长弦上沿 $x$ 轴正方向传播的振幅逐渐衰减的横波表达式为 $y(x, t)=A \\mathrm{e}^{-\\gamma x} \\cos \\left(\\omega t-\\frac{\\omega x}{u}+\\varphi\\right)$, 其中 $A$ 和 $\\varphi$ 分别为 $x=0$ 处振动的振幅和初相位。)", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一张紧的弦沿 $x$ 轴水平放置, 长度为 $L$ 。弦的左端位于坐标原点。弦可通过其左、右端与振源连接, 使弦产生沿 $y$ 方向的横向受迫振动, 振动传播的速度为 $u$ 。\n\n[图1]\n\n固定弦的右端 $P_{2}$, 将其左端 $P_{1}$ 与振源连接, 稳定时, 左端 $P_{1}$ 的振动表达式为 $y(x=0, t)=A_{0} \\cos (\\omega t)$, 其中 $A_{0}$ 为振幅, $\\omega$ 为圆频率。\n\n问题:\n已知弦上横波的振幅在传播方向上有衰减, 衰减常量为 $\\gamma(\\gamma>0)$, 求弦上各处振动的振幅; (已知: 在无限长弦上沿 $x$ 轴正方向传播的振幅逐渐衰减的横波表达式为 $y(x, t)=A \\mathrm{e}^{-\\gamma x} \\cos \\left(\\omega t-\\frac{\\omega x}{u}+\\varphi\\right)$, 其中 $A$ 和 $\\varphi$ 分别为 $x=0$ 处振动的振幅和初相位。)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_35bc41298eef336dfdafg-03.jpg?height=257&width=548&top_left_y=286&top_left_x=1231" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1563", "problem": "一个半径为 $r$ 、质量为 $m$ 的均质实心小圆柱被置于一个半径为 $R$ 、质量为 $M$ 的薄圆筒中，圆筒和小圆柱的中心轴均水平，横截面如图所示。重力加速度大小为 $g$ 。试在下述两种情形下,求小圆柱质心在其平衡位置附近做微振动的频率:圆筒可绕其固定的光滑中心细轴转动, 小圆柱仍在圆筒内底部附近作无滑滚动。\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一个半径为 $r$ 、质量为 $m$ 的均质实心小圆柱被置于一个半径为 $R$ 、质量为 $M$ 的薄圆筒中，圆筒和小圆柱的中心轴均水平，横截面如图所示。重力加速度大小为 $g$ 。试在下述两种情形下,求小圆柱质心在其平衡位置附近做微振动的频率:\n\n问题:\n圆筒可绕其固定的光滑中心细轴转动, 小圆柱仍在圆筒内底部附近作无滑滚动。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-01.jpg?height=411&width=417&top_left_y=434&top_left_x=1345" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_780", "problem": "Consider the following situations:\n\nI. You are standing in a lift and accelerating upwards\n\nII. You are standing in a lift and moving upwards at constant speed\n\nIII. You are standing in a lift which has experienced a cable failure, and is freely falling downwards\n\nIn which of these situations is your weight substantially different to when you are standing at rest on the surface of the Earth? Select one:\nA: None of the above\nB: I only\nC: I and III\nD: III only\nE: All of the above\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nConsider the following situations:\n\nI. You are standing in a lift and accelerating upwards\n\nII. You are standing in a lift and moving upwards at constant speed\n\nIII. You are standing in a lift which has experienced a cable failure, and is freely falling downwards\n\nIn which of these situations is your weight substantially different to when you are standing at rest on the surface of the Earth? Select one:\n\nA: None of the above\nB: I only\nC: I and III\nD: III only\nE: All of the above\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_244", "problem": "The characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part, analyze a situation when an angular momentum is not conserved. The system is the same as in the previous part with the only difference that the dipole is fixed and the charged small object with a mass $2 m$ is moving around the dipole. The electrostatic field of the dipole is easier to describe in the polar system of coordinates, which is defined with the distance $r$ from the center of the dipole, and angle $\\theta$ counted counterclockwise, as shown in Figure 3.\n\n[figure1]\n\nFigure 3: The system analyzed in Part 2. (Direction of the vector $\\mathbf{E}_{\\mathbf{n}}$ and $\\mathbf{E}_{\\mathbf{t}}$ could be wrong)\n\nCalculate the normal component of the velocity $v_{n 2}$ of the moving object.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part, analyze a situation when an angular momentum is not conserved. The system is the same as in the previous part with the only difference that the dipole is fixed and the charged small object with a mass $2 m$ is moving around the dipole. The electrostatic field of the dipole is easier to describe in the polar system of coordinates, which is defined with the distance $r$ from the center of the dipole, and angle $\\theta$ counted counterclockwise, as shown in Figure 3.\n\n[figure1]\n\nFigure 3: The system analyzed in Part 2. (Direction of the vector $\\mathbf{E}_{\\mathbf{n}}$ and $\\mathbf{E}_{\\mathbf{t}}$ could be wrong)\n\nCalculate the normal component of the velocity $v_{n 2}$ of the moving object.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ed4e92416bdbac30298dg-2.jpg?height=477&width=1442&top_left_y=1694&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_431", "problem": "This question explores issues around a commonplace domestic electrical device.\n\nModern sets of decorative lights, based on LED technology, are popular, efficient and safe. However, their forerunners had a poor reputation on safety grounds.\n\nFigure shows the principles of a set of 'fairy lights' used to decorate an indoor tree in, say, the 1960s. It consists of 20 identical $12 \\mathrm{~V}$ incandescent light bulbs (now illegal on energy grounds!) connected in series.\n\n[figure1]\n\nFigure: Decorative tree lights circuit of the 1960s.\n\nWhat is the p.d. between $\\mathbf{W}$ and $\\mathbf{X}$ when the system is functioning correctly?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThis question explores issues around a commonplace domestic electrical device.\n\nModern sets of decorative lights, based on LED technology, are popular, efficient and safe. However, their forerunners had a poor reputation on safety grounds.\n\nFigure shows the principles of a set of 'fairy lights' used to decorate an indoor tree in, say, the 1960s. It consists of 20 identical $12 \\mathrm{~V}$ incandescent light bulbs (now illegal on energy grounds!) connected in series.\n\n[figure1]\n\nFigure: Decorative tree lights circuit of the 1960s.\n\nWhat is the p.d. between $\\mathbf{W}$ and $\\mathbf{X}$ when the system is functioning correctly?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-6.jpg?height=403&width=1328&top_left_y=672&top_left_x=361" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_530", "problem": "A positive point charge $q$ is located inside a neutral hollow spherical conducting shell. The shell has inner radius $a$ and outer radius $b ; b-a$ is not negligible. The shell is centered on the origin.\n\n[figure1]\n\nAssume that the point charge $q$ is located at the origin in the very center of the shell.\n\nDetermine the magnitude of the electric field outside the conducting shell at $x=b$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA positive point charge $q$ is located inside a neutral hollow spherical conducting shell. The shell has inner radius $a$ and outer radius $b ; b-a$ is not negligible. The shell is centered on the origin.\n\n[figure1]\n\nAssume that the point charge $q$ is located at the origin in the very center of the shell.\n\nDetermine the magnitude of the electric field outside the conducting shell at $x=b$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0256c432f4019b26894dg-11.jpg?height=770&width=783&top_left_y=442&top_left_x=649" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_69", "problem": "A car is moving along a straight horizontal road at a speed of $20 \\mathrm{~m} / \\mathrm{s}$. The brakes are applied and a constant force of $5000 \\mathrm{~N}$ brings the car to a stop in $10 \\mathrm{~s}$. What is the mass of the car?\nA: $1250 \\mathrm{~kg}$\nB: $2500 \\mathrm{~kg}$\nC: $5000 \\mathrm{~kg}$\nD: $7500 \\mathrm{~kg}$\nE: $10,000 \\mathrm{~kg}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA car is moving along a straight horizontal road at a speed of $20 \\mathrm{~m} / \\mathrm{s}$. The brakes are applied and a constant force of $5000 \\mathrm{~N}$ brings the car to a stop in $10 \\mathrm{~s}$. What is the mass of the car?\n\nA: $1250 \\mathrm{~kg}$\nB: $2500 \\mathrm{~kg}$\nC: $5000 \\mathrm{~kg}$\nD: $7500 \\mathrm{~kg}$\nE: $10,000 \\mathrm{~kg}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1397", "problem": "潜艇从海水高密度区域驶入低密度区域, 浮力顿减, 潜艇如同汽车那样掉下悬崖, 称之为“掉深”, 曾有一些潜艇因此沉没。某潜艇总质量为 $3.0 \\times 10^{3} \\mathrm{t}$, 在高密度海水区域水下 $200 \\mathrm{~m}$ 沿水平方向缓慢潜航, 如图 12a 所示。当该潜艇驶入海水低密度区域时, 浮力突然降为 $2.4 \\times 10^{7} \\mathrm{~N}$; 10s 后，潜艇官兵迅速对潜艇减重（排水），此后潜艇以 $1.0 \\mathrm{~m} / \\mathrm{s}^{2}$ 的加速度匀减速下沉, 速度减为零后开始上浮, 到水\n\n下 $200 \\mathrm{~m}$ 处时立即对潜艇加重（加水）后使其缓慢匀减速上浮, 升到水面时速度恰好为零。取重力加速度\n\n为 $10 \\mathrm{~m} / \\mathrm{s}^{2}$, 不计潜艇加重和减重的时间和水的粘滞阻力。求\n\n[图1]\n\n图 12a潜艇“掉深”达到的最大深度（自海平面算起）;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n潜艇从海水高密度区域驶入低密度区域, 浮力顿减, 潜艇如同汽车那样掉下悬崖, 称之为“掉深”, 曾有一些潜艇因此沉没。某潜艇总质量为 $3.0 \\times 10^{3} \\mathrm{t}$, 在高密度海水区域水下 $200 \\mathrm{~m}$ 沿水平方向缓慢潜航, 如图 12a 所示。当该潜艇驶入海水低密度区域时, 浮力突然降为 $2.4 \\times 10^{7} \\mathrm{~N}$; 10s 后，潜艇官兵迅速对潜艇减重（排水），此后潜艇以 $1.0 \\mathrm{~m} / \\mathrm{s}^{2}$ 的加速度匀减速下沉, 速度减为零后开始上浮, 到水\n\n下 $200 \\mathrm{~m}$ 处时立即对潜艇加重（加水）后使其缓慢匀减速上浮, 升到水面时速度恰好为零。取重力加速度\n\n为 $10 \\mathrm{~m} / \\mathrm{s}^{2}$, 不计潜艇加重和减重的时间和水的粘滞阻力。求\n\n[图1]\n\n图 12a\n\n问题:\n潜艇“掉深”达到的最大深度（自海平面算起）;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~m}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-04.jpg?height=285&width=593&top_left_y=2496&top_left_x=1348" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_636", "problem": "A positive point charge $q$ is located inside a neutral hollow spherical conducting shell. The shell has inner radius $a$ and outer radius $b ; b-a$ is not negligible. The shell is centered on the origin.\n\n[figure1]\n\nAssume that the point charge $q$ is located at the origin in the very center of the shell.\n\nDetermine the electric potential at $x=a$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA positive point charge $q$ is located inside a neutral hollow spherical conducting shell. The shell has inner radius $a$ and outer radius $b ; b-a$ is not negligible. The shell is centered on the origin.\n\n[figure1]\n\nAssume that the point charge $q$ is located at the origin in the very center of the shell.\n\nDetermine the electric potential at $x=a$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0256c432f4019b26894dg-11.jpg?height=770&width=783&top_left_y=442&top_left_x=649" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1681", "problem": "我国 “玉兔号”月球车利用太阳能电池产生的电能进行驱动。月球车总质量为 $140 \\mathrm{~kg}$, 所安装的太阳能电池的电动势为 $45 \\mathrm{~V}$, 内阻为 $10 \\Omega$, 正常工作时电池的输出功率为 $45.0 \\mathrm{~W}$ 。月球车在某次正常工作时, 从静止出发沿直线行驶, 经过 $5.0 \\mathrm{~s}$ 后速度达到最大为 $0.50 \\mathrm{~m} / \\mathrm{s}$ 。假设此过程中月球车所受阻力恒定, 电池输出功率的 $80 \\%$ 转化为用于牵引月球车前进的机械功率。在此运动过程中,月球车所受阻力大小为 ___$\\mathrm{N}$, 前进的距离约为 ___ m。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n我国 “玉兔号”月球车利用太阳能电池产生的电能进行驱动。月球车总质量为 $140 \\mathrm{~kg}$, 所安装的太阳能电池的电动势为 $45 \\mathrm{~V}$, 内阻为 $10 \\Omega$, 正常工作时电池的输出功率为 $45.0 \\mathrm{~W}$ 。月球车在某次正常工作时, 从静止出发沿直线行驶, 经过 $5.0 \\mathrm{~s}$ 后速度达到最大为 $0.50 \\mathrm{~m} / \\mathrm{s}$ 。假设此过程中月球车所受阻力恒定, 电池输出功率的 $80 \\%$ 转化为用于牵引月球车前进的机械功率。在此运动过程中,月球车所受阻力大小为 ___$\\mathrm{N}$, 前进的距离约为 ___ m。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[在此运动过程中,月球车所受阻力大小, 在此运动过程中,月球车前进的距离]\n它们的单位依次是[$\\mathrm{N}$, $\\mathrm{m}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{N}$", "$\\mathrm{m}$" ], "answer_sequence": [ "在此运动过程中,月球车所受阻力大小", "在此运动过程中,月球车前进的距离" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_943", "problem": "A linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths $\\ell_{1}, \\ell_{2}, \\ell_{3}, \\ldots$ along the beam and with a fixed small separation $d$ between each tube. The tubes are connected to a high voltage, constant radio frequency $\\mathrm{AC}$ supply where the peak voltage of the $\\mathrm{AC}$ is $V_{0}$. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge $e$ and mass $m_{e}$ is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it \"drifts\" through the tube. It passes through a large potential difference between the tubes and, if the charged particle's motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10\n\n[figure1]\n\nFigure 9\n\n[figure2]\n\nFigure 10\n\nThe electron emerges from the end of the first tube with the same speed as it entered. What would be the speed of the electron as it leaves the second drift tube?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths $\\ell_{1}, \\ell_{2}, \\ell_{3}, \\ldots$ along the beam and with a fixed small separation $d$ between each tube. The tubes are connected to a high voltage, constant radio frequency $\\mathrm{AC}$ supply where the peak voltage of the $\\mathrm{AC}$ is $V_{0}$. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge $e$ and mass $m_{e}$ is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it \"drifts\" through the tube. It passes through a large potential difference between the tubes and, if the charged particle's motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10\n\n[figure1]\n\nFigure 9\n\n[figure2]\n\nFigure 10\n\nThe electron emerges from the end of the first tube with the same speed as it entered. What would be the speed of the electron as it leaves the second drift tube?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-10.jpg?height=300&width=454&top_left_y=818&top_left_x=296", "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-10.jpg?height=359&width=1128&top_left_y=757&top_left_x=795" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_365", "problem": "Eero Uustalu and Jaan Kalda. You are given the following tools: a retroreflective film an enlarged bottom view of which is shown in the first figure; stand, ruler, laser pointer, sreen, graph paper, protractor.\n\n[figure1]\n\nWhile the top surface of the film is flat, the bottom surface is a periodic array of slanted triangular faces. Six such faces are shown enlarged in the seond figure; the faces 1,3 a corner of a cube, and the faces 2, 4 nd 6 are also perpendicular to each other On the right of the second figure, a cross section of the film is shown. The film's material between the slanted faces and the flat surface form microprisms. The refracting angles of these microprisms are denoted by $\\alpha_{i}, i=1,2, \\ldots 6$ (the index numbers correspond to those of the faces). Among the an lges $\\alpha_{i}$, some may be equal to each other.\n\n[figure2]\n\ncross-section\n\nWhen light falls onto the flat surface close to perpendicular incidence, it undergoes total internal reflections on the slanted faces, and as a result, its direction of propagation is rotated by $180^{\\circ}$. However, the microprisms can also serve as prisms diverting a light beam by an angle $\\beta$. The angle $\\beta$ depends on the angle of incidence, and on the prism angle $\\alpha=\\alpha_{i}$. Let $\\beta_{i}$ denote the minimal deflection angle for a fixed prism angle $\\alpha_{i}$\n\nDetermine the minimal deflec tion angles $\\beta_{i}, i=1,2, \\ldots 6$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nEero Uustalu and Jaan Kalda. You are given the following tools: a retroreflective film an enlarged bottom view of which is shown in the first figure; stand, ruler, laser pointer, sreen, graph paper, protractor.\n\n[figure1]\n\nWhile the top surface of the film is flat, the bottom surface is a periodic array of slanted triangular faces. Six such faces are shown enlarged in the seond figure; the faces 1,3 a corner of a cube, and the faces 2, 4 nd 6 are also perpendicular to each other On the right of the second figure, a cross section of the film is shown. The film's material between the slanted faces and the flat surface form microprisms. The refracting angles of these microprisms are denoted by $\\alpha_{i}, i=1,2, \\ldots 6$ (the index numbers correspond to those of the faces). Among the an lges $\\alpha_{i}$, some may be equal to each other.\n\n[figure2]\n\ncross-section\n\nWhen light falls onto the flat surface close to perpendicular incidence, it undergoes total internal reflections on the slanted faces, and as a result, its direction of propagation is rotated by $180^{\\circ}$. However, the microprisms can also serve as prisms diverting a light beam by an angle $\\beta$. The angle $\\beta$ depends on the angle of incidence, and on the prism angle $\\alpha=\\alpha_{i}$. Let $\\beta_{i}$ denote the minimal deflection angle for a fixed prism angle $\\alpha_{i}$\n\nDetermine the minimal deflec tion angles $\\beta_{i}, i=1,2, \\ldots 6$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [$\\beta_{1}$, $\\beta_{2}$].\nTheir units are, in order, [^{\\circ}, ^{\\circ}], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_706aca6df357b4c9a255g-1.jpg?height=246&width=418&top_left_y=1593&top_left_x=1611", "https://cdn.mathpix.com/cropped/2024_03_06_706aca6df357b4c9a255g-1.jpg?height=164&width=373&top_left_y=448&top_left_x=2197" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "^{\\circ}", "^{\\circ}" ], "answer_sequence": [ "$\\beta_{1}$", "$\\beta_{2}$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_386", "problem": "The power radiated in gravitational waves by an orbiting binary system is given by $P\\left(r, m_{1}, m_{2}\\right)=\\frac{32}{5} \\frac{G^{4}}{c^{5}} \\frac{\\left(m_{1} m_{2}\\right)^{2}\\left(m_{1}+m_{2}\\right)}{r^{5}}$ where $r$ is the distance between the centers of the two orbiting masses $m_{1}$ and $m_{2}$. It is known that the most compact object is a black hole. The size of a black hole is defined by its Schwarzschild radius $r_{s}=\\frac{2 G m}{c^{2}}$, where $m$ is the mass of\n\nEstimate the upper limit of the power that can ever be emitted in gravitational waves by an orbiting binary system.\n\nThe gravitational wave detectors on Earth function by measuring the so called gravity wave strain $\\varepsilon(t)$ over time, which characterizes the deformation of spacetime. Data processing then yields the maximum strain $\\varepsilon$ and its corresponding wave frequency $f$. With the help of a theoretical spacetime model, the energy density $u$ associated with the wave can then be determined. We will use the analogy of linear elasticity to examine this model.\n\nDerive the energy density $u=$ $u(\\varepsilon, E)$ in a uniformly stretched elastic band in terms of the strain $\\varepsilon$ and the Elastic (Young's) modulus $E$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe power radiated in gravitational waves by an orbiting binary system is given by $P\\left(r, m_{1}, m_{2}\\right)=\\frac{32}{5} \\frac{G^{4}}{c^{5}} \\frac{\\left(m_{1} m_{2}\\right)^{2}\\left(m_{1}+m_{2}\\right)}{r^{5}}$ where $r$ is the distance between the centers of the two orbiting masses $m_{1}$ and $m_{2}$. It is known that the most compact object is a black hole. The size of a black hole is defined by its Schwarzschild radius $r_{s}=\\frac{2 G m}{c^{2}}$, where $m$ is the mass of\n\nEstimate the upper limit of the power that can ever be emitted in gravitational waves by an orbiting binary system.\n\nThe gravitational wave detectors on Earth function by measuring the so called gravity wave strain $\\varepsilon(t)$ over time, which characterizes the deformation of spacetime. Data processing then yields the maximum strain $\\varepsilon$ and its corresponding wave frequency $f$. With the help of a theoretical spacetime model, the energy density $u$ associated with the wave can then be determined. We will use the analogy of linear elasticity to examine this model.\n\nDerive the energy density $u=$ $u(\\varepsilon, E)$ in a uniformly stretched elastic band in terms of the strain $\\varepsilon$ and the Elastic (Young's) modulus $E$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of W, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "W" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_912", "problem": "The two-slit electron interference experiment was first performed by Möllenstedt et al, MerliMissiroli and Pozzi in 1974 and Tonomura et al in 1989. In the two-slit electron interference experiment, a monochromatic electron point source emits particles at $S$ that first passes through an electron \"biprism\" before impinging on an observational plane; $S_{1}$ and $S_{2}$ are virtual sources at distance $d$. In the diagram, the filament is pointing into the page. Note that it is a very thin filament (not drawn to scale in the diagram).\n\n[figure1]\n\nThe electron \"biprism\" consists of a grounded cylindrical wire mesh with a fine filament $F$ at the center. The distance between the source and the \"biprism\" is $\\ell$, and the distance between the distance between the \"biprism\" and the screen is $L$.\n\nBefore the point $S$, electrons are emitted from a field emission tip and accelerated through a potential $V_{0}$. Determine the wavelength of the electron in terms of the (rest) mass $m$, charge $e$ and $V_{0}$,\nassuming taking relativistic effects into consideration.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nThe two-slit electron interference experiment was first performed by Möllenstedt et al, MerliMissiroli and Pozzi in 1974 and Tonomura et al in 1989. In the two-slit electron interference experiment, a monochromatic electron point source emits particles at $S$ that first passes through an electron \"biprism\" before impinging on an observational plane; $S_{1}$ and $S_{2}$ are virtual sources at distance $d$. In the diagram, the filament is pointing into the page. Note that it is a very thin filament (not drawn to scale in the diagram).\n\n[figure1]\n\nThe electron \"biprism\" consists of a grounded cylindrical wire mesh with a fine filament $F$ at the center. The distance between the source and the \"biprism\" is $\\ell$, and the distance between the distance between the \"biprism\" and the screen is $L$.\n\nBefore the point $S$, electrons are emitted from a field emission tip and accelerated through a potential $V_{0}$. Determine the wavelength of the electron in terms of the (rest) mass $m$, charge $e$ and $V_{0}$,\nassuming taking relativistic effects into consideration.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_4f547aee877827e020bbg-1.jpg?height=1319&width=1091&top_left_y=1082&top_left_x=471" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_158", "problem": "A student makes an estimate of the acceleration due to gravity, $g$, by dropping a rock from a known height $h$ and measuring the time, $t$, it takes to hit the ground. Neglecting air resistance, which one of the following situations will lead to the smallest value of the relative uncertainty, $(\\Delta g) / g$, in the estimate?\nA: There is no uncertainty in $t$, and $h$ has a $10 \\%$ uncertainty. \nB: There is no uncertainty in $h$, and $t$ has a $10 \\%$ uncertainty.\nC: Both $t$ and $h$ have a $5 \\%$ uncertainty.\nD: (A) and (B) yield the same uncertainty, which is smaller than in (C).\nE: (A), (B), and (C) all yield the same uncertainty.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA student makes an estimate of the acceleration due to gravity, $g$, by dropping a rock from a known height $h$ and measuring the time, $t$, it takes to hit the ground. Neglecting air resistance, which one of the following situations will lead to the smallest value of the relative uncertainty, $(\\Delta g) / g$, in the estimate?\n\nA: There is no uncertainty in $t$, and $h$ has a $10 \\%$ uncertainty. \nB: There is no uncertainty in $h$, and $t$ has a $10 \\%$ uncertainty.\nC: Both $t$ and $h$ have a $5 \\%$ uncertainty.\nD: (A) and (B) yield the same uncertainty, which is smaller than in (C).\nE: (A), (B), and (C) all yield the same uncertainty.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_140", "problem": "A train, originally of mass $M$, is traveling on a frictionless straight horizontal track with constant speed $v$. Snow starts to fall vertically and sticks to the train at a rate of $\\rho$, where $\\rho$ has units of kilograms per second. The train's engine keeps the train moving at constant speed $v$ as snow accumulates on the train.\n\n7. The rate at which the kinetic energy of the train and snow increases is\nA: 0\nB: $M g v$\nC: $\\frac{1}{2} M v^{2}$\nD: $\\frac{1}{2} \\rho v^{2} $ \nE: $\\rho v^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA train, originally of mass $M$, is traveling on a frictionless straight horizontal track with constant speed $v$. Snow starts to fall vertically and sticks to the train at a rate of $\\rho$, where $\\rho$ has units of kilograms per second. The train's engine keeps the train moving at constant speed $v$ as snow accumulates on the train.\n\n7. The rate at which the kinetic energy of the train and snow increases is\n\nA: 0\nB: $M g v$\nC: $\\frac{1}{2} M v^{2}$\nD: $\\frac{1}{2} \\rho v^{2} $ \nE: $\\rho v^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_908", "problem": "When modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nA lumped element model of the cable is constructed by considering the inductance and capacitance of short sections of the cable. The inductance is assumed to be a property of the inner core, and the capacitance links the core with the shielding. A diagram of the lumped element model is shown below.\n\n[figure2]\n\nCircuit diagram of lumped element model of coaxial cable.\nShow that the impedance $Z_{0}$ of a semi-infinite length of cable is $Z_{0}=\\sqrt{L_{x} / C_{x}}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nWhen modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nA lumped element model of the cable is constructed by considering the inductance and capacitance of short sections of the cable. The inductance is assumed to be a property of the inner core, and the capacitance links the core with the shielding. A diagram of the lumped element model is shown below.\n\n[figure2]\n\nCircuit diagram of lumped element model of coaxial cable.\nShow that the impedance $Z_{0}$ of a semi-infinite length of cable is $Z_{0}=\\sqrt{L_{x} / C_{x}}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-1.jpg?height=358&width=844&top_left_y=1688&top_left_x=617", "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-2.jpg?height=260&width=834&top_left_y=1041&top_left_x=608" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_591", "problem": "A ship can be thought of as a symmetric arrangement of soft iron. In the presence of an external magnetic field, the soft iron will become magnetized, creating a second, weaker magnetic field. We want to examine the effect of the ship's field on the ship's compass, which will be located in the middle of the ship.\n\nLet the strength of the Earth's magnetic field near the ship be $B_{e}$, and the orientation of the field be horizontal, pointing directly toward true north.\n\nThe Earth's magnetic field $B_{e}$ will magnetize the ship, which will then create a second magnetic field $B_{s}$ in the vicinity of the ship's compass given by\n\n$$\n\\overrightarrow{\\mathbf{B}}_{s}=B_{e}\\left(-K_{b} \\cos \\theta \\hat{\\mathbf{b}}+K_{s} \\sin \\theta \\hat{\\mathbf{s}}\\right)\n$$\n\nwhere $K_{b}$ and $K_{s}$ are positive constants, $\\theta$ is the angle between the heading of the ship and magnetic north, measured clockwise, $\\hat{\\mathbf{b}}$ and $\\hat{\\mathbf{s}}$ are unit vectors pointing in the forward direction of the ship (bow) and directly right of the forward direction (starboard), respectively.\n\nBecause of the ship's magnetic field, the ship's compass will no longer necessarily point North.\n\nDerive an expression for the deviation of the compass, $\\delta \\theta$, from north as a function of $K_{b}$, $K_{s}$, and $\\theta$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA ship can be thought of as a symmetric arrangement of soft iron. In the presence of an external magnetic field, the soft iron will become magnetized, creating a second, weaker magnetic field. We want to examine the effect of the ship's field on the ship's compass, which will be located in the middle of the ship.\n\nLet the strength of the Earth's magnetic field near the ship be $B_{e}$, and the orientation of the field be horizontal, pointing directly toward true north.\n\nThe Earth's magnetic field $B_{e}$ will magnetize the ship, which will then create a second magnetic field $B_{s}$ in the vicinity of the ship's compass given by\n\n$$\n\\overrightarrow{\\mathbf{B}}_{s}=B_{e}\\left(-K_{b} \\cos \\theta \\hat{\\mathbf{b}}+K_{s} \\sin \\theta \\hat{\\mathbf{s}}\\right)\n$$\n\nwhere $K_{b}$ and $K_{s}$ are positive constants, $\\theta$ is the angle between the heading of the ship and magnetic north, measured clockwise, $\\hat{\\mathbf{b}}$ and $\\hat{\\mathbf{s}}$ are unit vectors pointing in the forward direction of the ship (bow) and directly right of the forward direction (starboard), respectively.\n\nBecause of the ship's magnetic field, the ship's compass will no longer necessarily point North.\n\nDerive an expression for the deviation of the compass, $\\delta \\theta$, from north as a function of $K_{b}$, $K_{s}$, and $\\theta$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_84", "problem": "Consider one horizontal circular path that a ball makes on the inside surface of a cone. In this situation, the normal force on the ball\nA: is mg.\nB: is always greater than $\\mathrm{mg}$.\nC: may be greater or less than $\\mathrm{mg}$.\nD: is always less than $\\mathrm{mg}$.\nE: depends upon the speed of the ball.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nConsider one horizontal circular path that a ball makes on the inside surface of a cone. In this situation, the normal force on the ball\n\nA: is mg.\nB: is always greater than $\\mathrm{mg}$.\nC: may be greater or less than $\\mathrm{mg}$.\nD: is always less than $\\mathrm{mg}$.\nE: depends upon the speed of the ball.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1337", "problem": "如图, 以 $a 、 b$ 为端点的线圈 1 的自感为 $L_{1}$, 以 $c 、 d$ 为端点的线圈 2 的自感为 $L_{2}$, 互感为 $M$ (线圈 1 中电流的变化在线圈 2 中产生的感应电动势与线圈 1 中\n[图1]\n电流随时间的变化率成正比, 比例系数称为互感 $M_{21}$; 且 $M_{12}=M_{21}=M$ )。若将两线圈 1 和 2 首尾相接 (顺接)而串联起来, 如图(a)所示, 则总自感为___ ; 若将两线圈 1 和 2 尾尾相接 (反接) 而串联起来, 如图(b)所示, 则总自感为___", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n如图, 以 $a 、 b$ 为端点的线圈 1 的自感为 $L_{1}$, 以 $c 、 d$ 为端点的线圈 2 的自感为 $L_{2}$, 互感为 $M$ (线圈 1 中电流的变化在线圈 2 中产生的感应电动势与线圈 1 中\n[图1]\n电流随时间的变化率成正比, 比例系数称为互感 $M_{21}$; 且 $M_{12}=M_{21}=M$ )。若将两线圈 1 和 2 首尾相接 (顺接)而串联起来, 如图(a)所示, 则总自感为___ ; 若将两线圈 1 和 2 尾尾相接 (反接) 而串联起来, 如图(b)所示, 则总自感为___\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[两线圈首尾相接的总自感, 两线圈尾尾相接的总自感]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_17b1131fe8d911867aa0g-03.jpg?height=308&width=996&top_left_y=266&top_left_x=858" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "两线圈首尾相接的总自感", "两线圈尾尾相接的总自感" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_819", "problem": "When modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nAt what speed do electromagnetic waves propagate in the co-axial cable?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nWhen modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nAt what speed do electromagnetic waves propagate in the co-axial cable?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-1.jpg?height=358&width=844&top_left_y=1688&top_left_x=617" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1574", "problem": "蹦极是年轻人喜爱的运动。为研究蹦极过程, 现将一长为 $L$ 、质量为 $m$ 、当仅受到绳本身重力时几乎不可伸长的均匀弹性绳的一端系在桥沿 $\\mathrm{b}$, 绳的另一端系一质量为 $M$ 的小物块（模拟蹦极者）; 假设 $M$ 比 $m$ 大很多, 以至于均匀弹性绳受到绳本身重力和蹦极者的重力向下拉时会显著伸长, 但仍在弹性限度内。在蹦极者从静止下落直至蹦极者到达最下端、但未向下拉紧绳之前的下落过程中, 不考虑水平运动和可能的能量损失。重力加速度大小为 $g$ 。求蹦极者从静止下落距离 $y(y2 \\pi R$, and a cylinder of radius $R$ is put through the loop. The coefficient of friction between the thread and the cylinder is $\\mu$. The free end of the thread is being pulled parallel to the axis of the cylinder (as shown by arrow in the photo) while keeping the cylinder at rest. If the length of the loop is longer than a critical value, $L>L_{0}$, the loop can slide along the cylinder without changing its shape, otherwise the friction \"locks\" it into a place and increasing the pulling force would eventually just break the thread. Find this critical value $L_{0}$. The weight of the thread is to be neglected; the thread will not twist when being pulled.\n\n[figure1]\n\nIt might be useful to know that:\n\n$$\n2 \\int \\sqrt{1+x^{2}} d x=x \\sqrt{1+x^{2}}+\\sinh ^{-1} x\n$$\n\nWhere:\n\n$$\n\\sinh ^{-1} x \\equiv \\ln \\left(x+\\sqrt{1+x^{2}}\\right)\n$$", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThread around a cylinder \n\nOne end of a thread is tied into a loop of length $L>2 \\pi R$, and a cylinder of radius $R$ is put through the loop. The coefficient of friction between the thread and the cylinder is $\\mu$. The free end of the thread is being pulled parallel to the axis of the cylinder (as shown by arrow in the photo) while keeping the cylinder at rest. If the length of the loop is longer than a critical value, $L>L_{0}$, the loop can slide along the cylinder without changing its shape, otherwise the friction \"locks\" it into a place and increasing the pulling force would eventually just break the thread. Find this critical value $L_{0}$. The weight of the thread is to be neglected; the thread will not twist when being pulled.\n\n[figure1]\n\nIt might be useful to know that:\n\n$$\n2 \\int \\sqrt{1+x^{2}} d x=x \\sqrt{1+x^{2}}+\\sinh ^{-1} x\n$$\n\nWhere:\n\n$$\n\\sinh ^{-1} x \\equiv \\ln \\left(x+\\sqrt{1+x^{2}}\\right)\n$$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8299572c51337db3fc92g-1.jpg?height=1005&width=1008&top_left_y=674&top_left_x=556" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_852", "problem": "In this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\n\nEstimate the SBH binary energy loss rate $d E / d t$. Estimate the radius variation rate $d a / d t$. Express it in terms of $a, \\rho, \\sigma, G$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\n\nEstimate the SBH binary energy loss rate $d E / d t$. Estimate the radius variation rate $d a / d t$. Express it in terms of $a, \\rho, \\sigma, G$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_148", "problem": "A $3.0 \\mathrm{~kg}$ mass moving at $30 \\mathrm{~m} / \\mathrm{s}$ to the right collides elastically with a $2.0 \\mathrm{~kg}$ mass traveling at $20 \\mathrm{~m} / \\mathrm{s}$ to the left. After the collision, the center of mass of the system is moving at a speed of\nA: $5 \\mathrm{~m} / \\mathrm{s}$\nB: $10 \\mathrm{~m} / \\mathrm{s} $ \nC: $20 \\mathrm{~m} / \\mathrm{s}$\nD: $24 \\mathrm{~m} / \\mathrm{s}$\nE: $26 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA $3.0 \\mathrm{~kg}$ mass moving at $30 \\mathrm{~m} / \\mathrm{s}$ to the right collides elastically with a $2.0 \\mathrm{~kg}$ mass traveling at $20 \\mathrm{~m} / \\mathrm{s}$ to the left. After the collision, the center of mass of the system is moving at a speed of\n\nA: $5 \\mathrm{~m} / \\mathrm{s}$\nB: $10 \\mathrm{~m} / \\mathrm{s} $ \nC: $20 \\mathrm{~m} / \\mathrm{s}$\nD: $24 \\mathrm{~m} / \\mathrm{s}$\nE: $26 \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1432", "problem": "假设对氨原子基态采用玻尔模型, 认为每个电子都在以氦核为中心的圆周上运动, 半径相同, 角动量为 $\\hbar, \\hbar=\\frac{h}{2 \\pi}$, 其中 $h$ 是普朗克常数。如果忽略电子问的相互作用, 璌原子的一级电离能是多少 $\\mathrm{eV}$ ?一级电离能是指把把其中的一个电子移到无限远所需要的能量。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n假设对氨原子基态采用玻尔模型, 认为每个电子都在以氦核为中心的圆周上运动, 半径相同, 角动量为 $\\hbar, \\hbar=\\frac{h}{2 \\pi}$, 其中 $h$ 是普朗克常数。\n\n问题:\n如果忽略电子问的相互作用, 璌原子的一级电离能是多少 $\\mathrm{eV}$ ?一级电离能是指把把其中的一个电子移到无限远所需要的能量。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以eV为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "eV" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_347", "problem": "The black box has three terminal wires: \"blue\", \"black\" and \"white\", and contains n a star configuration: a battery, a capacitor, an inductor in series with a diode. You may consider the diode to be \"ideal\" - it conducts current perfectly one way and not at all the other way. You may neglect internal resistance of the battery and capacitor, but the inductor has considerable internal resistance. The multimeter's internal resistance when measuring voltages is $R_{m}=10 \\mathrm{M} \\Omega$ and it displays a new reading every $t=0.4 \\mathrm{~s}$.\n\n Determine the electromotive force of the battery", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe black box has three terminal wires: \"blue\", \"black\" and \"white\", and contains n a star configuration: a battery, a capacitor, an inductor in series with a diode. You may consider the diode to be \"ideal\" - it conducts current perfectly one way and not at all the other way. You may neglect internal resistance of the battery and capacitor, but the inductor has considerable internal resistance. The multimeter's internal resistance when measuring voltages is $R_{m}=10 \\mathrm{M} \\Omega$ and it displays a new reading every $t=0.4 \\mathrm{~s}$.\n\n Determine the electromotive force of the battery\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of V, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "V" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1115", "problem": "Let us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nFor radii smaller than $r_{3}$ you may neglect heat loss due to radiation. Determine how the temperature $T$ of the ball depends on its radius for $r\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nFor radii smaller than $r_{3}$ you may neglect heat loss due to radiation. Determine how the temperature $T$ of the ball depends on its radius for $r1)$ 逸出; 第 $i$倍增级上逸出的电子有大部分 (百分比 $\\sigma$ ) 被第 $i+1$ 倍增级收集, 其他被阳极收集; 直至所有电子被阳极收集, 实现信号放大。已知电子电荷量绝对值为 $e$ 。为使尽可能多的电子从第 $i$ 倍增级直接到达第 $i+1$ 倍增级而非阳极, 早期的光电倍增管中,会施加垂直于电子运行轨迹所在平面（纸面）的匀强磁场。设倍增级的长度为 $a$ 且相邻倍增级间的几何位置如图 b 所示，倍增级间电势差引起的电场很小可忽略。所施加的匀强磁场应取什么方向以及磁感应强度大小为多少时, 才能使从第 $i$ 倍增级垂直出射的能量为 $E_{\\mathrm{e}}$ 的电子中直接打到第 $i+1$ 倍增级的电子最多? 磁感应强度大小为多少时, 可以保证在第 $i+1$ 倍增级上至少收集到一些从第 $i$ 倍增级垂直出\n\n[图2]\n\n图 b 射的能量为 $E_{\\mathrm{e}}$ 的电子?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n光电倍增管是用来将光信号转化为电信号并加以放大的装置，其结构如图 a 所示: 它主要由一个光阴极、 $n$ 个倍增级和一个阳极构成; 光阴极与第 1 倍增级、各相邻倍增级及第 $n$ 倍增级与阳极之间均有电势差 $V$; 从光阴极逸出的电子称为光电子，其中大部分（百分比 $\\eta$ ) 被收集到\n\n[图1]\n第 1 倍增级上, 余下的被直接收集到阳极上; 每个被收集到第 $i$ 倍增级 $(i=1, \\cdots, n)$ 的电子在该电极上又使得 $\\delta$ 个电子 $(\\delta>1)$ 逸出; 第 $i$倍增级上逸出的电子有大部分 (百分比 $\\sigma$ ) 被第 $i+1$ 倍增级收集, 其他被阳极收集; 直至所有电子被阳极收集, 实现信号放大。已知电子电荷量绝对值为 $e$ 。\n\n问题:\n为使尽可能多的电子从第 $i$ 倍增级直接到达第 $i+1$ 倍增级而非阳极, 早期的光电倍增管中,会施加垂直于电子运行轨迹所在平面（纸面）的匀强磁场。设倍增级的长度为 $a$ 且相邻倍增级间的几何位置如图 b 所示，倍增级间电势差引起的电场很小可忽略。所施加的匀强磁场应取什么方向以及磁感应强度大小为多少时, 才能使从第 $i$ 倍增级垂直出射的能量为 $E_{\\mathrm{e}}$ 的电子中直接打到第 $i+1$ 倍增级的电子最多? 磁感应强度大小为多少时, 可以保证在第 $i+1$ 倍增级上至少收集到一些从第 $i$ 倍增级垂直出\n\n[图2]\n\n图 b 射的能量为 $E_{\\mathrm{e}}$ 的电子?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[保证在第 $i+1$ 倍增级上至少收集到一些从第 $i$ 倍增级垂直出射的能量为 $E_{\\mathrm{e}}$ 的电子的磁感应强度大小, 使从第 $i$ 倍增级垂直出射的能量为 $E_{\\mathrm{e}}$ 的电子中直接打到第 $i+1$ 倍增级的电子最多的匀强磁场方向和磁感应强度]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_49158ed36459a540f197g-03.jpg?height=440&width=845&top_left_y=1002&top_left_x=1028", "https://cdn.mathpix.com/cropped/2024_03_31_49158ed36459a540f197g-03.jpg?height=468&width=856&top_left_y=1962&top_left_x=977" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "保证在第 $i+1$ 倍增级上至少收集到一些从第 $i$ 倍增级垂直出射的能量为 $E_{\\mathrm{e}}$ 的电子的磁感应强度大小", "使从第 $i$ 倍增级垂直出射的能量为 $E_{\\mathrm{e}}$ 的电子中直接打到第 $i+1$ 倍增级的电子最多的匀强磁场方向和磁感应强度" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1620", "problem": "1958 年穆斯堡尔发现的原子核无反冲共振吸收效应（即穆斯堡尔效应）可用于测量光子频率极微小的变化, 穆斯堡尔因此荣获 1961 年诺贝尔物理学奖。类似于原子的能级结构,原子核也具有分立的能级, 并能通过吸收或放出光子在能级间跃迁。原子核在吸收和放出光子时会有反冲, 部分能量转化为原子核的动能（即反冲能）。此外, 原子核的激发态相对于其基态的能量差并不是一个确定值, 而是在以 $E_{0}$ 为中心、宽度为 $2 \\Gamma$ 的范围内取值的。对于 ${ }^{57} \\mathrm{Fe}$ 从第一激发态到基态的跃迁, $E_{0}=2.31 \\times 10^{-15} \\mathrm{~J}, \\Gamma \\simeq 3.2 \\times 10^{-13} E_{0}$ 。已知质量 $m_{\\mathrm{Fe}} \\simeq 9.5 \\times 10^{-26} \\mathrm{~kg}$, 普朗克常量 $h=6.6 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 真空中的光速 $c=3.0 \\times 10^{8} \\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ 。忽略激发态的能级宽度, 求反冲能, 以及在考虑核反冲和不考虑核反冲的情形下, ${ }^{57} \\mathrm{Fe}$从基态跃迁到激发态吸收的光子的频率之差;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n1958 年穆斯堡尔发现的原子核无反冲共振吸收效应（即穆斯堡尔效应）可用于测量光子频率极微小的变化, 穆斯堡尔因此荣获 1961 年诺贝尔物理学奖。类似于原子的能级结构,原子核也具有分立的能级, 并能通过吸收或放出光子在能级间跃迁。原子核在吸收和放出光子时会有反冲, 部分能量转化为原子核的动能（即反冲能）。此外, 原子核的激发态相对于其基态的能量差并不是一个确定值, 而是在以 $E_{0}$ 为中心、宽度为 $2 \\Gamma$ 的范围内取值的。对于 ${ }^{57} \\mathrm{Fe}$ 从第一激发态到基态的跃迁, $E_{0}=2.31 \\times 10^{-15} \\mathrm{~J}, \\Gamma \\simeq 3.2 \\times 10^{-13} E_{0}$ 。已知质量 $m_{\\mathrm{Fe}} \\simeq 9.5 \\times 10^{-26} \\mathrm{~kg}$, 普朗克常量 $h=6.6 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 真空中的光速 $c=3.0 \\times 10^{8} \\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ 。\n\n问题:\n忽略激发态的能级宽度, 求反冲能, 以及在考虑核反冲和不考虑核反冲的情形下, ${ }^{57} \\mathrm{Fe}$从基态跃迁到激发态吸收的光子的频率之差;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~J}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~J}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_375", "problem": "The power radiated in gravitational waves by an orbiting binary system is given by $P\\left(r, m_{1}, m_{2}\\right)=\\frac{32}{5} \\frac{G^{4}}{c^{5}} \\frac{\\left(m_{1} m_{2}\\right)^{2}\\left(m_{1}+m_{2}\\right)}{r^{5}}$ where $r$ is the distance between the centers of the two orbiting masses $m_{1}$ and $m_{2}$. It is known that the most compact object is a black hole. The size of a black hole is defined by its Schwarzschild radius $r_{s}=\\frac{2 G m}{c^{2}}$, where $m$ is the mass of\n\nUse dimensional analysis to estimate the frequency-dependent elastic modulus of spacetime $E(f)$ in terms of the universal gravitational constant $G$, speed of light $c$ and gravitational wave frequency $f$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe power radiated in gravitational waves by an orbiting binary system is given by $P\\left(r, m_{1}, m_{2}\\right)=\\frac{32}{5} \\frac{G^{4}}{c^{5}} \\frac{\\left(m_{1} m_{2}\\right)^{2}\\left(m_{1}+m_{2}\\right)}{r^{5}}$ where $r$ is the distance between the centers of the two orbiting masses $m_{1}$ and $m_{2}$. It is known that the most compact object is a black hole. The size of a black hole is defined by its Schwarzschild radius $r_{s}=\\frac{2 G m}{c^{2}}$, where $m$ is the mass of\n\nUse dimensional analysis to estimate the frequency-dependent elastic modulus of spacetime $E(f)$ in terms of the universal gravitational constant $G$, speed of light $c$ and gravitational wave frequency $f$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_89", "problem": "A triangular glass prism $(n=1.6)$ is immersed in a liquid ( $\\mathrm{n}=1.1$ ) as shown. A laser light is incident as shown on face $A B$ making an angle of $20^{\\circ}$ with the normal $(N)$. Calculate the angle that the ray emerging from $A C$ makes with the ground when it leaves $A C$ and strikes the ground.\n\n[figure1]\nA: $28.1^{\\circ}$\nB: $30.3^{\\circ}$\nC: $33.8^{\\circ}$\nD: $36.1^{\\circ}$\nE: $18.9^{\\circ}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA triangular glass prism $(n=1.6)$ is immersed in a liquid ( $\\mathrm{n}=1.1$ ) as shown. A laser light is incident as shown on face $A B$ making an angle of $20^{\\circ}$ with the normal $(N)$. Calculate the angle that the ray emerging from $A C$ makes with the ground when it leaves $A C$ and strikes the ground.\n\n[figure1]\n\nA: $28.1^{\\circ}$\nB: $30.3^{\\circ}$\nC: $33.8^{\\circ}$\nD: $36.1^{\\circ}$\nE: $18.9^{\\circ}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_bd21c863b16c0f40a895g-09.jpg?height=382&width=655&top_left_y=1476&top_left_x=1250" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_795", "problem": "Alice wishes to measure the acceleration due to gravity for a falling object. However, unfortunately the only object she has on hand to drop is a tissue box. She wants to minimise the effect of air resistance, which creates a force in the opposite direction to an object's motion and therefore slows down the falling object. She films the object so error due to reaction time is negligible. Which of the following experimental set-ups is optimal?\nA: Drop an empty tissue box from a low height, around 1 metre, and time how long it takes to fall.\nB: Drop an empty tissue box from a higher height, around 10 metres, and time how long it takes to fall.\nC: Drop a full tissue box from a low height, around 1 metre, and time how long it takes to fall.\nD: Drop a full tissue box from a higher height, around 10 metres, and time how long it takes to fall.\nE: All the above methods will give the same result\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAlice wishes to measure the acceleration due to gravity for a falling object. However, unfortunately the only object she has on hand to drop is a tissue box. She wants to minimise the effect of air resistance, which creates a force in the opposite direction to an object's motion and therefore slows down the falling object. She films the object so error due to reaction time is negligible. Which of the following experimental set-ups is optimal?\n\nA: Drop an empty tissue box from a low height, around 1 metre, and time how long it takes to fall.\nB: Drop an empty tissue box from a higher height, around 10 metres, and time how long it takes to fall.\nC: Drop a full tissue box from a low height, around 1 metre, and time how long it takes to fall.\nD: Drop a full tissue box from a higher height, around 10 metres, and time how long it takes to fall.\nE: All the above methods will give the same result\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_246", "problem": "The characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part the peculiarity of the circular motion around the dipole is analyzed. Initially, the system is the same as in Part 2, with the exception that the charged object is connected to the center of the dipole with a light, rigid insulating rod with the length $L$. This rod easily rotates around the axis, which is perpendicular to the surface of the table. Thus, the charged object moves around the dipole along circular trajectory with radius $L$.\n\nWith what initial velocity $u_{c}$ should the charged object be launched, so that it will move around the dipole along a circular trajectory, even without the rigid rod?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a True or False question.\n\nproblem:\nThe characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part the peculiarity of the circular motion around the dipole is analyzed. Initially, the system is the same as in Part 2, with the exception that the charged object is connected to the center of the dipole with a light, rigid insulating rod with the length $L$. This rod easily rotates around the axis, which is perpendicular to the surface of the table. Thus, the charged object moves around the dipole along circular trajectory with radius $L$.\n\nWith what initial velocity $u_{c}$ should the charged object be launched, so that it will move around the dipole along a circular trajectory, even without the rigid rod?\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be either \"True\" or \"False\".", "figure_urls": null, "answer": null, "solution": null, "answer_type": "TF", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_864", "problem": "Most of the propellant is used during the ascent, however, after the payload has been separated from the stage, there still remains some fuel in its tank. Mass $m$ of residual fuel is negligible in comparison to the stage's mass $M$. Sloshing of the liquid fuel and viscous friction forces in the fuel tank result in energy losses, and after a transient process of irregular dynamics the energy reaches its minimum.\nFind the value $\\theta_{2}$ of angle $\\theta$ after the transient process for arbitrary initial values of $L$ and $\\theta(0)=\\theta_{1} \\in(0, \\pi / 2)$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nMost of the propellant is used during the ascent, however, after the payload has been separated from the stage, there still remains some fuel in its tank. Mass $m$ of residual fuel is negligible in comparison to the stage's mass $M$. Sloshing of the liquid fuel and viscous friction forces in the fuel tank result in energy losses, and after a transient process of irregular dynamics the energy reaches its minimum.\nFind the value $\\theta_{2}$ of angle $\\theta$ after the transient process for arbitrary initial values of $L$ and $\\theta(0)=\\theta_{1} \\in(0, \\pi / 2)$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1678", "problem": "假定月球绕地球作圆周运动, 地球绕太阳也作圆周运动, 且轨道都在同一平面内. 已知地球表面处的重力加速度 $g=9.80 \\mathrm{~m} / \\mathrm{s}^{2}$, 地球半径 $R_{\\mathrm{e}}=6.37 \\times 10^{6} \\mathrm{~m}$, 月球质量 $m_{\\mathrm{m}}=7.3 \\times 10^{22} \\mathrm{~kg}$, 月球半径 $R_{\\mathrm{m}}=1.7 \\times 10^{6} \\mathrm{~m}$, 引力恒量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} \\cdot \\mathrm{kg}^{-2}$, 月心地心间的距离约为 $r_{\\mathrm{em}}=3.84 \\times 10^{8} \\mathrm{~m}$.若忽略月球绕地球的运动, 设想从地球表面发射一枚火箭直接射向月球, 试估算火箭到达月球表面时的速度至少为多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n假定月球绕地球作圆周运动, 地球绕太阳也作圆周运动, 且轨道都在同一平面内. 已知地球表面处的重力加速度 $g=9.80 \\mathrm{~m} / \\mathrm{s}^{2}$, 地球半径 $R_{\\mathrm{e}}=6.37 \\times 10^{6} \\mathrm{~m}$, 月球质量 $m_{\\mathrm{m}}=7.3 \\times 10^{22} \\mathrm{~kg}$, 月球半径 $R_{\\mathrm{m}}=1.7 \\times 10^{6} \\mathrm{~m}$, 引力恒量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} \\cdot \\mathrm{kg}^{-2}$, 月心地心间的距离约为 $r_{\\mathrm{em}}=3.84 \\times 10^{8} \\mathrm{~m}$.\n\n问题:\n若忽略月球绕地球的运动, 设想从地球表面发射一枚火箭直接射向月球, 试估算火箭到达月球表面时的速度至少为多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$$ mathrm{~m} \\cdot \\mathrm{s}^{-1} $$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$$ mathrm{~m} \\cdot \\mathrm{s}^{-1} $$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_939", "problem": "A spherical shaped party balloon can be filled by blowing air into it. We observe that it is difficult to start the balloon expanding, but it becomes easier once the rubber has stretched a little. It is easier to inflate the balloon a second time. This behaviour is illustrated by the graph of Fig. 9 and is described by the equation,\n\n$$\nP_{\\text {in }}-P_{\\text {out }}=\\frac{C}{r_{0}^{2} r}\\left[1-\\left(\\frac{r_{0}}{r}\\right)^{6}\\right]\n$$\n\nwhere $P_{\\text {in }}$ is the pressure inside the balloon, $P_{\\text {out }}$ is the external atmospheric pressure, $r_{0}$ is the uninflated radius of the balloon, $r$ is the radius of the balloon, and $C$ is a constant.\n\n\n From equation (1) above, what is the relation between the uninflated radius $r_{0}$ and the radius at maximum pressure $r_{\\mathrm{p}}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA spherical shaped party balloon can be filled by blowing air into it. We observe that it is difficult to start the balloon expanding, but it becomes easier once the rubber has stretched a little. It is easier to inflate the balloon a second time. This behaviour is illustrated by the graph of Fig. 9 and is described by the equation,\n\n$$\nP_{\\text {in }}-P_{\\text {out }}=\\frac{C}{r_{0}^{2} r}\\left[1-\\left(\\frac{r_{0}}{r}\\right)^{6}\\right]\n$$\n\nwhere $P_{\\text {in }}$ is the pressure inside the balloon, $P_{\\text {out }}$ is the external atmospheric pressure, $r_{0}$ is the uninflated radius of the balloon, $r$ is the radius of the balloon, and $C$ is a constant.\n\n\n From equation (1) above, what is the relation between the uninflated radius $r_{0}$ and the radius at maximum pressure $r_{\\mathrm{p}}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_249", "problem": "Thermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nExobase is defined as the height above which a radially outward moving particle will suffer less than one backscattering collision on average. This means that the mean free path has to be equal to the scale height, which is defined as the height where the atmospheres density is $\\frac{1}{e}$ lower than on Earths surface $\\left(R_{E}=6.37 \\times 10^{6} \\mathrm{~m}\\right)$. The mean free path $\\lambda$ is the average distance covered by a moving particle in a gas (that we consider to be ideal) between two consecutive collisions and this can be expressed by the following equality:\n\n$$\n\\lambda(h)=\\frac{1}{\\sigma n_{V}(h)}\n$$\n\nwhere $\\sigma$ is the effective cross sectional area for the collision hydrogen atom-atmosphere $\\sigma=2 \\times$ $10^{-19} m^{2}$ and $n_{V}$ is the number of molecules per unit volume. Atmospheres density decreases with exponentially from altitude $250 \\mathrm{~km}$ :\n\n$$\nP(h)=P_{R e f} \\exp \\left(-\\frac{\\left(h-h_{R e f}\\right)}{H}\\right),\n$$\n\nwhere we know that at an altitude of $250 \\mathrm{~km}$, the pressure is $21 \\mu \\mathrm{Pa} . H$ is the scale height, and its value is $H=60 \\mathrm{~km}$.\n\nDetermine the air particles mean free path $\\lambda$ at the altitude of $250 \\mathrm{~km}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nExobase is defined as the height above which a radially outward moving particle will suffer less than one backscattering collision on average. This means that the mean free path has to be equal to the scale height, which is defined as the height where the atmospheres density is $\\frac{1}{e}$ lower than on Earths surface $\\left(R_{E}=6.37 \\times 10^{6} \\mathrm{~m}\\right)$. The mean free path $\\lambda$ is the average distance covered by a moving particle in a gas (that we consider to be ideal) between two consecutive collisions and this can be expressed by the following equality:\n\n$$\n\\lambda(h)=\\frac{1}{\\sigma n_{V}(h)}\n$$\n\nwhere $\\sigma$ is the effective cross sectional area for the collision hydrogen atom-atmosphere $\\sigma=2 \\times$ $10^{-19} m^{2}$ and $n_{V}$ is the number of molecules per unit volume. Atmospheres density decreases with exponentially from altitude $250 \\mathrm{~km}$ :\n\n$$\nP(h)=P_{R e f} \\exp \\left(-\\frac{\\left(h-h_{R e f}\\right)}{H}\\right),\n$$\n\nwhere we know that at an altitude of $250 \\mathrm{~km}$, the pressure is $21 \\mu \\mathrm{Pa} . H$ is the scale height, and its value is $H=60 \\mathrm{~km}$.\n\nDetermine the air particles mean free path $\\lambda$ at the altitude of $250 \\mathrm{~km}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of km, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_61a2ff399c33d9b3cd3bg-1.jpg?height=968&width=1044&top_left_y=1240&top_left_x=302" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "km" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_30", "problem": "A refrigerator has a mass of $150 \\mathrm{~kg}$ and rests in the open back end of a delivery truck. If the truck accelerates from rest at $1.5 \\mathrm{~m} / \\mathrm{s}^{2}$, what is the minimum coefficient of static friction between the refrigerator and the bed of the truck that is required to prevent the refrigerator from sliding off the back of the truck?\nA: 0.08\nB: 0.10\nC: 0.12\nD: 0.15\nE: 0.18\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA refrigerator has a mass of $150 \\mathrm{~kg}$ and rests in the open back end of a delivery truck. If the truck accelerates from rest at $1.5 \\mathrm{~m} / \\mathrm{s}^{2}$, what is the minimum coefficient of static friction between the refrigerator and the bed of the truck that is required to prevent the refrigerator from sliding off the back of the truck?\n\nA: 0.08\nB: 0.10\nC: 0.12\nD: 0.15\nE: 0.18\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_514", "problem": "A large block of mass $m_{b}$ is located on a horizontal frictionless surface. A second block of mass $m_{t}$ is located on top of the first block; the coefficient of friction (both static and kinetic) between the two blocks is given by $\\mu$. All surfaces are horizontal; all motion is effectively one dimensional. A spring with spring constant $k$ is connected to the top block only; the spring obeys Hooke's Law equally in both extension and compression. Assume that the top block never falls off of the bottom block; you may assume that the bottom block is very, very long. The top block is moved a distance $A$ away from the equilibrium position and then released from rest.\n\n[figure1]\n\nConsider now the scenario $A \\gg A_{c}$. In this scenario the amplitude of the oscillation of the top block as measured against the original equilibrium position will change with time. Determine the magnitude of the change in amplitude, $\\Delta A$, after one complete oscillation, as a function of any or all of $A, \\mu, g$, and the angular frequency of oscillation of the top block $\\omega_{t}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA large block of mass $m_{b}$ is located on a horizontal frictionless surface. A second block of mass $m_{t}$ is located on top of the first block; the coefficient of friction (both static and kinetic) between the two blocks is given by $\\mu$. All surfaces are horizontal; all motion is effectively one dimensional. A spring with spring constant $k$ is connected to the top block only; the spring obeys Hooke's Law equally in both extension and compression. Assume that the top block never falls off of the bottom block; you may assume that the bottom block is very, very long. The top block is moved a distance $A$ away from the equilibrium position and then released from rest.\n\n[figure1]\n\nConsider now the scenario $A \\gg A_{c}$. In this scenario the amplitude of the oscillation of the top block as measured against the original equilibrium position will change with time. Determine the magnitude of the change in amplitude, $\\Delta A$, after one complete oscillation, as a function of any or all of $A, \\mu, g$, and the angular frequency of oscillation of the top block $\\omega_{t}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-07.jpg?height=204&width=1160&top_left_y=684&top_left_x=474" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1401", "problem": "011 年 8 月中国发射的宇宙飞船 “嫦娥二号” 在完成探月任务后, 首次从绕月轨道飞向日地延长线上的拉格朗日点, 在该点, “嫦娥二号” 和地球一起同步绕太阳做圆周运动。已知太阳和地球的质量分别为 $M_{\\mathrm{S}}$ 和 $M_{\\mathrm{E}}$, 日地距离为 $R$ 。该拉格朗日点离地球的距离 $X$ 满足的方程为? \n\n由此解得 $x \\approx$?\n。\n\n（已知当 $\\lambda \\ll<1$ 时， $\\left.(1+\\lambda)^{n} \\approx 1+n \\lambda \\circ\\right)$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n011 年 8 月中国发射的宇宙飞船 “嫦娥二号” 在完成探月任务后, 首次从绕月轨道飞向日地延长线上的拉格朗日点, 在该点, “嫦娥二号” 和地球一起同步绕太阳做圆周运动。已知太阳和地球的质量分别为 $M_{\\mathrm{S}}$ 和 $M_{\\mathrm{E}}$, 日地距离为 $R$ 。该拉格朗日点离地球的距离 $X$ 满足的方程为? \n\n由此解得 $x \\approx$?\n。\n\n（已知当 $\\lambda \\ll<1$ 时， $\\left.(1+\\lambda)^{n} \\approx 1+n \\lambda \\circ\\right)$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[满足的方程, the value of x]\n它们的答案类型依次是[方程, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "满足的方程", "the value of x" ], "type_sequence": [ "EQ", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_260", "problem": "The characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part, analyze a situation when an angular momentum is not conserved. The system is the same as in the previous part with the only difference that the dipole is fixed and the charged small object with a mass $2 m$ is moving around the dipole. The electrostatic field of the dipole is easier to describe in the polar system of coordinates, which is defined with the distance $r$ from the center of the dipole, and angle $\\theta$ counted counterclockwise, as shown in Figure 3.\n\n[figure1]\n\nFigure 3: The system analyzed in Part 2. (Direction of the vector $\\mathbf{E}_{\\mathbf{n}}$ and $\\mathbf{E}_{\\mathbf{t}}$ could be wrong)\n\nWhat torque is applied to the moving object counting from the center of the dipole when the object is at the distance $r$ and angle $\\theta$ from the dipole?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part, analyze a situation when an angular momentum is not conserved. The system is the same as in the previous part with the only difference that the dipole is fixed and the charged small object with a mass $2 m$ is moving around the dipole. The electrostatic field of the dipole is easier to describe in the polar system of coordinates, which is defined with the distance $r$ from the center of the dipole, and angle $\\theta$ counted counterclockwise, as shown in Figure 3.\n\n[figure1]\n\nFigure 3: The system analyzed in Part 2. (Direction of the vector $\\mathbf{E}_{\\mathbf{n}}$ and $\\mathbf{E}_{\\mathbf{t}}$ could be wrong)\n\nWhat torque is applied to the moving object counting from the center of the dipole when the object is at the distance $r$ and angle $\\theta$ from the dipole?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ed4e92416bdbac30298dg-2.jpg?height=477&width=1442&top_left_y=1694&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1531", "problem": "制冷机是通过外界对机器做功, 把从低温吸取的热量连同外界对机器做功得到的能量一起送到高温处的机器。它能使低温处的温度降低, 高温处的温度升高。已知当制冷机工作在绝对温度为 $T_{1}$ 的高温处和绝对温度为 $T_{2}$ 的低温处之间时, 若制冷机从低温处吸取的热量为 $Q$,外界对制冷机做的功为 $W$, 则有\n\n$$\n\\frac{Q}{W} \\leq \\frac{T_{2}}{T_{1}-T_{2}}\n$$\n\n式中等号对应于理论上的理想情况。\n\n某制冷机在冬天作热永使用 (即取暖空调机), 在室外温度为 $-5.00^{\\circ} \\mathrm{C}$ 的情况下, 使某房间内的温度保持在 $20.00^{\\circ} \\mathrm{C}$ 。由于室内温度高于室外, 故将有热量从室内传递到室外。本题只考虑传导方式的传热, 它服从以下的规律: 设一块导热层, 其厚度为 $l$, 面积为 $S$, 两侧温度差的大小为 $\\Delta T$ ，则单位时间内通过导热层由高温处传导到低温处的热量为\n\n$$\nH=\\kappa \\frac{\\Delta T}{l} S\n$$\n\n其中 $\\kappa$ 为导热率, 取决于导热层材料的性质。假设该房问向外散热是由面向室外的面积为 $S=5.00 \\mathrm{~m}^{2}$ 、厚度为 $l=2.00 \\mathrm{~mm}$ 的玻璃引起的, 已知该玻璃的导热率为 $K=0.75 \\mathrm{~W} \\cdot \\mathrm{m}^{-1} \\cdot \\mathrm{k}^{-1}$, 电费为每度 0.50 元, 试求在理想情况下该热泉工作 12 小时需要多少度电?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n制冷机是通过外界对机器做功, 把从低温吸取的热量连同外界对机器做功得到的能量一起送到高温处的机器。它能使低温处的温度降低, 高温处的温度升高。已知当制冷机工作在绝对温度为 $T_{1}$ 的高温处和绝对温度为 $T_{2}$ 的低温处之间时, 若制冷机从低温处吸取的热量为 $Q$,外界对制冷机做的功为 $W$, 则有\n\n$$\n\\frac{Q}{W} \\leq \\frac{T_{2}}{T_{1}-T_{2}}\n$$\n\n式中等号对应于理论上的理想情况。\n\n某制冷机在冬天作热永使用 (即取暖空调机), 在室外温度为 $-5.00^{\\circ} \\mathrm{C}$ 的情况下, 使某房间内的温度保持在 $20.00^{\\circ} \\mathrm{C}$ 。由于室内温度高于室外, 故将有热量从室内传递到室外。本题只考虑传导方式的传热, 它服从以下的规律: 设一块导热层, 其厚度为 $l$, 面积为 $S$, 两侧温度差的大小为 $\\Delta T$ ，则单位时间内通过导热层由高温处传导到低温处的热量为\n\n$$\nH=\\kappa \\frac{\\Delta T}{l} S\n$$\n\n其中 $\\kappa$ 为导热率, 取决于导热层材料的性质。\n\n问题:\n假设该房问向外散热是由面向室外的面积为 $S=5.00 \\mathrm{~m}^{2}$ 、厚度为 $l=2.00 \\mathrm{~mm}$ 的玻璃引起的, 已知该玻璃的导热率为 $K=0.75 \\mathrm{~W} \\cdot \\mathrm{m}^{-1} \\cdot \\mathrm{k}^{-1}$, 电费为每度 0.50 元, 试求在理想情况下该热泉工作 12 小时需要多少度电?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以元为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "元" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_727", "problem": "If a stationary object was released near to the surface of the Sun, with approximately what acceleration would it fall?\n\n[figure1]\nA: $5 \\mathrm{~m} / \\mathrm{s}^{2}$\nB: $10 \\mathrm{~m} / \\mathrm{s}^{2}$\nC: $20 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: $250 \\mathrm{~m} / \\mathrm{s}^{2}$\nE: $1000 \\mathrm{~m} / \\mathrm{s}^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIf a stationary object was released near to the surface of the Sun, with approximately what acceleration would it fall?\n\n[figure1]\n\nA: $5 \\mathrm{~m} / \\mathrm{s}^{2}$\nB: $10 \\mathrm{~m} / \\mathrm{s}^{2}$\nC: $20 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: $250 \\mathrm{~m} / \\mathrm{s}^{2}$\nE: $1000 \\mathrm{~m} / \\mathrm{s}^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-03.jpg?height=382&width=366&top_left_y=1448&top_left_x=1299" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_408", "problem": "[figure1]\n\nA cylindrical stone disc (marked with ' 1 ' in the figure) of radius $R$, thickness $h$ and density $\\rho_{s}$ is pressed against the ceiling of a basin filled with water of density $\\rho_{w}$. Small bumps on the surface of the cei ing maintain a small gap of thickness $t \\ll$ $R$ between the ceiling and the surface of the disk. Water flows from a pipe (marked with ' 2 '; the outflow pipe ' 3 ' is far away) of radius $r \\ll R$ coaxially with the disk into the basin, see the figure. The radius of the pipe is much bigger than the gap between the disk and the ceiling, i.e. $r \\gg t$. What should be the mass flow rate $\\mu(\\mathrm{kg} / \\mathrm{s})$ from the pipe so as to keep the disk from fall ng down? The free fall acceleration is", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n[figure1]\n\nA cylindrical stone disc (marked with ' 1 ' in the figure) of radius $R$, thickness $h$ and density $\\rho_{s}$ is pressed against the ceiling of a basin filled with water of density $\\rho_{w}$. Small bumps on the surface of the cei ing maintain a small gap of thickness $t \\ll$ $R$ between the ceiling and the surface of the disk. Water flows from a pipe (marked with ' 2 '; the outflow pipe ' 3 ' is far away) of radius $r \\ll R$ coaxially with the disk into the basin, see the figure. The radius of the pipe is much bigger than the gap between the disk and the ceiling, i.e. $r \\gg t$. What should be the mass flow rate $\\mu(\\mathrm{kg} / \\mathrm{s})$ from the pipe so as to keep the disk from fall ng down? The free fall acceleration is\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_d54f6e2eba5a419814b7g-1.jpg?height=380&width=671&top_left_y=153&top_left_x=765" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_822", "problem": "[figure1]\n\nFig. 1 A monopole $q_{\\mathrm{m}}$ appears at a distance $h$ from a conducting thin film of thickness $d$. The origin of the coordinates is located on the upper surface.\n\nWe first focus on the initial response of the conducting thin film when at time $t=0$ a north monopole $q_{\\mathrm{m}}$ appears suddenly at the position $\\vec{r}_{\\mathrm{mp}}=h \\hat{z}(h>0)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nFor $t>0$, the total magnetic field $\\vec{B}$ becomes $\\vec{B}(\\vec{\\rho}, z ; t)=\\vec{B}_{\\mathrm{mp}}(\\vec{\\rho}, z)+\\vec{B}^{\\prime}(\\vec{\\rho}, z ; t)$, by superposition, with $\\vec{B}^{\\prime}(\\vec{\\rho}, z ; t)$ due to the induced electric current in the thin film. You are required below to obtain an equation for $B_{z}^{\\prime}(\\rho, z ; t)$ near the $z=0$ thin film surface. The time-evolution behavior of $B_{z}^{\\prime}$ would reveal a moving image-monopole picture for the description of the $\\vec{B}^{\\prime}$ field near $z \\approx 0$ in $t>0$.\n\nThe equation for $B_{z}^{\\prime}$ inside the thin film is given below,\n\n$$\n\\frac{\\partial^{2} B_{z}^{\\prime}(\\rho, z ; t)}{\\partial z^{2}}=\\mu_{0} \\sigma \\frac{\\partial B_{z}^{\\prime}(\\rho, z ; t)}{\\partial t} .\n$$\n\nThis equation has been obtained from imposing inside the thin film the Maxwell equation and the Ohmic behavior of the conducting thin film ( $\\vec{j}=\\sigma \\vec{E}$, where $\\sigma$ is the electrical conductivity) while neglecting the displacement-current effect. Term being neglected on the left-hand side of Eq.(2) is $\\frac{1}{\\rho} \\frac{\\partial}{\\partial \\rho}\\left(\\rho \\frac{\\partial B_{z}^{\\prime}}{\\partial \\rho}\\right)$, based on the $h \\gg d$ condition.\n\nObtain from Eq. ( 2 ) an equation of $B_{z}^{\\prime}(\\rho, z ; t)$ near $z \\approx 0$. The equation contains $0.6 \\mathrm{pt}$ first partial derivatives of $B_{z}^{\\prime}(\\rho, z ; t)$ with respect to $z$, and, separately, to $t$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\n[figure1]\n\nFig. 1 A monopole $q_{\\mathrm{m}}$ appears at a distance $h$ from a conducting thin film of thickness $d$. The origin of the coordinates is located on the upper surface.\n\nWe first focus on the initial response of the conducting thin film when at time $t=0$ a north monopole $q_{\\mathrm{m}}$ appears suddenly at the position $\\vec{r}_{\\mathrm{mp}}=h \\hat{z}(h>0)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nFor $t>0$, the total magnetic field $\\vec{B}$ becomes $\\vec{B}(\\vec{\\rho}, z ; t)=\\vec{B}_{\\mathrm{mp}}(\\vec{\\rho}, z)+\\vec{B}^{\\prime}(\\vec{\\rho}, z ; t)$, by superposition, with $\\vec{B}^{\\prime}(\\vec{\\rho}, z ; t)$ due to the induced electric current in the thin film. You are required below to obtain an equation for $B_{z}^{\\prime}(\\rho, z ; t)$ near the $z=0$ thin film surface. The time-evolution behavior of $B_{z}^{\\prime}$ would reveal a moving image-monopole picture for the description of the $\\vec{B}^{\\prime}$ field near $z \\approx 0$ in $t>0$.\n\nThe equation for $B_{z}^{\\prime}$ inside the thin film is given below,\n\n$$\n\\frac{\\partial^{2} B_{z}^{\\prime}(\\rho, z ; t)}{\\partial z^{2}}=\\mu_{0} \\sigma \\frac{\\partial B_{z}^{\\prime}(\\rho, z ; t)}{\\partial t} .\n$$\n\nThis equation has been obtained from imposing inside the thin film the Maxwell equation and the Ohmic behavior of the conducting thin film ( $\\vec{j}=\\sigma \\vec{E}$, where $\\sigma$ is the electrical conductivity) while neglecting the displacement-current effect. Term being neglected on the left-hand side of Eq.(2) is $\\frac{1}{\\rho} \\frac{\\partial}{\\partial \\rho}\\left(\\rho \\frac{\\partial B_{z}^{\\prime}}{\\partial \\rho}\\right)$, based on the $h \\gg d$ condition.\n\nObtain from Eq. ( 2 ) an equation of $B_{z}^{\\prime}(\\rho, z ; t)$ near $z \\approx 0$. The equation contains $0.6 \\mathrm{pt}$ first partial derivatives of $B_{z}^{\\prime}(\\rho, z ; t)$ with respect to $z$, and, separately, to $t$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d32b3b2f89cebe6f1c2ag-2.jpg?height=642&width=1244&top_left_y=296&top_left_x=194" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1624", "problem": "按如下原理制作一杆可直接测量液体密度的科, 称为密度科, 其外形和普通的杆秤差不多, 装秤钩的地方吊着一体积为 $1 \\mathrm{~cm}^{3}$的较重的合金块，杆上有表示液体密度数值的刻度.当科砣放在 $Q$ 点\n\n[图1]\n处时科杆恰好平衡, 如图所示. 当合金块完全浸没在待测密度的液体中时, 移动科砣的悬挂点, 直至秤杆恰好重新平衡, 便可直接在杆秤上读出液体的密度. 下列说法中错误的是\nA: 密度秤的零点刻度在 $Q$ 点\nB: 科杆上密度读数较大的刻度在较小的刻度的左边\nC: 密度秤的刻度都在 $Q$ 点的右侧\nD: 密度秤的刻度都在 $Q$ 点的左侧\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n按如下原理制作一杆可直接测量液体密度的科, 称为密度科, 其外形和普通的杆秤差不多, 装秤钩的地方吊着一体积为 $1 \\mathrm{~cm}^{3}$的较重的合金块，杆上有表示液体密度数值的刻度.当科砣放在 $Q$ 点\n\n[图1]\n处时科杆恰好平衡, 如图所示. 当合金块完全浸没在待测密度的液体中时, 移动科砣的悬挂点, 直至秤杆恰好重新平衡, 便可直接在杆秤上读出液体的密度. 下列说法中错误的是\n\nA: 密度秤的零点刻度在 $Q$ 点\nB: 科杆上密度读数较大的刻度在较小的刻度的左边\nC: 密度秤的刻度都在 $Q$ 点的右侧\nD: 密度秤的刻度都在 $Q$ 点的左侧\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_58b1fc45927d60138a23g-01.jpg?height=180&width=367&top_left_y=1255&top_left_x=1413" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_578", "problem": "In this problem we consider a simplified model of the electromagnetic radiation inside a cubical box of side length $L$. In this model, the electric field has spatial dependence\n\n$$\nE(x, y, z)=E_{0} \\sin \\left(k_{x} x\\right) \\sin \\left(k_{y} y\\right) \\sin \\left(k_{z} z\\right)\n$$\n\nwhere one corner of the box lies at the origin and the box is aligned with the $x, y$, and $z$ axes. Let $h$ be Planck's constant, $k_{B}$ be Boltzmann's constant, and $c$ be the speed of light.\n\nEach quantum state, in turn, may be occupied by photons with frequency $\\omega=\\frac{f}{2 \\pi}=c|\\mathbf{k}|$, where\n\n$$\n|\\mathbf{k}|=\\sqrt{k_{x}{ }^{2}+k_{y}{ }^{2}+k_{z}{ }^{2}}\n$$\n\nIn the model, if the temperature inside the box is $T$, no photon may have energy greater than $k_{B} T$. \n\nIs the shape of the region in state space corresponding to occupied states a ball of radius $k_{\\max }=k_{B} T / \\hbar c$ in state space centered at the origin.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a True or False question.\n\nproblem:\nIn this problem we consider a simplified model of the electromagnetic radiation inside a cubical box of side length $L$. In this model, the electric field has spatial dependence\n\n$$\nE(x, y, z)=E_{0} \\sin \\left(k_{x} x\\right) \\sin \\left(k_{y} y\\right) \\sin \\left(k_{z} z\\right)\n$$\n\nwhere one corner of the box lies at the origin and the box is aligned with the $x, y$, and $z$ axes. Let $h$ be Planck's constant, $k_{B}$ be Boltzmann's constant, and $c$ be the speed of light.\n\nEach quantum state, in turn, may be occupied by photons with frequency $\\omega=\\frac{f}{2 \\pi}=c|\\mathbf{k}|$, where\n\n$$\n|\\mathbf{k}|=\\sqrt{k_{x}{ }^{2}+k_{y}{ }^{2}+k_{z}{ }^{2}}\n$$\n\nIn the model, if the temperature inside the box is $T$, no photon may have energy greater than $k_{B} T$. \n\nIs the shape of the region in state space corresponding to occupied states a ball of radius $k_{\\max }=k_{B} T / \\hbar c$ in state space centered at the origin.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be either \"True\" or \"False\".", "figure_urls": null, "answer": null, "solution": null, "answer_type": "TF", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_322", "problem": "While in generic case, the dynamics of three gravitationally inter acting bodies is complicated and chaotic, there are special cases when the dynamics is regular. In particular, there are cases when bodies per form periodic motion. The simplest case of such periodic motion is when all three bodies, positioned at the vertices of an equilateral triangle, rotate as a rigid body. In what follows we consider a more complicated periodic motion.\n\nRelatively recently ${ }^{1}$, it was discovered that three equal point masses can move periodically along a common 8-shaped trajectory shown in the figure (arrow denotes the direction of motion). This figure is based on a computer simulation and has a correct shape. If needed, you can measure distances from the enlarged version of the figure (on a separate sheet) using a ruler.\n\n[figure1]\n\nLet us enumerate the three bodies with numbers 1,2 , and 3 , according to the order in which they pass the leftmost point $P$ shown in the fig ure. Let $O_{2}$ and $O_{3}$ denote the positions of the bodies 2 and 3 , respectively at that moment when the body 1 is passing the middle point $O$ Similarly, let $P_{2}$ and $P_{3}$ denote the positions of the bodies 2 and 3 , respectively at that moment when the body 1 is passing the leftmost point $P$. Let $T$ denote the full period of motion of each of the bodies along this 8-shaped trajectory.\n\nLet $\\vec{v}_{1}, \\vec{v}_{2}$, and $\\vec{v}_{3}$ denote the velocities of the three bodies at a certain moment of time. Write down an equality relating these three vectors to each other.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nWhile in generic case, the dynamics of three gravitationally inter acting bodies is complicated and chaotic, there are special cases when the dynamics is regular. In particular, there are cases when bodies per form periodic motion. The simplest case of such periodic motion is when all three bodies, positioned at the vertices of an equilateral triangle, rotate as a rigid body. In what follows we consider a more complicated periodic motion.\n\nRelatively recently ${ }^{1}$, it was discovered that three equal point masses can move periodically along a common 8-shaped trajectory shown in the figure (arrow denotes the direction of motion). This figure is based on a computer simulation and has a correct shape. If needed, you can measure distances from the enlarged version of the figure (on a separate sheet) using a ruler.\n\n[figure1]\n\nLet us enumerate the three bodies with numbers 1,2 , and 3 , according to the order in which they pass the leftmost point $P$ shown in the fig ure. Let $O_{2}$ and $O_{3}$ denote the positions of the bodies 2 and 3 , respectively at that moment when the body 1 is passing the middle point $O$ Similarly, let $P_{2}$ and $P_{3}$ denote the positions of the bodies 2 and 3 , respectively at that moment when the body 1 is passing the leftmost point $P$. Let $T$ denote the full period of motion of each of the bodies along this 8-shaped trajectory.\n\nLet $\\vec{v}_{1}, \\vec{v}_{2}$, and $\\vec{v}_{3}$ denote the velocities of the three bodies at a certain moment of time. Write down an equality relating these three vectors to each other.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_57fd2cb729ee4cd65883g-1.jpg?height=228&width=660&top_left_y=990&top_left_x=88" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_631", "problem": "A large block of mass $m_{b}$ is located on a horizontal frictionless surface. A second block of mass $m_{t}$ is located on top of the first block; the coefficient of friction (both static and kinetic) between the two blocks is given by $\\mu$. All surfaces are horizontal; all motion is effectively one dimensional. A spring with spring constant $k$ is connected to the top block only; the spring obeys Hooke's Law equally in both extension and compression. Assume that the top block never falls off of the bottom block; you may assume that the bottom block is very, very long. The top block is moved a distance $A$ away from the equilibrium position and then released from rest.\n\n[figure1]\n\nAssume that $A \\gg A_{c}$. What is the maximum speed of the bottom block during the first complete oscillation cycle of the upper block?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA large block of mass $m_{b}$ is located on a horizontal frictionless surface. A second block of mass $m_{t}$ is located on top of the first block; the coefficient of friction (both static and kinetic) between the two blocks is given by $\\mu$. All surfaces are horizontal; all motion is effectively one dimensional. A spring with spring constant $k$ is connected to the top block only; the spring obeys Hooke's Law equally in both extension and compression. Assume that the top block never falls off of the bottom block; you may assume that the bottom block is very, very long. The top block is moved a distance $A$ away from the equilibrium position and then released from rest.\n\n[figure1]\n\nAssume that $A \\gg A_{c}$. What is the maximum speed of the bottom block during the first complete oscillation cycle of the upper block?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-07.jpg?height=204&width=1160&top_left_y=684&top_left_x=474" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_731", "problem": "What is the voltage across the resistor $R_{1}$ in the circuit below?\n\n[figure1]\nA: $\\left(9 R_{5} /\\left(R_{5}+R_{4}\\right)\\right) * R_{1} /\\left(R_{1}+R_{2}+R 3\\right)$\nB: $\\left(9 R_{4} /\\left(R_{5}+R_{4}\\right)\\right) * R_{1} /\\left(R_{1}+R_{2}+R_{3}\\right)$\nC: 0\nD: $9 R_{1} /\\left(R_{1}+R_{2}+R_{3}\\right)$\nE: $\\left(36 R_{4} /\\left(R_{5}+R_{4}\\right)\\right) * R_{1} /\\left(R_{1}+R_{2}+R_{3}\\right)$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the voltage across the resistor $R_{1}$ in the circuit below?\n\n[figure1]\n\nA: $\\left(9 R_{5} /\\left(R_{5}+R_{4}\\right)\\right) * R_{1} /\\left(R_{1}+R_{2}+R 3\\right)$\nB: $\\left(9 R_{4} /\\left(R_{5}+R_{4}\\right)\\right) * R_{1} /\\left(R_{1}+R_{2}+R_{3}\\right)$\nC: 0\nD: $9 R_{1} /\\left(R_{1}+R_{2}+R_{3}\\right)$\nE: $\\left(36 R_{4} /\\left(R_{5}+R_{4}\\right)\\right) * R_{1} /\\left(R_{1}+R_{2}+R_{3}\\right)$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22e26a14ee6fdd9254b6g-05.jpg?height=629&width=552&top_left_y=1493&top_left_x=342" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_346", "problem": "A Zener diode is connected to a source of alternating current as shown in the figure. The current is sinus oidal $I=I_{0} \\cos \\omega t$ with a constant amplitude. The inductance $L$ of the inductor is such that $L \\omega I_{0} \\gg V_{1}, V_{2}$, where $V_{1}$ and $V_{2}$ are the break down voltages $\\left(V_{1}>V_{2}\\right)$. The current-voltage characteristic of the Zener diode is shown in the figure. In the following assume that a very long time has passed since the current source was first turned on.\n\n[figure1]\n\n Find the average current $\\langle I\\rangle$ through the inductor.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA Zener diode is connected to a source of alternating current as shown in the figure. The current is sinus oidal $I=I_{0} \\cos \\omega t$ with a constant amplitude. The inductance $L$ of the inductor is such that $L \\omega I_{0} \\gg V_{1}, V_{2}$, where $V_{1}$ and $V_{2}$ are the break down voltages $\\left(V_{1}>V_{2}\\right)$. The current-voltage characteristic of the Zener diode is shown in the figure. In the following assume that a very long time has passed since the current source was first turned on.\n\n[figure1]\n\n Find the average current $\\langle I\\rangle$ through the inductor.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e6c1582ce7f1c05fa0a6g-2.jpg?height=424&width=694&top_left_y=754&top_left_x=82" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_788", "problem": "Four particles, each of charge $+q$, are arranged symmetrically on the $x$-axis about the origin as shown. A fifth particle of charge $-Q$ is placed on the positive $y$-axis as shown. What is the direction of the net electrostatic force on the fifth particle?\n[figure1]\nA: $\\uparrow$\nB: $\\rightarrow$\nC: $\\downarrow$\nD: $\\leftarrow$\nE: The net electrostatic force on the particle of charge $-Q$ is zero.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nFour particles, each of charge $+q$, are arranged symmetrically on the $x$-axis about the origin as shown. A fifth particle of charge $-Q$ is placed on the positive $y$-axis as shown. What is the direction of the net electrostatic force on the fifth particle?\n[figure1]\n\nA: $\\uparrow$\nB: $\\rightarrow$\nC: $\\downarrow$\nD: $\\leftarrow$\nE: The net electrostatic force on the particle of charge $-Q$ is zero.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9bac30101a7c40be9bg-04.jpg?height=354&width=760&top_left_y=1505&top_left_x=959" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_902", "problem": "## How are aurora ignited by the solar wind?\n\n[figure1]\n\nFigure 1\n\nIt is well known that the Earth has a substantial magnetic field. The field lines defining the structure of the Earth's magnetic field is similar to that of a simple bar magnet, as shown in Figure 2. The Earth's magnetic field is disturbed by the solar wind, whichis a high-speed stream of hot plasma. (The plasma is the quasi-neutralionized gas.)The plasma blows outward from the Sun and varies in intensity with the amount of surface activity on the Sun. The solar wind compresses the Earth's magnetic field. On the other hand, the Earth's magnetic field shields the Earth from much of the solar wind. When the solar wind encounters the Earth's magnetic field, it is deflected like water around the bow of a ship, as illustrated in Figure 3.\n\n[figure2]\n\nFigure 2\n\n[figure3]\n\nFigure 3\n\nThe curved surface at which the solar wind is first deflected is called the bow shock. The corresponding region behind the bow shock andfront of the Earth's magnetic field is called themagnetosheath. The region surroundedby the solar wind is called the magnetosphere.The Earth's magnetic field largely prevents the solar wind from enteringthe magnetosphere. The contact region between the solar wind and the Earth's magnetic field is named the magnetopause. The location of the magnetopause is mainly determined by the intensity and the magnetic field direction of the solar wind. When the magnetic field in the solar wind is antiparallel to the Earth's magnetic field, magnetic reconnection as shown in Figure 4 takes place at the dayside magnetopause, which allows some charged particles ofthe solar wind in the region \"A\" to move into the magnetotail \"P\" on the night side as illustrated in Figure 5. A powerful solar wind can push the dayside magnetopause tovery close to the Earth, which could cause a high-orbit satellite (such as a geosynchronous satellite) to be fully exposed to the solar wind. The energetic particles in the solar wind could damage high-tech electronic components in a satellite.Therefore, it isimportant to study the motion of charged particles in magnetic fields, which will give an answer of the aurora generation and could help us to understand the mechanism of the interaction between the solar wind and the Earth's magnetic field.\n\n[figure4]\n\nFigure 4\n\n[figure5]\n\nFigure 5\n\nNumerical values of physical constants and the Earth's dipole magnetic field:\n\nSpeed of light in vacuum: $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$;\n\nPermittivity in vacuum: $\\varepsilon_{0}=8.9 \\times 10^{-12} \\mathrm{C}^{2} /\\left(\\mathrm{N} \\cdot \\mathrm{m}^{2}\\right)$;\n\nPermeability in vacuum: $\\mu_{0}=4 \\pi \\times 10^{-7} \\mathrm{~N} / \\mathrm{A}^{2}$;\n\nCharge of a proton: $e=1.6 \\times 10^{-19} \\mathrm{C}$;\n\nMass of an electron: $m=9.1 \\times 10^{-31} \\mathrm{~kg}$;\n\nMass of a proton: $m_{p}=1.67 \\times 10^{-27} \\mathrm{~kg}$;\n\nBoltzmann's constant: $k=1.38 \\times 10^{-23} \\mathrm{~J} / \\mathrm{K}$;\n\nGravitational acceleration: $g=9.8 \\mathrm{~m} / \\mathrm{s}^{2}$;\n\nPlanck's constant: $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$\n\nEarth's radius $R_{E}=6.4 \\times 10^{6} \\mathrm{~m}$.\n\nThe Earth's dipole magnetic field can be expressed as\n\n$\\vec{B}_{d}=\\frac{B_{0} R_{E}^{3}}{r^{5}}\\left[3 x z \\hat{x}-3 y z \\hat{y}+\\left(x^{2}+y^{2}-2 z^{2}\\right) \\hat{z}\\right] \\quad,\\left(r \\geq R_{E}\\right)$\n\nwhere $r=\\sqrt{x^{2}+y^{2}+z^{2}}, \\quad B_{0}=3.1 \\times 10^{-5} \\mathrm{~T}$, and $\\hat{x}, \\hat{y}, \\hat{z}$ are the unit vectors in the $x, y, z$ directions, respectively.\n\n\nBefore we study the motion of a charged particle in the Earth's dipole magnetic field, we first considerthe motion of an electron in a uniform magnetic field $\\vec{B}$. When the initial electron velocity $\\vec{v}$ is perpendicular to the uniform magnetic field as shown in Figure 6, please calculate the electron trajectory.The electron is initially located at $(x, y, z)=(0,0,0)$.\n\n[figure6]\n\nFigure 6", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\n## How are aurora ignited by the solar wind?\n\n[figure1]\n\nFigure 1\n\nIt is well known that the Earth has a substantial magnetic field. The field lines defining the structure of the Earth's magnetic field is similar to that of a simple bar magnet, as shown in Figure 2. The Earth's magnetic field is disturbed by the solar wind, whichis a high-speed stream of hot plasma. (The plasma is the quasi-neutralionized gas.)The plasma blows outward from the Sun and varies in intensity with the amount of surface activity on the Sun. The solar wind compresses the Earth's magnetic field. On the other hand, the Earth's magnetic field shields the Earth from much of the solar wind. When the solar wind encounters the Earth's magnetic field, it is deflected like water around the bow of a ship, as illustrated in Figure 3.\n\n[figure2]\n\nFigure 2\n\n[figure3]\n\nFigure 3\n\nThe curved surface at which the solar wind is first deflected is called the bow shock. The corresponding region behind the bow shock andfront of the Earth's magnetic field is called themagnetosheath. The region surroundedby the solar wind is called the magnetosphere.The Earth's magnetic field largely prevents the solar wind from enteringthe magnetosphere. The contact region between the solar wind and the Earth's magnetic field is named the magnetopause. The location of the magnetopause is mainly determined by the intensity and the magnetic field direction of the solar wind. When the magnetic field in the solar wind is antiparallel to the Earth's magnetic field, magnetic reconnection as shown in Figure 4 takes place at the dayside magnetopause, which allows some charged particles ofthe solar wind in the region \"A\" to move into the magnetotail \"P\" on the night side as illustrated in Figure 5. A powerful solar wind can push the dayside magnetopause tovery close to the Earth, which could cause a high-orbit satellite (such as a geosynchronous satellite) to be fully exposed to the solar wind. The energetic particles in the solar wind could damage high-tech electronic components in a satellite.Therefore, it isimportant to study the motion of charged particles in magnetic fields, which will give an answer of the aurora generation and could help us to understand the mechanism of the interaction between the solar wind and the Earth's magnetic field.\n\n[figure4]\n\nFigure 4\n\n[figure5]\n\nFigure 5\n\nNumerical values of physical constants and the Earth's dipole magnetic field:\n\nSpeed of light in vacuum: $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$;\n\nPermittivity in vacuum: $\\varepsilon_{0}=8.9 \\times 10^{-12} \\mathrm{C}^{2} /\\left(\\mathrm{N} \\cdot \\mathrm{m}^{2}\\right)$;\n\nPermeability in vacuum: $\\mu_{0}=4 \\pi \\times 10^{-7} \\mathrm{~N} / \\mathrm{A}^{2}$;\n\nCharge of a proton: $e=1.6 \\times 10^{-19} \\mathrm{C}$;\n\nMass of an electron: $m=9.1 \\times 10^{-31} \\mathrm{~kg}$;\n\nMass of a proton: $m_{p}=1.67 \\times 10^{-27} \\mathrm{~kg}$;\n\nBoltzmann's constant: $k=1.38 \\times 10^{-23} \\mathrm{~J} / \\mathrm{K}$;\n\nGravitational acceleration: $g=9.8 \\mathrm{~m} / \\mathrm{s}^{2}$;\n\nPlanck's constant: $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$\n\nEarth's radius $R_{E}=6.4 \\times 10^{6} \\mathrm{~m}$.\n\nThe Earth's dipole magnetic field can be expressed as\n\n$\\vec{B}_{d}=\\frac{B_{0} R_{E}^{3}}{r^{5}}\\left[3 x z \\hat{x}-3 y z \\hat{y}+\\left(x^{2}+y^{2}-2 z^{2}\\right) \\hat{z}\\right] \\quad,\\left(r \\geq R_{E}\\right)$\n\nwhere $r=\\sqrt{x^{2}+y^{2}+z^{2}}, \\quad B_{0}=3.1 \\times 10^{-5} \\mathrm{~T}$, and $\\hat{x}, \\hat{y}, \\hat{z}$ are the unit vectors in the $x, y, z$ directions, respectively.\n\n\nBefore we study the motion of a charged particle in the Earth's dipole magnetic field, we first considerthe motion of an electron in a uniform magnetic field $\\vec{B}$. When the initial electron velocity $\\vec{v}$ is perpendicular to the uniform magnetic field as shown in Figure 6, please calculate the electron trajectory.The electron is initially located at $(x, y, z)=(0,0,0)$.\n\n[figure6]\n\nFigure 6\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_5922cc4cab73c4b7533bg-1.jpg?height=737&width=1102&top_left_y=625&top_left_x=480", "https://cdn.mathpix.com/cropped/2024_03_14_5922cc4cab73c4b7533bg-1.jpg?height=522&width=1079&top_left_y=2092&top_left_x=494", "https://cdn.mathpix.com/cropped/2024_03_14_5922cc4cab73c4b7533bg-2.jpg?height=682&width=1241&top_left_y=570&top_left_x=316", "https://cdn.mathpix.com/cropped/2024_03_14_5922cc4cab73c4b7533bg-2.jpg?height=379&width=759&top_left_y=2272&top_left_x=657", "https://cdn.mathpix.com/cropped/2024_03_14_5922cc4cab73c4b7533bg-3.jpg?height=540&width=1171&top_left_y=615&top_left_x=451", "https://cdn.mathpix.com/cropped/2024_03_14_5922cc4cab73c4b7533bg-4.jpg?height=279&width=574&top_left_y=817&top_left_x=724" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_657", "problem": "An electron enters the space inside an infinite currentcarrying solenoid. The velocity of the electron is parallel to the solenoid's axis. The electron will:\nA: Deflect outwards\nB: Slow down\nC: Speed up\nD: Stay parallel to the axis of the solenoid\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn electron enters the space inside an infinite currentcarrying solenoid. The velocity of the electron is parallel to the solenoid's axis. The electron will:\n\nA: Deflect outwards\nB: Slow down\nC: Speed up\nD: Stay parallel to the axis of the solenoid\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_701", "problem": "An ideal gas goes through a process such that\n\n$$\nP V^{3}=\\text { const. }\n$$\n\nIf the volume doubles during this process the temperature:\nA: remains constant.\nB: increases four-fold.\nC: decreases four-fold.\nD: increases eight-fold.\nE: decreases eight-fold.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn ideal gas goes through a process such that\n\n$$\nP V^{3}=\\text { const. }\n$$\n\nIf the volume doubles during this process the temperature:\n\nA: remains constant.\nB: increases four-fold.\nC: decreases four-fold.\nD: increases eight-fold.\nE: decreases eight-fold.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1280", "problem": "一固体星球可近似看作半径为 $R$ (足够大) 的球形均匀的固体, 构成星球的物质的密度为 $\\rho$,引力常量为 $G$ 。考虑星球表面山体的高度。如果山高超出某一限度, 山基便发生流动（可认为是山基部分物质熔化的结果, 相当于超出山的最高限的那块固体物质从山顶移走了)，从而使山的高度减低。山在这种情况下其高度的小幅减低可视为一小块质量的物质从山顶移至山底。假设该小块物质重力势能的减小与其全部熔化所需要的能量相当, 山体由同一种物质构成, 该物质的熔化热为 $L$, 不考虑温度升到熔点所需要能量, 也不考虑压强对固体熔化热的影响。试估计由同一种物质构成的山体高度的上限。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一固体星球可近似看作半径为 $R$ (足够大) 的球形均匀的固体, 构成星球的物质的密度为 $\\rho$,引力常量为 $G$ 。\n\n问题:\n考虑星球表面山体的高度。如果山高超出某一限度, 山基便发生流动（可认为是山基部分物质熔化的结果, 相当于超出山的最高限的那块固体物质从山顶移走了)，从而使山的高度减低。山在这种情况下其高度的小幅减低可视为一小块质量的物质从山顶移至山底。假设该小块物质重力势能的减小与其全部熔化所需要的能量相当, 山体由同一种物质构成, 该物质的熔化热为 $L$, 不考虑温度升到熔点所需要能量, 也不考虑压强对固体熔化热的影响。试估计由同一种物质构成的山体高度的上限。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1323", "problem": "右图是某粒子穿过云室留下的径迹的照片。径迹在直面内，图的中间是一块与纸面垂直的铅板，外加的恒定的匀强磁场的方向垂直于纸面向里, 假设粒子的电荷的大小是一个基本电荷量 $e, e=1.60 \\times 10^{-19} \\mathrm{C}$, 铅板下部径迹的曲率半径 $r_{d}=210 \\mathrm{~mm}$, 铅板上部径迹的曲率半径 $\\mathrm{r}_{\\mathrm{u}}=76.0 \\mathrm{~mm}$, 铅板内的径迹与铅板法线成 $\\theta=15.0^{\\circ}$ 角。铅板厚度 $d=6.00 \\mathrm{~mm}$,\n\n[图1]\n磁感应强度 $B=1.00 \\mathrm{~T}$, 粒子质量 $m=9.11 \\times 10^{-31} \\mathrm{~kg}=0.511 \\mathrm{MeV} / \\mathrm{c}^{2}$, 不考虑云室中气体对粒子的阻力,假设射向铅板的不是一个粒子而是从加速器中流出的流量为 $j=5.00 \\times 10^{18} / \\mathrm{s}$ 的脉冲粒子流, 一个脉冲持续时间为 $\\tau=2.50 n s$, 试问铅板在此期间吸收的热量是多少 J?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n右图是某粒子穿过云室留下的径迹的照片。径迹在直面内，图的中间是一块与纸面垂直的铅板，外加的恒定的匀强磁场的方向垂直于纸面向里, 假设粒子的电荷的大小是一个基本电荷量 $e, e=1.60 \\times 10^{-19} \\mathrm{C}$, 铅板下部径迹的曲率半径 $r_{d}=210 \\mathrm{~mm}$, 铅板上部径迹的曲率半径 $\\mathrm{r}_{\\mathrm{u}}=76.0 \\mathrm{~mm}$, 铅板内的径迹与铅板法线成 $\\theta=15.0^{\\circ}$ 角。铅板厚度 $d=6.00 \\mathrm{~mm}$,\n\n[图1]\n磁感应强度 $B=1.00 \\mathrm{~T}$, 粒子质量 $m=9.11 \\times 10^{-31} \\mathrm{~kg}=0.511 \\mathrm{MeV} / \\mathrm{c}^{2}$, 不考虑云室中气体对粒子的阻力,\n\n问题:\n假设射向铅板的不是一个粒子而是从加速器中流出的流量为 $j=5.00 \\times 10^{18} / \\mathrm{s}$ 的脉冲粒子流, 一个脉冲持续时间为 $\\tau=2.50 n s$, 试问铅板在此期间吸收的热量是多少 J?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以J为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_1004b08dedac85274c96g-07.jpg?height=377&width=382&top_left_y=1239&top_left_x=1131" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "J" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_557", "problem": "Consider two objects with equal heat capacities $C$ and initial temperatures $T_{1}$ and $T_{2}$. A Carnot engine is run using these objects as its hot and cold reservoirs until they are at equal temperatures. Assume that the temperature changes of both the hot and cold reservoirs is very small compared to the temperature during any one cycle of the Carnot engine.\n\nNow consider three objects with equal and constant heat capacity at initial temperatures $T_{1}=100 \\mathrm{~K}, T_{2}=300 \\mathrm{~K}$, and $T_{3}=300 \\mathrm{~K}$. Suppose we wish to raise the temperature of the third object.\n\nTo do this, we could run a Carnot engine between the first and second objects, extracting work $W$. This work can then be dissipated as heat to raise the temperature of the third object. Even better, it can be stored and used to run a Carnot engine between the first and third object in reverse, which pumps heat into the third object.\n\nAssume that all work produced by running engines can be stored and used without dissipation.\n\nFind the maximum temperature $T_{H}$ to which the third object can be raised.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nConsider two objects with equal heat capacities $C$ and initial temperatures $T_{1}$ and $T_{2}$. A Carnot engine is run using these objects as its hot and cold reservoirs until they are at equal temperatures. Assume that the temperature changes of both the hot and cold reservoirs is very small compared to the temperature during any one cycle of the Carnot engine.\n\nNow consider three objects with equal and constant heat capacity at initial temperatures $T_{1}=100 \\mathrm{~K}, T_{2}=300 \\mathrm{~K}$, and $T_{3}=300 \\mathrm{~K}$. Suppose we wish to raise the temperature of the third object.\n\nTo do this, we could run a Carnot engine between the first and second objects, extracting work $W$. This work can then be dissipated as heat to raise the temperature of the third object. Even better, it can be stored and used to run a Carnot engine between the first and third object in reverse, which pumps heat into the third object.\n\nAssume that all work produced by running engines can be stored and used without dissipation.\n\nFind the maximum temperature $T_{H}$ to which the third object can be raised.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~K}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~K}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_664", "problem": "Knowing that Toronto's coordinates are $43.6532^{\\circ} \\mathrm{N}$, $79.3832^{\\circ} \\mathrm{W}$, what is the ratio of the solar power through a horizontal surface at noon on the shortest day of the year to the solar power through the same horizontal surface at noon on the longest day of the year? (The tilt of the Earth is about $23.4^{\\circ}$ )\nA: 0.21\nB: 0.33\nC: 0.42\nD: 0.62\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nKnowing that Toronto's coordinates are $43.6532^{\\circ} \\mathrm{N}$, $79.3832^{\\circ} \\mathrm{W}$, what is the ratio of the solar power through a horizontal surface at noon on the shortest day of the year to the solar power through the same horizontal surface at noon on the longest day of the year? (The tilt of the Earth is about $23.4^{\\circ}$ )\n\nA: 0.21\nB: 0.33\nC: 0.42\nD: 0.62\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_534", "problem": "Beloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nThe proton beam enters the electromagnet and is deflected by an angle $\\theta=10^{\\circ}$. Determine the magnetic field strength.\n\n[figure3]", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nBeloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nThe proton beam enters the electromagnet and is deflected by an angle $\\theta=10^{\\circ}$. Determine the magnetic field strength.\n\n[figure3]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~T}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=468&width=1592&top_left_y=438&top_left_x=259", "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=493&width=1268&top_left_y=1339&top_left_x=426", "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-21.jpg?height=339&width=656&top_left_y=362&top_left_x=840" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~T}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1103", "problem": "# A Three-body Problem and LISA \n\n[figure1]\n\nFIGURE 1 Coplanar orbits of three bodies.\n\nA third body of infinitesimal mass $\\mu$ is placed in a coplanar circular orbit about the same centre of mass so that $\\mu$ remains stationary relative to both $M$ and $m$ as shown in Figure 1. Assume that the infinitesimal mass is not collinear with $M$ and $m$. Find the values of the following parameters in terms of $R$ and $r$ :\n\ndistance from $\\mu$ to $m$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n# A Three-body Problem and LISA \n\n[figure1]\n\nFIGURE 1 Coplanar orbits of three bodies.\n\nA third body of infinitesimal mass $\\mu$ is placed in a coplanar circular orbit about the same centre of mass so that $\\mu$ remains stationary relative to both $M$ and $m$ as shown in Figure 1. Assume that the infinitesimal mass is not collinear with $M$ and $m$. Find the values of the following parameters in terms of $R$ and $r$ :\n\ndistance from $\\mu$ to $m$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_27986d152c1a1dded222g-1.jpg?height=918&width=919&top_left_y=506&top_left_x=619" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1278", "problem": "Gaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\n## Manzanares prototype\n\nThe prototype chimney built in Manzanares, Spain, had a height of $195 \\mathrm{~m}$, and a radius $5 \\mathrm{~m}$. The collector is circular with diameter of $244 \\mathrm{~m}$. The specific heat of the air under typical operational conditions of the prototype solar chimney is $1012 \\mathrm{~J} / \\mathrm{kg} \\mathrm{K}$, the density of the hot air is about $0.9 \\mathrm{~kg} / \\mathrm{m}^{3}$, and the typical temperature of the atmosphere $T_{\\text {Air }}=295 \\mathrm{~K}$. In Manzanares, the solar power per unit of horizontal surface is typically $150 \\mathrm{~W} / \\mathrm{m}^{2}$ during a sunny day.\n\nWhat is the efficiency of the prototype power plant? Write down the numerical estimate.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nGaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\n## Manzanares prototype\n\nThe prototype chimney built in Manzanares, Spain, had a height of $195 \\mathrm{~m}$, and a radius $5 \\mathrm{~m}$. The collector is circular with diameter of $244 \\mathrm{~m}$. The specific heat of the air under typical operational conditions of the prototype solar chimney is $1012 \\mathrm{~J} / \\mathrm{kg} \\mathrm{K}$, the density of the hot air is about $0.9 \\mathrm{~kg} / \\mathrm{m}^{3}$, and the typical temperature of the atmosphere $T_{\\text {Air }}=295 \\mathrm{~K}$. In Manzanares, the solar power per unit of horizontal surface is typically $150 \\mathrm{~W} / \\mathrm{m}^{2}$ during a sunny day.\n\nWhat is the efficiency of the prototype power plant? Write down the numerical estimate.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of % (percentage), but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_fd946cfac82ef740b1dag-1.jpg?height=977&width=1644&top_left_y=1453&top_left_x=206" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "% (percentage)" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1614", "problem": "卫星的运动可有地面的观测来决定, 而知道了卫星的运动, 又可以用空间的飞行体或地面上物体的运动, 这都涉及到时间和空间坐标的测定, 为简化分析和计算, 不考虑地球的自转和公转, 把它作惯性系。\n\n根据狭义相对论, 运动的钟比静止的钟慢, 钟在引力场中慢。现在来考虑在上述测量中\n相对论的这两种效应。已知天上卫星的钟与地面观测站的钟零点已对准, 假设卫星在离地面 $h=2.00 \\times 10^{4} \\mathrm{~m}$ 的圆形轨道上运行, 地球半径 $\\mathrm{R}$ 、光速 $\\mathrm{c}$ 和地球表面重力加速度 $\\mathrm{g}$ 取小题 2 中给的值。根据广义相对论, 钟在引力场中变慢的因子是 $\\left(1-2 \\phi / c^{2}\\right)^{1 / 2}, \\phi$ 是钟所在位置的引力势 (引力势能与所受引力作用的物体的质量之比, 取无限远处引力势为 0 ) 的大小, 试问地上的钟 24 小时后, 卫星上的钟的示数与地面上的钟的示数差多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n卫星的运动可有地面的观测来决定, 而知道了卫星的运动, 又可以用空间的飞行体或地面上物体的运动, 这都涉及到时间和空间坐标的测定, 为简化分析和计算, 不考虑地球的自转和公转, 把它作惯性系。\n\n根据狭义相对论, 运动的钟比静止的钟慢, 钟在引力场中慢。现在来考虑在上述测量中\n相对论的这两种效应。已知天上卫星的钟与地面观测站的钟零点已对准, 假设卫星在离地面 $h=2.00 \\times 10^{4} \\mathrm{~m}$ 的圆形轨道上运行, 地球半径 $\\mathrm{R}$ 、光速 $\\mathrm{c}$ 和地球表面重力加速度 $\\mathrm{g}$ 取小题 2 中给的值。\n\n问题:\n根据广义相对论, 钟在引力场中变慢的因子是 $\\left(1-2 \\phi / c^{2}\\right)^{1 / 2}, \\phi$ 是钟所在位置的引力势 (引力势能与所受引力作用的物体的质量之比, 取无限远处引力势为 0 ) 的大小, 试问地上的钟 24 小时后, 卫星上的钟的示数与地面上的钟的示数差多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$$\\mu s$$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$$\\mu s$$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_874", "problem": "The state of the SET is sensitive to electrical potentials created by nearby elements of the quantum circuit (such as quantum bits), and distinguishing between ON and OFF states provides a way to read out the information produced by the quantum computer. The SET in the ON state can be modelled by a resistance $R_{\\mathrm{ON}}=100 \\mathrm{k} \\Omega$ while in the OFF state we can assume the SET to be a complete insulator (neglecting any capacitative connection between the source and the drain via the SET). While it is possible to determine the state of the SET by measuring the response to an input signal through the source, it is faster to do so using RF reflectometry to measure both the amplitude and phase of the reflected signal, i.e. determined the reflectance $\\Gamma$.\n\nThe change in reflectance due to switching of an SET between ON and OFF states is\n\n$$\n\\Delta \\Gamma=\\left|\\Gamma_{\\mathrm{ON}}-\\Gamma_{\\mathrm{OFF}}\\right|,\n$$\n\nwhere $\\Gamma_{\\mathrm{ON}}$ and $\\Gamma_{\\mathrm{OFF}}$ are the reflectances in two different states.\n\n[figure1]\n\nCircuit diagram of transmission cable of impedance $Z_{0}$ connected to an SET.\n\nIn order to increase the change in reflectance, and hence the sensitivity of the RF reflectometry, the circuit is modified by inclusion of an inductor. The intrinsic capacitance due to the device geometry $C_{0} \\approx 0.4 \\mathrm{pF}$ is also taken into account. The RF reflectometry is conducted using a signal of angular frequency $\\omega_{\\mathrm{rf}}$.\n\n[figure2]\n\nModified SET circuit.\n\n Estimate the value of the inductance $L_{0}$ that can result in the change in re flection on the order of one. Calculate your estimate for $L_{0}$ numerically for $\\omega_{\\mathrm{rf}} /(2 \\pi)=100 \\mathrm{MHz}$ and compute the corresponding $\\Delta \\Gamma$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nThe state of the SET is sensitive to electrical potentials created by nearby elements of the quantum circuit (such as quantum bits), and distinguishing between ON and OFF states provides a way to read out the information produced by the quantum computer. The SET in the ON state can be modelled by a resistance $R_{\\mathrm{ON}}=100 \\mathrm{k} \\Omega$ while in the OFF state we can assume the SET to be a complete insulator (neglecting any capacitative connection between the source and the drain via the SET). While it is possible to determine the state of the SET by measuring the response to an input signal through the source, it is faster to do so using RF reflectometry to measure both the amplitude and phase of the reflected signal, i.e. determined the reflectance $\\Gamma$.\n\nThe change in reflectance due to switching of an SET between ON and OFF states is\n\n$$\n\\Delta \\Gamma=\\left|\\Gamma_{\\mathrm{ON}}-\\Gamma_{\\mathrm{OFF}}\\right|,\n$$\n\nwhere $\\Gamma_{\\mathrm{ON}}$ and $\\Gamma_{\\mathrm{OFF}}$ are the reflectances in two different states.\n\n[figure1]\n\nCircuit diagram of transmission cable of impedance $Z_{0}$ connected to an SET.\n\nIn order to increase the change in reflectance, and hence the sensitivity of the RF reflectometry, the circuit is modified by inclusion of an inductor. The intrinsic capacitance due to the device geometry $C_{0} \\approx 0.4 \\mathrm{pF}$ is also taken into account. The RF reflectometry is conducted using a signal of angular frequency $\\omega_{\\mathrm{rf}}$.\n\n[figure2]\n\nModified SET circuit.\n\n Estimate the value of the inductance $L_{0}$ that can result in the change in re flection on the order of one. Calculate your estimate for $L_{0}$ numerically for $\\omega_{\\mathrm{rf}} /(2 \\pi)=100 \\mathrm{MHz}$ and compute the corresponding $\\Delta \\Gamma$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the value of $\\Delta \\Gamma$, the value of $L_0$].\nTheir units are, in order, [None, \\mu H], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-5.jpg?height=451&width=817&top_left_y=845&top_left_x=631", "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-5.jpg?height=329&width=797&top_left_y=1840&top_left_x=641" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, "\\mu H" ], "answer_sequence": [ "the value of $\\Delta \\Gamma$", "the value of $L_0$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1365", "problem": "菲涅尔透镜又称同心圆阶梯透镜, 它是由很多个同轴环带套在一起构成的, 其迎光面是平面，折射面除中心是一个球冠外, 其它环带分别是属于不同球面的球台侧面, 其纵剖面如右图所示。这样的结构可以避免普通大口径球面透镜既厚又重的缺点。菲涅尔透镜的设计主要是确定每个环带的齿形 (即它所属球面的球半径和球心)，各环带都是一个独立的 (部分) 球面透镜, 它们的焦距不同, 但必须保证具有共同的焦点（即图中 F 点）。已知透镜材料的折射率为 $n$, 从透镜中心 $\\mathrm{O}$ (球冠的顶点) 到焦点 $\\mathrm{F}$ 的距离（焦距）为 $f$ (平行于光轴的平行光都能经环带折射后会聚到 $\\mathrm{F}$ 点), 相邻环带的间距为 $d$ ( $d$ 很小,可忽略同一带内的球面像差; $d$ 又不是非常小，可忽略衍射效应）。求\n\n[图1]每个环带所属球面的球半径和球心到焦点的距离;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n菲涅尔透镜又称同心圆阶梯透镜, 它是由很多个同轴环带套在一起构成的, 其迎光面是平面，折射面除中心是一个球冠外, 其它环带分别是属于不同球面的球台侧面, 其纵剖面如右图所示。这样的结构可以避免普通大口径球面透镜既厚又重的缺点。菲涅尔透镜的设计主要是确定每个环带的齿形 (即它所属球面的球半径和球心)，各环带都是一个独立的 (部分) 球面透镜, 它们的焦距不同, 但必须保证具有共同的焦点（即图中 F 点）。已知透镜材料的折射率为 $n$, 从透镜中心 $\\mathrm{O}$ (球冠的顶点) 到焦点 $\\mathrm{F}$ 的距离（焦距）为 $f$ (平行于光轴的平行光都能经环带折射后会聚到 $\\mathrm{F}$ 点), 相邻环带的间距为 $d$ ( $d$ 很小,可忽略同一带内的球面像差; $d$ 又不是非常小，可忽略衍射效应）。求\n\n[图1]\n\n问题:\n每个环带所属球面的球半径和球心到焦点的距离;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-04.jpg?height=317&width=322&top_left_y=1184&top_left_x=707", "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-22.jpg?height=429&width=690&top_left_y=759&top_left_x=1071", "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-22.jpg?height=315&width=781&top_left_y=1927&top_left_x=977" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1200", "problem": "## To Commemorate the Centenary of Rutherford's Atomic Nucleus: the Scattering of an Ion by a Neutral Atom\n\n[figure1]\n\nAn ion of mass $m$, charge $Q$, is moving with an initial non-relativistic speed $v_{0}$ from a great distance towards the vicinity of a neutral atom of mass $M>>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nIf the impact parameter $b$ is less than a critical value $b_{0}$, the ion will descend along a spiral to the atom. In such a case, the ion will be neutralized, and the atom is, in turn, charged. This process is known as the \"charge exchange\" interaction. What is the cross sectional area $A=\\pi b_{0}^{2}$ of this \"charge exchange\" collision of the atom as seen by the ion?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n## To Commemorate the Centenary of Rutherford's Atomic Nucleus: the Scattering of an Ion by a Neutral Atom\n\n[figure1]\n\nAn ion of mass $m$, charge $Q$, is moving with an initial non-relativistic speed $v_{0}$ from a great distance towards the vicinity of a neutral atom of mass $M>>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nIf the impact parameter $b$ is less than a critical value $b_{0}$, the ion will descend along a spiral to the atom. In such a case, the ion will be neutralized, and the atom is, in turn, charged. This process is known as the \"charge exchange\" interaction. What is the cross sectional area $A=\\pi b_{0}^{2}$ of this \"charge exchange\" collision of the atom as seen by the ion?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_51c4dc0e7c52a1226310g-1.jpg?height=462&width=1495&top_left_y=701&top_left_x=315" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_846", "problem": "The most outstanding fact in cosmology is that our universe is expanding. Space is continuously created as time lapses. The expansion of space indicates that, when the universe expands, the distance between objects in our universe also expands. It is convenient to use \"comoving\" coordinate system $\\vec{r}=(x, y, z)$ to label points in our expanding universe, in which the coordinate distance $\\Delta r=\\left|\\vec{r}_{2}-\\vec{r}_{1}\\right|=\\sqrt{\\left(x_{2}-x_{1}\\right)^{2}+\\left(y_{2}-y_{1}\\right)^{2}+\\left(z_{2}-z_{1}\\right)^{2}}$ between objects 1 and 2 does not change. (Here we assume no peculiar motion, i.e. no additional motion of those objects other than the motion following the expansion of the universe.) The situation is illustrated in the figure below (the figure has two space dimensions, but our universe actually has three space dimensions).\n\n[figure1]\n\nThe modern theory of cosmology is built upon Einstein's general relativity. However, under proper assumptions, a simplified understanding under the framework of Newton's theory of gravity is also possible. In the following questions, we shall work in the framework of Newton's gravity.\n\nTo measure the physical distance, a \"scale factor\" $a(t)$ is introduced such that the physical distance $\\Delta r_{\\mathrm{p}}$ between the comoving points $\\vec{r}_{1}$ and $\\vec{r}_{2}$ is \n$$\n\\Delta r_{\\mathrm{p}}=a(t) \\Delta r,\n$$\n\nThe expansion of the universe implies that $a(t)$ is an increasing function of time.\n\nOn large scales - scales much larger than galaxies and their clusters - our universe is approximately homogeneous and isotropic. So let us consider a toy model of our universe, which is filled with uniformly distributed particles. There are so many particles, such that we model them as a continuous fluid. Furthermore, we assume the number of particles is\nconserved.\n\nCurrently, our universe is dominated by non-relativistic matter, whose kinetic energy is negligible compared to its mass energy. Let $\\rho_{\\mathrm{m}}(t)$ be the physical energy density (i.e. energyper unit physical volume, which is dominated by mass energy for non-relativistic matter and the gravitational potential energy is not counted as part of the \"physical energy density\") of non-relativistic matter at time $t$. We use $t_{0}$ to denote the present time.\n\nBesides non-relativistic matter, there is also a small amount of radiation in our current universe, which is made of massless particles, for example, photons. The physical wavelength of massless particles increases with the universe expansion as $\\lambda_{\\mathrm{p}} \\propto a(t)$. Let the physical energy density of radiation be $\\rho_{\\mathrm{r}}(t)$.\n\nConsider a gas of non-interacting photons which has thermal equilibrium distribution. In this situation, the temperature of the photon depends on time as $T(t) \\propto[a(t)]^{\\gamma}$.\n\nConsider a star $\\mathrm{S}$. At the present time $t_{0}$, the star is at a physical distance $r_{\\mathrm{p}}=a\\left(t_{0}\\right) r$ away from us, where $r$ is the comoving distance. Here we ignore the peculiar motion, i.e. assume that both the star and us just follow the expansion of the universe without additional motion.\n\nThe star is emitting energy in the form of light at power $P_{\\mathrm{e}}$, which is isotropic in every direction. We use a telescope to observe its starlight. For simplicity, assume the telescope can observe all frequencies of light with $100 \\%$ efficiency. Let the area of the telescope lens be $A$.\n\nDerive the power received by the telescope $P_{\\mathrm{r}}$ from the star $\\mathrm{S}$, as a function of $r, A, P_{\\mathrm{e}}$, the scale factor $a\\left(t_{\\mathrm{e}}\\right)$ at the starlight emission time $t_{\\mathrm{e}}$, and the present (i.e. at the observation 4 points time) scale factor $a\\left(t_{0}\\right)$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nThe most outstanding fact in cosmology is that our universe is expanding. Space is continuously created as time lapses. The expansion of space indicates that, when the universe expands, the distance between objects in our universe also expands. It is convenient to use \"comoving\" coordinate system $\\vec{r}=(x, y, z)$ to label points in our expanding universe, in which the coordinate distance $\\Delta r=\\left|\\vec{r}_{2}-\\vec{r}_{1}\\right|=\\sqrt{\\left(x_{2}-x_{1}\\right)^{2}+\\left(y_{2}-y_{1}\\right)^{2}+\\left(z_{2}-z_{1}\\right)^{2}}$ between objects 1 and 2 does not change. (Here we assume no peculiar motion, i.e. no additional motion of those objects other than the motion following the expansion of the universe.) The situation is illustrated in the figure below (the figure has two space dimensions, but our universe actually has three space dimensions).\n\n[figure1]\n\nThe modern theory of cosmology is built upon Einstein's general relativity. However, under proper assumptions, a simplified understanding under the framework of Newton's theory of gravity is also possible. In the following questions, we shall work in the framework of Newton's gravity.\n\nTo measure the physical distance, a \"scale factor\" $a(t)$ is introduced such that the physical distance $\\Delta r_{\\mathrm{p}}$ between the comoving points $\\vec{r}_{1}$ and $\\vec{r}_{2}$ is \n$$\n\\Delta r_{\\mathrm{p}}=a(t) \\Delta r,\n$$\n\nThe expansion of the universe implies that $a(t)$ is an increasing function of time.\n\nOn large scales - scales much larger than galaxies and their clusters - our universe is approximately homogeneous and isotropic. So let us consider a toy model of our universe, which is filled with uniformly distributed particles. There are so many particles, such that we model them as a continuous fluid. Furthermore, we assume the number of particles is\nconserved.\n\nCurrently, our universe is dominated by non-relativistic matter, whose kinetic energy is negligible compared to its mass energy. Let $\\rho_{\\mathrm{m}}(t)$ be the physical energy density (i.e. energyper unit physical volume, which is dominated by mass energy for non-relativistic matter and the gravitational potential energy is not counted as part of the \"physical energy density\") of non-relativistic matter at time $t$. We use $t_{0}$ to denote the present time.\n\nBesides non-relativistic matter, there is also a small amount of radiation in our current universe, which is made of massless particles, for example, photons. The physical wavelength of massless particles increases with the universe expansion as $\\lambda_{\\mathrm{p}} \\propto a(t)$. Let the physical energy density of radiation be $\\rho_{\\mathrm{r}}(t)$.\n\nConsider a gas of non-interacting photons which has thermal equilibrium distribution. In this situation, the temperature of the photon depends on time as $T(t) \\propto[a(t)]^{\\gamma}$.\n\nConsider a star $\\mathrm{S}$. At the present time $t_{0}$, the star is at a physical distance $r_{\\mathrm{p}}=a\\left(t_{0}\\right) r$ away from us, where $r$ is the comoving distance. Here we ignore the peculiar motion, i.e. assume that both the star and us just follow the expansion of the universe without additional motion.\n\nThe star is emitting energy in the form of light at power $P_{\\mathrm{e}}$, which is isotropic in every direction. We use a telescope to observe its starlight. For simplicity, assume the telescope can observe all frequencies of light with $100 \\%$ efficiency. Let the area of the telescope lens be $A$.\n\nDerive the power received by the telescope $P_{\\mathrm{r}}$ from the star $\\mathrm{S}$, as a function of $r, A, P_{\\mathrm{e}}$, the scale factor $a\\left(t_{\\mathrm{e}}\\right)$ at the starlight emission time $t_{\\mathrm{e}}$, and the present (i.e. at the observation 4 points time) scale factor $a\\left(t_{0}\\right)$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_88fc0d3da9200bd5a832g-1.jpg?height=657&width=1455&top_left_y=1116&top_left_x=312" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1437", "problem": "库伦扭摆装置如图所示, 在细银丝下悬挂一根绝缘棒, 棒水平静止; 棒的两端各固定一相同的金属小球 $\\mathrm{a}$ 和 $\\mathrm{b}$, 另一相同的金属小球 $\\mathrm{c}$ 固定在插入的坚直杆上, 三个小球位于同一水平圆周上, 圆心为棒的悬点 $\\mathrm{O}$ 。细银丝自然悬挂时, $\\mathrm{a} 、 \\mathrm{c}$ 球对 $\\mathrm{O}$ 点的张角 $\\alpha=4^{\\circ}$ 。现在使 $\\mathrm{a}$ 和 $\\mathrm{c}$ 带相同电荷, 库伦力使细银丝扭转, 张角 $\\alpha$ 增大, 反向转动细银丝上端的旋钮可使张角 $\\alpha$ 变小; 若将旋钮缓慢反向转过角度 $\\beta=30^{\\circ}$, 可使小球 $\\mathrm{a}$ 最终回到原来位置, 这时细银丝的扭力矩与球 $\\mathrm{a}$ 所受球 $\\mathrm{c}$ 的静电力的力矩平衡。设细银丝的扭转回复力矩与银丝的转角 $\\beta$ 成正比。为使最后 $\\mathrm{a} 、 \\mathrm{c}$ 对 $\\mathrm{O}$ 点的张角 $\\alpha=2^{\\circ}$, 旋钮反向转过的角度应为 [\nA. $\\beta=45^{\\circ}$\nB. $\\beta=60^{\\circ}$\nC. $\\beta=90^{\\circ}$\nD. $\\beta=120^{\\circ}$\n\n[图1]\nA: $\\beta=45^{\\circ}$\nB: $\\beta=60^{\\circ}$\nC: $\\beta=90^{\\circ}$\nD: $\\beta=120^{\\circ}$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n库伦扭摆装置如图所示, 在细银丝下悬挂一根绝缘棒, 棒水平静止; 棒的两端各固定一相同的金属小球 $\\mathrm{a}$ 和 $\\mathrm{b}$, 另一相同的金属小球 $\\mathrm{c}$ 固定在插入的坚直杆上, 三个小球位于同一水平圆周上, 圆心为棒的悬点 $\\mathrm{O}$ 。细银丝自然悬挂时, $\\mathrm{a} 、 \\mathrm{c}$ 球对 $\\mathrm{O}$ 点的张角 $\\alpha=4^{\\circ}$ 。现在使 $\\mathrm{a}$ 和 $\\mathrm{c}$ 带相同电荷, 库伦力使细银丝扭转, 张角 $\\alpha$ 增大, 反向转动细银丝上端的旋钮可使张角 $\\alpha$ 变小; 若将旋钮缓慢反向转过角度 $\\beta=30^{\\circ}$, 可使小球 $\\mathrm{a}$ 最终回到原来位置, 这时细银丝的扭力矩与球 $\\mathrm{a}$ 所受球 $\\mathrm{c}$ 的静电力的力矩平衡。设细银丝的扭转回复力矩与银丝的转角 $\\beta$ 成正比。为使最后 $\\mathrm{a} 、 \\mathrm{c}$ 对 $\\mathrm{O}$ 点的张角 $\\alpha=2^{\\circ}$, 旋钮反向转过的角度应为 [\nA. $\\beta=45^{\\circ}$\nB. $\\beta=60^{\\circ}$\nC. $\\beta=90^{\\circ}$\nD. $\\beta=120^{\\circ}$\n\n[图1]\n\nA: $\\beta=45^{\\circ}$\nB: $\\beta=60^{\\circ}$\nC: $\\beta=90^{\\circ}$\nD: $\\beta=120^{\\circ}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-01.jpg?height=691&width=348&top_left_y=1682&top_left_x=1662" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_200", "problem": "A small rock is tied to a massless string of length $5 \\mathrm{~m}$. The density of the rock is twice the density of the water. The rock is lowered into the water, while the other end of the string is attached to a pivot. Neglect any resistive forces from the water. The rock oscillates like a pendulum with angular frequency of which of the following?\nA: $1 \\mathrm{rad} / \\mathrm{s} $ \nB: $0.7 \\mathrm{rad} / \\mathrm{s}$\nC: $0.5 \\mathrm{rad} / \\mathrm{s}$\nD: $1.4 \\mathrm{rad} / \\mathrm{s}$\nE: $2 \\mathrm{rad} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA small rock is tied to a massless string of length $5 \\mathrm{~m}$. The density of the rock is twice the density of the water. The rock is lowered into the water, while the other end of the string is attached to a pivot. Neglect any resistive forces from the water. The rock oscillates like a pendulum with angular frequency of which of the following?\n\nA: $1 \\mathrm{rad} / \\mathrm{s} $ \nB: $0.7 \\mathrm{rad} / \\mathrm{s}$\nC: $0.5 \\mathrm{rad} / \\mathrm{s}$\nD: $1.4 \\mathrm{rad} / \\mathrm{s}$\nE: $2 \\mathrm{rad} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_237", "problem": "The characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nStart with a case, where a point small charged object with a charge $+Q$ is fastened to the table. The center of the dipole is fixed at the distance $L$ from the charged object (see Figure 1). The dipole consists of two identical small balls fastened to the tiny, rigid rod with a length $d, d \\ll L$, so that the moment of inertia can be ignored. Each of the balls has a mass $m$ and have charge $+q$ and $-q$. The dipole can rotate around its center in a plane parallel to the surface of the smooth table.\n\n[figure1]\n\nFigure 1: Schematic representation of the system used in section 1.1\n\nFind this critical initial velocity $v_{c r}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nStart with a case, where a point small charged object with a charge $+Q$ is fastened to the table. The center of the dipole is fixed at the distance $L$ from the charged object (see Figure 1). The dipole consists of two identical small balls fastened to the tiny, rigid rod with a length $d, d \\ll L$, so that the moment of inertia can be ignored. Each of the balls has a mass $m$ and have charge $+q$ and $-q$. The dipole can rotate around its center in a plane parallel to the surface of the smooth table.\n\n[figure1]\n\nFigure 1: Schematic representation of the system used in section 1.1\n\nFind this critical initial velocity $v_{c r}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ed4e92416bdbac30298dg-1.jpg?height=179&width=1171&top_left_y=1187&top_left_x=477" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_105", "problem": "A block floats partially submerged in a container of liquid. When the entire container is accelerated upward, which of the following happens? Assume that both the liquid and the block are incompressible.\nA: The block descends down lower into the liquid.\nB: The block ascends up higher in the liquid.\nC: The block does not ascend nor descend in the liquid.\nD: The answer depends on the direction of motion of the container.\nE: The answer depends on the rate of change of the acceleration\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA block floats partially submerged in a container of liquid. When the entire container is accelerated upward, which of the following happens? Assume that both the liquid and the block are incompressible.\n\nA: The block descends down lower into the liquid.\nB: The block ascends up higher in the liquid.\nC: The block does not ascend nor descend in the liquid.\nD: The answer depends on the direction of motion of the container.\nE: The answer depends on the rate of change of the acceleration\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1405", "problem": "如图, 太空中有一由同心的内球和球壳构成的实验装置, 内球和球壳内表面之间为真空。内球半径为 $r=0.200 \\mathrm{~m}$, 温度保持恒定, 比辐射率为 $e=0.800$; 球壳的导热系数为 $\\kappa=1.00 \\times 10^{-2} \\mathrm{~J} \\cdot \\mathrm{m}^{-1} \\cdot \\mathrm{s}^{-1} \\cdot \\mathrm{K}^{-1}$, 内、外半径分别为 $R_{1}=0.900 \\mathrm{~m} 、 R_{2}=1.00 \\mathrm{~m}$ ，各表面的热表面可视为黑体; 该实验装置已处于热稳定状态, 此时球壳内表面比辐射率为 $E=0.800$ 。斯特藩常量为 $\\sigma=5.67 \\times 10^{-8} \\mathrm{~W} \\cdot \\mathrm{m}^{-2} \\cdot \\mathrm{K}^{-4}$, 宇宙微波背景辐射温度为 $T=2.73 \\mathrm{~K}$ 。若单位时间内由球壳内表面传递到球壳外表面的热量为 $Q=44.0 \\mathrm{~W}$, 求\n\n[图1]\n\n已知: 物体表面单位面积上的辐射功率与同温度下的黑体在该表面单位面积上的辐射功率之比称为比辐射率。当辐射照射到物体表面时, 物体表面单位面积吸收的辐射功率与照射到物体单位面积上的辐射功率之比称为吸收比。物体在某一温度下的吸收比等于其在同一温度下的比辐射率。当物体内某处在 $\\mathrm{z}$ 方向（热流方向）每单位距离温度的增量为 $\\frac{\\mathrm{d} T}{\\mathrm{~d} z}$ 时, 物体内该处单位时间在 $z$ 方向每单位面积流过的热量为 $-\\kappa \\frac{\\mathrm{d} T}{\\mathrm{~d} z}$, 此即傅里叶热传导定律。球壳内表面温度 $T_{1}$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 太空中有一由同心的内球和球壳构成的实验装置, 内球和球壳内表面之间为真空。内球半径为 $r=0.200 \\mathrm{~m}$, 温度保持恒定, 比辐射率为 $e=0.800$; 球壳的导热系数为 $\\kappa=1.00 \\times 10^{-2} \\mathrm{~J} \\cdot \\mathrm{m}^{-1} \\cdot \\mathrm{s}^{-1} \\cdot \\mathrm{K}^{-1}$, 内、外半径分别为 $R_{1}=0.900 \\mathrm{~m} 、 R_{2}=1.00 \\mathrm{~m}$ ，各表面的热表面可视为黑体; 该实验装置已处于热稳定状态, 此时球壳内表面比辐射率为 $E=0.800$ 。斯特藩常量为 $\\sigma=5.67 \\times 10^{-8} \\mathrm{~W} \\cdot \\mathrm{m}^{-2} \\cdot \\mathrm{K}^{-4}$, 宇宙微波背景辐射温度为 $T=2.73 \\mathrm{~K}$ 。若单位时间内由球壳内表面传递到球壳外表面的热量为 $Q=44.0 \\mathrm{~W}$, 求\n\n[图1]\n\n已知: 物体表面单位面积上的辐射功率与同温度下的黑体在该表面单位面积上的辐射功率之比称为比辐射率。当辐射照射到物体表面时, 物体表面单位面积吸收的辐射功率与照射到物体单位面积上的辐射功率之比称为吸收比。物体在某一温度下的吸收比等于其在同一温度下的比辐射率。当物体内某处在 $\\mathrm{z}$ 方向（热流方向）每单位距离温度的增量为 $\\frac{\\mathrm{d} T}{\\mathrm{~d} z}$ 时, 物体内该处单位时间在 $z$ 方向每单位面积流过的热量为 $-\\kappa \\frac{\\mathrm{d} T}{\\mathrm{~d} z}$, 此即傅里叶热传导定律。\n\n问题:\n球壳内表面温度 $T_{1}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~K}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-03.jpg?height=286&width=286&top_left_y=1276&top_left_x=1479" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~K}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_784", "problem": "The moon orbits the Earth. What is the Newton's Third Law reaction force to the gravitational force of the Earth on the Moon?\nA: The normal force of the Moon on the Earth\nB: The normal force of the Earth on the Moon\nC: The gravitational force of the Earth on the Moon\nD: The gravitational force of the Moon on the Earth\nE: There is no reaction force in this situation, as the objects are not in contact\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe moon orbits the Earth. What is the Newton's Third Law reaction force to the gravitational force of the Earth on the Moon?\n\nA: The normal force of the Moon on the Earth\nB: The normal force of the Earth on the Moon\nC: The gravitational force of the Earth on the Moon\nD: The gravitational force of the Moon on the Earth\nE: There is no reaction force in this situation, as the objects are not in contact\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_712", "problem": "A ball of mass $m=200 \\mathrm{~g}$ is hung in the middle of horizontal elastic cord of length $L=1.5 \\mathrm{~m}$, attached to the ceiling, as shown in figure below.\n\nThe elastic constant of the cord is $k=300 \\mathrm{~N} / \\mathrm{m}$.\n\n[figure1]\n\n What is the distance from the ceiling to the center of the ball, assuming that the cord was not stretched when it was horizontal?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA ball of mass $m=200 \\mathrm{~g}$ is hung in the middle of horizontal elastic cord of length $L=1.5 \\mathrm{~m}$, attached to the ceiling, as shown in figure below.\n\nThe elastic constant of the cord is $k=300 \\mathrm{~N} / \\mathrm{m}$.\n\n[figure1]\n\n What is the distance from the ceiling to the center of the ball, assuming that the cord was not stretched when it was horizontal?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of m, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-09.jpg?height=263&width=1657&top_left_y=855&top_left_x=234" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_861", "problem": "The state of the SET is sensitive to electrical potentials created by nearby elements of the quantum circuit (such as quantum bits), and distinguishing between ON and OFF states provides a way to read out the information produced by the quantum computer. The SET in the ON state can be modelled by a resistance $R_{\\mathrm{ON}}=100 \\mathrm{k} \\Omega$ while in the OFF state we can assume the SET to be a complete insulator (neglecting any capacitative connection between the source and the drain via the SET). While it is possible to determine the state of the SET by measuring the response to an input signal through the source, it is faster to do so using RF reflectometry to measure both the amplitude and phase of the reflected signal, i.e. determined the reflectance $\\Gamma$.\n\nThe change in reflectance due to switching of an SET between ON and OFF states is\n\n$$\n\\Delta \\Gamma=\\left|\\Gamma_{\\mathrm{ON}}-\\Gamma_{\\mathrm{OFF}}\\right|,\n$$\n\nwhere $\\Gamma_{\\mathrm{ON}}$ and $\\Gamma_{\\mathrm{OFF}}$ are the reflectances in two different states.\n\n[figure1]\n\nCircuit diagram of transmission cable of impedance $Z_{0}$ connected to an SET.\n\n Find the change in reflectance $\\Delta \\Gamma$ between the conductive and insulating states for a typical SET connected to a co-axial cable with impedance of $50 \\Omega$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe state of the SET is sensitive to electrical potentials created by nearby elements of the quantum circuit (such as quantum bits), and distinguishing between ON and OFF states provides a way to read out the information produced by the quantum computer. The SET in the ON state can be modelled by a resistance $R_{\\mathrm{ON}}=100 \\mathrm{k} \\Omega$ while in the OFF state we can assume the SET to be a complete insulator (neglecting any capacitative connection between the source and the drain via the SET). While it is possible to determine the state of the SET by measuring the response to an input signal through the source, it is faster to do so using RF reflectometry to measure both the amplitude and phase of the reflected signal, i.e. determined the reflectance $\\Gamma$.\n\nThe change in reflectance due to switching of an SET between ON and OFF states is\n\n$$\n\\Delta \\Gamma=\\left|\\Gamma_{\\mathrm{ON}}-\\Gamma_{\\mathrm{OFF}}\\right|,\n$$\n\nwhere $\\Gamma_{\\mathrm{ON}}$ and $\\Gamma_{\\mathrm{OFF}}$ are the reflectances in two different states.\n\n[figure1]\n\nCircuit diagram of transmission cable of impedance $Z_{0}$ connected to an SET.\n\n Find the change in reflectance $\\Delta \\Gamma$ between the conductive and insulating states for a typical SET connected to a co-axial cable with impedance of $50 \\Omega$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-5.jpg?height=451&width=817&top_left_y=845&top_left_x=631" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_696", "problem": "A positive charge $q$ is placed in front two semiinfinite conducting walls as shown below. The walls are grounded. Which arrow best represents the direction force on the charge $q$, if $d_{1}0\\right.$,但 $A_{0}$ 并不是已知量）。重力加速度大小为 $g$, 假设最大静摩擦力等于滑动摩擦力。\n\n[图1]如果小球只能完成 $n$ 次往返运动 (向右经过原点, 然后向左经过原点, 算 1 次往返),求小球最终静止的位置, 和此种情形下 $A_{0}$ 应满足的条件;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一劲度系数为 $k$ 的轻弹簧左端固定, 右端连一质量为 $m$ 的小球; 弹簧水平, 它处于自然状态时小球位于坐标原点 $O$; 小球可在水平地面上滑动, 它与地面之间的动摩擦因数为 $\\mu$ 。初始时小球速度为零, 将此时弹簧相对于其原长的伸长记为 $-A_{0}\\left(A_{0}>0\\right.$,但 $A_{0}$ 并不是已知量）。重力加速度大小为 $g$, 假设最大静摩擦力等于滑动摩擦力。\n\n[图1]\n\n问题:\n如果小球只能完成 $n$ 次往返运动 (向右经过原点, 然后向左经过原点, 算 1 次往返),求小球最终静止的位置, 和此种情形下 $A_{0}$ 应满足的条件;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-01.jpg?height=137&width=511&top_left_y=1362&top_left_x=1229" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_146", "problem": "A spring has a length of 1.0 meter when there is no tension on it. The spring is then stretched between two points 10 meters apart. A wave pulse travels between the two end points in the spring in a time of 1.0 seconds. The spring is now stretched between two points that are 20 meters apart. The new time it takes for a wave pulse to travel between the ends of the spring is closest to\nA: 0.5 seconds\nB: 0.7 seconds\nC: 1 second \nD: 1.4 seconds\nE: 2 seconds\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA spring has a length of 1.0 meter when there is no tension on it. The spring is then stretched between two points 10 meters apart. A wave pulse travels between the two end points in the spring in a time of 1.0 seconds. The spring is now stretched between two points that are 20 meters apart. The new time it takes for a wave pulse to travel between the ends of the spring is closest to\n\nA: 0.5 seconds\nB: 0.7 seconds\nC: 1 second \nD: 1.4 seconds\nE: 2 seconds\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_411", "problem": "A roller consists of a solid homogeneous cylinder of mass $M$ and radius $r$ it rests on a horizontal table and is attached to a wall via a helical spring of spring constant $k$ (see figure). The spring can be assumed to be of a negligible mass and ideal, i.e. the Hooke's law remains valid for arbitrarily large deformations.\n\n[figure1]\n\nAt first, let as assume that there is no friction between the cylinder and the table. The roller is pushed aside and released; find the period of oscillations $T_{0}$", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA roller consists of a solid homogeneous cylinder of mass $M$ and radius $r$ it rests on a horizontal table and is attached to a wall via a helical spring of spring constant $k$ (see figure). The spring can be assumed to be of a negligible mass and ideal, i.e. the Hooke's law remains valid for arbitrarily large deformations.\n\n[figure1]\n\nAt first, let as assume that there is no friction between the cylinder and the table. The roller is pushed aside and released; find the period of oscillations $T_{0}$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_706aca6df357b4c9a255g-1.jpg?height=142&width=511&top_left_y=1581&top_left_x=140" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1445", "problem": "从楼顶边缘以大小为 $v_{0}$ 的初速度坚直上抛一小球; 经过 $t_{0}$ 时间后在楼顶边缘从静止开始释放另一小球。若要求两小球同时落地, 忽略空气阻力, 则 $v_{0}$ 的取值范围和抛出点的高度 $h$ 应为\nA: $\\frac{1}{2} g t_{0} \\leq v_{0}0$, then the conductance of the SET is reduced (high-resistance OFF).\n\nFor the number of electrons on the quantum dot to remain well-defined, certain conditions need to be satisfied. Firstly, if electrons in the source or drain have thermal energies sufficient to move spontaneously onto the quantum dot, the contrast between the ON and OFF states will disappear.\n\nFind a condition on the temperature of the electrons so that electrons cannot move onto the quantum dot by thermal excitation.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA single electron transistor (SET) consists of a quantum dot, which is a small isolated conductor where electrons can be localised, and of several electrodes in its vicinity. The gate electrode couples capacitatively to the quantum dot, while the two other electrodes --- the source and the drain --- are connected via tunnel junctions, through which electrons can tunnel due to quantum mechanics. A simplified circuit diagram for an SET is shown in the figure.\n[figure1]\n\nCircuit diagram representation of an SET. QD is the quantum dot, $\\mathrm{S}$ is the source, $\\mathrm{D}$ is the drain and $\\mathrm{G}$ is the gate.\n\nThe capacitance of the gate is $C_{g}$ and the capacitance of the tunnel junctions is $C_{t} \\ll C_{g}$. Consider $C_{g}$ to be the total capacitance of the quantum dot. In this part of the problem, the source and the drain are held at zero potential, and the voltage on the gate electrode is fixed at $V_{g}$.\n\nConsider a state of the SET in which the quantum dot contains $n$ electrons.\n\nIf $\\Delta E_{\\mathcal{N}}=0$ then tunnelling of electrons does not require extra energy and SET is in a highly conductive ON state. If $\\Delta E_{\\mathcal{N}}>0$, then the conductance of the SET is reduced (high-resistance OFF).\n\nFor the number of electrons on the quantum dot to remain well-defined, certain conditions need to be satisfied. Firstly, if electrons in the source or drain have thermal energies sufficient to move spontaneously onto the quantum dot, the contrast between the ON and OFF states will disappear.\n\nFind a condition on the temperature of the electrons so that electrons cannot move onto the quantum dot by thermal excitation.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-3.jpg?height=582&width=868&top_left_y=2076&top_left_x=594" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1459", "problem": "Ioffe-Pritchard 磁阱可用来束缚原子的运动, 其主要部分如图所示。四根均通有恒定电流 $I$ 的长直导线 1、2、3、4 都垂直于 $x-y$ 平面, 它们与 $x-y$ 平面的交点是边长为 $2 a$ 、中心在原点 $O$ 的正方形的顶点, 导线 1、2 所在平面与 $x$ 轴平行, 各导线中电流方向已在图中标出。整个装置置于匀强磁场 $\\boldsymbol{B}_{0}=B_{0} \\boldsymbol{k}(\\boldsymbol{k}$ 为 $z$ 轴正方向单位矢量）中。已知真空磁导率为 $\\mu_{0}$ 。\n\n[图1]求电流在通电导线外产生的总磁场的空间分布;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\nIoffe-Pritchard 磁阱可用来束缚原子的运动, 其主要部分如图所示。四根均通有恒定电流 $I$ 的长直导线 1、2、3、4 都垂直于 $x-y$ 平面, 它们与 $x-y$ 平面的交点是边长为 $2 a$ 、中心在原点 $O$ 的正方形的顶点, 导线 1、2 所在平面与 $x$ 轴平行, 各导线中电流方向已在图中标出。整个装置置于匀强磁场 $\\boldsymbol{B}_{0}=B_{0} \\boldsymbol{k}(\\boldsymbol{k}$ 为 $z$ 轴正方向单位矢量）中。已知真空磁导率为 $\\mu_{0}$ 。\n\n[图1]\n\n问题:\n求电流在通电导线外产生的总磁场的空间分布;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-02.jpg?height=349&width=343&top_left_y=1119&top_left_x=1413" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1606", "problem": "一足球运动员 1 自 $\\mathrm{A}$ 点向球门的 $\\mathrm{B}$ 点踢出球, 已知 $\\mathrm{A} 、 \\mathrm{~B}$ 之间的距离为 $S$, 球自 $\\mathrm{A}$ 向 $\\mathrm{B}$ 的运动可视为水平地面上的匀速直线运动, 速率为 $u$ 。另一足球运动员 2 到 $\\mathrm{AB}$ 连线的距离为 $l$, 到 $\\mathrm{A}$ 、 $\\mathrm{B}$ 两点的距离相等。运动员 1 踢出球后, 运动员 2 以匀速 $v$ 沿直线去拦截该球。设运动员 2 开始出发去拦截球的时刻与球被运动员 1 踢出球的时刻相同。如果运动员 2 能拦截到球, 求运动员 2 开始出发去拦截球直至拦截到球的时间间隔、球被拦截时球到 $A$ 点的距离、球到运动员 2 出发点的距离和运动员 2 运动的方向与 $A 、 B$ 连线的夹角;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一足球运动员 1 自 $\\mathrm{A}$ 点向球门的 $\\mathrm{B}$ 点踢出球, 已知 $\\mathrm{A} 、 \\mathrm{~B}$ 之间的距离为 $S$, 球自 $\\mathrm{A}$ 向 $\\mathrm{B}$ 的运动可视为水平地面上的匀速直线运动, 速率为 $u$ 。另一足球运动员 2 到 $\\mathrm{AB}$ 连线的距离为 $l$, 到 $\\mathrm{A}$ 、 $\\mathrm{B}$ 两点的距离相等。运动员 1 踢出球后, 运动员 2 以匀速 $v$ 沿直线去拦截该球。设运动员 2 开始出发去拦截球的时刻与球被运动员 1 踢出球的时刻相同。\n\n问题:\n如果运动员 2 能拦截到球, 求运动员 2 开始出发去拦截球直至拦截到球的时间间隔、球被拦截时球到 $A$ 点的距离、球到运动员 2 出发点的距离和运动员 2 运动的方向与 $A 、 B$ 连线的夹角;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1152", "problem": "A point charge $q$ is placed in the vicinity of a grounded metallic sphere of radius $R$ [see Fig. 1(a)], and consequently a surface charge distribution is induced on the sphere. To calculate the electric field and potential from the distribution of the surface charge is a formidable task. However, the calculation can be considerably simplified by using the so called method of images. In this method, the electric field and potential produced by the charge distributed on the sphere can be represented as an electric field and potential of a single point charge $q$ ' placed inside the sphere (you do not have to prove it). **Note: The electric field of this image charge $q$ ' reproduces the electric field and the potential only outside the sphere (including its surface).**\n\n[figure1]\n\nThe symmetry of the problem dictates that the charge $q$ ' should be placed on the line connecting the point charge $q$ and the center of the sphere [see Fig. 1(b)].\n\nWhat is the value of the potential on the sphere?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA point charge $q$ is placed in the vicinity of a grounded metallic sphere of radius $R$ [see Fig. 1(a)], and consequently a surface charge distribution is induced on the sphere. To calculate the electric field and potential from the distribution of the surface charge is a formidable task. However, the calculation can be considerably simplified by using the so called method of images. In this method, the electric field and potential produced by the charge distributed on the sphere can be represented as an electric field and potential of a single point charge $q$ ' placed inside the sphere (you do not have to prove it). **Note: The electric field of this image charge $q$ ' reproduces the electric field and the potential only outside the sphere (including its surface).**\n\n[figure1]\n\nThe symmetry of the problem dictates that the charge $q$ ' should be placed on the line connecting the point charge $q$ and the center of the sphere [see Fig. 1(b)].\n\nWhat is the value of the potential on the sphere?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_843c51f18af1a9802d4bg-1.jpg?height=774&width=1627&top_left_y=1046&top_left_x=220" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_456", "problem": "The resultant amplitude of the waves from the slits arriving at the screen at a maxima (a bright fringe), where the incoming waves arrive in phase, is 2A.\n\nThe intensity at the maxima is 4I.This question relates to ways in which light energy may be concentrated in an interference pattern.\n\nFigure 1 shows wave fronts at normal incidence on a Young's double slit arrangement, illuminating them so that they both radiate in phase, each with an amplitude $A$ at the screen. One such slit alone will cause an intensity of illumination of $I$, where $I \\propto A^{2}$, in the central region of the screen where the Young's fringes pattern forms.\n\n[figure1]\nFigure: Young's double slits.\n\nWhat is the intensity at the minima？", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe resultant amplitude of the waves from the slits arriving at the screen at a maxima (a bright fringe), where the incoming waves arrive in phase, is 2A.\n\nThe intensity at the maxima is 4I.\n\nproblem:\nThis question relates to ways in which light energy may be concentrated in an interference pattern.\n\nFigure 1 shows wave fronts at normal incidence on a Young's double slit arrangement, illuminating them so that they both radiate in phase, each with an amplitude $A$ at the screen. One such slit alone will cause an intensity of illumination of $I$, where $I \\propto A^{2}$, in the central region of the screen where the Young's fringes pattern forms.\n\n[figure1]\nFigure: Young's double slits.\n\nWhat is the intensity at the minima？\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $I$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-3.jpg?height=522&width=859&top_left_y=584&top_left_x=610" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$I$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1470", "problem": "许多赛车上都装有可调节的导流翼片, 可以为水平道路上的赛车提供坚直向上或向下的附加压力。如果赛车速度的大小为 $v$, 则上述压力的大小为 $f_{B}=c_{B} v^{2}, c_{B}$ 为一常量。当导流翼片的前方上逃时，压力方向向上; 当导流翼片的后方上㪴时, 压力方向向下。赛车在运动过程中受到\n\n[图1]\n迎面空气的阻力, 阻力大小为 $f_{A}=c_{A} v^{2}, c_{A}$ 为一常量。已知赛车质量为 $m$, 轮胎与路面之间的静摩擦系数为 $\\mu_{\\mathrm{S}}\\left(\\mu_{\\mathrm{S}}<1\\right)$ 。赛车在水平直道上匀速行驶时, 考虑到在运动过程中轮胎的形变, 路面对赛车会形成阻力, 阻力大小与车对路面的正压力大小成正比, 比例系数为 $\\mu_{R}\\left(\\mu_{R}<\\mu_{S}\\right)$ 。若导流翼片被调至前方上翘, 求当赛车行驶速度大小为多大时, 赛车发动机输出的功率最大? 最大输出功率为多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n许多赛车上都装有可调节的导流翼片, 可以为水平道路上的赛车提供坚直向上或向下的附加压力。如果赛车速度的大小为 $v$, 则上述压力的大小为 $f_{B}=c_{B} v^{2}, c_{B}$ 为一常量。当导流翼片的前方上逃时，压力方向向上; 当导流翼片的后方上㪴时, 压力方向向下。赛车在运动过程中受到\n\n[图1]\n迎面空气的阻力, 阻力大小为 $f_{A}=c_{A} v^{2}, c_{A}$ 为一常量。已知赛车质量为 $m$, 轮胎与路面之间的静摩擦系数为 $\\mu_{\\mathrm{S}}\\left(\\mu_{\\mathrm{S}}<1\\right)$ 。\n\n问题:\n赛车在水平直道上匀速行驶时, 考虑到在运动过程中轮胎的形变, 路面对赛车会形成阻力, 阻力大小与车对路面的正压力大小成正比, 比例系数为 $\\mu_{R}\\left(\\mu_{R}<\\mu_{S}\\right)$ 。若导流翼片被调至前方上翘, 求当赛车行驶速度大小为多大时, 赛车发动机输出的功率最大? 最大输出功率为多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-03.jpg?height=340&width=506&top_left_y=1029&top_left_x=1249" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_649", "problem": "Deuterium is the heavy stable isotope of hydrogen, having a proton and neutron in its nucleus (protons and neutrons have approximately the same mass). Heavy water is a form of water that contains Deuterium and Oxygen. A heavy water nuclear reactor has heavy water between the nuclear fuel rods. Suppose that a neutron from the fuel rod has a head-on elastic collision with a Dueterium nucleus.\n\nHow many such collisions is needed to slow down the neutron from $20 \\mathrm{MeV}$ to $0.02 \\mathrm{eV}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nDeuterium is the heavy stable isotope of hydrogen, having a proton and neutron in its nucleus (protons and neutrons have approximately the same mass). Heavy water is a form of water that contains Deuterium and Oxygen. A heavy water nuclear reactor has heavy water between the nuclear fuel rods. Suppose that a neutron from the fuel rod has a head-on elastic collision with a Dueterium nucleus.\n\nHow many such collisions is needed to slow down the neutron from $20 \\mathrm{MeV}$ to $0.02 \\mathrm{eV}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_190", "problem": "A wheel of radius $R$ is rolling without slipping with angular velocity $\\omega$.\n\n[figure1]\n\nFor point $\\mathrm{A}$ on the wheel at an angle $\\theta$ with respect to the vertical, shown in the figure, what is the magnitude of its velocity with respect to the ground?\nA: $\\omega R$\nB: $\\omega R \\sin (|\\theta| / 2)$\nC: $\\sqrt{2} \\omega R \\sin (|\\theta| / 2)$\nD: $2 \\omega R \\sin (|\\theta|)$\nE: $2 \\omega R \\sin (|\\theta| / 2) \\mathrm{}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA wheel of radius $R$ is rolling without slipping with angular velocity $\\omega$.\n\n[figure1]\n\nFor point $\\mathrm{A}$ on the wheel at an angle $\\theta$ with respect to the vertical, shown in the figure, what is the magnitude of its velocity with respect to the ground?\n\nA: $\\omega R$\nB: $\\omega R \\sin (|\\theta| / 2)$\nC: $\\sqrt{2} \\omega R \\sin (|\\theta| / 2)$\nD: $2 \\omega R \\sin (|\\theta|)$\nE: $2 \\omega R \\sin (|\\theta| / 2) \\mathrm{}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-07.jpg?height=225&width=621&top_left_y=310&top_left_x=752" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1023", "problem": "Let us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nDuring much of the collapse, the gas is so transparent that any heat generated is immediately radiated away, i.e. the ball stays in thermodynamic equilibrium with its surroundings. What is the number of times, $n$, by which the pressure increases when the radius is halved to $r_{1}=0.5 r_{0}$ ? Assume that the gas density remains uniform.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nLet us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nDuring much of the collapse, the gas is so transparent that any heat generated is immediately radiated away, i.e. the ball stays in thermodynamic equilibrium with its surroundings. What is the number of times, $n$, by which the pressure increases when the radius is halved to $r_{1}=0.5 r_{0}$ ? Assume that the gas density remains uniform.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1443", "problem": "塞曼发现了钠光 $\\mathrm{D}$ 线在磁场中分裂成三条, 洛仑兹根据经典电磁理论对此做出了解释,他们因此荣获 1902 年诺贝尔物理学奖。假定原子中的价电子 (质量为 $m$, 电荷量为 $-e, e>0$ )受到一指向原子中心的等效线性回复力 $-m \\omega_{0}^{2} \\boldsymbol{r}$ ( $\\boldsymbol{r}$ 为价电子相对于原子中心的位矢）作用,做固有圆频率为 $\\omega_{0}$ 的简谐振动, 发出圆频率为 $\\omega_{0}$ 的光。现将该原子置于沿 $z$ 轴正方向的匀\n强磁场中, 磁感应强度大小为 $B$ (为方便起见, 将 $B$ 参数化为 $B=\\frac{2 m}{e} \\omega_{\\mathrm{L}}$ )。\n\n已知: 在转动角速度为 $\\omega$ 的转动参考系中, 运动电子受到的惯性力除惯性离心力外还受到科里奥利力作用, 当电子相对于转动参考系运动速度为 $v^{\\prime}$ 时, 作用于电子的科里奥利力为 $\\boldsymbol{f}_{\\mathrm{c}}=-2 m \\boldsymbol{\\omega} \\times \\boldsymbol{v}^{\\prime}$ 。证明在实验室参考系中原子发出的圆频率为 $\\omega_{0}$ 的谱线在磁场中一分为三; 并对弱磁场 (即 $\\omega_{\\mathrm{L}}<<\\omega_{0}$ ) 情形, 求出三条谱线的频率间隔。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n塞曼发现了钠光 $\\mathrm{D}$ 线在磁场中分裂成三条, 洛仑兹根据经典电磁理论对此做出了解释,他们因此荣获 1902 年诺贝尔物理学奖。假定原子中的价电子 (质量为 $m$, 电荷量为 $-e, e>0$ )受到一指向原子中心的等效线性回复力 $-m \\omega_{0}^{2} \\boldsymbol{r}$ ( $\\boldsymbol{r}$ 为价电子相对于原子中心的位矢）作用,做固有圆频率为 $\\omega_{0}$ 的简谐振动, 发出圆频率为 $\\omega_{0}$ 的光。现将该原子置于沿 $z$ 轴正方向的匀\n强磁场中, 磁感应强度大小为 $B$ (为方便起见, 将 $B$ 参数化为 $B=\\frac{2 m}{e} \\omega_{\\mathrm{L}}$ )。\n\n已知: 在转动角速度为 $\\omega$ 的转动参考系中, 运动电子受到的惯性力除惯性离心力外还受到科里奥利力作用, 当电子相对于转动参考系运动速度为 $v^{\\prime}$ 时, 作用于电子的科里奥利力为 $\\boldsymbol{f}_{\\mathrm{c}}=-2 m \\boldsymbol{\\omega} \\times \\boldsymbol{v}^{\\prime}$ 。\n\n问题:\n证明在实验室参考系中原子发出的圆频率为 $\\omega_{0}$ 的谱线在磁场中一分为三; 并对弱磁场 (即 $\\omega_{\\mathrm{L}}<<\\omega_{0}$ ) 情形, 求出三条谱线的频率间隔。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_159", "problem": "The coefficients of static and kinetic friction between a ball and an ramp are $\\mu_{s}=\\mu_{k}=\\mu$. The ball is released from rest at the top of the ramp. Which of the following graphs best shows the rotational acceleration of the ball about its center of mass as a function of the angle of the ramp?\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe coefficients of static and kinetic friction between a ball and an ramp are $\\mu_{s}=\\mu_{k}=\\mu$. The ball is released from rest at the top of the ramp. Which of the following graphs best shows the rotational acceleration of the ball about its center of mass as a function of the angle of the ramp?\n\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-18.jpg?height=398&width=483&top_left_y=408&top_left_x=211", "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-18.jpg?height=393&width=485&top_left_y=890&top_left_x=210", "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-18.jpg?height=390&width=478&top_left_y=412&top_left_x=1062", "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-18.jpg?height=393&width=482&top_left_y=888&top_left_x=1060", "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-18.jpg?height=396&width=486&top_left_y=1366&top_left_x=207" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_740", "problem": "A closed 5-litre cylinder containing $0.25 \\mathrm{~g}$ of a substance in solid and liquid forms was slowly heated using a constant power. A graph of the temperature as a function of time is shown below. The heat capacity of the liquid is $2.43 \\mathrm{~J} /(\\mathrm{g} \\cdot \\mathrm{K})$, the latent heat of melting is $105 \\mathrm{~J} / \\mathrm{g}$, and the molar density is $46 \\mathrm{~g} / \\mathrm{mol}$.\n\n[figure1]\n\nWhat is the specific heat of the gaseous state of the substance?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA closed 5-litre cylinder containing $0.25 \\mathrm{~g}$ of a substance in solid and liquid forms was slowly heated using a constant power. A graph of the temperature as a function of time is shown below. The heat capacity of the liquid is $2.43 \\mathrm{~J} /(\\mathrm{g} \\cdot \\mathrm{K})$, the latent heat of melting is $105 \\mathrm{~J} / \\mathrm{g}$, and the molar density is $46 \\mathrm{~g} / \\mathrm{mol}$.\n\n[figure1]\n\nWhat is the specific heat of the gaseous state of the substance?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\frac{J}{g K}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-08.jpg?height=1071&width=1612&top_left_y=825&top_left_x=251" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\frac{J}{g K}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_204", "problem": "A small bead is placed on the top of a frictionless glass sphere of diameter $D$ as shown. The bead is given a slight push and starts sliding down along the sphere. Find the speed $v$ of the bead at the point at which the which the bead leaves the sphere.\n\n[figure1]\nA: $v=\\sqrt{g D}$\nB: $v=\\sqrt{4 g D / 5}$\nC: $v=\\sqrt{2 g D / 3}$\nD: $v=\\sqrt{g D / 2}$\nE: $v=\\sqrt{g D / 3} $ \n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA small bead is placed on the top of a frictionless glass sphere of diameter $D$ as shown. The bead is given a slight push and starts sliding down along the sphere. Find the speed $v$ of the bead at the point at which the which the bead leaves the sphere.\n\n[figure1]\n\nA: $v=\\sqrt{g D}$\nB: $v=\\sqrt{4 g D / 5}$\nC: $v=\\sqrt{2 g D / 3}$\nD: $v=\\sqrt{g D / 2}$\nE: $v=\\sqrt{g D / 3} $ \n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-10.jpg?height=425&width=403&top_left_y=427&top_left_x=861" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_544", "problem": "The flow of heat through a material can be described via the thermal conductivity $\\kappa$. If the two faces of a slab of material with thermal conductivity $\\kappa$, area $A$, and thickness $d$ are held at temperatures differing by $\\Delta T$, the thermal power $P$ transferred through the slab is\n\n$$\nP=\\frac{\\kappa A \\Delta T}{d}\n$$\n\nA heat exchanger is a device which transfers heat between a hot fluid and a cold fluid; they are common in industrial applications such as power plants and heating systems. The heat exchanger shown below consists of two rectangular tubes of length $l$, width $w$, and height $h$. The tubes are separated by a metal wall of thickness $d$ and thermal conductivity $\\kappa$. Originally hot fluid flows through the lower tube at a speed $v$ from right to left, and originally cold fluid flows through the upper tube in the opposite direction (left to right) at the same speed. The heat capacity per unit volume of both fluids is $c$.\n\nThe hot fluid enters the heat exchanger at a higher temperature than the cold fluid; the difference between the temperatures of the entering fluids is $\\Delta T_{i}$. When the fluids exit the heat exchanger the difference has been reduced to $\\Delta T_{f}$. (It is possible for the exiting originally cold fluid to have a higher temperature than the exiting originally hot fluid, in which case $\\Delta T_{f}<0$.)\n\n[figure1]\n\nAssume that the temperature in each pipe depends only on the lengthwise position, and consider transfer of heat only due to conduction in the metal and due to the bulk movement of fluid. Under the assumptions in this problem, while the temperature of each fluid varies along the length of the exchanger, the temperature difference across the wall is the same everywhere. You need not prove this.\n\nFind $\\Delta T_{f}$ in terms of the other given parameters.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe flow of heat through a material can be described via the thermal conductivity $\\kappa$. If the two faces of a slab of material with thermal conductivity $\\kappa$, area $A$, and thickness $d$ are held at temperatures differing by $\\Delta T$, the thermal power $P$ transferred through the slab is\n\n$$\nP=\\frac{\\kappa A \\Delta T}{d}\n$$\n\nA heat exchanger is a device which transfers heat between a hot fluid and a cold fluid; they are common in industrial applications such as power plants and heating systems. The heat exchanger shown below consists of two rectangular tubes of length $l$, width $w$, and height $h$. The tubes are separated by a metal wall of thickness $d$ and thermal conductivity $\\kappa$. Originally hot fluid flows through the lower tube at a speed $v$ from right to left, and originally cold fluid flows through the upper tube in the opposite direction (left to right) at the same speed. The heat capacity per unit volume of both fluids is $c$.\n\nThe hot fluid enters the heat exchanger at a higher temperature than the cold fluid; the difference between the temperatures of the entering fluids is $\\Delta T_{i}$. When the fluids exit the heat exchanger the difference has been reduced to $\\Delta T_{f}$. (It is possible for the exiting originally cold fluid to have a higher temperature than the exiting originally hot fluid, in which case $\\Delta T_{f}<0$.)\n\n[figure1]\n\nAssume that the temperature in each pipe depends only on the lengthwise position, and consider transfer of heat only due to conduction in the metal and due to the bulk movement of fluid. Under the assumptions in this problem, while the temperature of each fluid varies along the length of the exchanger, the temperature difference across the wall is the same everywhere. You need not prove this.\n\nFind $\\Delta T_{f}$ in terms of the other given parameters.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_44c03ebf1ca2c0f19b5cg-03.jpg?height=597&width=1092&top_left_y=1241&top_left_x=511" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1019", "problem": "## To Commemorate the Centenary of Rutherford's Atomic Nucleus: the Scattering of an Ion by a Neutral Atom\n\n[figure1]\n\nAn ion of mass $m$, charge $Q$, is moving with an initial non-relativistic speed $v_{0}$ from a great distance towards the vicinity of a neutral atom of mass $M>>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nCalculate the electric field intensity $\\vec{E}_{p}$ at a distance $r$ from an ideal electric dipole $\\vec{p}$ at the origin $\\mathrm{O}$ along the direction of $\\vec{p}$ in Figure 2.\n\n$p=2 a q, \\quad r \\gg a$\n\n[figure2]\n\nFIGURE 2", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n## To Commemorate the Centenary of Rutherford's Atomic Nucleus: the Scattering of an Ion by a Neutral Atom\n\n[figure1]\n\nAn ion of mass $m$, charge $Q$, is moving with an initial non-relativistic speed $v_{0}$ from a great distance towards the vicinity of a neutral atom of mass $M>>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nCalculate the electric field intensity $\\vec{E}_{p}$ at a distance $r$ from an ideal electric dipole $\\vec{p}$ at the origin $\\mathrm{O}$ along the direction of $\\vec{p}$ in Figure 2.\n\n$p=2 a q, \\quad r \\gg a$\n\n[figure2]\n\nFIGURE 2\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_51c4dc0e7c52a1226310g-1.jpg?height=462&width=1495&top_left_y=701&top_left_x=315", "https://cdn.mathpix.com/cropped/2024_03_14_51c4dc0e7c52a1226310g-1.jpg?height=437&width=759&top_left_y=1844&top_left_x=648" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1413", "problem": "牛顿曾观察到一束细日光射到有灰尘的反射镜上面会产生干涉条纹。为了分析这一现象背后的物理, 考虑如图所示的简单实验。一平板玻璃的折射率为 $n$, 厚度为 $t$, 下表面涂有水银反射层,上表面撒有滑石粉（灰尘粒子）。观察者 $\\mathrm{O}$ 和单色点光源 $\\mathrm{L}$ （光线的波长为 $\\lambda$ ）的连线垂直于镜面 (垂足为 $\\mathrm{N}$ ), $\\mathrm{LN}=a, \\mathrm{ON}=b$ 。反射镜面上的某灰尘粒子 $\\mathrm{P}$ 与直线 $\\mathrm{ON}$ 的距离为 $r$ $(b>a>>r>t)$ 。观察者可以观察到明暗相间的环形条纹。$n=1.63, a=0.0495 \\mathrm{~m}, b=0.245 \\mathrm{~m}$,\n\n$t=1.1 \\times 10^{-5} \\mathrm{~m}, \\lambda=680 \\mathrm{~nm}$, 求最小亮环 $(m=1)$ 的半径。\n\n已知: $\\sin x \\approx x, \\sqrt{1+x} \\approx 1+\\frac{x}{2}$, 当 $x<<1$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n牛顿曾观察到一束细日光射到有灰尘的反射镜上面会产生干涉条纹。为了分析这一现象背后的物理, 考虑如图所示的简单实验。一平板玻璃的折射率为 $n$, 厚度为 $t$, 下表面涂有水银反射层,上表面撒有滑石粉（灰尘粒子）。观察者 $\\mathrm{O}$ 和单色点光源 $\\mathrm{L}$ （光线的波长为 $\\lambda$ ）的连线垂直于镜面 (垂足为 $\\mathrm{N}$ ), $\\mathrm{LN}=a, \\mathrm{ON}=b$ 。反射镜面上的某灰尘粒子 $\\mathrm{P}$ 与直线 $\\mathrm{ON}$ 的距离为 $r$ $(b>a>>r>t)$ 。观察者可以观察到明暗相间的环形条纹。\n\n问题:\n$n=1.63, a=0.0495 \\mathrm{~m}, b=0.245 \\mathrm{~m}$,\n\n$t=1.1 \\times 10^{-5} \\mathrm{~m}, \\lambda=680 \\mathrm{~nm}$, 求最小亮环 $(m=1)$ 的半径。\n\n已知: $\\sin x \\approx x, \\sqrt{1+x} \\approx 1+\\frac{x}{2}$, 当 $x<<1$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~m}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_884", "problem": "The fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nAs the magnetic field deviates from the exact filling $v=1 / n$ to a higher field, more vortices (whirlpools in the electron sea) are being created. They are not bound to electrons and behave like particles carrying effectively positive charges, hence known as quasiholes, compared to the negatively charged electrons. The amount of charge deficit in any of these quasiholes amounts to exactly $1 / n$ of an electronic charge. An analogous argument can be made for magnetic fields slightly below $v$ and the creation of quasielectrons of negative charge $e^{*}=-e / n$. At the quantized Hall plateau of $R_{\\mathrm{H}}=3 h / e^{2}$, calculate the amount of change in $B$ that corresponds to the introduction of exactly one fractionally charged quasihole. (When their density is low, the quasiparticles are confined by the random potential generated by impurities and imperfections, hence the Hall resistance remains quantized for a finite range of $B$.)", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nThe fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nAs the magnetic field deviates from the exact filling $v=1 / n$ to a higher field, more vortices (whirlpools in the electron sea) are being created. They are not bound to electrons and behave like particles carrying effectively positive charges, hence known as quasiholes, compared to the negatively charged electrons. The amount of charge deficit in any of these quasiholes amounts to exactly $1 / n$ of an electronic charge. An analogous argument can be made for magnetic fields slightly below $v$ and the creation of quasielectrons of negative charge $e^{*}=-e / n$. At the quantized Hall plateau of $R_{\\mathrm{H}}=3 h / e^{2}$, calculate the amount of change in $B$ that corresponds to the introduction of exactly one fractionally charged quasihole. (When their density is low, the quasiparticles are confined by the random potential generated by impurities and imperfections, hence the Hall resistance remains quantized for a finite range of $B$.)\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_835", "problem": "In the jets from AGN, we have populations of highly energetic electrons in regions with strong magnetic fields. This creates the conditions for the emission of high fluxes of synchrotron radiation. The electrons are often so highly energetic, that they can be described as ultra relativistic with $\\gamma \\gg 1$.\n \nAs the electron is accelerated due to the magnetic field it emits electromagnetic radiation. In a frame at which the electron is momentarily at rest, there is no preferred direction for the emission of the radiation. Half is emitted in the forward direction, and half in the backward direction. However, in the frame of the observer, for an electron moving at an ultra relativistic speed, with $\\gamma \\gg 1$, the radiation is concentrated in a forward cone with $\\theta \\lesssim 1 / \\gamma$ (so the total angle of cone is $2 / \\gamma$ ). As the electron is gyrating around the magnetic field, any observer will only see pulses of radiation as the forward cone sweeps through the line of sight.\n[figure1]\n\nFigure 3: The diagram on the left shows the distribution of power in radiation from an electron accelerating up the page in the frame at which the electron in momentarily at rest. The diagram on the right shows the distribution of power in radiation for the same electron in the observer's frame, where most radiation is emitted in the forward cone. In the observers frame, the direction of the electron's acceleration is shown by a vector labelled $\\mathbf{a}$ and the direction of its velocity is shown by a vector labelled $\\mathbf{v}$.\n\nHence, estimate the characteristic frequency, $\\nu_{\\mathrm{chr}}$, of the synchrotron radiation.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn the jets from AGN, we have populations of highly energetic electrons in regions with strong magnetic fields. This creates the conditions for the emission of high fluxes of synchrotron radiation. The electrons are often so highly energetic, that they can be described as ultra relativistic with $\\gamma \\gg 1$.\n \nAs the electron is accelerated due to the magnetic field it emits electromagnetic radiation. In a frame at which the electron is momentarily at rest, there is no preferred direction for the emission of the radiation. Half is emitted in the forward direction, and half in the backward direction. However, in the frame of the observer, for an electron moving at an ultra relativistic speed, with $\\gamma \\gg 1$, the radiation is concentrated in a forward cone with $\\theta \\lesssim 1 / \\gamma$ (so the total angle of cone is $2 / \\gamma$ ). As the electron is gyrating around the magnetic field, any observer will only see pulses of radiation as the forward cone sweeps through the line of sight.\n[figure1]\n\nFigure 3: The diagram on the left shows the distribution of power in radiation from an electron accelerating up the page in the frame at which the electron in momentarily at rest. The diagram on the right shows the distribution of power in radiation for the same electron in the observer's frame, where most radiation is emitted in the forward cone. In the observers frame, the direction of the electron's acceleration is shown by a vector labelled $\\mathbf{a}$ and the direction of its velocity is shown by a vector labelled $\\mathbf{v}$.\n\nHence, estimate the characteristic frequency, $\\nu_{\\mathrm{chr}}$, of the synchrotron radiation.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2416d49d47cb88c0a72bg-4.jpg?height=166&width=842&top_left_y=1502&top_left_x=618" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_293", "problem": "This question is about the possibility of large-scale energy storage in the form of gravitational potential energy of large masses. Several such systems are currently being developed and are often referred to as gravity batteries. These energy storage systems are designed to contribute energy to the national grid at short notice when demand is high.\n\nOne such system uses large masses lowered down disused mine shafts or purpose built shafts. At times of low demand, when excess electrical energy is available, electric winches raise the masses. At times of high demand, when extra energy is required by the national grid, the masses are allowed to descend with the winch systems acting as electrical generators.\n\nDomestic electrical energy is measured in units of kilowatt-hours ( $\\mathrm{kWh})$ where $1 \\mathrm{kWh}$ is the energy transferred at a rate of $1 \\mathrm{~kW}$ for 1 hour.\n\nThe gravity battery system uses a container of crushed rock as the large mass. The container is \ncylindrical with a diameter of 5.0 m and a depth of 4.0 m. It is filled with crushed rock with an \naverage density of 5200 kg/m\n. \nThe mass can be raised or lowered a total distance of 300 m in the shaft. \n\nCalculate the maximum energy that can be stored by the gravity battery in units of kWh.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThis question is about the possibility of large-scale energy storage in the form of gravitational potential energy of large masses. Several such systems are currently being developed and are often referred to as gravity batteries. These energy storage systems are designed to contribute energy to the national grid at short notice when demand is high.\n\nOne such system uses large masses lowered down disused mine shafts or purpose built shafts. At times of low demand, when excess electrical energy is available, electric winches raise the masses. At times of high demand, when extra energy is required by the national grid, the masses are allowed to descend with the winch systems acting as electrical generators.\n\nDomestic electrical energy is measured in units of kilowatt-hours ( $\\mathrm{kWh})$ where $1 \\mathrm{kWh}$ is the energy transferred at a rate of $1 \\mathrm{~kW}$ for 1 hour.\n\nThe gravity battery system uses a container of crushed rock as the large mass. The container is \ncylindrical with a diameter of 5.0 m and a depth of 4.0 m. It is filled with crushed rock with an \naverage density of 5200 kg/m\n. \nThe mass can be raised or lowered a total distance of 300 m in the shaft. \n\nCalculate the maximum energy that can be stored by the gravity battery in units of kWh.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of J, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "J" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_211", "problem": "The handle of a gallon of milk is plugged by a manufacturing defect. After removing the cap and pouring out some milk, the level of milk in the main part of the jug is lower than in the handle, as shown in the figure. Which statement is true of the gauge pressure $P$ of the milk at the bottom of the jug? $\\rho$ is the density of the milk.\n\n[figure1]\nA: $P=\\rho g h$\nB: $P=\\rho g H $ \nC: $\\rho g H\\rho g h$\nE: $P<\\rho g H$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe handle of a gallon of milk is plugged by a manufacturing defect. After removing the cap and pouring out some milk, the level of milk in the main part of the jug is lower than in the handle, as shown in the figure. Which statement is true of the gauge pressure $P$ of the milk at the bottom of the jug? $\\rho$ is the density of the milk.\n\n[figure1]\n\nA: $P=\\rho g h$\nB: $P=\\rho g H $ \nC: $\\rho g H\\rho g h$\nE: $P<\\rho g H$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f726c6cf4a23f08e0214g-05.jpg?height=317&width=417&top_left_y=1208&top_left_x=862" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_569", "problem": "Consider the electron in the first part, except now take into account radiation. Assume that the orbit remains circular and the orbital radius $r$ changes by an amount $|\\Delta r| \\ll r$.\n\nConsider a circular orbit for the electron where $r>R$. Determine the change in the orbital radius $\\Delta r$ during one orbit in terms of any or all of $r, R, Q, e$, and any necessary fundamental constants. Be very specific about the sign of $\\Delta r$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider the electron in the first part, except now take into account radiation. Assume that the orbit remains circular and the orbital radius $r$ changes by an amount $|\\Delta r| \\ll r$.\n\nConsider a circular orbit for the electron where $r>R$. Determine the change in the orbital radius $\\Delta r$ during one orbit in terms of any or all of $r, R, Q, e$, and any necessary fundamental constants. Be very specific about the sign of $\\Delta r$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1300", "problem": "在生产纺织品、纸张等绝缘材料过程中, 为了实时监控材料的厚度, 生产流水线上设置如图 5a 所示的传感器, 其中甲、乙为平行板电容器的上、下两个固定极板, 分别接在恒压直流电源的两极上。当通过极板间的材料厚度增大时, 下列说法正确的是\n\n[图1]\n\n图 5a\nA: 有负电荷从 $b$ 向 $a$ 流过灵敏电流计 $\\mathrm{G}$\nB: 甲、乙两板间材料内的电场强度不变\nC: 乙板上的电荷量变小\nD: 甲、乙平行板构成的电容器的电容增大\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n在生产纺织品、纸张等绝缘材料过程中, 为了实时监控材料的厚度, 生产流水线上设置如图 5a 所示的传感器, 其中甲、乙为平行板电容器的上、下两个固定极板, 分别接在恒压直流电源的两极上。当通过极板间的材料厚度增大时, 下列说法正确的是\n\n[图1]\n\n图 5a\n\nA: 有负电荷从 $b$ 向 $a$ 流过灵敏电流计 $\\mathrm{G}$\nB: 甲、乙两板间材料内的电场强度不变\nC: 乙板上的电荷量变小\nD: 甲、乙平行板构成的电容器的电容增大\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-02.jpg?height=317&width=482&top_left_y=2400&top_left_x=1461" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1304", "problem": "计算题 解答应写出必要的文字说明、方程式和重要的演算步骤, 只写出最后结果的不能得分,有数值计算的, 答案中必须明确写出数值和单位。\n\n某同特别喜爱造桥游戏, 如图 9 所示, 用 10 根一样的轻杆搭了一座桥, 杆长 $l$, 所有连接的地方均为铰链, 在 $\\mathrm{B}$ 点挂了一个重物, 重力大小为 $P, \\mathrm{~A} G$ 在同一水平高度。\n\n[图1]\n\n图 9求地面对 $\\mathrm{A} G$ 点作用力的坚直分量大小 $\\mathrm{N} L, \\mathrm{NR}$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n计算题 解答应写出必要的文字说明、方程式和重要的演算步骤, 只写出最后结果的不能得分,有数值计算的, 答案中必须明确写出数值和单位。\n\n某同特别喜爱造桥游戏, 如图 9 所示, 用 10 根一样的轻杆搭了一座桥, 杆长 $l$, 所有连接的地方均为铰链, 在 $\\mathrm{B}$ 点挂了一个重物, 重力大小为 $P, \\mathrm{~A} G$ 在同一水平高度。\n\n[图1]\n\n图 9\n\n问题:\n求地面对 $\\mathrm{A} G$ 点作用力的坚直分量大小 $\\mathrm{N} L, \\mathrm{NR}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$\\mathrm{N}_{L}$, $\\mathrm{N}_{R}$]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-05.jpg?height=345&width=762&top_left_y=1061&top_left_x=681" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$\\mathrm{N}_{L}$", "$\\mathrm{N}_{R}$" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_843", "problem": "Most of the propellant is used during the ascent, however, after the payload has been separated from the stage, there still remains some fuel in its tank. Mass $m$ of residual fuel is negligible in comparison to the stage's mass $M$. Sloshing of the liquid fuel and viscous friction forces in the fuel tank result in energy losses, and after a transient process of irregular dynamics the energy reaches its minimum.\n\n[figure1]\n\nAngle between the stage's angular velocity and the symmetry axis\n Calculate the value $\\omega_{2}$ of angular velocity $\\omega$ after the transient process, given that initial angular velocity $\\omega(0)=\\omega_{1}=1 \\mathrm{rad} / \\mathrm{s}$ makes an angle of $\\gamma(0)=\\gamma_{1}=30^{\\circ}$ with the stage's symmetry axis. The moments of inertia are $J_{x}=4200 \\mathrm{~kg} \\cdot \\mathrm{m}^{2}$ and $J_{y}=15000 \\mathrm{~kg} \\cdot \\mathrm{m}^{2}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nMost of the propellant is used during the ascent, however, after the payload has been separated from the stage, there still remains some fuel in its tank. Mass $m$ of residual fuel is negligible in comparison to the stage's mass $M$. Sloshing of the liquid fuel and viscous friction forces in the fuel tank result in energy losses, and after a transient process of irregular dynamics the energy reaches its minimum.\n\n[figure1]\n\nAngle between the stage's angular velocity and the symmetry axis\n Calculate the value $\\omega_{2}$ of angular velocity $\\omega$ after the transient process, given that initial angular velocity $\\omega(0)=\\omega_{1}=1 \\mathrm{rad} / \\mathrm{s}$ makes an angle of $\\gamma(0)=\\gamma_{1}=30^{\\circ}$ with the stage's symmetry axis. The moments of inertia are $J_{x}=4200 \\mathrm{~kg} \\cdot \\mathrm{m}^{2}$ and $J_{y}=15000 \\mathrm{~kg} \\cdot \\mathrm{m}^{2}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $rad/s$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d7ee880fa438d4cf18fag-3.jpg?height=369&width=742&top_left_y=1526&top_left_x=657" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$rad/s$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_550", "problem": "The square of the standard deviation is the variance, so\n\n$$\n(\\Delta r)^{2}=\\left\\langle r^{2}\\right\\rangle-\\langle r\\rangle^{2}\n$$\n\nComputing the average value of $r^{2}$ is mathematically the exact same thing as computing the moment of inertia of a uniform disk; we have\n\n$$\n\\left\\langle r^{2}\\right\\rangle=\\frac{1}{\\pi R^{2}} \\int_{0}^{R} r^{2}(2 \\pi r d r)=\\frac{1}{2} R^{2}\n$$\n\nSimilarly, the average value of $r$ is\n\n$$\n\\langle r\\rangle=\\frac{1}{\\pi R^{2}} \\int_{0}^{R} r(2 \\pi r d r)=\\frac{2}{3} R\n$$\n\nThen we have\n\n$$\n\\Delta r=\\frac{R}{\\sqrt{18}}\n$$In this problem, use a particle-like model of photons: they propagate in straight lines and obey the law of reflection, but are subject to the quantum uncertainty principle. You may use small-angle approximations throughout the problem.\n\nA photon with wavelength $\\lambda$ has traveled from a distant star to a telescope mirror, which has a circular cross-section with radius $R$ and a focal length $f \\gg R$. The path of the photon is nearly aligned to the axis of the mirror, but has some slight uncertainty $\\Delta \\theta$. The photon reflects off the mirror and travels to a detector, where it is absorbed by a particular pixel on a charge-coupled device (CCD).\n\nSuppose the telescope mirror is manufactured so that photons coming in parallel to each other are focused to the same pixel on the CCD, regardless of where they hit the mirror. Then all small cross-sectional areas of the mirror are equally likely to include the point of reflection for a photon.\n\nUse the uncertainty principle, $\\Delta r \\Delta p_{r} \\geq \\hbar / 2$, to place a bound on how accurately we can know the angle of the photon from the axis of the telescope. Give your answer in terms of $R$ and $\\lambda$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe square of the standard deviation is the variance, so\n\n$$\n(\\Delta r)^{2}=\\left\\langle r^{2}\\right\\rangle-\\langle r\\rangle^{2}\n$$\n\nComputing the average value of $r^{2}$ is mathematically the exact same thing as computing the moment of inertia of a uniform disk; we have\n\n$$\n\\left\\langle r^{2}\\right\\rangle=\\frac{1}{\\pi R^{2}} \\int_{0}^{R} r^{2}(2 \\pi r d r)=\\frac{1}{2} R^{2}\n$$\n\nSimilarly, the average value of $r$ is\n\n$$\n\\langle r\\rangle=\\frac{1}{\\pi R^{2}} \\int_{0}^{R} r(2 \\pi r d r)=\\frac{2}{3} R\n$$\n\nThen we have\n\n$$\n\\Delta r=\\frac{R}{\\sqrt{18}}\n$$\n\nproblem:\nIn this problem, use a particle-like model of photons: they propagate in straight lines and obey the law of reflection, but are subject to the quantum uncertainty principle. You may use small-angle approximations throughout the problem.\n\nA photon with wavelength $\\lambda$ has traveled from a distant star to a telescope mirror, which has a circular cross-section with radius $R$ and a focal length $f \\gg R$. The path of the photon is nearly aligned to the axis of the mirror, but has some slight uncertainty $\\Delta \\theta$. The photon reflects off the mirror and travels to a detector, where it is absorbed by a particular pixel on a charge-coupled device (CCD).\n\nSuppose the telescope mirror is manufactured so that photons coming in parallel to each other are focused to the same pixel on the CCD, regardless of where they hit the mirror. Then all small cross-sectional areas of the mirror are equally likely to include the point of reflection for a photon.\n\nUse the uncertainty principle, $\\Delta r \\Delta p_{r} \\geq \\hbar / 2$, to place a bound on how accurately we can know the angle of the photon from the axis of the telescope. Give your answer in terms of $R$ and $\\lambda$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1120", "problem": "Gaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\n## Solar power plant\n\nThe flow of gases in a chimney can be used to construct a particular kind of solar power plant (solar chimney). The idea is illustrated in Fig. 2. The Sun heats the air underneath the collector of area $S$ with an open periphery to allow the undisturbed inflow of air (see Fig. 2). As the heated air rises through the chimney (thin solid arrows), new cold air enters the collector from its surrounding (thick dotted arrows) enabling a continuous flow of air through the power plant. The flow of air through the chimney powers a turbine, resulting in the production of electrical energy. The energy of solar radiation per unit time per unit of horizontal area of the collector is $G$. Assume that all that energy can be used to heat the air in the collector (the mass heat capacity of the air is $c$, and one can neglect its dependence on the air temperature). We define the efficiency of the solar chimney as the ratio of the kinetic energy of the gas flow and the solar energy absorbed in heating of the air prior to its entry into the chimney.\n\n[figure2]\n\nWhat is the efficiency of the solar chimney power plant?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nGaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\n## Solar power plant\n\nThe flow of gases in a chimney can be used to construct a particular kind of solar power plant (solar chimney). The idea is illustrated in Fig. 2. The Sun heats the air underneath the collector of area $S$ with an open periphery to allow the undisturbed inflow of air (see Fig. 2). As the heated air rises through the chimney (thin solid arrows), new cold air enters the collector from its surrounding (thick dotted arrows) enabling a continuous flow of air through the power plant. The flow of air through the chimney powers a turbine, resulting in the production of electrical energy. The energy of solar radiation per unit time per unit of horizontal area of the collector is $G$. Assume that all that energy can be used to heat the air in the collector (the mass heat capacity of the air is $c$, and one can neglect its dependence on the air temperature). We define the efficiency of the solar chimney as the ratio of the kinetic energy of the gas flow and the solar energy absorbed in heating of the air prior to its entry into the chimney.\n\n[figure2]\n\nWhat is the efficiency of the solar chimney power plant?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_fd946cfac82ef740b1dag-1.jpg?height=977&width=1644&top_left_y=1453&top_left_x=206", "https://cdn.mathpix.com/cropped/2024_03_14_fd946cfac82ef740b1dag-3.jpg?height=1145&width=1627&top_left_y=296&top_left_x=220" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_830", "problem": "he schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\\pm \\varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale.\n\n[figure1]\n\n Assume that there is no wind velocity in the east-west direction around the point $\\mathrm{X}$. What is the expression for the east-west wind velocity $u_{Y}$ at the points Y? Convention: positive velocities point from west to east.\n\n(The angular velocity of the Earth about its spin axis is $\\Omega$, the radius of the Earth is $a$, and the thickness of the atmosphere is much smaller than $a$.)", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nhe schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\\pm \\varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale.\n\n[figure1]\n\n Assume that there is no wind velocity in the east-west direction around the point $\\mathrm{X}$. What is the expression for the east-west wind velocity $u_{Y}$ at the points Y? Convention: positive velocities point from west to east.\n\n(The angular velocity of the Earth about its spin axis is $\\Omega$, the radius of the Earth is $a$, and the thickness of the atmosphere is much smaller than $a$.)\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e7b9ba69a6dd20c5cfd1g-1.jpg?height=1148&width=1151&top_left_y=1096&top_left_x=384", "https://cdn.mathpix.com/cropped/2024_03_14_77160055aa935397f348g-02.jpg?height=517&width=651&top_left_y=855&top_left_x=520" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_898", "problem": "Besides capturing neutral atoms in the trap via the induced dipole potential energy, the oscillating electric field may cause a scattering force on atoms that arises from absorption and emission of light. The light scattering processes lead either to heating or to losses of atoms from the trap and may be characterized by the scattering rate, that is related to the number of photons scattered by an atom in unit time and is defined by $\\Gamma_{s c}(\\vec{r})=\\frac{\\left\\langle P_{a b s}(\\vec{r})\\right\\rangle}{\\hbar \\omega}$. Here, $\\left\\langle P_{a b s}(\\vec{r})\\right\\rangle$ is the time-averaged power absorbed from the laser field, and $\\hbar \\omega$ is the photon energy $(\\hbar=h / 2 \\pi)$.\n\nFind the scattering rate $\\Gamma_{s c}(\\vec{r})$ in term of $\\alpha, \\varphi, \\varepsilon_{0}, c, I(\\vec{r}), \\hbar$ and $\\omega$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nBesides capturing neutral atoms in the trap via the induced dipole potential energy, the oscillating electric field may cause a scattering force on atoms that arises from absorption and emission of light. The light scattering processes lead either to heating or to losses of atoms from the trap and may be characterized by the scattering rate, that is related to the number of photons scattered by an atom in unit time and is defined by $\\Gamma_{s c}(\\vec{r})=\\frac{\\left\\langle P_{a b s}(\\vec{r})\\right\\rangle}{\\hbar \\omega}$. Here, $\\left\\langle P_{a b s}(\\vec{r})\\right\\rangle$ is the time-averaged power absorbed from the laser field, and $\\hbar \\omega$ is the photon energy $(\\hbar=h / 2 \\pi)$.\n\nFind the scattering rate $\\Gamma_{s c}(\\vec{r})$ in term of $\\alpha, \\varphi, \\varepsilon_{0}, c, I(\\vec{r}), \\hbar$ and $\\omega$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_522", "problem": "Radiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe force of radiation on a spherical particle of radius $r$ is given by\n\n$$\nF=P Q \\pi r^{2}\n$$\n\nwhere $P$ is the radiation pressure and $Q$ is a dimensionless quality factor that depends on the relative size of the particle $r$ and the wavelength of light $\\lambda$. Throughout this problem assume that the sun emits a single wavelength $\\lambda_{\\max }$; unless told otherwise, leave your answers in terms of symbolic variables.\n\nThe quality factor is given by one of the following\n\n- If $r \\ll \\lambda, Q \\sim(r / \\lambda)^{2}$\n- If $r \\sim \\lambda, Q \\sim 1$.\n- If $r \\gg \\lambda, Q=1$\n\nConsidering the three possible particle sizes, which is most likely to be blown away by the solar radiation pressure?\nA: $r \\ll \\lambda$\nB: $r \\sim \\lambda$\nC: $r \\gg \\lambda$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nRadiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe force of radiation on a spherical particle of radius $r$ is given by\n\n$$\nF=P Q \\pi r^{2}\n$$\n\nwhere $P$ is the radiation pressure and $Q$ is a dimensionless quality factor that depends on the relative size of the particle $r$ and the wavelength of light $\\lambda$. Throughout this problem assume that the sun emits a single wavelength $\\lambda_{\\max }$; unless told otherwise, leave your answers in terms of symbolic variables.\n\nThe quality factor is given by one of the following\n\n- If $r \\ll \\lambda, Q \\sim(r / \\lambda)^{2}$\n- If $r \\sim \\lambda, Q \\sim 1$.\n- If $r \\gg \\lambda, Q=1$\n\nConsidering the three possible particle sizes, which is most likely to be blown away by the solar radiation pressure?\n\nA: $r \\ll \\lambda$\nB: $r \\sim \\lambda$\nC: $r \\gg \\lambda$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1573", "problem": "如图, 半径为 $R$ 、质量为 $M$ 的半球静置于光滑水平桌面上, 在半球顶点上有一质量为 $m$ 、半径为 $r$ 的匀质小球。某时刻, 小球受到微小的扰动后由静止开始沿半球表面运动。在运动过程中, 小球相对于半球的位置由角位置 $\\theta$ 描述, $\\theta$ 为两球心的连线与坚直方向之间的夹角。已知小球绕其对称轴的转动惯量为 $\\frac{2}{5} m r^{2}$, 小球与半球之间的动摩擦因数为 $\\mu$, 假定最大静摩擦力等于滑动摩擦力。重力加速度大小为 $g$ 。\n\n[图1]当小球纯滚动到角位置 $\\theta_{2}$ 时开始相对于半球滑动, 求 $\\theta_{2}$ 所满足的方程（可用半球速度大小 $V_{M}\\left(\\theta_{2}\\right)$ 和加速度大小 $a_{M}\\left(\\theta_{2}\\right)$ 以及题给条件表示）；", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 半径为 $R$ 、质量为 $M$ 的半球静置于光滑水平桌面上, 在半球顶点上有一质量为 $m$ 、半径为 $r$ 的匀质小球。某时刻, 小球受到微小的扰动后由静止开始沿半球表面运动。在运动过程中, 小球相对于半球的位置由角位置 $\\theta$ 描述, $\\theta$ 为两球心的连线与坚直方向之间的夹角。已知小球绕其对称轴的转动惯量为 $\\frac{2}{5} m r^{2}$, 小球与半球之间的动摩擦因数为 $\\mu$, 假定最大静摩擦力等于滑动摩擦力。重力加速度大小为 $g$ 。\n\n[图1]\n\n问题:\n当小球纯滚动到角位置 $\\theta_{2}$ 时开始相对于半球滑动, 求 $\\theta_{2}$ 所满足的方程（可用半球速度大小 $V_{M}\\left(\\theta_{2}\\right)$ 和加速度大小 $a_{M}\\left(\\theta_{2}\\right)$ 以及题给条件表示）；\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_35bc41298eef336dfdafg-01.jpg?height=337&width=486&top_left_y=351&top_left_x=1276" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_392", "problem": "The V-I-curve of a tunnel diode is depicted in the figure below, curve (a). In some parts of the problem, we use an idealized model curve (b).\n\n[figure1]\n\nWith the same settings as for task ii), how long will it take from the moment when the switch was closed until the diode voltage reaches $V_{2}=500 \\mathrm{mV}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe V-I-curve of a tunnel diode is depicted in the figure below, curve (a). In some parts of the problem, we use an idealized model curve (b).\n\n[figure1]\n\nWith the same settings as for task ii), how long will it take from the moment when the switch was closed until the diode voltage reaches $V_{2}=500 \\mathrm{mV}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_0b06edcc1e983d034aadg-1.jpg?height=414&width=690&top_left_y=35&top_left_x=744", "https://cdn.mathpix.com/cropped/2024_03_06_683c031a1f6fc545249eg-2.jpg?height=432&width=682&top_left_y=34&top_left_x=47" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_319", "problem": "ii) Now, let us study the effect of the self-inductance of the wires. In order to take into account such an inductance, the circuit needs to be modified as shown in circuit (b) let $L=500 \\mathrm{nH}$. The switch $K$ is kept open until the voltage is adjusted to $\\mathscr{E}=250 \\mathrm{mV}$ and is then closed. How long will it take for the current to reach $I_{1}=20 \\mathrm{~mA}$ ? Neglect henceforth (until otherwise instructed) the internal resistances of the battery and of the ammeter (put $r=0$ ), and use the idealized V-curve of the diode.The V-I-curve of a tunnel diode is depicted in the figure below, curve (a). In some parts of the problem, we use an idealized model curve (b).\n\n[figure1]\n\nWith the same setting as for task ii), find the period the current oscillations.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nii) Now, let us study the effect of the self-inductance of the wires. In order to take into account such an inductance, the circuit needs to be modified as shown in circuit (b) let $L=500 \\mathrm{nH}$. The switch $K$ is kept open until the voltage is adjusted to $\\mathscr{E}=250 \\mathrm{mV}$ and is then closed. How long will it take for the current to reach $I_{1}=20 \\mathrm{~mA}$ ? Neglect henceforth (until otherwise instructed) the internal resistances of the battery and of the ammeter (put $r=0$ ), and use the idealized V-curve of the diode.\n\nproblem:\nThe V-I-curve of a tunnel diode is depicted in the figure below, curve (a). In some parts of the problem, we use an idealized model curve (b).\n\n[figure1]\n\nWith the same setting as for task ii), find the period the current oscillations.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_0b06edcc1e983d034aadg-1.jpg?height=414&width=690&top_left_y=35&top_left_x=744", "https://cdn.mathpix.com/cropped/2024_03_06_683c031a1f6fc545249eg-2.jpg?height=432&width=698&top_left_y=34&top_left_x=740", "https://cdn.mathpix.com/cropped/2024_03_06_683c031a1f6fc545249eg-2.jpg?height=433&width=716&top_left_y=1063&top_left_x=742" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_168", "problem": "In the frame of the box, which of the following is a possible path followed by the particle?\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn the frame of the box, which of the following is a possible path followed by the particle?\n\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-07.jpg?height=390&width=393&top_left_y=973&top_left_x=324", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-07.jpg?height=390&width=396&top_left_y=973&top_left_x=867", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-07.jpg?height=400&width=558&top_left_y=962&top_left_x=1260", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-07.jpg?height=398&width=415&top_left_y=1411&top_left_x=324", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-07.jpg?height=399&width=401&top_left_y=1408&top_left_x=865" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_577", "problem": "Because the particle is moving, the radiation force is not directed directly away from the sun. Find the torque $\\tau$ on the particle because of radiation pressure. Assume that $v \\ll c$.\n\nWork in the reference frame of the particle. In this frame, the radiation hits the particle at an angle $\\theta=v / c$ from the radial direction. The particle then re-emits the radiation isotropically, contributing no additional radiation pressure. (We ignore relativistic effects because they occur at second order in $v / c$, while the effect we care about is first order.) The tangential component of the force is\n\nso\n\n$$\nF=\\frac{v}{c} \\frac{L_{\\odot}}{4 \\pi R^{2} c} Q \\pi r^{2}=\\frac{v}{c} \\frac{L_{\\odot}}{4 R^{2} c} Q r^{2}\n$$\n\n$$\n\\tau=-\\frac{v}{c} \\frac{L_{\\odot}}{4 R c} Q r^{2}\n$$\nwhere the negative sign is because this tends to decrease the angular momentum.\n\n Since $\\tau=d L / d t$, the angular momentum $L$ of the particle changes with time.\n\nThe angular momentum is\n\n$$\nL=m v R=\\frac{4}{3} \\pi \\rho r^{3} \\sqrt{\\frac{G M_{\\odot}}{R}} R=\\frac{4}{3} \\pi \\rho r^{3} \\sqrt{G M_{\\odot} R}\n$$\n\nDifferentiating both sides with respect to time,\n\n$$\n-\\frac{v}{c} \\frac{L_{\\odot}}{4 R c} Q r^{2}=\\frac{4}{3} \\pi \\rho r^{3} \\sqrt{\\frac{G M_{\\odot}}{R}} \\frac{1}{2} \\frac{d R}{d t}\n$$\n\nwhich simplifies to\n\n$$\n-\\frac{1}{c^{2}} \\frac{L_{\\odot}}{R} Q=\\frac{8}{3} \\pi \\rho r \\frac{d R}{d t}\n$$Radiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe Poynting-Robertson effect acts as another mechanism for cleaning out the solar system.\n\nDevelop an expression for the time required to remove particles of size $r \\approx 1 \\mathrm{~cm}$ and density $\\rho \\approx 1000 \\mathrm{~kg} / \\mathrm{m}^{3}$ originally in circular orbits at a distance $R=R_{\\text {earth }}$, and use the numbers below to simplify your expression.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nBecause the particle is moving, the radiation force is not directed directly away from the sun. Find the torque $\\tau$ on the particle because of radiation pressure. Assume that $v \\ll c$.\n\nWork in the reference frame of the particle. In this frame, the radiation hits the particle at an angle $\\theta=v / c$ from the radial direction. The particle then re-emits the radiation isotropically, contributing no additional radiation pressure. (We ignore relativistic effects because they occur at second order in $v / c$, while the effect we care about is first order.) The tangential component of the force is\n\nso\n\n$$\nF=\\frac{v}{c} \\frac{L_{\\odot}}{4 \\pi R^{2} c} Q \\pi r^{2}=\\frac{v}{c} \\frac{L_{\\odot}}{4 R^{2} c} Q r^{2}\n$$\n\n$$\n\\tau=-\\frac{v}{c} \\frac{L_{\\odot}}{4 R c} Q r^{2}\n$$\nwhere the negative sign is because this tends to decrease the angular momentum.\n\n Since $\\tau=d L / d t$, the angular momentum $L$ of the particle changes with time.\n\nThe angular momentum is\n\n$$\nL=m v R=\\frac{4}{3} \\pi \\rho r^{3} \\sqrt{\\frac{G M_{\\odot}}{R}} R=\\frac{4}{3} \\pi \\rho r^{3} \\sqrt{G M_{\\odot} R}\n$$\n\nDifferentiating both sides with respect to time,\n\n$$\n-\\frac{v}{c} \\frac{L_{\\odot}}{4 R c} Q r^{2}=\\frac{4}{3} \\pi \\rho r^{3} \\sqrt{\\frac{G M_{\\odot}}{R}} \\frac{1}{2} \\frac{d R}{d t}\n$$\n\nwhich simplifies to\n\n$$\n-\\frac{1}{c^{2}} \\frac{L_{\\odot}}{R} Q=\\frac{8}{3} \\pi \\rho r \\frac{d R}{d t}\n$$\n\nproblem:\nRadiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe Poynting-Robertson effect acts as another mechanism for cleaning out the solar system.\n\nDevelop an expression for the time required to remove particles of size $r \\approx 1 \\mathrm{~cm}$ and density $\\rho \\approx 1000 \\mathrm{~kg} / \\mathrm{m}^{3}$ originally in circular orbits at a distance $R=R_{\\text {earth }}$, and use the numbers below to simplify your expression.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{y}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{y}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1569", "problem": "一列简谐横波在均匀的介质中沿 $\\mathrm{x}$ 轴正向传播, 两质点 $P_{1}$ 和 $P_{2}$ 的平衡位置在 $x$ 轴上, 它们相距 $60 \\mathrm{~cm}$, 当 $P_{1}$ 质点在平衡位置处向上运动时, $P_{2}$ 质点处在波谷位置. 若波的传播速度为 $24 \\mathrm{~m} / \\mathrm{s}$, 则该波的频率可能为\nA: $50 \\mathrm{~Hz}$\nB: $60 \\mathrm{~Hz}$\nC: $400 \\mathrm{~Hz}$\nD: $410 \\mathrm{~Hz}$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n一列简谐横波在均匀的介质中沿 $\\mathrm{x}$ 轴正向传播, 两质点 $P_{1}$ 和 $P_{2}$ 的平衡位置在 $x$ 轴上, 它们相距 $60 \\mathrm{~cm}$, 当 $P_{1}$ 质点在平衡位置处向上运动时, $P_{2}$ 质点处在波谷位置. 若波的传播速度为 $24 \\mathrm{~m} / \\mathrm{s}$, 则该波的频率可能为\n\nA: $50 \\mathrm{~Hz}$\nB: $60 \\mathrm{~Hz}$\nC: $400 \\mathrm{~Hz}$\nD: $410 \\mathrm{~Hz}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_29", "problem": "The average density of blood is $1.06 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$. Imagine that you donate a pint of blood during a local blood drive. What mass of blood, in grams, have you donated? ( $1 \\mathrm{pt}=1 / 2 \\mathrm{~L}$, $1 \\mathrm{~L}=1000 \\mathrm{~cm}^{3}$.\nA: $530 \\mathrm{~g}$\nB: $0.530 \\mathrm{~g}$\nC: $5300 \\mathrm{~g}$\nD: $5.30 \\times 10^{5} \\mathrm{~g}$\nE: $53.0 \\mathrm{~g}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe average density of blood is $1.06 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$. Imagine that you donate a pint of blood during a local blood drive. What mass of blood, in grams, have you donated? ( $1 \\mathrm{pt}=1 / 2 \\mathrm{~L}$, $1 \\mathrm{~L}=1000 \\mathrm{~cm}^{3}$.\n\nA: $530 \\mathrm{~g}$\nB: $0.530 \\mathrm{~g}$\nC: $5300 \\mathrm{~g}$\nD: $5.30 \\times 10^{5} \\mathrm{~g}$\nE: $53.0 \\mathrm{~g}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_417", "problem": "The length of tight cord extracted at X in one revolution of the upper pulley is $2 \\pi R$.\n\nThe length of slack cord drawn in at Y in one revolution of the upper pulley is $2\\pi r$.A novel variant, known as the Weston Differential Pulley, which is able to lift very heavy loads, is shown in Figure.\n\n[figure1]\n\nFigure: Weston Differential Pulley.\n\nThe upper pulley consists of two parts, $\\mathbf{A}$ and $\\mathbf{B}$, of radii $R$ and $r$ respectively, rigidly fixed together so that they rotate as one body. There is sufficient friction to ensure that the cord does not slip on the composite pulley. As you did in part (a), assume that the mass of the lower block is negligible and that the pulleys are free-running.\n\nWhat is the change of height of the lower pulley and the load, $m$, during one revolution of the upper pulley?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe length of tight cord extracted at X in one revolution of the upper pulley is $2 \\pi R$.\n\nThe length of slack cord drawn in at Y in one revolution of the upper pulley is $2\\pi r$.\n\nproblem:\nA novel variant, known as the Weston Differential Pulley, which is able to lift very heavy loads, is shown in Figure.\n\n[figure1]\n\nFigure: Weston Differential Pulley.\n\nThe upper pulley consists of two parts, $\\mathbf{A}$ and $\\mathbf{B}$, of radii $R$ and $r$ respectively, rigidly fixed together so that they rotate as one body. There is sufficient friction to ensure that the cord does not slip on the composite pulley. As you did in part (a), assume that the mass of the lower block is negligible and that the pulleys are free-running.\n\nWhat is the change of height of the lower pulley and the load, $m$, during one revolution of the upper pulley?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-5.jpg?height=1093&width=466&top_left_y=1481&top_left_x=1389" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_538", "problem": "A vacuum system consists of a chamber of volume $V$ connected to a vacuum pump that is a cylinder with a piston that moves left and right. The minimum volume in the pump cylinder is $V_{0}$, and the maximum volume in the cylinder is $V_{0}+\\Delta V$. You should assume that $\\Delta V \\ll V$.\n\n[figure1]\n\nThe cylinder has two valves. The inlet valve opens when the pressure inside the cylinder is lower than the pressure in the chamber, but closes when the piston moves to the right. The outlet valve opens when the pressure inside the cylinder is greater than atmospheric pressure $P_{a}$, and closes when the piston moves to the left. A motor drives the piston to move back and forth. The piston moves at such a rate that heat is not conducted in or out of the gas contained in the cylinder during the pumping cycle. One complete cycle takes a time $\\Delta t$. You should assume that $\\Delta t$ is a very small quantity, but $\\Delta V / \\Delta t=R$ is finite. The gas in the chamber is ideal monatomic and remains at a fixed temperature of $T_{a}$.\n\nStart with assumption that $V_{0}=0$ and there are no leaks in the system.\n\nFind an expression for the temperature of the gas as it is emitted from the pump cylinder into the atmosphere. Your answer may depend on time.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA vacuum system consists of a chamber of volume $V$ connected to a vacuum pump that is a cylinder with a piston that moves left and right. The minimum volume in the pump cylinder is $V_{0}$, and the maximum volume in the cylinder is $V_{0}+\\Delta V$. You should assume that $\\Delta V \\ll V$.\n\n[figure1]\n\nThe cylinder has two valves. The inlet valve opens when the pressure inside the cylinder is lower than the pressure in the chamber, but closes when the piston moves to the right. The outlet valve opens when the pressure inside the cylinder is greater than atmospheric pressure $P_{a}$, and closes when the piston moves to the left. A motor drives the piston to move back and forth. The piston moves at such a rate that heat is not conducted in or out of the gas contained in the cylinder during the pumping cycle. One complete cycle takes a time $\\Delta t$. You should assume that $\\Delta t$ is a very small quantity, but $\\Delta V / \\Delta t=R$ is finite. The gas in the chamber is ideal monatomic and remains at a fixed temperature of $T_{a}$.\n\nStart with assumption that $V_{0}=0$ and there are no leaks in the system.\n\nFind an expression for the temperature of the gas as it is emitted from the pump cylinder into the atmosphere. Your answer may depend on time.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_0cd94795c1ff89067cedg-09.jpg?height=377&width=1222&top_left_y=489&top_left_x=446" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1698", "problem": "如气体压强-体积 $(P-V)$ 图所示, 摩尔数为 $v$ 的双原子理想气体构成的系统经历一正循环过程（正循环指沿图中箭头所示的循环), 其中自 A 到 B 为直线过程, 自 B 到 $A$ 为等温过程。双原子理想气体的定容摩尔热容量为 $\\frac{5}{2} R, R$ 为气体常量。\n\n[图1]求直线 $A B$ 过程中的最高温度;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如气体压强-体积 $(P-V)$ 图所示, 摩尔数为 $v$ 的双原子理想气体构成的系统经历一正循环过程（正循环指沿图中箭头所示的循环), 其中自 A 到 B 为直线过程, 自 B 到 $A$ 为等温过程。双原子理想气体的定容摩尔热容量为 $\\frac{5}{2} R, R$ 为气体常量。\n\n[图1]\n\n问题:\n求直线 $A B$ 过程中的最高温度;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-04.jpg?height=422&width=494&top_left_y=266&top_left_x=1249" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_711", "problem": "You hold your arms outstretched and level in front of you. You rest a meter stick on your hands, with the $0 \\mathrm{~cm}$ mark on your left hand and the $75 \\mathrm{~cm}$ mark on your right hand. Slowly and keeping your hands level, you bring your hands together. When your hands touch each other, they will be at the mark for\nA: $0 \\mathrm{~cm}$\nB: $25 \\mathrm{~cm}$\nC: $37.5 \\mathrm{~cm}$\nD: $50 \\mathrm{~cm}$\nE: $75 \\mathrm{~cm}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nYou hold your arms outstretched and level in front of you. You rest a meter stick on your hands, with the $0 \\mathrm{~cm}$ mark on your left hand and the $75 \\mathrm{~cm}$ mark on your right hand. Slowly and keeping your hands level, you bring your hands together. When your hands touch each other, they will be at the mark for\n\nA: $0 \\mathrm{~cm}$\nB: $25 \\mathrm{~cm}$\nC: $37.5 \\mathrm{~cm}$\nD: $50 \\mathrm{~cm}$\nE: $75 \\mathrm{~cm}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_226", "problem": "Earth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nWhen a proton rotates around a field line it generates a magnetic moment which remains constant along its path. When the proton reaches some latitude $\\lambda_{m}$ the parallel velocity becomes zero and the particle starts its path backwards. This latitude is known as the mirror point.\n\nDetermine the relation between the mirror point latitude $\\lambda_{m}$ and $\\alpha_{E q}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nEarth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nWhen a proton rotates around a field line it generates a magnetic moment which remains constant along its path. When the proton reaches some latitude $\\lambda_{m}$ the parallel velocity becomes zero and the particle starts its path backwards. This latitude is known as the mirror point.\n\nDetermine the relation between the mirror point latitude $\\lambda_{m}$ and $\\alpha_{E q}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=496&width=705&top_left_y=774&top_left_x=713", "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=220&width=504&top_left_y=1690&top_left_x=802" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_47", "problem": "Any atom that emits an alpha particle or a beta particle\nA: always becomes an atom of a different element.\nB: may become an atom of a different element.\nC: becomes a different isotope of the same element.\nD: increases its mass.\nE: becomes highly reactive.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAny atom that emits an alpha particle or a beta particle\n\nA: always becomes an atom of a different element.\nB: may become an atom of a different element.\nC: becomes a different isotope of the same element.\nD: increases its mass.\nE: becomes highly reactive.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1629", "problem": "通常电容器两极板间有多层电介质, 并有漏电现象,为了探究其规律性, 采用如图所示的简单模型. 电容器的两极板面积均为 $A$, 其间充有两层电介质 1 和 2 , 第 1 层电介质的介电常\n\n| $\\varepsilon_{1} \\sigma_{1}$ |\n| :---: |\n| $\\varepsilon_{2} \\sigma_{2}$ |\n\n数、电导率 (即电阻率的倒数) 和厚度分别为 $\\varepsilon_{1} 、 \\sigma_{1}$ 和 $d_{1}$, 第 2 层电介质的则为 $\\varepsilon_{2} 、 \\sigma_{2}$ 和 $d_{2}$. 现在两极板加一直流电压 $U$, 电容器处于稳定状态.计算两层电介质所损耗的功率", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n通常电容器两极板间有多层电介质, 并有漏电现象,为了探究其规律性, 采用如图所示的简单模型. 电容器的两极板面积均为 $A$, 其间充有两层电介质 1 和 2 , 第 1 层电介质的介电常\n\n| $\\varepsilon_{1} \\sigma_{1}$ |\n| :---: |\n| $\\varepsilon_{2} \\sigma_{2}$ |\n\n数、电导率 (即电阻率的倒数) 和厚度分别为 $\\varepsilon_{1} 、 \\sigma_{1}$ 和 $d_{1}$, 第 2 层电介质的则为 $\\varepsilon_{2} 、 \\sigma_{2}$ 和 $d_{2}$. 现在两极板加一直流电压 $U$, 电容器处于稳定状态.\n\n问题:\n计算两层电介质所损耗的功率\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1739", "problem": "图示为由粗细均匀的细玻璃管弯曲成的“双 U 形管”, a、b、c、d 为其中四段竖直的部分,其中 $a 、 d$ 上端是开口的, 处在大气中, 管中的水银把一段气柱密封在 $b 、 c$ 内, 达到平衡时,管内水银面的位置如图所示. 现缓慢地降低气柱中气体的温度, 若 $\\mathrm{c}$ 中的水银上升了一小段高度 $\\Delta h$, 则\n\n[图1]\nA: b 中的水银面也上升 $\\Delta h$\nB: $\\mathrm{b}$ 中的水银面也上升, 但上升的高度小于 $\\Delta h$\nC: 气柱中气体压强的减少量等于高为 $\\Delta h$ 的水银柱所产生的压强\nD: 气柱中气体压强的减少量等于高为 $2 \\Delta h$ 的水银柱所产生的压强 $\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n图示为由粗细均匀的细玻璃管弯曲成的“双 U 形管”, a、b、c、d 为其中四段竖直的部分,其中 $a 、 d$ 上端是开口的, 处在大气中, 管中的水银把一段气柱密封在 $b 、 c$ 内, 达到平衡时,管内水银面的位置如图所示. 现缓慢地降低气柱中气体的温度, 若 $\\mathrm{c}$ 中的水银上升了一小段高度 $\\Delta h$, 则\n\n[图1]\n\nA: b 中的水银面也上升 $\\Delta h$\nB: $\\mathrm{b}$ 中的水银面也上升, 但上升的高度小于 $\\Delta h$\nC: 气柱中气体压强的减少量等于高为 $\\Delta h$ 的水银柱所产生的压强\nD: 气柱中气体压强的减少量等于高为 $2 \\Delta h$ 的水银柱所产生的压强 $\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-02.jpg?height=349&width=277&top_left_y=1513&top_left_x=821" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_582", "problem": "Part A\n\nAssume that the point charge $q$ is located at the origin in the very center of the shell.\n\nWe apply Gauss's law for a sphere with radius $r>b$ centered about the origin. Since the shell is neutral, the enclosed charge is $q$, so by spherical symmetry\n\n$$\nE(r)=\\frac{q}{4 \\pi \\epsilon_{0} r^{2}}\n$$\n\noutside the shell. Just outside the shell, the field is $q / 4 \\pi \\epsilon_{0} b^{2}$.\n\nSince the shell is conducting, the electrostatic field is zero inside it. By Gauss's law, this is achieved by having a charge of $-q$ on the inner surface $r=a$ and a charge of $q$ on the outer surface $r=b$, both uniformly distributed.\n\nFor $rb$ centered about the origin. Since the shell is neutral, the enclosed charge is $q$, so by spherical symmetry\n\n$$\nE(r)=\\frac{q}{4 \\pi \\epsilon_{0} r^{2}}\n$$\n\noutside the shell. Just outside the shell, the field is $q / 4 \\pi \\epsilon_{0} b^{2}$.\n\nSince the shell is conducting, the electrostatic field is zero inside it. By Gauss's law, this is achieved by having a charge of $-q$ on the inner surface $r=a$ and a charge of $q$ on the outer surface $r=b$, both uniformly distributed.\n\nFor $r>1)$, 单个细孔面积为 $s$ 。运输车长度为 $l$, 质量为 $M$ 。气体的流动可认为遵从伯努利方程, 且温度不变, 细孔出口处气体的压强为较低的环境压强 $P_{\\text {low }}$ 。\n\n如图 c，在水平管道中固定有两条平行的水平光滑供电导轨（粗实线）, 运输车上固定有与导轨垂直的两根导线 (细实线) ; 导轨横截面为圆形, 半径为 $r_{\\mathrm{d}}$, 电阻率为 $\\rho_{\\mathrm{d}}$, 两导轨轴线间距为 $2\\left(D+r_{\\mathrm{d}}\\right)$; 两根导线的粗细可忽略, 间距为 $D$; 每根导线电阻是长度为 $D$ 的导轨电阻的 2 倍。两导线和导轨轴线均处于同一水平面内。导轨、导线电接触良好, 且所有接触电阻均可忽略。\n\n[图3]\n\n图c\n\n[图4]\n\n图d\n\n[图5]\n\n图e当运输车静止在水平光滑导轨上准备离站时, 在导线 2 后间距为 $D$ 处接通固定在导轨上电动势为 $\\varepsilon$ 的恒压直流电源 (如图 $\\mathrm{e}$ 所示)。设电源体积及其所连导线的电阻可忽略, 求刚接通电源时运输车的加速度大小。\n\n已知某恒流闭合回路中的一圆柱形直导线段, 电流沿横截面均匀分布, 如图 $\\mathrm{f}$ 所示, 其在空间中距导线轴线距离为 $r_{0}$ 的某点产生的磁感应强度方向垂直于此点和导线轴线构成的平面, 大小可用下式近似计算\n\n$$\nB=\\frac{\\mu_{0} I}{4 \\pi r_{0}}\\left(\\cos \\theta_{1}+\\cos \\theta_{2}\\right)\n$$\n\n其中 $I$ 为电流, $\\theta_{1} 、 \\theta_{2}$ 是此点与导线段轴线两端连线与导线轴线的夹角。\n\n可能会用到的积分公式:\n\n$$\n\\int_{a}^{b} \\frac{1}{x} \\frac{c}{\\sqrt{x^{2}+c^{2}}} \\mathrm{~d} x=\\ln \\left(\\frac{b}{a} \\frac{c+\\sqrt{c^{2}+a^{2}}}{c+\\sqrt{c^{2}+b^{2}}}\\right) \\text {, 其中 } a 、 b 、 c \\text { 均为正数。 }\n$$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\nHyperloop 是一款利用胶囊状的运输车在水平管道中的快速运动来实现超高速运输的系统（见图 a）。它采用了 “气垫” 技术和 “直线电动机” 原理。\n\n“气垫” 技术是将内部高压气体从水平放置的运输车下半部的\n\n[图1]\n\n图a 细孔快速喷出（见图 b），以至于整个运输车被托离管壁非常小的距离,\n\n[图2]\n从而可忽略摩擦。运输车横截面是半径为 $R$ 的圆, 运输车下半部壁上均匀分布有沿径向的大量细孔, 单位面积内细孔个数为 $n$ ( $n>>1)$, 单个细孔面积为 $s$ 。运输车长度为 $l$, 质量为 $M$ 。气体的流动可认为遵从伯努利方程, 且温度不变, 细孔出口处气体的压强为较低的环境压强 $P_{\\text {low }}$ 。\n\n如图 c，在水平管道中固定有两条平行的水平光滑供电导轨（粗实线）, 运输车上固定有与导轨垂直的两根导线 (细实线) ; 导轨横截面为圆形, 半径为 $r_{\\mathrm{d}}$, 电阻率为 $\\rho_{\\mathrm{d}}$, 两导轨轴线间距为 $2\\left(D+r_{\\mathrm{d}}\\right)$; 两根导线的粗细可忽略, 间距为 $D$; 每根导线电阻是长度为 $D$ 的导轨电阻的 2 倍。两导线和导轨轴线均处于同一水平面内。导轨、导线电接触良好, 且所有接触电阻均可忽略。\n\n[图3]\n\n图c\n\n[图4]\n\n图d\n\n[图5]\n\n图e\n\n问题:\n当运输车静止在水平光滑导轨上准备离站时, 在导线 2 后间距为 $D$ 处接通固定在导轨上电动势为 $\\varepsilon$ 的恒压直流电源 (如图 $\\mathrm{e}$ 所示)。设电源体积及其所连导线的电阻可忽略, 求刚接通电源时运输车的加速度大小。\n\n已知某恒流闭合回路中的一圆柱形直导线段, 电流沿横截面均匀分布, 如图 $\\mathrm{f}$ 所示, 其在空间中距导线轴线距离为 $r_{0}$ 的某点产生的磁感应强度方向垂直于此点和导线轴线构成的平面, 大小可用下式近似计算\n\n$$\nB=\\frac{\\mu_{0} I}{4 \\pi r_{0}}\\left(\\cos \\theta_{1}+\\cos \\theta_{2}\\right)\n$$\n\n其中 $I$ 为电流, $\\theta_{1} 、 \\theta_{2}$ 是此点与导线段轴线两端连线与导线轴线的夹角。\n\n可能会用到的积分公式:\n\n$$\n\\int_{a}^{b} \\frac{1}{x} \\frac{c}{\\sqrt{x^{2}+c^{2}}} \\mathrm{~d} x=\\ln \\left(\\frac{b}{a} \\frac{c+\\sqrt{c^{2}+a^{2}}}{c+\\sqrt{c^{2}+b^{2}}}\\right) \\text {, 其中 } a 、 b 、 c \\text { 均为正数。 }\n$$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-03.jpg?height=226&width=351&top_left_y=2194&top_left_x=1412", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-03.jpg?height=374&width=905&top_left_y=2486&top_left_x=884", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-04.jpg?height=268&width=420&top_left_y=914&top_left_x=430", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-04.jpg?height=263&width=440&top_left_y=919&top_left_x=842", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-04.jpg?height=263&width=380&top_left_y=919&top_left_x=1272", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-19.jpg?height=271&width=303&top_left_y=316&top_left_x=1456", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-20.jpg?height=185&width=328&top_left_y=684&top_left_x=1435" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_474", "problem": "Beloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nAssuming the $25 \\mu \\mathrm{A}$ proton beam is on, determine the surface charge density that must be sprayed onto the charging belt in order to maintain a steady charge of $500 \\mathrm{kV}$ on the dome.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nBeloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nAssuming the $25 \\mu \\mathrm{A}$ proton beam is on, determine the surface charge density that must be sprayed onto the charging belt in order to maintain a steady charge of $500 \\mathrm{kV}$ on the dome.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{C} / \\mathrm{m}^{2}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without any units and equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=468&width=1592&top_left_y=438&top_left_x=259", "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=493&width=1268&top_left_y=1339&top_left_x=426" ], "answer": null, "solution": null, "answer_type": "EX", "unit": [ "$\\mathrm{C} / \\mathrm{m}^{2}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_666", "problem": "It is desired that a flexible copper wire passes in a straight line between two supports. The wire has a mass $M$, a length $L$ and carries a current $I$. However, due to the force of gravity on the wire it will tend to sag between the two supports(Fig. 5). It is possible to correct for this sagging by adding a magnetic field to the region between the supports. To do this, the magnetic field should be\n\n[figure1]\nA: Into the page and of magnitude $\\frac{M g}{L I}$\nB: Out of the page and of magnitude $\\frac{M g}{L I}$\nC: Into the page and of magnitude $\\frac{M g}{2 L I}$\nD: Out of the page and of magnitude $\\frac{M g}{2 L I}$\nE: Into the page and of magnitude $\\frac{M L g}{2 I}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIt is desired that a flexible copper wire passes in a straight line between two supports. The wire has a mass $M$, a length $L$ and carries a current $I$. However, due to the force of gravity on the wire it will tend to sag between the two supports(Fig. 5). It is possible to correct for this sagging by adding a magnetic field to the region between the supports. To do this, the magnetic field should be\n\n[figure1]\n\nA: Into the page and of magnitude $\\frac{M g}{L I}$\nB: Out of the page and of magnitude $\\frac{M g}{L I}$\nC: Into the page and of magnitude $\\frac{M g}{2 L I}$\nD: Out of the page and of magnitude $\\frac{M g}{2 L I}$\nE: Into the page and of magnitude $\\frac{M L g}{2 I}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-07.jpg?height=451&width=802&top_left_y=1511&top_left_x=1095" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_612", "problem": "In general, whenever an electric and a magnetic field are at an angle to each other, energy is transferred; for example, this principle is the reason electromagnetic radiation transfers energy. The power transferred per unit area is given by the Poynting vector:\n\n$$\n\\vec{S}=\\frac{1}{\\mu_{0}} \\vec{E} \\times \\vec{B}\n$$\n\nIn each part of this problem, the last subpart asks you to verify that the rate of energy transfer agrees with the formula for the Poynting vector. Therefore, you should not use the formula for the Poynting vector before the last subpart!\n\nA long, insulating cylindrical rod has radius $R$ and carries a uniform volume charge density $\\rho$. A uniform external electric field $E$ exists in the direction of its axis. The rod moves in the direction of its axis at speed $v$.\n\nWhat is the magnetic field $B$ at the surface of the rod? Draw the direction on a diagram.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn general, whenever an electric and a magnetic field are at an angle to each other, energy is transferred; for example, this principle is the reason electromagnetic radiation transfers energy. The power transferred per unit area is given by the Poynting vector:\n\n$$\n\\vec{S}=\\frac{1}{\\mu_{0}} \\vec{E} \\times \\vec{B}\n$$\n\nIn each part of this problem, the last subpart asks you to verify that the rate of energy transfer agrees with the formula for the Poynting vector. Therefore, you should not use the formula for the Poynting vector before the last subpart!\n\nA long, insulating cylindrical rod has radius $R$ and carries a uniform volume charge density $\\rho$. A uniform external electric field $E$ exists in the direction of its axis. The rod moves in the direction of its axis at speed $v$.\n\nWhat is the magnetic field $B$ at the surface of the rod? Draw the direction on a diagram.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_602", "problem": "Part A\n\nAn infinite uniform sheet has a surface charge density $\\sigma$ and has an infinitesimal thickness. The sheet lies in the $x y$ plane.\n\ni. Assuming the sheet is at rest.\n\nBy symmetry, the fields above and below the sheet are equal in magnitude and directed away from the sheet. By Gauss's Law, using a cylinder of base area $A$,\n\n$$\n2 E A=\\frac{\\sigma A}{\\epsilon_{0}} \\Rightarrow E=\\frac{\\sigma}{2 \\epsilon_{0}}\n$$\n\npointing directly away from the sheet in the $z$ direction, or\n\n$$\n\\mathbf{E}=\\frac{\\sigma}{2 \\epsilon} \\times \\begin{cases}\\hat{\\mathbf{z}} & \\text { above the sheet } \\\\ -\\hat{\\mathbf{z}} & \\text { below the sheet. }\\end{cases}\n$$\n\nii. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{x}}$ (parallel to the sheet)\n\nThe motion does not affect the electric field, so the answer is the same as that of part (i).\n\niii. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{x}}$\n\nAssuming $v>0$, the right-hand rule indicates there is a magnetic field in the $-\\tilde{\\mathbf{y}}$ direction for $z>0$ and in the $+\\tilde{\\mathbf{y}}$ direction for $z<0$. From Ampere's law applied to a loop of length $l$ normal to the $\\tilde{\\mathbf{x}}$ direction,\n\n$$\n2 B l=\\mu_{0} \\sigma v l .\n$$\n\nTo get the right-hand side, note that in time $t$, an area $v t l$ moves through the loop, so a charge $\\sigma v t l$ moves through. Then the current through the loop is $\\sigma v l$.\n\nApplying symmetry, we have\n\n$$\n\\mathbf{B}=\\frac{\\mu_{0} \\sigma v}{2} \\times \\begin{cases}-\\hat{\\mathbf{y}} & \\text { above the sheet } \\\\ \\hat{\\mathbf{y}} & \\text { below the sheet. }\\end{cases}\n$$\n\niv. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ (perpendicular to the sheet)\n\nAgain the motion does not affect the electric field, so the answer is the same as that of part (i).\n\nv. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$\n\nApplying Ampere's law and symmetry, there is no magnetic field above and below the sheet.\n\nInterestingly, there's no magnetic field at the sheet either. Consider an Amperian loop of area $A$ in the $x y$ plane as the sheet passes through. The loop experiences a current of the form\n\n$$\nA \\sigma \\delta(t)\n$$\n\nBut the loop also experiences an oppositely directed change in flux of the form\n\n$$\nA \\frac{\\sigma}{\\epsilon_{0}} \\delta(t)\n$$\n\nso the right-hand side of Ampere's law, including the displacement current term, remains zero.In parts this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nIn a certain region there exists only an electric field $\\tilde{\\mathbf{E}}=E_{x} \\hat{\\mathbf{x}}+E_{y} \\hat{\\mathbf{y}}+E_{z} \\hat{\\mathbf{z}}$ (and no magnetic field) as measured by an observer at rest. The electric and magnetic fields $\\tilde{\\mathbf{E}}^{\\prime}$ and $\\tilde{\\mathbf{B}}^{\\prime}$ as measured by observers in motion can be determined entirely from the local value of $\\tilde{\\mathbf{E}}$, regardless of the charge configuration that may have produced it.\n\nWhat would be the observed electric field $\\tilde{\\mathbf{E}}^{\\prime}$ as measured by an observer moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nPart A\n\nAn infinite uniform sheet has a surface charge density $\\sigma$ and has an infinitesimal thickness. The sheet lies in the $x y$ plane.\n\ni. Assuming the sheet is at rest.\n\nBy symmetry, the fields above and below the sheet are equal in magnitude and directed away from the sheet. By Gauss's Law, using a cylinder of base area $A$,\n\n$$\n2 E A=\\frac{\\sigma A}{\\epsilon_{0}} \\Rightarrow E=\\frac{\\sigma}{2 \\epsilon_{0}}\n$$\n\npointing directly away from the sheet in the $z$ direction, or\n\n$$\n\\mathbf{E}=\\frac{\\sigma}{2 \\epsilon} \\times \\begin{cases}\\hat{\\mathbf{z}} & \\text { above the sheet } \\\\ -\\hat{\\mathbf{z}} & \\text { below the sheet. }\\end{cases}\n$$\n\nii. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{x}}$ (parallel to the sheet)\n\nThe motion does not affect the electric field, so the answer is the same as that of part (i).\n\niii. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{x}}$\n\nAssuming $v>0$, the right-hand rule indicates there is a magnetic field in the $-\\tilde{\\mathbf{y}}$ direction for $z>0$ and in the $+\\tilde{\\mathbf{y}}$ direction for $z<0$. From Ampere's law applied to a loop of length $l$ normal to the $\\tilde{\\mathbf{x}}$ direction,\n\n$$\n2 B l=\\mu_{0} \\sigma v l .\n$$\n\nTo get the right-hand side, note that in time $t$, an area $v t l$ moves through the loop, so a charge $\\sigma v t l$ moves through. Then the current through the loop is $\\sigma v l$.\n\nApplying symmetry, we have\n\n$$\n\\mathbf{B}=\\frac{\\mu_{0} \\sigma v}{2} \\times \\begin{cases}-\\hat{\\mathbf{y}} & \\text { above the sheet } \\\\ \\hat{\\mathbf{y}} & \\text { below the sheet. }\\end{cases}\n$$\n\niv. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ (perpendicular to the sheet)\n\nAgain the motion does not affect the electric field, so the answer is the same as that of part (i).\n\nv. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$\n\nApplying Ampere's law and symmetry, there is no magnetic field above and below the sheet.\n\nInterestingly, there's no magnetic field at the sheet either. Consider an Amperian loop of area $A$ in the $x y$ plane as the sheet passes through. The loop experiences a current of the form\n\n$$\nA \\sigma \\delta(t)\n$$\n\nBut the loop also experiences an oppositely directed change in flux of the form\n\n$$\nA \\frac{\\sigma}{\\epsilon_{0}} \\delta(t)\n$$\n\nso the right-hand side of Ampere's law, including the displacement current term, remains zero.\n\nproblem:\nIn parts this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nIn a certain region there exists only an electric field $\\tilde{\\mathbf{E}}=E_{x} \\hat{\\mathbf{x}}+E_{y} \\hat{\\mathbf{y}}+E_{z} \\hat{\\mathbf{z}}$ (and no magnetic field) as measured by an observer at rest. The electric and magnetic fields $\\tilde{\\mathbf{E}}^{\\prime}$ and $\\tilde{\\mathbf{B}}^{\\prime}$ as measured by observers in motion can be determined entirely from the local value of $\\tilde{\\mathbf{E}}$, regardless of the charge configuration that may have produced it.\n\nWhat would be the observed electric field $\\tilde{\\mathbf{E}}^{\\prime}$ as measured by an observer moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_896", "problem": "In the previous section, the density and the tensile strength of carbon nanotubes have been evaluated theoretically. These evaluated values indeed depend on the specific structure of carbon nanotubes. Nevertheless, the idea of space elevator construction is truly feasible. Now we will study a new space elevator design of the so-called tapered tower whose cross section varies with height in such a way that both the stress $\\sigma$ and mass density $\\rho$ are uniform over the entire tower length. The tower has axial symmetry and is positioned vertically at the equator; its height is greater than the height of the geostationary satellite orbit. Denote the cross sectional area of the tapered tower on the Earth surface by $A_{S}$ and at geostationary height $A_{G}$. \n\nThe taper ratio is defined as $A_{G} / A_{S}$. Find the taper ratio of the tower made of carbon nanotubes with tensile strength $130 \\mathrm{GPa}$ and density $1300 \\mathrm{~kg} / \\mathrm{m}^{3}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn the previous section, the density and the tensile strength of carbon nanotubes have been evaluated theoretically. These evaluated values indeed depend on the specific structure of carbon nanotubes. Nevertheless, the idea of space elevator construction is truly feasible. Now we will study a new space elevator design of the so-called tapered tower whose cross section varies with height in such a way that both the stress $\\sigma$ and mass density $\\rho$ are uniform over the entire tower length. The tower has axial symmetry and is positioned vertically at the equator; its height is greater than the height of the geostationary satellite orbit. Denote the cross sectional area of the tapered tower on the Earth surface by $A_{S}$ and at geostationary height $A_{G}$. \n\nThe taper ratio is defined as $A_{G} / A_{S}$. Find the taper ratio of the tower made of carbon nanotubes with tensile strength $130 \\mathrm{GPa}$ and density $1300 \\mathrm{~kg} / \\mathrm{m}^{3}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_253", "problem": "The characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part the peculiarity of the circular motion around the dipole is analyzed. Initially, the system is the same as in Part 2, with the exception that the charged object is connected to the center of the dipole with a light, rigid insulating rod with the length $L$. This rod easily rotates around the axis, which is perpendicular to the surface of the table. Thus, the charged object moves around the dipole along circular trajectory with radius $L$.\n\nWhat is the maximum and minimum velocity of the charged object $v_{\\max }$ and $v_{\\min }$ during circular motion around the dipole?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nThe characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part the peculiarity of the circular motion around the dipole is analyzed. Initially, the system is the same as in Part 2, with the exception that the charged object is connected to the center of the dipole with a light, rigid insulating rod with the length $L$. This rod easily rotates around the axis, which is perpendicular to the surface of the table. Thus, the charged object moves around the dipole along circular trajectory with radius $L$.\n\nWhat is the maximum and minimum velocity of the charged object $v_{\\max }$ and $v_{\\min }$ during circular motion around the dipole?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [$v_{\\max}$, $v_{\\min}$].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$v_{\\max}$", "$v_{\\min}$" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_510", "problem": "Suppose a domino stands upright on a table. It has height $h$, thickness $t$, width $w$ (as shown below), and mass $m$. The domino is free to rotate about its edges, but will not slide across the table.\n[figure1]\n\nA row of toppling dominoes can be considered to have a propagation speed of the length $l+t$ divided by the time between successive collisions. When the first domino is given a minimal push just large enough to topple and start a chain reaction of toppling dominoes, the speed increases with each domino, but approaches an asymptotic speed $v$.\n\n[figure2]\n\nSuppose there is a row of dominoes on another planet. These dominoes have the same density as the dominoes previously considered, but are twice as tall, wide, and thick, and placed with a spacing of $2 l$ between them. If this row of dominoes topples with the same asymptotic speed $v$ previously found, what is the local gravitational acceleration on this planet?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nSuppose a domino stands upright on a table. It has height $h$, thickness $t$, width $w$ (as shown below), and mass $m$. The domino is free to rotate about its edges, but will not slide across the table.\n[figure1]\n\nA row of toppling dominoes can be considered to have a propagation speed of the length $l+t$ divided by the time between successive collisions. When the first domino is given a minimal push just large enough to topple and start a chain reaction of toppling dominoes, the speed increases with each domino, but approaches an asymptotic speed $v$.\n\n[figure2]\n\nSuppose there is a row of dominoes on another planet. These dominoes have the same density as the dominoes previously considered, but are twice as tall, wide, and thick, and placed with a spacing of $2 l$ between them. If this row of dominoes topples with the same asymptotic speed $v$ previously found, what is the local gravitational acceleration on this planet?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-15.jpg?height=456&width=832&top_left_y=528&top_left_x=644", "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-18.jpg?height=249&width=358&top_left_y=1450&top_left_x=905" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_718", "problem": "The figure below shows two rectangular spaceships, one at rest and the other one moving at speed $v=\\sqrt{3} c / 2$. In the stationary frame, they are seen to have the same length $L$ and their two ends align at $t=0$. If we know that the clock at the left end of the moving spaceship is showing that the time is at $t^{\\prime}=0$, what time would the clock at its right end show?\n\n[figure1]\nA: $-\\sqrt{3} L / c$\nB: $\\sqrt{3} L / c$\nC: $-L / \\sqrt{3} c$\nD: $L / \\sqrt{3} c$\nE: 0\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe figure below shows two rectangular spaceships, one at rest and the other one moving at speed $v=\\sqrt{3} c / 2$. In the stationary frame, they are seen to have the same length $L$ and their two ends align at $t=0$. If we know that the clock at the left end of the moving spaceship is showing that the time is at $t^{\\prime}=0$, what time would the clock at its right end show?\n\n[figure1]\n\nA: $-\\sqrt{3} L / c$\nB: $\\sqrt{3} L / c$\nC: $-L / \\sqrt{3} c$\nD: $L / \\sqrt{3} c$\nE: 0\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22e26a14ee6fdd9254b6g-06.jpg?height=264&width=762&top_left_y=538&top_left_x=215" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1608", "problem": "充有水的连通软管常常用来检验建筑物的水平度。但软管中气泡会使得该软管两边管口水面不在同一水平面上。为了说明这一现象的物理原因，考虑如图所示的连通水管（由三段内径相同的 U 形管密接而成）, 其中封有一段空气（可视为理想气体), 与空气接触的四段水管均在坚直\n\n[图1]\n方向; 且两个有水的 U 形管两边水面分别等高。此时被封闭的空气柱的长度为 $L_{\\mathrm{a}}$ 。己知大气压强为 $P_{0}$ 、水的密度为 $\\rho$ 、重力加速度大小为 $g, L_{0} \\equiv P_{0} /(\\rho g)$ 。现由左管口添加体积为 $\\Delta V=x S$ 的水, $S$ 为水管的横截面积。当 $x<$ ). Which of the following light diagrams describe how an observer at the equator of the Earth can see the moon at night? Assume the Earth has no tilt, and the Earth and moon rotate in a flat plane around the sun. Note the figure is not to scale.\nA: ![]([figure1])\nB: ![]([figure2])\nC: ![]([figure3])\nD: ![]([figure4])\nE: ![]([figure5])\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe moon can be visible in the sky at night even though it does not produce any light itself. Instead, the sun produces light, and the moon reflects this light. We use light diagrams to show the path of light rays as arrows, from when they are produced as a source (tail of arrow, -) to when they are observed (head of arrow, $>$ ). Which of the following light diagrams describe how an observer at the equator of the Earth can see the moon at night? Assume the Earth has no tilt, and the Earth and moon rotate in a flat plane around the sun. Note the figure is not to scale.\n\nA: ![]([figure1])\nB: ![]([figure2])\nC: ![]([figure3])\nD: ![]([figure4])\nE: ![]([figure5])\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-05.jpg?height=359&width=371&top_left_y=1788&top_left_x=363", "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-05.jpg?height=359&width=394&top_left_y=1785&top_left_x=885", "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-05.jpg?height=369&width=391&top_left_y=1780&top_left_x=1369", "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-05.jpg?height=366&width=374&top_left_y=2170&top_left_x=630", "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-05.jpg?height=345&width=371&top_left_y=2192&top_left_x=1114" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_390", "problem": "A roller consists of a solid homogeneous cylinder of mass $M$ and radius $r$ it rests on a horizontal table and is attached to a wall via a helical spring of spring constant $k$ (see figure). The spring can be assumed to be of a negligible mass and ideal, i.e. the Hooke's law remains valid for arbitrarily large deformations.\n\n[figure1]\n\nIf the initial oscillation amplitude (measured as the deformation $x$ of the spring) is larger than a certain critical value $A_{\\star}$, the amplitude of oscillations starts decreasing in time. Express $A_{\\star}$ in terms of $k$ $M, r$, the free fall acceleration $g$,", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA roller consists of a solid homogeneous cylinder of mass $M$ and radius $r$ it rests on a horizontal table and is attached to a wall via a helical spring of spring constant $k$ (see figure). The spring can be assumed to be of a negligible mass and ideal, i.e. the Hooke's law remains valid for arbitrarily large deformations.\n\n[figure1]\n\nIf the initial oscillation amplitude (measured as the deformation $x$ of the spring) is larger than a certain critical value $A_{\\star}$, the amplitude of oscillations starts decreasing in time. Express $A_{\\star}$ in terms of $k$ $M, r$, the free fall acceleration $g$,\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_706aca6df357b4c9a255g-1.jpg?height=142&width=511&top_left_y=1581&top_left_x=140" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_424", "problem": "This question relates to ways in which light energy may be concentrated in an interference pattern.\n\nFigure 1 shows wave fronts at normal incidence on a Young's double slit arrangement, illuminating them so that they both radiate in phase, each with an amplitude $A$ at the screen. One such slit alone will cause an intensity of illumination of $I$, where $I \\propto A^{2}$, in the central region of the screen where the Young's fringes pattern forms.\n\n[figure1]\nFigure: Young's double slits.\n\nWhat is the resultant amplitude of the waves from the slits arriving at the screen at a maxima (a bright fringe), where the incoming waves arrive in phase?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThis question relates to ways in which light energy may be concentrated in an interference pattern.\n\nFigure 1 shows wave fronts at normal incidence on a Young's double slit arrangement, illuminating them so that they both radiate in phase, each with an amplitude $A$ at the screen. One such slit alone will cause an intensity of illumination of $I$, where $I \\propto A^{2}$, in the central region of the screen where the Young's fringes pattern forms.\n\n[figure1]\nFigure: Young's double slits.\n\nWhat is the resultant amplitude of the waves from the slits arriving at the screen at a maxima (a bright fringe), where the incoming waves arrive in phase?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-3.jpg?height=522&width=859&top_left_y=584&top_left_x=610" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_928", "problem": "he schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\\pm \\varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale.\n\n[figure1]\n\nAround the northern winter solstice, the rising branch of the Hadley circulation is located at the latitude $\\varphi_{r}$ and the descending branches are located at $\\varphi_{n}$ and $\\varphi_{s}$ as shown in the schematic below. Refer to this diagram for parts (c), (d) and (e).\n\n[figure2]\n\nAssume that there is no east-west wind velocity around the point $\\mathrm{Z}$. Given that $\\varphi_{r}=-8^{\\circ}, \\varphi_{n}=28^{\\circ}$ and $\\varphi_{s}=-20^{\\circ}$, what are the east-west wind velocities $u_{P}, u_{Q}$ and $u_{R}$ respectively at the points $\\mathrm{P}, \\mathrm{Q}$ and $\\mathrm{R}$ ?\n\n(The radius of the Earth is a $=6370 \\mathrm{~km}$.)\n\nHence, which hemisphere below has a stronger atmospheric jet stream?\nA: Winter Hemisphere\nB: Summer Hemisphere\nC: Both hemispheres have equally strong jet streams.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nhe schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\\pm \\varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale.\n\n[figure1]\n\nAround the northern winter solstice, the rising branch of the Hadley circulation is located at the latitude $\\varphi_{r}$ and the descending branches are located at $\\varphi_{n}$ and $\\varphi_{s}$ as shown in the schematic below. Refer to this diagram for parts (c), (d) and (e).\n\n[figure2]\n\nAssume that there is no east-west wind velocity around the point $\\mathrm{Z}$. Given that $\\varphi_{r}=-8^{\\circ}, \\varphi_{n}=28^{\\circ}$ and $\\varphi_{s}=-20^{\\circ}$, what are the east-west wind velocities $u_{P}, u_{Q}$ and $u_{R}$ respectively at the points $\\mathrm{P}, \\mathrm{Q}$ and $\\mathrm{R}$ ?\n\n(The radius of the Earth is a $=6370 \\mathrm{~km}$.)\n\nHence, which hemisphere below has a stronger atmospheric jet stream?\n\nA: Winter Hemisphere\nB: Summer Hemisphere\nC: Both hemispheres have equally strong jet streams.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e7b9ba69a6dd20c5cfd1g-1.jpg?height=1148&width=1151&top_left_y=1096&top_left_x=384", "https://cdn.mathpix.com/cropped/2024_03_14_e7b9ba69a6dd20c5cfd1g-2.jpg?height=1159&width=1160&top_left_y=1014&top_left_x=445" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_169", "problem": "A group of students wish to measure the acceleration of gravity with a simple pendulum. They take one length measurement of the pendulum to be $L=1.00 \\pm 0.05 \\mathrm{~m}$. They then measure the period of a single swing to be $T=2.00 \\pm 0.10 \\mathrm{~s}$. Assume that all uncertainties are Gaussian. The computed acceleration of gravity from this experiment illustrating the range of possible values should be recorded as\nA: $9.87 \\pm 0.10 \\mathrm{~m} / \\mathrm{s}^{2}$\nB: $9.87 \\pm 0.15 \\mathrm{~m} / \\mathrm{s}^{2}$\nC: $9.9 \\pm 0.25 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: $9.9 \\pm 1.1 \\mathrm{~m} / \\mathrm{s}^{2} $ \nE: $9.9 \\pm 1.5 \\mathrm{~m} / \\mathrm{s}^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA group of students wish to measure the acceleration of gravity with a simple pendulum. They take one length measurement of the pendulum to be $L=1.00 \\pm 0.05 \\mathrm{~m}$. They then measure the period of a single swing to be $T=2.00 \\pm 0.10 \\mathrm{~s}$. Assume that all uncertainties are Gaussian. The computed acceleration of gravity from this experiment illustrating the range of possible values should be recorded as\n\nA: $9.87 \\pm 0.10 \\mathrm{~m} / \\mathrm{s}^{2}$\nB: $9.87 \\pm 0.15 \\mathrm{~m} / \\mathrm{s}^{2}$\nC: $9.9 \\pm 0.25 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: $9.9 \\pm 1.1 \\mathrm{~m} / \\mathrm{s}^{2} $ \nE: $9.9 \\pm 1.5 \\mathrm{~m} / \\mathrm{s}^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_302", "problem": "A marathon runner crosses the start line of a race at a speed of $1 \\mathrm{~m} / \\mathrm{s}$ and accelerates at a constant rate of $2 \\mathrm{~m} / \\mathrm{s}^{2}$ for 2 seconds.\n\nWhich graph shows the relationship between displacement from the start line and time after crossing the start line?\n\n[figure1]\n\nA\n\n[figure2]\n\nB\n\n[figure3]\n\nC\n\n[figure4]\n\nD\nA: A\nB: B\nC: C\nD: D\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA marathon runner crosses the start line of a race at a speed of $1 \\mathrm{~m} / \\mathrm{s}$ and accelerates at a constant rate of $2 \\mathrm{~m} / \\mathrm{s}^{2}$ for 2 seconds.\n\nWhich graph shows the relationship between displacement from the start line and time after crossing the start line?\n\n[figure1]\n\nA\n\n[figure2]\n\nB\n\n[figure3]\n\nC\n\n[figure4]\n\nD\n\nA: A\nB: B\nC: C\nD: D\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-03.jpg?height=383&width=328&top_left_y=1756&top_left_x=247", "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-03.jpg?height=385&width=331&top_left_y=1755&top_left_x=654", "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-03.jpg?height=383&width=331&top_left_y=1759&top_left_x=1065", "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-03.jpg?height=383&width=345&top_left_y=1756&top_left_x=1458" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_393", "problem": "Equipment: a sheet of Fresnel prism, a sheet with purple and magenta stripes (see separate sheet), a piece of cardboard paper (can be used as a screen), ruler, measuring tape, stand, green laser $\\left(\\lambda_{0}=532 \\mathrm{~nm}\\right)$. NB! Avoid looking into direct or reflected laser light, it my damage your eyes! The intensity maximum for the scattered light from magenta stripes is $\\lambda_{m}=630 \\mathrm{~nm}$, and from cyan stripes $-\\lambda_{c}=$ $495 \\mathrm{~nm}$.\n\nFresnel prism is a transparent sheet with a periodic array of stripes; cross-section of such a sheet is shown in figure. The refraction index of the material from which the sheet is made $n=1.47$.\n\n[figure1]\n\nDetermine the angle $\\alpha$ of the prism.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nEquipment: a sheet of Fresnel prism, a sheet with purple and magenta stripes (see separate sheet), a piece of cardboard paper (can be used as a screen), ruler, measuring tape, stand, green laser $\\left(\\lambda_{0}=532 \\mathrm{~nm}\\right)$. NB! Avoid looking into direct or reflected laser light, it my damage your eyes! The intensity maximum for the scattered light from magenta stripes is $\\lambda_{m}=630 \\mathrm{~nm}$, and from cyan stripes $-\\lambda_{c}=$ $495 \\mathrm{~nm}$.\n\nFresnel prism is a transparent sheet with a periodic array of stripes; cross-section of such a sheet is shown in figure. The refraction index of the material from which the sheet is made $n=1.47$.\n\n[figure1]\n\nDetermine the angle $\\alpha$ of the prism.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_57fd2cb729ee4cd65883g-1.jpg?height=108&width=657&top_left_y=282&top_left_x=2174" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_922", "problem": "## Introduction\n\nActive galactic nuclei (AGN) are supermassive black holes which form the centres of galaxies, and emit large amounts of energy in radiation and particle flows. One feature of many AGN are jetted outflows, which can be observed through radio emission, and sometimes also in other parts of the electromagnetic spectrum, including $x$-rays. These jets are large flows of plasma at relativistic speeds, over lengths of order $10^{20} \\mathrm{~m}$, which is tens of thousands of light years. The $\\mathrm{x}$-ray emission from jets is usually dominated by synchrotron emission from relativistic electrons gyrating in the magnetic field of the jet.\n\n[figure1]\n\nFigure 1: X-ray image of the jet from the Centaurus A AGN. Darker regions represent regions of higher intensity $x$-rays. Brighter regions within the fainter jet are called knots. (Snios et al., 2019)\n\nA simple model of the flow of jets assumes that the flow is steady and directed radially away from the central AGN, so approximately one dimensional, and that the plasma in the jet is in pressure equilibrium with its surroundings. There is assumed to be a constant rate per volume of mass injected into the jet from stars which lose their outer layers as they move through their life cycle.\n\nThe jet is described in terms of the coordinate representing distance from the AGN, $s$, and the opening radius $r$ of the conical jet. These distances are measured in parsecs, where $1 \\mathrm{pc}=3.086 \\times 10^{16} \\mathrm{~m}$. The speed of the jet flow is assumed to be directed radially away from the central AGN, and be a function of $s$ only. The plasma in the jet is comprised of electrons, protons, and some heavier ionised nuclei. The average energy carried by each particle in the jet, in the reference frame of the bulk flow of the jet (which we will call the jet frame), is $\\epsilon_{\\mathrm{av}}=\\mu_{\\mathrm{pp}} c^{2}+h$, where the term $h$ includes all thermal kinetic energy and potential energies in terms of the pressure $P$ and $n$ is the number density of the plasma.\n\nAs the stars, which the jet flows past, move through their life cycles they can lose part of their atmosphere. This results in a uniform rate of injection of mass per unit volume $\\alpha$ into the jet, and the injected particles are assumed to be at rest relative to the AGN.\n\nThis model can be applied to the Centaurus A jet. Centaurus A is one of the nearest AGN, so it is possible to observe its jet at relatively high spatial resolution. The total power carried by the jet is estimated to be $P_{\\mathrm{j}}=1 \\times 10^{36} \\mathrm{~J} \\cdot \\mathrm{s}^{-1}$. See below for a diagram of a simple geometrical description of the Centaurus A jet, including measurements of some jet parameters. $s_{1}$ is the coordinate of the start of the jet, and $s_{2}$ the coordinate of the end of the jet. In Centuarus A the average mass per particle is $\\mu_{\\mathrm{pp}}=0.59 m_{\\mathrm{p}}$ and $h=\\frac{13}{4} P / n$. The pressure in the plasma surrounding the jet is $P(s)=5.7 \\times 10^{-12}\\left(\\frac{s}{s_{0}}\\right)^{-1.5} \\mathrm{~Pa}$, where $s_{0}=1 \\mathrm{kpc}$.\n\n[figure2]\n\nFigure 2: The Centaurus A jet, showing the geometry compared to the active galactic nucleus (AGN).\n\nThe jet is described by the following parameters, all of which depend on the distance $s$ from the AGN:\n\n- the opening radius of the jet $r(s)$ in the AGN frame\n- the cross sectional area of the jet $A(s)$ in the AGN frame\n- the speed of the jet $v(s)$ in the AGN frame\n- the lorentz gamma factor of the jet $\\gamma(s)$ in the AGN frame\n- the number density $n(s)$ in the frame of the jet\n\nFind the flux of particles, $F_{p}(s)$, across a cross section of the jet with area $A$, at $0.2 \\mathrm{pt}$ a distance $s$ from the AGN.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\n## Introduction\n\nActive galactic nuclei (AGN) are supermassive black holes which form the centres of galaxies, and emit large amounts of energy in radiation and particle flows. One feature of many AGN are jetted outflows, which can be observed through radio emission, and sometimes also in other parts of the electromagnetic spectrum, including $x$-rays. These jets are large flows of plasma at relativistic speeds, over lengths of order $10^{20} \\mathrm{~m}$, which is tens of thousands of light years. The $\\mathrm{x}$-ray emission from jets is usually dominated by synchrotron emission from relativistic electrons gyrating in the magnetic field of the jet.\n\n[figure1]\n\nFigure 1: X-ray image of the jet from the Centaurus A AGN. Darker regions represent regions of higher intensity $x$-rays. Brighter regions within the fainter jet are called knots. (Snios et al., 2019)\n\nA simple model of the flow of jets assumes that the flow is steady and directed radially away from the central AGN, so approximately one dimensional, and that the plasma in the jet is in pressure equilibrium with its surroundings. There is assumed to be a constant rate per volume of mass injected into the jet from stars which lose their outer layers as they move through their life cycle.\n\nThe jet is described in terms of the coordinate representing distance from the AGN, $s$, and the opening radius $r$ of the conical jet. These distances are measured in parsecs, where $1 \\mathrm{pc}=3.086 \\times 10^{16} \\mathrm{~m}$. The speed of the jet flow is assumed to be directed radially away from the central AGN, and be a function of $s$ only. The plasma in the jet is comprised of electrons, protons, and some heavier ionised nuclei. The average energy carried by each particle in the jet, in the reference frame of the bulk flow of the jet (which we will call the jet frame), is $\\epsilon_{\\mathrm{av}}=\\mu_{\\mathrm{pp}} c^{2}+h$, where the term $h$ includes all thermal kinetic energy and potential energies in terms of the pressure $P$ and $n$ is the number density of the plasma.\n\nAs the stars, which the jet flows past, move through their life cycles they can lose part of their atmosphere. This results in a uniform rate of injection of mass per unit volume $\\alpha$ into the jet, and the injected particles are assumed to be at rest relative to the AGN.\n\nThis model can be applied to the Centaurus A jet. Centaurus A is one of the nearest AGN, so it is possible to observe its jet at relatively high spatial resolution. The total power carried by the jet is estimated to be $P_{\\mathrm{j}}=1 \\times 10^{36} \\mathrm{~J} \\cdot \\mathrm{s}^{-1}$. See below for a diagram of a simple geometrical description of the Centaurus A jet, including measurements of some jet parameters. $s_{1}$ is the coordinate of the start of the jet, and $s_{2}$ the coordinate of the end of the jet. In Centuarus A the average mass per particle is $\\mu_{\\mathrm{pp}}=0.59 m_{\\mathrm{p}}$ and $h=\\frac{13}{4} P / n$. The pressure in the plasma surrounding the jet is $P(s)=5.7 \\times 10^{-12}\\left(\\frac{s}{s_{0}}\\right)^{-1.5} \\mathrm{~Pa}$, where $s_{0}=1 \\mathrm{kpc}$.\n\n[figure2]\n\nFigure 2: The Centaurus A jet, showing the geometry compared to the active galactic nucleus (AGN).\n\nThe jet is described by the following parameters, all of which depend on the distance $s$ from the AGN:\n\n- the opening radius of the jet $r(s)$ in the AGN frame\n- the cross sectional area of the jet $A(s)$ in the AGN frame\n- the speed of the jet $v(s)$ in the AGN frame\n- the lorentz gamma factor of the jet $\\gamma(s)$ in the AGN frame\n- the number density $n(s)$ in the frame of the jet\n\nFind the flux of particles, $F_{p}(s)$, across a cross section of the jet with area $A$, at $0.2 \\mathrm{pt}$ a distance $s$ from the AGN.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2416d49d47cb88c0a72bg-1.jpg?height=651&width=805&top_left_y=1045&top_left_x=631", "https://cdn.mathpix.com/cropped/2024_03_14_2416d49d47cb88c0a72bg-2.jpg?height=654&width=1356&top_left_y=821&top_left_x=356" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_672", "problem": "Two objects, A and B, appear the same length when A is stationary and $\\mathrm{B}$ is moving with speed $(3 / 5) c$ along its length. In the frame of reference where B is stationary and $\\mathrm{A}$ is moving, what is the ratio of their lengths?\nA: $L_{A} / L_{B}=5 / 4$\nB: $L_{A} / L_{B}=25 / 16$\nC: $L_{A} / L_{B}=4 / 5$\nD: $L_{A} / L_{B}=9 / 25$\nE: $L_{A} / L_{B}=16 / 25$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo objects, A and B, appear the same length when A is stationary and $\\mathrm{B}$ is moving with speed $(3 / 5) c$ along its length. In the frame of reference where B is stationary and $\\mathrm{A}$ is moving, what is the ratio of their lengths?\n\nA: $L_{A} / L_{B}=5 / 4$\nB: $L_{A} / L_{B}=25 / 16$\nC: $L_{A} / L_{B}=4 / 5$\nD: $L_{A} / L_{B}=9 / 25$\nE: $L_{A} / L_{B}=16 / 25$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_820", "problem": "Figure 1 shows a Fabry-Perot (F-P) etalon, in which air pressure is tunable. The F-P etalon consists of two glass plates with high-reflectivity inner surfaces. The two plates form a cavity in which light can be reflected back and forth. The outer surfaces of the plates are generally not parallel to the inner ones and do not affect the back-and-forth reflection. The air density in the etalon can be controlled. Light from a Sodium lamp is collimated by the lens L1 and then passes through the F-P etalon. The transmitivity of the etalon is given by $T=\\frac{1}{1+F \\sin ^{2}(\\delta / 2)}$, where $F=\\frac{4 R}{(1-R)^{2}}, \\mathrm{R}$ is the reflectivity of the inner surfaces, $\\delta=\\frac{4 \\pi n t \\cos \\theta}{\\lambda}$ is the phase shift of two neighboring rays, $\\mathrm{n}$ is the refractive index of the gas, $\\mathrm{t}$ is the spacing of inner surfaces, $\\theta$ is the incident angle, and $\\lambda$ is the light wavelength.\n\n[figure1]\n\nFigure 1\n\nThe Sodium lamp emits D1 $(\\lambda=589.6 \\mathrm{~nm})$ and D2 $(589 \\mathrm{~nm})$ spectral lines and is located in a tunable uniform magnetic field. For simplicity, an optical filter F1 is assumed to only allow the D1 line to pass through. The D1 line is then collimated to the F-P etalon by the lens L1. Circular interference fringes will be present on the focal plane of the lens L2 with a focal length $\\mathrm{f}=30 \\mathrm{~cm}$. Different fringes have the different incident angle $\\theta$. A microscope is used to observe the fringes. We take the reflectivity $\\mathrm{R}=90 \\%$ and the inner-surface spacing $\\mathrm{t}=1 \\mathrm{~cm}$.\n\nSome physical constants: $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}, e=1.6 \\times 10^{-19} \\mathrm{C}, m_{e}=9.1 \\times 10^{-31} \\mathrm{~kg}, c=3.0 \\times 10^{8} \\mathrm{~ms}^{-1}$.\n\nThe D1 line $(\\lambda=589.6 \\mathrm{~nm})$ is collimated to the F-P etalon. For the vacuum case ( $\\mathrm{n}=1.0$ ), please calculate (i) interference orders $m_{i}$, (ii) incidence angle $\\theta_{i}$ and (iii) diameter\n\n$D_{i}$ for the first three $(i=1,2,3)$ fringes from the center of the ring patterns on the focal plane.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nFigure 1 shows a Fabry-Perot (F-P) etalon, in which air pressure is tunable. The F-P etalon consists of two glass plates with high-reflectivity inner surfaces. The two plates form a cavity in which light can be reflected back and forth. The outer surfaces of the plates are generally not parallel to the inner ones and do not affect the back-and-forth reflection. The air density in the etalon can be controlled. Light from a Sodium lamp is collimated by the lens L1 and then passes through the F-P etalon. The transmitivity of the etalon is given by $T=\\frac{1}{1+F \\sin ^{2}(\\delta / 2)}$, where $F=\\frac{4 R}{(1-R)^{2}}, \\mathrm{R}$ is the reflectivity of the inner surfaces, $\\delta=\\frac{4 \\pi n t \\cos \\theta}{\\lambda}$ is the phase shift of two neighboring rays, $\\mathrm{n}$ is the refractive index of the gas, $\\mathrm{t}$ is the spacing of inner surfaces, $\\theta$ is the incident angle, and $\\lambda$ is the light wavelength.\n\n[figure1]\n\nFigure 1\n\nThe Sodium lamp emits D1 $(\\lambda=589.6 \\mathrm{~nm})$ and D2 $(589 \\mathrm{~nm})$ spectral lines and is located in a tunable uniform magnetic field. For simplicity, an optical filter F1 is assumed to only allow the D1 line to pass through. The D1 line is then collimated to the F-P etalon by the lens L1. Circular interference fringes will be present on the focal plane of the lens L2 with a focal length $\\mathrm{f}=30 \\mathrm{~cm}$. Different fringes have the different incident angle $\\theta$. A microscope is used to observe the fringes. We take the reflectivity $\\mathrm{R}=90 \\%$ and the inner-surface spacing $\\mathrm{t}=1 \\mathrm{~cm}$.\n\nSome physical constants: $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}, e=1.6 \\times 10^{-19} \\mathrm{C}, m_{e}=9.1 \\times 10^{-31} \\mathrm{~kg}, c=3.0 \\times 10^{8} \\mathrm{~ms}^{-1}$.\n\nThe D1 line $(\\lambda=589.6 \\mathrm{~nm})$ is collimated to the F-P etalon. For the vacuum case ( $\\mathrm{n}=1.0$ ), please calculate (i) interference orders $m_{i}$, (ii) incidence angle $\\theta_{i}$ and (iii) diameter\n\n$D_{i}$ for the first three $(i=1,2,3)$ fringes from the center of the ring patterns on the focal plane.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the value of $D_1$, the value of $D_2$, the value of $D_2=3$].\nTheir units are, in order, [mm, mm, mm], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_42bd302d6bc3274ac3b2g-1.jpg?height=691&width=1133&top_left_y=1236&top_left_x=473" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "mm", "mm", "mm" ], "answer_sequence": [ "the value of $D_1$", "the value of $D_2$", "the value of $D_2=3$" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1292", "problem": "如图所示, 1 和 2 是放在水平地面上的两个小物块(可视为质点), 与地面的滑动摩擦系数相同, 两物块间的距离 $d=170.00 \\mathrm{~m}$, 它们的质量分别为 $m_{1}=2.00 \\mathrm{~kg}, m_{2}=3.00 \\mathrm{~kg}$. 现令它们分别以初速度 $v_{1}=10.00 \\mathrm{~m} / \\mathrm{s}$ 和 $v_{2}=2.00 \\mathrm{~m} / \\mathrm{s}$ 迎向运动, 经过时间 $t=20.0 \\mathrm{~s}$, 两物块相碰,碰撞时间极短, 碰后两者粘在一起运动. 求从刚碰后到停止运动过程中损失的机械能.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n如图所示, 1 和 2 是放在水平地面上的两个小物块(可视为质点), 与地面的滑动摩擦系数相同, 两物块间的距离 $d=170.00 \\mathrm{~m}$, 它们的质量分别为 $m_{1}=2.00 \\mathrm{~kg}, m_{2}=3.00 \\mathrm{~kg}$. 现令它们分别以初速度 $v_{1}=10.00 \\mathrm{~m} / \\mathrm{s}$ 和 $v_{2}=2.00 \\mathrm{~m} / \\mathrm{s}$ 迎向运动, 经过时间 $t=20.0 \\mathrm{~s}$, 两物块相碰,碰撞时间极短, 碰后两者粘在一起运动. 求从刚碰后到停止运动过程中损失的机械能.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$$ J $$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$$ J $$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1048", "problem": "Gaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\n## Manzanares prototype\n\nThe prototype chimney built in Manzanares, Spain, had a height of $195 \\mathrm{~m}$, and a radius $5 \\mathrm{~m}$. The collector is circular with diameter of $244 \\mathrm{~m}$. The specific heat of the air under typical operational conditions of the prototype solar chimney is $1012 \\mathrm{~J} / \\mathrm{kg} \\mathrm{K}$, the density of the hot air is about $0.9 \\mathrm{~kg} / \\mathrm{m}^{3}$, and the typical temperature of the atmosphere $T_{\\text {Air }}=295 \\mathrm{~K}$. In Manzanares, the solar power per unit of horizontal surface is typically $150 \\mathrm{~W} / \\mathrm{m}^{2}$ during a sunny day.\n\nHow much energy could the power plant produce during a typical sunny day?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nGaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\n## Manzanares prototype\n\nThe prototype chimney built in Manzanares, Spain, had a height of $195 \\mathrm{~m}$, and a radius $5 \\mathrm{~m}$. The collector is circular with diameter of $244 \\mathrm{~m}$. The specific heat of the air under typical operational conditions of the prototype solar chimney is $1012 \\mathrm{~J} / \\mathrm{kg} \\mathrm{K}$, the density of the hot air is about $0.9 \\mathrm{~kg} / \\mathrm{m}^{3}$, and the typical temperature of the atmosphere $T_{\\text {Air }}=295 \\mathrm{~K}$. In Manzanares, the solar power per unit of horizontal surface is typically $150 \\mathrm{~W} / \\mathrm{m}^{2}$ during a sunny day.\n\nHow much energy could the power plant produce during a typical sunny day?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of kWh, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_fd946cfac82ef740b1dag-1.jpg?height=977&width=1644&top_left_y=1453&top_left_x=206" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "kWh" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_307", "problem": "Water waves on the surface of water (ripples) travel more slowly in shallow water than they do in deeper water.\n\nWater waves in a ripple tank travel from deeper water to shallower water.\n\nWhat are the corresponding changes in the wavelength and frequency of the water waves?\n\n| A | Wavelength decreases | Frequency decreases |\n| :---: | :--- | :--- |\n| B | Wavelength decreases | Frequency remains the same |\n| C | Wavelength stays the same | Frequency decreases |\n| D | Wavelength stays the same | Frequency remains the same |\nA: A\nB: B\nC: C\nD: D\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nWater waves on the surface of water (ripples) travel more slowly in shallow water than they do in deeper water.\n\nWater waves in a ripple tank travel from deeper water to shallower water.\n\nWhat are the corresponding changes in the wavelength and frequency of the water waves?\n\n| A | Wavelength decreases | Frequency decreases |\n| :---: | :--- | :--- |\n| B | Wavelength decreases | Frequency remains the same |\n| C | Wavelength stays the same | Frequency decreases |\n| D | Wavelength stays the same | Frequency remains the same |\n\nA: A\nB: B\nC: C\nD: D\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_856", "problem": "Initially the external field is turned off. Then the field magnitude is increased from zero to $E_{0}$ very slowly so that the electric field can be considered effectively time-independent in this question. The instantaneous value of the external field is denoted by $\\vec{E}=E \\hat{u}$, Find the total work done by the external field on the atom when the electric field is increased from zero to $E=E_{0}$. Hence deduce an expression for the induced\n\n$0.75 \\mathrm{pt}$ dipole potential energy $U_{\\text {induced }}$ in terms of $\\vec{E}_{o}$ and $\\vec{p}_{o}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nInitially the external field is turned off. Then the field magnitude is increased from zero to $E_{0}$ very slowly so that the electric field can be considered effectively time-independent in this question. The instantaneous value of the external field is denoted by $\\vec{E}=E \\hat{u}$, Find the total work done by the external field on the atom when the electric field is increased from zero to $E=E_{0}$. Hence deduce an expression for the induced\n\n$0.75 \\mathrm{pt}$ dipole potential energy $U_{\\text {induced }}$ in terms of $\\vec{E}_{o}$ and $\\vec{p}_{o}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_104", "problem": "An object starting from rest can roll without slipping down an incline.\n\n14. Which of the following four objects, each with mass $M$ and radius $R$, would have the largest acceleration down the incline?\nA: A uniform solid sphere \nB: A uniform solid disk\nC: A hollow spherical shell\nD: A hoop\nE: All objects would have the same acceleration.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn object starting from rest can roll without slipping down an incline.\n\n14. Which of the following four objects, each with mass $M$ and radius $R$, would have the largest acceleration down the incline?\n\nA: A uniform solid sphere \nB: A uniform solid disk\nC: A hollow spherical shell\nD: A hoop\nE: All objects would have the same acceleration.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_863", "problem": "The cooling power and the maximum temperature difference\n\nThe upper end of the thermocouple is a heat source with the initial temperature $T_{1}$. It is thermally isolated with ambient environment, and needs to be cooled. The lower ends of the thermocouple, A and $\\mathrm{B}$ bars are connected to a battery and are at the temperature $T_{2}$ of the heat sink. The sense of the electrical current is chosen so that the Peltier heat is absorbed at the upper junction and released to the heat sink at the lower junction.\n\n[figure1]\n\nFigure 5. Thermoelectric refrigerator. (1) Isolated heat source (temperature $T_{1}$ ); (2) Heat sink (temperature $T_{2}$ )\n\n\nFind the expression for the maximum temperature difference $\\Delta T_{\\max }=T_{2}- $T_{1 \\text { min }}$ in term of the figure of merit $Z$ of the thermocouple and the lowest temperature of the isolated heat source $T_{1 \\text { min }}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe cooling power and the maximum temperature difference\n\nThe upper end of the thermocouple is a heat source with the initial temperature $T_{1}$. It is thermally isolated with ambient environment, and needs to be cooled. The lower ends of the thermocouple, A and $\\mathrm{B}$ bars are connected to a battery and are at the temperature $T_{2}$ of the heat sink. The sense of the electrical current is chosen so that the Peltier heat is absorbed at the upper junction and released to the heat sink at the lower junction.\n\n[figure1]\n\nFigure 5. Thermoelectric refrigerator. (1) Isolated heat source (temperature $T_{1}$ ); (2) Heat sink (temperature $T_{2}$ )\n\n\nFind the expression for the maximum temperature difference $\\Delta T_{\\max }=T_{2}- $T_{1 \\text { min }}$ in term of the figure of merit $Z$ of the thermocouple and the lowest temperature of the isolated heat source $T_{1 \\text { min }}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_6e4e3efc8af7423242a0g-6.jpg?height=797&width=737&top_left_y=1509&top_left_x=662" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_100", "problem": "Water drips from the nozzle of a shower onto the floor $2.45 \\mathrm{~m}$ below. The drops fall at regular intervals of time, the first drop striking the floor at the instant the third drop begins to fall. Measured from the shower head, what is the location of the second drop when the first drop strikes the floor.\nA: $\\quad 0.313 \\mathrm{~m}$\nB: $0.613 \\mathrm{~m}$\nC: $0.938 \\mathrm{~m}$\nD: $1.25 \\mathrm{~m}$\nE: $1.563 \\mathrm{~m}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWater drips from the nozzle of a shower onto the floor $2.45 \\mathrm{~m}$ below. The drops fall at regular intervals of time, the first drop striking the floor at the instant the third drop begins to fall. Measured from the shower head, what is the location of the second drop when the first drop strikes the floor.\n\nA: $\\quad 0.313 \\mathrm{~m}$\nB: $0.613 \\mathrm{~m}$\nC: $0.938 \\mathrm{~m}$\nD: $1.25 \\mathrm{~m}$\nE: $1.563 \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_744", "problem": "Collision of cosmic-rays entering the upper atmosphere of the Earth produce large numbers of particles called muons. These muons move vertically through the atmosphere at very high speed (comparable to the speed of light, c), forming a beam of flux I. Because muons decay into other particles, the flux changes with time according to the exponential decay law $I=I_{0} e^{-\\frac{t}{\\tau}}$, where $\\tau=2.2$ $\\mu \\mathrm{s}$ is the avarage lifetime in the rest frame of the muon, $I_{0}$ is the muon flux at time $t=0$, and $t$ is measured in the rest frame of the muon. If we place a detector at a\nheight on $10 \\mathrm{~km}$ above the ground, we detect a muon flux of $I_{0}=1000 \\mathrm{part} /\\left(\\mathrm{s} \\cdot \\mathrm{m}^{2}\\right)$. If we place the same detector on the ground, we detect a muon flux of $I_{g}=49 \\mathrm{part} /$ (s. $\\mathrm{m}^{2}$ ). Assuming that the muons in the beam are all moving at the same relativistic speed, what is that speed?\nA: $v=0.81 c$\nB: $v=0.85 c$\nC: $v=0.90 c$\nD: $v=0.98 c$\nE: $v=c$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nCollision of cosmic-rays entering the upper atmosphere of the Earth produce large numbers of particles called muons. These muons move vertically through the atmosphere at very high speed (comparable to the speed of light, c), forming a beam of flux I. Because muons decay into other particles, the flux changes with time according to the exponential decay law $I=I_{0} e^{-\\frac{t}{\\tau}}$, where $\\tau=2.2$ $\\mu \\mathrm{s}$ is the avarage lifetime in the rest frame of the muon, $I_{0}$ is the muon flux at time $t=0$, and $t$ is measured in the rest frame of the muon. If we place a detector at a\nheight on $10 \\mathrm{~km}$ above the ground, we detect a muon flux of $I_{0}=1000 \\mathrm{part} /\\left(\\mathrm{s} \\cdot \\mathrm{m}^{2}\\right)$. If we place the same detector on the ground, we detect a muon flux of $I_{g}=49 \\mathrm{part} /$ (s. $\\mathrm{m}^{2}$ ). Assuming that the muons in the beam are all moving at the same relativistic speed, what is that speed?\n\nA: $v=0.81 c$\nB: $v=0.85 c$\nC: $v=0.90 c$\nD: $v=0.98 c$\nE: $v=c$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_426", "problem": "What is meant by the prefix nano ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nWhat is meant by the prefix nano ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_115", "problem": "A small bead slides from rest along a wire that is shaped like a vertical uniform helix (spring). Which graph below shows the magnitude of the acceleration $a$ as a function of time?\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA small bead slides from rest along a wire that is shaped like a vertical uniform helix (spring). Which graph below shows the magnitude of the acceleration $a$ as a function of time?\n\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-06.jpg?height=442&width=458&top_left_y=337&top_left_x=281", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-06.jpg?height=447&width=450&top_left_y=828&top_left_x=276", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-06.jpg?height=442&width=460&top_left_y=340&top_left_x=819", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-06.jpg?height=453&width=460&top_left_y=825&top_left_x=819", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-06.jpg?height=447&width=463&top_left_y=335&top_left_x=1357", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-07.jpg?height=390&width=526&top_left_y=366&top_left_x=686" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_646", "problem": "When two ends of a circular uniform wire are joined to the terminals of a battery, what is the strength of the magnetic field at the center of the circle?\nA: Zero\nB: Infinite\nC: Depends on the amount of e.m.f. applied d) Depends on the radius of the circle\nD: Both $\\mathrm{c}$ and $\\mathrm{d}$.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhen two ends of a circular uniform wire are joined to the terminals of a battery, what is the strength of the magnetic field at the center of the circle?\n\nA: Zero\nB: Infinite\nC: Depends on the amount of e.m.f. applied d) Depends on the radius of the circle\nD: Both $\\mathrm{c}$ and $\\mathrm{d}$.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1290", "problem": "火星大气可视为仅由很稀薄的 $\\mathrm{CO}_{2}$ 组成, 此大气的摩尔质量记为 $\\mu$, 且同一高度的大气可视为处于平衡态的理想气体。火星质量为 $M_{\\mathrm{m}}$ （远大于火星大气总质量）,半径为 $R_{\\mathrm{m}}$ 。假设火星大气的质量密度与距离火星表面的高度 $h$ 的关系为\n\n$$\n\\rho(h)=\\rho_{0}\\left(1+\\frac{h}{R_{\\mathrm{m}}}\\right)^{1-n}\n$$\n\n其中 $\\rho_{0}$ 为常量, $n(n>4)$ 为常数。求火星大气温度 $T(h)$ 随高度 $h$ 变化的表达式。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n火星大气可视为仅由很稀薄的 $\\mathrm{CO}_{2}$ 组成, 此大气的摩尔质量记为 $\\mu$, 且同一高度的大气可视为处于平衡态的理想气体。火星质量为 $M_{\\mathrm{m}}$ （远大于火星大气总质量）,半径为 $R_{\\mathrm{m}}$ 。假设火星大气的质量密度与距离火星表面的高度 $h$ 的关系为\n\n$$\n\\rho(h)=\\rho_{0}\\left(1+\\frac{h}{R_{\\mathrm{m}}}\\right)^{1-n}\n$$\n\n其中 $\\rho_{0}$ 为常量, $n(n>4)$ 为常数。\n\n问题:\n求火星大气温度 $T(h)$ 随高度 $h$ 变化的表达式。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_304", "problem": "A light ray from a ray box can be used to demonstrate refraction.\n\nWhen the light ray passes from the air into the glass the light ray is refracted towards the normal.\n\nWhich of the statements is not a valid explanation for refraction:\nA: The frequency of the light changes as it enters the glass\nB: The speed of the light changes as it enters the glass\nC: The wavelength of the light changes as it enters the glass\nD: The direction of the light changes as it enters the glass\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA light ray from a ray box can be used to demonstrate refraction.\n\nWhen the light ray passes from the air into the glass the light ray is refracted towards the normal.\n\nWhich of the statements is not a valid explanation for refraction:\n\nA: The frequency of the light changes as it enters the glass\nB: The speed of the light changes as it enters the glass\nC: The wavelength of the light changes as it enters the glass\nD: The direction of the light changes as it enters the glass\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_571", "problem": "A particle of mass $m$ moves under a force similar to that of an ideal spring, except that the force repels the particle from the origin:\n\n$$\nF=+m \\alpha^{2} x\n$$\n\nIn simple harmonic motion, the position of the particle as a function of time can be written\n\n$$\nx(t)=A \\cos \\omega t+B \\sin \\omega t\n$$\n\nLikewise, in the present case we have\n\n$$\nx(t)=A f_{1}(t)+B f_{2}(t)\n$$\n\nfor some appropriate functions $f_{1}$ and $f_{2}$.\n\n$f_{1}(t)$ and $f_{2}(t)$ can be chosen to have the form $e^{r t}$. What are the two appropriate values of $r$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA particle of mass $m$ moves under a force similar to that of an ideal spring, except that the force repels the particle from the origin:\n\n$$\nF=+m \\alpha^{2} x\n$$\n\nIn simple harmonic motion, the position of the particle as a function of time can be written\n\n$$\nx(t)=A \\cos \\omega t+B \\sin \\omega t\n$$\n\nLikewise, in the present case we have\n\n$$\nx(t)=A f_{1}(t)+B f_{2}(t)\n$$\n\nfor some appropriate functions $f_{1}$ and $f_{2}$.\n\n$f_{1}(t)$ and $f_{2}(t)$ can be chosen to have the form $e^{r t}$. What are the two appropriate values of $r$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_764", "problem": "A roller-coaster cart full of water is moving at a constant speed along a horizontal, frictionless length of track. Suddenly, a plug in the bottom of the cart is removed, and the water starts to flow downwards out of the cart. What happens to the speed of the cart while the water is flowing? Ignore air resistance in your answer.\nA: The cart speeds up.\nB: The cart slows down.\nC: The cart speeds up until half the water is gone, then it slows down.\nD: The cart slows down until half the water is gone, then it speeds up.\nE: The cart's speed does not change.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA roller-coaster cart full of water is moving at a constant speed along a horizontal, frictionless length of track. Suddenly, a plug in the bottom of the cart is removed, and the water starts to flow downwards out of the cart. What happens to the speed of the cart while the water is flowing? Ignore air resistance in your answer.\n\nA: The cart speeds up.\nB: The cart slows down.\nC: The cart speeds up until half the water is gone, then it slows down.\nD: The cart slows down until half the water is gone, then it speeds up.\nE: The cart's speed does not change.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_11", "problem": "If the sun collapsed suddenly to become a black hole, the Earth would\nA: continue in its present orbit.\nB: fly off tangential to the current orbital path.\nC: likely be pulled into the black hole.\nD: be pulled apart by tidal forces.\nE: Both C \\& D.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIf the sun collapsed suddenly to become a black hole, the Earth would\n\nA: continue in its present orbit.\nB: fly off tangential to the current orbital path.\nC: likely be pulled into the black hole.\nD: be pulled apart by tidal forces.\nE: Both C \\& D.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_339", "problem": "Equipment: a sheet of Fresnel prism, a sheet with purple and magenta stripes (see separate sheet), a piece of cardboard paper (can be used as a screen), ruler, measuring tape, stand, green laser $\\left(\\lambda_{0}=532 \\mathrm{~nm}\\right)$. NB! Avoid looking into direct or reflected laser light, it my damage your eyes! The intensity maximum for the scattered light from magenta stripes is $\\lambda_{m}=630 \\mathrm{~nm}$, and from cyan stripes $-\\lambda_{c}=$ $495 \\mathrm{~nm}$.\n\nFresnel prism is a transparent sheet with a periodic array of stripes; cross-section of such a sheet is shown in figure. The refraction index of the material from which the sheet is made $n=1.47$.\n\n[figure1]\n\nAssuming that in the range of visible light, the refraction index $n=n(\\lambda)$ of the Fresnel prism material is a linear function of wavelength $\\lambda$, determine the chromatic dispersion $\\frac{\\mathrm{d} n}{\\mathrm{~d} \\lambda}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nEquipment: a sheet of Fresnel prism, a sheet with purple and magenta stripes (see separate sheet), a piece of cardboard paper (can be used as a screen), ruler, measuring tape, stand, green laser $\\left(\\lambda_{0}=532 \\mathrm{~nm}\\right)$. NB! Avoid looking into direct or reflected laser light, it my damage your eyes! The intensity maximum for the scattered light from magenta stripes is $\\lambda_{m}=630 \\mathrm{~nm}$, and from cyan stripes $-\\lambda_{c}=$ $495 \\mathrm{~nm}$.\n\nFresnel prism is a transparent sheet with a periodic array of stripes; cross-section of such a sheet is shown in figure. The refraction index of the material from which the sheet is made $n=1.47$.\n\n[figure1]\n\nAssuming that in the range of visible light, the refraction index $n=n(\\lambda)$ of the Fresnel prism material is a linear function of wavelength $\\lambda$, determine the chromatic dispersion $\\frac{\\mathrm{d} n}{\\mathrm{~d} \\lambda}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_57fd2cb729ee4cd65883g-1.jpg?height=108&width=657&top_left_y=282&top_left_x=2174" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_605", "problem": "For this problem, assume the existence of a hypothetical particle known as a magnetic monopole. Such a particle would have a \"magnetic charge\" $q_{m}$, and in analogy to an electrically charged particle would produce a radially directed magnetic field of magnitude\n\n$$\nB=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{r^{2}}\n$$\n\nand be subject to a force (in the absence of electric fields)\n\n$$\nF=q_{m} B\n$$\n\nA magnetic monopole of mass $m$ and magnetic charge $q_{m}$ is constrained to move on a vertical, nonmagnetic, insulating, frictionless U-shaped track. At the bottom of the track is a wire loop whose radius $b$ is much smaller than the width of the \"U\" of the track. The section of track near the loop can thus be approximated as a long straight line. The wire that makes up the loop has radius $a \\ll b$ and resistivity $\\rho$. The monopole is released from rest a height $H$ above the bottom of the track.\n\nIgnore the self-inductance of the loop, and assume that the monopole passes through the loop many times before coming to a rest.\n\nSuppose the monopole is a distance $x$ from the center of the loop. What is the magnetic flux $\\Phi_{B}$ through the loop?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nFor this problem, assume the existence of a hypothetical particle known as a magnetic monopole. Such a particle would have a \"magnetic charge\" $q_{m}$, and in analogy to an electrically charged particle would produce a radially directed magnetic field of magnitude\n\n$$\nB=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{r^{2}}\n$$\n\nand be subject to a force (in the absence of electric fields)\n\n$$\nF=q_{m} B\n$$\n\nA magnetic monopole of mass $m$ and magnetic charge $q_{m}$ is constrained to move on a vertical, nonmagnetic, insulating, frictionless U-shaped track. At the bottom of the track is a wire loop whose radius $b$ is much smaller than the width of the \"U\" of the track. The section of track near the loop can thus be approximated as a long straight line. The wire that makes up the loop has radius $a \\ll b$ and resistivity $\\rho$. The monopole is released from rest a height $H$ above the bottom of the track.\n\nIgnore the self-inductance of the loop, and assume that the monopole passes through the loop many times before coming to a rest.\n\nSuppose the monopole is a distance $x$ from the center of the loop. What is the magnetic flux $\\Phi_{B}$ through the loop?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_455", "problem": "The figure shows a more complex system, known as a block and tackle, consisting of two light pulley blocks and a light cord.\n\n[figure1]\nFigure: Two light pulley blocks and a light cord.\n\nWhat length of cord must be extracted at the free end, $\\mathbf{X}$, to raise the mass through a distance $h$?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe figure shows a more complex system, known as a block and tackle, consisting of two light pulley blocks and a light cord.\n\n[figure1]\nFigure: Two light pulley blocks and a light cord.\n\nWhat length of cord must be extracted at the free end, $\\mathbf{X}$, to raise the mass through a distance $h$?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-4.jpg?height=1074&width=514&top_left_y=1593&top_left_x=1356" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_619", "problem": "A particle of mass $m$ moves under a force similar to that of an ideal spring, except that the force repels the particle from the origin:\n\n$$\nF=+m \\alpha^{2} x\n$$\n\nIn simple harmonic motion, the position of the particle as a function of time can be written\n\n$$\nx(t)=A \\cos \\omega t+B \\sin \\omega t\n$$\n\nLikewise, in the present case we have\n\n$$\nx(t)=A f_{1}(t)+B f_{2}(t)\n$$\n\nfor some appropriate functions $f_{1}$ and $f_{2}$.\n\nSuppose that the particle begins at position $x(0)=x_{0}$ and with velocity $v(0)=0$. What is $x(t)$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA particle of mass $m$ moves under a force similar to that of an ideal spring, except that the force repels the particle from the origin:\n\n$$\nF=+m \\alpha^{2} x\n$$\n\nIn simple harmonic motion, the position of the particle as a function of time can be written\n\n$$\nx(t)=A \\cos \\omega t+B \\sin \\omega t\n$$\n\nLikewise, in the present case we have\n\n$$\nx(t)=A f_{1}(t)+B f_{2}(t)\n$$\n\nfor some appropriate functions $f_{1}$ and $f_{2}$.\n\nSuppose that the particle begins at position $x(0)=x_{0}$ and with velocity $v(0)=0$. What is $x(t)$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1360", "problem": "某机场候机楼外景如图 a 所示。该候机楼结构简化图如图 b 所示: 候机楼侧壁是倾斜的,用钢索将两边斜壁系住, 在钢索上有许多坚直短钢棒将屋面支撑在钢索上。假设每边斜壁的质量为 $m$, 质量分布均匀; 钢索与屋面 (包括短钢棒) 的总质量为 $\\frac{m}{2}$, 在地面处用铰链与水平地面连接, 钢索固定于斜壁上端以支撑整个屋面, 钢索上端与斜壁的夹角为 $30^{\\circ}$; 整个系统左右对称。求\n\n[图1]\n\n[图2]\n\n图a\n\n图 b斜壁与地面的夹角", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某机场候机楼外景如图 a 所示。该候机楼结构简化图如图 b 所示: 候机楼侧壁是倾斜的,用钢索将两边斜壁系住, 在钢索上有许多坚直短钢棒将屋面支撑在钢索上。假设每边斜壁的质量为 $m$, 质量分布均匀; 钢索与屋面 (包括短钢棒) 的总质量为 $\\frac{m}{2}$, 在地面处用铰链与水平地面连接, 钢索固定于斜壁上端以支撑整个屋面, 钢索上端与斜壁的夹角为 $30^{\\circ}$; 整个系统左右对称。求\n\n[图1]\n\n[图2]\n\n图a\n\n图 b\n\n问题:\n斜壁与地面的夹角\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以\\circ为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-04.jpg?height=403&width=944&top_left_y=2508&top_left_x=245", "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-05.jpg?height=277&width=686&top_left_y=267&top_left_x=408", "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-07.jpg?height=360&width=720&top_left_y=2416&top_left_x=1202" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\circ" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_261", "problem": "The specific heat capacity of copper is $385 \\mathrm{~J} /\\left(\\mathrm{kg}{ }^{\\circ} \\mathrm{C}\\right)$ which means that 385 Joules of thermal energy are needed to raise the temperature of $1 \\mathrm{~kg}$ of copper by $1^{\\circ} \\mathrm{C}$. The melting point of copper is $1085^{\\circ} \\mathrm{C}$.\n\nHow much thermal energy is needed to raise 50.0 grams of copper wire to its melting point when it is initially at room temperature of $20.0^{\\circ} \\mathrm{C}$ ?\nA: $\\quad 20.9 \\mathrm{MJ}$\nB: $\\quad 20.5 \\mathrm{MJ}$\nC: $\\quad 20.9 \\mathrm{~kJ}$\nD: $\\quad 20.5 \\mathrm{~kJ}$\nE: $\\quad 385 \\mathrm{~J}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nThe specific heat capacity of copper is $385 \\mathrm{~J} /\\left(\\mathrm{kg}{ }^{\\circ} \\mathrm{C}\\right)$ which means that 385 Joules of thermal energy are needed to raise the temperature of $1 \\mathrm{~kg}$ of copper by $1^{\\circ} \\mathrm{C}$. The melting point of copper is $1085^{\\circ} \\mathrm{C}$.\n\nHow much thermal energy is needed to raise 50.0 grams of copper wire to its melting point when it is initially at room temperature of $20.0^{\\circ} \\mathrm{C}$ ?\n\nA: $\\quad 20.9 \\mathrm{MJ}$\nB: $\\quad 20.5 \\mathrm{MJ}$\nC: $\\quad 20.9 \\mathrm{~kJ}$\nD: $\\quad 20.5 \\mathrm{~kJ}$\nE: $\\quad 385 \\mathrm{~J}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_516", "problem": "A cube of side length $a$ consists of 6 such pyramids. Then we simply compute $0.1894 / 6$ for\n\n$$\n\\Phi_{p}(a, \\rho) \\approx \\frac{0.0316 \\rho a^{2}}{\\epsilon_{0}}\n$$\n\nLet the potential due to such a square be $\\Phi_{s}(a, \\sigma)$. Note that adding a square plate of infinitesimal thickness $d z$ and side length $a$ to a square pyramid with base side length $a$ and height $a / 2$ yields a square pyramid with base side length $a+2 d z$ and height $a / 2+d z$.\n\nThe surface charge density of a square plate with thickness $d z$ and volume charge density $\\rho$ is $\\sigma=\\rho d z$. Then by the principle of superposition,\n\n$$\n\\Phi_{s}(a, \\rho d z)=\\Phi_{p}(a+2 d z, \\rho)-\\Phi_{p}(a, \\rho) \\approx \\frac{0.0316 \\rho\\left((a+2 d z)^{2}-a^{2}\\right)}{\\epsilon_{0}}=\\frac{0.126 a \\rho d z}{\\epsilon_{0}}\n$$\n\nso we have\n\n$$\n\\Phi_{s}(a, \\sigma) \\approx \\frac{0.126 a \\sigma}{\\epsilon_{0}}\n$$The electric potential at the center of a cube with uniform charge density $\\rho$ and side length $a$ is\n\n$$\n\\Phi \\approx \\frac{0.1894 \\rho a^{2}}{\\epsilon_{0}}\n$$\n\nYou do not need to derive this. ${ }^{1}$\n\nFor the entirety of this problem, any computed numerical constants should be written to three significant figures.\n\nLet $E(z)$ be the electric field at a height $z$ above the center of a square with charge density $\\sigma$ and side length $a$. If the electric potential at the center of the square is approximately $\\frac{0.281 a \\sigma}{\\epsilon_{0}}$, estimate $E(a / 2)$ by assuming that $E(z)$ is linear in $z$ for $00$, the right-hand rule indicates there is a magnetic field in the $-\\tilde{\\mathbf{y}}$ direction for $z>0$ and in the $+\\tilde{\\mathbf{y}}$ direction for $z<0$. From Ampere's law applied to a loop of length $l$ normal to the $\\tilde{\\mathbf{x}}$ direction,\n\n$$\n2 B l=\\mu_{0} \\sigma v l .\n$$\n\nTo get the right-hand side, note that in time $t$, an area $v t l$ moves through the loop, so a charge $\\sigma v t l$ moves through. Then the current through the loop is $\\sigma v l$.\n\nApplying symmetry, we have\n\n$$\n\\mathbf{B}=\\frac{\\mu_{0} \\sigma v}{2} \\times \\begin{cases}-\\hat{\\mathbf{y}} & \\text { above the sheet } \\\\ \\hat{\\mathbf{y}} & \\text { below the sheet. }\\end{cases}\n$$\n\niv. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ (perpendicular to the sheet)\n\nAgain the motion does not affect the electric field, so the answer is the same as that of part (i).\n\nv. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$\n\nApplying Ampere's law and symmetry, there is no magnetic field above and below the sheet.\n\nInterestingly, there's no magnetic field at the sheet either. Consider an Amperian loop of area $A$ in the $x y$ plane as the sheet passes through. The loop experiences a current of the form\n\n$$\nA \\sigma \\delta(t)\n$$\n\nBut the loop also experiences an oppositely directed change in flux of the form\n\n$$\nA \\frac{\\sigma}{\\epsilon_{0}} \\delta(t)\n$$\n\nso the right-hand side of Ampere's law, including the displacement current term, remains zero.In parts this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nIn a certain region there exists only an electric field $\\tilde{\\mathbf{E}}=E_{x} \\hat{\\mathbf{x}}+E_{y} \\hat{\\mathbf{y}}+E_{z} \\hat{\\mathbf{z}}$ (and no magnetic field) as measured by an observer at rest. The electric and magnetic fields $\\tilde{\\mathbf{E}}^{\\prime}$ and $\\tilde{\\mathbf{B}}^{\\prime}$ as measured by observers in motion can be determined entirely from the local value of $\\tilde{\\mathbf{E}}$, regardless of the charge configuration that may have produced it.\n\nWhat would be the observed magnetic field $\\tilde{\\mathbf{B}^{\\prime}}$ as measured by an observer moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nPart A\n\nAn infinite uniform sheet has a surface charge density $\\sigma$ and has an infinitesimal thickness. The sheet lies in the $x y$ plane.\n\ni. Assuming the sheet is at rest.\n\nBy symmetry, the fields above and below the sheet are equal in magnitude and directed away from the sheet. By Gauss's Law, using a cylinder of base area $A$,\n\n$$\n2 E A=\\frac{\\sigma A}{\\epsilon_{0}} \\Rightarrow E=\\frac{\\sigma}{2 \\epsilon_{0}}\n$$\n\npointing directly away from the sheet in the $z$ direction, or\n\n$$\n\\mathbf{E}=\\frac{\\sigma}{2 \\epsilon} \\times \\begin{cases}\\hat{\\mathbf{z}} & \\text { above the sheet } \\\\ -\\hat{\\mathbf{z}} & \\text { below the sheet. }\\end{cases}\n$$\n\nii. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{x}}$ (parallel to the sheet)\n\nThe motion does not affect the electric field, so the answer is the same as that of part (i).\n\niii. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{x}}$\n\nAssuming $v>0$, the right-hand rule indicates there is a magnetic field in the $-\\tilde{\\mathbf{y}}$ direction for $z>0$ and in the $+\\tilde{\\mathbf{y}}$ direction for $z<0$. From Ampere's law applied to a loop of length $l$ normal to the $\\tilde{\\mathbf{x}}$ direction,\n\n$$\n2 B l=\\mu_{0} \\sigma v l .\n$$\n\nTo get the right-hand side, note that in time $t$, an area $v t l$ moves through the loop, so a charge $\\sigma v t l$ moves through. Then the current through the loop is $\\sigma v l$.\n\nApplying symmetry, we have\n\n$$\n\\mathbf{B}=\\frac{\\mu_{0} \\sigma v}{2} \\times \\begin{cases}-\\hat{\\mathbf{y}} & \\text { above the sheet } \\\\ \\hat{\\mathbf{y}} & \\text { below the sheet. }\\end{cases}\n$$\n\niv. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ (perpendicular to the sheet)\n\nAgain the motion does not affect the electric field, so the answer is the same as that of part (i).\n\nv. Assuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$\n\nApplying Ampere's law and symmetry, there is no magnetic field above and below the sheet.\n\nInterestingly, there's no magnetic field at the sheet either. Consider an Amperian loop of area $A$ in the $x y$ plane as the sheet passes through. The loop experiences a current of the form\n\n$$\nA \\sigma \\delta(t)\n$$\n\nBut the loop also experiences an oppositely directed change in flux of the form\n\n$$\nA \\frac{\\sigma}{\\epsilon_{0}} \\delta(t)\n$$\n\nso the right-hand side of Ampere's law, including the displacement current term, remains zero.\n\nproblem:\nIn parts this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nIn a certain region there exists only an electric field $\\tilde{\\mathbf{E}}=E_{x} \\hat{\\mathbf{x}}+E_{y} \\hat{\\mathbf{y}}+E_{z} \\hat{\\mathbf{z}}$ (and no magnetic field) as measured by an observer at rest. The electric and magnetic fields $\\tilde{\\mathbf{E}}^{\\prime}$ and $\\tilde{\\mathbf{B}}^{\\prime}$ as measured by observers in motion can be determined entirely from the local value of $\\tilde{\\mathbf{E}}$, regardless of the charge configuration that may have produced it.\n\nWhat would be the observed magnetic field $\\tilde{\\mathbf{B}^{\\prime}}$ as measured by an observer moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_332", "problem": "A wire loop is hovering in outer space (weightless vacuum) with its plane parallel to the $x y$-plane. In $x<0$ there is a homogeneous magnetic field parallel to the $z$-axis. The rigid rectangular loop is $l=10 \\mathrm{~cm}$ wide and $h=30 \\mathrm{~cm}$ long. The loop is made of copper wire with a circular cross section (radius $r=1.0 \\mathrm{~mm}$ ). At $t=0 \\mathrm{~s}$ the external magnetic field starts to decrease at a rate of $0.025 \\mathrm{~T} / \\mathrm{s}$.\n\nFind the acceleration of the loop ight after $t=0 \\mathrm{~s}$. The magnetic flux density is initially $B=2.0 \\mathrm{~T}$ and the loop is immersed $d=12 \\mathrm{~cm}$ into the external field with its shorter side parallel to the $y$-axis.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA wire loop is hovering in outer space (weightless vacuum) with its plane parallel to the $x y$-plane. In $x<0$ there is a homogeneous magnetic field parallel to the $z$-axis. The rigid rectangular loop is $l=10 \\mathrm{~cm}$ wide and $h=30 \\mathrm{~cm}$ long. The loop is made of copper wire with a circular cross section (radius $r=1.0 \\mathrm{~mm}$ ). At $t=0 \\mathrm{~s}$ the external magnetic field starts to decrease at a rate of $0.025 \\mathrm{~T} / \\mathrm{s}$.\n\nFind the acceleration of the loop ight after $t=0 \\mathrm{~s}$. The magnetic flux density is initially $B=2.0 \\mathrm{~T}$ and the loop is immersed $d=12 \\mathrm{~cm}$ into the external field with its shorter side parallel to the $y$-axis.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $m^2/s$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$m^2/s$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1366", "problem": "激光瞄准系统的设计需考虑空气折射率的变化。由于受到地表状况、海拔高度、气温、湿度和空气密度等多种因素的影响, 空气的折射率在大气层中的分布是不均匀的,因而激光的传播路径并不是直线。为简化起见, 假设某地的空气折射率随高度 $y$ 的变化如下式所示\n\n$$\nn^{2}=n_{0}^{2}+\\alpha^{2} y,\n$$\n\n式中 $n_{0}$ 是 $y=0$ 处 (地面) 空气的折射率, $n_{0}$\n\n[图1]\n和 $\\alpha$ 均为大于零的已知常量。激光本身的传播时间可忽略。激光发射器位于坐标原点 O，如图。假定目标 A 位于第 I 象限。当目标 A 的高度为 $y_{a}$ 时, 求激光发射器可照射到的目标 A 的最大 $x$-坐标值 $x_{a \\text { max }}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n激光瞄准系统的设计需考虑空气折射率的变化。由于受到地表状况、海拔高度、气温、湿度和空气密度等多种因素的影响, 空气的折射率在大气层中的分布是不均匀的,因而激光的传播路径并不是直线。为简化起见, 假设某地的空气折射率随高度 $y$ 的变化如下式所示\n\n$$\nn^{2}=n_{0}^{2}+\\alpha^{2} y,\n$$\n\n式中 $n_{0}$ 是 $y=0$ 处 (地面) 空气的折射率, $n_{0}$\n\n[图1]\n和 $\\alpha$ 均为大于零的已知常量。激光本身的传播时间可忽略。激光发射器位于坐标原点 O，如图。\n\n问题:\n假定目标 A 位于第 I 象限。当目标 A 的高度为 $y_{a}$ 时, 求激光发射器可照射到的目标 A 的最大 $x$-坐标值 $x_{a \\text { max }}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_07aa406e17d01fd01b36g-05.jpg?height=634&width=737&top_left_y=1105&top_left_x=1019" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_762", "problem": "A cockatoo and a sparrow are flying towards each other. Both birds have the same kinetic energy, but the cockatoo has four times the mass of the sparrow. Which one of the following statements is true? Select one:\nA: The magnitude of the cockatoo's momentum is four times larger than the sparrow's\nB: The magnitude of the cockatoo's momentum is twice as large as the sparrow's\nC: The magnitude of the cockatoo's momentum is the same as the sparrow's\nD: The magnitude of the cockatoo's momentum is half that of the sparrow's\nE: The magnitude of the cockatoo's momentum is one quarter that of the sparrow's\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA cockatoo and a sparrow are flying towards each other. Both birds have the same kinetic energy, but the cockatoo has four times the mass of the sparrow. Which one of the following statements is true? Select one:\n\nA: The magnitude of the cockatoo's momentum is four times larger than the sparrow's\nB: The magnitude of the cockatoo's momentum is twice as large as the sparrow's\nC: The magnitude of the cockatoo's momentum is the same as the sparrow's\nD: The magnitude of the cockatoo's momentum is half that of the sparrow's\nE: The magnitude of the cockatoo's momentum is one quarter that of the sparrow's\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1597", "problem": "如图, 一焦距为 $20 \\mathrm{~cm}$ 的薄透镜位于 $x=0$ 平面上, 光心位于坐标原点 $\\mathrm{O}$,光轴与 $x$ 轴重合。在 $z=0$ 平面内的一束平行光入射到该透镜上, 入射方向与光轴的夹角为 $30^{\\circ}$ 。该光束通过透镜后汇聚点的位置坐标为", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图, 一焦距为 $20 \\mathrm{~cm}$ 的薄透镜位于 $x=0$ 平面上, 光心位于坐标原点 $\\mathrm{O}$,光轴与 $x$ 轴重合。在 $z=0$ 平面内的一束平行光入射到该透镜上, 入射方向与光轴的夹角为 $30^{\\circ}$ 。该光束通过透镜后汇聚点的位置坐标为\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_910", "problem": "If the medium is isotropic, we have $\\vec{P}=\\chi \\epsilon_{0} \\vec{E}$ and $\\vec{D}=\\epsilon \\vec{E}$, with $\\chi$ and $\\epsilon=\\epsilon_{0}(1+\\chi)$ being the electric susceptibility and permittivity, respectively, of the medium. For a light wave of angular frequency $\\omega$ in such a medium, a given phase will propagate in the direction $\\vec{k}$ with a velocity (called phase velocity) $v_{p}=c / n$. Here $c$ is the speed of light in vacuum and $n$ is the refractive index of the medium. One can also use rays to represent a train of light waves. The propagation of a light ray is characterized by the direction and speed $v_{r}$ of the electromagnetic energy flow.\n\nConsider a plane wave of light with angular frequency $\\omega$ and wave vector $\\vec{k}$ in a homogeneous isotropic dielectric medium.\n\nWhat is the refractive index $n$ of the dielectric medium for the wave?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIf the medium is isotropic, we have $\\vec{P}=\\chi \\epsilon_{0} \\vec{E}$ and $\\vec{D}=\\epsilon \\vec{E}$, with $\\chi$ and $\\epsilon=\\epsilon_{0}(1+\\chi)$ being the electric susceptibility and permittivity, respectively, of the medium. For a light wave of angular frequency $\\omega$ in such a medium, a given phase will propagate in the direction $\\vec{k}$ with a velocity (called phase velocity) $v_{p}=c / n$. Here $c$ is the speed of light in vacuum and $n$ is the refractive index of the medium. One can also use rays to represent a train of light waves. The propagation of a light ray is characterized by the direction and speed $v_{r}$ of the electromagnetic energy flow.\n\nConsider a plane wave of light with angular frequency $\\omega$ and wave vector $\\vec{k}$ in a homogeneous isotropic dielectric medium.\n\nWhat is the refractive index $n$ of the dielectric medium for the wave?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_730", "problem": "A conical, metal beam is placed between two constant temperature surfaces. The surfaces are at 20 and 100 degrees, as shown in the picture. Which one of the graphs below best represents the temperature along the center of the beam at equilibrium?\n\n[figure1]\n\na)\n\n[figure2]\n\nc)\n\n[figure3]\n\nb)\n\n[figure4]\n\nd)\n\n[figure5]\nA: A\nB: B\nC: C\nD: D\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA conical, metal beam is placed between two constant temperature surfaces. The surfaces are at 20 and 100 degrees, as shown in the picture. Which one of the graphs below best represents the temperature along the center of the beam at equilibrium?\n\n[figure1]\n\na)\n\n[figure2]\n\nc)\n\n[figure3]\n\nb)\n\n[figure4]\n\nd)\n\n[figure5]\n\nA: A\nB: B\nC: C\nD: D\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-06.jpg?height=368&width=388&top_left_y=2123&top_left_x=366", "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-06.jpg?height=208&width=277&top_left_y=273&top_left_x=1112", "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-06.jpg?height=211&width=279&top_left_y=503&top_left_x=1114", "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-06.jpg?height=206&width=259&top_left_y=274&top_left_x=1481", "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-06.jpg?height=214&width=274&top_left_y=501&top_left_x=1482" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_325", "problem": "Consider a spacecraft shaped as a homogeneous pipe, which is closed at both ends. The spacecraft is rotating around its center of mass with angular velocity $\\omega$, around an axis perpendicular to the pipe, in order to simulate gravity. The spacecraft is filled with air of molar mass $\\mu$, which has pressure $p_{0}$ at the rota- tion axis. The diameter of the spacecraft is much smaller than its length. Added during the competition: the temperature is $T$.\n\n As a comparison, consider a (nonrotating) tower in a constant gravitational field of strength $g$, filled with the same gas. If the ground level pressure is $p_{0}$, what is the pressure $p$ as a function of height $h$ above the ground in such a tower?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nConsider a spacecraft shaped as a homogeneous pipe, which is closed at both ends. The spacecraft is rotating around its center of mass with angular velocity $\\omega$, around an axis perpendicular to the pipe, in order to simulate gravity. The spacecraft is filled with air of molar mass $\\mu$, which has pressure $p_{0}$ at the rota- tion axis. The diameter of the spacecraft is much smaller than its length. Added during the competition: the temperature is $T$.\n\n As a comparison, consider a (nonrotating) tower in a constant gravitational field of strength $g$, filled with the same gas. If the ground level pressure is $p_{0}$, what is the pressure $p$ as a function of height $h$ above the ground in such a tower?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1717", "problem": "一半径为 $R$ 、内侧光滑的半球面固定在地面上, 开口水平且朝上. 一小滑块在半球面内侧最高点处获得沿球面的水平速度, 其大小为 $v_{0}\\left(v_{0} \\neq 0\\right)$. 求滑块在整个运动过程中可能达到的最大速率. 重力加速度大小为 $g$.\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n一半径为 $R$ 、内侧光滑的半球面固定在地面上, 开口水平且朝上. 一小滑块在半球面内侧最高点处获得沿球面的水平速度, 其大小为 $v_{0}\\left(v_{0} \\neq 0\\right)$. 求滑块在整个运动过程中可能达到的最大速率. 重力加速度大小为 $g$.\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_734a14c1a8de5e19f8ecg-01.jpg?height=354&width=488&top_left_y=448&top_left_x=1572", "https://cdn.mathpix.com/cropped/2024_03_31_734a14c1a8de5e19f8ecg-06.jpg?height=323&width=434&top_left_y=307&top_left_x=1322" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_213", "problem": "A water hose is at ground level against the base of a large wall. By aiming the hose at some angle, and squirting water at a speed $v$, one wets a region on the wall, as shown below. If the speed of the water is doubled, what is the new region that can be wetted? Ignore the effect of water splashing beyond the point of contact.\n\nIn the answer choices, the dotted line marks the initial wetted region.\n\n[figure1]\nA: [figure2]\nB: [figure3] four times height, double width\nC: [figure4] double height, four times width\nD: [figure5] double height, double width\nE: [figure6]\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA water hose is at ground level against the base of a large wall. By aiming the hose at some angle, and squirting water at a speed $v$, one wets a region on the wall, as shown below. If the speed of the water is doubled, what is the new region that can be wetted? Ignore the effect of water splashing beyond the point of contact.\n\nIn the answer choices, the dotted line marks the initial wetted region.\n\n[figure1]\n\nA: [figure2]\nB: [figure3] four times height, double width\nC: [figure4] double height, four times width\nD: [figure5] double height, double width\nE: [figure6]\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-10.jpg?height=358&width=423&top_left_y=526&top_left_x=843", "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-10.jpg?height=350&width=447&top_left_y=974&top_left_x=384", "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-10.jpg?height=355&width=407&top_left_y=1468&top_left_x=401", "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-10.jpg?height=312&width=420&top_left_y=2153&top_left_x=392", "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-10.jpg?height=307&width=437&top_left_y=974&top_left_x=1191", "https://i.postimg.cc/wBBFYjBy/image.png" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_64", "problem": "A fisherman watches a dolphin leap out of the water at an angle of $35^{\\circ}$ above the horizontal. The horizontal component of the dolphin's velocity is $7.7 \\mathrm{~m} / \\mathrm{s}$. Find the magnitude of the vertical component of the velocity.\nA: $4.4 \\mathrm{~m} / \\mathrm{s}$\nB: $6.3 \\mathrm{~m} / \\mathrm{s}$\nC: $11 \\mathrm{~m} / \\mathrm{s}$\nD: $5.4 \\mathrm{~m} / \\mathrm{s}$\nE: $3.2 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA fisherman watches a dolphin leap out of the water at an angle of $35^{\\circ}$ above the horizontal. The horizontal component of the dolphin's velocity is $7.7 \\mathrm{~m} / \\mathrm{s}$. Find the magnitude of the vertical component of the velocity.\n\nA: $4.4 \\mathrm{~m} / \\mathrm{s}$\nB: $6.3 \\mathrm{~m} / \\mathrm{s}$\nC: $11 \\mathrm{~m} / \\mathrm{s}$\nD: $5.4 \\mathrm{~m} / \\mathrm{s}$\nE: $3.2 \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1438", "problem": "如图, 一张紧的弦沿 $x$ 轴水平放置, 长度为 $L$ 。弦的左端位于坐标原点。弦可通过其左、右端与振源连接, 使弦产生沿 $y$ 方向的横向受迫振动, 振动传播的速度为 $u$ 。\n\n[图1]将 $P_{1} 、 P_{2}$ 都与振源连接, $P_{1} 、 P_{2}$ 处的振动表达式分别为: $y(x=0, t)=A_{0} \\cos \\omega t$ 、 $y(x=L, t)=A_{0} \\cos \\left(\\omega t+\\varphi_{0}\\right)$, 其中 $\\varphi_{0}$ 为常量。忽略波的振幅在传播方向上的衰减, 分别计算 $\\varphi_{0}=0$ 和 $\\varphi_{0}=\\pi$ 情形下弦上各处振动的表达式以及共振时圆频率 $\\omega$ 应满足的条件。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一张紧的弦沿 $x$ 轴水平放置, 长度为 $L$ 。弦的左端位于坐标原点。弦可通过其左、右端与振源连接, 使弦产生沿 $y$ 方向的横向受迫振动, 振动传播的速度为 $u$ 。\n\n[图1]\n\n问题:\n将 $P_{1} 、 P_{2}$ 都与振源连接, $P_{1} 、 P_{2}$ 处的振动表达式分别为: $y(x=0, t)=A_{0} \\cos \\omega t$ 、 $y(x=L, t)=A_{0} \\cos \\left(\\omega t+\\varphi_{0}\\right)$, 其中 $\\varphi_{0}$ 为常量。忽略波的振幅在传播方向上的衰减, 分别计算 $\\varphi_{0}=0$ 和 $\\varphi_{0}=\\pi$ 情形下弦上各处振动的表达式以及共振时圆频率 $\\omega$ 应满足的条件。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[计算 $\\varphi_{0}=0$情形下弦上各处振动的表达式, 计算 $\\varphi_{0}=0$情形下共振时圆频率 $\\omega$ 应满足的条件, 计算 $\\varphi_{0}=\\pi$情形下弦上各处振动的表达式, 计算 $\\varphi_{0}=\\pi$情形下共振时圆频率 $\\omega$ 应满足的条件。]\n它们的答案类型依次是[方程, 方程, 方程, 方程]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_35bc41298eef336dfdafg-03.jpg?height=257&width=548&top_left_y=286&top_left_x=1231" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null, null ], "answer_sequence": [ "计算 $\\varphi_{0}=0$情形下弦上各处振动的表达式", "计算 $\\varphi_{0}=0$情形下共振时圆频率 $\\omega$ 应满足的条件", "计算 $\\varphi_{0}=\\pi$情形下弦上各处振动的表达式", "计算 $\\varphi_{0}=\\pi$情形下共振时圆频率 $\\omega$ 应满足的条件。" ], "type_sequence": [ "EQ", "EQ", "EQ", "EQ" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1581", "problem": "如图, 在磁感应强度大小为 $B$ 、方向坚直向上的匀强磁场中, 有一均质刚性导电的正方形线框 abcd, 线框质量为 $m$ ，边长为 $l$, 总电阻为 $R$ 。线框可绕通过 ad 边和 bc 边中点的光滑轴 $\\mathrm{OO}^{\\prime}$ 转动。 $\\mathrm{P} 、 \\mathrm{Q}$ 点是线框引线的两端, $\\mathrm{OO}^{\\prime}$ 轴和 X 轴位于同一水平面内, 且相互垂直。不考虑线框自感。\n\n[图1]求线框绕 $\\mathrm{OO}^{\\prime}$ 轴的转动惯量 $J$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 在磁感应强度大小为 $B$ 、方向坚直向上的匀强磁场中, 有一均质刚性导电的正方形线框 abcd, 线框质量为 $m$ ，边长为 $l$, 总电阻为 $R$ 。线框可绕通过 ad 边和 bc 边中点的光滑轴 $\\mathrm{OO}^{\\prime}$ 转动。 $\\mathrm{P} 、 \\mathrm{Q}$ 点是线框引线的两端, $\\mathrm{OO}^{\\prime}$ 轴和 X 轴位于同一水平面内, 且相互垂直。不考虑线框自感。\n\n[图1]\n\n问题:\n求线框绕 $\\mathrm{OO}^{\\prime}$ 轴的转动惯量 $J$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_a47de6806e8da0a0f86dg-02.jpg?height=440&width=674&top_left_y=1322&top_left_x=1222" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_775", "problem": "Which of the following graphs best describes the total mechanical energy of both the birds as a function of time for the birds before and after collision.\nA: ![]([figure1])\nB: ![]([figure2])\nC: ![]([figure3])\nD: ![]([figure4])\nE: ![]([figure5])\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhich of the following graphs best describes the total mechanical energy of both the birds as a function of time for the birds before and after collision.\n\nA: ![]([figure1])\nB: ![]([figure2])\nC: ![]([figure3])\nD: ![]([figure4])\nE: ![]([figure5])\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-04.jpg?height=386&width=494&top_left_y=1623&top_left_x=244", "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-04.jpg?height=383&width=511&top_left_y=1622&top_left_x=790", "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-04.jpg?height=386&width=504&top_left_y=1623&top_left_x=1324", "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-04.jpg?height=388&width=508&top_left_y=2076&top_left_x=543", "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-04.jpg?height=388&width=505&top_left_y=2076&top_left_x=1118" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_97", "problem": "Two cranes are used to lift loads vertically at constant speed from the ground to the top of a building. Crane 1 lifts an $8000-\\mathrm{N}$ load a height of $40 \\mathrm{~m}$ in a time of 20 minutes. Crane 2 lifts a $6400-\\mathrm{N}$ load a height of $50 \\mathrm{~m}$ in a time of 16 minutes. Which statement below correctly compares the mechanical power output of each crane?\nA: The cranes have the same power because they perform the same amount of work.\nB: The cranes have the same power because the ratio of force exerted to time is the same.\nC: Crane 1 has a greater power because it exerts more force to lift the load.\nD: Crane 1 has a greater power because it raises the load over a longer time.\nE: Crane 2 has a greater power because it performs the same amount of work in less time.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo cranes are used to lift loads vertically at constant speed from the ground to the top of a building. Crane 1 lifts an $8000-\\mathrm{N}$ load a height of $40 \\mathrm{~m}$ in a time of 20 minutes. Crane 2 lifts a $6400-\\mathrm{N}$ load a height of $50 \\mathrm{~m}$ in a time of 16 minutes. Which statement below correctly compares the mechanical power output of each crane?\n\nA: The cranes have the same power because they perform the same amount of work.\nB: The cranes have the same power because the ratio of force exerted to time is the same.\nC: Crane 1 has a greater power because it exerts more force to lift the load.\nD: Crane 1 has a greater power because it raises the load over a longer time.\nE: Crane 2 has a greater power because it performs the same amount of work in less time.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_767", "problem": "The pigeon sees its friend, an eagle, in the sky, and flies to meet the eagle who is gliding in the opposite direction. The eagle has a mass of $2 \\mathrm{~kg}$. If the pigeon and eagle have the same kinetic energy, which of the following statements are correct.\n[figure1]\nA: The pigeon is moving at a speed two times that of the eagle.\nB: The pigeon is moving at a speed four times that of the eagle.\nC: The pigeon is moving at a speed half that of the eagle.\nD: The pigeon is moving at a speed one quarter that of the eagle.\nE: The pigeon and eagle are moving at the same speed.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe pigeon sees its friend, an eagle, in the sky, and flies to meet the eagle who is gliding in the opposite direction. The eagle has a mass of $2 \\mathrm{~kg}$. If the pigeon and eagle have the same kinetic energy, which of the following statements are correct.\n[figure1]\n\nA: The pigeon is moving at a speed two times that of the eagle.\nB: The pigeon is moving at a speed four times that of the eagle.\nC: The pigeon is moving at a speed half that of the eagle.\nD: The pigeon is moving at a speed one quarter that of the eagle.\nE: The pigeon and eagle are moving at the same speed.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_57a3ef546cf8883a23edg-03.jpg?height=542&width=1356&top_left_y=1585&top_left_x=356" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_540", "problem": "Part A\n\nA \"Gilbert\" dipole consists of a pair of magnetic monopoles each with a magnitude $q_{m}$ but opposite magnetic charges separated by a distance $d$, where $d$ is small. In this case, assume that $-q_{m}$ is located at $z=0$ and $+q_{m}$ is located at $z=d$.\n\n[figure1]\n\nAssume that magnetic monopoles behave like electric monopoles according to a coulomb-like force\n\nand the magnetic field obeys\n\n$$\nF=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m 1} q_{m 2}}{r^{2}}\n$$\n\n$$\nB=F / q_{m} .\n$$\n\nBy the second expression, $q_{m}$ must be measured in Newtons per Tesla. But since Tesla are also Newtons per Ampere per meter, then $q_{m}$ is also measured in Ampere meters.\n\nAdding the two terms,\n\n$$\nB(z)=-\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{z^{2}}+\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{(z+d)^{2}}\n$$\n\nEvaluate this expression in the limit as $d \\rightarrow 0$, assuming that the product $q_{m} d=p_{m}$ is kept constant, keeping only the lowest non-zero term.\n\nSimplifying our previous expression,\n\n$$\nB(z)=\\frac{\\mu_{0}}{4 \\pi} q_{m} d\\left(\\frac{2+d / z}{z(z+d)^{2}}\\right) \\text {. }\n$$\n\nThus in the limit $d \\rightarrow 0$ we have\n\n$$\nB(z)=\\frac{\\mu_{0}}{2 \\pi} \\frac{q_{m} d}{z^{3}}=\\frac{\\mu_{0}}{2 \\pi} \\frac{p_{m}}{z^{3}}\n$$\n\nPart B\n\nAn \"Ampre\" dipole is a magnetic dipole produced by a current loop $I$ around a circle of radius $r$, where $r$ is small. Assume the that the $z$ axis is the axis of rotational symmetry for the circular loop, and the loop lies in the $x y$ plane at $z=0$.\n\n[figure2]\n\nApplying the Biot-Savart law, with $\\mathbf{s}$ the vector from the point on the loop to the point on the $z$ axis,\n\n$$\nB(z)=\\frac{\\mu_{0} I}{4 \\pi} \\oint \\frac{d \\mathbf{l} \\times \\mathbf{s}}{s^{3}}=\\frac{\\mu_{0} I}{4 \\pi} \\frac{2 \\pi r}{r^{2}+z^{2}} \\sin \\theta\n$$\n\nwhere $\\theta$ is the angle between the point on the loop and the center of the loop as measured by the point on the $z$ axis, so\n\n$$\n\\sin \\theta=\\frac{r}{\\sqrt{r^{2}+z^{2}}}\n$$\n\nThen we have\n\n$$\nB(z)=\\frac{\\mu_{0} I}{4 \\pi} \\frac{2 \\pi r^{2}}{\\left(r^{2}+z^{2}\\right)^{3 / 2}}\n$$\n\nLet $k I r^{\\gamma}$ have dimensions equal to that of the quantity $p_{m}$ defined above in Part A.\n\nEvaluate the expression in Part bi in the limit as $r \\rightarrow 0$, assuming that the product $k I r^{\\gamma}=p_{m}^{\\prime}$ is kept constant, keeping only the lowest non-zero term.\n\n$$\nB(z)=\\frac{\\mu_{0} I}{4 \\pi} \\frac{2 \\pi r^{2}}{\\left(r^{2}+z^{2}\\right)^{3 / 2}} \\approx \\frac{\\mu_{0} I}{2 \\pi} \\frac{\\pi r^{2}}{z^{3}}=\\frac{\\mu_{0}}{2 \\pi} \\frac{\\pi}{k} \\frac{p_{m}^{\\prime}}{z^{3}}\n$$Now we try to compare the two approaches if we model a physical magnet as being composed of densely packed microscopic dipoles.\n\n[figure3]\n\nA cylinder of this uniform magnetic material has a radius $R$ and a length $L$. It is composed of $N$ magnetic dipoles that could be either all Ampre type or all Gilbert type. $N$ is a very large number. The axis of rotation of the cylinder and all of the dipoles are all aligned with the $z$ axis and all point in the same direction as defined above so that the magnetic field outside the cylinder is the same in either dipole case as you previously determined. Below is a picture of the two dipole models; they are cubes of side $d<v_{h}$. What is the maximum height reached by the particle, in terms of $v_{0}, z_{0}, g$ and/or $k$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA particle is constrained to move on the inner surface of a frictionless parabolic bowl whose crosssection has equation $z=k r^{2}$. The particle begins at a height $z_{0}$ above the bottom of the bowl with a horizontal velocity $v_{0}$ along the surface of the bowl. The acceleration due to gravity is $g$.\n[figure1]\n\nSuppose that the initial horizontal velocity is now $v_{0}>v_{h}$. What is the maximum height reached by the particle, in terms of $v_{0}, z_{0}, g$ and/or $k$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_3899026513eb55709c81g-13.jpg?height=392&width=1266&top_left_y=476&top_left_x=428" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_665", "problem": "One mole of an ideal gas $\\left(C_{p}=5 R / 2\\right)$ in a closed cylindrical piston is expanded from $T_{i}=300 \\mathrm{~K}, P_{i}=0.5$ $\\mathrm{MPa}$ to $P_{f}=0.1 \\mathrm{MPa}$ by an adiabatic pathway. The energy supplied to the system $\\Delta Q$, change in internal energy $\\Delta U$, and work done by the gas $\\Delta W$ are:\nA: $\\Delta U=0 \\mathrm{~J}, \\Delta Q=4014 \\mathrm{~J}, \\Delta W=-4014 \\mathrm{~J}$\nB: $\\Delta U=-4270 \\mathrm{~J}, \\Delta Q=0 \\mathrm{~J}, \\Delta W=4270 \\mathrm{~J}$\nC: $\\Delta U=4270 \\mathrm{~J}, \\Delta Q=0 \\mathrm{~J}, \\Delta W=-4270 \\mathrm{~J}$\nD: $\\Delta U=-5487 \\mathrm{~J}, \\Delta Q=0 \\mathrm{~J}, \\Delta W=-5487 \\mathrm{~J}$\nE: $\\Delta U=-5487 \\mathrm{~J}, \\Delta Q=-5487 \\mathrm{~J}, \\Delta W=0 \\mathrm{~J}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOne mole of an ideal gas $\\left(C_{p}=5 R / 2\\right)$ in a closed cylindrical piston is expanded from $T_{i}=300 \\mathrm{~K}, P_{i}=0.5$ $\\mathrm{MPa}$ to $P_{f}=0.1 \\mathrm{MPa}$ by an adiabatic pathway. The energy supplied to the system $\\Delta Q$, change in internal energy $\\Delta U$, and work done by the gas $\\Delta W$ are:\n\nA: $\\Delta U=0 \\mathrm{~J}, \\Delta Q=4014 \\mathrm{~J}, \\Delta W=-4014 \\mathrm{~J}$\nB: $\\Delta U=-4270 \\mathrm{~J}, \\Delta Q=0 \\mathrm{~J}, \\Delta W=4270 \\mathrm{~J}$\nC: $\\Delta U=4270 \\mathrm{~J}, \\Delta Q=0 \\mathrm{~J}, \\Delta W=-4270 \\mathrm{~J}$\nD: $\\Delta U=-5487 \\mathrm{~J}, \\Delta Q=0 \\mathrm{~J}, \\Delta W=-5487 \\mathrm{~J}$\nE: $\\Delta U=-5487 \\mathrm{~J}, \\Delta Q=-5487 \\mathrm{~J}, \\Delta W=0 \\mathrm{~J}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1462", "problem": "双星系统是一类重要的天文观测对象。假设某两星体均可视为质点, 其质量分别为 $M$ 和 $m$,一起围绕它们的质心做圆周运动,\n\n构成一双星系统, 观测到该系统的转动周期为 $T_{0}$ 。在某一时刻, $M$ 星突然发生爆炸而失去质量 $\\Delta M$ 。假设爆炸是瞬时的、相对于 $M$ 星是各向同性的，因而爆炸后 $M$ 星的残余体 $M^{\\prime}$ $\\left(M^{\\prime}=M-\\Delta M\\right)$ 星的瞬间速度与爆炸前瞬间 $M$ 星的速度相同,且爆炸过程和抛射物质 $\\Delta M$ 都对 $m$ 星没有影响。已知引力常量为 $G$ ，不考虑相对论效应。\n\n[图1]若爆炸后 $M^{\\prime}$ 星和 $m$ 星仍然做周期运动, 求该运动的周期 $T_{1}$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n双星系统是一类重要的天文观测对象。假设某两星体均可视为质点, 其质量分别为 $M$ 和 $m$,一起围绕它们的质心做圆周运动,\n\n构成一双星系统, 观测到该系统的转动周期为 $T_{0}$ 。在某一时刻, $M$ 星突然发生爆炸而失去质量 $\\Delta M$ 。假设爆炸是瞬时的、相对于 $M$ 星是各向同性的，因而爆炸后 $M$ 星的残余体 $M^{\\prime}$ $\\left(M^{\\prime}=M-\\Delta M\\right)$ 星的瞬间速度与爆炸前瞬间 $M$ 星的速度相同,且爆炸过程和抛射物质 $\\Delta M$ 都对 $m$ 星没有影响。已知引力常量为 $G$ ，不考虑相对论效应。\n\n[图1]\n\n问题:\n若爆炸后 $M^{\\prime}$ 星和 $m$ 星仍然做周期运动, 求该运动的周期 $T_{1}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_a47de6806e8da0a0f86dg-01.jpg?height=537&width=419&top_left_y=1171&top_left_x=1361" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_460", "problem": "The ship starts out in a circular orbit around the sun very near the Earth and has a goal of moving to a circular orbit around the Sun that is very close to Mars. It will make this transfer in an elliptical orbit as shown in bold in the diagram below. This is accomplished with an initial velocity boost near the Earth $\\Delta v_{1}$ and then a second velocity boost near Mars $\\Delta v_{2}$. Assume that both of these boosts are from instantaneous impulses, and ignore mass changes in the rocket as well as gravitational attraction to either Earth or Mars. Don't ignore the\n\nSun! Assume that the Earth and Mars are both in circular orbits around the Sun of radii $R_{E}$ and $R_{M}=R_{E} / \\alpha$ respectively. The orbital speeds are $v_{E}$ and $v_{M}$ respectively.\n\n[figure1]\n\nDerive an expression for the velocity boost $\\Delta v_{2}$ to change the orbit from elliptical to circular. Express your answer in terms of $v_{E}$ and $\\alpha$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe ship starts out in a circular orbit around the sun very near the Earth and has a goal of moving to a circular orbit around the Sun that is very close to Mars. It will make this transfer in an elliptical orbit as shown in bold in the diagram below. This is accomplished with an initial velocity boost near the Earth $\\Delta v_{1}$ and then a second velocity boost near Mars $\\Delta v_{2}$. Assume that both of these boosts are from instantaneous impulses, and ignore mass changes in the rocket as well as gravitational attraction to either Earth or Mars. Don't ignore the\n\nSun! Assume that the Earth and Mars are both in circular orbits around the Sun of radii $R_{E}$ and $R_{M}=R_{E} / \\alpha$ respectively. The orbital speeds are $v_{E}$ and $v_{M}$ respectively.\n\n[figure1]\n\nDerive an expression for the velocity boost $\\Delta v_{2}$ to change the orbit from elliptical to circular. Express your answer in terms of $v_{E}$ and $\\alpha$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-16.jpg?height=658&width=656&top_left_y=365&top_left_x=778" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1186", "problem": "Let us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nAssuming that the pressure remains negligible, find the time $t_{r \\rightarrow 0}$ needed for the ball to collapse from $r_{0}$ down to a much smaller radius, using Kepler's Laws.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nLet us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nAssuming that the pressure remains negligible, find the time $t_{r \\rightarrow 0}$ needed for the ball to collapse from $r_{0}$ down to a much smaller radius, using Kepler's Laws.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_553", "problem": "The nature of magnetic dipoles.\n\nA \"Gilbert\" dipole consists of a pair of magnetic monopoles each with a magnitude $q_{m}$ but opposite magnetic charges separated by a distance $d$, where $d$ is small. In this case, assume that $-q_{m}$ is located at $z=0$ and $+q_{m}$ is located at $z=d$.\n\n[figure1]\n\nAssume that magnetic monopoles behave like electric monopoles according to a coulomb-like force\n\nand the magnetic field obeys\n\n$$\nF=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m 1} q_{m 2}}{r^{2}}\n$$\n\n$$\nB=F / q_{m} .\n$$\n\nWrite an exact expression for the magnetic field strength $B(z)$ along the $z$ axis as a function of $z$ for $z>d$. Write your answer in terms of $q_{m}, d, z$, and any necessary fundamental constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe nature of magnetic dipoles.\n\nA \"Gilbert\" dipole consists of a pair of magnetic monopoles each with a magnitude $q_{m}$ but opposite magnetic charges separated by a distance $d$, where $d$ is small. In this case, assume that $-q_{m}$ is located at $z=0$ and $+q_{m}$ is located at $z=d$.\n\n[figure1]\n\nAssume that magnetic monopoles behave like electric monopoles according to a coulomb-like force\n\nand the magnetic field obeys\n\n$$\nF=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m 1} q_{m 2}}{r^{2}}\n$$\n\n$$\nB=F / q_{m} .\n$$\n\nWrite an exact expression for the magnetic field strength $B(z)$ along the $z$ axis as a function of $z$ for $z>d$. Write your answer in terms of $q_{m}, d, z$, and any necessary fundamental constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-18.jpg?height=147&width=653&top_left_y=569&top_left_x=777" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_923", "problem": "A Tippe top is a special kind of top that can spontaneously invert once it has been set spinning. One can model a Tippe top as a sphere of radius $R$ that is truncated, with a stem added. It has rotational symmetry about an axis through the stem, which is at angle $\\theta$ from the vertical. As shown in Figure 1(a), its centre of mass $C$ is offset from its geometric centre $O$ by $\\alpha R$ along its symmetry axis. The Tippe top makes contact with the surface it rests on at point $A$; we assume this surface is planar, and refer to it as the floor. Given certain geometrical constraints and if spun fast enough initially, the Tippe top will tip so that the stem points increasingly downwards, until it starts to spin on in its stem, and eventually comes to a stop.\n[figure1]\n\nFigure 1. Views of the Tippe top (a) from the side and (b) from above\n\nLet $x y z$ be the rotating reference frame defined such that $\\hat{\\mathbf{z}}$ is stationary and upwards, and the top's symmetry axis is within the $x z$-plane. Two views of the Tippe top are shown in Figure 1: from the side, and from above. As shown in Figure 1(b), the top's symmetry axis is aligned with the $x$-axis when viewed from above.\n\nFigure 2 shows the top's motion at several phases after it is started spinning:\n\n(a) phase I: immediately after it is initially set spinning, with $\\theta \\sim 0$\n\n(b) phase II: soon after, having tipped to angle $0<\\theta<\\frac{\\pi}{2}$\n\n(c) phase III: when the stem first touches the floor, with $\\theta>\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nShow that the components of the angular momentum $\\mathbf{L}$ and angular velocity $\\omega$ that are perpendicular to the $\\hat{\\mathbf{3}}$ direction are proportional, i.e.\n\n$$\n\\mathbf{L} \\times \\hat{\\mathbf{3}}=k(\\boldsymbol{\\omega} \\times \\hat{\\mathbf{3}}),\n$$\n\nand find the proportionality constant $k$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA Tippe top is a special kind of top that can spontaneously invert once it has been set spinning. One can model a Tippe top as a sphere of radius $R$ that is truncated, with a stem added. It has rotational symmetry about an axis through the stem, which is at angle $\\theta$ from the vertical. As shown in Figure 1(a), its centre of mass $C$ is offset from its geometric centre $O$ by $\\alpha R$ along its symmetry axis. The Tippe top makes contact with the surface it rests on at point $A$; we assume this surface is planar, and refer to it as the floor. Given certain geometrical constraints and if spun fast enough initially, the Tippe top will tip so that the stem points increasingly downwards, until it starts to spin on in its stem, and eventually comes to a stop.\n[figure1]\n\nFigure 1. Views of the Tippe top (a) from the side and (b) from above\n\nLet $x y z$ be the rotating reference frame defined such that $\\hat{\\mathbf{z}}$ is stationary and upwards, and the top's symmetry axis is within the $x z$-plane. Two views of the Tippe top are shown in Figure 1: from the side, and from above. As shown in Figure 1(b), the top's symmetry axis is aligned with the $x$-axis when viewed from above.\n\nFigure 2 shows the top's motion at several phases after it is started spinning:\n\n(a) phase I: immediately after it is initially set spinning, with $\\theta \\sim 0$\n\n(b) phase II: soon after, having tipped to angle $0<\\theta<\\frac{\\pi}{2}$\n\n(c) phase III: when the stem first touches the floor, with $\\theta>\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nShow that the components of the angular momentum $\\mathbf{L}$ and angular velocity $\\omega$ that are perpendicular to the $\\hat{\\mathbf{3}}$ direction are proportional, i.e.\n\n$$\n\\mathbf{L} \\times \\hat{\\mathbf{3}}=k(\\boldsymbol{\\omega} \\times \\hat{\\mathbf{3}}),\n$$\n\nand find the proportionality constant $k$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-1.jpg?height=522&width=1332&top_left_y=1121&top_left_x=360", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=578&width=1778&top_left_y=316&top_left_x=184", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=417&width=545&top_left_y=1296&top_left_x=527", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=431&width=397&top_left_y=1298&top_left_x=1189" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_388", "problem": "In this problem, we analyse the working principle of a speed camera. The transmitter of the speed camera emits an electomagnetic wave of frequency $f_{0}=24 \\mathrm{GHz}$ having waveform $\\cos \\left(2 \\pi f_{0} t\\right)$. The wave gets reflected from an approaching car moving with speed $v$. The reflected wave is recorded by the receiver of the speed camera.\n\nExpress the freqency $f_{1}$ of the reflected wave.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn this problem, we analyse the working principle of a speed camera. The transmitter of the speed camera emits an electomagnetic wave of frequency $f_{0}=24 \\mathrm{GHz}$ having waveform $\\cos \\left(2 \\pi f_{0} t\\right)$. The wave gets reflected from an approaching car moving with speed $v$. The reflected wave is recorded by the receiver of the speed camera.\n\nExpress the freqency $f_{1}$ of the reflected wave.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_372", "problem": "When a body moves in a fluid, the effective inertial mass appears to be larger than the mass of the body itself, because to accelerate the body some of the fluid needs to be accelerated as well. This increase is called added mass or virtual mass. Measure the amount of virtual mass $m_{v}$ that is added to the ball as it moves through water. The diameter of the ball is $d=72.0 \\mathrm{~mm}$. You don't have to calculate the uncertainty of your results, however the accuracy of your methods and results is important and will be graded.\n\nEquipment: Ball fixed to a spring, stand, stopwatch, ruler, container with water.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nWhen a body moves in a fluid, the effective inertial mass appears to be larger than the mass of the body itself, because to accelerate the body some of the fluid needs to be accelerated as well. This increase is called added mass or virtual mass. Measure the amount of virtual mass $m_{v}$ that is added to the ball as it moves through water. The diameter of the ball is $d=72.0 \\mathrm{~mm}$. You don't have to calculate the uncertainty of your results, however the accuracy of your methods and results is important and will be graded.\n\nEquipment: Ball fixed to a spring, stand, stopwatch, ruler, container with water.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of g, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "g" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_130", "problem": "A block on a ramp is given an initial velocity $v_{0}$ upward along the ramp. It slides upward for a time $t_{u}$, traveling some distance, and then slides downward for a time $t_{d}$ until it returns to its original position. What is $t_{d}$ in terms of $t_{u}$ ?\n\nThe height of the incline is 0.6 times its length and the coefficient of kinetic friction between the block and the incline is 0.5 .\n\n[figure1]\nA: $t_{0} / 5$\nB: $t_{0} / \\sqrt{5}$\nC: $\\sqrt{5} t_{0} $ \nD: $2 t_{0}$\nE: $5 t_{0}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA block on a ramp is given an initial velocity $v_{0}$ upward along the ramp. It slides upward for a time $t_{u}$, traveling some distance, and then slides downward for a time $t_{d}$ until it returns to its original position. What is $t_{d}$ in terms of $t_{u}$ ?\n\nThe height of the incline is 0.6 times its length and the coefficient of kinetic friction between the block and the incline is 0.5 .\n\n[figure1]\n\nA: $t_{0} / 5$\nB: $t_{0} / \\sqrt{5}$\nC: $\\sqrt{5} t_{0} $ \nD: $2 t_{0}$\nE: $5 t_{0}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-17.jpg?height=466&width=656&top_left_y=491&top_left_x=732" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_93", "problem": "An object starting from rest moves on a circular path with a radius $40 \\mathrm{~cm}$ and a constant tangential acceleration of $10 \\mathrm{~cm} / \\mathrm{s}^{2}$. How much time is needed after the motion begins for the centripetal acceleration of the object to be equal to the tangential acceleration?\nA: $0.2 \\mathrm{~s}$\nB: $1.0 \\mathrm{~s}$\nC: $1.2 \\mathrm{~s}$\nD: $1.8 \\mathrm{~s}$\nE: $2.0 \\mathrm{~s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn object starting from rest moves on a circular path with a radius $40 \\mathrm{~cm}$ and a constant tangential acceleration of $10 \\mathrm{~cm} / \\mathrm{s}^{2}$. How much time is needed after the motion begins for the centripetal acceleration of the object to be equal to the tangential acceleration?\n\nA: $0.2 \\mathrm{~s}$\nB: $1.0 \\mathrm{~s}$\nC: $1.2 \\mathrm{~s}$\nD: $1.8 \\mathrm{~s}$\nE: $2.0 \\mathrm{~s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_461", "problem": "A thin, uniform rod of length $L$ and mass $M=0.258 \\mathrm{~kg}$ is suspended from a point a distance $R$ away from its center of mass. When the end of the rod is displaced slightly and released it executes simple harmonic oscillation. The period, $T$, of the oscillation is timed using an electronic timer. The following data is recorded for the period as a function of $R$. What is the local value of $g$ ? Do not assume it is the canonical value of $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$. What is the length, $L$, of the rod? No estimation\nof error in either value is required. The moment of inertia of a rod about its center of mass is $(1 / 12) M L^{2}$.\n\n| $R$ | $T$ |\n| :---: | :---: |\n| $(\\mathrm{~m})$ | $(\\mathrm{s})$ |\n| 0.050 | 3.842 |\n| 0.075 | 3.164 |\n| 0.102 | 2.747 |\n| 0.156 | 2.301 |\n| 0.198 | 2.115 |\n\n\n| $R$
$(\\mathrm{~m})$ | $T$
$(\\mathrm{~s})$ |\n| :---: | :---: |\n| 0.211 | 2.074 |\n| 0.302 | 1.905 |\n| 0.387 | 1.855 |\n| 0.451 | 1.853 |\n| 0.588 | 1.900 |\n\nYou must show your work to obtain full credit. If you use graphical techniques then you must plot the graph; if you use linear regression techniques then you must show all of the formulae and associated workings used to obtain your result.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nA thin, uniform rod of length $L$ and mass $M=0.258 \\mathrm{~kg}$ is suspended from a point a distance $R$ away from its center of mass. When the end of the rod is displaced slightly and released it executes simple harmonic oscillation. The period, $T$, of the oscillation is timed using an electronic timer. The following data is recorded for the period as a function of $R$. What is the local value of $g$ ? Do not assume it is the canonical value of $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$. What is the length, $L$, of the rod? No estimation\nof error in either value is required. The moment of inertia of a rod about its center of mass is $(1 / 12) M L^{2}$.\n\n| $R$ | $T$ |\n| :---: | :---: |\n| $(\\mathrm{~m})$ | $(\\mathrm{s})$ |\n| 0.050 | 3.842 |\n| 0.075 | 3.164 |\n| 0.102 | 2.747 |\n| 0.156 | 2.301 |\n| 0.198 | 2.115 |\n\n\n| $R$
$(\\mathrm{~m})$ | $T$
$(\\mathrm{~s})$ |\n| :---: | :---: |\n| 0.211 | 2.074 |\n| 0.302 | 1.905 |\n| 0.387 | 1.855 |\n| 0.451 | 1.853 |\n| 0.588 | 1.900 |\n\nYou must show your work to obtain full credit. If you use graphical techniques then you must plot the graph; if you use linear regression techniques then you must show all of the formulae and associated workings used to obtain your result.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [ the local value of $g$, the length, $L$, of the rod].\nTheir units are, in order, [$\\mathrm{~m} / \\mathrm{s}^{2}$, $\\mathrm{~m}$], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_3899026513eb55709c81g-06.jpg?height=686&width=781&top_left_y=286&top_left_x=669" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{~m} / \\mathrm{s}^{2}$", "$\\mathrm{~m}$" ], "answer_sequence": [ " the local value of $g$", "the length, $L$, of the rod" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_674", "problem": "Newtons law of cooling can be used to show that the temperature as a function of time takes the form $T(t)=A\\left(1-e^{-B t}\\right)$. Measurements of the temperature of an object which has been removed from a freezer and placed in a warm room are shown in Fig. 6. Which of the following are the best estimates for the values of $A$ and $B$ ?\n\n[figure1]\nA: $A=20^{\\circ} \\mathrm{C}$ and $B=1 / 5 \\mathrm{~min}^{-1}$\nB: $A=20^{\\circ} \\mathrm{C}$ and $B=1 / 2 \\mathrm{~min}^{-1}$\nC: $A=15^{\\circ} \\mathrm{C}$ and $B=5 \\mathrm{~min}^{-1}$\nD: $A=15^{\\circ} \\mathrm{C}$ and $B=3 \\mathrm{~min}^{-1}$\nE: $A=15^{\\circ} \\mathrm{C}$ and $B=1 / 3 \\mathrm{~min}^{-1}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nNewtons law of cooling can be used to show that the temperature as a function of time takes the form $T(t)=A\\left(1-e^{-B t}\\right)$. Measurements of the temperature of an object which has been removed from a freezer and placed in a warm room are shown in Fig. 6. Which of the following are the best estimates for the values of $A$ and $B$ ?\n\n[figure1]\n\nA: $A=20^{\\circ} \\mathrm{C}$ and $B=1 / 5 \\mathrm{~min}^{-1}$\nB: $A=20^{\\circ} \\mathrm{C}$ and $B=1 / 2 \\mathrm{~min}^{-1}$\nC: $A=15^{\\circ} \\mathrm{C}$ and $B=5 \\mathrm{~min}^{-1}$\nD: $A=15^{\\circ} \\mathrm{C}$ and $B=3 \\mathrm{~min}^{-1}$\nE: $A=15^{\\circ} \\mathrm{C}$ and $B=1 / 3 \\mathrm{~min}^{-1}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-05.jpg?height=499&width=880&top_left_y=1561&top_left_x=151" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_463", "problem": "An AC power line cable transmits electrical power using a sinusoidal waveform with frequency $60 \\mathrm{~Hz}$. The load receives an RMS voltage of $500 \\mathrm{kV}$ and requires $1000 \\mathrm{MW}$ of average power. For this problem, consider only the cable carrying current in one of the two directions, and ignore effects due to capacitance or inductance between the cable and with the ground.\n\nA local rancher thinks he might be able to extract electrical power from the cable using electromagnetic induction. The rancher constructs a rectangular loop of length $a$ and width $b0\\right)$, the velocity imparted to the fluid particle is in the direction of propagation, while for an extensional wave $\\left(\\Delta P_{\\mathrm{s}}<0\\right)$, the velocity imparted is in the opposite direction of propagation.\n\nEqs. (a2) and (a4) can be combined to give\n\n$$\n\\Delta P_{\\mathrm{s}}=\\rho_{0} c^{2}\\left(1+\\frac{v_{0}}{c}\\right)^{2} \\frac{\\Delta \\rho}{\\rho_{1}}\n$$\n\nFrom the definition of the bulk modulus $B$, which is assumed to be constant, it follows\n\n$$\n\\Delta P_{\\mathrm{s}}=B \\frac{V_{0}-V_{1}}{V_{0}}=B \\frac{1 / \\rho_{0}-1 / \\rho_{1}}{1 / \\rho_{0}}=B \\frac{\\Delta \\rho}{\\rho_{1}}\n$$\n\nFrom Eqs. (a6) and (a7), we obtain\n\n$$\n\\rho_{0} c^{2}\\left(1+\\frac{v_{0}}{c}\\right)^{2}=B\n$$\n\nThus\n\n$$\nc=\\sqrt{\\frac{B}{\\rho_{0}}}-v_{0} \\quad \\Rightarrow \\gamma=1 \\quad \\beta=-v_{0}\n$$In a uniform cylindrical pipe of length $L$, water is flowing steadily along the $+x$ direction with horizontal velocity $v_{0}$, density $\\rho_{0}$, and pressure $P_{0}$. As shown in Fig. 1 , the pipe is connected to a reservoir at a depth $h$ and opens into the atmosphere at pressure $P_{\\mathrm{a}}$.\n\nSuppose the flow-control valve $T$ at the end of the pipe is then shut instantly so that the oncoming liquid element next to the valve suffers both a pressure change $\\Delta P_{\\mathrm{s}} \\equiv P_{1}-P_{0}$ and a velocity change $\\Delta v=v_{1}-v_{0}$ with $v_{1} \\leq 0$. This causes a longitudinal wave of excess pressure $\\Delta P_{\\mathrm{s}}$ to travel upstream in the $-x$ direction with a speed of propagation $c$.\n\n[figure1]\n\nFig. 1: Steady flow in a uniform pipe.\n\nCalculate values of $c$ and $\\Delta P_{\\mathrm{s}}$ for the case of water flow with $v_{0}=4.0 \\mathrm{~m} / \\mathrm{s}$ and $v_{1}=0$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\nHere is some context information for this question, which might assist you in solving it:\nWhen the valve opening is suddenly blocked, fluid pressure at the valve jumps from $P_{0}$ to $P_{1}=P_{0}+\\Delta P_{\\mathrm{s}}$, thus sending a pressure wave traveling upstream (to the left) with speed $c$ and amplitude $\\Delta P_{\\mathrm{s}}$. Taking positive $x$ direction as pointing to the right, the velocity of fluid particles next to the valve changes from $v_{0}$ to $v_{1}$ $\\left(v_{1} \\leq 0\\right)$. Thus the velocity change is $\\Delta v=v_{1}-v_{0}$.\n\nIn a frame moving to left (along $-x$ direction) with speed $c$, i.e., riding on the wave (see Fig. S1), velocity of fluid in the pressure wave is $c+v_{1}$, while that of the incoming fluid in the steady flow ahead of the wave is $c+v_{0}$. Let $\\rho_{1}$ be the density of fluid in the pressure wave. From conservation of mass, i.e., equation of continuity, we have\n\n$$\n\\rho_{0}\\left(c+v_{0}\\right)=\\rho_{1}\\left(c+v_{1}\\right)\n$$\n\nor, by letting $\\Delta \\rho \\equiv \\rho_{1}-\\rho_{0}$,\n\n$$\n\\frac{\\Delta \\rho}{\\rho_{1}}=1-\\frac{\\rho_{0}}{\\rho_{1}}=\\frac{v_{0}-v_{1}}{c+v_{0}}=\\frac{-\\Delta v}{c+v_{0}}\n$$\n\nMoreover, impulse imparted to the fluid must equal its momentum change. Thus, in a short time interval $\\tau$ after the valve is closed, we must have\n\n$$\n\\rho_{0}\\left(c+v_{0}\\right) \\tau\\left[\\left(c+v_{1}\\right)-\\left(c+v_{0}\\right)\\right]=-\\tau \\Delta P=\\left(P_{0}-P_{1}\\right) \\tau\n$$\n\nor\n\n$$\n\\Delta P_{\\mathrm{s}}=-\\rho_{0} c\\left(1+\\frac{v_{0}}{c}\\right)\\left(v_{1}-v_{0}\\right)=-\\rho_{0} c\\left(1+\\frac{v_{0}}{c}\\right) \\Delta v \\Rightarrow \\alpha=-\\left(1+\\frac{v_{0}}{c}\\right)\n$$\n\nIf $v_{0} / c \\ll 1$, we have\n\n$$\n\\Delta P_{\\mathrm{s}}=-\\rho_{0} c \\Delta v\n$$\n\nNote that the negative sign in Eqs. (a4) and (a5) follows from the fact that the direction of propagation is opposite to the positive direction for $x$ axis (and velocity). Otherwise the sign should be positive. Note also that for a compressional wave\n$\\left(\\Delta P_{\\mathrm{s}}>0\\right)$, the velocity imparted to the fluid particle is in the direction of propagation, while for an extensional wave $\\left(\\Delta P_{\\mathrm{s}}<0\\right)$, the velocity imparted is in the opposite direction of propagation.\n\nEqs. (a2) and (a4) can be combined to give\n\n$$\n\\Delta P_{\\mathrm{s}}=\\rho_{0} c^{2}\\left(1+\\frac{v_{0}}{c}\\right)^{2} \\frac{\\Delta \\rho}{\\rho_{1}}\n$$\n\nFrom the definition of the bulk modulus $B$, which is assumed to be constant, it follows\n\n$$\n\\Delta P_{\\mathrm{s}}=B \\frac{V_{0}-V_{1}}{V_{0}}=B \\frac{1 / \\rho_{0}-1 / \\rho_{1}}{1 / \\rho_{0}}=B \\frac{\\Delta \\rho}{\\rho_{1}}\n$$\n\nFrom Eqs. (a6) and (a7), we obtain\n\n$$\n\\rho_{0} c^{2}\\left(1+\\frac{v_{0}}{c}\\right)^{2}=B\n$$\n\nThus\n\n$$\nc=\\sqrt{\\frac{B}{\\rho_{0}}}-v_{0} \\quad \\Rightarrow \\gamma=1 \\quad \\beta=-v_{0}\n$$\n\nproblem:\nIn a uniform cylindrical pipe of length $L$, water is flowing steadily along the $+x$ direction with horizontal velocity $v_{0}$, density $\\rho_{0}$, and pressure $P_{0}$. As shown in Fig. 1 , the pipe is connected to a reservoir at a depth $h$ and opens into the atmosphere at pressure $P_{\\mathrm{a}}$.\n\nSuppose the flow-control valve $T$ at the end of the pipe is then shut instantly so that the oncoming liquid element next to the valve suffers both a pressure change $\\Delta P_{\\mathrm{s}} \\equiv P_{1}-P_{0}$ and a velocity change $\\Delta v=v_{1}-v_{0}$ with $v_{1} \\leq 0$. This causes a longitudinal wave of excess pressure $\\Delta P_{\\mathrm{s}}$ to travel upstream in the $-x$ direction with a speed of propagation $c$.\n\n[figure1]\n\nFig. 1: Steady flow in a uniform pipe.\n\nCalculate values of $c$ and $\\Delta P_{\\mathrm{s}}$ for the case of water flow with $v_{0}=4.0 \\mathrm{~m} / \\mathrm{s}$ and $v_{1}=0$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the value of $ c$, the value of $\\Delta P_{\\mathrm{s}}$].\nTheir units are, in order, [$\\mathrm{~m} / \\mathrm{s} $, $\\mathrm{MPa}$], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_04e7339923135da91bb4g-1.jpg?height=542&width=1445&top_left_y=1825&top_left_x=243" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{~m} / \\mathrm{s} $", "$\\mathrm{MPa}$" ], "answer_sequence": [ "the value of $ c$", "the value of $\\Delta P_{\\mathrm{s}}$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_199", "problem": "A planet orbits around a star $\\mathrm{S}$, as shown in the figure. The semi-major axis of the orbit is $a$. The perigee, namely the shortest distance between the planet and the star is $0.5 a$. When the planet passes point $\\mathrm{P}$ (on the line through the star and perpendicular to the major axis), its speed is $v_{1}$. What is its speed $v_{2}$ when it passes the perigee?\n\n[figure1]\nA: $v_{2}=\\frac{3}{\\sqrt{5}} v_{1} . $ \nB: $v_{2}=\\frac{3}{\\sqrt{7}} v_{1}$.\nC: $v_{2}=\\frac{2}{\\sqrt{3}} v_{1}$.\nD: $v_{2}=\\frac{\\sqrt{7}}{\\sqrt{3}} v_{1}$.\nE: $v_{2}=4 v_{1}$.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA planet orbits around a star $\\mathrm{S}$, as shown in the figure. The semi-major axis of the orbit is $a$. The perigee, namely the shortest distance between the planet and the star is $0.5 a$. When the planet passes point $\\mathrm{P}$ (on the line through the star and perpendicular to the major axis), its speed is $v_{1}$. What is its speed $v_{2}$ when it passes the perigee?\n\n[figure1]\n\nA: $v_{2}=\\frac{3}{\\sqrt{5}} v_{1} . $ \nB: $v_{2}=\\frac{3}{\\sqrt{7}} v_{1}$.\nC: $v_{2}=\\frac{2}{\\sqrt{3}} v_{1}$.\nD: $v_{2}=\\frac{\\sqrt{7}}{\\sqrt{3}} v_{1}$.\nE: $v_{2}=4 v_{1}$.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f726c6cf4a23f08e0214g-10.jpg?height=425&width=591&top_left_y=1083&top_left_x=775" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_501", "problem": "Beloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nAssuming the proton beam is off, determine the time constant for the accelerating dome (the time it takes for the charge on the dome to decrease to $1 / e \\approx 1 / 3$ of the initial value.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nBeloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nAssuming the proton beam is off, determine the time constant for the accelerating dome (the time it takes for the charge on the dome to decrease to $1 / e \\approx 1 / 3$ of the initial value.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~s}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=468&width=1592&top_left_y=438&top_left_x=259", "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=493&width=1268&top_left_y=1339&top_left_x=426" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_267", "problem": "In a science experiment an insulated block of metal is heated using an electric heater for 10 minutes. The mass of the block is $900 \\mathrm{~g}$. The initial temperature of the block is measured to be $17{ }^{\\circ} \\mathrm{C}$ and the final temperature is measured as $43^{\\circ} \\mathrm{C}$.\n\nThe experiment is repeated using the same metal block and electric heater. This time the starting temperature is $19{ }^{\\circ} \\mathrm{C}$ and the block is heated for 15 minutes.\n\nThe final temperature of the metal block in the second experiment will be approximately:\nA: $\\quad 39^{\\circ} \\mathrm{C}$\nB: $\\quad 45^{\\circ} \\mathrm{C}$\nC: $\\quad 58^{\\circ} \\mathrm{C}$\nD: $65^{\\circ} \\mathrm{C}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nIn a science experiment an insulated block of metal is heated using an electric heater for 10 minutes. The mass of the block is $900 \\mathrm{~g}$. The initial temperature of the block is measured to be $17{ }^{\\circ} \\mathrm{C}$ and the final temperature is measured as $43^{\\circ} \\mathrm{C}$.\n\nThe experiment is repeated using the same metal block and electric heater. This time the starting temperature is $19{ }^{\\circ} \\mathrm{C}$ and the block is heated for 15 minutes.\n\nThe final temperature of the metal block in the second experiment will be approximately:\n\nA: $\\quad 39^{\\circ} \\mathrm{C}$\nB: $\\quad 45^{\\circ} \\mathrm{C}$\nC: $\\quad 58^{\\circ} \\mathrm{C}$\nD: $65^{\\circ} \\mathrm{C}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_181", "problem": ". An empty freight car on a level railroad track has a mass $M$. A chute above the freight car opens and grain falls down into the car at a rate of $r$, measured in kilograms per second. The grain falls a vertical distance $h$ before hitting the bed of the freight car without bouncing up. How much normal force is exerted on the freight car from the rails at a time $t$ after the grain begins to hit the bed of the car? Assume the grain starts from rest.\nA: $M g+r t g$\nB: $M g+r \\sqrt{g h}$\nC: $M g+r \\sqrt{2 g h}$\nD: $M g+r \\sqrt{g h}+r t g$\nE: $M g+r \\sqrt{2 g h}+r t g $ \n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\n. An empty freight car on a level railroad track has a mass $M$. A chute above the freight car opens and grain falls down into the car at a rate of $r$, measured in kilograms per second. The grain falls a vertical distance $h$ before hitting the bed of the freight car without bouncing up. How much normal force is exerted on the freight car from the rails at a time $t$ after the grain begins to hit the bed of the car? Assume the grain starts from rest.\n\nA: $M g+r t g$\nB: $M g+r \\sqrt{g h}$\nC: $M g+r \\sqrt{2 g h}$\nD: $M g+r \\sqrt{g h}+r t g$\nE: $M g+r \\sqrt{2 g h}+r t g $ \n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_161", "problem": "A ball of radius $R$ and mass $m$ is magically put inside a thin shell of the same mass and radius $2 R$. The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface?\n\n[figure1]\nA: $R$.\nB: $R / 2 . $ \nC: $R / 4$.\nD: $3 R / 8$.\nE: $R / 8$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA ball of radius $R$ and mass $m$ is magically put inside a thin shell of the same mass and radius $2 R$. The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface?\n\n[figure1]\n\nA: $R$.\nB: $R / 2 . $ \nC: $R / 4$.\nD: $3 R / 8$.\nE: $R / 8$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f726c6cf4a23f08e0214g-02.jpg?height=336&width=399&top_left_y=1510&top_left_x=863" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1421", "problem": "图 a 是基于全内反射原理制备的折射率阶跃型光纤及其耦合光路示意图. 光纤内芯直径 $50 \\mu \\mathrm{m}$,折射率 $\\mathrm{n}_{1}=1.46$; 它的玻璃外包层的外径为 $125 \\mu \\mathrm{m}$, 折射率 $\\mathrm{n}_{2}=1.45$. 氦氛激光器输出一圆柱形平行光束, 为了将该激光束有效地耦合进入光纤传输, 可以在光纤前端放置一微球透镜进行聚焦和耦合, 微球透镜的直径 $\\mathrm{D}=3.00 \\mathrm{~mm}$, 折射率 $\\mathrm{n}=1.50$. 已知激光束中心轴通过微球透镜中心, 且与光纤对称轴重合. 空气折射率 $\\mathrm{n}_{0}=1.00$.\n\n[图1]\n\n图 a. 激光束经微球透镜耦合进入光纤示意图若光束在透镜聚焦过程中满足近轴条件, 为了使平行激光束刚好聚焦于光纤端面 (与光纤对称轴垂直)处, 微球透镜后表面中心顶点 0 与光纤端面距离应为多大?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n图 a 是基于全内反射原理制备的折射率阶跃型光纤及其耦合光路示意图. 光纤内芯直径 $50 \\mu \\mathrm{m}$,折射率 $\\mathrm{n}_{1}=1.46$; 它的玻璃外包层的外径为 $125 \\mu \\mathrm{m}$, 折射率 $\\mathrm{n}_{2}=1.45$. 氦氛激光器输出一圆柱形平行光束, 为了将该激光束有效地耦合进入光纤传输, 可以在光纤前端放置一微球透镜进行聚焦和耦合, 微球透镜的直径 $\\mathrm{D}=3.00 \\mathrm{~mm}$, 折射率 $\\mathrm{n}=1.50$. 已知激光束中心轴通过微球透镜中心, 且与光纤对称轴重合. 空气折射率 $\\mathrm{n}_{0}=1.00$.\n\n[图1]\n\n图 a. 激光束经微球透镜耦合进入光纤示意图\n\n问题:\n若光束在透镜聚焦过程中满足近轴条件, 为了使平行激光束刚好聚焦于光纤端面 (与光纤对称轴垂直)处, 微球透镜后表面中心顶点 0 与光纤端面距离应为多大?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~mm}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-05.jpg?height=411&width=1077&top_left_y=271&top_left_x=472" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~mm}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_743", "problem": "A person biking down a $5^{\\circ}$ slope without pedaling or braking noticed that her speed is constant at $25 \\mathrm{~km} / \\mathrm{h}$. Approximately what power does she need to bike up this hill at the same speed if her weight including the bike is $65 \\mathrm{~kg}$ ?\nA: Nobody can bike down the hill at constant speed without braking.\nB: $800 \\mathrm{~W}$\nC: $1400 \\mathrm{~W}$\nD: $400 \\mathrm{~W}$\nE: $2800 \\mathrm{~W}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA person biking down a $5^{\\circ}$ slope without pedaling or braking noticed that her speed is constant at $25 \\mathrm{~km} / \\mathrm{h}$. Approximately what power does she need to bike up this hill at the same speed if her weight including the bike is $65 \\mathrm{~kg}$ ?\n\nA: Nobody can bike down the hill at constant speed without braking.\nB: $800 \\mathrm{~W}$\nC: $1400 \\mathrm{~W}$\nD: $400 \\mathrm{~W}$\nE: $2800 \\mathrm{~W}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1723", "problem": "如图, $O$ 点是小球平抛运动抛出点; 在 $O$ 点有一个频闪点光源, 闪光频率为 $30 \\mathrm{~Hz}$; 在抛出点的正前方, 坚直放置一块毛玻璃, 小球初速度与毛玻璃平面垂直。在小球抛出时点光源开始闪光。当点光源闪光时，在毛玻璃上有小球的一个投影点。已知图中 $\\mathrm{O}$ 点与毛玻璃水平距离 $L=1.20 \\mathrm{~m}$ ， 测得第一、二个投影点之间的距离为 $0.05 \\mathrm{~m}$ 。取重力\n\n[图1]\n加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。下列说法正确的是\nA: 小球平抛运动的初速度为 $4 \\mathrm{~m} / \\mathrm{s}$\nB: 小球平抛运动过程中，在相等时间内的动量变化不相等\nC: 小球投影点的速度在相等时间内的变化量越来越大\nD: 小球第二、三个投影点之间的距离 $0.15 \\mathrm{~m}$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图, $O$ 点是小球平抛运动抛出点; 在 $O$ 点有一个频闪点光源, 闪光频率为 $30 \\mathrm{~Hz}$; 在抛出点的正前方, 坚直放置一块毛玻璃, 小球初速度与毛玻璃平面垂直。在小球抛出时点光源开始闪光。当点光源闪光时，在毛玻璃上有小球的一个投影点。已知图中 $\\mathrm{O}$ 点与毛玻璃水平距离 $L=1.20 \\mathrm{~m}$ ， 测得第一、二个投影点之间的距离为 $0.05 \\mathrm{~m}$ 。取重力\n\n[图1]\n加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。下列说法正确的是\n\nA: 小球平抛运动的初速度为 $4 \\mathrm{~m} / \\mathrm{s}$\nB: 小球平抛运动过程中，在相等时间内的动量变化不相等\nC: 小球投影点的速度在相等时间内的变化量越来越大\nD: 小球第二、三个投影点之间的距离 $0.15 \\mathrm{~m}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-02.jpg?height=337&width=334&top_left_y=420&top_left_x=1449" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1394", "problem": "一线膨胀系数为 $\\alpha$ 的正立方体物块, 当膨胀量较小时, 其体膨胀系数等于\nA: $\\alpha$\nB: $\\alpha^{1 / 3}$\nC: $\\alpha^{3}$\nD: $3 \\alpha$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n一线膨胀系数为 $\\alpha$ 的正立方体物块, 当膨胀量较小时, 其体膨胀系数等于\n\nA: $\\alpha$\nB: $\\alpha^{1 / 3}$\nC: $\\alpha^{3}$\nD: $3 \\alpha$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_535", "problem": "A block of mass $M$ has a hole drilled through it so that a ball of mass $m$ can enter horizontally and then pass through the block and exit vertically upward. The ball and block are located on a frictionless surface; the block is originally at rest.\n\n[figure1]\n\nfrictionless horizontal surface\n\nConsider the scenario where the ball is traveling horizontally with a speed $v_{0}$. The ball enters the block and is ejected out the top of the block. Assume there are no frictional losses as the ball passes through the block, and the ball rises to a height much higher than the dimensions of the block. The ball then returns to the level of the block, where it enters the top hole and then is ejected from the side hole. Determine the time $t$ for the ball to return to the position where the original collision occurs in terms of the mass ratio $\\beta=M / m$, speed $v_{0}$, and acceleration of free fall $g$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA block of mass $M$ has a hole drilled through it so that a ball of mass $m$ can enter horizontally and then pass through the block and exit vertically upward. The ball and block are located on a frictionless surface; the block is originally at rest.\n\n[figure1]\n\nfrictionless horizontal surface\n\nConsider the scenario where the ball is traveling horizontally with a speed $v_{0}$. The ball enters the block and is ejected out the top of the block. Assume there are no frictional losses as the ball passes through the block, and the ball rises to a height much higher than the dimensions of the block. The ball then returns to the level of the block, where it enters the top hole and then is ejected from the side hole. Determine the time $t$ for the ball to return to the position where the original collision occurs in terms of the mass ratio $\\beta=M / m$, speed $v_{0}$, and acceleration of free fall $g$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0256c432f4019b26894dg-17.jpg?height=415&width=1027&top_left_y=562&top_left_x=538" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_891", "problem": "Consider a gas of ultra relativistic electrons $(\\gamma \\gg 1)$, with an isotropic distribution of velocities (does not depend on direction). The proper number density of particles with energies between $\\epsilon$ and $\\epsilon+d \\epsilon$ is given by $f(\\epsilon) d \\epsilon$, where $\\epsilon$ is the energy per particle. Consider also a wall of area $\\Delta A$, which is in contact with the gas.\n\nFind an expression for the total rate of change in momentum $\\Delta p_{\\mathrm{z}} / \\Delta t$ of the gas, $\\quad 0.8 \\mathrm{pt}$ in the z-direction which is normal to the wall, due to collisions with the wall.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider a gas of ultra relativistic electrons $(\\gamma \\gg 1)$, with an isotropic distribution of velocities (does not depend on direction). The proper number density of particles with energies between $\\epsilon$ and $\\epsilon+d \\epsilon$ is given by $f(\\epsilon) d \\epsilon$, where $\\epsilon$ is the energy per particle. Consider also a wall of area $\\Delta A$, which is in contact with the gas.\n\nFind an expression for the total rate of change in momentum $\\Delta p_{\\mathrm{z}} / \\Delta t$ of the gas, $\\quad 0.8 \\mathrm{pt}$ in the z-direction which is normal to the wall, due to collisions with the wall.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1370", "problem": "海平面能将无线电波全反射, 反射波与入射波之间存在由于反射造成的半个波长的相位突变。一艘船在其离海平面高度为 $25 \\mathrm{~m}$ 的桅杆上装有发射天线, 向位于海岸高处的山顶接收站发射波长在 2 4 m 范围内的无线电波。当船驶至与接收站的水平距离 $L$ 越接近 $2000 \\mathrm{~m}$, 山顶接收站所接收到的信号越弱; 当 $L=2000 \\mathrm{~m}$ 时失去无线电联系。山顶接收站海拔高度为 $150 \\mathrm{~m}$ 。船上天线发出的无线电波中有一部分直接传播到接收站, 另一部分经海平面反射后传播到接收站, 两列波的几何波程差为 ___$\\mathrm{m}$ ，该无线电波的实际波长为 ___$\\mathrm{m}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n海平面能将无线电波全反射, 反射波与入射波之间存在由于反射造成的半个波长的相位突变。一艘船在其离海平面高度为 $25 \\mathrm{~m}$ 的桅杆上装有发射天线, 向位于海岸高处的山顶接收站发射波长在 2 4 m 范围内的无线电波。当船驶至与接收站的水平距离 $L$ 越接近 $2000 \\mathrm{~m}$, 山顶接收站所接收到的信号越弱; 当 $L=2000 \\mathrm{~m}$ 时失去无线电联系。山顶接收站海拔高度为 $150 \\mathrm{~m}$ 。船上天线发出的无线电波中有一部分直接传播到接收站, 另一部分经海平面反射后传播到接收站, 两列波的几何波程差为 ___$\\mathrm{m}$ ，该无线电波的实际波长为 ___$\\mathrm{m}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[船上天线发出的无线电波中有一部分直接传播到接收站, 另一部分经海平面反射后传播到接收站, 两列波的几何波程差, 该无线电波的实际波长]\n它们的单位依次是[$\\mathrm{m}$, $\\mathrm{m}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{m}$", "$\\mathrm{m}$" ], "answer_sequence": [ "船上天线发出的无线电波中有一部分直接传播到接收站, 另一部分经海平面反射后传播到接收站, 两列波的几何波程差", "该无线电波的实际波长" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1313", "problem": "爱因斯坦等效原理可表述为: 在有引力作用的情况下的物理规律和没有引力但有适当加速度的参考系中的物理规律是相同的。作为一个例子，考察下面两种情况：根据等效原理, 试比较 (i) 和（ii）的结果, 要使物理规律在（i）和（ii）中的情况下相同，则（ii）中的 $a$ 应为多大?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n爱因斯坦等效原理可表述为: 在有引力作用的情况下的物理规律和没有引力但有适当加速度的参考系中的物理规律是相同的。作为一个例子，考察下面两种情况：\n\n问题:\n根据等效原理, 试比较 (i) 和（ii）的结果, 要使物理规律在（i）和（ii）中的情况下相同，则（ii）中的 $a$ 应为多大?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_165", "problem": "If the magnitude of the acceleration of the box is chosen correctly, the launched particle will follow a path that returns to the point that it was launched. In the frame of the box, which path is followed by the particle?\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIf the magnitude of the acceleration of the box is chosen correctly, the launched particle will follow a path that returns to the point that it was launched. In the frame of the box, which path is followed by the particle?\n\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-08.jpg?height=399&width=415&top_left_y=386&top_left_x=321", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-08.jpg?height=401&width=399&top_left_y=385&top_left_x=863", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-08.jpg?height=390&width=399&top_left_y=388&top_left_x=1405", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-08.jpg?height=401&width=401&top_left_y=824&top_left_x=865" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_469", "problem": "In general, whenever an electric and a magnetic field are at an angle to each other, energy is transferred; for example, this principle is the reason electromagnetic radiation transfers energy. The power transferred per unit area is given by the Poynting vector:\n\n$$\n\\vec{S}=\\frac{1}{\\mu_{0}} \\vec{E} \\times \\vec{B}\n$$\n\nIn each part of this problem, the last subpart asks you to verify that the rate of energy transfer agrees with the formula for the Poynting vector. Therefore, you should not use the formula for the Poynting vector before the last subpart!\n\nA long, insulating cylindrical rod has radius $R$ and carries a uniform volume charge density $\\rho$. A uniform external electric field $E$ exists in the direction of its axis. The rod moves in the direction of its axis at speed $v$.\n\nWhat is the power per unit length $\\mathcal{P}$ delivered to the $\\operatorname{rod}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn general, whenever an electric and a magnetic field are at an angle to each other, energy is transferred; for example, this principle is the reason electromagnetic radiation transfers energy. The power transferred per unit area is given by the Poynting vector:\n\n$$\n\\vec{S}=\\frac{1}{\\mu_{0}} \\vec{E} \\times \\vec{B}\n$$\n\nIn each part of this problem, the last subpart asks you to verify that the rate of energy transfer agrees with the formula for the Poynting vector. Therefore, you should not use the formula for the Poynting vector before the last subpart!\n\nA long, insulating cylindrical rod has radius $R$ and carries a uniform volume charge density $\\rho$. A uniform external electric field $E$ exists in the direction of its axis. The rod moves in the direction of its axis at speed $v$.\n\nWhat is the power per unit length $\\mathcal{P}$ delivered to the $\\operatorname{rod}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_233", "problem": "Thermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nThermal atmospheric escape is one of the processes that explain why some gases are present in the atmosphere and some others are not. Currently the atmospheric pressure is approximately $P_{0}=10^{5} \\mathrm{~Pa}$ and a fraction of $\\chi_{H}=5.5 \\times 10^{-5 \\%}$ of the atmosphere molecules are hydrogen molecules. When these molecules reach a certain height (lower than $h_{E B}$ ) they split into two atoms due to solar radiation. Concentration of hydrogen atoms in the exobase can be considered constant over time.\n\nFind out how much time would it take for half of the hydrogen atom to escape the Earths atmosphere.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nThermal atmospheric escape is one of the processes that explain why some gases are present in the atmosphere and some others are not. Currently the atmospheric pressure is approximately $P_{0}=10^{5} \\mathrm{~Pa}$ and a fraction of $\\chi_{H}=5.5 \\times 10^{-5 \\%}$ of the atmosphere molecules are hydrogen molecules. When these molecules reach a certain height (lower than $h_{E B}$ ) they split into two atoms due to solar radiation. Concentration of hydrogen atoms in the exobase can be considered constant over time.\n\nFind out how much time would it take for half of the hydrogen atom to escape the Earths atmosphere.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of years, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_61a2ff399c33d9b3cd3bg-1.jpg?height=968&width=1044&top_left_y=1240&top_left_x=302" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "years" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_287", "problem": "A firework uses a chemical reaction to create a thrust force. This thrust force does work on the rocket to change the velocity and height above the ground.\n\nIgnoring air resistance, the relationship between the work done (WD) by the thrust force, the change in kinetic energy ( $\\triangle K E)$ of the rocket and the change in gravitational potential energy ( $\\triangle \\mathrm{GPE}$ ) of the rocket is:\nA: $\\quad \\triangle \\mathrm{GPE}=\\mathrm{WD}+\\Delta \\mathrm{KE}$\nB: $\\Delta \\mathrm{KE}=\\mathrm{WD}+\\Delta \\mathrm{GPE}$\nC: $\\quad W D=\\triangle G P E+\\triangle K E$\nD: $W D=\\triangle G P E-\\triangle K E$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA firework uses a chemical reaction to create a thrust force. This thrust force does work on the rocket to change the velocity and height above the ground.\n\nIgnoring air resistance, the relationship between the work done (WD) by the thrust force, the change in kinetic energy ( $\\triangle K E)$ of the rocket and the change in gravitational potential energy ( $\\triangle \\mathrm{GPE}$ ) of the rocket is:\n\nA: $\\quad \\triangle \\mathrm{GPE}=\\mathrm{WD}+\\Delta \\mathrm{KE}$\nB: $\\Delta \\mathrm{KE}=\\mathrm{WD}+\\Delta \\mathrm{GPE}$\nC: $\\quad W D=\\triangle G P E+\\triangle K E$\nD: $W D=\\triangle G P E-\\triangle K E$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1342", "problem": "如图, 球心在坐标原点 $\\mathrm{O}$ 的球面上有三个彼此绝缘的金属环, 它们分别与 $x-y$ 平面、 $y-z$ 平面、 $z-x$ 平面与球面的交线（大圆）重合，各自通有大小相等的电流，电流的流向如图中箭头所示。坐标原点处的磁场方向与 $x$ 轴、 $y$ 轴、 $z$ 轴的夹角分别是 [ ]\n\n[图1]\nA: $-\\arccos \\frac{1}{\\sqrt{3}},-\\arccos \\frac{1}{\\sqrt{3}},-\\arccos \\frac{1}{\\sqrt{3}}$\nB: $\\arcsin \\frac{1}{\\sqrt{3}},-\\arcsin \\frac{1}{\\sqrt{3}},-\\arcsin \\frac{1}{\\sqrt{3}}$\nC: $\\arcsin \\frac{1}{\\sqrt{3}}, \\arcsin \\frac{1}{\\sqrt{3}},-\\arcsin \\frac{1}{\\sqrt{3}}$\nD: $\\arccos \\frac{1}{\\sqrt{3}}, \\arccos \\frac{1}{\\sqrt{3}}, \\arccos \\frac{1}{\\sqrt{3}}$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n如图, 球心在坐标原点 $\\mathrm{O}$ 的球面上有三个彼此绝缘的金属环, 它们分别与 $x-y$ 平面、 $y-z$ 平面、 $z-x$ 平面与球面的交线（大圆）重合，各自通有大小相等的电流，电流的流向如图中箭头所示。坐标原点处的磁场方向与 $x$ 轴、 $y$ 轴、 $z$ 轴的夹角分别是 [ ]\n\n[图1]\n\nA: $-\\arccos \\frac{1}{\\sqrt{3}},-\\arccos \\frac{1}{\\sqrt{3}},-\\arccos \\frac{1}{\\sqrt{3}}$\nB: $\\arcsin \\frac{1}{\\sqrt{3}},-\\arcsin \\frac{1}{\\sqrt{3}},-\\arcsin \\frac{1}{\\sqrt{3}}$\nC: $\\arcsin \\frac{1}{\\sqrt{3}}, \\arcsin \\frac{1}{\\sqrt{3}},-\\arcsin \\frac{1}{\\sqrt{3}}$\nD: $\\arccos \\frac{1}{\\sqrt{3}}, \\arccos \\frac{1}{\\sqrt{3}}, \\arccos \\frac{1}{\\sqrt{3}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_17b1131fe8d911867aa0g-01.jpg?height=331&width=363&top_left_y=820&top_left_x=1486" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_415", "problem": "In the future, as the miniaturisation of microprocessors continues, an obvious limit to the technology is when the size of one active unit in a device approaches the size of an atom. We will explore, speculatively, an aspect of passing a current into and out of the structure of graphene.\n\nThe following Figures represent part of an infinite graphene layer for which we wish to find the measured resistance between a pair of atoms located at points $\\mathbf{A}$ and $\\mathbf{B}$. Each straight line (bond) between two nodes represents a resistance $R$.\n\nFigure: The single layer hexagonal structure of graphene.\n(a)\n[figure1]\n\n(b)\n[figure2]\n\n(c)\n[figure3]\n\nNow, with current $I$ flowing into the network at $\\mathbf{A}$ and the same current flowing out of the network at $\\mathbf{B}$, state the current flowing from $\\mathbf{A}$ to $\\mathbf{B}$ by all other routes through the infinite network (i.e. not through resistor $\\mathbf{A B}$ ).", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn the future, as the miniaturisation of microprocessors continues, an obvious limit to the technology is when the size of one active unit in a device approaches the size of an atom. We will explore, speculatively, an aspect of passing a current into and out of the structure of graphene.\n\nThe following Figures represent part of an infinite graphene layer for which we wish to find the measured resistance between a pair of atoms located at points $\\mathbf{A}$ and $\\mathbf{B}$. Each straight line (bond) between two nodes represents a resistance $R$.\n\nFigure: The single layer hexagonal structure of graphene.\n(a)\n[figure1]\n\n(b)\n[figure2]\n\n(c)\n[figure3]\n\nNow, with current $I$ flowing into the network at $\\mathbf{A}$ and the same current flowing out of the network at $\\mathbf{B}$, state the current flowing from $\\mathbf{A}$ to $\\mathbf{B}$ by all other routes through the infinite network (i.e. not through resistor $\\mathbf{A B}$ ).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-7.jpg?height=360&width=366&top_left_y=188&top_left_x=434", "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-7.jpg?height=363&width=366&top_left_y=187&top_left_x=868", "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-7.jpg?height=365&width=349&top_left_y=183&top_left_x=1276" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_655", "problem": "A spaceship moves around a planet in a circular orbit as shown in Fig. 2. At this instant, the spaceship's engines turn on for a time much shorter than the orbital period of the spaceship, and accelerate the spaceship in the direction of its velocity vector. The new orbit of the spaceship is most like:\n\na)\n\n[figure1]\n\nb)\n\n[figure2]\n\nc)\n\n[figure3]\n\nd)\n\n[figure4]\nA: A\nB: B\nC: C\nD: D\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA spaceship moves around a planet in a circular orbit as shown in Fig. 2. At this instant, the spaceship's engines turn on for a time much shorter than the orbital period of the spaceship, and accelerate the spaceship in the direction of its velocity vector. The new orbit of the spaceship is most like:\n\na)\n\n[figure1]\n\nb)\n\n[figure2]\n\nc)\n\n[figure3]\n\nd)\n\n[figure4]\n\nA: A\nB: B\nC: C\nD: D\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-03.jpg?height=232&width=414&top_left_y=2277&top_left_x=1141", "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-04.jpg?height=308&width=234&top_left_y=297&top_left_x=266", "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-04.jpg?height=351&width=352&top_left_y=704&top_left_x=241", "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-04.jpg?height=422&width=302&top_left_y=1174&top_left_x=246" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_936", "problem": "A circuit of two resistors $R$ and $R_{\\mathrm{C}}$ in series is connected to a supply as shown in Fig. 5. The potentials at three points are marked as $0 \\mathrm{~V}, V_{\\mathrm{A}}, V_{\\mathrm{B}}$. The current $I$ in the circuit depends upon the value of $R_{\\mathrm{C}}$.\n\n[figure1]\n\n Obtain a relation between $V_{\\mathrm{A}}, V_{\\mathrm{B}}, I$ and $R$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA circuit of two resistors $R$ and $R_{\\mathrm{C}}$ in series is connected to a supply as shown in Fig. 5. The potentials at three points are marked as $0 \\mathrm{~V}, V_{\\mathrm{A}}, V_{\\mathrm{B}}$. The current $I$ in the circuit depends upon the value of $R_{\\mathrm{C}}$.\n\n[figure1]\n\n Obtain a relation between $V_{\\mathrm{A}}, V_{\\mathrm{B}}, I$ and $R$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-07.jpg?height=334&width=642&top_left_y=1880&top_left_x=707" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_525", "problem": "For large $V_{0}$, the velocity of the positive charges is determined by a strong drag force, so that\n\n$$\nv=\\mu E\n$$\n\nwhere $E$ is the local electric field and $\\mu$ is the charge mobility.\n\nIn the steady state, the current is the same everywhere. Consider the region $(x, x+d x)$. The time it takes for the charge in the second region to leave is $\\frac{\\mathrm{d} x}{v(x)}$. The amount of charge that leaves is $\\rho A \\mathrm{~d} x$. The current is thus given by $\\rho A v$, so $\\rho v$ is constant. Alternatively, one can write this as\n\n$$\nv \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}+\\rho \\frac{\\mathrm{d} v}{\\mathrm{~d} x}=0\n$$\n\nThe position $x$ is effectively in between two uniform sheets of charge density. The sheet on the left has charge density $\\int_{0}^{x} \\rho \\mathrm{d} x+\\sigma_{0}$, where $\\sigma_{0}$ is the charge density on the left plate, and the sheet on the right has charge density $\\int_{x}^{d} \\rho \\mathrm{d} x+\\sigma_{d}$, where $\\sigma_{d}$ is the charge density on the left plate. Then, the electric field is given by\n\n$$\nE=\\sigma_{0} /\\left(2 \\epsilon_{0}\\right)+\\int_{0}^{x} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\int_{x}^{d} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\sigma_{d} /\\left(2 \\epsilon_{0}\\right)\n$$\n\nThen, by the Fundamental Theorem of Calculus\n\n$$\n\\frac{d E}{d x}=\\frac{\\rho}{\\epsilon_{0}}\n$$\nso\n\n$$\n\\frac{d^{2} V}{d x^{2}}=-\\frac{\\rho}{\\epsilon_{0}}\n$$\n\nWe have that\n\n$$\n\\rho \\frac{d v}{d x}+v \\frac{d \\rho}{d x}=0\n$$\n\nand now that $v=-\\mu \\frac{d V}{d x}$, so substituting in Poisson's equation gives us that\n\n$$\n\\left(\\frac{d^{2} V}{d x^{2}}\\right)^{2}+\\frac{d V}{d x}\\left(\\frac{d^{3} V}{d x^{3}}\\right)=0\n$$\n\nUsing $V(x)=-V_{0}(x / d)^{b}$ gives\n\n$$\nb(b-1) b(b-1)=-b b(b-1)(b-2) \\text {. }\n$$\n\nThe solution with $b=0$ cannot satisfy the boundary conditions, while $b=1$ has zero current. Assuming $b$ is neither of these values, we have $b-1=-(b-2)$, so $b=3 / 2$. Substituting gives\n\n$$\nv=-\\frac{3 V_{0} \\mu x^{1 / 2}}{2 d^{3 / 2}}\n$$\n\nand\n\n$$\n\\rho=-\\frac{3 V_{0} \\epsilon_{0}}{4 d^{3 / 2} x^{1 / 2}}\n$$\n\nso\n\n$$\nI=\\rho A v=\\frac{9 \\epsilon_{0} \\mu A V_{0}^{2}}{8 d^{3}}\n$$\n\nwith the current flowing from left to right.Two large parallel plates of area $A$ are placed at $x=0$ and $x=d \\ll \\sqrt{A}$ in a semiconductor medium. The plate at $x=0$ is grounded, and the plate at $x=d$ is at a fixed potential $-V_{0}$, where $V_{0}>0$. Particles of positive charge $q$ flow between the two plates. You may neglect any dielectric effects of the medium.\n\nFor small $V_{0}$, the positive charges move by diffusion. The current due to diffusion is given by Fick's Law,\n\n$$\nI=-A D \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}\n$$\n\nHere, $D$ is the diffusion constant, which you can assume to be described by the Einstein relation\n\n$$\nD=\\frac{\\mu k_{B} T}{q}\n$$\n\nwhere $T$ is the temperature of the system.\n\nAt roughly what voltage $V_{0}$ does the system transition from this regime to the high voltage regime of the previous part?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nFor large $V_{0}$, the velocity of the positive charges is determined by a strong drag force, so that\n\n$$\nv=\\mu E\n$$\n\nwhere $E$ is the local electric field and $\\mu$ is the charge mobility.\n\nIn the steady state, the current is the same everywhere. Consider the region $(x, x+d x)$. The time it takes for the charge in the second region to leave is $\\frac{\\mathrm{d} x}{v(x)}$. The amount of charge that leaves is $\\rho A \\mathrm{~d} x$. The current is thus given by $\\rho A v$, so $\\rho v$ is constant. Alternatively, one can write this as\n\n$$\nv \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}+\\rho \\frac{\\mathrm{d} v}{\\mathrm{~d} x}=0\n$$\n\nThe position $x$ is effectively in between two uniform sheets of charge density. The sheet on the left has charge density $\\int_{0}^{x} \\rho \\mathrm{d} x+\\sigma_{0}$, where $\\sigma_{0}$ is the charge density on the left plate, and the sheet on the right has charge density $\\int_{x}^{d} \\rho \\mathrm{d} x+\\sigma_{d}$, where $\\sigma_{d}$ is the charge density on the left plate. Then, the electric field is given by\n\n$$\nE=\\sigma_{0} /\\left(2 \\epsilon_{0}\\right)+\\int_{0}^{x} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\int_{x}^{d} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\sigma_{d} /\\left(2 \\epsilon_{0}\\right)\n$$\n\nThen, by the Fundamental Theorem of Calculus\n\n$$\n\\frac{d E}{d x}=\\frac{\\rho}{\\epsilon_{0}}\n$$\nso\n\n$$\n\\frac{d^{2} V}{d x^{2}}=-\\frac{\\rho}{\\epsilon_{0}}\n$$\n\nWe have that\n\n$$\n\\rho \\frac{d v}{d x}+v \\frac{d \\rho}{d x}=0\n$$\n\nand now that $v=-\\mu \\frac{d V}{d x}$, so substituting in Poisson's equation gives us that\n\n$$\n\\left(\\frac{d^{2} V}{d x^{2}}\\right)^{2}+\\frac{d V}{d x}\\left(\\frac{d^{3} V}{d x^{3}}\\right)=0\n$$\n\nUsing $V(x)=-V_{0}(x / d)^{b}$ gives\n\n$$\nb(b-1) b(b-1)=-b b(b-1)(b-2) \\text {. }\n$$\n\nThe solution with $b=0$ cannot satisfy the boundary conditions, while $b=1$ has zero current. Assuming $b$ is neither of these values, we have $b-1=-(b-2)$, so $b=3 / 2$. Substituting gives\n\n$$\nv=-\\frac{3 V_{0} \\mu x^{1 / 2}}{2 d^{3 / 2}}\n$$\n\nand\n\n$$\n\\rho=-\\frac{3 V_{0} \\epsilon_{0}}{4 d^{3 / 2} x^{1 / 2}}\n$$\n\nso\n\n$$\nI=\\rho A v=\\frac{9 \\epsilon_{0} \\mu A V_{0}^{2}}{8 d^{3}}\n$$\n\nwith the current flowing from left to right.\n\nproblem:\nTwo large parallel plates of area $A$ are placed at $x=0$ and $x=d \\ll \\sqrt{A}$ in a semiconductor medium. The plate at $x=0$ is grounded, and the plate at $x=d$ is at a fixed potential $-V_{0}$, where $V_{0}>0$. Particles of positive charge $q$ flow between the two plates. You may neglect any dielectric effects of the medium.\n\nFor small $V_{0}$, the positive charges move by diffusion. The current due to diffusion is given by Fick's Law,\n\n$$\nI=-A D \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}\n$$\n\nHere, $D$ is the diffusion constant, which you can assume to be described by the Einstein relation\n\n$$\nD=\\frac{\\mu k_{B} T}{q}\n$$\n\nwhere $T$ is the temperature of the system.\n\nAt roughly what voltage $V_{0}$ does the system transition from this regime to the high voltage regime of the previous part?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_678", "problem": "Charged particles with mass $m$ and charge $Q$ are injected with speed $v$ into a region with a constant magnetic field. The particles are injected in a direction perpendicular to the magnetic field lines and are found to move in circular orbits with period $T$. If the experiment is repeated for particles with mass $2 m$, charge $Q / 4$, while the magnetic field strength is tripled, what is the orbital period of the particles in the new experiment?\nA: $8 \\mathrm{~T} / 3$\nB: $3 \\mathrm{~T} / 2$\nC: $4 \\mathrm{~T} / 3$\nD: $2 \\mathrm{~T}$\nE: $\\mathrm{T} / 3$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nCharged particles with mass $m$ and charge $Q$ are injected with speed $v$ into a region with a constant magnetic field. The particles are injected in a direction perpendicular to the magnetic field lines and are found to move in circular orbits with period $T$. If the experiment is repeated for particles with mass $2 m$, charge $Q / 4$, while the magnetic field strength is tripled, what is the orbital period of the particles in the new experiment?\n\nA: $8 \\mathrm{~T} / 3$\nB: $3 \\mathrm{~T} / 2$\nC: $4 \\mathrm{~T} / 3$\nD: $2 \\mathrm{~T}$\nE: $\\mathrm{T} / 3$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_659", "problem": "An inflated balloon, filled with a gas of temperature $\\mathrm{T}$, is held in a room with the same temperature. If we make a small hole on the side of the balloon and let the gas escape, which statement is true about the temperature T' of the gas right after leaving the balloon?\nA: $T^{\\prime}>T$\nB: $T^{\\prime}=T$\nC: $T^{\\prime}T$\nB: $T^{\\prime}=T$\nC: $T^{\\prime}a_{m}$, 而\n\n$$\n\\alpha_{m}=\\left(\\frac{G M_{e}}{\\omega_{e}^{2} R_{e}^{3}}\\right)^{1 / 3}-1\n$$\n\n$M_{e}$ 和 $W_{e}$ 分别为地球的质量和自转角速度, $G$ 为引力常数. 设想从供应站到飞行器有一根用于运送物资的刚性、管壁匀质、质量为 $m_{p}$ 的坚直输送管, 输送管下端固定在地面上, 并设法保持输送管与地面始终垂直. 推送物资时, 把物资放进输送管下端内的平底托盘上, 沿管壁向上推进, 并保持托盘运行速度不致过大. 忽略托盘与管壁之间的摩擦力, 考虑地球的自转, 但不考虑地球的公转. 设某次所推送物资和托盘的总质量为 $m$.在把物资从供应站送到飞行器的过程中, 外推力至少需要做多少正功?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n太空中有一飞行器靠其自身动力维持在地球赤道的正上方 $L=a R_{e}$ 处, 相对于赤道上的一地面物资供应站保持静止. 这里, $R_{e}$ 为地球的半径, $a$ 为常数, $a>a_{m}$, 而\n\n$$\n\\alpha_{m}=\\left(\\frac{G M_{e}}{\\omega_{e}^{2} R_{e}^{3}}\\right)^{1 / 3}-1\n$$\n\n$M_{e}$ 和 $W_{e}$ 分别为地球的质量和自转角速度, $G$ 为引力常数. 设想从供应站到飞行器有一根用于运送物资的刚性、管壁匀质、质量为 $m_{p}$ 的坚直输送管, 输送管下端固定在地面上, 并设法保持输送管与地面始终垂直. 推送物资时, 把物资放进输送管下端内的平底托盘上, 沿管壁向上推进, 并保持托盘运行速度不致过大. 忽略托盘与管壁之间的摩擦力, 考虑地球的自转, 但不考虑地球的公转. 设某次所推送物资和托盘的总质量为 $m$.\n\n问题:\n在把物资从供应站送到飞行器的过程中, 外推力至少需要做多少正功?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_320", "problem": "A space station at a geostationary orbit has a form of a cylinder of length $L=100 \\mathrm{~km}$ and radius $R=1 \\mathrm{~km}$ is filled with air (molar mass $M=29 \\mathrm{~g} / \\mathrm{mol}$ ) at the atmospheric pressure and temperatur $T=295 \\mathrm{~K}$ and the cylindrical walls serve as ground for the people living inside. It rotates around its axis so as to create normal gravity $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$ at the \"ground\"\n\nAssuming that at the point $A$ the rope meets the \"ground\" at an angle $\\alpha$, de termine the ratio of the tension forces $T_{A} / T_{C}$", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA space station at a geostationary orbit has a form of a cylinder of length $L=100 \\mathrm{~km}$ and radius $R=1 \\mathrm{~km}$ is filled with air (molar mass $M=29 \\mathrm{~g} / \\mathrm{mol}$ ) at the atmospheric pressure and temperatur $T=295 \\mathrm{~K}$ and the cylindrical walls serve as ground for the people living inside. It rotates around its axis so as to create normal gravity $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$ at the \"ground\"\n\nAssuming that at the point $A$ the rope meets the \"ground\" at an angle $\\alpha$, de termine the ratio of the tension forces $T_{A} / T_{C}$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_756", "problem": "Superconducting mesh \n\nConsider a mesh made from a flat superconducting sheet by drilling a dense grid of small holes into it. Initially the sheet is in a non-superconducting state, and a magnetic dipole of dipole moment $m$ is at a distance $a$ from the mesh pointing perpendicularly towards the mesh. Now the mesh is cooled so that it becomes superconducting. Next, the dipole is displaced perpendicularly to the surface of the mesh so that its new distance from the mesh is $b$. Find the force between the mesh and the dipole. The pitch of the grid of holes is much smaller than both $a$ and $b$, and the linear size of the sheet is much larger than both $a$ and $b$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nSuperconducting mesh \n\nConsider a mesh made from a flat superconducting sheet by drilling a dense grid of small holes into it. Initially the sheet is in a non-superconducting state, and a magnetic dipole of dipole moment $m$ is at a distance $a$ from the mesh pointing perpendicularly towards the mesh. Now the mesh is cooled so that it becomes superconducting. Next, the dipole is displaced perpendicularly to the surface of the mesh so that its new distance from the mesh is $b$. Find the force between the mesh and the dipole. The pitch of the grid of holes is much smaller than both $a$ and $b$, and the linear size of the sheet is much larger than both $a$ and $b$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_948", "problem": "Gamma radiation such as that from a Co-60 source is a penetrating radiation which requires shielding for safety purposes. The radiation is reduced in intensity when it passes through a material by a factor\n\n$$\nS=2^{-\\frac{x}{a}}\n$$\n\nwhere $x$ is the distance travelled through the material and $a$ is a constant which depends on the material and gamma ray energy.\n\nWhat thickness $x$ of lead will reduce the intensity of the same gamma rays to $\\frac{1}{8}^{\\text {th }}$ that of concrete of thickness $y=1.0 \\mathrm{~m}$ ?\n\nFor gamma rays produced by cobalt-60:", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nGamma radiation such as that from a Co-60 source is a penetrating radiation which requires shielding for safety purposes. The radiation is reduced in intensity when it passes through a material by a factor\n\n$$\nS=2^{-\\frac{x}{a}}\n$$\n\nwhere $x$ is the distance travelled through the material and $a$ is a constant which depends on the material and gamma ray energy.\n\nWhat thickness $x$ of lead will reduce the intensity of the same gamma rays to $\\frac{1}{8}^{\\text {th }}$ that of concrete of thickness $y=1.0 \\mathrm{~m}$ ?\n\nFor gamma rays produced by cobalt-60:\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of mm, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "mm" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1709", "problem": "光在物体表面反射或者被吸收时, 光子将其动量传给物体, 产生光的辐射压力（光压）。利用光压, 可实现对某些微小粒子的精确操控 (光镊)。设在 $|y| \\geq L$ 的区域有一匀强激光场,沿 $z$ 轴的负方向入射, 其强度 (单位时间内通过单位横截面积的光能) 为 $I$; 在 $y \\in(-L, L)$ 之间没有光场, 其横截面如图所示。一个密度为 $\\rho$ 的三棱柱形小物体 (其横截面是底边长为 $2 L$ 、底角为 $\\theta$ 的等腰三角形) 被置于光滑的水平面 $x y$ 上, 其朝上的A和B两面涂有特殊反射层,能完全反射入射光。小物体初始时静止, 位置如图。\n\n[图1]假定光子的反射角等于入射角, 且反射前后光子的频率不变。小物体从初始位置向 $y$ 轴负方向移动 $3 L / 2$ 的距离所需的时间", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n光在物体表面反射或者被吸收时, 光子将其动量传给物体, 产生光的辐射压力（光压）。利用光压, 可实现对某些微小粒子的精确操控 (光镊)。设在 $|y| \\geq L$ 的区域有一匀强激光场,沿 $z$ 轴的负方向入射, 其强度 (单位时间内通过单位横截面积的光能) 为 $I$; 在 $y \\in(-L, L)$ 之间没有光场, 其横截面如图所示。一个密度为 $\\rho$ 的三棱柱形小物体 (其横截面是底边长为 $2 L$ 、底角为 $\\theta$ 的等腰三角形) 被置于光滑的水平面 $x y$ 上, 其朝上的A和B两面涂有特殊反射层,能完全反射入射光。小物体初始时静止, 位置如图。\n\n[图1]\n\n问题:\n假定光子的反射角等于入射角, 且反射前后光子的频率不变。小物体从初始位置向 $y$ 轴负方向移动 $3 L / 2$ 的距离所需的时间\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_07aa406e17d01fd01b36g-02.jpg?height=380&width=1176&top_left_y=2540&top_left_x=400" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_75", "problem": "Before testing the circuit shown at right, and taking the current and voltage readings, a student is asked to predict what the meters will read at each location indicated. What will be the ammeter reading at $A_{2}$ ?\n[figure1]\nA: $2.0 \\mathrm{~A}$\nB: $12 \\mathrm{~A}$\nC: $8.0 \\mathrm{~A}$\nD: $6.0 \\mathrm{~A}$\nE: $4.0 \\mathrm{~A}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nBefore testing the circuit shown at right, and taking the current and voltage readings, a student is asked to predict what the meters will read at each location indicated. What will be the ammeter reading at $A_{2}$ ?\n[figure1]\n\nA: $2.0 \\mathrm{~A}$\nB: $12 \\mathrm{~A}$\nC: $8.0 \\mathrm{~A}$\nD: $6.0 \\mathrm{~A}$\nE: $4.0 \\mathrm{~A}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ae7e25be7efc2df26f6eg-07.jpg?height=393&width=648&top_left_y=714&top_left_x=1237" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_895", "problem": "Calculation in the previous part shows that in order to build the space elevator, it is neccessary to have light materials with very high tensile strength. Carbon nanotubes are materials that meet such requirements because of strong chemical bondings between very light atoms. Two natural polymorphs of carbon are diamond and graphite. In diamond every carbon atom is surrounded by four nearest neighbor (NN) atoms to form a tetrahedron. Graphite has a layer structure. In each layer, carbon atoms are arranged in a hexagonal plane lattice with three NNs. Although diamond is known as the hardest materials, covalent bondings between carbon atoms in hexagonal layers of graphite is stronger than those between carbon atoms in diamond tetrahedra. Graphite is much softer than diamond because of the van der Waals bonding between carbon atoms of different layers, which is much weaker than covalent bonding.\n\n[figure1]\n\nFigure 2. Graphite structure\n(a)\n[figure2]\n\nFigure 3. Graphene (a) and carbon nanotube (b).\n\nA monatomic layer in graphite is called graphene and has monoatomic thickness. Isolated graphene sheet is not stable and has a tendency to roll up to form carbon spheres or carbon nanotubes. The hexagonal crystal lattice of graphene is depicted in Fig. 4. The distance between two NN carbon atoms is $a=0.142 \\mathrm{~nm}$ and the distance between two closest parallel bondings is $b=0.246 \\mathrm{~nm}$. Because the covalent bondings between carbon atoms in graphene are very strong, mechanical properties of carbon nanotubes are very special. They have an extremely large Young's modulus and tensile strength, as well as a very light density. Young's modulus is defined as the ratio of the stress along an axis to the strain (ratio of deformation over initial length) along that axis in the range of stress in which Hooke's law holds.\n\n[figure3]\n\nFigure 4. Graphene.\n\n## Theory\n\n[figure4]\n\nFigure 5. An illustration of a carbon nanotube with 9 carbon-carbon parallel bondings. Note: In this problem, there are 27 carbon-carbon parallel bondings. (1) parallel bond; (2) slanted bond; (3) tube axis.\n\nNow we examine some mechanical properties of a carbon nanotube having 27 carbon-carbon bondings parallel to the tube axis (for an illustration, see Figure 5). The bonding between two carbon atoms can be described by the Morse potential $V(x)=V_{0}\\left(e^{-4 \\frac{x}{a}}-2 e^{-2 \\frac{x}{a}}\\right)$. Here $a=0.142 \\mathrm{~nm}$ is the equilibrium distance between two $\\mathrm{NN}$ carbon atoms, $V_{0}=4.93 \\mathrm{eV}$ is the bonding energy, and $x$ is the displacement of the atom from the equilibrium position. Hereafter, we approximate the Morse potential by a quadratic potential $V(x)=P+Q x^{2}$. All non-nearest-neighbor interactions are neglected. In this approximation, one can propose that carbon atoms are bonded through \"springs\" with the spring constant $k$. Changes in angles between bonds are neglected.\n\nCalculate the value of the spring constant $k$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nCalculation in the previous part shows that in order to build the space elevator, it is neccessary to have light materials with very high tensile strength. Carbon nanotubes are materials that meet such requirements because of strong chemical bondings between very light atoms. Two natural polymorphs of carbon are diamond and graphite. In diamond every carbon atom is surrounded by four nearest neighbor (NN) atoms to form a tetrahedron. Graphite has a layer structure. In each layer, carbon atoms are arranged in a hexagonal plane lattice with three NNs. Although diamond is known as the hardest materials, covalent bondings between carbon atoms in hexagonal layers of graphite is stronger than those between carbon atoms in diamond tetrahedra. Graphite is much softer than diamond because of the van der Waals bonding between carbon atoms of different layers, which is much weaker than covalent bonding.\n\n[figure1]\n\nFigure 2. Graphite structure\n(a)\n[figure2]\n\nFigure 3. Graphene (a) and carbon nanotube (b).\n\nA monatomic layer in graphite is called graphene and has monoatomic thickness. Isolated graphene sheet is not stable and has a tendency to roll up to form carbon spheres or carbon nanotubes. The hexagonal crystal lattice of graphene is depicted in Fig. 4. The distance between two NN carbon atoms is $a=0.142 \\mathrm{~nm}$ and the distance between two closest parallel bondings is $b=0.246 \\mathrm{~nm}$. Because the covalent bondings between carbon atoms in graphene are very strong, mechanical properties of carbon nanotubes are very special. They have an extremely large Young's modulus and tensile strength, as well as a very light density. Young's modulus is defined as the ratio of the stress along an axis to the strain (ratio of deformation over initial length) along that axis in the range of stress in which Hooke's law holds.\n\n[figure3]\n\nFigure 4. Graphene.\n\n## Theory\n\n[figure4]\n\nFigure 5. An illustration of a carbon nanotube with 9 carbon-carbon parallel bondings. Note: In this problem, there are 27 carbon-carbon parallel bondings. (1) parallel bond; (2) slanted bond; (3) tube axis.\n\nNow we examine some mechanical properties of a carbon nanotube having 27 carbon-carbon bondings parallel to the tube axis (for an illustration, see Figure 5). The bonding between two carbon atoms can be described by the Morse potential $V(x)=V_{0}\\left(e^{-4 \\frac{x}{a}}-2 e^{-2 \\frac{x}{a}}\\right)$. Here $a=0.142 \\mathrm{~nm}$ is the equilibrium distance between two $\\mathrm{NN}$ carbon atoms, $V_{0}=4.93 \\mathrm{eV}$ is the bonding energy, and $x$ is the displacement of the atom from the equilibrium position. Hereafter, we approximate the Morse potential by a quadratic potential $V(x)=P+Q x^{2}$. All non-nearest-neighbor interactions are neglected. In this approximation, one can propose that carbon atoms are bonded through \"springs\" with the spring constant $k$. Changes in angles between bonds are neglected.\n\nCalculate the value of the spring constant $k$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $N m^{-1}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-2.jpg?height=549&width=525&top_left_y=1850&top_left_x=731", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-3.jpg?height=430&width=900&top_left_y=710&top_left_x=518", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-3.jpg?height=451&width=691&top_left_y=1856&top_left_x=660", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-4.jpg?height=944&width=1144&top_left_y=172&top_left_x=184" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$N m^{-1}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1670", "problem": "电磁驱动是与炮弹发射、航空母舰上飞机弹射起飞有关的一种新型驱动方式. 电磁驱动的原理如图所示, 当直\n\n[图1]\n流电流突然加到一固定线圈上, 可以将置于线圈上的环弹射出去. 现在同一个固定线圈上,先后置有分别用铜、铝和硅制成的形状、大小和横截面积均相同的三种环; 当电流突然接通时, 它们所受到的推力分别为 $F_{1} 、 F_{2}$ 和 $F_{3}$. 若环的重力可忽略, 下列说法正确的是\nA: $F_{1}>F_{2}>F_{3}$\nB: $F_{2}>F_{3}>F_{1}$\nC: $F_{3}>F_{2}>F_{1}$\nD: $F_{1}=F_{2}=F_{3}$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n电磁驱动是与炮弹发射、航空母舰上飞机弹射起飞有关的一种新型驱动方式. 电磁驱动的原理如图所示, 当直\n\n[图1]\n流电流突然加到一固定线圈上, 可以将置于线圈上的环弹射出去. 现在同一个固定线圈上,先后置有分别用铜、铝和硅制成的形状、大小和横截面积均相同的三种环; 当电流突然接通时, 它们所受到的推力分别为 $F_{1} 、 F_{2}$ 和 $F_{3}$. 若环的重力可忽略, 下列说法正确的是\n\nA: $F_{1}>F_{2}>F_{3}$\nB: $F_{2}>F_{3}>F_{1}$\nC: $F_{3}>F_{2}>F_{1}$\nD: $F_{1}=F_{2}=F_{3}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_58b1fc45927d60138a23g-01.jpg?height=315&width=440&top_left_y=2558&top_left_x=1348" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1435", "problem": "在太阳内部存在两个主要的核聚变反应过程:环和质子-质子循环；其中碳循环是贝蒂在 1938 年提出的,环反应过程如图所示。图中 $\\mathrm{P} 、 \\mathrm{e}^{+}$和 $v_{\\mathrm{e}}$ 分别表示质子、正电电子型中微子; 粗箭头表示循环反应进行的先后次序。当从图顶端开始, 质子 $\\mathrm{p}$ 与 ${ }^{12} \\mathrm{C}$ 核发生反应生成 ${ }^{13} \\mathrm{~N}$ 核, 反应按粗所示的次序进行, 直到完成一个循环后, 重新开始下一个循知 $\\mathrm{e}^{+} 、 \\mathrm{p}$ 和 $\\mathrm{He}$ 核的质量分别为 $0.511 \\mathrm{MeV} / \\mathrm{c}^{2} 、 1.0078 \\mathrm{u}$ 和\n\n[图1]\n\n碳 循碳 循子 和循 环箭 头环。已 4.0026 $\\mathrm{u}\\left(1 \\mathrm{u} \\approx 931.494 \\mathrm{MeV} / \\mathrm{c}^{2}\\right)$, 电子型中微子 $v_{\\mathrm{e}}$ 的质量可以忽略。计算完成一个碳循环过程释放的核能。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n在太阳内部存在两个主要的核聚变反应过程:环和质子-质子循环；其中碳循环是贝蒂在 1938 年提出的,环反应过程如图所示。图中 $\\mathrm{P} 、 \\mathrm{e}^{+}$和 $v_{\\mathrm{e}}$ 分别表示质子、正电电子型中微子; 粗箭头表示循环反应进行的先后次序。当从图顶端开始, 质子 $\\mathrm{p}$ 与 ${ }^{12} \\mathrm{C}$ 核发生反应生成 ${ }^{13} \\mathrm{~N}$ 核, 反应按粗所示的次序进行, 直到完成一个循环后, 重新开始下一个循知 $\\mathrm{e}^{+} 、 \\mathrm{p}$ 和 $\\mathrm{He}$ 核的质量分别为 $0.511 \\mathrm{MeV} / \\mathrm{c}^{2} 、 1.0078 \\mathrm{u}$ 和\n\n[图1]\n\n碳 循碳 循子 和循 环箭 头环。已 4.0026 $\\mathrm{u}\\left(1 \\mathrm{u} \\approx 931.494 \\mathrm{MeV} / \\mathrm{c}^{2}\\right)$, 电子型中微子 $v_{\\mathrm{e}}$ 的质量可以忽略。\n\n问题:\n计算完成一个碳循环过程释放的核能。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以MeV为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e3a9fdbbef225ad3aefbg-01.jpg?height=534&width=437&top_left_y=732&top_left_x=1301" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "MeV" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1486", "problem": "熟练的荡秋千的人能够通过在秋千板上适时站起和蹲下使秋千越荡越高。一质量为 $m$ 的人荡一架底板和摆杆均为刚性的秋千，底板和摆杆的质量均可忽略，假定人的质量集中在其质心。人在秋千上每次完全站起时其质心距悬点 $\\mathrm{O}$ 的距离为 $l$, 完全蹲下时此距离变为 $l+d$ 。实际上,人在秋千上站起和蹲下过程都是在一段时间内完成的。作为一个简单的模型，假设人在第 1 个最高点 $\\mathrm{A}$ 点从完全站立的姿势迅速完全下蹲, 然后荡至最低点 $\\mathrm{B}, \\mathrm{A}$ 与 $\\mathrm{B}$ 的高度差为 $h_{1}$; 随后他在 $\\mathrm{B}$ 点迅速完全站起 (且最终径向速度为零), 继而随秋千荡至第 2 个最高点 $\\mathrm{C}$, 这一过程中该人质心运动的轨迹如图所示。此后人以同样的方式回荡, 重复前述过程, 荡向第 3、4 等最高点。假设人在站起和蹲下的过程中, 人与秋千的相互作用力始终与摆杆平行。以最低点 $\\mathrm{B}$ 为重力势能零点。\n\n[图1]\n\n假定在始终完全蹲下和始终完全站立过程中的机械能损失 $\\Delta E$ 与过程前后高度差的绝对值 $\\Delta h$ 的关系分别为\n\n$$\n\\begin{array}{lll}\n\\Delta E=k_{1} m g\\left(h_{0}+\\Delta h\\right), & 0M$. Thus if $\\beta \\leq 1$, the time $t$ is infinite; the ball never returns to its starting position.\n\nAssuming that $\\beta>1$, the ball must move a distance $x$ towards its original collision point, where we have neglected the time taken for the collisions themselves. Thus the time for the ball to return to its original position horizontally is\n\n$$\nt_{2}=-\\frac{x}{v_{3}}=\\frac{2 v_{0}}{g} \\sqrt{\\frac{m^{2} M}{(M+m)(M-m)^{2}}}\n$$\n\nThe total time since the first collision is\n\n$$\nt=t_{1}+t_{2}=\\frac{2 v_{0}}{g} \\sqrt{\\frac{M}{m+M}}\\left(\\frac{M}{M-m}\\right)=\\frac{2 v_{0}}{g} \\sqrt{\\frac{\\beta}{1+\\beta}}\\left(\\frac{\\beta}{\\beta-1}\\right)\n$$\n\nwhere $\\beta>1$.A block of mass $M$ has a hole drilled through it so that a ball of mass $m$ can enter horizontally and then pass through the block and exit vertically upward. The ball and block are located on a frictionless surface; the block is originally at rest.\n\n[figure1]\n\nfrictionless horizontal surface\n\nNow consider friction. The ball has moment of inertia $I=\\frac{2}{5} m r^{2}$ and is originally not rotating. When it enters the hole in the block it rubs against one surface so that when it is ejected upwards the ball is rolling without slipping. To what height does the ball rise above the block?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nConsider the scenario where the ball is traveling horizontally with a speed $v_{0}$. The ball enters the block and is ejected out the top of the block. Assume there are no frictional losses as the ball passes through the block, and the ball rises to a height much higher than the dimensions of the block. The ball then returns to the level of the block, where it enters the top hole and then is ejected from the side hole. Determine the time $t$ for the ball to return to the position where the original collision occurs in terms of the mass ratio $\\beta=M / m$, speed $v_{0}$, and acceleration of free fall $g$.\n\nAfter the collision, the ball and block have the same horizontal velocity $v_{1}$. Since the horizontal momentum is conserved,\n\n$$\nv_{1}=\\frac{m}{m+M} v_{0}\n$$\n\nLet $v_{2}$ be the vertical component of the velocity of the ball immediately after the collision. Since there are no frictional losses, conservation of energy yields\n\n$$\n\\frac{1}{2} m v_{0}^{2}=\\frac{1}{2} M v_{1}^{2}+\\frac{1}{2} m\\left(v_{1}^{2}+v_{2}^{2}\\right)\n$$\n\nSince the ball rises to a height much higher than the height of the block, the gravitational potential energy is negligible, so we have ignored it. This assumption also means we can ignore the duration of the collision itself in our calculations below.\n\nPlugging in our result for $v_{1}$,\n\n$$\nm v_{0}^{2}-(M+m)\\left(\\frac{m}{m+M}\\right)^{2} v_{0}^{2}=m v_{2}^{2} \\quad \\Rightarrow \\quad v_{2}=\\sqrt{\\frac{M}{m+M}} v_{0}\n$$\n\nThe time spent by the ball in the air is\n\n$$\nt_{1}=2 v_{2} / g\n$$\n\nThe distance traveled horizontally by the ball while it is in the air is\n\n$$\nx=v_{1} t_{1}=\\frac{2 v_{1} v_{2}}{g}=\\frac{2 v_{0}^{2}}{g} \\sqrt{\\frac{m^{2} M}{(m+M)^{3}}}\n$$\n\nThe ball then falls back into the block and is ejected horizontally.\n\nSince energy and momentum are conserved from before the first collision and after the second, the final horizontal velocity $v_{3}$ of the ball is given by the result for a perfectly elastic collision,\n\n$$\nv_{3}=v_{0} \\frac{m-M}{m+M}\n$$\n\nas can be derived from the conservation laws. Note that $v_{3}$ is positive for $m>M$. Thus if $\\beta \\leq 1$, the time $t$ is infinite; the ball never returns to its starting position.\n\nAssuming that $\\beta>1$, the ball must move a distance $x$ towards its original collision point, where we have neglected the time taken for the collisions themselves. Thus the time for the ball to return to its original position horizontally is\n\n$$\nt_{2}=-\\frac{x}{v_{3}}=\\frac{2 v_{0}}{g} \\sqrt{\\frac{m^{2} M}{(M+m)(M-m)^{2}}}\n$$\n\nThe total time since the first collision is\n\n$$\nt=t_{1}+t_{2}=\\frac{2 v_{0}}{g} \\sqrt{\\frac{M}{m+M}}\\left(\\frac{M}{M-m}\\right)=\\frac{2 v_{0}}{g} \\sqrt{\\frac{\\beta}{1+\\beta}}\\left(\\frac{\\beta}{\\beta-1}\\right)\n$$\n\nwhere $\\beta>1$.\n\nproblem:\nA block of mass $M$ has a hole drilled through it so that a ball of mass $m$ can enter horizontally and then pass through the block and exit vertically upward. The ball and block are located on a frictionless surface; the block is originally at rest.\n\n[figure1]\n\nfrictionless horizontal surface\n\nNow consider friction. The ball has moment of inertia $I=\\frac{2}{5} m r^{2}$ and is originally not rotating. When it enters the hole in the block it rubs against one surface so that when it is ejected upwards the ball is rolling without slipping. To what height does the ball rise above the block?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0256c432f4019b26894dg-17.jpg?height=415&width=1027&top_left_y=562&top_left_x=538" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1288", "problem": "一圆盘沿顺时针方向绕过圆盘中心 $O$ 并与盘面垂直的固定水平转轴以匀角速度 $\\omega=4.43 \\mathrm{rad} / \\mathrm{s}$ 转动. 圆盘半径 $r=1.00 \\mathrm{~m}$, 圆盘正上方有一水平天花板. 设圆盘边缘各处始终有水滴被甩出. 现发现天花板上只有一点处有水. 取重力加速度大小 $g=9.80 \\mathrm{~m} / \\mathrm{s}^{2}$. 求天花板上有水的那一点的位置坐标", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个元组。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一圆盘沿顺时针方向绕过圆盘中心 $O$ 并与盘面垂直的固定水平转轴以匀角速度 $\\omega=4.43 \\mathrm{rad} / \\mathrm{s}$ 转动. 圆盘半径 $r=1.00 \\mathrm{~m}$, 圆盘正上方有一水平天花板. 设圆盘边缘各处始终有水滴被甩出. 现发现天花板上只有一点处有水. 取重力加速度大小 $g=9.80 \\mathrm{~m} / \\mathrm{s}^{2}$. 求\n\n问题:\n天花板上有水的那一点的位置坐标\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以m为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含任何单位的元组，例如ANSWER=(3, 5)", "figure_urls": null, "answer": null, "solution": null, "answer_type": "TUP", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_2", "problem": "An ice skater is rotating with her arms extended. When she pulls in her arms, her rate of rotation increases. No external torques act on the skater. Which of the following statements is true?\nA: Her moment of inertia increased.\nB: Her kinetic energy is conserved.\nC: Her angular momentum decreased.\nD: She does work when pulling in her arms.\nE: Her angular momentum increased.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn ice skater is rotating with her arms extended. When she pulls in her arms, her rate of rotation increases. No external torques act on the skater. Which of the following statements is true?\n\nA: Her moment of inertia increased.\nB: Her kinetic energy is conserved.\nC: Her angular momentum decreased.\nD: She does work when pulling in her arms.\nE: Her angular momentum increased.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_792", "problem": "As part of a recent charity fundraising event, people were asked to donate five-cent coins by placing them onto the surface of a large circle (radius $3 \\mathrm{~m}$ ). At the end of the day, the circle was completely covered by a single layer of coins. Approximately how much money was raised?\nA: Five hundred dollars.\nB: Five thousand dollars.\nC: Fifty thousand dollars.\nD: Five hundred thousand dollars.\nE: Five million dollars.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAs part of a recent charity fundraising event, people were asked to donate five-cent coins by placing them onto the surface of a large circle (radius $3 \\mathrm{~m}$ ). At the end of the day, the circle was completely covered by a single layer of coins. Approximately how much money was raised?\n\nA: Five hundred dollars.\nB: Five thousand dollars.\nC: Fifty thousand dollars.\nD: Five hundred thousand dollars.\nE: Five million dollars.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_477", "problem": "A newly discovered subatomic particle, the $S$ meson, has a mass $M$. When at rest, it lives for exactly $\\tau=3 \\times 10^{-8}$ seconds before decaying into two identical particles called $P$ mesons (peons?) that each have a mass of $\\alpha M$.\n\nIn a reference frame where the $\\mathrm{S}$ meson is at rest, determine the kinetic energy of each $\\mathrm{P}$ meson particle in terms of $M, \\alpha$, the speed of light $c$, and any numerical constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA newly discovered subatomic particle, the $S$ meson, has a mass $M$. When at rest, it lives for exactly $\\tau=3 \\times 10^{-8}$ seconds before decaying into two identical particles called $P$ mesons (peons?) that each have a mass of $\\alpha M$.\n\nIn a reference frame where the $\\mathrm{S}$ meson is at rest, determine the kinetic energy of each $\\mathrm{P}$ meson particle in terms of $M, \\alpha$, the speed of light $c$, and any numerical constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_42", "problem": "A cyclist, using a power meter while on a training ride, checks and sees that she is doing work at the rate of $500 \\mathrm{~W}$. How much average force does her foot exert on the pedals when she is traveling at $8.0 \\mathrm{~m} / \\mathrm{s}$ ?\nA: $31 \\mathrm{~N}$\nB: $63 \\mathrm{~N}$\nC: $80 \\mathrm{~N}$\nD: $320 \\mathrm{~N}$\nE: $710 \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA cyclist, using a power meter while on a training ride, checks and sees that she is doing work at the rate of $500 \\mathrm{~W}$. How much average force does her foot exert on the pedals when she is traveling at $8.0 \\mathrm{~m} / \\mathrm{s}$ ?\n\nA: $31 \\mathrm{~N}$\nB: $63 \\mathrm{~N}$\nC: $80 \\mathrm{~N}$\nD: $320 \\mathrm{~N}$\nE: $710 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1572", "problem": "如图, 左边试管由一段 $24 \\mathrm{~cm}$ 长的水银柱封住一段高为 $60 \\mathrm{~cm}$ 、温度为 $300 \\mathrm{~K}$ 的理想气体柱，上水银面与管侧面小孔相距 $16 \\mathrm{~cm}$ ，小孔右边用一软管连接一空的试管。一控温系统可持续升高或降低被封住的气体柱的温度,当气体温度升高到一定值时水银会从左边试管通过小孔溢出到右边试管中。左边试管坚直放置，右边试管可上下移动，上移时可使右边试管中的水银回流到左边试管内，从而控制左边试管中水银柱的高度。大气压强为 $76 \\mathrm{cmHg}$ 。\n\n[图1]在左边试管中水银上表面与小孔平齐的条件下, 求被封住的气体平衡态的温度 $T$ 与水银柱高度 $x$ 的关系式, 以及该气体平衡态可能的最高温度;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 左边试管由一段 $24 \\mathrm{~cm}$ 长的水银柱封住一段高为 $60 \\mathrm{~cm}$ 、温度为 $300 \\mathrm{~K}$ 的理想气体柱，上水银面与管侧面小孔相距 $16 \\mathrm{~cm}$ ，小孔右边用一软管连接一空的试管。一控温系统可持续升高或降低被封住的气体柱的温度,当气体温度升高到一定值时水银会从左边试管通过小孔溢出到右边试管中。左边试管坚直放置，右边试管可上下移动，上移时可使右边试管中的水银回流到左边试管内，从而控制左边试管中水银柱的高度。大气压强为 $76 \\mathrm{cmHg}$ 。\n\n[图1]\n\n问题:\n在左边试管中水银上表面与小孔平齐的条件下, 求被封住的气体平衡态的温度 $T$ 与水银柱高度 $x$ 的关系式, 以及该气体平衡态可能的最高温度;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[被封住的气体平衡态的温度 $T$ 与水银柱高度 $x$ 的关系式, 该气体平衡态可能的最高温度]\n它们的单位依次是[None, $\\mathrm{~K}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[方程, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-05.jpg?height=545&width=488&top_left_y=1201&top_left_x=1521" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, "$\\mathrm{~K}$" ], "answer_sequence": [ "被封住的气体平衡态的温度 $T$ 与水银柱高度 $x$ 的关系式", "该气体平衡态可能的最高温度" ], "type_sequence": [ "EQ", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1695", "problem": "光纤光栅是一种介质折射率周期性变化的光学器件. 设一光纤光栅的纤芯基体材料折射率为 $n_{1}=1.51$; 在光纤中周期性地改变纤芯材料的折射率, 其改变了的部分的材料折射率为 $n_{2}=1.55$; 折射率分别为 $n_{2}$ 和 $n_{1}$ 、厚度分别为 $d_{2}$ 和 $d_{1}$ 的介质层相间排布, 总层数为 $N$, 其纵向剖面图如图(a)所示. 在该器件设计过程中, 一般只考虑每层界面的单次反射, 忽略光在介质传播过程中的吸收损耗. 假设入射光在真空中的波长为 $\\lambda=1.06 \\mu m$, 当反射光相干叠加加强时, 则每层的厚度 $d_{1}$ 和 $d_{2}$ 最小应分别为多少? 若要求器件反射率达到 $8 \\%$, 则总层数 $N$ 至少为多少?提示: 如图(b)所示, 当光从折射率 $n_{1}$ 介质垂直入射到 $n_{2}$ 介质时, 界面上产生反射和透射, 有: $\\frac{\\text { 反射光电场强度 }}{\\text { 入射光电场强度 }}=\\frac{n_{1}-n_{2}}{n_{1}+n_{2}}, \\frac{\\text { 透射光电场强度 }}{\\text { 入射光电场强度 }}=\\frac{2 n_{1}}{n_{1}+n_{2}}$, 反射率 $=\\left|\\frac{\\text { 反射光电场强度 }}{\\text { 入射光电场强度 }}\\right|^{2}$,\n\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n光纤光栅是一种介质折射率周期性变化的光学器件. 设一光纤光栅的纤芯基体材料折射率为 $n_{1}=1.51$; 在光纤中周期性地改变纤芯材料的折射率, 其改变了的部分的材料折射率为 $n_{2}=1.55$; 折射率分别为 $n_{2}$ 和 $n_{1}$ 、厚度分别为 $d_{2}$ 和 $d_{1}$ 的介质层相间排布, 总层数为 $N$, 其纵向剖面图如图(a)所示. 在该器件设计过程中, 一般只考虑每层界面的单次反射, 忽略光在介质传播过程中的吸收损耗. 假设入射光在真空中的波长为 $\\lambda=1.06 \\mu m$, 当反射光相干叠加加强时, 则每层的厚度 $d_{1}$ 和 $d_{2}$ 最小应分别为多少? 若要求器件反射率达到 $8 \\%$, 则总层数 $N$ 至少为多少?提示: 如图(b)所示, 当光从折射率 $n_{1}$ 介质垂直入射到 $n_{2}$ 介质时, 界面上产生反射和透射, 有: $\\frac{\\text { 反射光电场强度 }}{\\text { 入射光电场强度 }}=\\frac{n_{1}-n_{2}}{n_{1}+n_{2}}, \\frac{\\text { 透射光电场强度 }}{\\text { 入射光电场强度 }}=\\frac{2 n_{1}}{n_{1}+n_{2}}$, 反射率 $=\\left|\\frac{\\text { 反射光电场强度 }}{\\text { 入射光电场强度 }}\\right|^{2}$,\n\n[图1]\n\n图(a)\n\n[图2]\n\n图(b)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-03.jpg?height=360&width=645&top_left_y=2104&top_left_x=317", "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-03.jpg?height=237&width=443&top_left_y=2166&top_left_x=1018" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_354", "problem": "DRAGON (5 points) - Aigar Vaigu. Here is a photo of a dragon under water (there is a larger photo on a separate sheet). The length of the dragon is $l=8 \\mathrm{~cm}$ and its height is $h=3 \\mathrm{~cm}$. The diameter of the bottom of the bowl is $d=$ $10 \\mathrm{~cm}$ and the angle between the table and the side of the bowl is $\\alpha=60^{\\circ}$. The refractive index of water is $n=1.33$. The photo has been taken so that the camera was pointing directly along the water surface. In the following questions the angle between the horizon when looking at some point on the image of the dragon is defined as the angle between the horizontal water surface (or any other horizontal surface) and the straight line from the eye to that point on the image.\n\n[figure1]\n\nWhat is the highest angle above the horizon from which this reflection can be seen?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nDRAGON (5 points) - Aigar Vaigu. Here is a photo of a dragon under water (there is a larger photo on a separate sheet). The length of the dragon is $l=8 \\mathrm{~cm}$ and its height is $h=3 \\mathrm{~cm}$. The diameter of the bottom of the bowl is $d=$ $10 \\mathrm{~cm}$ and the angle between the table and the side of the bowl is $\\alpha=60^{\\circ}$. The refractive index of water is $n=1.33$. The photo has been taken so that the camera was pointing directly along the water surface. In the following questions the angle between the horizon when looking at some point on the image of the dragon is defined as the angle between the horizontal water surface (or any other horizontal surface) and the straight line from the eye to that point on the image.\n\n[figure1]\n\nWhat is the highest angle above the horizon from which this reflection can be seen?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $^{\\circ}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e6c1582ce7f1c05fa0a6g-1.jpg?height=381&width=663&top_left_y=865&top_left_x=90", "https://cdn.mathpix.com/cropped/2024_03_14_83cc90b265723e4db811g-1.jpg?height=228&width=690&top_left_y=1460&top_left_x=774" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$^{\\circ}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1359", "problem": "如图所示, 一绝缘容器内部为立方体空腔, 其长和宽分别为 $a$ 和 $b$, 厚度为 $d$, 其两侧等高处装有两根与大气相通的玻璃管 (可用来测量液体两侧的压强差). 容器内装满密度为 $\\rho$ 的导电液体, 容器上下两端装有铂电极 $A$ 和 $C$, 这样就构成了一个液体电阻. 该液体电阻置于一方向与容器的厚度方向平行的均匀恒磁场 $B$ 中, 并通过开关 $K$ 接在一电动势为\n\n[图1]\n$\\mathscr{E}$ 、内阻为 $r$ 的电池的两端. 闭合开关. 若稳定时两侧玻璃管中液面的高度差为 $h$, 求导电液体的电导率 $\\sigma$. 重力加速度大小为 $g$.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 一绝缘容器内部为立方体空腔, 其长和宽分别为 $a$ 和 $b$, 厚度为 $d$, 其两侧等高处装有两根与大气相通的玻璃管 (可用来测量液体两侧的压强差). 容器内装满密度为 $\\rho$ 的导电液体, 容器上下两端装有铂电极 $A$ 和 $C$, 这样就构成了一个液体电阻. 该液体电阻置于一方向与容器的厚度方向平行的均匀恒磁场 $B$ 中, 并通过开关 $K$ 接在一电动势为\n\n[图1]\n$\\mathscr{E}$ 、内阻为 $r$ 的电池的两端. 闭合开关. 若稳定时两侧玻璃管中液面的高度差为 $h$, 求导电液体的电导率 $\\sigma$. 重力加速度大小为 $g$.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_58b1fc45927d60138a23g-05.jpg?height=497&width=625&top_left_y=1696&top_left_x=1184" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_765", "problem": "A box of mass $5 \\mathrm{~kg}$ sits at rest on a horizontal floor. What is the Newton's third law (reaction) force to the normal force of the floor on the box (action)?\nA: The weight of the box.\nB: The normal force of the box on the floor.\nC: The gravitational force of the Earth on the box.\nD: The gravitational force of the box on the Earth.\nE: There is no reaction force in this situation.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA box of mass $5 \\mathrm{~kg}$ sits at rest on a horizontal floor. What is the Newton's third law (reaction) force to the normal force of the floor on the box (action)?\n\nA: The weight of the box.\nB: The normal force of the box on the floor.\nC: The gravitational force of the Earth on the box.\nD: The gravitational force of the box on the Earth.\nE: There is no reaction force in this situation.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_607", "problem": "The electric potential at the center of a cube with uniform charge density $\\rho$ and side length $a$ is\n\n$$\n\\Phi \\approx \\frac{0.1894 \\rho a^{2}}{\\epsilon_{0}}\n$$\n\nYou do not need to derive this. ${ }^{1}$\n\nFor the entirety of this problem, any computed numerical constants should be written to three significant figures.\n\nWhat is the electric potential at the tip of a pyramid with a square base of side length $a$, height $a / 2$, and uniform charge density $\\rho$ ? Write your answer in terms of $\\rho, a, \\epsilon_{0}$, and any necessary numerical constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe electric potential at the center of a cube with uniform charge density $\\rho$ and side length $a$ is\n\n$$\n\\Phi \\approx \\frac{0.1894 \\rho a^{2}}{\\epsilon_{0}}\n$$\n\nYou do not need to derive this. ${ }^{1}$\n\nFor the entirety of this problem, any computed numerical constants should be written to three significant figures.\n\nWhat is the electric potential at the tip of a pyramid with a square base of side length $a$, height $a / 2$, and uniform charge density $\\rho$ ? Write your answer in terms of $\\rho, a, \\epsilon_{0}$, and any necessary numerical constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_728", "problem": "While the ball in the previous question is still going upwards, the magnitude of the net force acting on the ball in the previous problem is:\nA: Increasing\nB: Decreasing\nC: Remains constant with the height of the ball\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhile the ball in the previous question is still going upwards, the magnitude of the net force acting on the ball in the previous problem is:\n\nA: Increasing\nB: Decreasing\nC: Remains constant with the height of the ball\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1517", "problem": "两根质量均匀分布的杆 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 质量均为 $m$ ，长均为 $l, \\mathrm{~A}$ 端被光滑较接到一固定点 (即 $\\mathrm{AB}$ 杆可在坚直平面内绕 $\\mathrm{A}$ 点无摩擦转动）。开始时 $\\mathrm{C}$ 点有外力保持两杆静止, $\\mathrm{A} 、 \\mathrm{C}$ 在同一水平线 $\\mathrm{AD}$ 上, $\\mathrm{A}$ 、 $B 、 C$ 三点都在同一坚直平面内, $\\angle A B C=60^{\\circ}$ 。某时刻撤去外力后两杆始终在坚直平面内运动。\n\n若两杆在 B 点固结在一起, 求初始时两杆的角加速度;\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n两根质量均匀分布的杆 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 质量均为 $m$ ，长均为 $l, \\mathrm{~A}$ 端被光滑较接到一固定点 (即 $\\mathrm{AB}$ 杆可在坚直平面内绕 $\\mathrm{A}$ 点无摩擦转动）。开始时 $\\mathrm{C}$ 点有外力保持两杆静止, $\\mathrm{A} 、 \\mathrm{C}$ 在同一水平线 $\\mathrm{AD}$ 上, $\\mathrm{A}$ 、 $B 、 C$ 三点都在同一坚直平面内, $\\angle A B C=60^{\\circ}$ 。某时刻撤去外力后两杆始终在坚直平面内运动。\n\n若两杆在 B 点固结在一起, 求\n\n问题:\n初始时两杆的角加速度;\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_49158ed36459a540f197g-04.jpg?height=549&width=688&top_left_y=248&top_left_x=1164" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1348", "problem": "图中 $L$ 是绕在铁心上的线图, 它与电阻 $R 、 R_{0}$ 、电链和电池 $E$ 可构成闭合回路. 线圈上的箭头表示线圈中电流的正方向, 当电流的流向与箭头所示的方向相同, 该电流为正, 否则为负. 电链 $\\mathrm{K}_{1}$ 和 $\\mathrm{K}_{2}$ 都处于断开状态. 设在 $t=0$ 时刻, 接通电键 $\\mathrm{K}_{1}$, 经过一段时间, 在 $t=t_{1}$时刻, 再接通电链 $\\mathrm{K}_{2}$, 则能较正确在表示 $L$ 中的电流 $I$ 随时间 $t$ 的变化的图线是下面给出的四个图中的哪个图?\n\n[图1]\n\n[图2]\n\n图\n\n[图3]\n\n图\n\n[图4]\n\n图\n\n[图5]\n\n图\nA: 图 1\nB: 图 2\nC: 图 3\nD: 图 4\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n图中 $L$ 是绕在铁心上的线图, 它与电阻 $R 、 R_{0}$ 、电链和电池 $E$ 可构成闭合回路. 线圈上的箭头表示线圈中电流的正方向, 当电流的流向与箭头所示的方向相同, 该电流为正, 否则为负. 电链 $\\mathrm{K}_{1}$ 和 $\\mathrm{K}_{2}$ 都处于断开状态. 设在 $t=0$ 时刻, 接通电键 $\\mathrm{K}_{1}$, 经过一段时间, 在 $t=t_{1}$时刻, 再接通电链 $\\mathrm{K}_{2}$, 则能较正确在表示 $L$ 中的电流 $I$ 随时间 $t$ 的变化的图线是下面给出的四个图中的哪个图?\n\n[图1]\n\n[图2]\n\n图\n\n[图3]\n\n图\n\n[图4]\n\n图\n\n[图5]\n\n图\n\nA: 图 1\nB: 图 2\nC: 图 3\nD: 图 4\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-02.jpg?height=288&width=306&top_left_y=2192&top_left_x=798", "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-03.jpg?height=165&width=286&top_left_y=360&top_left_x=334", "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-03.jpg?height=177&width=302&top_left_y=348&top_left_x=637", "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-03.jpg?height=160&width=289&top_left_y=363&top_left_x=969", "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-03.jpg?height=165&width=303&top_left_y=360&top_left_x=1276" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1718", "problem": "图中 $A 、 B$ 为两块金属板, 分别与高压直流电源的正负极相连。一个电荷量为 $q$ 、质量为 $m$ 的带正电的点电荷自贴在近 $A$ 板处静止释放（不计重力作用）。已知当 $A 、 B$ 两板平行、两板的面积很大且两板间的距离很小时, 它刚到达 $B$ 板时的速度为 $u_{0}$, 在下列情况下以 $u$ 表示点电葏到达 $B$ 板时的速度, 则[ ]\n\n[图1]\nA: 若 $A 、 B$ 两板不平行, 则 $u $10^{-8} \\mathrm{~m}$ | Density
$10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ |\n| :--- | :--- | :--- |\n| Iron | 9.71 | 7.87 |\n| Copper | 1.67 | 8.96 |\n| Aluminum | 2.65 | 2.70 |\n| Lithium | 8.55 | 0.53 |\nA: Iron\nB: Copper\nC: Aluminum\nD: Lithium\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA wire loop is hovering in outer space (weightless vacuum) with its plane parallel to the $x y$-plane. In $x<0$ there is a homogeneous magnetic field parallel to the $z$-axis. The rigid rectangular loop is $l=10 \\mathrm{~cm}$ wide and $h=30 \\mathrm{~cm}$ long. The loop is made of copper wire with a circular cross section (radius $r=1.0 \\mathrm{~mm}$ ). At $t=0 \\mathrm{~s}$ the external magnetic field starts to decrease at a rate of $0.025 \\mathrm{~T} / \\mathrm{s}$.\n\nWe can try to increase the acceleration in many ways. How does the result in i) change if the loop is made of a different metal? Which of the metals given in the table gives the best result?\n\n| Metal | Resistivity
$10^{-8} \\mathrm{~m}$ | Density
$10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ |\n| :--- | :--- | :--- |\n| Iron | 9.71 | 7.87 |\n| Copper | 1.67 | 8.96 |\n| Aluminum | 2.65 | 2.70 |\n| Lithium | 8.55 | 0.53 |\n\nA: Iron\nB: Copper\nC: Aluminum\nD: Lithium\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1607", "problem": "找到两块很大的金属平面, 如图 11 所示摆成 $\\theta_{0}=\\frac{\\pi}{6}$ 角, 角的顶点为 $\\mathrm{O}$ 点, 两块板之间在电压大小为 $\\mathrm{V}_{0}$ 的电源, 金属板和 $\\mathrm{O}$ 点比较靠近, 以至于在角内的电场线几乎为圆弧, $\\mathrm{A}$ 位于角内, $|O A|$ $=\\rho, \\mathrm{OA}$ 和下面的平面夹角为 $\\theta$ 。\n\n[图1]\n\n图 11有一个质量为 $m$, 电量为 $q$ 的小电荷开始在很靠近下平面的某点静止释放, 电荷很小以至于几乎不改变空间电场的分布, 经过时间 $t$ 后电荷运动到了上平面上的 $\\mathrm{B}$ 点, $|O B|=l$; 求出点电荷到达 $\\mathrm{B}$ 点的速度大小 $v_{\\mathrm{B}}$, 以及此时速度方向和上平面之间的夹角 $\\theta_{\\mathrm{B}}$ 。 ( $\\theta_{\\mathrm{B}}$ 取锐角, 由于径向运动比较复杂, 此题中只考虑角动量定理和能量守恒, 不考虑重力)", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n找到两块很大的金属平面, 如图 11 所示摆成 $\\theta_{0}=\\frac{\\pi}{6}$ 角, 角的顶点为 $\\mathrm{O}$ 点, 两块板之间在电压大小为 $\\mathrm{V}_{0}$ 的电源, 金属板和 $\\mathrm{O}$ 点比较靠近, 以至于在角内的电场线几乎为圆弧, $\\mathrm{A}$ 位于角内, $|O A|$ $=\\rho, \\mathrm{OA}$ 和下面的平面夹角为 $\\theta$ 。\n\n[图1]\n\n图 11\n\n问题:\n有一个质量为 $m$, 电量为 $q$ 的小电荷开始在很靠近下平面的某点静止释放, 电荷很小以至于几乎不改变空间电场的分布, 经过时间 $t$ 后电荷运动到了上平面上的 $\\mathrm{B}$ 点, $|O B|=l$; 求出点电荷到达 $\\mathrm{B}$ 点的速度大小 $v_{\\mathrm{B}}$, 以及此时速度方向和上平面之间的夹角 $\\theta_{\\mathrm{B}}$ 。 ( $\\theta_{\\mathrm{B}}$ 取锐角, 由于径向运动比较复杂, 此题中只考虑角动量定理和能量守恒, 不考虑重力)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-06.jpg?height=377&width=576&top_left_y=1513&top_left_x=777" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_333", "problem": "The glass plate has width $L \\gg$ $h_{0}$, thickness $t \\ll L$, density $\\rho_{g}$, and its length (into the depth of the figure) is much bigger than $L$. How will the angular speed of the plate depend on $h$ during its subsequent motion if the density of air is $\\rho_{a}$ ? Neglect the ravity as well as the viscosity and compressbility of the air. Assume that the air flow re mains everywhere laminar.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe glass plate has width $L \\gg$ $h_{0}$, thickness $t \\ll L$, density $\\rho_{g}$, and its length (into the depth of the figure) is much bigger than $L$. How will the angular speed of the plate depend on $h$ during its subsequent motion if the density of air is $\\rho_{a}$ ? Neglect the ravity as well as the viscosity and compressbility of the air. Assume that the air flow re mains everywhere laminar.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1562", "problem": "爱因斯坦等效原理可表述为: 在有引力作用的情况下的物理规律和没有引力但有适当加速度的参考系中的物理规律是相同的。作为一个例子，考察下面两种情况：当一束光从引力势比较低的地方传播到引力势比较高的地方时, 其波长变长, 这个现象称为引力红移。如果在某质量分布均匀的球形星体表面附近的 $\\mathrm{A}$ 处坚直向上发射波长为 $\\lambda_{0}$ 的光, 在 $\\mathrm{A}$ 处坚直上方高度为 $L$ 的 $\\mathrm{B}$ 处放置一固定接收器, 求 $\\mathrm{B}$ 处接收器接收到的光的波长 $\\lambda^{\\prime}$ 。已知该星体质量为 $M$, 半径为 $R(R>>)$; 引力场满足弱场条件, 可应用牛顿引力理论; 真空中的光速为 $c$, 引力常量为 $G$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n爱因斯坦等效原理可表述为: 在有引力作用的情况下的物理规律和没有引力但有适当加速度的参考系中的物理规律是相同的。作为一个例子，考察下面两种情况：\n\n问题:\n当一束光从引力势比较低的地方传播到引力势比较高的地方时, 其波长变长, 这个现象称为引力红移。如果在某质量分布均匀的球形星体表面附近的 $\\mathrm{A}$ 处坚直向上发射波长为 $\\lambda_{0}$ 的光, 在 $\\mathrm{A}$ 处坚直上方高度为 $L$ 的 $\\mathrm{B}$ 处放置一固定接收器, 求 $\\mathrm{B}$ 处接收器接收到的光的波长 $\\lambda^{\\prime}$ 。已知该星体质量为 $M$, 半径为 $R(R>>)$; 引力场满足弱场条件, 可应用牛顿引力理论; 真空中的光速为 $c$, 引力常量为 $G$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_106", "problem": "An object of mass $m$ is attached to the end of a massless rod of length $L$. The other end of the rod is attached to a frictionless pivot. The object is raised so that its height is $0.8 L$ above the pivot, as shown in the figure. After the object is released from rest, what is the tension in the rod when it is horizontal?\n\n[figure1]\nA: $0.6 \\mathrm{mg}$\nB: $1.6 \\mathrm{mg} $ \nC: $2.6 \\mathrm{mg}$\nD: $3.6 \\mathrm{mg}$\nE: $5.36 \\mathrm{mg}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn object of mass $m$ is attached to the end of a massless rod of length $L$. The other end of the rod is attached to a frictionless pivot. The object is raised so that its height is $0.8 L$ above the pivot, as shown in the figure. After the object is released from rest, what is the tension in the rod when it is horizontal?\n\n[figure1]\n\nA: $0.6 \\mathrm{mg}$\nB: $1.6 \\mathrm{mg} $ \nC: $2.6 \\mathrm{mg}$\nD: $3.6 \\mathrm{mg}$\nE: $5.36 \\mathrm{mg}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-05.jpg?height=374&width=479&top_left_y=1458&top_left_x=823" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_124", "problem": "A coin of mass $m$ is dropped straight down from the top of a very tall building. As the coin approaches terminal speed, which is true of the net force on the coin?\nA: The net force on the coin is upward.\nB: The net force on the coin is 0 .\nC: The net force on the coin is downward, with a magnitude less than $m g $\nD: The net force on the coin is downward, with a magnitude equal to $m g$.\nE: The net force on the coin is downward, with a magnitude greater than $m g$.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA coin of mass $m$ is dropped straight down from the top of a very tall building. As the coin approaches terminal speed, which is true of the net force on the coin?\n\nA: The net force on the coin is upward.\nB: The net force on the coin is 0 .\nC: The net force on the coin is downward, with a magnitude less than $m g $\nD: The net force on the coin is downward, with a magnitude equal to $m g$.\nE: The net force on the coin is downward, with a magnitude greater than $m g$.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_400", "problem": "A drone is pulling a cuboid with a rope as shown in the sketch; the cuboid is sliding slowly, with a constant speed, on the hori zontal floor. The cuboid is made from an ho mogeneous material. You may take measure ments from the sketch (on a separate page) assuming that the dimensions and distances on it are correct within an unknown scale factor. In order to help you in case you don't have access to a printer, and need to read the problem texts directly from the com puter screen, some auxiliary dashed lines are shown in the diagram (which might or might not be useful).\n\nNext we will be studying the flight of a drone in adiabatic atmosphere. In adiabatic atmosphere, air parcels are moving continuously up or down, and while doing so expand or contract adiabatically. It can be shown that in adiabatic atmosphere, the tem perature is a linear function of the height $z$ $T=T_{0}-z g / c_{p}$, where $T_{0}=293 \\mathrm{~K}$ is the tem perature on the ground, $c_{p}=1.00 \\mathrm{Jg}^{-1} \\mathrm{~K}^{-1}$ is the specific heat of air at constant pressure and $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$. Find the dependence of air density $\\rho$ as a function of height, in terms of the density $\\rho_{0}$ at the ground level $(z=0)$ specific heat of air at constant volume $c_{v}=$ $0.718 \\mathrm{~J} \\mathrm{~g}^{-1} \\mathrm{~K}^{-1}$, and other previously defined", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA drone is pulling a cuboid with a rope as shown in the sketch; the cuboid is sliding slowly, with a constant speed, on the hori zontal floor. The cuboid is made from an ho mogeneous material. You may take measure ments from the sketch (on a separate page) assuming that the dimensions and distances on it are correct within an unknown scale factor. In order to help you in case you don't have access to a printer, and need to read the problem texts directly from the com puter screen, some auxiliary dashed lines are shown in the diagram (which might or might not be useful).\n\nNext we will be studying the flight of a drone in adiabatic atmosphere. In adiabatic atmosphere, air parcels are moving continuously up or down, and while doing so expand or contract adiabatically. It can be shown that in adiabatic atmosphere, the tem perature is a linear function of the height $z$ $T=T_{0}-z g / c_{p}$, where $T_{0}=293 \\mathrm{~K}$ is the tem perature on the ground, $c_{p}=1.00 \\mathrm{Jg}^{-1} \\mathrm{~K}^{-1}$ is the specific heat of air at constant pressure and $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$. Find the dependence of air density $\\rho$ as a function of height, in terms of the density $\\rho_{0}$ at the ground level $(z=0)$ specific heat of air at constant volume $c_{v}=$ $0.718 \\mathrm{~J} \\mathrm{~g}^{-1} \\mathrm{~K}^{-1}$, and other previously defined\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1341", "problem": "国际上已规定 ${ }^{133} \\mathrm{Cs}$ 原子的频率 $f=9192631770 \\mathrm{~Hz}$ （没有误差）。这样秒的定义。\n\n国际上已规定一个公认的光速值 $c=299792458 \\mathrm{~m} / \\mathrm{s}$ （没有误差）。长度单位由时间单位导出, 则米定义为", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n国际上已规定 ${ }^{133} \\mathrm{Cs}$ 原子的频率 $f=9192631770 \\mathrm{~Hz}$ （没有误差）。这样秒的定义。\n\n国际上已规定一个公认的光速值 $c=299792458 \\mathrm{~m} / \\mathrm{s}$ （没有误差）。长度单位由时间单位导出, 则米定义为\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[秒的定义, 米定义]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "秒的定义", "米定义" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_48", "problem": "A car and a truck are both traveling with a constant speed of $20 \\mathrm{~m} / \\mathrm{s}$. The car is $10 \\mathrm{~m}$ behind the truck. The truck driver suddenly applies his brakes, causing the truck to slow to a stop at the constant rate of $2 \\mathrm{~m} / \\mathrm{s}^{2}$. Two seconds later, the driver of the car applies their brakes and just manages to avoid a rear-end collision. Determine the constant rate at which the car slowed.\nA: $\\ 3.33 \\mathrm{~m} / \\mathrm{s}^{2}$\nB: $4.33 \\mathrm{~m} / \\mathrm{s}^{2}$\nC: $1.33 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: $3.03 \\mathrm{~m} / \\mathrm{s}^{2}$\nE: $3.93 \\mathrm{~m} / \\mathrm{s}^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA car and a truck are both traveling with a constant speed of $20 \\mathrm{~m} / \\mathrm{s}$. The car is $10 \\mathrm{~m}$ behind the truck. The truck driver suddenly applies his brakes, causing the truck to slow to a stop at the constant rate of $2 \\mathrm{~m} / \\mathrm{s}^{2}$. Two seconds later, the driver of the car applies their brakes and just manages to avoid a rear-end collision. Determine the constant rate at which the car slowed.\n\nA: $\\ 3.33 \\mathrm{~m} / \\mathrm{s}^{2}$\nB: $4.33 \\mathrm{~m} / \\mathrm{s}^{2}$\nC: $1.33 \\mathrm{~m} / \\mathrm{s}^{2}$\nD: $3.03 \\mathrm{~m} / \\mathrm{s}^{2}$\nE: $3.93 \\mathrm{~m} / \\mathrm{s}^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_837", "problem": "Calculation in the previous part shows that in order to build the space elevator, it is neccessary to have light materials with very high tensile strength. Carbon nanotubes are materials that meet such requirements because of strong chemical bondings between very light atoms. Two natural polymorphs of carbon are diamond and graphite. In diamond every carbon atom is surrounded by four nearest neighbor (NN) atoms to form a tetrahedron. Graphite has a layer structure. In each layer, carbon atoms are arranged in a hexagonal plane lattice with three NNs. Although diamond is known as the hardest materials, covalent bondings between carbon atoms in hexagonal layers of graphite is stronger than those between carbon atoms in diamond tetrahedra. Graphite is much softer than diamond because of the van der Waals bonding between carbon atoms of different layers, which is much weaker than covalent bonding.\n\n[figure1]\n\nFigure 2. Graphite structure\n(a)\n[figure2]\n\nFigure 3. Graphene (a) and carbon nanotube (b).\n\nA monatomic layer in graphite is called graphene and has monoatomic thickness. Isolated graphene sheet is not stable and has a tendency to roll up to form carbon spheres or carbon nanotubes. The hexagonal crystal lattice of graphene is depicted in Fig. 4. The distance between two NN carbon atoms is $a=0.142 \\mathrm{~nm}$ and the distance between two closest parallel bondings is $b=0.246 \\mathrm{~nm}$. Because the covalent bondings between carbon atoms in graphene are very strong, mechanical properties of carbon nanotubes are very special. They have an extremely large Young's modulus and tensile strength, as well as a very light density. Young's modulus is defined as the ratio of the stress along an axis to the strain (ratio of deformation over initial length) along that axis in the range of stress in which Hooke's law holds.\n\n[figure3]\n\nFigure 4. Graphene.\n\n## Theory\n\n[figure4]\n\nFigure 5. An illustration of a carbon nanotube with 9 carbon-carbon parallel bondings. Note: In this problem, there are 27 carbon-carbon parallel bondings. (1) parallel bond; (2) slanted bond; (3) tube axis.\n\nNow we examine some mechanical properties of a carbon nanotube having 27 carbon-carbon bondings parallel to the tube axis (for an illustration, see Figure 5). The bonding between two carbon atoms can be described by the Morse potential $V(x)=V_{0}\\left(e^{-4 \\frac{x}{a}}-2 e^{-2 \\frac{x}{a}}\\right)$. Here $a=0.142 \\mathrm{~nm}$ is the equilibrium distance between two $\\mathrm{NN}$ carbon atoms, $V_{0}=4.93 \\mathrm{eV}$ is the bonding energy, and $x$ is the displacement of the atom from the equilibrium position. Hereafter, we approximate the Morse potential by a quadratic potential $V(x)=P+Q x^{2}$. All non-nearest-neighbor interactions are neglected. In this approximation, one can propose that carbon atoms are bonded through \"springs\" with the spring constant $k$. Changes in angles between bonds are neglected.\n\nEstimate the tensile strength $\\sigma_{0}$ of the carbon nanotube.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nCalculation in the previous part shows that in order to build the space elevator, it is neccessary to have light materials with very high tensile strength. Carbon nanotubes are materials that meet such requirements because of strong chemical bondings between very light atoms. Two natural polymorphs of carbon are diamond and graphite. In diamond every carbon atom is surrounded by four nearest neighbor (NN) atoms to form a tetrahedron. Graphite has a layer structure. In each layer, carbon atoms are arranged in a hexagonal plane lattice with three NNs. Although diamond is known as the hardest materials, covalent bondings between carbon atoms in hexagonal layers of graphite is stronger than those between carbon atoms in diamond tetrahedra. Graphite is much softer than diamond because of the van der Waals bonding between carbon atoms of different layers, which is much weaker than covalent bonding.\n\n[figure1]\n\nFigure 2. Graphite structure\n(a)\n[figure2]\n\nFigure 3. Graphene (a) and carbon nanotube (b).\n\nA monatomic layer in graphite is called graphene and has monoatomic thickness. Isolated graphene sheet is not stable and has a tendency to roll up to form carbon spheres or carbon nanotubes. The hexagonal crystal lattice of graphene is depicted in Fig. 4. The distance between two NN carbon atoms is $a=0.142 \\mathrm{~nm}$ and the distance between two closest parallel bondings is $b=0.246 \\mathrm{~nm}$. Because the covalent bondings between carbon atoms in graphene are very strong, mechanical properties of carbon nanotubes are very special. They have an extremely large Young's modulus and tensile strength, as well as a very light density. Young's modulus is defined as the ratio of the stress along an axis to the strain (ratio of deformation over initial length) along that axis in the range of stress in which Hooke's law holds.\n\n[figure3]\n\nFigure 4. Graphene.\n\n## Theory\n\n[figure4]\n\nFigure 5. An illustration of a carbon nanotube with 9 carbon-carbon parallel bondings. Note: In this problem, there are 27 carbon-carbon parallel bondings. (1) parallel bond; (2) slanted bond; (3) tube axis.\n\nNow we examine some mechanical properties of a carbon nanotube having 27 carbon-carbon bondings parallel to the tube axis (for an illustration, see Figure 5). The bonding between two carbon atoms can be described by the Morse potential $V(x)=V_{0}\\left(e^{-4 \\frac{x}{a}}-2 e^{-2 \\frac{x}{a}}\\right)$. Here $a=0.142 \\mathrm{~nm}$ is the equilibrium distance between two $\\mathrm{NN}$ carbon atoms, $V_{0}=4.93 \\mathrm{eV}$ is the bonding energy, and $x$ is the displacement of the atom from the equilibrium position. Hereafter, we approximate the Morse potential by a quadratic potential $V(x)=P+Q x^{2}$. All non-nearest-neighbor interactions are neglected. In this approximation, one can propose that carbon atoms are bonded through \"springs\" with the spring constant $k$. Changes in angles between bonds are neglected.\n\nEstimate the tensile strength $\\sigma_{0}$ of the carbon nanotube.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{GPa}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-2.jpg?height=549&width=525&top_left_y=1850&top_left_x=731", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-3.jpg?height=430&width=900&top_left_y=710&top_left_x=518", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-3.jpg?height=451&width=691&top_left_y=1856&top_left_x=660", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-4.jpg?height=944&width=1144&top_left_y=172&top_left_x=184" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{GPa}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_23", "problem": "A high-voltage power transmission line is to supply power at $104 \\mathrm{~kW}$. The electricity travels through $30 \\mathrm{~km}$ of cable that has a resistance $0.7 \\frac{\\Omega}{\\mathrm{km}}$. Find the rate of energy loss if the power is transmitted at $100 \\mathrm{kV}$.\nA: $4.2 \\%$\nB: $4.9 \\%$\nC: $2.1 \\%$\nD: $9.8 \\%$\nE: $6.4 \\%$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA high-voltage power transmission line is to supply power at $104 \\mathrm{~kW}$. The electricity travels through $30 \\mathrm{~km}$ of cable that has a resistance $0.7 \\frac{\\Omega}{\\mathrm{km}}$. Find the rate of energy loss if the power is transmitted at $100 \\mathrm{kV}$.\n\nA: $4.2 \\%$\nB: $4.9 \\%$\nC: $2.1 \\%$\nD: $9.8 \\%$\nE: $6.4 \\%$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_512", "problem": "Because the load is purely resistive, the average power is simply\n\n$$\nP=I_{\\mathrm{rms}} V_{\\mathrm{rms}}\n$$\n\nso\n\n$$\nI_{\\mathrm{rms}}=\\frac{P}{V_{\\mathrm{rms}}}=2000 \\mathrm{~A}\n$$An AC power line cable transmits electrical power using a sinusoidal waveform with frequency $60 \\mathrm{~Hz}$. The load receives an RMS voltage of $500 \\mathrm{kV}$ and requires $1000 \\mathrm{MW}$ of average power. For this problem, consider only the cable carrying current in one of the two directions, and ignore effects due to capacitance or inductance between the cable and with the ground.\n\nSuppose that the load on the power line cable is a residential area that behaves like a pure resistor.\n\nThe cable has diameter $3 \\mathrm{~cm}$, is $500 \\mathrm{~km}$ long, and is made of aluminum with resistivity $2.8 \\times 10^{-8} \\Omega \\cdot \\mathrm{m}$. How much power is lost in the wire?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nBecause the load is purely resistive, the average power is simply\n\n$$\nP=I_{\\mathrm{rms}} V_{\\mathrm{rms}}\n$$\n\nso\n\n$$\nI_{\\mathrm{rms}}=\\frac{P}{V_{\\mathrm{rms}}}=2000 \\mathrm{~A}\n$$\n\nproblem:\nAn AC power line cable transmits electrical power using a sinusoidal waveform with frequency $60 \\mathrm{~Hz}$. The load receives an RMS voltage of $500 \\mathrm{kV}$ and requires $1000 \\mathrm{MW}$ of average power. For this problem, consider only the cable carrying current in one of the two directions, and ignore effects due to capacitance or inductance between the cable and with the ground.\n\nSuppose that the load on the power line cable is a residential area that behaves like a pure resistor.\n\nThe cable has diameter $3 \\mathrm{~cm}$, is $500 \\mathrm{~km}$ long, and is made of aluminum with resistivity $2.8 \\times 10^{-8} \\Omega \\cdot \\mathrm{m}$. How much power is lost in the wire?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{MW}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{MW}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1247", "problem": "Although atomic nuclei are quantum objects, a number of phenomenological laws for their basic properties (like radius or binding energy) can be deduced from simple assumptions: (i) nuclei are built from nucleons (i.e. protons and neutrons); (ii) strong nuclear interaction holding these nucleons together has a very short range (it acts only between neighboring nucleons); (iii) the number of protons $(Z)$ in a given nucleus is approximately equal to the number of neutrons $(N)$, i.e. $Z \\approx N \\approx A / 2$, where $A$ is the total number of nucleons $(A \\gg 1)$.\n\n## Electrostatic (Coulomb) effects on the binding energy\n\nThe electrostatic energy of a homogeneously charged ball (with radius $R$ and total charge $Q_{0}$ ) is $U_{c}=\\frac{3 Q_{0}^{2}}{20 \\pi \\varepsilon_{0} R}$, where $\\varepsilon_{0}=8.85 \\cdot 10^{-12} \\mathrm{C}^{2} N^{-1} m^{-2}$.\n\nApply this formula to get the electrostatic energy of a nucleus. In a nucleus, each proton is not acting upon itself (by Coulomb force), but only upon the rest of the protons. One can take this into account by replacing $Z^{2} \\rightarrow Z(Z-1)$ in the obtained formula. Use this correction in subsequent tasks.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nAlthough atomic nuclei are quantum objects, a number of phenomenological laws for their basic properties (like radius or binding energy) can be deduced from simple assumptions: (i) nuclei are built from nucleons (i.e. protons and neutrons); (ii) strong nuclear interaction holding these nucleons together has a very short range (it acts only between neighboring nucleons); (iii) the number of protons $(Z)$ in a given nucleus is approximately equal to the number of neutrons $(N)$, i.e. $Z \\approx N \\approx A / 2$, where $A$ is the total number of nucleons $(A \\gg 1)$.\n\n## Electrostatic (Coulomb) effects on the binding energy\n\nThe electrostatic energy of a homogeneously charged ball (with radius $R$ and total charge $Q_{0}$ ) is $U_{c}=\\frac{3 Q_{0}^{2}}{20 \\pi \\varepsilon_{0} R}$, where $\\varepsilon_{0}=8.85 \\cdot 10^{-12} \\mathrm{C}^{2} N^{-1} m^{-2}$.\n\nApply this formula to get the electrostatic energy of a nucleus. In a nucleus, each proton is not acting upon itself (by Coulomb force), but only upon the rest of the protons. One can take this into account by replacing $Z^{2} \\rightarrow Z(Z-1)$ in the obtained formula. Use this correction in subsequent tasks.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_368", "problem": "The black box has three terminal wires: \"blue\", \"black\" and \"white\", and contains n a star configuration: a battery, a capacitor, an inductor in series with a diode. You may consider the diode to be \"ideal\" - it conducts current perfectly one way and not at all the other way. You may neglect internal resistance of the battery and capacitor, but the inductor has considerable internal resistance. The multimeter's internal resistance when measuring voltages is $R_{m}=10 \\mathrm{M} \\Omega$ and it displays a new reading every $t=0.4 \\mathrm{~s}$.\n\nEstimate the value of the capacitance $C$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe black box has three terminal wires: \"blue\", \"black\" and \"white\", and contains n a star configuration: a battery, a capacitor, an inductor in series with a diode. You may consider the diode to be \"ideal\" - it conducts current perfectly one way and not at all the other way. You may neglect internal resistance of the battery and capacitor, but the inductor has considerable internal resistance. The multimeter's internal resistance when measuring voltages is $R_{m}=10 \\mathrm{M} \\Omega$ and it displays a new reading every $t=0.4 \\mathrm{~s}$.\n\nEstimate the value of the capacitance $C$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{uF}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{uF}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_934", "problem": "A rod of mass $m_{1}$ is constrained to move vertically by a pair of guides, as shown in Fig. 4. The rod is in contact with a smooth wedge of mass $m_{2}$ and angle $\\theta$, which itself sits on a smooth horizontal surface. At time $t=0$ the rod is released and moves downwards, whilst the wedge accelerates to the right.\n\nNow obtain an expression for the speed of the wedge $v$ in terms of $m_{1}, m_{2}, g, h$ and $\\theta$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA rod of mass $m_{1}$ is constrained to move vertically by a pair of guides, as shown in Fig. 4. The rod is in contact with a smooth wedge of mass $m_{2}$ and angle $\\theta$, which itself sits on a smooth horizontal surface. At time $t=0$ the rod is released and moves downwards, whilst the wedge accelerates to the right.\n\nNow obtain an expression for the speed of the wedge $v$ in terms of $m_{1}, m_{2}, g, h$ and $\\theta$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1617", "problem": "假定月球绕地球作圆周运动, 地球绕太阳也作圆周运动, 且轨道都在同一平面内. 已知地球表面处的重力加速度 $g=9.80 \\mathrm{~m} / \\mathrm{s}^{2}$, 地球半径 $R_{\\mathrm{e}}=6.37 \\times 10^{6} \\mathrm{~m}$, 月球质量 $m_{\\mathrm{m}}=7.3 \\times 10^{22} \\mathrm{~kg}$, 月球半径 $R_{\\mathrm{m}}=1.7 \\times 10^{6} \\mathrm{~m}$, 引力恒量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} \\cdot \\mathrm{kg}^{-2}$, 月心地心间的距离约为 $r_{\\mathrm{em}}=3.84 \\times 10^{8} \\mathrm{~m}$.地球上的观察者相继两次看到满月需多少天?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n假定月球绕地球作圆周运动, 地球绕太阳也作圆周运动, 且轨道都在同一平面内. 已知地球表面处的重力加速度 $g=9.80 \\mathrm{~m} / \\mathrm{s}^{2}$, 地球半径 $R_{\\mathrm{e}}=6.37 \\times 10^{6} \\mathrm{~m}$, 月球质量 $m_{\\mathrm{m}}=7.3 \\times 10^{22} \\mathrm{~kg}$, 月球半径 $R_{\\mathrm{m}}=1.7 \\times 10^{6} \\mathrm{~m}$, 引力恒量 $G=6.67 \\times 10^{-11} \\mathrm{~N} \\cdot \\mathrm{m}^{2} \\cdot \\mathrm{kg}^{-2}$, 月心地心间的距离约为 $r_{\\mathrm{em}}=3.84 \\times 10^{8} \\mathrm{~m}$.\n\n问题:\n地球上的观察者相继两次看到满月需多少天?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以天为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "天" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_776", "problem": "The pigeon sees its friend, an eagle, in the sky, and flies to meet the eagle who is gliding in the opposite direction. The eagle has a mass of $2 \\mathrm{~kg}$. If the pigeon and eagle have the same kinetic energy, which of the following statements are correct.Unfortunately, in their excitement to see each other, the pigeon and eagle collide mid-air while flying horizontally, directly at each other. Their wings get tangled up and so they are unable to fly. In the instant after their collision, which direction do the tangled-up birds go?\nA: In the direction the pigeon was flying\nB: In the direction the eagle was flying\nC: The birds stop dead in the air\nD: The birds fall straight downwards\nE: More information is needed to answer this question\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\nHere is some context information for this question, which might assist you in solving it:\nThe pigeon sees its friend, an eagle, in the sky, and flies to meet the eagle who is gliding in the opposite direction. The eagle has a mass of $2 \\mathrm{~kg}$. If the pigeon and eagle have the same kinetic energy, which of the following statements are correct.\n\nproblem:\nUnfortunately, in their excitement to see each other, the pigeon and eagle collide mid-air while flying horizontally, directly at each other. Their wings get tangled up and so they are unable to fly. In the instant after their collision, which direction do the tangled-up birds go?\n\nA: In the direction the pigeon was flying\nB: In the direction the eagle was flying\nC: The birds stop dead in the air\nD: The birds fall straight downwards\nE: More information is needed to answer this question\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_223", "problem": "Thermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nParticles in the exobase with enough outwards velocity will escape gravitational attraction.\n\nDetermine the hydrogen atoms flux (number of particle per unit area and per unit time) $\\Phi$ that will escape the atmosphere, knowing that the concentration of hydrogen atoms in the exobase is $n_{H}=10^{11} m^{-3}$. Be careful with the dimensions.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nParticles in the exobase with enough outwards velocity will escape gravitational attraction.\n\nDetermine the hydrogen atoms flux (number of particle per unit area and per unit time) $\\Phi$ that will escape the atmosphere, knowing that the concentration of hydrogen atoms in the exobase is $n_{H}=10^{11} m^{-3}$. Be careful with the dimensions.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\frac{1}{\\mathrm{~m}^{2} \\mathrm{~s}}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_61a2ff399c33d9b3cd3bg-1.jpg?height=968&width=1044&top_left_y=1240&top_left_x=302" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\frac{1}{\\mathrm{~m}^{2} \\mathrm{~s}}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1440", "problem": "一转速测量和控制装置的原理如图所示. 在 $O$ 点有电量为 $Q$ 的正电荷, 内壁光滑的轻质绝缘细管可绕通过 $\\mathrm{O}$ 点的坚直轴在水平面内转动, 在管内距离 $O$ 为 $L$ 处有一光电触发控制开关\n\n[图1]\n$A$, 在 $O$ 端固定有一自由长度为 $L / 4$ 的轻质绝缘弹簧, 弹簧另一端与一质量为 $m$ 、带有正电荷 $q$ 的小球相连接.\n\n开始时, 系统处于静态平衡. 细管在外力矩作用下, 作定轴转动, 小球可在细管内运动. 当细管转速 $\\omega$ 逐渐变大时, 小球到达细管的 A 处刚好相对于细管径向平衡, 并触发控制开关,外力矩瞬时变为零, 从而限制转速过大; 同时 $\\mathrm{O}$ 点的电荷变为等量负电荷 $-\\mathrm{Q}$. 通过测量此后小球相对于细管径向平衡点的位置 $\\mathrm{B}$, 可测定转速. 若测得 $\\mathrm{OB}$ 的距离为 $\\mathrm{L} / 2$, 求弹簧系数 $k_{0}$ 及小球在 $\\mathrm{B}$ 处时细管的转速", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一转速测量和控制装置的原理如图所示. 在 $O$ 点有电量为 $Q$ 的正电荷, 内壁光滑的轻质绝缘细管可绕通过 $\\mathrm{O}$ 点的坚直轴在水平面内转动, 在管内距离 $O$ 为 $L$ 处有一光电触发控制开关\n\n[图1]\n$A$, 在 $O$ 端固定有一自由长度为 $L / 4$ 的轻质绝缘弹簧, 弹簧另一端与一质量为 $m$ 、带有正电荷 $q$ 的小球相连接.\n\n开始时, 系统处于静态平衡. 细管在外力矩作用下, 作定轴转动, 小球可在细管内运动. 当细管转速 $\\omega$ 逐渐变大时, 小球到达细管的 A 处刚好相对于细管径向平衡, 并触发控制开关,外力矩瞬时变为零, 从而限制转速过大; 同时 $\\mathrm{O}$ 点的电荷变为等量负电荷 $-\\mathrm{Q}$. 通过测量此后小球相对于细管径向平衡点的位置 $\\mathrm{B}$, 可测定转速. 若测得 $\\mathrm{OB}$ 的距离为 $\\mathrm{L} / 2$, 求\n\n问题:\n弹簧系数 $k_{0}$ 及小球在 $\\mathrm{B}$ 处时细管的转速\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[弹簧系数 $k_{0}$, $\\mathrm{B}$ 处时细管的转速]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-01.jpg?height=283&width=571&top_left_y=447&top_left_x=1139" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "弹簧系数 $k_{0}$", "$\\mathrm{B}$ 处时细管的转速" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_353", "problem": "Consider two absolutely elastic dielec tric balls of radius $r$ and mass $m$ one of which carries isotropically distributed charge $-q$, and the other $-+q$. There is so strong homogen eous magnetic field $B$, parallel to the axis $z$, tha electrostatic interaction of the two charges can be neglected; neglect also gravity and friction forces. The first ball (negatively charged) moves with speed $v$ and collides with the second ball which had been resting at the origin. The collision is central, and immediately before the im pact, the velocity of the first ball was parallel to the $x$-axis.\n\n What is the average velocity (magnitude and direction) of the balls during their subsequent motion?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider two absolutely elastic dielec tric balls of radius $r$ and mass $m$ one of which carries isotropically distributed charge $-q$, and the other $-+q$. There is so strong homogen eous magnetic field $B$, parallel to the axis $z$, tha electrostatic interaction of the two charges can be neglected; neglect also gravity and friction forces. The first ball (negatively charged) moves with speed $v$ and collides with the second ball which had been resting at the origin. The collision is central, and immediately before the im pact, the velocity of the first ball was parallel to the $x$-axis.\n\n What is the average velocity (magnitude and direction) of the balls during their subsequent motion?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_841", "problem": "The main application of space elevator is the use of the tower's rotational energy to launch payload into orbit or send spacecraft to the other planets. It is very easy to get payload into space: we simply have to make it ride up the elevator to an altitude $r$ and release it from rest. For simplicity in the calculations, let us assume that the motion of the tower occurs in the plane of Earth's orbit.\n\nFind the critical height $r_{C}$ up the tower, measured from Earth's center, at which the object would have to be released from rest to escape Earth's gravity.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe main application of space elevator is the use of the tower's rotational energy to launch payload into orbit or send spacecraft to the other planets. It is very easy to get payload into space: we simply have to make it ride up the elevator to an altitude $r$ and release it from rest. For simplicity in the calculations, let us assume that the motion of the tower occurs in the plane of Earth's orbit.\n\nFind the critical height $r_{C}$ up the tower, measured from Earth's center, at which the object would have to be released from rest to escape Earth's gravity.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of km, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "km" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1490", "problem": "太空中有一飞行器靠其自身动力维持在地球赤道的正上方 $L=a R_{e}$ 处, 相对于赤道上的一地面物资供应站保持静止. 这里, $R_{e}$ 为地球的半径, $a$ 为常数, $a>a_{m}$, 而\n\n$$\n\\alpha_{m}=\\left(\\frac{G M_{e}}{\\omega_{e}^{2} R_{e}^{3}}\\right)^{1 / 3}-1\n$$\n\n$M_{e}$ 和 $W_{e}$ 分别为地球的质量和自转角速度, $G$ 为引力常数. 设想从供应站到飞行器有一根用于运送物资的刚性、管壁匀质、质量为 $m_{p}$ 的坚直输送管, 输送管下端固定在地面上, 并设法保持输送管与地面始终垂直. 推送物资时, 把物资放进输送管下端内的平底托盘上, 沿管壁向上推进, 并保持托盘运行速度不致过大. 忽略托盘与管壁之间的摩擦力, 考虑地球的自转, 但不考虑地球的公转. 设某次所推送物资和托盘的总质量为 $m$.在把物资从供应站送到飞行器的过程中, 地球引力和惯性离心力做的功分别是多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n太空中有一飞行器靠其自身动力维持在地球赤道的正上方 $L=a R_{e}$ 处, 相对于赤道上的一地面物资供应站保持静止. 这里, $R_{e}$ 为地球的半径, $a$ 为常数, $a>a_{m}$, 而\n\n$$\n\\alpha_{m}=\\left(\\frac{G M_{e}}{\\omega_{e}^{2} R_{e}^{3}}\\right)^{1 / 3}-1\n$$\n\n$M_{e}$ 和 $W_{e}$ 分别为地球的质量和自转角速度, $G$ 为引力常数. 设想从供应站到飞行器有一根用于运送物资的刚性、管壁匀质、质量为 $m_{p}$ 的坚直输送管, 输送管下端固定在地面上, 并设法保持输送管与地面始终垂直. 推送物资时, 把物资放进输送管下端内的平底托盘上, 沿管壁向上推进, 并保持托盘运行速度不致过大. 忽略托盘与管壁之间的摩擦力, 考虑地球的自转, 但不考虑地球的公转. 设某次所推送物资和托盘的总质量为 $m$.\n\n问题:\n在把物资从供应站送到飞行器的过程中, 地球引力和惯性离心力做的功分别是多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1311", "problem": "某电磁轨道炮的简化模型如图 a 所示, 两圆柱形固定导轨相互平行, 其对称轴所在平面与水平面的夹角为 $\\theta$, 两导轨的长均为 $\\mathrm{L}$ 、半径均为 $\\mathrm{b}$ 、每单位长度的电阻均为 $\\lambda$, 两导轨之间的最近距离为 $\\mathrm{d}(\\mathrm{d}$很小). 一弹丸质量为 $\\mathrm{m}$ ( $\\mathrm{m}$ 较小) 的金属弹丸 (可视为薄片) 置于两导轨之间, 弹丸直径为 $\\mathrm{d}$ 、电阻为 $\\mathrm{R}$, 与导轨保持良好接触. 两导轨下端横截面共面, 下端 (通过两根与相应导轨同轴的、较长的硬导线) 与一电流为 I 的理想恒流源 (恒流源内部的能量损耗可不计) 相连, 不考虑空气阻力和摩擦阻力, 重力加速度大小图 a. 某电磁轨道炮的简化模型为 $g$, 真空磁导率为 $\\mu_{0}$. 考虑一弹丸自导轨下端从静止开始被磁场加速直至射出的过程.\n\n[图1]\n\n图 a. 某电磁轨道炮的简化模型求弹丸的出射速度;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某电磁轨道炮的简化模型如图 a 所示, 两圆柱形固定导轨相互平行, 其对称轴所在平面与水平面的夹角为 $\\theta$, 两导轨的长均为 $\\mathrm{L}$ 、半径均为 $\\mathrm{b}$ 、每单位长度的电阻均为 $\\lambda$, 两导轨之间的最近距离为 $\\mathrm{d}(\\mathrm{d}$很小). 一弹丸质量为 $\\mathrm{m}$ ( $\\mathrm{m}$ 较小) 的金属弹丸 (可视为薄片) 置于两导轨之间, 弹丸直径为 $\\mathrm{d}$ 、电阻为 $\\mathrm{R}$, 与导轨保持良好接触. 两导轨下端横截面共面, 下端 (通过两根与相应导轨同轴的、较长的硬导线) 与一电流为 I 的理想恒流源 (恒流源内部的能量损耗可不计) 相连, 不考虑空气阻力和摩擦阻力, 重力加速度大小图 a. 某电磁轨道炮的简化模型为 $g$, 真空磁导率为 $\\mu_{0}$. 考虑一弹丸自导轨下端从静止开始被磁场加速直至射出的过程.\n\n[图1]\n\n图 a. 某电磁轨道炮的简化模型\n\n问题:\n求弹丸的出射速度;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-03.jpg?height=539&width=808&top_left_y=250&top_left_x=521" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1632", "problem": "如图 7 所示, 有一个坚直放置的导热气缸, 用一个轻质活塞密封, 活塞可以自由上下移动, 面积为 $\\mathrm{S}_{0}$, 初态气缸内封有体积为 $\\mathrm{V}_{0}$, 压强等于大气压 $p_{0}$, 温度和环境温度相同的单原子理想气体,缓慢在活塞上面堆放细沙（每次堆上的细沙都放在活塞所在的位置）, 结果活塞下降, 使得密封的气体体积变小到 $x \\mathrm{~V}_{0}$, 重力加速度 $g$, 求出细沙的质量 $m_{0}=$ ___；把导热气缸换成绝热气缸，其他条件不变, 求出这个过程中活塞对体系做功 $W_{0}=$ ____；普适气体常量为 $R$, 单原子理想气体的定体摩尔热容量为 $\\mathrm{Cv}=\\frac{3}{2} R$ 。\n\n[图1]\n\n图 7", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n如图 7 所示, 有一个坚直放置的导热气缸, 用一个轻质活塞密封, 活塞可以自由上下移动, 面积为 $\\mathrm{S}_{0}$, 初态气缸内封有体积为 $\\mathrm{V}_{0}$, 压强等于大气压 $p_{0}$, 温度和环境温度相同的单原子理想气体,缓慢在活塞上面堆放细沙（每次堆上的细沙都放在活塞所在的位置）, 结果活塞下降, 使得密封的气体体积变小到 $x \\mathrm{~V}_{0}$, 重力加速度 $g$, 求出细沙的质量 $m_{0}=$ ___；把导热气缸换成绝热气缸，其他条件不变, 求出这个过程中活塞对体系做功 $W_{0}=$ ____；普适气体常量为 $R$, 单原子理想气体的定体摩尔热容量为 $\\mathrm{Cv}=\\frac{3}{2} R$ 。\n\n[图1]\n\n图 7\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[细沙的质量 $m_{0}$, 其他条件不变, 求出这个过程中活塞对体系做功 $W_{0}$]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-04.jpg?height=291&width=240&top_left_y=1311&top_left_x=951" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "细沙的质量 $m_{0}$", "其他条件不变, 求出这个过程中活塞对体系做功 $W_{0}$" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1724", "problem": "某根水平固定的滑竿上有 $n(n \\geq 3)$ 个质量均为 $m$ 的相同滑扣（即可以滑动的圆环），每相邻的两个滑扣（极薄)之间有不可伸长的柔软轻质细线相连, 细线长度均为 $L$, 滑扣在滑竿上滑行的阻力恒为滑扣对滑竿正压力的 $\\mu$ 倍。开始时所有滑扣可近似地看成挨在一起（但未相互挤压）; 当给第 1 个滑扣一个初速度使其在滑等上开始滑行 (平动) ; 在滑扣滑行的过程中, 前、后滑扣之间的细线拉紧后都以共同的速度向前滑行, 但最后一个 (即第 $n$ 个) 滑扣固定在滑竿边缘。已知从第 1 个滑扣开始的 $(n-1)$ 个滑扣都依次拉紧, 继续滑行距离 $l(0 \\leq l\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nFind the total angular velocity $\\omega$ of the rotating top about its centre of mass $C$ in terms of the time derivatives of the Euler angles: $\\dot{\\theta}=\\frac{d \\theta}{d t}, \\dot{\\phi}=\\frac{d \\phi}{d t}$, and $\\dot{\\psi}=\\frac{d \\psi}{d t}$. Use Figure 3 if this is helpful. Give your answer in the $x y z$ frame, and in the 123 frame.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\n$$\n\\begin{aligned}\n\\mathbf{F}_{\\text {ext }} & =(N-m g) \\hat{z}+\\mathbf{F}_{f} \\quad(\\text { sufficient for full marks }) \\\\\n& =(N-m g) \\hat{z}-\\frac{\\mu_{k} N}{\\left|v_{A}\\right|} \\mathbf{v}_{\\mathbf{A}}\n\\end{aligned}\n$$\n\n$$ \\begin{aligned} \\boldsymbol{\\tau}_{\\text {ext }} & =\\mathbf{a} \\times\\left(N \\hat{\\mathbf{z}}+\\mathbf{F}_{f}\\right) \\\\ & =(\\alpha R \\hat{\\mathbf{3}}-R \\hat{\\mathbf{z}}) \\times\\left(N \\hat{\\mathbf{z}}+F_{f, x} \\hat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}\\right) \\\\ & =\\alpha R N \\hat{\\mathbf{3}} \\times \\hat{\\mathbf{z}}+\\alpha R(\\sin \\theta \\hat{\\mathbf{x}}+\\cos \\theta \\hat{\\mathbf{z}}) \\times\\left(F_{f, x} \\hat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}\\right)-R \\hat{\\mathbf{z}} \\times\\left(F_{f, x} \\hat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}\\right) \\\\ & =-\\alpha R N \\sin \\theta \\hat{\\mathbf{y}}+\\alpha R \\sin \\theta F_{f, y} \\hat{\\mathbf{z}}+\\alpha R \\cos \\theta F_{f, x} \\hat{\\mathbf{y}}-\\alpha R \\cos \\theta F_{f, y} \\hat{\\mathbf{x}}-R F_{f, x} \\hat{\\mathbf{y}}+R F_{f, x} \\hat{\\mathbf{x}} \\\\ & =R F_{f, y}(1-\\alpha \\cos \\theta) \\hat{\\mathbf{x}}+\\left[R F_{f, x}(\\alpha \\cos \\theta-1)-\\alpha R N \\sin \\theta\\right] \\hat{\\mathbf{y}}+\\alpha R \\sin \\theta F_{f, y} \\hat{\\mathbf{z}} \\end{aligned} $$\n\nproblem:\nA Tippe top is a special kind of top that can spontaneously invert once it has been set spinning. One can model a Tippe top as a sphere of radius $R$ that is truncated, with a stem added. It has rotational symmetry about an axis through the stem, which is at angle $\\theta$ from the vertical. As shown in Figure 1(a), its centre of mass $C$ is offset from its geometric centre $O$ by $\\alpha R$ along its symmetry axis. The Tippe top makes contact with the surface it rests on at point $A$; we assume this surface is planar, and refer to it as the floor. Given certain geometrical constraints and if spun fast enough initially, the Tippe top will tip so that the stem points increasingly downwards, until it starts to spin on in its stem, and eventually comes to a stop.\n[figure1]\n\nFigure 1. Views of the Tippe top (a) from the side and (b) from above\n\nLet $x y z$ be the rotating reference frame defined such that $\\hat{\\mathbf{z}}$ is stationary and upwards, and the top's symmetry axis is within the $x z$-plane. Two views of the Tippe top are shown in Figure 1: from the side, and from above. As shown in Figure 1(b), the top's symmetry axis is aligned with the $x$-axis when viewed from above.\n\nFigure 2 shows the top's motion at several phases after it is started spinning:\n\n(a) phase I: immediately after it is initially set spinning, with $\\theta \\sim 0$\n\n(b) phase II: soon after, having tipped to angle $0<\\theta<\\frac{\\pi}{2}$\n\n(c) phase III: when the stem first touches the floor, with $\\theta>\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nFind the total angular velocity $\\omega$ of the rotating top about its centre of mass $C$ in terms of the time derivatives of the Euler angles: $\\dot{\\theta}=\\frac{d \\theta}{d t}, \\dot{\\phi}=\\frac{d \\phi}{d t}$, and $\\dot{\\psi}=\\frac{d \\psi}{d t}$. Use Figure 3 if this is helpful. Give your answer in the $x y z$ frame, and in the 123 frame.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-1.jpg?height=522&width=1332&top_left_y=1121&top_left_x=360", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=578&width=1778&top_left_y=316&top_left_x=184", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=417&width=545&top_left_y=1296&top_left_x=527", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=431&width=397&top_left_y=1298&top_left_x=1189" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_632", "problem": "If the sphere immediately begins to roll without slipping, we can calculate the frictional impulse independently of the normal impulse. We have\n\n$$\nm v \\sin \\theta-p_{F}=m u\n$$\n\nThe frictional impulse is responsible for the sphere's rotation, so its angular momentum about its center of mass is $L=p_{F} R$. But we also know that\n\n$$\nL=\\beta m R^{2} \\omega=\\beta m R u\n$$\n\nThen\n\n$$\np_{F}=\\beta m u\n$$\n\nSubstituting into the previous expression gives\n\n$$\nm v \\sin \\theta=(1+\\beta) m u \\quad \\Rightarrow \\quad u=\\frac{v \\sin \\theta}{1+\\beta}\n$$Suppose you drop a sphere with mass $m$, radius $R$ and moment of inertia $\\beta m R^{2}$ vertically onto the same fixed ramp such that it reaches the ramp with speed $v$.\n\nSuppose the sphere immediately begins to roll without slipping. What is the minimum coefficient of friction such that the sphere rolls without slipping immediately after the collision?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIf the sphere immediately begins to roll without slipping, we can calculate the frictional impulse independently of the normal impulse. We have\n\n$$\nm v \\sin \\theta-p_{F}=m u\n$$\n\nThe frictional impulse is responsible for the sphere's rotation, so its angular momentum about its center of mass is $L=p_{F} R$. But we also know that\n\n$$\nL=\\beta m R^{2} \\omega=\\beta m R u\n$$\n\nThen\n\n$$\np_{F}=\\beta m u\n$$\n\nSubstituting into the previous expression gives\n\n$$\nm v \\sin \\theta=(1+\\beta) m u \\quad \\Rightarrow \\quad u=\\frac{v \\sin \\theta}{1+\\beta}\n$$\n\nproblem:\nSuppose you drop a sphere with mass $m$, radius $R$ and moment of inertia $\\beta m R^{2}$ vertically onto the same fixed ramp such that it reaches the ramp with speed $v$.\n\nSuppose the sphere immediately begins to roll without slipping. What is the minimum coefficient of friction such that the sphere rolls without slipping immediately after the collision?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_74", "problem": "A thick glass plate has parallel sides. A beam of white light is incident on one side at an angle between $0^{\\circ}$ and $90^{\\circ}$ with the normal. Which color emerges from the other side first?\nA: All of them\nB: red\nC: green\nD: violet\nE: None of them\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA thick glass plate has parallel sides. A beam of white light is incident on one side at an angle between $0^{\\circ}$ and $90^{\\circ}$ with the normal. Which color emerges from the other side first?\n\nA: All of them\nB: red\nC: green\nD: violet\nE: None of them\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_693", "problem": "A level (a device for establishing a horizontal plane, which consists of a small sealed transparent tube containing liquid and an air bubble) was pushed on the table. When the level was accelerated, the bubble:\nA: did not move relative to the glass\nB: moved in the direction of acceleration\nC: moved in the direction opposite to the acceleration\nD: moved to the side of the tube\nE: was pushed deeper into the liquid\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA level (a device for establishing a horizontal plane, which consists of a small sealed transparent tube containing liquid and an air bubble) was pushed on the table. When the level was accelerated, the bubble:\n\nA: did not move relative to the glass\nB: moved in the direction of acceleration\nC: moved in the direction opposite to the acceleration\nD: moved to the side of the tube\nE: was pushed deeper into the liquid\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1281", "problem": "质量均为 $m$ 的小球 1 和 2 由一质量可忽略、长度为 $l$ 的刚性轻杆连接,坚直地靠在墙角, 小球 1 在杆的上端, 如图所示. 假设墙和地面都是光滑的. 初始时给小球 2 一个微小的向右初速度. 问在系统运动过程中,当杆与坚直墙面之间的夹角等于何值时, 小球 1 开始离开坚直墙面?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n质量均为 $m$ 的小球 1 和 2 由一质量可忽略、长度为 $l$ 的刚性轻杆连接,坚直地靠在墙角, 小球 1 在杆的上端, 如图所示. 假设墙和地面都是光滑的. 初始时给小球 2 一个微小的向右初速度. 问在系统运动过程中,当杆与坚直墙面之间的夹角等于何值时, 小球 1 开始离开坚直墙面?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以degree为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e680ccd11e7de3ee63f1g-09.jpg?height=365&width=369&top_left_y=2025&top_left_x=1483" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "degree" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1258", "problem": "## Air flow around a wing\n\nFor this part of the problem, the following information may be useful. For a flow of liquid or gas in a tube along a streamline, $p+\\rho g h+\\frac{1}{2} \\rho v^{2}=$ const., assuming that the velocity $v$ is much less than the speed of sound. Here $\\rho$ is the density, $h$ is the height, $g$ is free fall acceleration and $p$ is hydrostatic pressure. Streamlines are defined as the trajectories of fluid particles (assuming that the flow pattern is stationary). Note that the term $\\frac{1}{2} \\rho v^{2}$ is called the dynamic pressure.\n\nIn the fig. shown below, a cross-section of an aircraft wing is depicted together with streamlines of the air flow around the wing, as seen in the wing's reference frame. Assume that (a) the air flow is purely two-dimensional (i.e. that the velocity vectors of air lie in the plane of the figure); (b) the streamline pattern is independent of the aircraft speed; (c) there is no wind; (d) the dynamic pressure is much smaller than the atmospheric pressure, $p_{0}=1.0 \\times 10^{5} \\mathrm{~Pa}$.\n\nYou can use a ruler to take measurements from the fig. on the answer sheet.\n\n[figure1]\n\nEstimate the critical speed $v_{\\text {crit }}$ using the following data: relative humidity of the air is $r=90 \\%$, specific heat capacity of air at constant pressure $c_{p}=1.00 \\times 10^{3} \\mathrm{~J} / \\mathrm{kg} \\cdot \\mathrm{K}$, pressure of saturated water vapour: $p_{s a}=2.31 \\mathrm{kPa}$ at the temperature of the unperturbed air $T_{a}=293 \\mathrm{~K}$ and $p_{s b}=2.46 \\mathrm{kPa}$ at $T_{b}=294 \\mathrm{~K}$. Depending on your approximations, you may also need the specific heat capacity of air at constant volume $c_{V}=0.717 \\times 10^{3} \\mathrm{~J} / \\mathrm{kg} \\cdot \\mathrm{K}$. Note that the relative humidity is defined as the ratio of the vapour pressure to the saturated vapour pressure at the given temperature. Saturated vapour pressure is defined as the vapour pressure by which vapour is in equilibrium with the liquid.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n## Air flow around a wing\n\nFor this part of the problem, the following information may be useful. For a flow of liquid or gas in a tube along a streamline, $p+\\rho g h+\\frac{1}{2} \\rho v^{2}=$ const., assuming that the velocity $v$ is much less than the speed of sound. Here $\\rho$ is the density, $h$ is the height, $g$ is free fall acceleration and $p$ is hydrostatic pressure. Streamlines are defined as the trajectories of fluid particles (assuming that the flow pattern is stationary). Note that the term $\\frac{1}{2} \\rho v^{2}$ is called the dynamic pressure.\n\nIn the fig. shown below, a cross-section of an aircraft wing is depicted together with streamlines of the air flow around the wing, as seen in the wing's reference frame. Assume that (a) the air flow is purely two-dimensional (i.e. that the velocity vectors of air lie in the plane of the figure); (b) the streamline pattern is independent of the aircraft speed; (c) there is no wind; (d) the dynamic pressure is much smaller than the atmospheric pressure, $p_{0}=1.0 \\times 10^{5} \\mathrm{~Pa}$.\n\nYou can use a ruler to take measurements from the fig. on the answer sheet.\n\n[figure1]\n\nEstimate the critical speed $v_{\\text {crit }}$ using the following data: relative humidity of the air is $r=90 \\%$, specific heat capacity of air at constant pressure $c_{p}=1.00 \\times 10^{3} \\mathrm{~J} / \\mathrm{kg} \\cdot \\mathrm{K}$, pressure of saturated water vapour: $p_{s a}=2.31 \\mathrm{kPa}$ at the temperature of the unperturbed air $T_{a}=293 \\mathrm{~K}$ and $p_{s b}=2.46 \\mathrm{kPa}$ at $T_{b}=294 \\mathrm{~K}$. Depending on your approximations, you may also need the specific heat capacity of air at constant volume $c_{V}=0.717 \\times 10^{3} \\mathrm{~J} / \\mathrm{kg} \\cdot \\mathrm{K}$. Note that the relative humidity is defined as the ratio of the vapour pressure to the saturated vapour pressure at the given temperature. Saturated vapour pressure is defined as the vapour pressure by which vapour is in equilibrium with the liquid.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_acce87fa3c24869ce68cg-1.jpg?height=260&width=948&top_left_y=1269&top_left_x=1039" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_551", "problem": "Scientists have recently detected a new star, the MAR-Kappa. The star is almost a perfect blackbody, and its measured light spectrum is shown below.\n\n[figure1]\n\nThe total measured light intensity from MAR-Kappa is $I=1.12 \\times 10^{-8} \\mathrm{~W} / \\mathrm{m}^{2}$. The mass of MAR-Kappa is estimated to be $3.5 \\times 10^{30} \\mathrm{~kg}$. It is stationary relative to the sun. You may find the Stefan-Boltzmann law useful, which states the power emitted by a blackbody with area $A$ is $\\sigma A T^{4}$.\n\nThe spectrum of wavelengths $\\lambda$ emitted from a blackbody only depends on $h, c, k_{B}, \\lambda$, and $T$. \n\nThe \"lines\" in the spectrum result from atoms in the star absorbing specific wavelengths of the emitted light. One contribution to the width of the spectral lines is the Doppler shift associated with the thermal motion of the atoms in the star. The spectral line at $\\lambda=389 \\mathrm{~nm}$ is due to helium. Estimate to within an order of magnitude the thermal broadening $\\Delta \\lambda$ of this line. The mass of a helium atom is $6.65 \\times 10^{-27} \\mathrm{~kg}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nScientists have recently detected a new star, the MAR-Kappa. The star is almost a perfect blackbody, and its measured light spectrum is shown below.\n\n[figure1]\n\nThe total measured light intensity from MAR-Kappa is $I=1.12 \\times 10^{-8} \\mathrm{~W} / \\mathrm{m}^{2}$. The mass of MAR-Kappa is estimated to be $3.5 \\times 10^{30} \\mathrm{~kg}$. It is stationary relative to the sun. You may find the Stefan-Boltzmann law useful, which states the power emitted by a blackbody with area $A$ is $\\sigma A T^{4}$.\n\nThe spectrum of wavelengths $\\lambda$ emitted from a blackbody only depends on $h, c, k_{B}, \\lambda$, and $T$. \n\nThe \"lines\" in the spectrum result from atoms in the star absorbing specific wavelengths of the emitted light. One contribution to the width of the spectral lines is the Doppler shift associated with the thermal motion of the atoms in the star. The spectral line at $\\lambda=389 \\mathrm{~nm}$ is due to helium. Estimate to within an order of magnitude the thermal broadening $\\Delta \\lambda$ of this line. The mass of a helium atom is $6.65 \\times 10^{-27} \\mathrm{~kg}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~nm}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_fed953f9e38b72bf8bd7g-17.jpg?height=640&width=1051&top_left_y=507&top_left_x=537" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~nm}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1361", "problem": "在固体材料中, 考虑相互作用后, 可以利用“准粒子”的概念研究材料的物理性质。准粒子的动能与动量之间的关系可能与真实粒子的不同。当外加电场或磁场时, 准粒子的运动往往可以用经典力学的方法来处理。在某种二维界面结构中, 存在电量为 $q$ 、有效质量为 $m$ 的准粒子, 它只能在 $x$ - $y$ 平面内运动, 其动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}+\\alpha p$,其中 $\\alpha$ 为正的常量。将该二维界面结构放置在匀强电场中, 准粒子可能在垂直于电场的方向上产生加速度。如果电场沿 $x$ 轴正方向, 电场强度大小为 $E$ 。当准粒子的速度大小为 $v(v \\neq \\alpha)$ 、方向与 $x$ 轴正方向成 $\\theta$ 角时, 求其运动的加速度的分量 $a_{x}$ 和 $a_{y}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n在固体材料中, 考虑相互作用后, 可以利用“准粒子”的概念研究材料的物理性质。准粒子的动能与动量之间的关系可能与真实粒子的不同。当外加电场或磁场时, 准粒子的运动往往可以用经典力学的方法来处理。在某种二维界面结构中, 存在电量为 $q$ 、有效质量为 $m$ 的准粒子, 它只能在 $x$ - $y$ 平面内运动, 其动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}+\\alpha p$,其中 $\\alpha$ 为正的常量。\n\n问题:\n将该二维界面结构放置在匀强电场中, 准粒子可能在垂直于电场的方向上产生加速度。如果电场沿 $x$ 轴正方向, 电场强度大小为 $E$ 。当准粒子的速度大小为 $v(v \\neq \\alpha)$ 、方向与 $x$ 轴正方向成 $\\theta$ 角时, 求其运动的加速度的分量 $a_{x}$ 和 $a_{y}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[求其运动的加速度的分量 $a_{x}$ , 求其运动的加速度的分量 $a_{y}$ ]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "求其运动的加速度的分量 $a_{x}$ ", "求其运动的加速度的分量 $a_{y}$ " ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_932", "problem": "An alternative hypothetical transmission line is shown in the diagram below. The input signal is sent through a very thin conductor of radius $a$, which is a distance $d \\gg a$ from a highly conductive grounded plane. The material surrounding the conductor has dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$. The return current flows along the grounded plane.\n\n[figure1]\n\nDiagram of a hypothetical transmission line showing $\\mathrm{C}$ - the conductor of radius $a$, at a distance $d \\gg a$ from $\\mathrm{P}$ - the grounded conducting plane. The conductor is embedded in a material with dimensionless relative permeability $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permittivity $\\mu_{\\mathrm{r}}$.\n Find an expression for the characteristic impedance of this hypothetical transmission line.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nAn alternative hypothetical transmission line is shown in the diagram below. The input signal is sent through a very thin conductor of radius $a$, which is a distance $d \\gg a$ from a highly conductive grounded plane. The material surrounding the conductor has dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$. The return current flows along the grounded plane.\n\n[figure1]\n\nDiagram of a hypothetical transmission line showing $\\mathrm{C}$ - the conductor of radius $a$, at a distance $d \\gg a$ from $\\mathrm{P}$ - the grounded conducting plane. The conductor is embedded in a material with dimensionless relative permeability $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permittivity $\\mu_{\\mathrm{r}}$.\n Find an expression for the characteristic impedance of this hypothetical transmission line.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-2.jpg?height=294&width=848&top_left_y=1955&top_left_x=610" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_103", "problem": "An object starting from rest can roll without slipping down an incline.\nWhich of the following four objects, each a uniform solid sphere released from rest, would have the largest speed after the center of mass has moved through a vertical distance $h$ ?\nA: A sphere of mass $M$ and radius $R$.\nB: A sphere of mass $2 M$ and radius $\\frac{1}{2} R$.\nC: A sphere of mass $M / 2$ and radius $2 R$.\nD: A sphere of mass $3 M$ and radius $3 R$.\nE: All objects would have the same speed. \n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn object starting from rest can roll without slipping down an incline.\nWhich of the following four objects, each a uniform solid sphere released from rest, would have the largest speed after the center of mass has moved through a vertical distance $h$ ?\n\nA: A sphere of mass $M$ and radius $R$.\nB: A sphere of mass $2 M$ and radius $\\frac{1}{2} R$.\nC: A sphere of mass $M / 2$ and radius $2 R$.\nD: A sphere of mass $3 M$ and radius $3 R$.\nE: All objects would have the same speed. \n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1735", "problem": "有一根长为 $6.00 \\mathrm{~cm}$ 、内外半径分别为 $0.500 \\mathrm{~mm}$ 和 $5.00 \\mathrm{~mm}$ 的玻璃毛细管。\n\n已知玻璃的密度是水的 2 倍, 水的密度为 $1.00 \\times 10^{3} \\mathrm{~kg} \\cdot \\mathrm{m}^{-3}$, 水的表面张力系数为 $7.27 \\times 10^{-2} \\mathrm{~N} \\cdot \\mathrm{m}^{-1}$, 水与玻璃的接触角 $\\theta$ 可视为零, 重力加速度取 $9.80 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$ 。毛细管坚直悬空固定放置, 注入水后, 在管的下端中央形成一悬挂的水滴, 管中水柱表面中心相对于水滴底部的高度为 $3.50 \\mathrm{~cm}$, 求水滴底部表面的曲率半径 $a$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n有一根长为 $6.00 \\mathrm{~cm}$ 、内外半径分别为 $0.500 \\mathrm{~mm}$ 和 $5.00 \\mathrm{~mm}$ 的玻璃毛细管。\n\n已知玻璃的密度是水的 2 倍, 水的密度为 $1.00 \\times 10^{3} \\mathrm{~kg} \\cdot \\mathrm{m}^{-3}$, 水的表面张力系数为 $7.27 \\times 10^{-2} \\mathrm{~N} \\cdot \\mathrm{m}^{-1}$, 水与玻璃的接触角 $\\theta$ 可视为零, 重力加速度取 $9.80 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$ 。\n\n问题:\n毛细管坚直悬空固定放置, 注入水后, 在管的下端中央形成一悬挂的水滴, 管中水柱表面中心相对于水滴底部的高度为 $3.50 \\mathrm{~cm}$, 求水滴底部表面的曲率半径 $a$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~m}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_123", "problem": "A launcher is designed to shoot objects horizontally across an ice rink. It consists of two boards of negligible mass connected via a spring-loaded hinge, which exerts a constant torque $\\tau$ on each board to keep them together. For both problems, neglect friction with either the ice or the launch boards.\n\nThe disc is removed and replaced with a pie-shaped wedge of the same mass $m$, so that the hinge is still initially held open at an angle $\\theta$, as shown in the following figure. If the wedge is released from rest, what is its speed after it exits the launcher?\n\n[figure1]\nA: $\\sqrt{\\frac{\\tau}{2 m \\theta}}$\nB: $\\sqrt{\\frac{4 \\tau \\theta}{m}}$\nC: $\\sqrt{\\frac{\\tau \\theta^{2}}{2 m}}$\nD: $\\sqrt{\\frac{2 \\tau \\theta}{m}} $ \nE: $\\sqrt{\\frac{2 \\tau}{m \\theta}}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA launcher is designed to shoot objects horizontally across an ice rink. It consists of two boards of negligible mass connected via a spring-loaded hinge, which exerts a constant torque $\\tau$ on each board to keep them together. For both problems, neglect friction with either the ice or the launch boards.\n\nThe disc is removed and replaced with a pie-shaped wedge of the same mass $m$, so that the hinge is still initially held open at an angle $\\theta$, as shown in the following figure. If the wedge is released from rest, what is its speed after it exits the launcher?\n\n[figure1]\n\nA: $\\sqrt{\\frac{\\tau}{2 m \\theta}}$\nB: $\\sqrt{\\frac{4 \\tau \\theta}{m}}$\nC: $\\sqrt{\\frac{\\tau \\theta^{2}}{2 m}}$\nD: $\\sqrt{\\frac{2 \\tau \\theta}{m}} $ \nE: $\\sqrt{\\frac{2 \\tau}{m \\theta}}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-15.jpg?height=317&width=512&top_left_y=400&top_left_x=812" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_276", "problem": "A bicycle tyre has a volume of $1800 \\mathrm{~cm}$ and contains compressed air at pressure of $36 \\mathrm{psi}$. The air in the tyre behaves as an ideal gas.\n\nThe air in the tyre is released slowly so that it is at a pressure of 1 atmosphere with no change in temperature.\n\n- $1 \\mathrm{psi}=6900 \\mathrm{~Pa}$\n- 1 atmosphere $=100000 \\mathrm{~Pa}$\n\nWhat volume does the air from the tyre occupy after it is released?\nA: $\\quad 0.0045 \\mathrm{~m}^{3}$\nB: $\\quad 0.45 \\mathrm{~m}^{3}$\nC: $\\quad 45 \\mathrm{~m}^{3}$\nD: $\\quad 4500 \\mathrm{~m}^{3}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA bicycle tyre has a volume of $1800 \\mathrm{~cm}$ and contains compressed air at pressure of $36 \\mathrm{psi}$. The air in the tyre behaves as an ideal gas.\n\nThe air in the tyre is released slowly so that it is at a pressure of 1 atmosphere with no change in temperature.\n\n- $1 \\mathrm{psi}=6900 \\mathrm{~Pa}$\n- 1 atmosphere $=100000 \\mathrm{~Pa}$\n\nWhat volume does the air from the tyre occupy after it is released?\n\nA: $\\quad 0.0045 \\mathrm{~m}^{3}$\nB: $\\quad 0.45 \\mathrm{~m}^{3}$\nC: $\\quad 45 \\mathrm{~m}^{3}$\nD: $\\quad 4500 \\mathrm{~m}^{3}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_174", "problem": "A cylinder has a radius $R$ and weight $G$. You try to roll it over a step of height $h0)$ 。因发生紧急情况, 卡车突然制动。已知卡车车轮与地面间的动摩擦因数和最大静摩擦因数均为 $\\mu_{1}$, 重物与车厢底板间的动摩擦因数和最大静摩擦因数均为 $\\mu_{2}\\left(\\mu_{2}<\\mu_{1}\\right)$ 。若重物与车厢前壁发生碰撞, 则假定碰撞时间极短, 碰后重物与车厢前壁不分开。重力加速度大小为 $g$ 。\n\n[图1]若重物和车厢前壁发生碰撞, 求卡车从制动开始到卡车和重物都停止的过程所经历的时间、卡车走过的路程、以及碰撞过程中重物对车厢前壁的冲量。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一质量为 $M$ 的载重卡车 $\\mathrm{A}$ 的水平车板上载有一质量为 $m$ 的重物 $\\mathrm{B}$, 在水平直公路上以速度 $v_{0}$ 做匀速直线运动, 重物与车厢前壁间的距离为 $L(L>0)$ 。因发生紧急情况, 卡车突然制动。已知卡车车轮与地面间的动摩擦因数和最大静摩擦因数均为 $\\mu_{1}$, 重物与车厢底板间的动摩擦因数和最大静摩擦因数均为 $\\mu_{2}\\left(\\mu_{2}<\\mu_{1}\\right)$ 。若重物与车厢前壁发生碰撞, 则假定碰撞时间极短, 碰后重物与车厢前壁不分开。重力加速度大小为 $g$ 。\n\n[图1]\n\n问题:\n若重物和车厢前壁发生碰撞, 求卡车从制动开始到卡车和重物都停止的过程所经历的时间、卡车走过的路程、以及碰撞过程中重物对车厢前壁的冲量。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[重物在卡车停下后与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程所经历的时间, 重物在卡车停下后与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程卡车走过的路程, 重物在卡车停下后与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程碰撞过程中重物对车厢前壁的冲量。, 重物在卡车停下前与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程碰撞过程中所经历的时间, 重物在卡车停下前与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程碰撞过程中卡车走过的路程, 重物在卡车停下前与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程碰撞过程中重物对车厢前壁的冲量]\n它们的答案类型依次是[表达式, 表达式, 表达式, 表达式, 表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-02.jpg?height=319&width=751&top_left_y=286&top_left_x=1007" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null, null, null, null ], "answer_sequence": [ "重物在卡车停下后与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程所经历的时间", "重物在卡车停下后与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程卡车走过的路程", "重物在卡车停下后与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程碰撞过程中重物对车厢前壁的冲量。", "重物在卡车停下前与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程碰撞过程中所经历的时间", "重物在卡车停下前与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程碰撞过程中卡车走过的路程", "重物在卡车停下前与车厢前壁发生碰撞情况下卡车从制动开始到卡车和重物都停止的过程碰撞过程中重物对车厢前壁的冲量" ], "type_sequence": [ "EX", "EX", "EX", "EX", "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_716", "problem": "People were watching a movie showing a car driving inside a big cylinder on a vertical wall (see below a frame from this video obtained from www.aol.co.uk/2012/04/05). How would you explain this?\n\n[figure1]\nA: Impossible - some kind of film trick\nB: The only way to do it would be to add some very strong magnets in the wheels and drive on iron wall. The attraction force between the magnets and the iron wall results in the friction force to be bigger than the force of gravity pushing the car down.\nC: Most likely, there is a huge vacuum pump under the car, which sucks it to the wall and results in a friction force bigger than the force of gravity pushing the car down.\nD: The car is moving at high speed so a centripetal force forcing it to drive in circle inside the cylinder results in the force pushing the car towards the wall. This force results in the friction force bigger than the force of gravity pushing the car down.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nPeople were watching a movie showing a car driving inside a big cylinder on a vertical wall (see below a frame from this video obtained from www.aol.co.uk/2012/04/05). How would you explain this?\n\n[figure1]\n\nA: Impossible - some kind of film trick\nB: The only way to do it would be to add some very strong magnets in the wheels and drive on iron wall. The attraction force between the magnets and the iron wall results in the friction force to be bigger than the force of gravity pushing the car down.\nC: Most likely, there is a huge vacuum pump under the car, which sucks it to the wall and results in a friction force bigger than the force of gravity pushing the car down.\nD: The car is moving at high speed so a centripetal force forcing it to drive in circle inside the cylinder results in the force pushing the car towards the wall. This force results in the friction force bigger than the force of gravity pushing the car down.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22e26a14ee6fdd9254b6g-04.jpg?height=372&width=712&top_left_y=405&top_left_x=1151" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_532", "problem": "A unicyclist of total height $h$ goes around a circular track of radius $R$ while leaning inward at an angle $\\theta$ to the vertical. The acceleration due to gravity is $g$.\n\nSuppose $h \\ll R$. What angular velocity $\\omega$ must the unicyclist sustain?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA unicyclist of total height $h$ goes around a circular track of radius $R$ while leaning inward at an angle $\\theta$ to the vertical. The acceleration due to gravity is $g$.\n\nSuppose $h \\ll R$. What angular velocity $\\omega$ must the unicyclist sustain?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1595", "problem": "1914 年，弗兰克-赫兹用电子碰撞原子的方法使原子从低能级激发到高能级，从而证明了原子能级的存在。加速电子碰撞自由的氢原子, 使某氢原子从基态激发到激发态。该氢原子仅能发出一条可见光波长范围 ( 400nm 760nm ) 内的光谱线。仅考虑一维正碰。\n\n已知 $h c=1240 \\mathrm{~nm} \\cdot \\mathrm{eV}$, 其中 $h$ 为普朗克常量, $c$ 为真空中的光速; 质子质量近似为电子质量的 1836 倍, 氢原子在碰撞前的速度可忽略。如果将电子改为质子, 求加速质子的加速电压的范围。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个区间。\n这里是一些可能会帮助你解决问题的先验信息提示：\n1914 年，弗兰克-赫兹用电子碰撞原子的方法使原子从低能级激发到高能级，从而证明了原子能级的存在。加速电子碰撞自由的氢原子, 使某氢原子从基态激发到激发态。该氢原子仅能发出一条可见光波长范围 ( 400nm 760nm ) 内的光谱线。仅考虑一维正碰。\n\n已知 $h c=1240 \\mathrm{~nm} \\cdot \\mathrm{eV}$, 其中 $h$ 为普朗克常量, $c$ 为真空中的光速; 质子质量近似为电子质量的 1836 倍, 氢原子在碰撞前的速度可忽略。\n\n问题:\n如果将电子改为质子, 求加速质子的加速电压的范围。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~V}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含任何单位的区间，例如ANSWER=(1,2] \\cup[7,+\\infty)", "figure_urls": null, "answer": null, "solution": null, "answer_type": "IN", "unit": [ "$\\mathrm{~V}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_608", "problem": "In the steady state, the current is the same everywhere. Consider the region $(x, x+d x)$. The time it takes for the charge in the second region to leave is $\\frac{\\mathrm{d} x}{v(x)}$. The amount of charge that leaves is $\\rho A \\mathrm{~d} x$. The current is thus given by $\\rho A v$, so $\\rho v$ is constant. Alternatively, one can write this as\n\n$$\nv \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}+\\rho \\frac{\\mathrm{d} v}{\\mathrm{~d} x}=0\n$$\n\nThe position $x$ is effectively in between two uniform sheets of charge density. The sheet on the left has charge density $\\int_{0}^{x} \\rho \\mathrm{d} x+\\sigma_{0}$, where $\\sigma_{0}$ is the charge density on the left plate, and the sheet on the right has charge density $\\int_{x}^{d} \\rho \\mathrm{d} x+\\sigma_{d}$, where $\\sigma_{d}$ is the charge density on the left plate. Then, the electric field is given by\n\n$$\nE=\\sigma_{0} /\\left(2 \\epsilon_{0}\\right)+\\int_{0}^{x} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\int_{x}^{d} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\sigma_{d} /\\left(2 \\epsilon_{0}\\right)\n$$\n\nThen, by the Fundamental Theorem of Calculus\n\n$$\n\\frac{d E}{d x}=\\frac{\\rho}{\\epsilon_{0}}\n$$\nso\n\n$$\n\\frac{d^{2} V}{d x^{2}}=-\\frac{\\rho}{\\epsilon_{0}}\n$$Two large parallel plates of area $A$ are placed at $x=0$ and $x=d \\ll \\sqrt{A}$ in a semiconductor medium. The plate at $x=0$ is grounded, and the plate at $x=d$ is at a fixed potential $-V_{0}$, where $V_{0}>0$. Particles of positive charge $q$ flow between the two plates. You may neglect any dielectric effects of the medium.\n\nFor large $V_{0}$, the velocity of the positive charges is determined by a strong drag force, so that\n\n$$\nv=\\mu E\n$$\n\nwhere $E$ is the local electric field and $\\mu$ is the charge mobility.\n\nIn the steady state, there is a nonzero but time-independent density of charges between the two plates. Let the charge density at position $x$ be $\\rho(x)$. Let $V(x)$ be the electric potential at $x$. \n\nSuppose that in the steady state, conditions have been established so that $V(x)$ is proportional to $x^{b}$, where $b$ is an exponent you must find, and the current is nonzero. Derive an expression for the current in terms of $V_{0}$ and the other given parameters.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nIn the steady state, the current is the same everywhere. Consider the region $(x, x+d x)$. The time it takes for the charge in the second region to leave is $\\frac{\\mathrm{d} x}{v(x)}$. The amount of charge that leaves is $\\rho A \\mathrm{~d} x$. The current is thus given by $\\rho A v$, so $\\rho v$ is constant. Alternatively, one can write this as\n\n$$\nv \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}+\\rho \\frac{\\mathrm{d} v}{\\mathrm{~d} x}=0\n$$\n\nThe position $x$ is effectively in between two uniform sheets of charge density. The sheet on the left has charge density $\\int_{0}^{x} \\rho \\mathrm{d} x+\\sigma_{0}$, where $\\sigma_{0}$ is the charge density on the left plate, and the sheet on the right has charge density $\\int_{x}^{d} \\rho \\mathrm{d} x+\\sigma_{d}$, where $\\sigma_{d}$ is the charge density on the left plate. Then, the electric field is given by\n\n$$\nE=\\sigma_{0} /\\left(2 \\epsilon_{0}\\right)+\\int_{0}^{x} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\int_{x}^{d} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\sigma_{d} /\\left(2 \\epsilon_{0}\\right)\n$$\n\nThen, by the Fundamental Theorem of Calculus\n\n$$\n\\frac{d E}{d x}=\\frac{\\rho}{\\epsilon_{0}}\n$$\nso\n\n$$\n\\frac{d^{2} V}{d x^{2}}=-\\frac{\\rho}{\\epsilon_{0}}\n$$\n\nproblem:\nTwo large parallel plates of area $A$ are placed at $x=0$ and $x=d \\ll \\sqrt{A}$ in a semiconductor medium. The plate at $x=0$ is grounded, and the plate at $x=d$ is at a fixed potential $-V_{0}$, where $V_{0}>0$. Particles of positive charge $q$ flow between the two plates. You may neglect any dielectric effects of the medium.\n\nFor large $V_{0}$, the velocity of the positive charges is determined by a strong drag force, so that\n\n$$\nv=\\mu E\n$$\n\nwhere $E$ is the local electric field and $\\mu$ is the charge mobility.\n\nIn the steady state, there is a nonzero but time-independent density of charges between the two plates. Let the charge density at position $x$ be $\\rho(x)$. Let $V(x)$ be the electric potential at $x$. \n\nSuppose that in the steady state, conditions have been established so that $V(x)$ is proportional to $x^{b}$, where $b$ is an exponent you must find, and the current is nonzero. Derive an expression for the current in terms of $V_{0}$ and the other given parameters.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1293", "problem": "[图1]\n\n亚毫米细丝直径的双光束干涉测量装置如图(a)所示，其中 $M_{1} 、 M_{2} 、 M_{3}$ 为全反射镜, 相机为 CCD (电荷耦合装置)相机. 来自钒酸铱自倍频激光器的激光束被分束器分成两束, 一束经全反镜 $M_{1}$ 反射后从上侧入射到细丝上; 另一束经全反镜 $M_{3}$ 和 $M_{2}$ 相继反射后从下侧入射到细丝上. 图(b)给出了两条反射光线产生干涉的光路, 其中 $q_{1} 、 q_{2}$ 分别为上、下两侧的入射角, $D$ 为细丝轴线到观察屏 (即相机感光片)的距离, $P$ 为两条反射线在观察屏上的交点. 已建立这样的直角坐标系: 坐标原点 $O$ 位于细丝的轴线上, $x$ 轴(未画出)沿细丝轴线、指向纸面内, $y$轴与入射到细丝上的光线平行 ( $y$ 轴的正向向上), $z$ 轴指向观察屏并与其垂直. 已知光波长为 $\\lambda$, 屏上干涉条纹的间距为 $a$. 由于 $D$ 远大于细丝直径和观察屏的尺寸, 可假设投射到屏上的只有非常接近平行于 $z$ 轴的细光束.\n\n[图2]\n\n图(a)\n\n[图3]\n\n图(b)求细丝的直径 $d$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n[图1]\n\n亚毫米细丝直径的双光束干涉测量装置如图(a)所示，其中 $M_{1} 、 M_{2} 、 M_{3}$ 为全反射镜, 相机为 CCD (电荷耦合装置)相机. 来自钒酸铱自倍频激光器的激光束被分束器分成两束, 一束经全反镜 $M_{1}$ 反射后从上侧入射到细丝上; 另一束经全反镜 $M_{3}$ 和 $M_{2}$ 相继反射后从下侧入射到细丝上. 图(b)给出了两条反射光线产生干涉的光路, 其中 $q_{1} 、 q_{2}$ 分别为上、下两侧的入射角, $D$ 为细丝轴线到观察屏 (即相机感光片)的距离, $P$ 为两条反射线在观察屏上的交点. 已建立这样的直角坐标系: 坐标原点 $O$ 位于细丝的轴线上, $x$ 轴(未画出)沿细丝轴线、指向纸面内, $y$轴与入射到细丝上的光线平行 ( $y$ 轴的正向向上), $z$ 轴指向观察屏并与其垂直. 已知光波长为 $\\lambda$, 屏上干涉条纹的间距为 $a$. 由于 $D$ 远大于细丝直径和观察屏的尺寸, 可假设投射到屏上的只有非常接近平行于 $z$ 轴的细光束.\n\n[图2]\n\n图(a)\n\n[图3]\n\n图(b)\n\n问题:\n求细丝的直径 $d$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e680ccd11e7de3ee63f1g-06.jpg?height=94&width=346&top_left_y=244&top_left_x=244", "https://cdn.mathpix.com/cropped/2024_03_31_e680ccd11e7de3ee63f1g-06.jpg?height=371&width=506&top_left_y=934&top_left_x=521", "https://cdn.mathpix.com/cropped/2024_03_31_e680ccd11e7de3ee63f1g-06.jpg?height=417&width=586&top_left_y=931&top_left_x=1089" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_904", "problem": "In fact the energy damping rate $\\gamma_{\\omega}$ is independent of the electron orbits. Therefore we will adopt another simple model where the electron cloud center performs a circular motion in the absence of the laser field but with the frequency $\\omega$ and speed $v$. Being accelerated, the electron radiates an electromagnectic wave with power given by the Larmor formula $P_{L}=\\frac{1}{6 \\pi \\varepsilon_{0}} \\frac{e^{2} a^{2}}{c^{3}}$ with $a$ denoting acceleration. The damping force is supposed to be related to the damping rate $\\gamma_{\\omega}$ as $F_{d}=-m_{e} \\gamma_{\\omega} v$. We also assume that the total energy of the electron is large compared with the energy loss per cycle.\n\n Find the energy damping rate $\\gamma_{\\omega}$ in term of $e, \\epsilon_{0}, c, m_{e}$, and $\\omega$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn fact the energy damping rate $\\gamma_{\\omega}$ is independent of the electron orbits. Therefore we will adopt another simple model where the electron cloud center performs a circular motion in the absence of the laser field but with the frequency $\\omega$ and speed $v$. Being accelerated, the electron radiates an electromagnectic wave with power given by the Larmor formula $P_{L}=\\frac{1}{6 \\pi \\varepsilon_{0}} \\frac{e^{2} a^{2}}{c^{3}}$ with $a$ denoting acceleration. The damping force is supposed to be related to the damping rate $\\gamma_{\\omega}$ as $F_{d}=-m_{e} \\gamma_{\\omega} v$. We also assume that the total energy of the electron is large compared with the energy loss per cycle.\n\n Find the energy damping rate $\\gamma_{\\omega}$ in term of $e, \\epsilon_{0}, c, m_{e}$, and $\\omega$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1322", "problem": "根据广义相对论, 光线在星体的引力场中会发生弯曲, 在包含引力中心的平面内是一条在引力中心附近微弯的曲线, 它距离引力中心的最近的点, 称为光线的近星点。通过近星点与引力中心的直线, 是光线的对称轴。若在光线所在的平面内选择引力中心为平面极坐标 $(r, \\phi)$ 的原点, 选取光线的对称轴为极坐标轴, 则光线方程 (光子的轨迹方程) 为\n\n$$\nr=\\frac{G M / c^{2}}{a \\cos \\phi+a^{2}\\left(1+\\sin ^{2} \\phi\\right)}\n$$\n\n$\\mathrm{G}$ 是万有引力恒量, $\\mathrm{M}$ 是星体质量, $\\mathrm{C}$ 是光速, $\\mathrm{a}$ 是绝对值远小于 1 的参数。现在假设离地球 80.0 光年处有一星体, 在它与地球球心的连线的中点处有一白矮星。如果经过该白矮星两侧的星光对地球上的观察者所张的视角是 $1.80 \\times 10^{-7}$ 弧度, 试问此白矮星的质量是多少千克? 已知 $G=6.673 \\times 10^{-11} \\mathrm{~m}^{3} /\\left(\\mathrm{kg} \\cdot \\mathrm{s}^{2}\\right)$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n根据广义相对论, 光线在星体的引力场中会发生弯曲, 在包含引力中心的平面内是一条在引力中心附近微弯的曲线, 它距离引力中心的最近的点, 称为光线的近星点。通过近星点与引力中心的直线, 是光线的对称轴。若在光线所在的平面内选择引力中心为平面极坐标 $(r, \\phi)$ 的原点, 选取光线的对称轴为极坐标轴, 则光线方程 (光子的轨迹方程) 为\n\n$$\nr=\\frac{G M / c^{2}}{a \\cos \\phi+a^{2}\\left(1+\\sin ^{2} \\phi\\right)}\n$$\n\n$\\mathrm{G}$ 是万有引力恒量, $\\mathrm{M}$ 是星体质量, $\\mathrm{C}$ 是光速, $\\mathrm{a}$ 是绝对值远小于 1 的参数。现在假设离地球 80.0 光年处有一星体, 在它与地球球心的连线的中点处有一白矮星。如果经过该白矮星两侧的星光对地球上的观察者所张的视角是 $1.80 \\times 10^{-7}$ 弧度, 试问此白矮星的质量是多少千克? 已知 $G=6.673 \\times 10^{-11} \\mathrm{~m}^{3} /\\left(\\mathrm{kg} \\cdot \\mathrm{s}^{2}\\right)$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以kg为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_1004b08dedac85274c96g-18.jpg?height=488&width=180&top_left_y=681&top_left_x=1349" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "kg" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_786", "problem": "The cockatoo and sparrow become tangled together during the collision into a single feathery object. What happens in the instant immediately after the collision?\n\nSelect one:\nA: The feathery object moves in the same direction as the cockatoo's original motion.\nB: The feathery object moves in the same direction as the sparrow's original motion.\nC: The feathery object stops dead in the air.\nD: As soon as they collide they move directly downward.\nE: More information is needed to answer this question.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe cockatoo and sparrow become tangled together during the collision into a single feathery object. What happens in the instant immediately after the collision?\n\nSelect one:\n\nA: The feathery object moves in the same direction as the cockatoo's original motion.\nB: The feathery object moves in the same direction as the sparrow's original motion.\nC: The feathery object stops dead in the air.\nD: As soon as they collide they move directly downward.\nE: More information is needed to answer this question.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_416", "problem": "The figure shows a more complex system, known as a block and tackle, consisting of two light pulley blocks and a light cord.\n\n[figure1]\nFigure: Two light pulley blocks and a light cord.\n\nHow much work is done by the external agent pulling cord towards $\\mathbf{X}$?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe figure shows a more complex system, known as a block and tackle, consisting of two light pulley blocks and a light cord.\n\n[figure1]\nFigure: Two light pulley blocks and a light cord.\n\nHow much work is done by the external agent pulling cord towards $\\mathbf{X}$?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of %, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-4.jpg?height=1074&width=514&top_left_y=1593&top_left_x=1356" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "%" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_493", "problem": "For this problem, assume the existence of a hypothetical particle known as a magnetic monopole. Such a particle would have a \"magnetic charge\" $q_{m}$, and in analogy to an electrically charged particle would produce a radially directed magnetic field of magnitude\n\n$$\nB=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{r^{2}}\n$$\n\nand be subject to a force (in the absence of electric fields)\n\n$$\nF=q_{m} B\n$$\n\nA magnetic monopole of mass $m$ and magnetic charge $q_{m}$ is constrained to move on a vertical, nonmagnetic, insulating, frictionless U-shaped track. At the bottom of the track is a wire loop whose radius $b$ is much smaller than the width of the \"U\" of the track. The section of track near the loop can thus be approximated as a long straight line. The wire that makes up the loop has radius $a \\ll b$ and resistivity $\\rho$. The monopole is released from rest a height $H$ above the bottom of the track.\n\nIgnore the self-inductance of the loop, and assume that the monopole passes through the loop many times before coming to a rest.\n\nHow many times does the monopole pass through the loop before coming to a rest?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nFor this problem, assume the existence of a hypothetical particle known as a magnetic monopole. Such a particle would have a \"magnetic charge\" $q_{m}$, and in analogy to an electrically charged particle would produce a radially directed magnetic field of magnitude\n\n$$\nB=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{r^{2}}\n$$\n\nand be subject to a force (in the absence of electric fields)\n\n$$\nF=q_{m} B\n$$\n\nA magnetic monopole of mass $m$ and magnetic charge $q_{m}$ is constrained to move on a vertical, nonmagnetic, insulating, frictionless U-shaped track. At the bottom of the track is a wire loop whose radius $b$ is much smaller than the width of the \"U\" of the track. The section of track near the loop can thus be approximated as a long straight line. The wire that makes up the loop has radius $a \\ll b$ and resistivity $\\rho$. The monopole is released from rest a height $H$ above the bottom of the track.\n\nIgnore the self-inductance of the loop, and assume that the monopole passes through the loop many times before coming to a rest.\n\nHow many times does the monopole pass through the loop before coming to a rest?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_150", "problem": "A massless spring hangs from the ceiling, and a mass is hung from the bottom of it. The mass is supported so that initially the tension in the spring is zero. The mass is then suddenly released. At the bottom of its trajectory, the mass is 5 centimeters from its original position. Find its oscillation period.\nA: $0.05 \\mathrm{~s}$\nB: $0.07 \\mathrm{~s}$\nC: $0.31 \\mathrm{~s} $ \nD: $0.44 \\mathrm{~s}$\nE: Not enough information is given.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA massless spring hangs from the ceiling, and a mass is hung from the bottom of it. The mass is supported so that initially the tension in the spring is zero. The mass is then suddenly released. At the bottom of its trajectory, the mass is 5 centimeters from its original position. Find its oscillation period.\n\nA: $0.05 \\mathrm{~s}$\nB: $0.07 \\mathrm{~s}$\nC: $0.31 \\mathrm{~s} $ \nD: $0.44 \\mathrm{~s}$\nE: Not enough information is given.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_101", "problem": "The depth of a well, $d$, is measured by dropping a stone into it and measuring the time $t$ until the splash is heard at the bottom. What is the smallest value of $d$ for which ignoring the time for the sound to travel gives less than a $5 \\%$ error in the depth measurement? The speed of sound in air is $330 \\mathrm{~m} / \\mathrm{s}$.\nA: $3.5 \\mathrm{~m}$\nB: $7 \\mathrm{~m}$\nC: $14 \\mathrm{~m} $ \nD: $54 \\mathrm{~m}$\nE: $330 \\mathrm{~m}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe depth of a well, $d$, is measured by dropping a stone into it and measuring the time $t$ until the splash is heard at the bottom. What is the smallest value of $d$ for which ignoring the time for the sound to travel gives less than a $5 \\%$ error in the depth measurement? The speed of sound in air is $330 \\mathrm{~m} / \\mathrm{s}$.\n\nA: $3.5 \\mathrm{~m}$\nB: $7 \\mathrm{~m}$\nC: $14 \\mathrm{~m} $ \nD: $54 \\mathrm{~m}$\nE: $330 \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1702", "problem": "如图, 一质量为 $M$ 、长为 $l$ 的匀质细杆 $\\mathrm{AB}$ 自由悬挂于通过坐标原点 $O$ 点的水平光滑转轴上 (此时, 杆的上端 $\\mathrm{A}$ 未在图中标出, 可视为与 $O$ 点重合), 杆可绕通过 $O$ 点的轴在坚直平面 (即 $x-y$ 平面, $x$ 轴正方向水平向右）内转动; $O$ 点相对于地面足够高, 初始时杆自然下垂; 一质量为 $m$ 的弹丸以大小为 $v_{0}$ 的水平速度撞击杆的打击中心（打击过程中轴对杆的水平作用力为零）并很快嵌入杆中。在杆转半圈至坚直状态时立即撤除转轴。重力加速度大小为 $g$ 。\n\n[图1]以撤除转轴的瞬间为计时零点, 求撤除转轴后直至杆着地前, 杆端 $B$ 的位置随时间 $t$ 变化的表达式 $x_{\\mathrm{B}}(t)$ 和 $y_{\\mathrm{B}}(t)$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一质量为 $M$ 、长为 $l$ 的匀质细杆 $\\mathrm{AB}$ 自由悬挂于通过坐标原点 $O$ 点的水平光滑转轴上 (此时, 杆的上端 $\\mathrm{A}$ 未在图中标出, 可视为与 $O$ 点重合), 杆可绕通过 $O$ 点的轴在坚直平面 (即 $x-y$ 平面, $x$ 轴正方向水平向右）内转动; $O$ 点相对于地面足够高, 初始时杆自然下垂; 一质量为 $m$ 的弹丸以大小为 $v_{0}$ 的水平速度撞击杆的打击中心（打击过程中轴对杆的水平作用力为零）并很快嵌入杆中。在杆转半圈至坚直状态时立即撤除转轴。重力加速度大小为 $g$ 。\n\n[图1]\n\n问题:\n以撤除转轴的瞬间为计时零点, 求撤除转轴后直至杆着地前, 杆端 $B$ 的位置随时间 $t$ 变化的表达式 $x_{\\mathrm{B}}(t)$ 和 $y_{\\mathrm{B}}(t)$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-01.jpg?height=483&width=374&top_left_y=2437&top_left_x=1412", "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-15.jpg?height=274&width=311&top_left_y=2470&top_left_x=1409" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_247", "problem": "Earth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nWhen a proton rotates around a field line it generates a magnetic moment which remains constant along its path. When the proton reaches some latitude $\\lambda_{m}$ the parallel velocity becomes zero and the particle starts its path backwards. This latitude is known as the mirror point.\n\nDetermine the magnetic moment $\\mu$ created by a proton when it rotates around a magnetic field line as a function of $L, W$ and the pitch angle of a particle at Equator $\\alpha_{E q}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nEarth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nWhen a proton rotates around a field line it generates a magnetic moment which remains constant along its path. When the proton reaches some latitude $\\lambda_{m}$ the parallel velocity becomes zero and the particle starts its path backwards. This latitude is known as the mirror point.\n\nDetermine the magnetic moment $\\mu$ created by a proton when it rotates around a magnetic field line as a function of $L, W$ and the pitch angle of a particle at Equator $\\alpha_{E q}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=496&width=705&top_left_y=774&top_left_x=713", "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=220&width=504&top_left_y=1690&top_left_x=802" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1731", "problem": "电子感应加速器 (betatron) 的基本原理如下: 一个圆环真空室处于分布在圆柱形体积内的磁场中, 磁场方向沿圆柱的轴线, 圆柱的轴线过圆环的圆心并与环面垂直。圆中两个同心的实线圆代表圆环的边界, 与实线圆同心的虚线圆为电子在加速过程中运行的轨道。已知磁场的磁感应强度 $\\mathrm{B}$ 随时间 $\\mathrm{t}$ 的变化规律为 $B=B_{0} \\cos (2 \\pi t / T)$, 其中 $\\mathrm{T}$\n\n[图1]\n为磁场变化的周期。 $\\mathrm{B}_{0}$ 为大于 0 的常量。当 $\\mathrm{B}$ 为正时, 磁场的方向垂直于纸面指向纸外。若持续地将初速度为 $\\mathrm{v}_{0}$ 的电子沿虚线圆的切线方向注入到环内 (如图), 则电子在该磁场变化的一个周期内可能被加速的时间是从 $\\mathrm{t}=$到 $\\mathrm{t}=$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n电子感应加速器 (betatron) 的基本原理如下: 一个圆环真空室处于分布在圆柱形体积内的磁场中, 磁场方向沿圆柱的轴线, 圆柱的轴线过圆环的圆心并与环面垂直。圆中两个同心的实线圆代表圆环的边界, 与实线圆同心的虚线圆为电子在加速过程中运行的轨道。已知磁场的磁感应强度 $\\mathrm{B}$ 随时间 $\\mathrm{t}$ 的变化规律为 $B=B_{0} \\cos (2 \\pi t / T)$, 其中 $\\mathrm{T}$\n\n[图1]\n为磁场变化的周期。 $\\mathrm{B}_{0}$ 为大于 0 的常量。当 $\\mathrm{B}$ 为正时, 磁场的方向垂直于纸面指向纸外。若持续地将初速度为 $\\mathrm{v}_{0}$ 的电子沿虚线圆的切线方向注入到环内 (如图), 则电子在该磁场变化的一个周期内可能被加速的时间是从 $\\mathrm{t}=$到 $\\mathrm{t}=$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[可能被加速的时间起始, 可能被加速的时间结束]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_40a5e1e69014b22d267bg-01.jpg?height=325&width=328&top_left_y=1522&top_left_x=1298" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "可能被加速的时间起始", "可能被加速的时间结束" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_472", "problem": "A vacuum system consists of a chamber of volume $V$ connected to a vacuum pump that is a cylinder with a piston that moves left and right. The minimum volume in the pump cylinder is $V_{0}$, and the maximum volume in the cylinder is $V_{0}+\\Delta V$. You should assume that $\\Delta V \\ll V$.\n\n[figure1]\n\nThe cylinder has two valves. The inlet valve opens when the pressure inside the cylinder is lower than the pressure in the chamber, but closes when the piston moves to the right. The outlet valve opens when the pressure inside the cylinder is greater than atmospheric pressure $P_{a}$, and closes when the piston moves to the left. A motor drives the piston to move back and forth. The piston moves at such a rate that heat is not conducted in or out of the gas contained in the cylinder during the pumping cycle. One complete cycle takes a time $\\Delta t$. You should assume that $\\Delta t$ is a very small quantity, but $\\Delta V / \\Delta t=R$ is finite. The gas in the chamber is ideal monatomic and remains at a fixed temperature of $T_{a}$.\n\nStart with assumption that $V_{0}=0$ and there are no leaks in the system.\n\nAt $t=0$ the pressure inside the chamber is $P_{a}$. Find an equation for the pressure at a later time $t$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA vacuum system consists of a chamber of volume $V$ connected to a vacuum pump that is a cylinder with a piston that moves left and right. The minimum volume in the pump cylinder is $V_{0}$, and the maximum volume in the cylinder is $V_{0}+\\Delta V$. You should assume that $\\Delta V \\ll V$.\n\n[figure1]\n\nThe cylinder has two valves. The inlet valve opens when the pressure inside the cylinder is lower than the pressure in the chamber, but closes when the piston moves to the right. The outlet valve opens when the pressure inside the cylinder is greater than atmospheric pressure $P_{a}$, and closes when the piston moves to the left. A motor drives the piston to move back and forth. The piston moves at such a rate that heat is not conducted in or out of the gas contained in the cylinder during the pumping cycle. One complete cycle takes a time $\\Delta t$. You should assume that $\\Delta t$ is a very small quantity, but $\\Delta V / \\Delta t=R$ is finite. The gas in the chamber is ideal monatomic and remains at a fixed temperature of $T_{a}$.\n\nStart with assumption that $V_{0}=0$ and there are no leaks in the system.\n\nAt $t=0$ the pressure inside the chamber is $P_{a}$. Find an equation for the pressure at a later time $t$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_0cd94795c1ff89067cedg-09.jpg?height=377&width=1222&top_left_y=489&top_left_x=446" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_329", "problem": "The black box has three terminal wires: \"blue\", \"black\" and \"white\", and contains n a star configuration: a battery, a capacitor, an inductor in series with a diode. You may consider the diode to be \"ideal\" - it conducts current perfectly one way and not at all the other way. You may neglect internal resistance of the battery and capacitor, but the inductor has considerable internal resistance. The multimeter's internal resistance when measuring voltages is $R_{m}=10 \\mathrm{M} \\Omega$ and it displays a new reading every $t=0.4 \\mathrm{~s}$.\n\nEstimate the value of the inductance L", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a range interval.\n\nproblem:\nThe black box has three terminal wires: \"blue\", \"black\" and \"white\", and contains n a star configuration: a battery, a capacitor, an inductor in series with a diode. You may consider the diode to be \"ideal\" - it conducts current perfectly one way and not at all the other way. You may neglect internal resistance of the battery and capacitor, but the inductor has considerable internal resistance. The multimeter's internal resistance when measuring voltages is $R_{m}=10 \\mathrm{M} \\Omega$ and it displays a new reading every $t=0.4 \\mathrm{~s}$.\n\nEstimate the value of the inductance L\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{mH}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an interval without any units, e.g. ANSWER=(1,2] \\cup[7,+\\infty)", "figure_urls": null, "answer": null, "solution": null, "answer_type": "IN", "unit": [ "$\\mathrm{mH}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_445", "problem": "The figure shows a load of mass, $m$, supported by a simple pulley system with a tension $T$ in the cord. \n\n[figure1]\nFigure: Two light pulleys and a light cord.\n\nState the increase in gravitational potential energy of the mass, and comment on the mechanical efficiency of this arrangement.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe figure shows a load of mass, $m$, supported by a simple pulley system with a tension $T$ in the cord. \n\n[figure1]\nFigure: Two light pulleys and a light cord.\n\nState the increase in gravitational potential energy of the mass, and comment on the mechanical efficiency of this arrangement.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of %, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-4.jpg?height=497&width=391&top_left_y=742&top_left_x=838" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "%" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1411", "problem": "一轻杆通过分别连在其两端的轻质弹簧 $\\mathrm{A}$ 和 B 悬挂在天花板下, 一物块 $\\mathrm{D}$ 通过轻质弹簧 C 连在轻杆上; A、B 和 C 的劲度系数分别为 $k_{1} 、 k_{2}$ 和 $k_{3}, \\mathrm{D}$ 的质量为 $m, \\mathrm{C}$ 与轻杆的连接点到 $\\mathrm{A}$ 和 $\\mathrm{B}$ 的水平距离分别为 $a$ 和 $b$; 整个系统的平衡时, 轻杆接近水平, 如图所示。假设物块 $\\mathrm{D}$ 在坚直方向做微小振动, $\\mathrm{A} 、 \\mathrm{~B}$ 始终可视为坚直, 忽略空气阻力。\n\n[图1]当 $a$ 和 $b$ 满足什么条件时，物块 $\\mathrm{D}$ 的固有频率最大? 并求出该固有频率的最大值。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一轻杆通过分别连在其两端的轻质弹簧 $\\mathrm{A}$ 和 B 悬挂在天花板下, 一物块 $\\mathrm{D}$ 通过轻质弹簧 C 连在轻杆上; A、B 和 C 的劲度系数分别为 $k_{1} 、 k_{2}$ 和 $k_{3}, \\mathrm{D}$ 的质量为 $m, \\mathrm{C}$ 与轻杆的连接点到 $\\mathrm{A}$ 和 $\\mathrm{B}$ 的水平距离分别为 $a$ 和 $b$; 整个系统的平衡时, 轻杆接近水平, 如图所示。假设物块 $\\mathrm{D}$ 在坚直方向做微小振动, $\\mathrm{A} 、 \\mathrm{~B}$ 始终可视为坚直, 忽略空气阻力。\n\n[图1]\n\n问题:\n当 $a$ 和 $b$ 满足什么条件时，物块 $\\mathrm{D}$ 的固有频率最大? 并求出该固有频率的最大值。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$a$ 和 $b$ 满足什么条件时，物块 $\\mathrm{D}$ 的固有频率最大?, 固有频率的最大值]\n它们的答案类型依次是[方程, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_07aa406e17d01fd01b36g-01.jpg?height=428&width=457&top_left_y=937&top_left_x=1299" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$a$ 和 $b$ 满足什么条件时，物块 $\\mathrm{D}$ 的固有频率最大?", "固有频率的最大值" ], "type_sequence": [ "EQ", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_576", "problem": "Single bubble sonoluminescence occurs when sound waves cause a bubble suspended in a fluid to collapse so that the gas trapped inside increases in temperature enough to emit light. The bubble actually undergoes a series of expansions and collapses caused by the sound wave pressure variations.\n\nWe now consider a simplified model of a bubble undergoing sonoluminescence. Assume the bubble is originally at atmospheric pressure $P_{0}=101 \\mathrm{kPa}$. When the pressure in the fluid surrounding the bubble is decreased, the bubble expands isothermally to a radius of $36.0 \\mu \\mathrm{m}$. When the pressure increases again, the bubble collapses to a radius of $4.50 \\mu \\mathrm{m}$ so quickly that no heat can escape. Between the collapse and subsequent expansion, the bubble undergoes isochoric (constant volume) cooling back to its original pressure and temperature. For a bubble containing a monatomic gas, suspended in water of $T=293 \\mathrm{~K}$, find the number of moles of gas in the bubble.\n\nYou may find the following useful: the specific heat capacity at constant volume is $C_{V}=3 R / 2$ and the ratio of specific heat at constant pressure to constant volume is $\\gamma=5 / 3$ for a monatomic gas.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nSingle bubble sonoluminescence occurs when sound waves cause a bubble suspended in a fluid to collapse so that the gas trapped inside increases in temperature enough to emit light. The bubble actually undergoes a series of expansions and collapses caused by the sound wave pressure variations.\n\nWe now consider a simplified model of a bubble undergoing sonoluminescence. Assume the bubble is originally at atmospheric pressure $P_{0}=101 \\mathrm{kPa}$. When the pressure in the fluid surrounding the bubble is decreased, the bubble expands isothermally to a radius of $36.0 \\mu \\mathrm{m}$. When the pressure increases again, the bubble collapses to a radius of $4.50 \\mu \\mathrm{m}$ so quickly that no heat can escape. Between the collapse and subsequent expansion, the bubble undergoes isochoric (constant volume) cooling back to its original pressure and temperature. For a bubble containing a monatomic gas, suspended in water of $T=293 \\mathrm{~K}$, find the number of moles of gas in the bubble.\n\nYou may find the following useful: the specific heat capacity at constant volume is $C_{V}=3 R / 2$ and the ratio of specific heat at constant pressure to constant volume is $\\gamma=5 / 3$ for a monatomic gas.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~mol}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~mol}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1736", "problem": "一足球运动员 1 自 $\\mathrm{A}$ 点向球门的 $\\mathrm{B}$ 点踢出球, 已知 $\\mathrm{A} 、 \\mathrm{~B}$ 之间的距离为 $S$, 球自 $\\mathrm{A}$ 向 $\\mathrm{B}$ 的运动可视为水平地面上的匀速直线运动, 速率为 $u$ 。另一足球运动员 2 到 $\\mathrm{AB}$ 连线的距离为 $l$, 到 $\\mathrm{A}$ 、 $\\mathrm{B}$ 两点的距离相等。运动员 1 踢出球后, 运动员 2 以匀速 $v$ 沿直线去拦截该球。设运动员 2 开始出发去拦截球的时刻与球被运动员 1 踢出球的时刻相同。求为了使运动员 2 能拦截到球， $u 、 v 、 s$ 和 $l$ 应当满足的条件。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一足球运动员 1 自 $\\mathrm{A}$ 点向球门的 $\\mathrm{B}$ 点踢出球, 已知 $\\mathrm{A} 、 \\mathrm{~B}$ 之间的距离为 $S$, 球自 $\\mathrm{A}$ 向 $\\mathrm{B}$ 的运动可视为水平地面上的匀速直线运动, 速率为 $u$ 。另一足球运动员 2 到 $\\mathrm{AB}$ 连线的距离为 $l$, 到 $\\mathrm{A}$ 、 $\\mathrm{B}$ 两点的距离相等。运动员 1 踢出球后, 运动员 2 以匀速 $v$ 沿直线去拦截该球。设运动员 2 开始出发去拦截球的时刻与球被运动员 1 踢出球的时刻相同。\n\n问题:\n求为了使运动员 2 能拦截到球， $u 、 v 、 s$ 和 $l$ 应当满足的条件。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[u满足条件, t满足条件, $s_{1}$满足条件, $s_{2}$满足条件]\n它们的答案类型依次是[数值, 数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null, null ], "answer_sequence": [ "u满足条件", "t满足条件", "$s_{1}$满足条件", "$s_{2}$满足条件" ], "type_sequence": [ "NV", "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_313", "problem": "A space station at a geostationary orbit has a form of a cylinder of length $L=100 \\mathrm{~km}$ and radius $R=1 \\mathrm{~km}$ is filled with air (molar mass $M=29 \\mathrm{~g} / \\mathrm{mol}$ ) at the atmospheric pressure and temperatur $T=295 \\mathrm{~K}$ and the cylindrical walls serve as ground for the people living inside. It rotates around its axis so as to create normal gravity $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$ at the \"ground\"\n\n\nWhat is the rotation period $\\tau$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA space station at a geostationary orbit has a form of a cylinder of length $L=100 \\mathrm{~km}$ and radius $R=1 \\mathrm{~km}$ is filled with air (molar mass $M=29 \\mathrm{~g} / \\mathrm{mol}$ ) at the atmospheric pressure and temperatur $T=295 \\mathrm{~K}$ and the cylindrical walls serve as ground for the people living inside. It rotates around its axis so as to create normal gravity $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$ at the \"ground\"\n\n\nWhat is the rotation period $\\tau$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1521", "problem": "如图, 导热性能良好的气缸 A 和 B 高度均为 $h$ (已除开活塞的厚度）, 横截面积不同, 坚直浸没在温度为 $T_{0}$ 的恒温槽内, 它们的底部由一细管连通（细管容积可忽略）。两气缸内各有一个活塞, 质量\n\n[图1]\n分别为 $m_{\\mathrm{A}}=2 m$ 和 $m_{\\mathrm{B}}=m$, 活塞与气缸之间无摩擦, 两活塞的下方为理想气体, 上方为真空。当两活塞下方气体处于平衡状态时, 两活塞底面相对于气缸底的高度均为 $\\frac{h}{2}$ 。现保持恒温槽温度不变, 在两活塞上面同时各缓慢加上同样大小的压力, 让压力从零缓慢增加, 直至其大小等于 $2 m g$ ( $g$ 为重力加速度) 为止, 并一直保持两活塞上的压力不变; 系统再次达到平衡后, 缓慢升高恒温慒的温度, 对气体加热, 直至气缸 B 中活塞底面恰好回到高度为 $\\frac{h}{2}$ 处。求两个活塞的横截面积之比 $S_{\\mathrm{A}}: S_{\\mathrm{B}}$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 导热性能良好的气缸 A 和 B 高度均为 $h$ (已除开活塞的厚度）, 横截面积不同, 坚直浸没在温度为 $T_{0}$ 的恒温槽内, 它们的底部由一细管连通（细管容积可忽略）。两气缸内各有一个活塞, 质量\n\n[图1]\n分别为 $m_{\\mathrm{A}}=2 m$ 和 $m_{\\mathrm{B}}=m$, 活塞与气缸之间无摩擦, 两活塞的下方为理想气体, 上方为真空。当两活塞下方气体处于平衡状态时, 两活塞底面相对于气缸底的高度均为 $\\frac{h}{2}$ 。现保持恒温槽温度不变, 在两活塞上面同时各缓慢加上同样大小的压力, 让压力从零缓慢增加, 直至其大小等于 $2 m g$ ( $g$ 为重力加速度) 为止, 并一直保持两活塞上的压力不变; 系统再次达到平衡后, 缓慢升高恒温慒的温度, 对气体加热, 直至气缸 B 中活塞底面恰好回到高度为 $\\frac{h}{2}$ 处。求\n\n问题:\n两个活塞的横截面积之比 $S_{\\mathrm{A}}: S_{\\mathrm{B}}$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-06.jpg?height=242&width=343&top_left_y=530&top_left_x=1413" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_644", "problem": "Draw a free-body diagram for both blocks and we can find that the acceleration of $A$ points down along the incline: $a_{A}=g \\sin \\theta-\\mu_{A} g \\cos \\theta=\\frac{1}{2} g-\\frac{1}{4} g=2.5 \\mathrm{~m} / \\mathrm{s}^{2}$. Similarly, $a_{B}=g \\sin \\theta-\\mu_{B} g \\cos \\theta=0$.\n\nLet's first look at a qualitative picture of the collisions: When $A$ slides down the incline before colliding with $B$, it moves with acceleration. When the blocks collide, the total momentum of the system is conserved. Because $m_{A}=m_{B}$, the blocks exchange velocity, and thus $B$ slides down with constant velocity. After a momentary stop, $A$ will again accelerate down the incline, and catches up with $B$, and another collision occurs.\n\nQuantitatively, the first collision happens when $A$ travels $\\ell=5 \\mathrm{~cm}=0.05 \\mathrm{~m}$.\n\n$$\nt_{1}=\\sqrt{2 \\ell / a_{A}}=0.2 \\mathrm{~s} ; \\quad v_{A 1}=a_{A} t_{1}=0.5 \\mathrm{~m} / \\mathrm{s}\n$$\n\nThis is when $A$ and $B$ first collide. Then $B$ moves down the incline at constant velocity $v_{B}=v_{A_{1}}=0.5 \\mathrm{~m} / \\mathrm{s}$, while $A$ starts from rest and accelerates down the incline with $a_{A}=$ $2.5 \\mathrm{~m} / \\mathrm{s}^{2}$, until catches up with $B$ at $t_{2}=0.6 \\mathrm{~s}$. At that point,\n\n$$\nv_{A 2}=a_{A}\\left(t_{2}-t_{1}\\right)=1 \\mathrm{~m} / \\mathrm{s}\n$$\n\nUsing a similar approach, we can find that at $t_{3}=1 \\mathrm{~s}$.\n\nGraphically, the $v_{A / B}(t)$ graphs during the first second are included below.\n\n[figure1]Two blocks, $A$ and $B$, of the same mass are on a fixed inclined plane, which makes a $30^{\\circ}$ angle with the horizontal. At time $t=0, A$ is a distance $\\ell=5 \\mathrm{~cm}$ along the incline above $B$, and both blocks are at rest. Suppose the coefficients of static and kinetic friction between the blocks and the incline are\n\n$$\n\\mu_{A}=\\frac{\\sqrt{3}}{6}, \\quad \\mu_{B}=\\frac{\\sqrt{3}}{3}\n$$\n\nand that the blocks collide perfectly elastically. Let $v_{A}(t)$ and $v_{B}(t)$ be the speeds of the blocks down the incline. For this problem, use $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, assume both blocks stay on the incline for the entire time, and neglect the sizes of the blocks.\n\nDerive an expression for the total distance block $A$ has moved from its original position right after its $n^{\\text {th }}$ collision, in terms of $\\ell$ and $n$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nDraw a free-body diagram for both blocks and we can find that the acceleration of $A$ points down along the incline: $a_{A}=g \\sin \\theta-\\mu_{A} g \\cos \\theta=\\frac{1}{2} g-\\frac{1}{4} g=2.5 \\mathrm{~m} / \\mathrm{s}^{2}$. Similarly, $a_{B}=g \\sin \\theta-\\mu_{B} g \\cos \\theta=0$.\n\nLet's first look at a qualitative picture of the collisions: When $A$ slides down the incline before colliding with $B$, it moves with acceleration. When the blocks collide, the total momentum of the system is conserved. Because $m_{A}=m_{B}$, the blocks exchange velocity, and thus $B$ slides down with constant velocity. After a momentary stop, $A$ will again accelerate down the incline, and catches up with $B$, and another collision occurs.\n\nQuantitatively, the first collision happens when $A$ travels $\\ell=5 \\mathrm{~cm}=0.05 \\mathrm{~m}$.\n\n$$\nt_{1}=\\sqrt{2 \\ell / a_{A}}=0.2 \\mathrm{~s} ; \\quad v_{A 1}=a_{A} t_{1}=0.5 \\mathrm{~m} / \\mathrm{s}\n$$\n\nThis is when $A$ and $B$ first collide. Then $B$ moves down the incline at constant velocity $v_{B}=v_{A_{1}}=0.5 \\mathrm{~m} / \\mathrm{s}$, while $A$ starts from rest and accelerates down the incline with $a_{A}=$ $2.5 \\mathrm{~m} / \\mathrm{s}^{2}$, until catches up with $B$ at $t_{2}=0.6 \\mathrm{~s}$. At that point,\n\n$$\nv_{A 2}=a_{A}\\left(t_{2}-t_{1}\\right)=1 \\mathrm{~m} / \\mathrm{s}\n$$\n\nUsing a similar approach, we can find that at $t_{3}=1 \\mathrm{~s}$.\n\nGraphically, the $v_{A / B}(t)$ graphs during the first second are included below.\n\n[figure1]\n\nproblem:\nTwo blocks, $A$ and $B$, of the same mass are on a fixed inclined plane, which makes a $30^{\\circ}$ angle with the horizontal. At time $t=0, A$ is a distance $\\ell=5 \\mathrm{~cm}$ along the incline above $B$, and both blocks are at rest. Suppose the coefficients of static and kinetic friction between the blocks and the incline are\n\n$$\n\\mu_{A}=\\frac{\\sqrt{3}}{6}, \\quad \\mu_{B}=\\frac{\\sqrt{3}}{3}\n$$\n\nand that the blocks collide perfectly elastically. Let $v_{A}(t)$ and $v_{B}(t)$ be the speeds of the blocks down the incline. For this problem, use $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, assume both blocks stay on the incline for the entire time, and neglect the sizes of the blocks.\n\nDerive an expression for the total distance block $A$ has moved from its original position right after its $n^{\\text {th }}$ collision, in terms of $\\ell$ and $n$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_fed953f9e38b72bf8bd7g-05.jpg?height=694&width=835&top_left_y=236&top_left_x=669" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_12", "problem": "Light is refracted when it passes from air into glass. Which of these wave properties changes?\nA: Speed only\nB: Wavelength only\nC: Frequency only\nD: Both speed and wavelength\nE: Both speed and frequency\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nLight is refracted when it passes from air into glass. Which of these wave properties changes?\n\nA: Speed only\nB: Wavelength only\nC: Frequency only\nD: Both speed and wavelength\nE: Both speed and frequency\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_717", "problem": "A photocell emits electrons when it is illuminated with green light. We are uncertain whether it emits electrons when it is illuminated with:\nA: ultra-violet radiation\nB: X-rays\nC: red light\nD: blue light\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA photocell emits electrons when it is illuminated with green light. We are uncertain whether it emits electrons when it is illuminated with:\n\nA: ultra-violet radiation\nB: X-rays\nC: red light\nD: blue light\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_382", "problem": "In this problem we will look at the phase diagram of water (see graphs on a separate page). The first figure shows the phase diagram in a region close to the triple point [(s) - solid, (l) - liquid, (g) - gas], while the second figure shows the melting curve. When two phases $\\alpha$ and $\\beta$ are in equilibrium, the phase transition curve follows the law of Clausius-Clapeyron:$$\n\\frac{\\mathrm{d} p}{\\mathrm{~d} T}=\\frac{1}{T} \\frac{H_{\\beta}-H_{\\alpha}}{V_{\\beta}-V_{\\alpha}} $$\n\nwhere $H_{\\alpha}$ is the specific enthalpy (enthalpy per mass) of phase $\\alpha$, and $V_{\\alpha}$ is the specific volume (volume per mass).\n\nUsing that $V_{g} \\gg V_{l}$, find an expression for the liquid-gas transition curve $p(T)$ in terms of the latent heat of evaporation $\\Delta H_{l g} \\equiv H_{l}-H_{g}$, the pressure $p_{0}$ at any reference point along the curve, the gas constant $R$ and the molar mass $\\mu$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn this problem we will look at the phase diagram of water (see graphs on a separate page). The first figure shows the phase diagram in a region close to the triple point [(s) - solid, (l) - liquid, (g) - gas], while the second figure shows the melting curve. When two phases $\\alpha$ and $\\beta$ are in equilibrium, the phase transition curve follows the law of Clausius-Clapeyron:$$\n\\frac{\\mathrm{d} p}{\\mathrm{~d} T}=\\frac{1}{T} \\frac{H_{\\beta}-H_{\\alpha}}{V_{\\beta}-V_{\\alpha}} $$\n\nwhere $H_{\\alpha}$ is the specific enthalpy (enthalpy per mass) of phase $\\alpha$, and $V_{\\alpha}$ is the specific volume (volume per mass).\n\nUsing that $V_{g} \\gg V_{l}$, find an expression for the liquid-gas transition curve $p(T)$ in terms of the latent heat of evaporation $\\Delta H_{l g} \\equiv H_{l}-H_{g}$, the pressure $p_{0}$ at any reference point along the curve, the gas constant $R$ and the molar mass $\\mu$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1491", "problem": "如图, 一固定的坚直长导线载有恒定电流 $I$, 其旁边有一正方形导线框, 导线框可围绕过对边中心的坚直轴 $\\mathrm{O}_{1} \\mathrm{O}_{2}$ 转动, 转轴到长直导线的距离为 $b$ 。已知导线框的边长为 $2 a$ $(a0)$ 。现将一厚度为 $t$ 、面积为 $S / 2$\n\n[图1]\n\n（宽度和原来的极板相同, 长度是原来极板的一半）的金属片在上极板的正下方平行插入电容器, 将电容器分成如图所示的 $1 、 2 、 3$ 三部分。不考虑边缘效应。静电力常量为 $k$ 。试求插入金属片以后电容器的总电容;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n真空中平行板电容器两极板的面积均为 $S$, 相距 $d$, 上、下极\n\n板所带电量分别为 $Q$ 和 $-Q(Q>0)$ 。现将一厚度为 $t$ 、面积为 $S / 2$\n\n[图1]\n\n（宽度和原来的极板相同, 长度是原来极板的一半）的金属片在上极板的正下方平行插入电容器, 将电容器分成如图所示的 $1 、 2 、 3$ 三部分。不考虑边缘效应。静电力常量为 $k$ 。试求\n\n问题:\n插入金属片以后电容器的总电容;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://i.postimg.cc/c4YQHfdj/2016-CPho-Q13.png" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_536", "problem": "Consider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nA second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height $H$, we can write\n\n$$\n2 \\int_{0}^{H} p d x=\\left(n+\\frac{1}{2}\\right) h\n$$\n\nwhere $p$ is the momentum as a function of height $x$ above the ground, $n$ is a non-negative integer, and $h$ is Planck's constant.\n\nDetermine the bounce height of one of these minimum energy neutrons.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nConsider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nA second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height $H$, we can write\n\n$$\n2 \\int_{0}^{H} p d x=\\left(n+\\frac{1}{2}\\right) h\n$$\n\nwhere $p$ is the momentum as a function of height $x$ above the ground, $n$ is a non-negative integer, and $h$ is Planck's constant.\n\nDetermine the bounce height of one of these minimum energy neutrons.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mu \\mathrm{m}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mu \\mathrm{m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_634", "problem": "Consider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nA second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height $H$, we can write\n\n$$\n2 \\int_{0}^{H} p d x=\\left(n+\\frac{1}{2}\\right) h\n$$\n\nwhere $p$ is the momentum as a function of height $x$ above the ground, $n$ is a non-negative integer, and $h$ is Planck's constant.\n\nNumerically determine the minimum energy of a bouncing neutron. The mass of a neutron is $m_{n}=1.675 \\times 10^{-27} \\mathrm{~kg}=940 \\mathrm{MeV} / \\mathrm{c}^{2}$; you may express your answer in either Joules or eV.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nConsider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nA second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height $H$, we can write\n\n$$\n2 \\int_{0}^{H} p d x=\\left(n+\\frac{1}{2}\\right) h\n$$\n\nwhere $p$ is the momentum as a function of height $x$ above the ground, $n$ is a non-negative integer, and $h$ is Planck's constant.\n\nNumerically determine the minimum energy of a bouncing neutron. The mass of a neutron is $m_{n}=1.675 \\times 10^{-27} \\mathrm{~kg}=940 \\mathrm{MeV} / \\mathrm{c}^{2}$; you may express your answer in either Joules or eV.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{eV}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{eV}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_373", "problem": "The V-I-curve of a tunnel diode is depicted in the figure below, curve (a). In some parts of the problem, we use an idealized model curve (b).\n\n[figure1]\nThus far we have assumed that the diode is an ideal device; in reality, it has a small parasitic capacitance, let it be $C=30 \\mathrm{pF}$. Taking this into account, our circuit should be drawn as shown in diagram (c). Now we assume the ammeter, again, to be non-ideal, with internal resistance $r=2 \\Omega$. Let us assume that after closing the switch, the voltage was slowly increased from $\\mathscr{E}=0 \\mathrm{mV}$ to $\\mathscr{E}=150 \\mathrm{mV}$ so that a sta tionary (oscillations-free) operation regime $V(t) \\equiv V_{0}$ and $I(t) \\equiv I_{0}$ has been achieved Suppose there is a small perturbation to the diode current and voltage: $I=I_{0}+\\delta I(t)$ and $V=V_{0}+\\delta V(t)$, where $I_{0}$ and $V_{0}$ are the cur rent and voltage in the stationary operationa regime. For small perturbation amplitudes, the V-I-curve of the diode can be linearized resulting in $\\delta V=R_{d} \\delta I$, where $R_{d}$ is the differ ential resistance of the diode. Determine the value of $R_{d}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe V-I-curve of a tunnel diode is depicted in the figure below, curve (a). In some parts of the problem, we use an idealized model curve (b).\n\n[figure1]\nThus far we have assumed that the diode is an ideal device; in reality, it has a small parasitic capacitance, let it be $C=30 \\mathrm{pF}$. Taking this into account, our circuit should be drawn as shown in diagram (c). Now we assume the ammeter, again, to be non-ideal, with internal resistance $r=2 \\Omega$. Let us assume that after closing the switch, the voltage was slowly increased from $\\mathscr{E}=0 \\mathrm{mV}$ to $\\mathscr{E}=150 \\mathrm{mV}$ so that a sta tionary (oscillations-free) operation regime $V(t) \\equiv V_{0}$ and $I(t) \\equiv I_{0}$ has been achieved Suppose there is a small perturbation to the diode current and voltage: $I=I_{0}+\\delta I(t)$ and $V=V_{0}+\\delta V(t)$, where $I_{0}$ and $V_{0}$ are the cur rent and voltage in the stationary operationa regime. For small perturbation amplitudes, the V-I-curve of the diode can be linearized resulting in $\\delta V=R_{d} \\delta I$, where $R_{d}$ is the differ ential resistance of the diode. Determine the value of $R_{d}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\Omega, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_0b06edcc1e983d034aadg-1.jpg?height=414&width=690&top_left_y=35&top_left_x=744" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\Omega" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_485", "problem": "The Doppler effect for a source moving relative to a stationary observer is described by\n\n$$\nf=\\frac{f_{0}}{1-(v / c) \\cos \\theta}\n$$\n\nwhere $f$ is the frequency measured by the observer, $f_{0}$ is the frequency emitted by the source, $v$ is the speed of the source, $c$ is the wave speed, and $\\theta$ is the angle between the source velocity and the line between the source and observer. (Thus $\\theta=0$ when the source is moving directly towards the observer and $\\theta=\\pi$ when moving directly away.)\n\nA sound source of constant frequency travels at a constant velocity past an observer, and the observed frequency is plotted as a function of time:\n\n[figure1]\n\nThe experiment happens in room temperature air, so the speed of sound is $340 \\mathrm{~m} / \\mathrm{s}$.\n\nWhat is the speed of the source?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe Doppler effect for a source moving relative to a stationary observer is described by\n\n$$\nf=\\frac{f_{0}}{1-(v / c) \\cos \\theta}\n$$\n\nwhere $f$ is the frequency measured by the observer, $f_{0}$ is the frequency emitted by the source, $v$ is the speed of the source, $c$ is the wave speed, and $\\theta$ is the angle between the source velocity and the line between the source and observer. (Thus $\\theta=0$ when the source is moving directly towards the observer and $\\theta=\\pi$ when moving directly away.)\n\nA sound source of constant frequency travels at a constant velocity past an observer, and the observed frequency is plotted as a function of time:\n\n[figure1]\n\nThe experiment happens in room temperature air, so the speed of sound is $340 \\mathrm{~m} / \\mathrm{s}$.\n\nWhat is the speed of the source?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~m} / \\mathrm{s}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_33ea9fa74c8b34628eedg-03.jpg?height=1261&width=1569&top_left_y=936&top_left_x=278" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m} / \\mathrm{s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_476", "problem": "Suppose that, after some amount of work is done by the ion pumps, the charges on the outer and inner surfaces are $Q$ and $-Q$, respectively.\n\nOne charge layer by itself creates an electric field $E_{1}=Q /(2 \\epsilon A)$ in each direction. So the force between the two sides of the membrane $F_{E}=Q E_{1}=Q^{2} /(2 \\epsilon A)$.\n\nThis electric force is balanced by the spring force $F_{s}=k x$, where $x=d_{0}-d$. Equating these two forces and solving for $d$ gives gives\n\n$$\nd=d_{0}-\\frac{Q^{2}}{2 \\epsilon A k} .\n$$\n\nThe electric field inside the membrane (as produced by both the left and right plates) is $E=Q /\\left(\\epsilon_{0} \\kappa A\\right)$. So the voltage between them is\n\n$$\nV=E d=\\frac{Q}{\\epsilon A} d\n$$\n\nInserting the expression for $Q$ from part (a) gives\n\n$$\nV=\\frac{Q}{\\epsilon A}\\left(d_{0}-\\frac{Q^{2}}{2 \\epsilon A k}\\right)\n$$\n\nThis equation implies that as the charge $Q$ is increased, the voltage first increases and then decreases again.[figure1]\n\nThe wall of a neuron is made from an elastic membrane, which resists compression in the same way as a spring. It has an effective spring constant $k$ and an equilibrium thickness $d_{0}$. Assume that the membrane has a very large area $A$ and negligible curvature.\n\nThe neuron has \"ion pumps\" that can move ions across the membrane. In the resulting charged state, positive and negative ionic charge is arranged uniformly along the outer and inner surfaces of the membrane, respectively. The permittivity of the membrane is $\\epsilon$.\n\nSuppose that the ion pumps are first turned on in the uncharged state, and the membrane is charged very slowly (quasistatically). The pumps will only turn off when the voltage difference across the membrane becomes larger than a particular value $V_{\\text {th }}$. How large must the spring constant $k$ be so that the ion pumps turn off before the membrane collapses?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nSuppose that, after some amount of work is done by the ion pumps, the charges on the outer and inner surfaces are $Q$ and $-Q$, respectively.\n\nOne charge layer by itself creates an electric field $E_{1}=Q /(2 \\epsilon A)$ in each direction. So the force between the two sides of the membrane $F_{E}=Q E_{1}=Q^{2} /(2 \\epsilon A)$.\n\nThis electric force is balanced by the spring force $F_{s}=k x$, where $x=d_{0}-d$. Equating these two forces and solving for $d$ gives gives\n\n$$\nd=d_{0}-\\frac{Q^{2}}{2 \\epsilon A k} .\n$$\n\nThe electric field inside the membrane (as produced by both the left and right plates) is $E=Q /\\left(\\epsilon_{0} \\kappa A\\right)$. So the voltage between them is\n\n$$\nV=E d=\\frac{Q}{\\epsilon A} d\n$$\n\nInserting the expression for $Q$ from part (a) gives\n\n$$\nV=\\frac{Q}{\\epsilon A}\\left(d_{0}-\\frac{Q^{2}}{2 \\epsilon A k}\\right)\n$$\n\nThis equation implies that as the charge $Q$ is increased, the voltage first increases and then decreases again.\n\nproblem:\n[figure1]\n\nThe wall of a neuron is made from an elastic membrane, which resists compression in the same way as a spring. It has an effective spring constant $k$ and an equilibrium thickness $d_{0}$. Assume that the membrane has a very large area $A$ and negligible curvature.\n\nThe neuron has \"ion pumps\" that can move ions across the membrane. In the resulting charged state, positive and negative ionic charge is arranged uniformly along the outer and inner surfaces of the membrane, respectively. The permittivity of the membrane is $\\epsilon$.\n\nSuppose that the ion pumps are first turned on in the uncharged state, and the membrane is charged very slowly (quasistatically). The pumps will only turn off when the voltage difference across the membrane becomes larger than a particular value $V_{\\text {th }}$. How large must the spring constant $k$ be so that the ion pumps turn off before the membrane collapses?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_fed953f9e38b72bf8bd7g-14.jpg?height=358&width=927&top_left_y=469&top_left_x=596" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_77", "problem": "A $0.200 \\mathrm{~kg}$ mass attached to the end of a spring moves up and down through 10 cycles in $6.50 \\mathrm{~s}$. What is the force constant of the spring\nA: $\\quad 18.7 \\mathrm{~N} / \\mathrm{m}$\nB: $4.67 \\mathrm{~N} / \\mathrm{m}$\nC: $22.8 \\mathrm{~N} / \\mathrm{m}$\nD: $5.95 \\mathrm{~N} / \\mathrm{m}$\nE: $12.0 \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA $0.200 \\mathrm{~kg}$ mass attached to the end of a spring moves up and down through 10 cycles in $6.50 \\mathrm{~s}$. What is the force constant of the spring\n\nA: $\\quad 18.7 \\mathrm{~N} / \\mathrm{m}$\nB: $4.67 \\mathrm{~N} / \\mathrm{m}$\nC: $22.8 \\mathrm{~N} / \\mathrm{m}$\nD: $5.95 \\mathrm{~N} / \\mathrm{m}$\nE: $12.0 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_753", "problem": "Solenoid and loop \n\nA closed circular loop of radius $r$ consists of an ideal battery of electromotive force $\\xi$ and a wire of resistance $R$. A long thin air-core solenoid is aligned with the axis of the loop ( $z$-axis). Its length is\n\n$l \\gg r$, cross-sectional area is $A(r \\gg \\sqrt{A})$, and the number of turns is $N$. The solenoid is powered by a constant current $I$ provided by an ideal current source. The directions of the currents in the solenoid and in the loop are the same (clockwise in the figure).\n\n[figure1]\n\nFind the force $F_{1}$ acting on the solenoid when its head $O_{1}$ is positioned in the loop centre $O$. What is the force $F_{2}$ acting on the solenoid when its tail $O_{2}$ is located in the centre of the loop?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nSolenoid and loop \n\nA closed circular loop of radius $r$ consists of an ideal battery of electromotive force $\\xi$ and a wire of resistance $R$. A long thin air-core solenoid is aligned with the axis of the loop ( $z$-axis). Its length is\n\n$l \\gg r$, cross-sectional area is $A(r \\gg \\sqrt{A})$, and the number of turns is $N$. The solenoid is powered by a constant current $I$ provided by an ideal current source. The directions of the currents in the solenoid and in the loop are the same (clockwise in the figure).\n\n[figure1]\n\nFind the force $F_{1}$ acting on the solenoid when its head $O_{1}$ is positioned in the loop centre $O$. What is the force $F_{2}$ acting on the solenoid when its tail $O_{2}$ is located in the centre of the loop?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the force $F_{1}$ acting on the solenoid when its head $O_{1}$ is positioned in the loop centre $O$., the force $F_{2}$ acting on the solenoid when its tail $O_{2}$ is located in the centre of the loop].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_dd4414e209d4a417111bg-1.jpg?height=526&width=1109&top_left_y=645&top_left_x=535" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "the force $F_{1}$ acting on the solenoid when its head $O_{1}$ is positioned in the loop centre $O$.", "the force $F_{2}$ acting on the solenoid when its tail $O_{2}$ is located in the centre of the loop" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_684", "problem": "A pulsar is a highly magnetized, rotating neutron star (or white dwarf), with high density and short, regular rotating period. You have just discovered a new pulsar that spins around its axis 20 times per second. Assuming that the object is spherical, what is its minimum density?\nA: $\\rho=5.6 \\times 10^{13} \\mathrm{~kg} / \\mathrm{m}^{3}$\nB: $\\rho=7.4 \\times 10^{8} \\mathrm{~kg} / \\mathrm{m}^{3}$\nC: $\\rho=6.1 \\times 10^{10} \\mathrm{~kg} / \\mathrm{m}^{3}$\nD: $\\rho=2.2 \\times 10^{8} \\mathrm{~kg} / \\mathrm{m}^{3}$\nE: $\\rho=1.7 \\times 10^{13} \\mathrm{~kg} / \\mathrm{m}^{3}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA pulsar is a highly magnetized, rotating neutron star (or white dwarf), with high density and short, regular rotating period. You have just discovered a new pulsar that spins around its axis 20 times per second. Assuming that the object is spherical, what is its minimum density?\n\nA: $\\rho=5.6 \\times 10^{13} \\mathrm{~kg} / \\mathrm{m}^{3}$\nB: $\\rho=7.4 \\times 10^{8} \\mathrm{~kg} / \\mathrm{m}^{3}$\nC: $\\rho=6.1 \\times 10^{10} \\mathrm{~kg} / \\mathrm{m}^{3}$\nD: $\\rho=2.2 \\times 10^{8} \\mathrm{~kg} / \\mathrm{m}^{3}$\nE: $\\rho=1.7 \\times 10^{13} \\mathrm{~kg} / \\mathrm{m}^{3}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1326", "problem": "相对于站立在地面的李同学, 张同学以相对论速率 $v$ 向右运动, 王同学以同样的速率 $v$ 向左运动. 当张同学和王同学相遇时, 三位同学各自把自己时钟的读数调整到零. 当张同学和王同学之间的距离为 $L$ 时(在地面参考系中观察), 张同学拍一下手.已知张同学和王同学之间的相对速率为\n\n$$\nv_{r}=\\frac{2 \\beta c}{1+\\beta^{2}}\n$$\n\n其中 $\\beta=\\frac{v}{c}, c$ 为真空中的光速.求张同学拍手时其随身携带的时钟的读数", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n相对于站立在地面的李同学, 张同学以相对论速率 $v$ 向右运动, 王同学以同样的速率 $v$ 向左运动. 当张同学和王同学相遇时, 三位同学各自把自己时钟的读数调整到零. 当张同学和王同学之间的距离为 $L$ 时(在地面参考系中观察), 张同学拍一下手.已知张同学和王同学之间的相对速率为\n\n$$\nv_{r}=\\frac{2 \\beta c}{1+\\beta^{2}}\n$$\n\n其中 $\\beta=\\frac{v}{c}, c$ 为真空中的光速.\n\n问题:\n求张同学拍手时其随身携带的时钟的读数\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_906", "problem": "In this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\n\n Let us denote the initial radius of the system as $a_{1}$. Estimate the time $T_{S S}$ for the radius to decrease by a factor of 2 due to \"gravitational slingshot\". Calculate $T_{S S}$ for $\\sigma=200 \\mathrm{~km} / \\mathrm{s}, a_{1}=1 \\mathrm{pc}, \\rho=10^{4} M_{S} / \\mathrm{pc}^{3}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\n\n Let us denote the initial radius of the system as $a_{1}$. Estimate the time $T_{S S}$ for the radius to decrease by a factor of 2 due to \"gravitational slingshot\". Calculate $T_{S S}$ for $\\sigma=200 \\mathrm{~km} / \\mathrm{s}, a_{1}=1 \\mathrm{pc}, \\rho=10^{4} M_{S} / \\mathrm{pc}^{3}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_54", "problem": "As an airplane is flying, it is determined that the airflow speed past the lower surface of the wing is $100 \\mathrm{~m} / \\mathrm{s}$. What speed of airflow over the upper surface of the wing would give a pressure difference of $1000 \\mathrm{~Pa}$ ? $1.293 \\frac{\\mathrm{kg}}{\\mathrm{m}^{3}}$ is the density of the air.\nA: $102 \\mathrm{~m} / \\mathrm{s}$\nB: $103 \\mathrm{~m} / \\mathrm{s}$\nC: $105 \\mathrm{~m} / \\mathrm{s}$\nD: $108 \\mathrm{~m} / \\mathrm{s}$\nE: $111 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAs an airplane is flying, it is determined that the airflow speed past the lower surface of the wing is $100 \\mathrm{~m} / \\mathrm{s}$. What speed of airflow over the upper surface of the wing would give a pressure difference of $1000 \\mathrm{~Pa}$ ? $1.293 \\frac{\\mathrm{kg}}{\\mathrm{m}^{3}}$ is the density of the air.\n\nA: $102 \\mathrm{~m} / \\mathrm{s}$\nB: $103 \\mathrm{~m} / \\mathrm{s}$\nC: $105 \\mathrm{~m} / \\mathrm{s}$\nD: $108 \\mathrm{~m} / \\mathrm{s}$\nE: $111 \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_227", "problem": "Earth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nIf the mirror point lies not far from the surface of the Earth the proton collides with the particles of the atmosphere. This distance is small compared with Earth radius $\\left(R_{E}=6400 \\mathrm{~km}\\right)$, so we will assume that the particles will collide when the mirror point is in the surface of the Earth.\n\nDetermine the minimum equatorial pitch angle $\\alpha_{l}$, below which a proton would collide with the surface of the Earth as a function of $L$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nEarth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nIf the mirror point lies not far from the surface of the Earth the proton collides with the particles of the atmosphere. This distance is small compared with Earth radius $\\left(R_{E}=6400 \\mathrm{~km}\\right)$, so we will assume that the particles will collide when the mirror point is in the surface of the Earth.\n\nDetermine the minimum equatorial pitch angle $\\alpha_{l}$, below which a proton would collide with the surface of the Earth as a function of $L$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=496&width=705&top_left_y=774&top_left_x=713", "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=220&width=504&top_left_y=1690&top_left_x=802" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1467", "problem": "一种拉伸传感器的示意图如图 a 所示: 它由一半径为 $r_{2}$ 的圆柱形塑料棒和在上面紧密缠绕 $N(N>>1)$圈的一层细绳组成; 绳柔软绝缘, 半径为 $r_{1}$, 外表面均匀涂有厚度为 $t\\left(t<>1)$圈的一层细绳组成; 绳柔软绝缘, 半径为 $r_{1}$, 外表面均匀涂有厚度为 $t\\left(t<0)$ 。因发生紧急情况, 卡车突然制动。已知卡车车轮与地面间的动摩擦因数和最大静摩擦因数均为 $\\mu_{1}$, 重物与车厢底板间的动摩擦因数和最大静摩擦因数均为 $\\mu_{2}\\left(\\mu_{2}<\\mu_{1}\\right)$ 。若重物与车厢前壁发生碰撞, 则假定碰撞时间极短, 碰后重物与车厢前壁不分开。重力加速度大小为 $g$ 。\n\n[图1]若重物和车厢前壁不发生碰撞, 求卡车从制动开始到卡车停止的过程所花的时间和走过的路程、重物从制动开始到重物停止的过程所花的时间和走过的路程, 并导出重物 B 与车厢前壁不发生碰撞的条件;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一质量为 $M$ 的载重卡车 $\\mathrm{A}$ 的水平车板上载有一质量为 $m$ 的重物 $\\mathrm{B}$, 在水平直公路上以速度 $v_{0}$ 做匀速直线运动, 重物与车厢前壁间的距离为 $L(L>0)$ 。因发生紧急情况, 卡车突然制动。已知卡车车轮与地面间的动摩擦因数和最大静摩擦因数均为 $\\mu_{1}$, 重物与车厢底板间的动摩擦因数和最大静摩擦因数均为 $\\mu_{2}\\left(\\mu_{2}<\\mu_{1}\\right)$ 。若重物与车厢前壁发生碰撞, 则假定碰撞时间极短, 碰后重物与车厢前壁不分开。重力加速度大小为 $g$ 。\n\n[图1]\n\n问题:\n若重物和车厢前壁不发生碰撞, 求卡车从制动开始到卡车停止的过程所花的时间和走过的路程、重物从制动开始到重物停止的过程所花的时间和走过的路程, 并导出重物 B 与车厢前壁不发生碰撞的条件;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[卡车从制动开始到静止时所用的时间, 卡车从制动开始到静止时所用的路程, 从卡车制动开始到重物对地面速度为零时所用的时间, 从卡车制动开始到重物对地面速度为零时重物移动的距离, 重物 B 与车厢前壁不发生碰撞的条件]\n它们的答案类型依次是[表达式, 表达式, 表达式, 表达式, 区间]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-02.jpg?height=319&width=751&top_left_y=286&top_left_x=1007" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null, null, null ], "answer_sequence": [ "卡车从制动开始到静止时所用的时间", "卡车从制动开始到静止时所用的路程", "从卡车制动开始到重物对地面速度为零时所用的时间", "从卡车制动开始到重物对地面速度为零时重物移动的距离", "重物 B 与车厢前壁不发生碰撞的条件" ], "type_sequence": [ "EX", "EX", "EX", "EX", "IN" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1414", "problem": "如图所示, 一细长的圆柱形均匀玻璃棒, 其一个端面是平面(垂直于轴线), 另一个端面是球面, 球心位于轴线上. 现有一根很细的光束沿平行于轴线方向且很靠近轴线入射. 当光从平端面射入棒内时, 光线从另一端面射出后与轴线的交点到球面的距离为 $a$; 当光线从球形端面射入棒内时, 光线在棒内与轴线的的交点到球面的距离为 $b$. 试近似地求出玻璃的折射率 $n$.\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n\n问题:\n如图所示, 一细长的圆柱形均匀玻璃棒, 其一个端面是平面(垂直于轴线), 另一个端面是球面, 球心位于轴线上. 现有一根很细的光束沿平行于轴线方向且很靠近轴线入射. 当光从平端面射入棒内时, 光线从另一端面射出后与轴线的交点到球面的距离为 $a$; 当光线从球形端面射入棒内时, 光线在棒内与轴线的的交点到球面的距离为 $b$. 试近似地求出玻璃的折射率 $n$.\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-08.jpg?height=212&width=1259&top_left_y=585&top_left_x=204", "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-16.jpg?height=462&width=1363&top_left_y=1942&top_left_x=198" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1591", "problem": "如图 8 所示, 在水平面内有一个光滑匀质圆环, 圆环总电阻为 $R_{0}$, 半径为 $r$, 质量为 $m$,初速度 $v_{0}$ 向右，右半空间有均匀的稳定的垂直于面的磁场，大小为 $\\mathrm{B}$ ，结果圆环进入磁场后恰好静止，整个过程中圆环中通过的电量大小 $Q_{0}=$ ___；如果保持圆环单位长度的质量和电阻大小不变，但是把半径变为原来的两倍, 为了使得圆环进入磁场后仍然恰好静止, 则 $v_{0}$ 应当变为原来的___倍。\n\n[图1]\n\n图 8", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n如图 8 所示, 在水平面内有一个光滑匀质圆环, 圆环总电阻为 $R_{0}$, 半径为 $r$, 质量为 $m$,初速度 $v_{0}$ 向右，右半空间有均匀的稳定的垂直于面的磁场，大小为 $\\mathrm{B}$ ，结果圆环进入磁场后恰好静止，整个过程中圆环中通过的电量大小 $Q_{0}=$ ___；如果保持圆环单位长度的质量和电阻大小不变，但是把半径变为原来的两倍, 为了使得圆环进入磁场后仍然恰好静止, 则 $v_{0}$ 应当变为原来的___倍。\n\n[图1]\n\n图 8\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[整个过程中圆环中通过的电量大小 $Q_{0}$, 把半径变为原来的两倍, 为了使得圆环进入磁场后仍然恰好静止, 则 $v_{0}$ 应当变为原来的几倍]\n它们的答案类型依次是[表达式, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-05.jpg?height=340&width=394&top_left_y=207&top_left_x=868" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "整个过程中圆环中通过的电量大小 $Q_{0}$", "把半径变为原来的两倍, 为了使得圆环进入磁场后仍然恰好静止, 则 $v_{0}$ 应当变为原来的几倍" ], "type_sequence": [ "EX", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_766", "problem": "The amount of chocolate of chocolate Kate consumes in a day is given by\n\n$$\nC=0.05 \\mathrm{~kg}+\\sum_{i} T_{i} A_{i}^{2}\n$$\n\nWhere $i$ is the number of annoying people Kate has to deal with in the day, $\\sum_{i}$ means that you add a copy of $T_{i} A_{i}^{2}$ for each of the people. $T_{i}$ is the amount of time measured in hours that Kate spends with the $i^{\\text {th }}$ annoying person and $A_{i}$ is the \"annoyance factor\" of the $i^{\\text {th }}$ annoying person. What are the units of the annoyance factor, $\\mathrm{A}$ ?\nA: $\\mathrm{kg}$\nB: $\\mathrm{kg} / \\mathrm{s}$\nC: $\\mathrm{kg} / \\mathrm{h}$\nD: $\\mathrm{kg}^{2} / \\mathrm{h}^{2}$\nE: $\\mathrm{kg}^{1 / 2} / \\mathrm{h}^{1 / 2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe amount of chocolate of chocolate Kate consumes in a day is given by\n\n$$\nC=0.05 \\mathrm{~kg}+\\sum_{i} T_{i} A_{i}^{2}\n$$\n\nWhere $i$ is the number of annoying people Kate has to deal with in the day, $\\sum_{i}$ means that you add a copy of $T_{i} A_{i}^{2}$ for each of the people. $T_{i}$ is the amount of time measured in hours that Kate spends with the $i^{\\text {th }}$ annoying person and $A_{i}$ is the \"annoyance factor\" of the $i^{\\text {th }}$ annoying person. What are the units of the annoyance factor, $\\mathrm{A}$ ?\n\nA: $\\mathrm{kg}$\nB: $\\mathrm{kg} / \\mathrm{s}$\nC: $\\mathrm{kg} / \\mathrm{h}$\nD: $\\mathrm{kg}^{2} / \\mathrm{h}^{2}$\nE: $\\mathrm{kg}^{1 / 2} / \\mathrm{h}^{1 / 2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_585", "problem": "An AC power line cable transmits electrical power using a sinusoidal waveform with frequency $60 \\mathrm{~Hz}$. The load receives an RMS voltage of $500 \\mathrm{kV}$ and requires $1000 \\mathrm{MW}$ of average power. For this problem, consider only the cable carrying current in one of the two directions, and ignore effects due to capacitance or inductance between the cable and with the ground.\n\nThe load at the end of the power line cable changes to include a manufacturing plant with a large number of electric motors. While the average power consumed remains the same, it now behaves like a resistor in parallel with a $0.25 \\mathrm{H}$ inductor.\n\nDoes the power lost in the power line cable increase, decrease, or stay the same? (You need not calculate the new value explicitly, but you should show some work to defend your answer.)\nA: increase\nB: decrease\nC: stay the same\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn AC power line cable transmits electrical power using a sinusoidal waveform with frequency $60 \\mathrm{~Hz}$. The load receives an RMS voltage of $500 \\mathrm{kV}$ and requires $1000 \\mathrm{MW}$ of average power. For this problem, consider only the cable carrying current in one of the two directions, and ignore effects due to capacitance or inductance between the cable and with the ground.\n\nThe load at the end of the power line cable changes to include a manufacturing plant with a large number of electric motors. While the average power consumed remains the same, it now behaves like a resistor in parallel with a $0.25 \\mathrm{H}$ inductor.\n\nDoes the power lost in the power line cable increase, decrease, or stay the same? (You need not calculate the new value explicitly, but you should show some work to defend your answer.)\n\nA: increase\nB: decrease\nC: stay the same\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1305", "problem": "如图, 一边长为 $L$ 的正方形铜线框 abcd 可绕水平轴 ab 自由转动, 一坚直向上的外力 $F$ 作用在 $\\mathrm{cd}$ 边的中点, 整个线框置于方向坚直向上的均匀磁场中, 磁感应强度大小随时间变化。已知该方形线框铜线的电导率为 $\\sigma$, 铜线的半径为 $r_{0}$,质量密度 $\\rho$, 重力加速度大小为 $g$ 。\n[图1]当框平面与水平面 abef 的夹角为 $\\theta$ 时, 框平面恰好处于平衡状态。求此时线框中 $\\mathrm{Cd}$ 边所受到的磁场 $B$ 的作用力的大小与外力的大小 $F$ 之间的关系。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一边长为 $L$ 的正方形铜线框 abcd 可绕水平轴 ab 自由转动, 一坚直向上的外力 $F$ 作用在 $\\mathrm{cd}$ 边的中点, 整个线框置于方向坚直向上的均匀磁场中, 磁感应强度大小随时间变化。已知该方形线框铜线的电导率为 $\\sigma$, 铜线的半径为 $r_{0}$,质量密度 $\\rho$, 重力加速度大小为 $g$ 。\n[图1]\n\n问题:\n当框平面与水平面 abef 的夹角为 $\\theta$ 时, 框平面恰好处于平衡状态。求此时线框中 $\\mathrm{Cd}$ 边所受到的磁场 $B$ 的作用力的大小与外力的大小 $F$ 之间的关系。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_d716ce15f03757bb482eg-03.jpg?height=300&width=309&top_left_y=1523&top_left_x=1610", "https://cdn.mathpix.com/cropped/2024_03_31_d716ce15f03757bb482eg-08.jpg?height=246&width=392&top_left_y=636&top_left_x=1326" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1601", "problem": "如图, 一个球冠形光滑凹槽深度 $h=0.050 \\mathrm{~m}$, 球半径为 $20 \\mathrm{~m}$ 。现将一质量为 $0.10 \\mathrm{~kg}$ 的小球放在凹槽边缘从静止释放。重力加速度大小为 $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$ 。小球由凹槽最高点滑到最低点所用时间为___S。\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n如图, 一个球冠形光滑凹槽深度 $h=0.050 \\mathrm{~m}$, 球半径为 $20 \\mathrm{~m}$ 。现将一质量为 $0.10 \\mathrm{~kg}$ 的小球放在凹槽边缘从静止释放。重力加速度大小为 $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$ 。小球由凹槽最高点滑到最低点所用时间为___S。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_d716ce15f03757bb482eg-02.jpg?height=200&width=503&top_left_y=1191&top_left_x=1482" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_831", "problem": "The fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\n In Tsui et al. experiment,\n\nthe magnetic field corresponding to the center of the quantized Hall plateau\n\n$$\nR_{\\mathrm{H}}=3 h / e^{2}, B_{1 / 3}=15 \\text { Tesla, }\n$$\n\nthe effective mass of an electron in GaAs, $m^{*}=0.067 m_{e}$,\n\nthe electron mass, $m_{e}=9.1 \\times 10^{-31} \\mathrm{~kg}$,\n\nCoulomb's constant, $k=9.0 \\times 10^{9} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{C}^{2}$,\n\nthe vacuum permittivity, $\\varepsilon_{0}=1 / 4 \\pi k=8.854 \\times 10^{-12} \\mathrm{~F} / \\mathrm{m}$,\n\nthe relative permittivity (the ratio of the permittivity of a substance to the vacuum permittivity) of GaAs, $\\varepsilon_{r}=13$,\n\nthe elementary charge, $e=1.6 \\times 10^{-19} \\mathrm{C}$,\n\nPlanck's constant, $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, and\n\nBoltzmann's constant, $k_{\\mathrm{B}}=1.38 \\times 10^{-23} \\mathrm{~J} / \\mathrm{K}$.\n\nIn our analysis, we have neglected several factors, whose corresponding energy scales, compared to $\\Delta U(B)$ discussed in (d), are either too large to excite or too small to be relevant.\n\n The electrons spatially confined in the whirlpools (or vortices) have a large kinetic energy. Using the uncertainty relation, estimate the order of magnitude of the kinetic energy. (This amount would also be the additional energy penalty if we put two electrons in the same whirlpool, instead of in two separate whirlpools, due to Pauli exclusion principle.)", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\n In Tsui et al. experiment,\n\nthe magnetic field corresponding to the center of the quantized Hall plateau\n\n$$\nR_{\\mathrm{H}}=3 h / e^{2}, B_{1 / 3}=15 \\text { Tesla, }\n$$\n\nthe effective mass of an electron in GaAs, $m^{*}=0.067 m_{e}$,\n\nthe electron mass, $m_{e}=9.1 \\times 10^{-31} \\mathrm{~kg}$,\n\nCoulomb's constant, $k=9.0 \\times 10^{9} \\mathrm{~N} \\cdot \\mathrm{m}^{2} / \\mathrm{C}^{2}$,\n\nthe vacuum permittivity, $\\varepsilon_{0}=1 / 4 \\pi k=8.854 \\times 10^{-12} \\mathrm{~F} / \\mathrm{m}$,\n\nthe relative permittivity (the ratio of the permittivity of a substance to the vacuum permittivity) of GaAs, $\\varepsilon_{r}=13$,\n\nthe elementary charge, $e=1.6 \\times 10^{-19} \\mathrm{C}$,\n\nPlanck's constant, $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, and\n\nBoltzmann's constant, $k_{\\mathrm{B}}=1.38 \\times 10^{-23} \\mathrm{~J} / \\mathrm{K}$.\n\nIn our analysis, we have neglected several factors, whose corresponding energy scales, compared to $\\Delta U(B)$ discussed in (d), are either too large to excite or too small to be relevant.\n\n The electrons spatially confined in the whirlpools (or vortices) have a large kinetic energy. Using the uncertainty relation, estimate the order of magnitude of the kinetic energy. (This amount would also be the additional energy penalty if we put two electrons in the same whirlpool, instead of in two separate whirlpools, due to Pauli exclusion principle.)\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~J}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~J}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1566", "problem": "某秋天清晨, 气温为 $4.0^{\\circ} \\mathrm{C}$,一加水员到实验园区给一内径为 $2.00 \\mathrm{~m}$ 、高为 $2.00 \\mathrm{~m}$ 的圆柱形不锈钢蒸馏水罐加水。罐体导热良好。罐外有一内径为 $4.00 \\mathrm{~cm}$的透明圆柱形观察柱，底部与罐相连（连接处很短）,顶部与大气相通，如图所示。加完水后，加水员在水面上覆盖一层轻质防蒸发膜（不溶于水，与罐壁无摩擦），并密闭了罐顶的加水口。此时加水员通过观察柱上的刻度看到罐内水高为 $1.00 \\mathrm{~m}$ 。\n\n[图1]从密闭水罐后至中午, 罐内空气对外做的功和吸收的热量分别为多少? 求这个过程中罐内空气的热容量。\n已知罐外气压始终为标准大气压 $p_{0}=1.01 \\times 10^{5} \\mathrm{~Pa}$, 水在 $4.0^{\\circ} \\mathrm{C}$ 时的密度为 $\\rho_{0}=1.00 \\times 10^{3} \\mathrm{~kg} \\cdot \\mathrm{m}^{-3}$, 水在温度变化过程中的平均体积膨胀系数为 $\\kappa=3.03 \\times 10^{-4} \\mathrm{~K}^{-1}$, 重力加速度大小为 $g=9.80 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$, 绝对零度为 $-273.15^{\\circ} \\mathrm{C}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某秋天清晨, 气温为 $4.0^{\\circ} \\mathrm{C}$,一加水员到实验园区给一内径为 $2.00 \\mathrm{~m}$ 、高为 $2.00 \\mathrm{~m}$ 的圆柱形不锈钢蒸馏水罐加水。罐体导热良好。罐外有一内径为 $4.00 \\mathrm{~cm}$的透明圆柱形观察柱，底部与罐相连（连接处很短）,顶部与大气相通，如图所示。加完水后，加水员在水面上覆盖一层轻质防蒸发膜（不溶于水，与罐壁无摩擦），并密闭了罐顶的加水口。此时加水员通过观察柱上的刻度看到罐内水高为 $1.00 \\mathrm{~m}$ 。\n\n[图1]\n\n问题:\n从密闭水罐后至中午, 罐内空气对外做的功和吸收的热量分别为多少? 求这个过程中罐内空气的热容量。\n已知罐外气压始终为标准大气压 $p_{0}=1.01 \\times 10^{5} \\mathrm{~Pa}$, 水在 $4.0^{\\circ} \\mathrm{C}$ 时的密度为 $\\rho_{0}=1.00 \\times 10^{3} \\mathrm{~kg} \\cdot \\mathrm{m}^{-3}$, 水在温度变化过程中的平均体积膨胀系数为 $\\kappa=3.03 \\times 10^{-4} \\mathrm{~K}^{-1}$, 重力加速度大小为 $g=9.80 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$, 绝对零度为 $-273.15^{\\circ} \\mathrm{C}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~J}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_49158ed36459a540f197g-01.jpg?height=588&width=671&top_left_y=1822&top_left_x=1161" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~J}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_511", "problem": "Single bubble sonoluminescence occurs when sound waves cause a bubble suspended in a fluid to collapse so that the gas trapped inside increases in temperature enough to emit light. The bubble actually undergoes a series of expansions and collapses caused by the sound wave pressure variations.\n\nWe now consider a simplified model of a bubble undergoing sonoluminescence. Assume the bubble is originally at atmospheric pressure $P_{0}=101 \\mathrm{kPa}$. When the pressure in the fluid surrounding the bubble is decreased, the bubble expands isothermally to a radius of $36.0 \\mu \\mathrm{m}$. When the pressure increases again, the bubble collapses to a radius of $4.50 \\mu \\mathrm{m}$ so quickly that no heat can escape. Between the collapse and subsequent expansion, the bubble undergoes isochoric (constant volume) cooling back to its original pressure and temperature. For a bubble containing a monatomic gas, suspended in water of $T=293 \\mathrm{~K}$, find the pressure after collapse.\n\nYou may find the following useful: the specific heat capacity at constant volume is $C_{V}=3 R / 2$ and the ratio of specific heat at constant pressure to constant volume is $\\gamma=5 / 3$ for a monatomic gas.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nSingle bubble sonoluminescence occurs when sound waves cause a bubble suspended in a fluid to collapse so that the gas trapped inside increases in temperature enough to emit light. The bubble actually undergoes a series of expansions and collapses caused by the sound wave pressure variations.\n\nWe now consider a simplified model of a bubble undergoing sonoluminescence. Assume the bubble is originally at atmospheric pressure $P_{0}=101 \\mathrm{kPa}$. When the pressure in the fluid surrounding the bubble is decreased, the bubble expands isothermally to a radius of $36.0 \\mu \\mathrm{m}$. When the pressure increases again, the bubble collapses to a radius of $4.50 \\mu \\mathrm{m}$ so quickly that no heat can escape. Between the collapse and subsequent expansion, the bubble undergoes isochoric (constant volume) cooling back to its original pressure and temperature. For a bubble containing a monatomic gas, suspended in water of $T=293 \\mathrm{~K}$, find the pressure after collapse.\n\nYou may find the following useful: the specific heat capacity at constant volume is $C_{V}=3 R / 2$ and the ratio of specific heat at constant pressure to constant volume is $\\gamma=5 / 3$ for a monatomic gas.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~Pa}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~Pa}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_7", "problem": "There are $6.02 \\times 10^{23}$ carbon atoms (Avogadro's number) in $12.0 \\mathrm{~g}$ of carbon. If you could count one atom each second, how much time would it take to count the atoms in $1.0 \\mathrm{~g}$ of carbon?\nA: $1.59 \\times 10^{15}$ days\nB: $1.59 \\times 10^{15}$ weeks\nC: $1.59 \\mathrm{X} 10^{15}$ years\nD: $1.59 \\times 10^{18}$ weeks\nE: $1.59 \\times 10^{18}$ years\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThere are $6.02 \\times 10^{23}$ carbon atoms (Avogadro's number) in $12.0 \\mathrm{~g}$ of carbon. If you could count one atom each second, how much time would it take to count the atoms in $1.0 \\mathrm{~g}$ of carbon?\n\nA: $1.59 \\times 10^{15}$ days\nB: $1.59 \\times 10^{15}$ weeks\nC: $1.59 \\mathrm{X} 10^{15}$ years\nD: $1.59 \\times 10^{18}$ weeks\nE: $1.59 \\times 10^{18}$ years\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1489", "problem": "具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：给出金纳米球颗粒的质量 $m$ 和它绕其对称轴的转动惯量 $J$ 的值。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：\n\n问题:\n给出金纳米球颗粒的质量 $m$ 和它绕其对称轴的转动惯量 $J$ 的值。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[金纳米球颗粒的质量 $m$ , 金纳米球颗粒绕其对称轴的转动惯量 $J$ ]\n它们的单位依次是[$\\mathrm{~kg}$, $\\mathrm{~kg} \\cdot \\mathrm{m}^{2}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-02.jpg?height=450&width=450&top_left_y=1883&top_left_x=1292" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{~kg}$", "$\\mathrm{~kg} \\cdot \\mathrm{m}^{2}$" ], "answer_sequence": [ "金纳米球颗粒的质量 $m$ ", "金纳米球颗粒绕其对称轴的转动惯量 $J$ " ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1523", "problem": "宇宙空间某区域有一磁感应强度大小为 $B=1.0 \\times 10^{-9} \\mathrm{~T}$ 的均匀磁场, 现有一电子绕磁力线做螺旋运动. 该电子绕磁力线旋转一圈所需的时间间隔为 ? $\\mathrm{s}$; \n\n若该电子沿磁场方向的运动速度为 $1.0 \\times 10^{-2} c$ ( $c$ 为真空中光速的大小), 则它在沿磁场方向前进 $1.0 \\times 10^{-3}$ 光年的过程中, 绕磁力线转了?圈. \n\n已知电子电荷量为 $1.60 \\times 10^{-19} \\mathrm{C}$, 电子质量为 $9.11 \\times 10^{-31} \\mathrm{~kg}$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个集合。\n\n问题:\n宇宙空间某区域有一磁感应强度大小为 $B=1.0 \\times 10^{-9} \\mathrm{~T}$ 的均匀磁场, 现有一电子绕磁力线做螺旋运动. 该电子绕磁力线旋转一圈所需的时间间隔为 ? $\\mathrm{s}$; \n\n若该电子沿磁场方向的运动速度为 $1.0 \\times 10^{-2} c$ ( $c$ 为真空中光速的大小), 则它在沿磁场方向前进 $1.0 \\times 10^{-3}$ 光年的过程中, 绕磁力线转了?圈. \n\n已知电子电荷量为 $1.60 \\times 10^{-19} \\mathrm{C}$, 电子质量为 $9.11 \\times 10^{-31} \\mathrm{~kg}$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是所有不同答案的集合，例如ANSWER={3, 4, 5}", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SET", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1640", "problem": "$1 \\mathrm{~mol}$ 的理想气体经历一循环过程 1-2-3-1,如 $p-T$ 图示所示. 过程 1-2 是等压过程, 过程 3-1 是通过 $p-T$ 图原点的直线上的一段, 描述过程 2-3 的方程为\n\n$$\nc_{1} p^{2}+c_{2} p=T\n$$\n\n式中 $c_{1}$ 和 $c_{2}$ 都是待定的常量, $p$ 和 $T$ 分别是气体的压强和绝对温度. 已知, 气体在状态 1 的压强、绝对温度分别为 $p_{1}$\n\n[图1]\n和 $T_{1}$, 气体在状态 2 的绝对温度以及在状态 3 的压强和绝对温度分别为 $T_{2}$ 以及 $p_{3}$ 和 $T_{3}$. 气体常量 $R$ 也是已知的.求该气体在一次循环过程中对外做的总功.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n$1 \\mathrm{~mol}$ 的理想气体经历一循环过程 1-2-3-1,如 $p-T$ 图示所示. 过程 1-2 是等压过程, 过程 3-1 是通过 $p-T$ 图原点的直线上的一段, 描述过程 2-3 的方程为\n\n$$\nc_{1} p^{2}+c_{2} p=T\n$$\n\n式中 $c_{1}$ 和 $c_{2}$ 都是待定的常量, $p$ 和 $T$ 分别是气体的压强和绝对温度. 已知, 气体在状态 1 的压强、绝对温度分别为 $p_{1}$\n\n[图1]\n和 $T_{1}$, 气体在状态 2 的绝对温度以及在状态 3 的压强和绝对温度分别为 $T_{2}$ 以及 $p_{3}$ 和 $T_{3}$. 气体常量 $R$ 也是已知的.\n\n问题:\n求该气体在一次循环过程中对外做的总功.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_58b1fc45927d60138a23g-06.jpg?height=385&width=503&top_left_y=624&top_left_x=1299" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_268", "problem": "For the circuit in question 3, the power dissipated by resistor $\\mathbf{R 3}$ is:\nA: The same power as dissipated by resistor $\\mathrm{R} 1$\nB: $2 x$ the power dissipated by resistor $\\mathrm{R} 1$\nC: $3 x$ the power dissipated by resistor $\\mathrm{R} 1$\nD: $4 x$ the power dissipated by resistor $\\mathrm{R} 1$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nFor the circuit in question 3, the power dissipated by resistor $\\mathbf{R 3}$ is:\n\nA: The same power as dissipated by resistor $\\mathrm{R} 1$\nB: $2 x$ the power dissipated by resistor $\\mathrm{R} 1$\nC: $3 x$ the power dissipated by resistor $\\mathrm{R} 1$\nD: $4 x$ the power dissipated by resistor $\\mathrm{R} 1$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_798", "problem": "A car is travelling at $120 \\mathrm{~km} / \\mathrm{h}$, when the driver sees a herd of cows on the road ahead and slams on the brakes. The performance of the car's brakes is such that the car comes to a stop in a distance $D$ metres. Assuming that the acceleration of the car under braking is independent of the car's speed, what distance would the car require to come to a stop if it were travelling at $40 \\mathrm{~km} / \\mathrm{h}$ instead?\nA: $D / 9$ metres\nB: $D / 6$ metres\nC: $D / 4$ metres\nD: $D / 3$ metres\nE: $D$ meters\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA car is travelling at $120 \\mathrm{~km} / \\mathrm{h}$, when the driver sees a herd of cows on the road ahead and slams on the brakes. The performance of the car's brakes is such that the car comes to a stop in a distance $D$ metres. Assuming that the acceleration of the car under braking is independent of the car's speed, what distance would the car require to come to a stop if it were travelling at $40 \\mathrm{~km} / \\mathrm{h}$ instead?\n\nA: $D / 9$ metres\nB: $D / 6$ metres\nC: $D / 4$ metres\nD: $D / 3$ metres\nE: $D$ meters\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_129", "problem": "A motorcycle rides on the vertical walls around the perimeter of a large circular room. The friction coefficient between the motorcycle tires and the walls is $\\mu$. How does the minimum $\\mu$ needed to prevent the motorcycle from slipping downwards change with the motorcycle's speed, $s$ ?\nA: $\\mu \\propto s^{0}$\nB: $\\mu \\propto s^{-1 / 2}$\nC: $\\mu \\propto s^{-1}$\nD: $\\mu \\propto s^{-2} $ \nE: none of these\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA motorcycle rides on the vertical walls around the perimeter of a large circular room. The friction coefficient between the motorcycle tires and the walls is $\\mu$. How does the minimum $\\mu$ needed to prevent the motorcycle from slipping downwards change with the motorcycle's speed, $s$ ?\n\nA: $\\mu \\propto s^{0}$\nB: $\\mu \\propto s^{-1 / 2}$\nC: $\\mu \\propto s^{-1}$\nD: $\\mu \\propto s^{-2} $ \nE: none of these\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_817", "problem": "Free vortices move about in space with the flow 2 . In other words each element of the filament moves with the velocity $\\vec{v}$ of the fluid at the position of that element.[^1]\n\nAs an example, consider a pair of counter-rotating straight vortices placed initially at distance $r_{0}$ from each other, see Fig. 3. Each vortex produces velocity $v_{0}=\\kappa / r_{0}$ at the axis of another. As a result, the vortex pair moves rectilinearly with constant speed $v_{0}=\\kappa / r_{0}$ so that the distance between them remains unchanged.\n\n[figure1]\n\nFig. 3: Parallel vortex filaments with opposite circulations.\n\nWork out the \"smoothed out\" (omitting the lattice structure) free helium surface shape $z(\\vec{r})$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nFree vortices move about in space with the flow 2 . In other words each element of the filament moves with the velocity $\\vec{v}$ of the fluid at the position of that element.[^1]\n\nAs an example, consider a pair of counter-rotating straight vortices placed initially at distance $r_{0}$ from each other, see Fig. 3. Each vortex produces velocity $v_{0}=\\kappa / r_{0}$ at the axis of another. As a result, the vortex pair moves rectilinearly with constant speed $v_{0}=\\kappa / r_{0}$ so that the distance between them remains unchanged.\n\n[figure1]\n\nFig. 3: Parallel vortex filaments with opposite circulations.\n\nWork out the \"smoothed out\" (omitting the lattice structure) free helium surface shape $z(\\vec{r})$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_61fc31149c0f627b45f3g-3.jpg?height=369&width=531&top_left_y=775&top_left_x=768" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_933", "problem": "A Tippe top is a special kind of top that can spontaneously invert once it has been set spinning. One can model a Tippe top as a sphere of radius $R$ that is truncated, with a stem added. It has rotational symmetry about an axis through the stem, which is at angle $\\theta$ from the vertical. As shown in Figure 1(a), its centre of mass $C$ is offset from its geometric centre $O$ by $\\alpha R$ along its symmetry axis. The Tippe top makes contact with the surface it rests on at point $A$; we assume this surface is planar, and refer to it as the floor. Given certain geometrical constraints and if spun fast enough initially, the Tippe top will tip so that the stem points increasingly downwards, until it starts to spin on in its stem, and eventually comes to a stop.\n[figure1]\n\nFigure 1. Views of the Tippe top (a) from the side and (b) from above\n\nLet $x y z$ be the rotating reference frame defined such that $\\hat{\\mathbf{z}}$ is stationary and upwards, and the top's symmetry axis is within the $x z$-plane. Two views of the Tippe top are shown in Figure 1: from the side, and from above. As shown in Figure 1(b), the top's symmetry axis is aligned with the $x$-axis when viewed from above.\n\nFigure 2 shows the top's motion at several phases after it is started spinning:\n\n(a) phase I: immediately after it is initially set spinning, with $\\theta \\sim 0$\n\n(b) phase II: soon after, having tipped to angle $0<\\theta<\\frac{\\pi}{2}$\n\n(c) phase III: when the stem first touches the floor, with $\\theta>\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nFind the total external force $\\mathbf{F}_{\\text {ext }}$ on the Tippe top. Draw a free body diagram of $1 \\mathrm{pt}$ the top, projected onto each of the $x z$ - and $x y$-planes. Indicate the direction of $\\mathbf{v}_{A}$ in the space provided, on your diagram in the $x y$-plane.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA Tippe top is a special kind of top that can spontaneously invert once it has been set spinning. One can model a Tippe top as a sphere of radius $R$ that is truncated, with a stem added. It has rotational symmetry about an axis through the stem, which is at angle $\\theta$ from the vertical. As shown in Figure 1(a), its centre of mass $C$ is offset from its geometric centre $O$ by $\\alpha R$ along its symmetry axis. The Tippe top makes contact with the surface it rests on at point $A$; we assume this surface is planar, and refer to it as the floor. Given certain geometrical constraints and if spun fast enough initially, the Tippe top will tip so that the stem points increasingly downwards, until it starts to spin on in its stem, and eventually comes to a stop.\n[figure1]\n\nFigure 1. Views of the Tippe top (a) from the side and (b) from above\n\nLet $x y z$ be the rotating reference frame defined such that $\\hat{\\mathbf{z}}$ is stationary and upwards, and the top's symmetry axis is within the $x z$-plane. Two views of the Tippe top are shown in Figure 1: from the side, and from above. As shown in Figure 1(b), the top's symmetry axis is aligned with the $x$-axis when viewed from above.\n\nFigure 2 shows the top's motion at several phases after it is started spinning:\n\n(a) phase I: immediately after it is initially set spinning, with $\\theta \\sim 0$\n\n(b) phase II: soon after, having tipped to angle $0<\\theta<\\frac{\\pi}{2}$\n\n(c) phase III: when the stem first touches the floor, with $\\theta>\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nFind the total external force $\\mathbf{F}_{\\text {ext }}$ on the Tippe top. Draw a free body diagram of $1 \\mathrm{pt}$ the top, projected onto each of the $x z$ - and $x y$-planes. Indicate the direction of $\\mathbf{v}_{A}$ in the space provided, on your diagram in the $x y$-plane.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-1.jpg?height=522&width=1332&top_left_y=1121&top_left_x=360", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=578&width=1778&top_left_y=316&top_left_x=184", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=417&width=545&top_left_y=1296&top_left_x=527", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=431&width=397&top_left_y=1298&top_left_x=1189", "https://cdn.mathpix.com/cropped/2024_03_14_d0f8b9fffcdcdf2f565bg-2.jpg?height=834&width=1694&top_left_y=471&top_left_x=273" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1350", "problem": "球磨机利用旋转圆筒驱动锰钢球对矿石颗粒进行冲击和剥磨。如图 14a, 某球磨机圆筒半径为 $R$, 绕其 (水平)对称轴匀速旋转。球磨机内装有矿石颗粒和一个质量为 $m$ 的锰钢小球, 钢球与筒壁之间摩擦系数足够大。若圆筒转速较低,球磨机内的钢球达到一定高度后会因为其本身的重量沿圆筒内壁滑滚下落 (被称为处于江落状态), 此时矿石被钢球剥磨;若圆筒旋转的角速度超过某临界值, 钢球随着圆筒旋转而不下落（被称为处于离心状态）, 球磨机研磨作用停止; 若圆筒的角速度介于上述两情形之间, 钢球沿圆筒内壁上升至某一点后会脱离圆筒落下（被称为处于抛落状态）冲击筒中的矿石粉, 此时矿石被冲磨。重力加速度大小为 $g$ 。求\n\n[图1]能使钢球对矿石的冲击作用最大时的圆筒转动角速度以及钢球对矿石的最大冲击功。可利用不等式: 设 $a_{1}, a_{2}, \\cdots, a_{n}$ 均为正数, 则\n\n$$\n\\sqrt[n]{a_{1} a_{2} \\cdots a_{n}} \\leq \\frac{a_{1}+a_{2}+\\cdots+a_{n}}{n}\n$$\n\n等号当且仅当 $a_{1}=a_{2}=\\cdots=a_{n}$ 时成立。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n球磨机利用旋转圆筒驱动锰钢球对矿石颗粒进行冲击和剥磨。如图 14a, 某球磨机圆筒半径为 $R$, 绕其 (水平)对称轴匀速旋转。球磨机内装有矿石颗粒和一个质量为 $m$ 的锰钢小球, 钢球与筒壁之间摩擦系数足够大。若圆筒转速较低,球磨机内的钢球达到一定高度后会因为其本身的重量沿圆筒内壁滑滚下落 (被称为处于江落状态), 此时矿石被钢球剥磨;若圆筒旋转的角速度超过某临界值, 钢球随着圆筒旋转而不下落（被称为处于离心状态）, 球磨机研磨作用停止; 若圆筒的角速度介于上述两情形之间, 钢球沿圆筒内壁上升至某一点后会脱离圆筒落下（被称为处于抛落状态）冲击筒中的矿石粉, 此时矿石被冲磨。重力加速度大小为 $g$ 。求\n\n[图1]\n\n问题:\n能使钢球对矿石的冲击作用最大时的圆筒转动角速度以及钢球对矿石的最大冲击功。可利用不等式: 设 $a_{1}, a_{2}, \\cdots, a_{n}$ 均为正数, 则\n\n$$\n\\sqrt[n]{a_{1} a_{2} \\cdots a_{n}} \\leq \\frac{a_{1}+a_{2}+\\cdots+a_{n}}{n}\n$$\n\n等号当且仅当 $a_{1}=a_{2}=\\cdots=a_{n}$ 时成立。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-05.jpg?height=665&width=688&top_left_y=2189&top_left_x=1232", "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-12.jpg?height=711&width=737&top_left_y=1501&top_left_x=1025" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_404", "problem": "A space station at a geostationary orbit has a form of a cylinder of length $L=100 \\mathrm{~km}$ and radius $R=1 \\mathrm{~km}$ is filled with air (molar mass $M=29 \\mathrm{~g} / \\mathrm{mol}$ ) at the atmospheric pressure and temperatur $T=295 \\mathrm{~K}$ and the cylindrical walls serve as ground for the people living inside. It rotates around its axis so as to create normal gravity $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$ at the \"ground\"\n\nAssuming that the height of the point $C$ above the \"ground\" is $h$, determine $T_{A}-T_{C}$, the difference of the tension forces in the rope at the points $A$ and $C$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA space station at a geostationary orbit has a form of a cylinder of length $L=100 \\mathrm{~km}$ and radius $R=1 \\mathrm{~km}$ is filled with air (molar mass $M=29 \\mathrm{~g} / \\mathrm{mol}$ ) at the atmospheric pressure and temperatur $T=295 \\mathrm{~K}$ and the cylindrical walls serve as ground for the people living inside. It rotates around its axis so as to create normal gravity $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$ at the \"ground\"\n\nAssuming that the height of the point $C$ above the \"ground\" is $h$, determine $T_{A}-T_{C}$, the difference of the tension forces in the rope at the points $A$ and $C$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1701", "problem": "$\\mu^{-}$子与电子的性质相似, 其电量与电子相同, 而质量 $m_{\\mu}$ 约为电子的 206.8 倍。用 $\\mu^{-}$子代替氢原子中的电子就形成 $\\mu$ 子-氢原子, $\\mu^{-}$子-氢原子的线状光谱与氢原子具有相似的规律。 $\\mu^{-}$子-氢原子基态的电离能为___ $\\mathrm{eV}, \\mu$ 子-氢原子从第二激发态跃迁到第一激发态发出的光子的波长为 ___ $\\AA$ 。已知质子质量 $m_{\\mathrm{p}}$ 是电子的 1836 倍, 氢原子基态的电离能为 $13.605 \\mathrm{eV}$; 光在真空中的速度为 $2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 普朗克常量为 $4.136 \\times 10^{-15} \\mathrm{eV} \\cdot \\mathrm{s}$ 。（按玻尔理论计算时, 在 $\\mu$ 子-氢原子中若仍将质子视为不动, 则 $\\mu^{-}$子相当于质量为 $\\frac{m_{\\mu} m_{\\mathrm{p}}}{m_{\\mu}+m_{\\mathrm{p}}}$ 的带电粒子。)", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n$\\mu^{-}$子与电子的性质相似, 其电量与电子相同, 而质量 $m_{\\mu}$ 约为电子的 206.8 倍。用 $\\mu^{-}$子代替氢原子中的电子就形成 $\\mu$ 子-氢原子, $\\mu^{-}$子-氢原子的线状光谱与氢原子具有相似的规律。 $\\mu^{-}$子-氢原子基态的电离能为___ $\\mathrm{eV}, \\mu$ 子-氢原子从第二激发态跃迁到第一激发态发出的光子的波长为 ___ $\\AA$ 。已知质子质量 $m_{\\mathrm{p}}$ 是电子的 1836 倍, 氢原子基态的电离能为 $13.605 \\mathrm{eV}$; 光在真空中的速度为 $2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 普朗克常量为 $4.136 \\times 10^{-15} \\mathrm{eV} \\cdot \\mathrm{s}$ 。（按玻尔理论计算时, 在 $\\mu$ 子-氢原子中若仍将质子视为不动, 则 $\\mu^{-}$子相当于质量为 $\\frac{m_{\\mu} m_{\\mathrm{p}}}{m_{\\mu}+m_{\\mathrm{p}}}$ 的带电粒子。)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$\\mu^{-}$子-氢原子基态的电离能, $\\mu$ 子-氢原子从第二激发态跃迁到第一激发态发出的光子的波长]\n它们的单位依次是[$\\mathrm{eV}$, $\\AA$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{eV}$", "$\\AA$" ], "answer_sequence": [ "$\\mu^{-}$子-氢原子基态的电离能", "$\\mu$ 子-氢原子从第二激发态跃迁到第一激发态发出的光子的波长" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_405", "problem": "Find the mass flow rate $\\mu$ of the reated steam jet, as well as the relative mas content $r$ of liquid-phase-water in it. Assume that while flowing into and in the channel the expansion of water vapours is reversible (i.e. heat conductivity can be neglected, and that there is always equilibrium between the liquid and gaseous phases); the adiabatic in dex of water vapours $\\gamma=4 / 3$", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nFind the mass flow rate $\\mu$ of the reated steam jet, as well as the relative mas content $r$ of liquid-phase-water in it. Assume that while flowing into and in the channel the expansion of water vapours is reversible (i.e. heat conductivity can be neglected, and that there is always equilibrium between the liquid and gaseous phases); the adiabatic in dex of water vapours $\\gamma=4 / 3$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~g} / \\mathrm{s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~g} / \\mathrm{s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_897", "problem": "The most outstanding fact in cosmology is that our universe is expanding. Space is continuously created as time lapses. The expansion of space indicates that, when the universe expands, the distance between objects in our universe also expands. It is convenient to use \"comoving\" coordinate system $\\vec{r}=(x, y, z)$ to label points in our expanding universe, in which the coordinate distance $\\Delta r=\\left|\\vec{r}_{2}-\\vec{r}_{1}\\right|=\\sqrt{\\left(x_{2}-x_{1}\\right)^{2}+\\left(y_{2}-y_{1}\\right)^{2}+\\left(z_{2}-z_{1}\\right)^{2}}$ between objects 1 and 2 does not change. (Here we assume no peculiar motion, i.e. no additional motion of those objects other than the motion following the expansion of the universe.) The situation is illustrated in the figure below (the figure has two space dimensions, but our universe actually has three space dimensions).\n\n[figure1]\n\nThe modern theory of cosmology is built upon Einstein's general relativity. However, under proper assumptions, a simplified understanding under the framework of Newton's theory of gravity is also possible. In the following questions, we shall work in the framework of Newton's gravity.\n\nTo measure the physical distance, a \"scale factor\" $a(t)$ is introduced such that the physical distance $\\Delta r_{\\mathrm{p}}$ between the comoving points $\\vec{r}_{1}$ and $\\vec{r}_{2}$ is \n$$\n\\Delta r_{\\mathrm{p}}=a(t) \\Delta r,\n$$\n\nThe expansion of the universe implies that $a(t)$ is an increasing function of time.\n\nOn large scales - scales much larger than galaxies and their clusters - our universe is approximately homogeneous and isotropic. So let us consider a toy model of our universe, which is filled with uniformly distributed particles. There are so many particles, such that we model them as a continuous fluid. Furthermore, we assume the number of particles is\nconserved.\n\nCurrently, our universe is dominated by non-relativistic matter, whose kinetic energy is negligible compared to its mass energy. Let $\\rho_{\\mathrm{m}}(t)$ be the physical energy density (i.e. energyper unit physical volume, which is dominated by mass energy for non-relativistic matter and the gravitational potential energy is not counted as part of the \"physical energy density\") of non-relativistic matter at time $t$. We use $t_{0}$ to denote the present time.\n\nDerive the expression of $\\rho_{\\mathrm{m}}(t)$ at time $t$ in terms of $a(t), a\\left(t_{0}\\right)$ and $\\rho_{\\mathrm{m}}\\left(t_{0}\\right)$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nThe most outstanding fact in cosmology is that our universe is expanding. Space is continuously created as time lapses. The expansion of space indicates that, when the universe expands, the distance between objects in our universe also expands. It is convenient to use \"comoving\" coordinate system $\\vec{r}=(x, y, z)$ to label points in our expanding universe, in which the coordinate distance $\\Delta r=\\left|\\vec{r}_{2}-\\vec{r}_{1}\\right|=\\sqrt{\\left(x_{2}-x_{1}\\right)^{2}+\\left(y_{2}-y_{1}\\right)^{2}+\\left(z_{2}-z_{1}\\right)^{2}}$ between objects 1 and 2 does not change. (Here we assume no peculiar motion, i.e. no additional motion of those objects other than the motion following the expansion of the universe.) The situation is illustrated in the figure below (the figure has two space dimensions, but our universe actually has three space dimensions).\n\n[figure1]\n\nThe modern theory of cosmology is built upon Einstein's general relativity. However, under proper assumptions, a simplified understanding under the framework of Newton's theory of gravity is also possible. In the following questions, we shall work in the framework of Newton's gravity.\n\nTo measure the physical distance, a \"scale factor\" $a(t)$ is introduced such that the physical distance $\\Delta r_{\\mathrm{p}}$ between the comoving points $\\vec{r}_{1}$ and $\\vec{r}_{2}$ is \n$$\n\\Delta r_{\\mathrm{p}}=a(t) \\Delta r,\n$$\n\nThe expansion of the universe implies that $a(t)$ is an increasing function of time.\n\nOn large scales - scales much larger than galaxies and their clusters - our universe is approximately homogeneous and isotropic. So let us consider a toy model of our universe, which is filled with uniformly distributed particles. There are so many particles, such that we model them as a continuous fluid. Furthermore, we assume the number of particles is\nconserved.\n\nCurrently, our universe is dominated by non-relativistic matter, whose kinetic energy is negligible compared to its mass energy. Let $\\rho_{\\mathrm{m}}(t)$ be the physical energy density (i.e. energyper unit physical volume, which is dominated by mass energy for non-relativistic matter and the gravitational potential energy is not counted as part of the \"physical energy density\") of non-relativistic matter at time $t$. We use $t_{0}$ to denote the present time.\n\nDerive the expression of $\\rho_{\\mathrm{m}}(t)$ at time $t$ in terms of $a(t), a\\left(t_{0}\\right)$ and $\\rho_{\\mathrm{m}}\\left(t_{0}\\right)$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_88fc0d3da9200bd5a832g-1.jpg?height=657&width=1455&top_left_y=1116&top_left_x=312" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_669", "problem": "A very heavy box of mass $M$ is being lifted upwards by a crane at a constant velocity $u$. A ball of mass $m \\ll M$ is thrown at the box with velocity $v$ at angle $\\alpha$.\n[figure1]\n\nIf the ball hits the box at some time $t$, what are the $x$ and $y$ components of the velocity of the ball $\\left(v_{x}(t), v_{y}(t)\\right)$ after the collision? Assume a perfectly elastic collision.\nA: $(-v \\cos \\alpha,|v \\sin \\alpha-g t|)$\nB: $(-v \\cos \\alpha,-|v \\sin \\alpha-g t|)$\nC: $(-v \\cos \\alpha, 2 u+|v \\sin \\alpha-g t|)$\nD: $(-v \\cos \\alpha, 2 u-|v \\sin \\alpha-g t|)$\nE: Either a or b.\nF: Either c or d.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA very heavy box of mass $M$ is being lifted upwards by a crane at a constant velocity $u$. A ball of mass $m \\ll M$ is thrown at the box with velocity $v$ at angle $\\alpha$.\n[figure1]\n\nIf the ball hits the box at some time $t$, what are the $x$ and $y$ components of the velocity of the ball $\\left(v_{x}(t), v_{y}(t)\\right)$ after the collision? Assume a perfectly elastic collision.\n\nA: $(-v \\cos \\alpha,|v \\sin \\alpha-g t|)$\nB: $(-v \\cos \\alpha,-|v \\sin \\alpha-g t|)$\nC: $(-v \\cos \\alpha, 2 u+|v \\sin \\alpha-g t|)$\nD: $(-v \\cos \\alpha, 2 u-|v \\sin \\alpha-g t|)$\nE: Either a or b.\nF: Either c or d.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E, F].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_a0d0ab960474d1b0609cg-03.jpg?height=560&width=708&top_left_y=1122&top_left_x=1142" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_507", "problem": "Beloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\n There are $N=24$ coils stacked on top of each other. Tap water with an initial temperature of $T_{c}=18^{\\circ} \\mathrm{C}$ enters the spiral through the copper pipe to keep it from over heating; the water exits at a temperature of $T_{h}=31^{\\circ} \\mathrm{C}$. The copper pipe carries a direct $45 \\mathrm{Amp}$ current in order to generate the necessary magnetic field. At what rate must the cooling water flow be provided to the electromagnet? Express your answer in liters per second with only one significant digit. The specific heat capacity of water is $4200 \\mathrm{~J} /{ }^{\\circ} \\mathrm{C} \\cdot \\mathrm{kg}$; the density of water is $1000 \\mathrm{~kg} / \\mathrm{m}^{3}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nBeloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\n There are $N=24$ coils stacked on top of each other. Tap water with an initial temperature of $T_{c}=18^{\\circ} \\mathrm{C}$ enters the spiral through the copper pipe to keep it from over heating; the water exits at a temperature of $T_{h}=31^{\\circ} \\mathrm{C}$. The copper pipe carries a direct $45 \\mathrm{Amp}$ current in order to generate the necessary magnetic field. At what rate must the cooling water flow be provided to the electromagnet? Express your answer in liters per second with only one significant digit. The specific heat capacity of water is $4200 \\mathrm{~J} /{ }^{\\circ} \\mathrm{C} \\cdot \\mathrm{kg}$; the density of water is $1000 \\mathrm{~kg} / \\mathrm{m}^{3}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{l} / \\mathrm{s}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=468&width=1592&top_left_y=438&top_left_x=259", "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=493&width=1268&top_left_y=1339&top_left_x=426" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{l} / \\mathrm{s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_876", "problem": "he schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\\pm \\varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale.\n\n[figure1]\n\nWhich of the following explains ultimately why angular momentum is not conserved along the lower branches of the Hadley circulation?\nA: There is friction from the Earth's surface.\nB: There is turbulence in the lower atmosphere, where different layers of air are mixed\nC: The air is denser lower down and so inertia slows down the motion around the spin axis of the Earth.\nD: The air is moist at the lower levels causing retardation to the wind velocity.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nhe schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\\pm \\varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale.\n\n[figure1]\n\nWhich of the following explains ultimately why angular momentum is not conserved along the lower branches of the Hadley circulation?\n\nA: There is friction from the Earth's surface.\nB: There is turbulence in the lower atmosphere, where different layers of air are mixed\nC: The air is denser lower down and so inertia slows down the motion around the spin axis of the Earth.\nD: The air is moist at the lower levels causing retardation to the wind velocity.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e7b9ba69a6dd20c5cfd1g-1.jpg?height=1148&width=1151&top_left_y=1096&top_left_x=384" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_79", "problem": "A brick is moving at a speed of $3 \\mathrm{~m} / \\mathrm{s}$ and a pebble is moving at a speed of $5 \\mathrm{~m} / \\mathrm{s}$. If both objects have the same kinetic energy, what is the ratio of the brick's mass to the pebble's mass?\nA: $25: 9$\nB: $5: 3$\nC: $4: 1$\nD: $3: 1$\nE: $\\sqrt{5}: \\sqrt{3}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA brick is moving at a speed of $3 \\mathrm{~m} / \\mathrm{s}$ and a pebble is moving at a speed of $5 \\mathrm{~m} / \\mathrm{s}$. If both objects have the same kinetic energy, what is the ratio of the brick's mass to the pebble's mass?\n\nA: $25: 9$\nB: $5: 3$\nC: $4: 1$\nD: $3: 1$\nE: $\\sqrt{5}: \\sqrt{3}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1726", "problem": "飞行时间质谱仪 (TOFMS) 的基本原理如图 1 所示, 主要由离子源区、漂移区和探测器三部分组成。带正电的离子在离子源中形成后被电场 $E_{\\mathrm{s}}$ 加速, 经过漂移区（真空无场）, 到达离子探测器。设离子在离子源区加速的距离为 $S$, 在漂移区漂移的距离为 $L$ 。通过记录离子到达探测器的时间, 可以把不同的离子按质荷比 $m / q$ 的大小进行分离, 这里 $m$ 和 $q$ 分别\n\n[图1]\n\n图1\n\n表示离子的质量和电量。分辨率是 TOFMS 最重要的性能指标, 本题将在不同情况下进行讨论或计算。忽略重力的影响。对于理想的 TOFMS, 不同离子在离子源 X 轴方向同一位置、同一时刻 $t=0$ 产生,且初速度为 0 。探测器可以测定离子到达探测器的时刻 $t$, 其最小分辨时间为 $\\Delta t$ (即探测器所测时刻 $t$ 的误差）。定义仪器的分辨率为 $R=m / \\Delta m$, 其中 $\\Delta m$ 为最小分辨时间 $\\Delta t$ 对应的最小分辨质量。此种情形下, $R$ 完全由 $\\Delta t$ 决定，试推导 $R$ 与 $t$ 和 $\\Delta t$ 之间的关系。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n飞行时间质谱仪 (TOFMS) 的基本原理如图 1 所示, 主要由离子源区、漂移区和探测器三部分组成。带正电的离子在离子源中形成后被电场 $E_{\\mathrm{s}}$ 加速, 经过漂移区（真空无场）, 到达离子探测器。设离子在离子源区加速的距离为 $S$, 在漂移区漂移的距离为 $L$ 。通过记录离子到达探测器的时间, 可以把不同的离子按质荷比 $m / q$ 的大小进行分离, 这里 $m$ 和 $q$ 分别\n\n[图1]\n\n图1\n\n表示离子的质量和电量。分辨率是 TOFMS 最重要的性能指标, 本题将在不同情况下进行讨论或计算。忽略重力的影响。\n\n问题:\n对于理想的 TOFMS, 不同离子在离子源 X 轴方向同一位置、同一时刻 $t=0$ 产生,且初速度为 0 。探测器可以测定离子到达探测器的时刻 $t$, 其最小分辨时间为 $\\Delta t$ (即探测器所测时刻 $t$ 的误差）。定义仪器的分辨率为 $R=m / \\Delta m$, 其中 $\\Delta m$ 为最小分辨时间 $\\Delta t$ 对应的最小分辨质量。此种情形下, $R$ 完全由 $\\Delta t$ 决定，试推导 $R$ 与 $t$ 和 $\\Delta t$ 之间的关系。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_07aa406e17d01fd01b36g-04.jpg?height=311&width=1077&top_left_y=938&top_left_x=478" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1559", "problem": "如图, 一质量为 $M$ 、长为 $l$ 的匀质细杆 $\\mathrm{AB}$ 自由悬挂于通过坐标原点 $O$ 点的水平光滑转轴上 (此时, 杆的上端 $\\mathrm{A}$ 未在图中标出, 可视为与 $O$ 点重合), 杆可绕通过 $O$ 点的轴在坚直平面 (即 $x-y$ 平面, $x$ 轴正方向水平向右）内转动; $O$ 点相对于地面足够高, 初始时杆自然下垂; 一质量为 $m$ 的弹丸以大小为 $v_{0}$ 的水平速度撞击杆的打击中心（打击过程中轴对杆的水平作用力为零）并很快嵌入杆中。在杆转半圈至坚直状态时立即撤除转轴。重力加速度大小为 $g$ 。\n\n[图1]求杆的打击中心到 $O$ 点的距离;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一质量为 $M$ 、长为 $l$ 的匀质细杆 $\\mathrm{AB}$ 自由悬挂于通过坐标原点 $O$ 点的水平光滑转轴上 (此时, 杆的上端 $\\mathrm{A}$ 未在图中标出, 可视为与 $O$ 点重合), 杆可绕通过 $O$ 点的轴在坚直平面 (即 $x-y$ 平面, $x$ 轴正方向水平向右）内转动; $O$ 点相对于地面足够高, 初始时杆自然下垂; 一质量为 $m$ 的弹丸以大小为 $v_{0}$ 的水平速度撞击杆的打击中心（打击过程中轴对杆的水平作用力为零）并很快嵌入杆中。在杆转半圈至坚直状态时立即撤除转轴。重力加速度大小为 $g$ 。\n\n[图1]\n\n问题:\n求杆的打击中心到 $O$ 点的距离;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-01.jpg?height=483&width=374&top_left_y=2437&top_left_x=1412" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_794", "problem": "Beatrice has decided to use a dynamic method to find $k$ for her spring. She measures the period, $T$, of oscillation for a mass, $m$, on a spring for a series of different masses. The equation that relates period to mass is: $T=2 \\pi \\sqrt{\\frac{m}{k}}$. If Beatrice plots a graph of $T$ vs $m$, which of the following graphs will her plot look like? [figure1]\nA: a\nB: b\nC: c\nD: d\nE: e\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nBeatrice has decided to use a dynamic method to find $k$ for her spring. She measures the period, $T$, of oscillation for a mass, $m$, on a spring for a series of different masses. The equation that relates period to mass is: $T=2 \\pi \\sqrt{\\frac{m}{k}}$. If Beatrice plots a graph of $T$ vs $m$, which of the following graphs will her plot look like? [figure1]\n\nA: a\nB: b\nC: c\nD: d\nE: e\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_bdd0a782567902ad78d8g-05.jpg?height=690&width=1194&top_left_y=588&top_left_x=474" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1284", "problem": "光子被电子散射时, 如果初态电子具有足够的动能, 以至于在散射过程中有能量从电子转移到光子, 则该散射被称为逆康普顿散射. 当低能光子与高能电子发生对头碰撞时, 就会出现逆康普顿散射. 已知电子静止质量为 $m_{e}$, 真空中的光速为 $c$. 若能量为 $E_{e}$ 的电子与能量为 $E_{\\gamma}$ 的光子相向对碰求散射后光子的能量", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n光子被电子散射时, 如果初态电子具有足够的动能, 以至于在散射过程中有能量从电子转移到光子, 则该散射被称为逆康普顿散射. 当低能光子与高能电子发生对头碰撞时, 就会出现逆康普顿散射. 已知电子静止质量为 $m_{e}$, 真空中的光速为 $c$. 若能量为 $E_{e}$ 的电子与能量为 $E_{\\gamma}$ 的光子相向对碰\n\n问题:\n求散射后光子的能量\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1479", "problem": "有一根长为 $6.00 \\mathrm{~cm}$ 、内外半径分别为 $0.500 \\mathrm{~mm}$ 和 $5.00 \\mathrm{~mm}$ 的玻璃毛细管。\n\n已知玻璃的密度是水的 2 倍, 水的密度为 $1.00 \\times 10^{3} \\mathrm{~kg} \\cdot \\mathrm{m}^{-3}$, 水的表面张力系数为 $7.27 \\times 10^{-2} \\mathrm{~N} \\cdot \\mathrm{m}^{-1}$, 水与玻璃的接触角 $\\theta$ 可视为零, 重力加速度取 $9.80 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$ 。若将该毛细管长度的三分之一坚直浸入水中, 问需要多大向上的力才能使该毛细管保持不动?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n有一根长为 $6.00 \\mathrm{~cm}$ 、内外半径分别为 $0.500 \\mathrm{~mm}$ 和 $5.00 \\mathrm{~mm}$ 的玻璃毛细管。\n\n已知玻璃的密度是水的 2 倍, 水的密度为 $1.00 \\times 10^{3} \\mathrm{~kg} \\cdot \\mathrm{m}^{-3}$, 水的表面张力系数为 $7.27 \\times 10^{-2} \\mathrm{~N} \\cdot \\mathrm{m}^{-1}$, 水与玻璃的接触角 $\\theta$ 可视为零, 重力加速度取 $9.80 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$ 。\n\n问题:\n若将该毛细管长度的三分之一坚直浸入水中, 问需要多大向上的力才能使该毛细管保持不动?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~N}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~N}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_653", "problem": "Suppose that a positron (charge $+e$, mass $m_{e}$ ) is fired at a non-relativistic velocity $v_{0}$ towards a proton (charge $+e$, mass $m_{p}$ ) at rest. What is the minimum separation between the two particles as they approach each other?\nA: $\\frac{e^{2}}{2 \\pi \\epsilon_{0} m_{e} v_{0}^{2}}$\nB: $\\frac{e}{2 \\pi \\epsilon_{0} m_{e} v_{0}^{2}}$\nC: $\\left(\\frac{e^{2}}{2 \\pi \\epsilon_{0} m_{e} v_{0}^{2}}\\right)^{1 / 2}$\nD: $\\frac{e^{2}}{2 \\pi \\epsilon_{0} v_{0}^{2}} \\frac{m_{e}+m_{p}}{m_{e} m_{p}}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nSuppose that a positron (charge $+e$, mass $m_{e}$ ) is fired at a non-relativistic velocity $v_{0}$ towards a proton (charge $+e$, mass $m_{p}$ ) at rest. What is the minimum separation between the two particles as they approach each other?\n\nA: $\\frac{e^{2}}{2 \\pi \\epsilon_{0} m_{e} v_{0}^{2}}$\nB: $\\frac{e}{2 \\pi \\epsilon_{0} m_{e} v_{0}^{2}}$\nC: $\\left(\\frac{e^{2}}{2 \\pi \\epsilon_{0} m_{e} v_{0}^{2}}\\right)^{1 / 2}$\nD: $\\frac{e^{2}}{2 \\pi \\epsilon_{0} v_{0}^{2}} \\frac{m_{e}+m_{p}}{m_{e} m_{p}}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_808", "problem": "Consider an arbitrary initial rotation of the stage with angular momentum $L$ (Fig. 2), where $\\theta$ is the angle between the symmetry axis and the direction of angular momentum. Fuel tank at this point is assumed to be empty. No forces or torques act upon the stage.\n\n[figure1]\n\nFig. 2: Rocket stage rotation\n\nFind the projections of angular velocity $\\vec{\\omega}$ on $x$ and $y$, given that $\\vec{L}=J_{x} \\omega_{x} \\vec{e}_{x}+J_{y} \\omega_{y} \\vec{e}_{y}$ for material symmetry axes $x$ and $y$ with unit vectors $\\vec{e}_{x}$ and $\\vec{e}_{y}$. Provide the answer in terms of $L=|\\vec{L}|$, angle $\\theta$, and inertia moments $J_{x}, J_{y}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nConsider an arbitrary initial rotation of the stage with angular momentum $L$ (Fig. 2), where $\\theta$ is the angle between the symmetry axis and the direction of angular momentum. Fuel tank at this point is assumed to be empty. No forces or torques act upon the stage.\n\n[figure1]\n\nFig. 2: Rocket stage rotation\n\nFind the projections of angular velocity $\\vec{\\omega}$ on $x$ and $y$, given that $\\vec{L}=J_{x} \\omega_{x} \\vec{e}_{x}+J_{y} \\omega_{y} \\vec{e}_{y}$ for material symmetry axes $x$ and $y$ with unit vectors $\\vec{e}_{x}$ and $\\vec{e}_{y}$. Provide the answer in terms of $L=|\\vec{L}|$, angle $\\theta$, and inertia moments $J_{x}, J_{y}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [$\\omega_{y}$, $\\omega_{x}$].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d7ee880fa438d4cf18fag-2.jpg?height=356&width=763&top_left_y=553&top_left_x=652" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$\\omega_{y}$", "$\\omega_{x}$" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_301", "problem": "An electromagnet is formed when a current flows through a coil of wire.\n\nWhich of the following changes on its own does not necessarily increase the strength of an electromagnet?\nA: Using thicker wire\nB: Using a higher current\nC: Adding an iron core\nD: Using more turns of wire\nE: Making the turns more tightly packed\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nAn electromagnet is formed when a current flows through a coil of wire.\n\nWhich of the following changes on its own does not necessarily increase the strength of an electromagnet?\n\nA: Using thicker wire\nB: Using a higher current\nC: Adding an iron core\nD: Using more turns of wire\nE: Making the turns more tightly packed\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_321", "problem": "DRAGON (5 points) - Aigar Vaigu. Here is a photo of a dragon under water (there is a larger photo on a separate sheet). The length of the dragon is $l=8 \\mathrm{~cm}$ and its height is $h=3 \\mathrm{~cm}$. The diameter of the bottom of the bowl is $d=$ $10 \\mathrm{~cm}$ and the angle between the table and the side of the bowl is $\\alpha=60^{\\circ}$. The refractive index of water is $n=1.33$. The photo has been taken so that the camera was pointing directly along the water surface. In the following questions the angle between the horizon when looking at some point on the image of the dragon is defined as the angle between the horizontal water surface (or any other horizontal surface) and the straight line from the eye to that point on the image.\n\n[figure1]\n\nWhat is the largest angle below the horizon from which the reflection of the dragon from the water surface can be seen?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nDRAGON (5 points) - Aigar Vaigu. Here is a photo of a dragon under water (there is a larger photo on a separate sheet). The length of the dragon is $l=8 \\mathrm{~cm}$ and its height is $h=3 \\mathrm{~cm}$. The diameter of the bottom of the bowl is $d=$ $10 \\mathrm{~cm}$ and the angle between the table and the side of the bowl is $\\alpha=60^{\\circ}$. The refractive index of water is $n=1.33$. The photo has been taken so that the camera was pointing directly along the water surface. In the following questions the angle between the horizon when looking at some point on the image of the dragon is defined as the angle between the horizontal water surface (or any other horizontal surface) and the straight line from the eye to that point on the image.\n\n[figure1]\n\nWhat is the largest angle below the horizon from which the reflection of the dragon from the water surface can be seen?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $^{\\circ}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e6c1582ce7f1c05fa0a6g-1.jpg?height=381&width=663&top_left_y=865&top_left_x=90" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$^{\\circ}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1532", "problem": "中性粒子分析器 (Neutral-Particle Analyser) 是核聚变研究中测量快离子温度及其能量分布的重要设备. 其基本原理如图所示, 通过对高能量 (200eV 30KeV) 中性原子（它们容易穿透探测区中的电磁区域) 的能量和动量的测量, 可诊断曾与这些中性原子充分碰撞过的粒子的性质. 为了测量中性原子的能量分布, 首先让中性原子电离然后让离子束以 $\\theta$ 角入射到间距为 $d$ 、电压为 $V$ 的平行板电极组成的区域, 经电场偏转后离开电场区域, 在保证所测量离子不碰到上极板的前提下, 通过测量入射孔 $\\mathrm{A}$ 和出射孔 $\\mathrm{B}$ 间平行于极板方向的距离 1 来决定离子的能量. 设 $\\mathrm{A}$ 与下极板的距离为 $h_{1}, \\mathrm{~B}$ 与下极板的距离为 $h_{2}$, 已知离子所带电荷为 $q$.\n\n[图1]被测离子束一般具有发散角 $\\Delta \\alpha(\\Delta \\alpha<<\\theta)$. 为了提高测量的精度, 要求具有相同能量 $E$,但入射方向在 $\\Delta \\alpha$ 范围内变化的离子在同一小孔 $\\mathrm{B}$ 处射出, 求 $h_{2}$ 的表达式; 并给出此时能量 $E$与 $l$ 的关系", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n中性粒子分析器 (Neutral-Particle Analyser) 是核聚变研究中测量快离子温度及其能量分布的重要设备. 其基本原理如图所示, 通过对高能量 (200eV 30KeV) 中性原子（它们容易穿透探测区中的电磁区域) 的能量和动量的测量, 可诊断曾与这些中性原子充分碰撞过的粒子的性质. 为了测量中性原子的能量分布, 首先让中性原子电离然后让离子束以 $\\theta$ 角入射到间距为 $d$ 、电压为 $V$ 的平行板电极组成的区域, 经电场偏转后离开电场区域, 在保证所测量离子不碰到上极板的前提下, 通过测量入射孔 $\\mathrm{A}$ 和出射孔 $\\mathrm{B}$ 间平行于极板方向的距离 1 来决定离子的能量. 设 $\\mathrm{A}$ 与下极板的距离为 $h_{1}, \\mathrm{~B}$ 与下极板的距离为 $h_{2}$, 已知离子所带电荷为 $q$.\n\n[图1]\n\n问题:\n被测离子束一般具有发散角 $\\Delta \\alpha(\\Delta \\alpha<<\\theta)$. 为了提高测量的精度, 要求具有相同能量 $E$,但入射方向在 $\\Delta \\alpha$ 范围内变化的离子在同一小孔 $\\mathrm{B}$ 处射出, 求 $h_{2}$ 的表达式; 并给出此时能量 $E$与 $l$ 的关系\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-04.jpg?height=471&width=829&top_left_y=1706&top_left_x=996" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_737", "problem": "In the cube shown below, each edge is a wire with resistance $R$. What is the equivalent resistance between points $\\mathrm{A}$ and $\\mathrm{B}$ ?\n\n[figure1]\nA: $R$\nB: $\\frac{1}{3} R$\nC: $\\frac{2}{3} R$\nD: $\\frac{1}{2} R$\nE: $\\frac{5}{6} R$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn the cube shown below, each edge is a wire with resistance $R$. What is the equivalent resistance between points $\\mathrm{A}$ and $\\mathrm{B}$ ?\n\n[figure1]\n\nA: $R$\nB: $\\frac{1}{3} R$\nC: $\\frac{2}{3} R$\nD: $\\frac{1}{2} R$\nE: $\\frac{5}{6} R$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-03.jpg?height=417&width=374&top_left_y=1746&top_left_x=648" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_345", "problem": "In this problem we will look at the phase diagram of water (see graphs on a separate page). The first figure shows the phase diagram in a region close to the triple point [(s) - solid, (l) - liquid, (g) - gas], while the second figure shows the melting curve. When two phases $\\alpha$ and $\\beta$ are in equilibrium, the phase transition curve follows the law of Clausius-Clapeyron:$$\n\\frac{\\mathrm{d} p}{\\mathrm{~d} T}=\\frac{1}{T} \\frac{H_{\\beta}-H_{\\alpha}}{V_{\\beta}-V_{\\alpha}} $$\n\nwhere $H_{\\alpha}$ is the specific enthalpy (enthalpy per mass) of phase $\\alpha$, and $V_{\\alpha}$ is the specific volume (volume per mass).\n\nApproximate the Earth as a system of a homogeneous atmosphere, consisting of air and water vapor, in equilibrium with a sea of liquid water. If the atmospheric temperature rises by $3^{\\circ} \\mathrm{C}$, by what percentage does the water vapor pressure rise? (The current temperature of the Earth is $15^{\\circ} \\mathrm{C}$.) You may need the values $R=$ $8.314 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$ and $\\mu=18.015 \\mathrm{~g} \\mathrm{~mol}^{-1}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn this problem we will look at the phase diagram of water (see graphs on a separate page). The first figure shows the phase diagram in a region close to the triple point [(s) - solid, (l) - liquid, (g) - gas], while the second figure shows the melting curve. When two phases $\\alpha$ and $\\beta$ are in equilibrium, the phase transition curve follows the law of Clausius-Clapeyron:$$\n\\frac{\\mathrm{d} p}{\\mathrm{~d} T}=\\frac{1}{T} \\frac{H_{\\beta}-H_{\\alpha}}{V_{\\beta}-V_{\\alpha}} $$\n\nwhere $H_{\\alpha}$ is the specific enthalpy (enthalpy per mass) of phase $\\alpha$, and $V_{\\alpha}$ is the specific volume (volume per mass).\n\nApproximate the Earth as a system of a homogeneous atmosphere, consisting of air and water vapor, in equilibrium with a sea of liquid water. If the atmospheric temperature rises by $3^{\\circ} \\mathrm{C}$, by what percentage does the water vapor pressure rise? (The current temperature of the Earth is $15^{\\circ} \\mathrm{C}$.) You may need the values $R=$ $8.314 \\mathrm{~J} \\mathrm{~mol}^{-1} \\mathrm{~K}^{-1}$ and $\\mu=18.015 \\mathrm{~g} \\mathrm{~mol}^{-1}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of %(percentage), but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "%(percentage)" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_259", "problem": "The characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part, analyze a situation when an angular momentum is not conserved. The system is the same as in the previous part with the only difference that the dipole is fixed and the charged small object with a mass $2 m$ is moving around the dipole. The electrostatic field of the dipole is easier to describe in the polar system of coordinates, which is defined with the distance $r$ from the center of the dipole, and angle $\\theta$ counted counterclockwise, as shown in Figure 3.\n\n[figure1]\n\nFigure 3: The system analyzed in Part 2. (Direction of the vector $\\mathbf{E}_{\\mathbf{n}}$ and $\\mathbf{E}_{\\mathbf{t}}$ could be wrong)\n\nDetermine the electrostatic potential $\\varphi$ at a distance $r \\gg d$ from the dipole, as a function of $\\theta$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part, analyze a situation when an angular momentum is not conserved. The system is the same as in the previous part with the only difference that the dipole is fixed and the charged small object with a mass $2 m$ is moving around the dipole. The electrostatic field of the dipole is easier to describe in the polar system of coordinates, which is defined with the distance $r$ from the center of the dipole, and angle $\\theta$ counted counterclockwise, as shown in Figure 3.\n\n[figure1]\n\nFigure 3: The system analyzed in Part 2. (Direction of the vector $\\mathbf{E}_{\\mathbf{n}}$ and $\\mathbf{E}_{\\mathbf{t}}$ could be wrong)\n\nDetermine the electrostatic potential $\\varphi$ at a distance $r \\gg d$ from the dipole, as a function of $\\theta$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ed4e92416bdbac30298dg-2.jpg?height=477&width=1442&top_left_y=1694&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1465", "problem": "如图 15a, 间距为 $L$ 的两根平行光滑金属导轨 $\\mathrm{MN} 、 \\mathrm{PQ}$ 放置于同一水平面内, 导轨左端连接一阻值为 $R$ 的定值电阻, 导体棒 $a$ 垂直于导轨放置在导轨上, 在 $a$棒左侧和导轨间存在坚直向下的匀强磁场, 磁感应强度大小为 $B$, 在 $a$ 棒右侧有一绝缘棒 $b, b$ 棒与 $a$ 棒平行, 且与固定在墙上的轻弹簧接触但不相连, 弹簧处于压缩状态且被锁定。现解除锁定, $b$ 棒在弹簧的作用下向左移动, 脱离弹簧后以速度 $v_{0}$ 与 $a$ 棒碰撞并粘在一起。已知 $a 、 b$ 棒的质量分别为 $m 、 M$, 碰撞前后, 两棒始终垂直于导轨, $a$ 棒在两导轨之间的部分的电阻为 $r$, 导轨电阻、接触电阻以及空气阻力均忽略不计, $a 、 b$ 棒总是保持与导轨接触良好。不计电路中感应电流的磁场, 求\n\n[图1]\n\n图 15a在 $a$ 棒从开向左滑行直至滑行距离为 $x$ 的过程中通过定值电阻的电量 $q$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图 15a, 间距为 $L$ 的两根平行光滑金属导轨 $\\mathrm{MN} 、 \\mathrm{PQ}$ 放置于同一水平面内, 导轨左端连接一阻值为 $R$ 的定值电阻, 导体棒 $a$ 垂直于导轨放置在导轨上, 在 $a$棒左侧和导轨间存在坚直向下的匀强磁场, 磁感应强度大小为 $B$, 在 $a$ 棒右侧有一绝缘棒 $b, b$ 棒与 $a$ 棒平行, 且与固定在墙上的轻弹簧接触但不相连, 弹簧处于压缩状态且被锁定。现解除锁定, $b$ 棒在弹簧的作用下向左移动, 脱离弹簧后以速度 $v_{0}$ 与 $a$ 棒碰撞并粘在一起。已知 $a 、 b$ 棒的质量分别为 $m 、 M$, 碰撞前后, 两棒始终垂直于导轨, $a$ 棒在两导轨之间的部分的电阻为 $r$, 导轨电阻、接触电阻以及空气阻力均忽略不计, $a 、 b$ 棒总是保持与导轨接触良好。不计电路中感应电流的磁场, 求\n\n[图1]\n\n图 15a\n\n问题:\n在 $a$ 棒从开向左滑行直至滑行距离为 $x$ 的过程中通过定值电阻的电量 $q$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-06.jpg?height=362&width=739&top_left_y=1161&top_left_x=1184" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_964", "problem": "A linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths $\\ell_{1}, \\ell_{2}, \\ell_{3}, \\ldots$ along the beam and with a fixed small separation $d$ between each tube. The tubes are connected to a high voltage, constant radio frequency $\\mathrm{AC}$ supply where the peak voltage of the $\\mathrm{AC}$ is $V_{0}$. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge $e$ and mass $m_{e}$ is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it \"drifts\" through the tube. It passes through a large potential difference between the tubes and, if the charged particle's motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10\n\n[figure1]\n\nFigure 9\n\n[figure2]\n\nFigure 10\n\n\n To generate $2 \\mathrm{~W}$, how many electrons flow through in a second?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths $\\ell_{1}, \\ell_{2}, \\ell_{3}, \\ldots$ along the beam and with a fixed small separation $d$ between each tube. The tubes are connected to a high voltage, constant radio frequency $\\mathrm{AC}$ supply where the peak voltage of the $\\mathrm{AC}$ is $V_{0}$. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge $e$ and mass $m_{e}$ is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it \"drifts\" through the tube. It passes through a large potential difference between the tubes and, if the charged particle's motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10\n\n[figure1]\n\nFigure 9\n\n[figure2]\n\nFigure 10\n\n\n To generate $2 \\mathrm{~W}$, how many electrons flow through in a second?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-10.jpg?height=300&width=454&top_left_y=818&top_left_x=296", "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-10.jpg?height=359&width=1128&top_left_y=757&top_left_x=795" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_575", "problem": "A particle is constrained to move on the inner surface of a frictionless parabolic bowl whose crosssection has equation $z=k r^{2}$. The particle begins at a height $z_{0}$ above the bottom of the bowl with a horizontal velocity $v_{0}$ along the surface of the bowl. The acceleration due to gravity is $g$.\n[figure1]\n\nSuppose that the particle now begins at a height $z_{0}$ above the bottom of the bowl with an initial velocity $v_{0}=0$.\n\nAssuming that $z_{0}$ is small enough so that the motion can be approximated as simple harmonic, find the period of the motion in terms any or all of the mass of the particle $m, g, z_{0}$, and/or $k$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA particle is constrained to move on the inner surface of a frictionless parabolic bowl whose crosssection has equation $z=k r^{2}$. The particle begins at a height $z_{0}$ above the bottom of the bowl with a horizontal velocity $v_{0}$ along the surface of the bowl. The acceleration due to gravity is $g$.\n[figure1]\n\nSuppose that the particle now begins at a height $z_{0}$ above the bottom of the bowl with an initial velocity $v_{0}=0$.\n\nAssuming that $z_{0}$ is small enough so that the motion can be approximated as simple harmonic, find the period of the motion in terms any or all of the mass of the particle $m, g, z_{0}$, and/or $k$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_3899026513eb55709c81g-13.jpg?height=392&width=1266&top_left_y=476&top_left_x=428" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1318", "problem": "如图 2 所示, 某同学经过一段时间练习, 掌握了利用在瓶中装不同高度的水, 在瓶口吹出不同频率声音, 以演奏乐曲的技巧。以下说法中正确的有\n\n[图1]\n\n图 2\nA: 若瓶中水柱高度之比为 $2: 3: 4$, 则吹出来的声音频率之比也为 $2: 3: 4$\nB: 吹出来声音频率主要由在空气柱中声波形成的驻波频率来决定\nC: 空气驻波在水面附近是波节, 在瓶口附近是波腹\nD: 空气柱越长, 发出的声音频率越高\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图 2 所示, 某同学经过一段时间练习, 掌握了利用在瓶中装不同高度的水, 在瓶口吹出不同频率声音, 以演奏乐曲的技巧。以下说法中正确的有\n\n[图1]\n\n图 2\n\nA: 若瓶中水柱高度之比为 $2: 3: 4$, 则吹出来的声音频率之比也为 $2: 3: 4$\nB: 吹出来声音频率主要由在空气柱中声波形成的驻波频率来决定\nC: 空气驻波在水面附近是波节, 在瓶口附近是波腹\nD: 空气柱越长, 发出的声音频率越高\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-02.jpg?height=377&width=374&top_left_y=517&top_left_x=881" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_915", "problem": "## Introduction\n\nActive galactic nuclei (AGN) are supermassive black holes which form the centres of galaxies, and emit large amounts of energy in radiation and particle flows. One feature of many AGN are jetted outflows, which can be observed through radio emission, and sometimes also in other parts of the electromagnetic spectrum, including $x$-rays. These jets are large flows of plasma at relativistic speeds, over lengths of order $10^{20} \\mathrm{~m}$, which is tens of thousands of light years. The $\\mathrm{x}$-ray emission from jets is usually dominated by synchrotron emission from relativistic electrons gyrating in the magnetic field of the jet.\n\n[figure1]\n\nFigure 1: X-ray image of the jet from the Centaurus A AGN. Darker regions represent regions of higher intensity $x$-rays. Brighter regions within the fainter jet are called knots. (Snios et al., 2019)\n\nA simple model of the flow of jets assumes that the flow is steady and directed radially away from the central AGN, so approximately one dimensional, and that the plasma in the jet is in pressure equilibrium with its surroundings. There is assumed to be a constant rate per volume of mass injected into the jet from stars which lose their outer layers as they move through their life cycle.\n\nThe jet is described in terms of the coordinate representing distance from the AGN, $s$, and the opening radius $r$ of the conical jet. These distances are measured in parsecs, where $1 \\mathrm{pc}=3.086 \\times 10^{16} \\mathrm{~m}$. The speed of the jet flow is assumed to be directed radially away from the central AGN, and be a function of $s$ only. The plasma in the jet is comprised of electrons, protons, and some heavier ionised nuclei. The average energy carried by each particle in the jet, in the reference frame of the bulk flow of the jet (which we will call the jet frame), is $\\epsilon_{\\mathrm{av}}=\\mu_{\\mathrm{pp}} c^{2}+h$, where the term $h$ includes all thermal kinetic energy and potential energies in terms of the pressure $P$ and $n$ is the number density of the plasma.\n\nAs the stars, which the jet flows past, move through their life cycles they can lose part of their atmosphere. This results in a uniform rate of injection of mass per unit volume $\\alpha$ into the jet, and the injected particles are assumed to be at rest relative to the AGN.\n\nThis model can be applied to the Centaurus A jet. Centaurus A is one of the nearest AGN, so it is possible to observe its jet at relatively high spatial resolution. The total power carried by the jet is estimated to be $P_{\\mathrm{j}}=1 \\times 10^{36} \\mathrm{~J} \\cdot \\mathrm{s}^{-1}$. See below for a diagram of a simple geometrical description of the Centaurus A jet, including measurements of some jet parameters. $s_{1}$ is the coordinate of the start of the jet, and $s_{2}$ the coordinate of the end of the jet. In Centuarus A the average mass per particle is $\\mu_{\\mathrm{pp}}=0.59 m_{\\mathrm{p}}$ and $h=\\frac{13}{4} P / n$. The pressure in the plasma surrounding the jet is $P(s)=5.7 \\times 10^{-12}\\left(\\frac{s}{s_{0}}\\right)^{-1.5} \\mathrm{~Pa}$, where $s_{0}=1 \\mathrm{kpc}$.\n\n[figure2]\n\nFigure 2: The Centaurus A jet, showing the geometry compared to the active galactic nucleus (AGN).\n\nThe jet is described by the following parameters, all of which depend on the distance $s$ from the AGN:\n\n- the opening radius of the jet $r(s)$ in the AGN frame\n- the cross sectional area of the jet $A(s)$ in the AGN frame\n- the speed of the jet $v(s)$ in the AGN frame\n- the lorentz gamma factor of the jet $\\gamma(s)$ in the AGN frame\n- the number density $n(s)$ in the frame of the jet\n\nThe power carried by a jet is defined to be the sum of the total bulk kinetic energy flux and the total thermal energy flux, so\n\n$$\nP_{\\mathrm{j}}(s)=F_{\\mathrm{E}}(s)-\\dot{M} c^{2}\n$$\n\nwhere $F_{\\mathrm{E}}(s)$ is the flux of energy through the cross section of the jet at $s$, and $\\dot{M}$ is the mass flux through the jet cross section at the same distance $s$ from the AGN.\n\nFind an expression for the total momentum flux, $\\Pi$, into the Centaurus $A$ jet. $0.5 \\mathrm{pt}$ Also numerically evaluate this expression.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\n## Introduction\n\nActive galactic nuclei (AGN) are supermassive black holes which form the centres of galaxies, and emit large amounts of energy in radiation and particle flows. One feature of many AGN are jetted outflows, which can be observed through radio emission, and sometimes also in other parts of the electromagnetic spectrum, including $x$-rays. These jets are large flows of plasma at relativistic speeds, over lengths of order $10^{20} \\mathrm{~m}$, which is tens of thousands of light years. The $\\mathrm{x}$-ray emission from jets is usually dominated by synchrotron emission from relativistic electrons gyrating in the magnetic field of the jet.\n\n[figure1]\n\nFigure 1: X-ray image of the jet from the Centaurus A AGN. Darker regions represent regions of higher intensity $x$-rays. Brighter regions within the fainter jet are called knots. (Snios et al., 2019)\n\nA simple model of the flow of jets assumes that the flow is steady and directed radially away from the central AGN, so approximately one dimensional, and that the plasma in the jet is in pressure equilibrium with its surroundings. There is assumed to be a constant rate per volume of mass injected into the jet from stars which lose their outer layers as they move through their life cycle.\n\nThe jet is described in terms of the coordinate representing distance from the AGN, $s$, and the opening radius $r$ of the conical jet. These distances are measured in parsecs, where $1 \\mathrm{pc}=3.086 \\times 10^{16} \\mathrm{~m}$. The speed of the jet flow is assumed to be directed radially away from the central AGN, and be a function of $s$ only. The plasma in the jet is comprised of electrons, protons, and some heavier ionised nuclei. The average energy carried by each particle in the jet, in the reference frame of the bulk flow of the jet (which we will call the jet frame), is $\\epsilon_{\\mathrm{av}}=\\mu_{\\mathrm{pp}} c^{2}+h$, where the term $h$ includes all thermal kinetic energy and potential energies in terms of the pressure $P$ and $n$ is the number density of the plasma.\n\nAs the stars, which the jet flows past, move through their life cycles they can lose part of their atmosphere. This results in a uniform rate of injection of mass per unit volume $\\alpha$ into the jet, and the injected particles are assumed to be at rest relative to the AGN.\n\nThis model can be applied to the Centaurus A jet. Centaurus A is one of the nearest AGN, so it is possible to observe its jet at relatively high spatial resolution. The total power carried by the jet is estimated to be $P_{\\mathrm{j}}=1 \\times 10^{36} \\mathrm{~J} \\cdot \\mathrm{s}^{-1}$. See below for a diagram of a simple geometrical description of the Centaurus A jet, including measurements of some jet parameters. $s_{1}$ is the coordinate of the start of the jet, and $s_{2}$ the coordinate of the end of the jet. In Centuarus A the average mass per particle is $\\mu_{\\mathrm{pp}}=0.59 m_{\\mathrm{p}}$ and $h=\\frac{13}{4} P / n$. The pressure in the plasma surrounding the jet is $P(s)=5.7 \\times 10^{-12}\\left(\\frac{s}{s_{0}}\\right)^{-1.5} \\mathrm{~Pa}$, where $s_{0}=1 \\mathrm{kpc}$.\n\n[figure2]\n\nFigure 2: The Centaurus A jet, showing the geometry compared to the active galactic nucleus (AGN).\n\nThe jet is described by the following parameters, all of which depend on the distance $s$ from the AGN:\n\n- the opening radius of the jet $r(s)$ in the AGN frame\n- the cross sectional area of the jet $A(s)$ in the AGN frame\n- the speed of the jet $v(s)$ in the AGN frame\n- the lorentz gamma factor of the jet $\\gamma(s)$ in the AGN frame\n- the number density $n(s)$ in the frame of the jet\n\nThe power carried by a jet is defined to be the sum of the total bulk kinetic energy flux and the total thermal energy flux, so\n\n$$\nP_{\\mathrm{j}}(s)=F_{\\mathrm{E}}(s)-\\dot{M} c^{2}\n$$\n\nwhere $F_{\\mathrm{E}}(s)$ is the flux of energy through the cross section of the jet at $s$, and $\\dot{M}$ is the mass flux through the jet cross section at the same distance $s$ from the AGN.\n\nFind an expression for the total momentum flux, $\\Pi$, into the Centaurus $A$ jet. $0.5 \\mathrm{pt}$ Also numerically evaluate this expression.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $ \\mathrm{~kg} \\mathrm{~m} \\mathrm{~s}^{-2}$., but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2416d49d47cb88c0a72bg-1.jpg?height=651&width=805&top_left_y=1045&top_left_x=631", "https://cdn.mathpix.com/cropped/2024_03_14_2416d49d47cb88c0a72bg-2.jpg?height=654&width=1356&top_left_y=821&top_left_x=356" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$ \\mathrm{~kg} \\mathrm{~m} \\mathrm{~s}^{-2}$." ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_505", "problem": "A graduated cylinder is partially filled with water; a rubber duck floats at the surface. Oil is poured into the graduated cylinder at a slow, constant rate, and the volume marks corresponding to the surface of the water and the surface of the oil are recorded as a function of time.\n\n[figure1]\n\nWater has a density of $1.00 \\mathrm{~g} / \\mathrm{mL}$; the density of air is negligible, as are surface effects. Find the density of the oil.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA graduated cylinder is partially filled with water; a rubber duck floats at the surface. Oil is poured into the graduated cylinder at a slow, constant rate, and the volume marks corresponding to the surface of the water and the surface of the oil are recorded as a function of time.\n\n[figure1]\n\nWater has a density of $1.00 \\mathrm{~g} / \\mathrm{mL}$; the density of air is negligible, as are surface effects. Find the density of the oil.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~g} / \\mathrm{mL}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_44c03ebf1ca2c0f19b5cg-07.jpg?height=1122&width=1176&top_left_y=1214&top_left_x=469", "https://cdn.mathpix.com/cropped/2024_03_14_44c03ebf1ca2c0f19b5cg-08.jpg?height=1117&width=1171&top_left_y=1317&top_left_x=474" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~g} / \\mathrm{mL}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_637", "problem": "The Doppler effect for a source moving relative to a stationary observer is described by\n\n$$\nf=\\frac{f_{0}}{1-(v / c) \\cos \\theta}\n$$\n\nwhere $f$ is the frequency measured by the observer, $f_{0}$ is the frequency emitted by the source, $v$ is the speed of the source, $c$ is the wave speed, and $\\theta$ is the angle between the source velocity and the line between the source and observer. (Thus $\\theta=0$ when the source is moving directly towards the observer and $\\theta=\\pi$ when moving directly away.)\n\nA sound source of constant frequency travels at a constant velocity past an observer, and the observed frequency is plotted as a function of time:\n\n[figure1]\n\nThe experiment happens in room temperature air, so the speed of sound is $340 \\mathrm{~m} / \\mathrm{s}$.\n\nWhat is the smallest distance between the source and the observer?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe Doppler effect for a source moving relative to a stationary observer is described by\n\n$$\nf=\\frac{f_{0}}{1-(v / c) \\cos \\theta}\n$$\n\nwhere $f$ is the frequency measured by the observer, $f_{0}$ is the frequency emitted by the source, $v$ is the speed of the source, $c$ is the wave speed, and $\\theta$ is the angle between the source velocity and the line between the source and observer. (Thus $\\theta=0$ when the source is moving directly towards the observer and $\\theta=\\pi$ when moving directly away.)\n\nA sound source of constant frequency travels at a constant velocity past an observer, and the observed frequency is plotted as a function of time:\n\n[figure1]\n\nThe experiment happens in room temperature air, so the speed of sound is $340 \\mathrm{~m} / \\mathrm{s}$.\n\nWhat is the smallest distance between the source and the observer?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~m}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_33ea9fa74c8b34628eedg-03.jpg?height=1261&width=1569&top_left_y=936&top_left_x=278" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_134", "problem": "A stream of sand is dropped out of a helicopter initially moving at a constant speed $v$ to the right. The helicopter suddenly turns and begins moving a constant speed $v$ to the left. Neglecting air resistance on the sand, what is the shape of the stream of sand, as viewed from the ground? The black dot represents the helicopter.\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA stream of sand is dropped out of a helicopter initially moving at a constant speed $v$ to the right. The helicopter suddenly turns and begins moving a constant speed $v$ to the left. Neglecting air resistance on the sand, what is the shape of the stream of sand, as viewed from the ground? The black dot represents the helicopter.\n\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-14.jpg?height=395&width=556&top_left_y=453&top_left_x=210", "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-14.jpg?height=428&width=553&top_left_y=968&top_left_x=211", "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-14.jpg?height=396&width=553&top_left_y=1553&top_left_x=211", "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-14.jpg?height=431&width=567&top_left_y=413&top_left_x=1061", "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-14.jpg?height=426&width=561&top_left_y=969&top_left_x=1064" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1534", "problem": "激光瞄准系统的设计需考虑空气折射率的变化。由于受到地表状况、海拔高度、气温、湿度和空气密度等多种因素的影响, 空气的折射率在大气层中的分布是不均匀的,因而激光的传播路径并不是直线。为简化起见, 假设某地的空气折射率随高度 $y$ 的变化如下式所示\n\n$$\nn^{2}=n_{0}^{2}+\\alpha^{2} y,\n$$\n\n式中 $n_{0}$ 是 $y=0$ 处 (地面) 空气的折射率, $n_{0}$\n\n[图1]\n和 $\\alpha$ 均为大于零的已知常量。激光本身的传播时间可忽略。激光发射器位于坐标原点 O，如图。激光发射器的攻击通常遵从安全击毁的原则, 即既要击毁目标飞行器 A, 又必须使目标飞行器 $\\mathrm{A}$ 水平投出的所有炸弹, 都不能炸到激光发射器（炸弹在投出时相对于 $\\mathrm{A}$ 静止）。假定 A 一旦进入激光发射器可攻击范围, 激光发射器便立即用激光照射它。已知水平飞行\n目标 $\\mathrm{A}$ 的高度为 $y_{a}$, 击毁 $\\mathrm{A}$ 需要激光持续照射的时间为 $t_{a}$, 且位于坐标原点 $\\mathrm{O}$ 的激光器能安全击毁它; 试求 $\\mathrm{A}$ 的速度范围。不考虑空气阻力, 重力加速度大小为 $g$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个区间。\n这里是一些可能会帮助你解决问题的先验信息提示：\n激光瞄准系统的设计需考虑空气折射率的变化。由于受到地表状况、海拔高度、气温、湿度和空气密度等多种因素的影响, 空气的折射率在大气层中的分布是不均匀的,因而激光的传播路径并不是直线。为简化起见, 假设某地的空气折射率随高度 $y$ 的变化如下式所示\n\n$$\nn^{2}=n_{0}^{2}+\\alpha^{2} y,\n$$\n\n式中 $n_{0}$ 是 $y=0$ 处 (地面) 空气的折射率, $n_{0}$\n\n[图1]\n和 $\\alpha$ 均为大于零的已知常量。激光本身的传播时间可忽略。激光发射器位于坐标原点 O，如图。\n\n问题:\n激光发射器的攻击通常遵从安全击毁的原则, 即既要击毁目标飞行器 A, 又必须使目标飞行器 $\\mathrm{A}$ 水平投出的所有炸弹, 都不能炸到激光发射器（炸弹在投出时相对于 $\\mathrm{A}$ 静止）。假定 A 一旦进入激光发射器可攻击范围, 激光发射器便立即用激光照射它。已知水平飞行\n目标 $\\mathrm{A}$ 的高度为 $y_{a}$, 击毁 $\\mathrm{A}$ 需要激光持续照射的时间为 $t_{a}$, 且位于坐标原点 $\\mathrm{O}$ 的激光器能安全击毁它; 试求 $\\mathrm{A}$ 的速度范围。不考虑空气阻力, 重力加速度大小为 $g$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个区间，例如ANSWER=(1,2] \\cup[7,+\\infty)", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_07aa406e17d01fd01b36g-05.jpg?height=634&width=737&top_left_y=1105&top_left_x=1019" ], "answer": null, "solution": null, "answer_type": "IN", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_813", "problem": "The cooling power and the maximum temperature difference\n\nThe upper end of the thermocouple is a heat source with the initial temperature $T_{1}$. It is thermally isolated with ambient environment, and needs to be cooled. The lower ends of the thermocouple, A and $\\mathrm{B}$ bars are connected to a battery and are at the temperature $T_{2}$ of the heat sink. The sense of the electrical current is chosen so that the Peltier heat is absorbed at the upper junction and released to the heat sink at the lower junction.\n\n[figure1]\n\nFigure 5. Thermoelectric refrigerator. (1) Isolated heat source (temperature $T_{1}$ ); (2) Heat sink (temperature $T_{2}$ )\n\n\nThe thermocouple made from materials $\\mathrm{A}$ and $\\mathrm{B}$ with best value of figure of merit $Z_{m}$ found in part $\\mathrm{A}$ is used for the refrigerator.\n\n Calculate the numerical value of the minimum temperature of the isolated heat source $T_{1 \\text { min }}$ if the temperature of the heat sink is $T_{2}=300 \\mathrm{~K}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe cooling power and the maximum temperature difference\n\nThe upper end of the thermocouple is a heat source with the initial temperature $T_{1}$. It is thermally isolated with ambient environment, and needs to be cooled. The lower ends of the thermocouple, A and $\\mathrm{B}$ bars are connected to a battery and are at the temperature $T_{2}$ of the heat sink. The sense of the electrical current is chosen so that the Peltier heat is absorbed at the upper junction and released to the heat sink at the lower junction.\n\n[figure1]\n\nFigure 5. Thermoelectric refrigerator. (1) Isolated heat source (temperature $T_{1}$ ); (2) Heat sink (temperature $T_{2}$ )\n\n\nThe thermocouple made from materials $\\mathrm{A}$ and $\\mathrm{B}$ with best value of figure of merit $Z_{m}$ found in part $\\mathrm{A}$ is used for the refrigerator.\n\n Calculate the numerical value of the minimum temperature of the isolated heat source $T_{1 \\text { min }}$ if the temperature of the heat sink is $T_{2}=300 \\mathrm{~K}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $ \\mathrm{~K}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_6e4e3efc8af7423242a0g-6.jpg?height=797&width=737&top_left_y=1509&top_left_x=662" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$ \\mathrm{~K}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_389", "problem": "solar powered satellite with starting velocity $v_{0}$ is launched from Earth onto an elliptical heliocentric orbit with the intent of gathering as much solar energy as possible. The angle of departure can be freely changed.\n\n What is the maximal averag solar irradiance the satellite can gather and what is the needed launch angle relative to Earth's motion?\n\nThe Sun's mass is $M_{\\odot}$, Earth's orbital radius $\\boldsymbol{R}_{\\oplus}$, free fall acceleration on Earth $g$, Earth's radius $r_{\\oplus}$ and Sun's luminosity $L_{\\odot}$", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nsolar powered satellite with starting velocity $v_{0}$ is launched from Earth onto an elliptical heliocentric orbit with the intent of gathering as much solar energy as possible. The angle of departure can be freely changed.\n\n What is the maximal averag solar irradiance the satellite can gather and what is the needed launch angle relative to Earth's motion?\n\nThe Sun's mass is $M_{\\odot}$, Earth's orbital radius $\\boldsymbol{R}_{\\oplus}$, free fall acceleration on Earth $g$, Earth's radius $r_{\\oplus}$ and Sun's luminosity $L_{\\odot}$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_31", "problem": "Atoms in a solid are not stationary. They vibrate about their equilibrium positions. Typically, the frequency of vibration is about $2.0 \\times 10^{12} \\mathrm{~Hz}$, and the amplitude is about $1.1 \\mathrm{X} 10^{-11} \\mathrm{~m}$. For a typical atom, what is its maximum speed?\nA: $108 \\mathrm{~m} / \\mathrm{s}$\nB: $10.8 \\mathrm{~m} / \\mathrm{s}$\nC: $128 \\mathrm{~m} / \\mathrm{s}$\nD: $12.8 \\mathrm{~m} / \\mathrm{s}$\nE: $138 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAtoms in a solid are not stationary. They vibrate about their equilibrium positions. Typically, the frequency of vibration is about $2.0 \\times 10^{12} \\mathrm{~Hz}$, and the amplitude is about $1.1 \\mathrm{X} 10^{-11} \\mathrm{~m}$. For a typical atom, what is its maximum speed?\n\nA: $108 \\mathrm{~m} / \\mathrm{s}$\nB: $10.8 \\mathrm{~m} / \\mathrm{s}$\nC: $128 \\mathrm{~m} / \\mathrm{s}$\nD: $12.8 \\mathrm{~m} / \\mathrm{s}$\nE: $138 \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_144", "problem": "A vertical pole has two massless strings, both of length $L$, attached a distance $L$ apart. The other ends of the strings are attached to a mass $M$. The mass is rotated around the pole with an angular speed $\\omega$. Which of the following graphs best gives the ratio of the tension in the bottom string to the tension in the top string as a function of $\\omega$ ?\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA vertical pole has two massless strings, both of length $L$, attached a distance $L$ apart. The other ends of the strings are attached to a mass $M$. The mass is rotated around the pole with an angular speed $\\omega$. Which of the following graphs best gives the ratio of the tension in the bottom string to the tension in the top string as a function of $\\omega$ ?\n\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-19.jpg?height=447&width=548&top_left_y=476&top_left_x=317", "https://cdn.mathpix.com/cropped/2024_03_06_fed58f7ed40ad305c449g-10.jpg?height=452&width=550&top_left_y=1140&top_left_x=316", "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-19.jpg?height=447&width=550&top_left_y=1815&top_left_x=316", "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-19.jpg?height=455&width=547&top_left_y=469&top_left_x=1128", "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-19.jpg?height=453&width=550&top_left_y=1145&top_left_x=1129" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1627", "problem": "光子被电子散射时, 如果初态电子具有足够的动能, 以至于在散射过程中有能量从电子转移到光子, 则该散射被称为逆康普顿散射. 当低能光子与高能电子发生对头碰撞时, 就会出现逆康普顿散射. 已知电子静止质量为 $m_{e}$, 真空中的光速为 $c$. 若能量为 $E_{e}$ 的电子与能量为 $E_{\\gamma}$ 的光子相向对碰求逆康普顿散射能够发生的条件", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n光子被电子散射时, 如果初态电子具有足够的动能, 以至于在散射过程中有能量从电子转移到光子, 则该散射被称为逆康普顿散射. 当低能光子与高能电子发生对头碰撞时, 就会出现逆康普顿散射. 已知电子静止质量为 $m_{e}$, 真空中的光速为 $c$. 若能量为 $E_{e}$ 的电子与能量为 $E_{\\gamma}$ 的光子相向对碰\n\n问题:\n求逆康普顿散射能够发生的条件\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1325", "problem": "一质量为 $m$ 的小球在距水平地面 $h$ 高处以水平速度 $\\sqrt{2 g h}$ 抛出, 空气阻力不计. 小球每次落地反弹时水平速度不变, 坚直速度大小按同样的比率减小. 若自第一次反弹开始小球的运动轨迹与其在地面的投影之间所包围的面积总和为 $\\frac{8}{21} h^{2}$, 求小球在各次与地面碰撞过程中所受到的总冲量.\n\n提示: 小球每次做斜抛运动（从水平地面射出又落至地面）的轨迹与其在地面的投影之间所包围的面积等于其最大高度和水平射程乘积的 $\\frac{2}{3}$.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n一质量为 $m$ 的小球在距水平地面 $h$ 高处以水平速度 $\\sqrt{2 g h}$ 抛出, 空气阻力不计. 小球每次落地反弹时水平速度不变, 坚直速度大小按同样的比率减小. 若自第一次反弹开始小球的运动轨迹与其在地面的投影之间所包围的面积总和为 $\\frac{8}{21} h^{2}$, 求小球在各次与地面碰撞过程中所受到的总冲量.\n\n提示: 小球每次做斜抛运动（从水平地面射出又落至地面）的轨迹与其在地面的投影之间所包围的面积等于其最大高度和水平射程乘积的 $\\frac{2}{3}$.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_70", "problem": "A pendulum has a length of $2.00 \\mathrm{~m}$ and has a mass attached to its end. The mass is pulled back $60.0^{\\circ}$ from the vertical before being released and allowed to swing downward. If the string breaks under a tension of more than $20 \\mathrm{~N}$, what is the maximum amount of mass that can be attached to string to allow the mass to swing without breaking the string?\nA: $0.8 \\mathrm{~kg}$\nB: $1.0 \\mathrm{~kg}$\nC: $1.2 \\mathrm{~kg}$\nD: $1.4 \\mathrm{~kg}$\nE: $1.6 \\mathrm{~kg}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA pendulum has a length of $2.00 \\mathrm{~m}$ and has a mass attached to its end. The mass is pulled back $60.0^{\\circ}$ from the vertical before being released and allowed to swing downward. If the string breaks under a tension of more than $20 \\mathrm{~N}$, what is the maximum amount of mass that can be attached to string to allow the mass to swing without breaking the string?\n\nA: $0.8 \\mathrm{~kg}$\nB: $1.0 \\mathrm{~kg}$\nC: $1.2 \\mathrm{~kg}$\nD: $1.4 \\mathrm{~kg}$\nE: $1.6 \\mathrm{~kg}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_845", "problem": "[figure1]\n\nFig. 1 A monopole $q_{\\mathrm{m}}$ appears at a distance $h$ from a conducting thin film of thickness $d$. The origin of the coordinates is located on the upper surface.\n\nWe first focus on the initial response of the conducting thin film when at time $t=0$ a north monopole $q_{\\mathrm{m}}$ appears suddenly at the position $\\vec{r}_{\\mathrm{mp}}=h \\hat{z}(h>0)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nObtain the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in $z \\geq 0$ at $t_{0}=0$. $0.4 \\mathrm{pt}$", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\n[figure1]\n\nFig. 1 A monopole $q_{\\mathrm{m}}$ appears at a distance $h$ from a conducting thin film of thickness $d$. The origin of the coordinates is located on the upper surface.\n\nWe first focus on the initial response of the conducting thin film when at time $t=0$ a north monopole $q_{\\mathrm{m}}$ appears suddenly at the position $\\vec{r}_{\\mathrm{mp}}=h \\hat{z}(h>0)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nObtain the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in $z \\geq 0$ at $t_{0}=0$. $0.4 \\mathrm{pt}$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d32b3b2f89cebe6f1c2ag-2.jpg?height=642&width=1244&top_left_y=296&top_left_x=194" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_257", "problem": "Thermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nThermal atmospheric escape is one of the processes that explain why some gases are present in the atmosphere and some others are not. Currently the atmospheric pressure is approximately $P_{0}=10^{5} \\mathrm{~Pa}$ and a fraction of $\\chi_{H}=5.5 \\times 10^{-5 \\%}$ of the atmosphere molecules are hydrogen molecules. When these molecules reach a certain height (lower than $h_{E B}$ ) they split into two atoms due to solar radiation. Concentration of hydrogen atoms in the exobase can be considered constant over time.\n\nFind also this time for Helium atoms on Earth, knowing that their concentration in the exobase is $n_{H e}=2.5 \\times 10^{12} m^{-3}$. Helium atoms are the $5 \\times 10^{-4} \\%$ of the atmosphere. (1.5 points) Jeans escape is not the only atmospheric escape process and there are also other reactions on the surface on Earth that may produce atmospheric gases. Yet this calculations should show the difference in orders of magnitude of the escape of different gases", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nThermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nThermal atmospheric escape is one of the processes that explain why some gases are present in the atmosphere and some others are not. Currently the atmospheric pressure is approximately $P_{0}=10^{5} \\mathrm{~Pa}$ and a fraction of $\\chi_{H}=5.5 \\times 10^{-5 \\%}$ of the atmosphere molecules are hydrogen molecules. When these molecules reach a certain height (lower than $h_{E B}$ ) they split into two atoms due to solar radiation. Concentration of hydrogen atoms in the exobase can be considered constant over time.\n\nFind also this time for Helium atoms on Earth, knowing that their concentration in the exobase is $n_{H e}=2.5 \\times 10^{12} m^{-3}$. Helium atoms are the $5 \\times 10^{-4} \\%$ of the atmosphere. (1.5 points) Jeans escape is not the only atmospheric escape process and there are also other reactions on the surface on Earth that may produce atmospheric gases. Yet this calculations should show the difference in orders of magnitude of the escape of different gases\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [$\\tau_{H}$, $\\tau_{He}$].\nTheir units are, in order, [years, years], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_61a2ff399c33d9b3cd3bg-1.jpg?height=968&width=1044&top_left_y=1240&top_left_x=302" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "years", "years" ], "answer_sequence": [ "$\\tau_{H}$", "$\\tau_{He}$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_749", "problem": "Solenoid \n\nA solenoid of length $l=20 \\mathrm{~cm}$ is wound around a vertical, cylindrical test tube made of glass, filled with water. The solenoid is thermally insulated from the water. The height of the water level is approximately $20 \\mathrm{~cm}$ above the upper end of the solenoid, the diameter of the test tube is $1 \\mathrm{~cm}$, the number of turns of the coil is $N=6000$.\n\n[figure1]\n\nThe atmospheric pressure is $p_{0}=101 \\mathrm{kPa}$, the temperature of water is $293 \\mathrm{~K}$. Magnetic susceptibility of water is $\\chi \\equiv \\mu_{r}-1=-9.04 \\cdot 10^{-6}$. Vacuum permeability $\\mu_{0}=12.57 \\cdot 10^{-7} \\mathrm{H} / \\mathrm{m}$. The current in the solenoid is slowly increased until the water starts boiling inside the coil. At which current strength would this happen? Make reasonable approximations when needed. Note that the required current strength might be slightly too large for the present technology.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nSolenoid \n\nA solenoid of length $l=20 \\mathrm{~cm}$ is wound around a vertical, cylindrical test tube made of glass, filled with water. The solenoid is thermally insulated from the water. The height of the water level is approximately $20 \\mathrm{~cm}$ above the upper end of the solenoid, the diameter of the test tube is $1 \\mathrm{~cm}$, the number of turns of the coil is $N=6000$.\n\n[figure1]\n\nThe atmospheric pressure is $p_{0}=101 \\mathrm{kPa}$, the temperature of water is $293 \\mathrm{~K}$. Magnetic susceptibility of water is $\\chi \\equiv \\mu_{r}-1=-9.04 \\cdot 10^{-6}$. Vacuum permeability $\\mu_{0}=12.57 \\cdot 10^{-7} \\mathrm{H} / \\mathrm{m}$. The current in the solenoid is slowly increased until the water starts boiling inside the coil. At which current strength would this happen? Make reasonable approximations when needed. Note that the required current strength might be slightly too large for the present technology.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of kA, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_3358f62225913bd79896g-1.jpg?height=978&width=401&top_left_y=606&top_left_x=843" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "kA" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1434", "problem": "如图, 一边长为 $L$ 的正方形铜线框 abcd 可绕水平轴 ab 自由转动, 一坚直向上的外力 $F$ 作用在 $\\mathrm{cd}$ 边的中点, 整个线框置于方向坚直向上的均匀磁场中, 磁感应强度大小随时间变化。已知该方形线框铜线的电导率为 $\\sigma$, 铜线的半径为 $r_{0}$,质量密度 $\\rho$, 重力加速度大小为 $g$ 。\n[图1]当框平面与水平面 abef 的夹角为 $\\theta$ 时, 求该方形线框所受到的重力矩。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一边长为 $L$ 的正方形铜线框 abcd 可绕水平轴 ab 自由转动, 一坚直向上的外力 $F$ 作用在 $\\mathrm{cd}$ 边的中点, 整个线框置于方向坚直向上的均匀磁场中, 磁感应强度大小随时间变化。已知该方形线框铜线的电导率为 $\\sigma$, 铜线的半径为 $r_{0}$,质量密度 $\\rho$, 重力加速度大小为 $g$ 。\n[图1]\n\n问题:\n当框平面与水平面 abef 的夹角为 $\\theta$ 时, 求该方形线框所受到的重力矩。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_d716ce15f03757bb482eg-03.jpg?height=300&width=309&top_left_y=1523&top_left_x=1610" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_126", "problem": "A light board of length $\\ell$ is hinged to a vertical wall. A solid ball of weight $G$ and radius $R$ is held between the wall and the board by a force $F$ applied perpendicular at the end of the board, as shown in the figure below. Both the wall and the board are frictionless. The angle between the board and the wall is $60^{\\circ}$. What is the magnitude of $F$ ?\n\n[figure1]\nA: $\\frac{2}{\\sqrt{3}} G$\nB: $\\frac{2 R}{\\ell} G $ \nC: $\\frac{2 R}{\\sqrt{3} \\ell} G$\nD: $\\frac{4 R}{\\sqrt{3} \\ell} G$\nE: $\\frac{\\sqrt{3} R}{\\ell} G$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA light board of length $\\ell$ is hinged to a vertical wall. A solid ball of weight $G$ and radius $R$ is held between the wall and the board by a force $F$ applied perpendicular at the end of the board, as shown in the figure below. Both the wall and the board are frictionless. The angle between the board and the wall is $60^{\\circ}$. What is the magnitude of $F$ ?\n\n[figure1]\n\nA: $\\frac{2}{\\sqrt{3}} G$\nB: $\\frac{2 R}{\\ell} G $ \nC: $\\frac{2 R}{\\sqrt{3} \\ell} G$\nD: $\\frac{4 R}{\\sqrt{3} \\ell} G$\nE: $\\frac{\\sqrt{3} R}{\\ell} G$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-06.jpg?height=282&width=358&top_left_y=1317&top_left_x=881" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1673", "problem": "Hyperloop 是一款利用胶囊状的运输车在水平管道中的快速运动来实现超高速运输的系统（见图 a）。它采用了 “气垫” 技术和 “直线电动机” 原理。\n\n“气垫” 技术是将内部高压气体从水平放置的运输车下半部的\n\n[图1]\n\n图a 细孔快速喷出（见图 b），以至于整个运输车被托离管壁非常小的距离,\n\n[图2]\n从而可忽略摩擦。运输车横截面是半径为 $R$ 的圆, 运输车下半部壁上均匀分布有沿径向的大量细孔, 单位面积内细孔个数为 $n$ ( $n>>1)$, 单个细孔面积为 $s$ 。运输车长度为 $l$, 质量为 $M$ 。气体的流动可认为遵从伯努利方程, 且温度不变, 细孔出口处气体的压强为较低的环境压强 $P_{\\text {low }}$ 。\n\n如图 c，在水平管道中固定有两条平行的水平光滑供电导轨（粗实线）, 运输车上固定有与导轨垂直的两根导线 (细实线) ; 导轨横截面为圆形, 半径为 $r_{\\mathrm{d}}$, 电阻率为 $\\rho_{\\mathrm{d}}$, 两导轨轴线间距为 $2\\left(D+r_{\\mathrm{d}}\\right)$; 两根导线的粗细可忽略, 间距为 $D$; 每根导线电阻是长度为 $D$ 的导轨电阻的 2 倍。两导线和导轨轴线均处于同一水平面内。导轨、导线电接触良好, 且所有接触电阻均可忽略。\n\n[图3]\n\n图c\n\n[图4]\n\n图d\n\n[图5]\n\n图e求内部高压气体压强 $P$ 为多大时运输车才刚好能被托起?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\nHyperloop 是一款利用胶囊状的运输车在水平管道中的快速运动来实现超高速运输的系统（见图 a）。它采用了 “气垫” 技术和 “直线电动机” 原理。\n\n“气垫” 技术是将内部高压气体从水平放置的运输车下半部的\n\n[图1]\n\n图a 细孔快速喷出（见图 b），以至于整个运输车被托离管壁非常小的距离,\n\n[图2]\n从而可忽略摩擦。运输车横截面是半径为 $R$ 的圆, 运输车下半部壁上均匀分布有沿径向的大量细孔, 单位面积内细孔个数为 $n$ ( $n>>1)$, 单个细孔面积为 $s$ 。运输车长度为 $l$, 质量为 $M$ 。气体的流动可认为遵从伯努利方程, 且温度不变, 细孔出口处气体的压强为较低的环境压强 $P_{\\text {low }}$ 。\n\n如图 c，在水平管道中固定有两条平行的水平光滑供电导轨（粗实线）, 运输车上固定有与导轨垂直的两根导线 (细实线) ; 导轨横截面为圆形, 半径为 $r_{\\mathrm{d}}$, 电阻率为 $\\rho_{\\mathrm{d}}$, 两导轨轴线间距为 $2\\left(D+r_{\\mathrm{d}}\\right)$; 两根导线的粗细可忽略, 间距为 $D$; 每根导线电阻是长度为 $D$ 的导轨电阻的 2 倍。两导线和导轨轴线均处于同一水平面内。导轨、导线电接触良好, 且所有接触电阻均可忽略。\n\n[图3]\n\n图c\n\n[图4]\n\n图d\n\n[图5]\n\n图e\n\n问题:\n求内部高压气体压强 $P$ 为多大时运输车才刚好能被托起?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-03.jpg?height=226&width=351&top_left_y=2194&top_left_x=1412", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-03.jpg?height=374&width=905&top_left_y=2486&top_left_x=884", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-04.jpg?height=268&width=420&top_left_y=914&top_left_x=430", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-04.jpg?height=263&width=440&top_left_y=919&top_left_x=842", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-04.jpg?height=263&width=380&top_left_y=919&top_left_x=1272", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-17.jpg?height=389&width=354&top_left_y=1970&top_left_x=1391" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1494", "problem": "如图 10 所示, 在 $\\mathrm{O}$ 点有一个物屏, 上面有一个发光小物体, 垂直于物屏有光轴, 共轴第 5 页 / 共 20 页\n放置一个焦距为 $f$ 的透镜, 光心为 $\\mathrm{C}$, 在距离 $\\mathrm{O}$ 点 $L$ 处共轴放置一个平面镜, 当 $\\mathrm{OC}$ 距离为 $x_{1}$ 时, 发现经过透镜透射, 平面镜反射, 再经过透镜透射, 发光物体在物屏处成了清晰像, 向右移动 $\\mathrm{C}$ 点到 $x_{2}$ 处, 再次在物屏处成了清晰像, 继续向右移动 $\\mathrm{C}$ 点到 $x_{3}$ 处, 又在物屏处成了清晰像。\n\n[图1]\n\n图 10求出 $x_{1}, x_{2}, x_{3}$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图 10 所示, 在 $\\mathrm{O}$ 点有一个物屏, 上面有一个发光小物体, 垂直于物屏有光轴, 共轴第 5 页 / 共 20 页\n放置一个焦距为 $f$ 的透镜, 光心为 $\\mathrm{C}$, 在距离 $\\mathrm{O}$ 点 $L$ 处共轴放置一个平面镜, 当 $\\mathrm{OC}$ 距离为 $x_{1}$ 时, 发现经过透镜透射, 平面镜反射, 再经过透镜透射, 发光物体在物屏处成了清晰像, 向右移动 $\\mathrm{C}$ 点到 $x_{2}$ 处, 再次在物屏处成了清晰像, 继续向右移动 $\\mathrm{C}$ 点到 $x_{3}$ 处, 又在物屏处成了清晰像。\n\n[图1]\n\n图 10\n\n问题:\n求出 $x_{1}, x_{2}, x_{3}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$x_{1}$, $x_{2}$, $x_{3}$]\n它们的答案类型依次是[表达式, 表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-06.jpg?height=357&width=594&top_left_y=444&top_left_x=777", "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-17.jpg?height=320&width=711&top_left_y=2010&top_left_x=707" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "$x_{1}$", "$x_{2}$", "$x_{3}$" ], "type_sequence": [ "EX", "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_524", "problem": "Scientists have recently detected a new star, the MAR-Kappa. The star is almost a perfect blackbody, and its measured light spectrum is shown below.\n\n[figure1]\n\nThe total measured light intensity from MAR-Kappa is $I=1.12 \\times 10^{-8} \\mathrm{~W} / \\mathrm{m}^{2}$. The mass of MAR-Kappa is estimated to be $3.5 \\times 10^{30} \\mathrm{~kg}$. It is stationary relative to the sun. You may find the Stefan-Boltzmann law useful, which states the power emitted by a blackbody with area $A$ is $\\sigma A T^{4}$.\n\nThe spectrum of wavelengths $\\lambda$ emitted from a blackbody only depends on $h, c, k_{B}, \\lambda$, and $T$. Given that the sun has a surface temperature of $5778 \\mathrm{~K}$ and peak emission at $500 \\mathrm{~nm}$, what is the approximate surface temperature of MAR-Kappa?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nScientists have recently detected a new star, the MAR-Kappa. The star is almost a perfect blackbody, and its measured light spectrum is shown below.\n\n[figure1]\n\nThe total measured light intensity from MAR-Kappa is $I=1.12 \\times 10^{-8} \\mathrm{~W} / \\mathrm{m}^{2}$. The mass of MAR-Kappa is estimated to be $3.5 \\times 10^{30} \\mathrm{~kg}$. It is stationary relative to the sun. You may find the Stefan-Boltzmann law useful, which states the power emitted by a blackbody with area $A$ is $\\sigma A T^{4}$.\n\nThe spectrum of wavelengths $\\lambda$ emitted from a blackbody only depends on $h, c, k_{B}, \\lambda$, and $T$. Given that the sun has a surface temperature of $5778 \\mathrm{~K}$ and peak emission at $500 \\mathrm{~nm}$, what is the approximate surface temperature of MAR-Kappa?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $ \\mathrm{~K}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_fed953f9e38b72bf8bd7g-17.jpg?height=640&width=1051&top_left_y=507&top_left_x=537" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$ \\mathrm{~K}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1584", "problem": "如图, 一质量为 $M$ 、足够长的长方形薄平板静止于光滑的水平地面上; 平板左端与劲度系数为 $k$ 的水平轻弹簧相连, 弹簧的另一端固定在墙面上; 平板上面有一质量为 $m$ 的小物块。在 $t=0$ 时刻, 平板静止, 而小物块以某一初速度向右开始运动。已知小物块与平板之间的动摩擦因数为 $\\mu$, 重力加速度大小为 $g$ 。\n\n若平板从 $t=0$ 时刻开始运动至四分之一周期时, 恰好小物块第一次相对于平板静止。求\n\n[图1]小物块初速度的值;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一质量为 $M$ 、足够长的长方形薄平板静止于光滑的水平地面上; 平板左端与劲度系数为 $k$ 的水平轻弹簧相连, 弹簧的另一端固定在墙面上; 平板上面有一质量为 $m$ 的小物块。在 $t=0$ 时刻, 平板静止, 而小物块以某一初速度向右开始运动。已知小物块与平板之间的动摩擦因数为 $\\mu$, 重力加速度大小为 $g$ 。\n\n若平板从 $t=0$ 时刻开始运动至四分之一周期时, 恰好小物块第一次相对于平板静止。求\n\n[图1]\n\n问题:\n小物块初速度的值;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-04.jpg?height=140&width=648&top_left_y=815&top_left_x=1269" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1609", "problem": "如图, 一质量为 $M$ 、足够长的长方形薄平板静止于光滑的水平地面上; 平板左端与劲度系数为 $k$ 的水平轻弹簧相连, 弹簧的另一端固定在墙面上; 平板上面有一质量为 $m$ 的小物块。在 $t=0$ 时刻, 平板静止, 而小物块以某一初速度向右开始运动。已知小物块与平板之间的动摩擦因数为 $\\mu$, 重力加速度大小为 $g$ 。\n\n若平板从 $t=0$ 时刻开始运动至四分之一周期时, 恰好小物块第一次相对于平板静止。求\n\n[图1]平板在运动四分之一周期的过程中产生的热量。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一质量为 $M$ 、足够长的长方形薄平板静止于光滑的水平地面上; 平板左端与劲度系数为 $k$ 的水平轻弹簧相连, 弹簧的另一端固定在墙面上; 平板上面有一质量为 $m$ 的小物块。在 $t=0$ 时刻, 平板静止, 而小物块以某一初速度向右开始运动。已知小物块与平板之间的动摩擦因数为 $\\mu$, 重力加速度大小为 $g$ 。\n\n若平板从 $t=0$ 时刻开始运动至四分之一周期时, 恰好小物块第一次相对于平板静止。求\n\n[图1]\n\n问题:\n平板在运动四分之一周期的过程中产生的热量。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-04.jpg?height=140&width=648&top_left_y=815&top_left_x=1269" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_790", "problem": "Tanh is doing an experiment to measure the resistance of an unmarked mystery resistor. She knows that the potential difference across a resistor is given by $V=I R$ where $I$ is the current through the resistor and $R$ is the resistance of the resistor. She has a variable power supply, a voltmeter (to measure potential difference) and an ammeter (to measure current). The best method for her to use is:\nA: Use a fixed potential difference $(V)$, and measure the current $(I)$ through the resistor. Put this pair of values into the equation to calculate $R$.\nB: Use two different values of potential difference $(V)$, and measure the current $(I)$ through the resistor for each. Put these two pairs of values into the equation and average the results.\nC: Use at least three different values of potential difference $(V)$, and measure the current (I) through the resistor for each. Put these pairs of values into the equation and average the results.\nD: Use at least three different values of potential difference $(V)$, and measure the current (I) through the resistor for each. Plot a graph of $V$ vs $I$ and find the value of $R$ from the gradient of a line of best fit for the data.\nE: Use at least three different values of potential difference $(V)$, and measure the current (I) through the resistor for each. Plot a graph of $V$ vs $I$ and find the value of $R$ from the gradient of a line of best fit for the data that also passes through the origin.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTanh is doing an experiment to measure the resistance of an unmarked mystery resistor. She knows that the potential difference across a resistor is given by $V=I R$ where $I$ is the current through the resistor and $R$ is the resistance of the resistor. She has a variable power supply, a voltmeter (to measure potential difference) and an ammeter (to measure current). The best method for her to use is:\n\nA: Use a fixed potential difference $(V)$, and measure the current $(I)$ through the resistor. Put this pair of values into the equation to calculate $R$.\nB: Use two different values of potential difference $(V)$, and measure the current $(I)$ through the resistor for each. Put these two pairs of values into the equation and average the results.\nC: Use at least three different values of potential difference $(V)$, and measure the current (I) through the resistor for each. Put these pairs of values into the equation and average the results.\nD: Use at least three different values of potential difference $(V)$, and measure the current (I) through the resistor for each. Plot a graph of $V$ vs $I$ and find the value of $R$ from the gradient of a line of best fit for the data.\nE: Use at least three different values of potential difference $(V)$, and measure the current (I) through the resistor for each. Plot a graph of $V$ vs $I$ and find the value of $R$ from the gradient of a line of best fit for the data that also passes through the origin.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1568", "problem": "如图 13a, 一半径 $r=50 \\mathrm{~cm}$ 的球形薄壁玻璃鱼缸内充满水, 水中有一条小鱼。玻璃和水的折射率都是 $n=4 / 3$ 。观察者在不同位置和不同角度对玻璃鱼缸里的鱼进行观察。\n\n[图1]\n\n图 13a当鱼位于鱼缸的中心时, 求观察者看到的鱼的表观位置和横向放大率。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图 13a, 一半径 $r=50 \\mathrm{~cm}$ 的球形薄壁玻璃鱼缸内充满水, 水中有一条小鱼。玻璃和水的折射率都是 $n=4 / 3$ 。观察者在不同位置和不同角度对玻璃鱼缸里的鱼进行观察。\n\n[图1]\n\n图 13a\n\n问题:\n当鱼位于鱼缸的中心时, 求观察者看到的鱼的表观位置和横向放大率。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[当鱼位于鱼缸的中心时, 求观察者看到的鱼的表观位置, 当鱼位于鱼缸的中心时, 求观察者看到的鱼的横向放大率]\n它们的单位依次是[$\\mathrm{~cm}$, None]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-05.jpg?height=388&width=397&top_left_y=1048&top_left_x=1595", "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-10.jpg?height=388&width=428&top_left_y=1528&top_left_x=1308" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{~cm}$", null ], "answer_sequence": [ "当鱼位于鱼缸的中心时, 求观察者看到的鱼的表观位置", "当鱼位于鱼缸的中心时, 求观察者看到的鱼的横向放大率" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1575", "problem": "如图, 两根内径相同的绝缘细管 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 连接成倒 $\\mathrm{V}$ 字形, 坚直放置,连接点 $\\mathrm{B}$ 处可视为一段很短的圆弧; 两管长度均为 $l=2.0 \\mathrm{~m}$, 倾角 $\\alpha=37^{\\circ}$,处于方向坚直向下的匀强电场中, 场强大小 $E=10000 \\mathrm{~V} / \\mathrm{m}$ 。一质量 $m=1.0 \\times 10^{-4} \\mathrm{~kg}$ 、带电量 $-q=-2.0 \\times 10^{-7} \\mathrm{C}$ 的小球 (小球直径比细管内径稍小,可视为质点), 从 $\\mathrm{A}$ 点由静止开始在管内运动, 小球与 $\\mathrm{AB}$ 管壁间的动摩擦因数为 $\\mu_{1}=0.50$,小球与 $\\mathrm{BC}$ 管壁间的动摩擦因数为 $\\mu_{2}=0.25$ 。小球在运动过程中带电量保持不变。已知重力加速度大小 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}, \\sin 37^{\\circ}=\\frac{3}{5}$ 。求\n\n[图1]小球第一次速度为零的位置与B点之间的距离 $S_{1}$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 两根内径相同的绝缘细管 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 连接成倒 $\\mathrm{V}$ 字形, 坚直放置,连接点 $\\mathrm{B}$ 处可视为一段很短的圆弧; 两管长度均为 $l=2.0 \\mathrm{~m}$, 倾角 $\\alpha=37^{\\circ}$,处于方向坚直向下的匀强电场中, 场强大小 $E=10000 \\mathrm{~V} / \\mathrm{m}$ 。一质量 $m=1.0 \\times 10^{-4} \\mathrm{~kg}$ 、带电量 $-q=-2.0 \\times 10^{-7} \\mathrm{C}$ 的小球 (小球直径比细管内径稍小,可视为质点), 从 $\\mathrm{A}$ 点由静止开始在管内运动, 小球与 $\\mathrm{AB}$ 管壁间的动摩擦因数为 $\\mu_{1}=0.50$,小球与 $\\mathrm{BC}$ 管壁间的动摩擦因数为 $\\mu_{2}=0.25$ 。小球在运动过程中带电量保持不变。已知重力加速度大小 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}, \\sin 37^{\\circ}=\\frac{3}{5}$ 。求\n\n[图1]\n\n问题:\n小球第一次速度为零的位置与B点之间的距离 $S_{1}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~m}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-04.jpg?height=197&width=414&top_left_y=1735&top_left_x=1569" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_650", "problem": "A rotating mirror is reflecting a laser light on a screen. What would the speed of the spot illuminated by the laser light be at the instance shown in the picture below, if the angular speed of the rotation of the mirror is $\\omega$ ?\n\n[figure1]\nA: $\\omega r^{\\prime}$\nB: $2 \\omega r^{\\prime}$\nC: $\\omega\\left(r+r^{\\prime}\\right)$\nD: $\\omega r$\nE: $2 \\omega r$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA rotating mirror is reflecting a laser light on a screen. What would the speed of the spot illuminated by the laser light be at the instance shown in the picture below, if the angular speed of the rotation of the mirror is $\\omega$ ?\n\n[figure1]\n\nA: $\\omega r^{\\prime}$\nB: $2 \\omega r^{\\prime}$\nC: $\\omega\\left(r+r^{\\prime}\\right)$\nD: $\\omega r$\nE: $2 \\omega r$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-08.jpg?height=729&width=765&top_left_y=475&top_left_x=200" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_698", "problem": "Two charged spheres $X$ and $Y$ are suspended from the ceiling by identical strings as shown below. When the system is in equilibrium, the angles between string OX and the vertical is $i$ and that between OY and the vertical is $j$. If $i>j$, which of the following statement must be correct?\n\n[figure1]\nA: Both $X$ and $Y$ carry positive charges.\nB: The charges on $\\mathrm{X}$ and $\\mathrm{Y}$ are opposite.\nC: The magnitude of charges on $Y$ is greater than that on $\\mathrm{X}$.\nD: The mass of $Y$ is larger than that of $X$.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo charged spheres $X$ and $Y$ are suspended from the ceiling by identical strings as shown below. When the system is in equilibrium, the angles between string OX and the vertical is $i$ and that between OY and the vertical is $j$. If $i>j$, which of the following statement must be correct?\n\n[figure1]\n\nA: Both $X$ and $Y$ carry positive charges.\nB: The charges on $\\mathrm{X}$ and $\\mathrm{Y}$ are opposite.\nC: The magnitude of charges on $Y$ is greater than that on $\\mathrm{X}$.\nD: The mass of $Y$ is larger than that of $X$.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_a0d0ab960474d1b0609cg-04.jpg?height=426&width=569&top_left_y=1348&top_left_x=323" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_349", "problem": "There are three beams between two absolutely rigid plates. The weight of the plates and the beams can be neglected. The coefficient of thermal ex- pansion of the beams is $\\alpha=1.0 \\times 10^{-5} \\mathrm{~K}^{-1}$. The maximum strain (relative length change compared to the no load case) before permanent inelastic deformations of the material of the beam is $\\beta=0.40 \\%$. The beams can support a maximum amount of weight on the top plate, at which point permanent deformations start taking place in some of the beams.\n\n[figure1]\n\nAll of the beams are originally at temperature $T_{0}=0^{\\circ} \\mathrm{C}$. A load of $20 \\%$ of the maximal load is placed on the top plate. Keeping the same load on, to what temperature can the beam in the middle be heated such that no permanent deformations occur? This time the elastic modulus of the material changes depending on the temperature as shown in the figure below (a larger figure is given on an extra sheet).\n\nHint (added during the competition): the elastic modulus or Young modulus $E$ is defined by the formula $F / A=E \\Delta l / l$, where $F$ is the force, $A$ is the area and $\\Delta l / l$ is the relative lengthening.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThere are three beams between two absolutely rigid plates. The weight of the plates and the beams can be neglected. The coefficient of thermal ex- pansion of the beams is $\\alpha=1.0 \\times 10^{-5} \\mathrm{~K}^{-1}$. The maximum strain (relative length change compared to the no load case) before permanent inelastic deformations of the material of the beam is $\\beta=0.40 \\%$. The beams can support a maximum amount of weight on the top plate, at which point permanent deformations start taking place in some of the beams.\n\n[figure1]\n\nAll of the beams are originally at temperature $T_{0}=0^{\\circ} \\mathrm{C}$. A load of $20 \\%$ of the maximal load is placed on the top plate. Keeping the same load on, to what temperature can the beam in the middle be heated such that no permanent deformations occur? This time the elastic modulus of the material changes depending on the temperature as shown in the figure below (a larger figure is given on an extra sheet).\n\nHint (added during the competition): the elastic modulus or Young modulus $E$ is defined by the formula $F / A=E \\Delta l / l$, where $F$ is the force, $A$ is the area and $\\Delta l / l$ is the relative lengthening.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of ^{\\circ} \\mathrm{C}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e6c1582ce7f1c05fa0a6g-2.jpg?height=358&width=582&top_left_y=429&top_left_x=824", "https://cdn.mathpix.com/cropped/2024_03_14_83cc90b265723e4db811g-4.jpg?height=395&width=642&top_left_y=1089&top_left_x=794" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "^{\\circ} \\mathrm{C}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1691", "problem": "具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：取光开始照到处于静止状态的金纳米颗粒的瞬间为计时零点 $t_{0}=0$, 求在任意时刻 $t$ $(t>0)$ 该颗粒转速的表达式 $f(t)$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：\n\n问题:\n取光开始照到处于静止状态的金纳米颗粒的瞬间为计时零点 $t_{0}=0$, 求在任意时刻 $t$ $(t>0)$ 该颗粒转速的表达式 $f(t)$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-02.jpg?height=450&width=450&top_left_y=1883&top_left_x=1292" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1380", "problem": "如图 15a, 间距为 $L$ 的两根平行光滑金属导轨 $\\mathrm{MN} 、 \\mathrm{PQ}$ 放置于同一水平面内, 导轨左端连接一阻值为 $R$ 的定值电阻, 导体棒 $a$ 垂直于导轨放置在导轨上, 在 $a$棒左侧和导轨间存在坚直向下的匀强磁场, 磁感应强度大小为 $B$, 在 $a$ 棒右侧有一绝缘棒 $b, b$ 棒与 $a$ 棒平行, 且与固定在墙上的轻弹簧接触但不相连, 弹簧处于压缩状态且被锁定。现解除锁定, $b$ 棒在弹簧的作用下向左移动, 脱离弹簧后以速度 $v_{0}$ 与 $a$ 棒碰撞并粘在一起。已知 $a 、 b$ 棒的质量分别为 $m 、 M$, 碰撞前后, 两棒始终垂直于导轨, $a$ 棒在两导轨之间的部分的电阻为 $r$, 导轨电阻、接触电阻以及空气阻力均忽略不计, $a 、 b$ 棒总是保持与导轨接触良好。不计电路中感应电流的磁场, 求\n\n[图1]\n\n图 15a弹簧初始时的弹性势能和 $a$ 棒中电流的方向;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图 15a, 间距为 $L$ 的两根平行光滑金属导轨 $\\mathrm{MN} 、 \\mathrm{PQ}$ 放置于同一水平面内, 导轨左端连接一阻值为 $R$ 的定值电阻, 导体棒 $a$ 垂直于导轨放置在导轨上, 在 $a$棒左侧和导轨间存在坚直向下的匀强磁场, 磁感应强度大小为 $B$, 在 $a$ 棒右侧有一绝缘棒 $b, b$ 棒与 $a$ 棒平行, 且与固定在墙上的轻弹簧接触但不相连, 弹簧处于压缩状态且被锁定。现解除锁定, $b$ 棒在弹簧的作用下向左移动, 脱离弹簧后以速度 $v_{0}$ 与 $a$ 棒碰撞并粘在一起。已知 $a 、 b$ 棒的质量分别为 $m 、 M$, 碰撞前后, 两棒始终垂直于导轨, $a$ 棒在两导轨之间的部分的电阻为 $r$, 导轨电阻、接触电阻以及空气阻力均忽略不计, $a 、 b$ 棒总是保持与导轨接触良好。不计电路中感应电流的磁场, 求\n\n[图1]\n\n图 15a\n\n问题:\n弹簧初始时的弹性势能和 $a$ 棒中电流的方向;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-06.jpg?height=362&width=739&top_left_y=1161&top_left_x=1184" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1429", "problem": "卫星的运动可有地面的观测来决定, 而知道了卫星的运动, 又可以用空间的飞行体或地面上物体的运动, 这都涉及到时间和空间坐标的测定, 为简化分析和计算, 不考虑地球的自转和公转, 把它作惯性系。\n\n1. 先来考虑卫星的测定, 设不考虑相对论效应, 在卫星上装有发射电波的装置和高精度的原子钟。假设在卫星上每次发出的电波信号, 都包含该信号发出的时刻这一信息。地面观测系统（包含若干个观测站）可利用从电波中接收到的这一信息, 并根据自己所处的已知位置和自己的时钟来确定卫星每一时刻的位置, 从而确定卫星的运动。这种测量系统至少需要包含几个地面观测站？列出可以确定卫星位置的方程。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n卫星的运动可有地面的观测来决定, 而知道了卫星的运动, 又可以用空间的飞行体或地面上物体的运动, 这都涉及到时间和空间坐标的测定, 为简化分析和计算, 不考虑地球的自转和公转, 把它作惯性系。\n\n1. 先来考虑卫星的测定, 设不考虑相对论效应, 在卫星上装有发射电波的装置和高精度的原子钟。假设在卫星上每次发出的电波信号, 都包含该信号发出的时刻这一信息。\n\n问题:\n地面观测系统（包含若干个观测站）可利用从电波中接收到的这一信息, 并根据自己所处的已知位置和自己的时钟来确定卫星每一时刻的位置, 从而确定卫星的运动。这种测量系统至少需要包含几个地面观测站？列出可以确定卫星位置的方程。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_317", "problem": "Consider two absolutely elastic dielec tric balls of radius $r$ and mass $m$ one of which carries isotropically distributed charge $-q$, and the other $-+q$. There is so strong homogen eous magnetic field $B$, parallel to the axis $z$, tha electrostatic interaction of the two charges can be neglected; neglect also gravity and friction forces. The first ball (negatively charged) moves with speed $v$ and collides with the second ball which had been resting at the origin. The collision is central, and immediately before the im pact, the velocity of the first ball was parallel to the $x$-axis.\n\nConsider the same situation as before, except that the there are three differences in the assumptions: the both balls have now identical positive charge $+q$; electrostatic repulsion between the balls is no longer negligible; the collision is not necessarily central (but the balls move still at the same value of $z$ so that the collision will not induce any motion in the direction of the $z$-axis). Let $P_{i}$ denote the poin where the surfaces of the two balls are in contact during the $i$-th collision. What is the maximal distance between $P_{i}$ and $P_{j}$ (maximize over all the values $i, j=1, \\ldots \\infty$, and over all the impact parameters of the collisions for fixed values of $B$, $m$, and $q)$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider two absolutely elastic dielec tric balls of radius $r$ and mass $m$ one of which carries isotropically distributed charge $-q$, and the other $-+q$. There is so strong homogen eous magnetic field $B$, parallel to the axis $z$, tha electrostatic interaction of the two charges can be neglected; neglect also gravity and friction forces. The first ball (negatively charged) moves with speed $v$ and collides with the second ball which had been resting at the origin. The collision is central, and immediately before the im pact, the velocity of the first ball was parallel to the $x$-axis.\n\nConsider the same situation as before, except that the there are three differences in the assumptions: the both balls have now identical positive charge $+q$; electrostatic repulsion between the balls is no longer negligible; the collision is not necessarily central (but the balls move still at the same value of $z$ so that the collision will not induce any motion in the direction of the $z$-axis). Let $P_{i}$ denote the poin where the surfaces of the two balls are in contact during the $i$-th collision. What is the maximal distance between $P_{i}$ and $P_{j}$ (maximize over all the values $i, j=1, \\ldots \\infty$, and over all the impact parameters of the collisions for fixed values of $B$, $m$, and $q)$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1525", "problem": "爱因斯坦引力理论预言物质分布的变化会导致时空几何结构的波动一一引力波。为简明起见, 考虑沿 $z$ 轴传播的平面引力波。对于任意给定的 $z$, 在 $x-y$ 二维空间中两个无限邻近点 $(x, y)$ 和 $(x+\\mathrm{d} x, y+\\mathrm{d} y)$ 之间距离 $\\mathrm{d} r$ 的表达式为\n\n$$\n\\mathrm{d} r=\\sqrt{\\left(1+f_{1}\\right)(\\mathrm{d} x)^{2}+f_{2}(\\mathrm{~d} x \\mathrm{~d} y+\\mathrm{d} y \\mathrm{~d} x)+\\left(1-f_{1}\\right)(\\mathrm{d} y)^{2}}\n$$\n\n引力波体现为 $f_{1}$ 和 $f_{2}$ 的变化 (波动)。\n\n假设一列平面引力波传来时, $f_{1}$ 和 $f_{2}$ 可表示为\n\n$$\nf_{1}=A \\sin \\left[\\omega\\left(t-\\frac{Z}{c}\\right)\\right], \\quad f_{2}=0 ; \\quad 00)$ 的离子, 由静止开始被电场加速, 经狭缝中的 $\\mathrm{O}$ 点进入磁场区域, $\\mathrm{O}$ 点到极板右端的距离为 $D$, 到出射孔 $\\mathrm{P}$ 的距离为 $b D$ (常数 $b$ 为大于 2 的自然数）。已知磁感应强度大小在零到 $B_{\\text {max }}$ 之间可调, 离子从离子源上方的 $\\mathrm{O}$ 点射入磁场区域, 最终只能从出射孔 P 射出。假设如果离子打到器壁或离子源外壁便即被吸收。忽略相对论效应。求\n[图1]\n\n图a \n\n[图2]\n\n图 b磁感应强度 $B$ 的其它所有可能值;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某种回旋加速器的设计方案如俯视图 a 所示, 图中粗黑线段为两个正对的极板，其间仅在带电粒子经过的过程中存在匀强电场, 两极板间电势差为 $U$ 。两个极板的板面中部各有一狭缝(沿 OP 方向的狭长区域），带电粒子可通过狭缝穿越极板（见图 b）；两细虚线间（除开两极板之间的区域）既无电场也无磁场；其它部分存在匀强磁场, 磁感应强度方向垂直于纸面。在离子源 $\\mathrm{S}$ 中产生的质量为 $\\mathrm{m}$ 、带电量为 $q(q>0)$ 的离子, 由静止开始被电场加速, 经狭缝中的 $\\mathrm{O}$ 点进入磁场区域, $\\mathrm{O}$ 点到极板右端的距离为 $D$, 到出射孔 $\\mathrm{P}$ 的距离为 $b D$ (常数 $b$ 为大于 2 的自然数）。已知磁感应强度大小在零到 $B_{\\text {max }}$ 之间可调, 离子从离子源上方的 $\\mathrm{O}$ 点射入磁场区域, 最终只能从出射孔 P 射出。假设如果离子打到器壁或离子源外壁便即被吸收。忽略相对论效应。求\n[图1]\n\n图a \n\n[图2]\n\n图 b\n\n问题:\n磁感应强度 $B$ 的其它所有可能值;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-03.jpg?height=896&width=894&top_left_y=243&top_left_x=889", "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-03.jpg?height=331&width=371&top_left_y=1248&top_left_x=1345" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1344", "problem": "用光照射处在基态的氢原子, 有可能使氢原子电离, 下列说法中正确的是\nA: 只要光的光强足够大, 就一定可以使氢原子电离\nB: 只要光的频率足够高, 就一定可以使氢原子电离\nC: 只要光子的能量足够大, 就一定可以使氢原子电离\nD: 只要光照的时间足够长，就一定可以使氢原子电离\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n用光照射处在基态的氢原子, 有可能使氢原子电离, 下列说法中正确的是\n\nA: 只要光的光强足够大, 就一定可以使氢原子电离\nB: 只要光的频率足够高, 就一定可以使氢原子电离\nC: 只要光子的能量足够大, 就一定可以使氢原子电离\nD: 只要光照的时间足够长，就一定可以使氢原子电离\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_73", "problem": "All of the following are base units of the SI system except:\nA: kilogram\nB: kelvin\nC: meter\nD: volt\nE: candela\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAll of the following are base units of the SI system except:\n\nA: kilogram\nB: kelvin\nC: meter\nD: volt\nE: candela\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_827", "problem": "A single electron transistor (SET) consists of a quantum dot, which is a small isolated conductor where electrons can be localised, and of several electrodes in its vicinity. The gate electrode couples capacitatively to the quantum dot, while the two other electrodes --- the source and the drain --- are connected via tunnel junctions, through which electrons can tunnel due to quantum mechanics. A simplified circuit diagram for an SET is shown in the figure.\n[figure1]\n\nCircuit diagram representation of an SET. QD is the quantum dot, $\\mathrm{S}$ is the source, $\\mathrm{D}$ is the drain and $\\mathrm{G}$ is the gate.\n\nThe capacitance of the gate is $C_{g}$ and the capacitance of the tunnel junctions is $C_{t} \\ll C_{g}$. Consider $C_{g}$ to be the total capacitance of the quantum dot. In this part of the problem, the source and the drain are held at zero potential, and the voltage on the gate electrode is fixed at $V_{g}$.\n\nConsider a state of the SET in which the quantum dot contains $n$ electrons.\n\nTunnelling of electrons onto or off the dot limits the lifetime of their energy states. This tunnelling can be modelled using an effective resistance of the tunnel junction with the characteristic tunnelling time equal to the characteristic time for charging or discharging the quantum dot through the junction.\n\nFind a condition on the effective resistance $R_{t}$ so that the electrons in the quantum dot retain sufficiently well-defined energy for the ON and OFF states to remain distinct.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA single electron transistor (SET) consists of a quantum dot, which is a small isolated conductor where electrons can be localised, and of several electrodes in its vicinity. The gate electrode couples capacitatively to the quantum dot, while the two other electrodes --- the source and the drain --- are connected via tunnel junctions, through which electrons can tunnel due to quantum mechanics. A simplified circuit diagram for an SET is shown in the figure.\n[figure1]\n\nCircuit diagram representation of an SET. QD is the quantum dot, $\\mathrm{S}$ is the source, $\\mathrm{D}$ is the drain and $\\mathrm{G}$ is the gate.\n\nThe capacitance of the gate is $C_{g}$ and the capacitance of the tunnel junctions is $C_{t} \\ll C_{g}$. Consider $C_{g}$ to be the total capacitance of the quantum dot. In this part of the problem, the source and the drain are held at zero potential, and the voltage on the gate electrode is fixed at $V_{g}$.\n\nConsider a state of the SET in which the quantum dot contains $n$ electrons.\n\nTunnelling of electrons onto or off the dot limits the lifetime of their energy states. This tunnelling can be modelled using an effective resistance of the tunnel junction with the characteristic tunnelling time equal to the characteristic time for charging or discharging the quantum dot through the junction.\n\nFind a condition on the effective resistance $R_{t}$ so that the electrons in the quantum dot retain sufficiently well-defined energy for the ON and OFF states to remain distinct.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-3.jpg?height=582&width=868&top_left_y=2076&top_left_x=594" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_108", "problem": "To test the speed of a model car, you time the car with a stopwatch as it travels a distance of $100 \\mathrm{~m}$. You record a time of $5.0 \\mathrm{~s}$, and your measurement has an uncertainty of $0.2 \\mathrm{~s}$. What is the uncertainty in your estimate of the car's speed? Assume that the car travels at a constant speed and the distance of $100 \\mathrm{~m}$ is known very precisely.\nA: $v=20 \\pm 0.16 \\mathrm{~m} / \\mathrm{s}$\nB: $v=20 \\pm 0.8 \\mathrm{~m} / \\mathrm{s} $ \nC: $v=20 \\pm 1.0 \\mathrm{~m} / \\mathrm{s}$\nD: $v=20 \\pm 1.25 \\mathrm{~m} / \\mathrm{s}$\nE: $v=20 \\pm 4.0 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTo test the speed of a model car, you time the car with a stopwatch as it travels a distance of $100 \\mathrm{~m}$. You record a time of $5.0 \\mathrm{~s}$, and your measurement has an uncertainty of $0.2 \\mathrm{~s}$. What is the uncertainty in your estimate of the car's speed? Assume that the car travels at a constant speed and the distance of $100 \\mathrm{~m}$ is known very precisely.\n\nA: $v=20 \\pm 0.16 \\mathrm{~m} / \\mathrm{s}$\nB: $v=20 \\pm 0.8 \\mathrm{~m} / \\mathrm{s} $ \nC: $v=20 \\pm 1.0 \\mathrm{~m} / \\mathrm{s}$\nD: $v=20 \\pm 1.25 \\mathrm{~m} / \\mathrm{s}$\nE: $v=20 \\pm 4.0 \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_279", "problem": "Note: In this question all of the diagrams are not to scale. The planets and the sun are very small compared to the distances between them.\n\nIn 1619 Johannes Kepler published his $3^{\\text {rd }}$ Law of planetary motion. The $3^{\\text {rd }}$ Law stated that the square of the orbital period of a planet is proportional to the cube of the mean orbital radius around the Sun. This is written as:\n\n$$\nT^{2} \\propto R^{3} \\quad T=\\text { period of orbit } \\quad R=\\text { mean radius of orbit }\n$$\n\nThe mean orbital radius of the Earth is defined as 1 Astronomical Unit (1 AU) i.e. the distance between the Earth and the Sun is 1 AU.\n\nThe orbital period of Venus is 225 days.\n\nCalculate the mean orbital radius of Venus in astronomical units.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nNote: In this question all of the diagrams are not to scale. The planets and the sun are very small compared to the distances between them.\n\nIn 1619 Johannes Kepler published his $3^{\\text {rd }}$ Law of planetary motion. The $3^{\\text {rd }}$ Law stated that the square of the orbital period of a planet is proportional to the cube of the mean orbital radius around the Sun. This is written as:\n\n$$\nT^{2} \\propto R^{3} \\quad T=\\text { period of orbit } \\quad R=\\text { mean radius of orbit }\n$$\n\nThe mean orbital radius of the Earth is defined as 1 Astronomical Unit (1 AU) i.e. the distance between the Earth and the Sun is 1 AU.\n\nThe orbital period of Venus is 225 days.\n\nCalculate the mean orbital radius of Venus in astronomical units.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of AU, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "AU" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_941", "problem": "In a modern electric car, the most important reason the batteries are lithium ion rather than lead acid is\nA: lithium cells are easier to recycle\nB: lithium is much cheaper than lead\nC: lithium is less dense so the batteries are much lighter\nD: the energy density of the lithium battery is much greater\nE: lead batteries are full of dangerous acid\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn a modern electric car, the most important reason the batteries are lithium ion rather than lead acid is\n\nA: lithium cells are easier to recycle\nB: lithium is much cheaper than lead\nC: lithium is less dense so the batteries are much lighter\nD: the energy density of the lithium battery is much greater\nE: lead batteries are full of dangerous acid\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_888", "problem": "For a scalable quantum computing architecture, the number of wires reaching each individual quantum bit need to be minimized. A promising alternative to an SET for charge sensing in silicon quantum computing is a Single Lead Quantum Dot (SLQD). In many ways it is similar to an SET, but does not have the source and drain leads. The gate is the only electrode, through which the electron energy states of the quantum dot are controlled and also through which RF reflectometry is conducted.\n\nLike an SET, a SLQD has an OFF in which the SLQD behaves as a total insulator. In contrast to an SET, the ON state of the SLQD is capacitive, with capacitance $C_{\\mathrm{q}}$. In order to maximize the difference in reflectance $\\Delta \\Gamma$ of the SLQD, the following circuit is constructed. The parasitic capacitance $C_{0} \\approx 0.4 \\mathrm{pF}$ is fixed by circuit geometry, but the value of $L_{0}$ and the operating frequency can be changed to optimize the performance. The characteristic impedance of the transmission line is $Z_{0}=50 \\Omega$.\n\n[figure1]\n\nCircuit diagram of the SLQD readout circuit connected to the transmission line.\n\nOptimal values of $L_{0}$ are relatively large and not always technically feasible. Hence, other types of circuit elements may be needed to improve sensitivity of the reflectometry readout circuit.\n\nAssume that $L_{0}$ (and hence $Z_{C}$ ) is fixed. Draw a circuit diagram showing where $0.5 \\mathrm{pt}$ to place an additional element in the SLQD readout circuit and specify the parameter(s) of this element such that $\\Delta \\Gamma \\sim 1$ can still be achieved without requiring a large inductance.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nFor a scalable quantum computing architecture, the number of wires reaching each individual quantum bit need to be minimized. A promising alternative to an SET for charge sensing in silicon quantum computing is a Single Lead Quantum Dot (SLQD). In many ways it is similar to an SET, but does not have the source and drain leads. The gate is the only electrode, through which the electron energy states of the quantum dot are controlled and also through which RF reflectometry is conducted.\n\nLike an SET, a SLQD has an OFF in which the SLQD behaves as a total insulator. In contrast to an SET, the ON state of the SLQD is capacitive, with capacitance $C_{\\mathrm{q}}$. In order to maximize the difference in reflectance $\\Delta \\Gamma$ of the SLQD, the following circuit is constructed. The parasitic capacitance $C_{0} \\approx 0.4 \\mathrm{pF}$ is fixed by circuit geometry, but the value of $L_{0}$ and the operating frequency can be changed to optimize the performance. The characteristic impedance of the transmission line is $Z_{0}=50 \\Omega$.\n\n[figure1]\n\nCircuit diagram of the SLQD readout circuit connected to the transmission line.\n\nOptimal values of $L_{0}$ are relatively large and not always technically feasible. Hence, other types of circuit elements may be needed to improve sensitivity of the reflectometry readout circuit.\n\nAssume that $L_{0}$ (and hence $Z_{C}$ ) is fixed. Draw a circuit diagram showing where $0.5 \\mathrm{pt}$ to place an additional element in the SLQD readout circuit and specify the parameter(s) of this element such that $\\Delta \\Gamma \\sim 1$ can still be achieved without requiring a large inductance.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-6.jpg?height=343&width=808&top_left_y=1002&top_left_x=635" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_914", "problem": "An electron with charge $-e(e>0)$ experiences the Lorentz force due to the perpendicular magnetic field and the electric force\n\n$$\nm^{*} \\frac{d \\vec{v}}{d t}=-e(\\vec{v} \\times \\vec{B}+\\vec{E})\n$$\n\nwhere $\\vec{v}$ is the velocity of the electron.\n\nIn the stationary regime, the acceleration vanishes. Hence\n\n$$\n\\vec{v}_{s} \\times \\vec{B}+\\vec{E}=0\n$$\n\nThe velocity can be expressed as\n\n$$\n\\vec{v}_{s}=\\frac{\\vec{E} \\times \\vec{B}}{B^{2}}\n$$\n\nwhose magnitude is simply $v_{s}=E / B$.The fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nThe Hall resistance is defined as $R_{\\mathrm{H}}=V_{\\mathrm{H}} / I$. In the classical model, find $R_{\\mathrm{H}}$ as a function of the number of the electrons $N$ and the magnetic flux $\\phi=B A=B W L$, where $A$ is the area of the sample, and $W$ and $L$ the effective width and length of the sample, respectively.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\nHere is some context information for this question, which might assist you in solving it:\nAn electron with charge $-e(e>0)$ experiences the Lorentz force due to the perpendicular magnetic field and the electric force\n\n$$\nm^{*} \\frac{d \\vec{v}}{d t}=-e(\\vec{v} \\times \\vec{B}+\\vec{E})\n$$\n\nwhere $\\vec{v}$ is the velocity of the electron.\n\nIn the stationary regime, the acceleration vanishes. Hence\n\n$$\n\\vec{v}_{s} \\times \\vec{B}+\\vec{E}=0\n$$\n\nThe velocity can be expressed as\n\n$$\n\\vec{v}_{s}=\\frac{\\vec{E} \\times \\vec{B}}{B^{2}}\n$$\n\nwhose magnitude is simply $v_{s}=E / B$.\n\nproblem:\nThe fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nThe Hall resistance is defined as $R_{\\mathrm{H}}=V_{\\mathrm{H}} / I$. In the classical model, find $R_{\\mathrm{H}}$ as a function of the number of the electrons $N$ and the magnetic flux $\\phi=B A=B W L$, where $A$ is the area of the sample, and $W$ and $L$ the effective width and length of the sample, respectively.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_810", "problem": "Consider an arbitrary initial rotation of the stage with angular momentum $L$ (Fig. 2), where $\\theta$ is the angle between the symmetry axis and the direction of angular momentum. Fuel tank at this point is assumed to be empty. No forces or torques act upon the stage.\n\n[figure1]\n\nFig. 2: Rocket stage rotation\n\nind the rotational energy $E_{x}$ associated with rotation $\\omega_{x}$ and $E_{y}$ associated with $0.4 \\mathrm{pt}$ rotation $\\omega_{y}$. Find total rotational kinetic energy $E=E_{x}+E_{y}$ of the stage as a function of the angular momentum $L$ and $\\cos \\theta$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider an arbitrary initial rotation of the stage with angular momentum $L$ (Fig. 2), where $\\theta$ is the angle between the symmetry axis and the direction of angular momentum. Fuel tank at this point is assumed to be empty. No forces or torques act upon the stage.\n\n[figure1]\n\nFig. 2: Rocket stage rotation\n\nind the rotational energy $E_{x}$ associated with rotation $\\omega_{x}$ and $E_{y}$ associated with $0.4 \\mathrm{pt}$ rotation $\\omega_{y}$. Find total rotational kinetic energy $E=E_{x}+E_{y}$ of the stage as a function of the angular momentum $L$ and $\\cos \\theta$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d7ee880fa438d4cf18fag-2.jpg?height=356&width=763&top_left_y=553&top_left_x=652" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1598", "problem": "一厚度为 $t$ 的薄金属盘悬吊在温度为 $300.0 \\mathrm{~K}$ 的空气中，其上表面受太阳直射，温度为 $360.0 \\mathrm{~K}$, 下表面的温度为 $340.0 \\mathrm{~K}$. 空气的温度保持不变, 单位时间内金属盘每个表面散失到空气中的能量与此表面和空气的温度差以及此表面的面积成正比,忽略金属盘侧面的能量损失. 若金属盘的厚度变为原来的 2 倍, 求金属盘上、下表面的温度.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n一厚度为 $t$ 的薄金属盘悬吊在温度为 $300.0 \\mathrm{~K}$ 的空气中，其上表面受太阳直射，温度为 $360.0 \\mathrm{~K}$, 下表面的温度为 $340.0 \\mathrm{~K}$. 空气的温度保持不变, 单位时间内金属盘每个表面散失到空气中的能量与此表面和空气的温度差以及此表面的面积成正比,忽略金属盘侧面的能量损失. 若金属盘的厚度变为原来的 2 倍, 求金属盘上、下表面的温度.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[金属盘上温度, 金属盘下温度]\n它们的单位依次是[K, K]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "K", "K" ], "answer_sequence": [ "金属盘上温度", "金属盘下温度" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_757", "problem": "Staircase \n\nThe equilibrium shape of bodies in zero-gravity is determined by the minimum of their surface energy. Thus, for example, the equilibrium shape of a water droplet turns out to be spherical: the sphere has the smallest surface area among bodies of the same volume. At low temperature, the equilibrium shape of crystals may have flat facets. The parts of the crystal surface that have a small angle $\\phi$ with the facet are in fact staircases of rare steps on this facet. The height of such steps is equal to the period of the crystal lattice $h$.\n[figure1]\n\nEquilibrium surface profile $y(x)$ of a certain crystal and the corresponding microscopic staircase are shown schematically in the figure, where $\\mathrm{n}$ denotes the step number, counting from $x=0$. The profile shape at $x>0$ can be approximated as $y(x)=-\\left(\\frac{x}{\\lambda}\\right)^{\\frac{3}{2}} h$, where $\\lambda=45 \\mu \\mathrm{m}$ and $h=0.3 \\mathrm{~nm}$.\n\nExpress the distance $d_{n}$ between two adjacent steps as a function of $n$ for $n \\gg 1$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nStaircase \n\nThe equilibrium shape of bodies in zero-gravity is determined by the minimum of their surface energy. Thus, for example, the equilibrium shape of a water droplet turns out to be spherical: the sphere has the smallest surface area among bodies of the same volume. At low temperature, the equilibrium shape of crystals may have flat facets. The parts of the crystal surface that have a small angle $\\phi$ with the facet are in fact staircases of rare steps on this facet. The height of such steps is equal to the period of the crystal lattice $h$.\n[figure1]\n\nEquilibrium surface profile $y(x)$ of a certain crystal and the corresponding microscopic staircase are shown schematically in the figure, where $\\mathrm{n}$ denotes the step number, counting from $x=0$. The profile shape at $x>0$ can be approximated as $y(x)=-\\left(\\frac{x}{\\lambda}\\right)^{\\frac{3}{2}} h$, where $\\lambda=45 \\mu \\mathrm{m}$ and $h=0.3 \\mathrm{~nm}$.\n\nExpress the distance $d_{n}$ between two adjacent steps as a function of $n$ for $n \\gg 1$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $$\\mu \\mathrm{m}$$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without any units and equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2878f1ea9876cd74ba9g-1.jpg?height=598&width=1418&top_left_y=684&top_left_x=344" ], "answer": null, "solution": null, "answer_type": "EX", "unit": [ "$$\\mu \\mathrm{m}$$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1652", "problem": "相对于站立在地面的李同学, 张同学以相对论速率 $v$ 向右运动, 王同学以同样的速率 $v$ 向左运动. 当张同学和王同学相遇时, 三位同学各自把自己时钟的读数调整到零. 当张同学和王同学之间的距离为 $L$ 时(在地面参考系中观察), 张同学拍一下手.已知张同学和王同学之间的相对速率为\n\n$$\nv_{r}=\\frac{2 \\beta c}{1+\\beta^{2}}\n$$\n\n其中 $\\beta=\\frac{v}{c}, c$ 为真空中的光速.从王同学自身静止的参考系看, 在张同学拍手这一事件发生的时刻, 王同学也拍一下手. 从张同学自身静止的参考系看, 在王同学拍手这一事件发生的时刻, 张同学第二次拍一下手. 从王同学自身静止的参考系看, 在张同学第二次拍手这一事件发生的时刻, 王同学第二次拍一下手. 照此继续下去. 求当张同学第 $n$ 次拍手时地面参考系中张、王同学之间的距离", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n相对于站立在地面的李同学, 张同学以相对论速率 $v$ 向右运动, 王同学以同样的速率 $v$ 向左运动. 当张同学和王同学相遇时, 三位同学各自把自己时钟的读数调整到零. 当张同学和王同学之间的距离为 $L$ 时(在地面参考系中观察), 张同学拍一下手.已知张同学和王同学之间的相对速率为\n\n$$\nv_{r}=\\frac{2 \\beta c}{1+\\beta^{2}}\n$$\n\n其中 $\\beta=\\frac{v}{c}, c$ 为真空中的光速.\n\n问题:\n从王同学自身静止的参考系看, 在张同学拍手这一事件发生的时刻, 王同学也拍一下手. 从张同学自身静止的参考系看, 在王同学拍手这一事件发生的时刻, 张同学第二次拍一下手. 从王同学自身静止的参考系看, 在张同学第二次拍手这一事件发生的时刻, 王同学第二次拍一下手. 照此继续下去. 求当张同学第 $n$ 次拍手时地面参考系中张、王同学之间的距离\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_298", "problem": "A radioisotope power source of the type used to provide electrical energy for space missions uses the radioactive isotope Plutonium- $238\\left(\\mathrm{Pu}^{238}\\right)$. Plutonium-238 decays by alpha decay.\n\nThe activity of 1 gram of Plutonium-238 is $6.3 \\times 10^{11} \\mathrm{~Bq}$.\n\nThe energy released by each alpha decay is $9.0 \\times 10^{-13} \\mathrm{~J}$.\n\nThe mass of Plutonium-238 needed to provide a power of $100 \\mathrm{~W}$ is approximately:\nA: $\\quad 100 \\mathrm{~g}$\nB: $\\quad 140 \\mathrm{~g}$\nC: $\\quad 180 \\mathrm{~g}$\nD: $\\quad 238 \\mathrm{~g}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA radioisotope power source of the type used to provide electrical energy for space missions uses the radioactive isotope Plutonium- $238\\left(\\mathrm{Pu}^{238}\\right)$. Plutonium-238 decays by alpha decay.\n\nThe activity of 1 gram of Plutonium-238 is $6.3 \\times 10^{11} \\mathrm{~Bq}$.\n\nThe energy released by each alpha decay is $9.0 \\times 10^{-13} \\mathrm{~J}$.\n\nThe mass of Plutonium-238 needed to provide a power of $100 \\mathrm{~W}$ is approximately:\n\nA: $\\quad 100 \\mathrm{~g}$\nB: $\\quad 140 \\mathrm{~g}$\nC: $\\quad 180 \\mathrm{~g}$\nD: $\\quad 238 \\mathrm{~g}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_920", "problem": "Free vortices move about in space with the flow 2 . In other words each element of the filament moves with the velocity $\\vec{v}$ of the fluid at the position of that element.[^1]\n\nAs an example, consider a pair of counter-rotating straight vortices placed initially at distance $r_{0}$ from each other, see Fig. 3. Each vortex produces velocity $v_{0}=\\kappa / r_{0}$ at the axis of another. As a result, the vortex pair moves rectilinearly with constant speed $v_{0}=\\kappa / r_{0}$ so that the distance between them remains unchanged.\n\n[figure1]\n\nFig. 3: Parallel vortex filaments with opposite circulations.\n\nFind velocity $v(\\vec{r})$ of a vortex positioned at $\\vec{r}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nFree vortices move about in space with the flow 2 . In other words each element of the filament moves with the velocity $\\vec{v}$ of the fluid at the position of that element.[^1]\n\nAs an example, consider a pair of counter-rotating straight vortices placed initially at distance $r_{0}$ from each other, see Fig. 3. Each vortex produces velocity $v_{0}=\\kappa / r_{0}$ at the axis of another. As a result, the vortex pair moves rectilinearly with constant speed $v_{0}=\\kappa / r_{0}$ so that the distance between them remains unchanged.\n\n[figure1]\n\nFig. 3: Parallel vortex filaments with opposite circulations.\n\nFind velocity $v(\\vec{r})$ of a vortex positioned at $\\vec{r}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_61fc31149c0f627b45f3g-3.jpg?height=369&width=531&top_left_y=775&top_left_x=768" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_44", "problem": "LaGrange Point 2 is a location in space where the gravitational attraction of the sun is equal to the gravitational attraction of the Earth. The sun is $1.5 \\mathrm{X} 10^{8} \\mathrm{~km}$ from Earth, and it has a mass that is $3.24 \\times 10^{5}$ times that of Earth. How far from Earth, along a line to the sun, must an object be located to be in this location?\nA: $1.50 \\times 10^{6} \\mathrm{~km}$\nB: $1.83 \\times 10^{6} \\mathrm{~km}$\nC: $1.53 \\times 10^{6} \\mathrm{~km}$\nD: $3.26 \\times 10^{6} \\mathrm{~km}$\nE: $3.06 \\times 10^{6} \\mathrm{~km}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nLaGrange Point 2 is a location in space where the gravitational attraction of the sun is equal to the gravitational attraction of the Earth. The sun is $1.5 \\mathrm{X} 10^{8} \\mathrm{~km}$ from Earth, and it has a mass that is $3.24 \\times 10^{5}$ times that of Earth. How far from Earth, along a line to the sun, must an object be located to be in this location?\n\nA: $1.50 \\times 10^{6} \\mathrm{~km}$\nB: $1.83 \\times 10^{6} \\mathrm{~km}$\nC: $1.53 \\times 10^{6} \\mathrm{~km}$\nD: $3.26 \\times 10^{6} \\mathrm{~km}$\nE: $3.06 \\times 10^{6} \\mathrm{~km}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_6", "problem": "A hot air balloon is moving up at $10 \\mathrm{~m} / \\mathrm{s}$. At a height of $100 \\mathrm{~m}$ above the ground, a package is released from the balloon. How much time does it take the package to reach the ground after being released?\nA: $4.58 \\mathrm{~s}$\nB: $5.58 \\mathrm{~s}$\nC: $4.24 \\mathrm{~s}$\nD: $5.24 \\mathrm{~s}$\nE: $4.64 \\mathrm{~s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA hot air balloon is moving up at $10 \\mathrm{~m} / \\mathrm{s}$. At a height of $100 \\mathrm{~m}$ above the ground, a package is released from the balloon. How much time does it take the package to reach the ground after being released?\n\nA: $4.58 \\mathrm{~s}$\nB: $5.58 \\mathrm{~s}$\nC: $4.24 \\mathrm{~s}$\nD: $5.24 \\mathrm{~s}$\nE: $4.64 \\mathrm{~s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_528", "problem": "Assuming that $z_{0}$ is small enough so that the motion can be approximated as simple harmonic, find the period of the motion in terms any or all of the mass of the particle $m, g, z_{0}$, and/or $k$.\n\nWe present a force-based approach and an energy-based approach. In each case, let $r$ be the radial position of the particle, so that $z=k r^{2}$ is the height of the particle above the bottom of the bowl.\n\nLet the angle of the bowl's surface to the horizontal be $\\theta$. Because $z_{0}$ is small,\n\n$$\n\\sin \\theta \\approx \\theta \\approx \\tan \\theta=\\frac{d z}{d r}=2 k r\n$$\n\nand $\\cos \\theta \\approx 1$. Moreover, since $z_{0}$ is small, the centripetal acceleration is negligible, so we can consider only the tangential acceleration. Since the force tangential to the bowl is $m g \\sin \\theta$, this is $a=g \\sin \\theta$. The radial acceleration $a_{r}$ is\n\n$$\na_{r}=-a \\cos \\theta=-g \\cos \\theta \\sin \\theta\n$$\n\nThen in the small- $z$ approximation,\n\n$$\na_{r} \\approx-g \\tan \\theta=-2 k r g .\n$$\n\nThis is simple harmonic motion with $\\omega=\\sqrt{2 k g}$ and hence period\n\n$$\nT=\\frac{2 \\pi}{\\sqrt{2 k g}}\n$$\n\nThe energy-based approach begins with the total energy\n\n$$\nE=\\frac{1}{2} m v^{2}+m g z\n$$\n\nThe velocity $v$ is given by\n\n$$\nv^{2}=\\left(\\frac{d r}{d t}\\right)^{2}+\\left(\\frac{d z}{d t}\\right)^{2}\n$$\n\nBecause $z$ is small, $\\frac{d z}{d t} \\ll \\frac{d r}{d t}$, and we conclude that\n\n$$\nE=\\frac{1}{2} m\\left(\\frac{d r}{d t}\\right)^{2}+m g k r^{2}\n$$\n\nBy conservation of energy,\n\n$$\n0=\\frac{d E}{d t}=m \\frac{d r}{d t} \\frac{d^{2} r}{d t^{2}}+2 m g k r \\frac{d r}{d t}\n$$\n\nwhich implies that\n\n$$\n0=\\frac{d^{2} r}{d t^{2}}+2 k r g\n$$\n\nwhich is the same equation as before.A particle is constrained to move on the inner surface of a frictionless parabolic bowl whose crosssection has equation $z=k r^{2}$. The particle begins at a height $z_{0}$ above the bottom of the bowl with a horizontal velocity $v_{0}$ along the surface of the bowl. The acceleration due to gravity is $g$.\n[figure1]\n\nSuppose that the particle now begins at a height $z_{0}$ above the bottom of the bowl with an initial velocity $v_{0}=0$.\n\nAssuming that $z_{0}$ is not small, will the actual period of motion be greater than, less than, or equal to your simple harmonic approximation above? (You need not calculate the new value explicitly, but you should show some work to defend your answer.)\nA: greater than\nB: less than\nC: equal to\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\nHere is some context information for this question, which might assist you in solving it:\nAssuming that $z_{0}$ is small enough so that the motion can be approximated as simple harmonic, find the period of the motion in terms any or all of the mass of the particle $m, g, z_{0}$, and/or $k$.\n\nWe present a force-based approach and an energy-based approach. In each case, let $r$ be the radial position of the particle, so that $z=k r^{2}$ is the height of the particle above the bottom of the bowl.\n\nLet the angle of the bowl's surface to the horizontal be $\\theta$. Because $z_{0}$ is small,\n\n$$\n\\sin \\theta \\approx \\theta \\approx \\tan \\theta=\\frac{d z}{d r}=2 k r\n$$\n\nand $\\cos \\theta \\approx 1$. Moreover, since $z_{0}$ is small, the centripetal acceleration is negligible, so we can consider only the tangential acceleration. Since the force tangential to the bowl is $m g \\sin \\theta$, this is $a=g \\sin \\theta$. The radial acceleration $a_{r}$ is\n\n$$\na_{r}=-a \\cos \\theta=-g \\cos \\theta \\sin \\theta\n$$\n\nThen in the small- $z$ approximation,\n\n$$\na_{r} \\approx-g \\tan \\theta=-2 k r g .\n$$\n\nThis is simple harmonic motion with $\\omega=\\sqrt{2 k g}$ and hence period\n\n$$\nT=\\frac{2 \\pi}{\\sqrt{2 k g}}\n$$\n\nThe energy-based approach begins with the total energy\n\n$$\nE=\\frac{1}{2} m v^{2}+m g z\n$$\n\nThe velocity $v$ is given by\n\n$$\nv^{2}=\\left(\\frac{d r}{d t}\\right)^{2}+\\left(\\frac{d z}{d t}\\right)^{2}\n$$\n\nBecause $z$ is small, $\\frac{d z}{d t} \\ll \\frac{d r}{d t}$, and we conclude that\n\n$$\nE=\\frac{1}{2} m\\left(\\frac{d r}{d t}\\right)^{2}+m g k r^{2}\n$$\n\nBy conservation of energy,\n\n$$\n0=\\frac{d E}{d t}=m \\frac{d r}{d t} \\frac{d^{2} r}{d t^{2}}+2 m g k r \\frac{d r}{d t}\n$$\n\nwhich implies that\n\n$$\n0=\\frac{d^{2} r}{d t^{2}}+2 k r g\n$$\n\nwhich is the same equation as before.\n\nproblem:\nA particle is constrained to move on the inner surface of a frictionless parabolic bowl whose crosssection has equation $z=k r^{2}$. The particle begins at a height $z_{0}$ above the bottom of the bowl with a horizontal velocity $v_{0}$ along the surface of the bowl. The acceleration due to gravity is $g$.\n[figure1]\n\nSuppose that the particle now begins at a height $z_{0}$ above the bottom of the bowl with an initial velocity $v_{0}=0$.\n\nAssuming that $z_{0}$ is not small, will the actual period of motion be greater than, less than, or equal to your simple harmonic approximation above? (You need not calculate the new value explicitly, but you should show some work to defend your answer.)\n\nA: greater than\nB: less than\nC: equal to\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_3899026513eb55709c81g-13.jpg?height=392&width=1266&top_left_y=476&top_left_x=428" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_629", "problem": "A solid round object of radius $R$ can roll down an incline that makes an angle $\\theta$ with the horizontal. Assume that the rotational inertia about an axis through the center of mass is given by $I=\\beta m R^{2}$. The coefficient of kinetic and static friction between the object and the incline is $\\mu$. The object moves from rest through a vertical distance $h$.\n\nIf the angle of the incline is sufficiently large, then the object will slip and roll; if the angle of the incline is sufficiently small, then the object with roll without slipping. Determine the angle $\\theta_{c}$ that separates the two types of motion.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA solid round object of radius $R$ can roll down an incline that makes an angle $\\theta$ with the horizontal. Assume that the rotational inertia about an axis through the center of mass is given by $I=\\beta m R^{2}$. The coefficient of kinetic and static friction between the object and the incline is $\\mu$. The object moves from rest through a vertical distance $h$.\n\nIf the angle of the incline is sufficiently large, then the object will slip and roll; if the angle of the incline is sufficiently small, then the object with roll without slipping. Determine the angle $\\theta_{c}$ that separates the two types of motion.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_269", "problem": "[figure1]\n\nThe graph shows\nA: Current vs Potential difference for a fixed resistor\nB: Kinetic energy vs velocity for a fixed mass\nC: Pressure vs Volume at a constant temperature for a fixed mass of gas\nD: Mass of a beaker containing water vs volume of water in the beaker at a fixed temperature\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\n[figure1]\n\nThe graph shows\n\nA: Current vs Potential difference for a fixed resistor\nB: Kinetic energy vs velocity for a fixed mass\nC: Pressure vs Volume at a constant temperature for a fixed mass of gas\nD: Mass of a beaker containing water vs volume of water in the beaker at a fixed temperature\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ae10da91e53998b18280g-03.jpg?height=525&width=520&top_left_y=414&top_left_x=768" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_80", "problem": "The Joule (J) is a derived unit of energy in the International System of Units. It is equal to which of the following\nA: $\\operatorname{kg}\\left(\\frac{m}{s}\\right)$\nB: $k g\\left(\\frac{m}{s^{2}}\\right)$\nC: $k g\\left(\\frac{m^{2}}{s}\\right)$\nD: $\\operatorname{kg}\\left(\\frac{m^{2}}{s^{2}}\\right)$\nE: $\\operatorname{kg}\\left(\\frac{m^{2}}{s^{3}}\\right)$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe Joule (J) is a derived unit of energy in the International System of Units. It is equal to which of the following\n\nA: $\\operatorname{kg}\\left(\\frac{m}{s}\\right)$\nB: $k g\\left(\\frac{m}{s^{2}}\\right)$\nC: $k g\\left(\\frac{m^{2}}{s}\\right)$\nD: $\\operatorname{kg}\\left(\\frac{m^{2}}{s^{2}}\\right)$\nE: $\\operatorname{kg}\\left(\\frac{m^{2}}{s^{3}}\\right)$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1549", "problem": "温度开关用厚度均为 $0.20 \\mathrm{~mm}$ 的钢片和青铜片作感温元件; 在温度为 $20^{\\circ} \\mathrm{C}$ 时, 将它们紧贴, 两端焊接在一起, 成为等长的平直双金属片. 若钢和青铜的线膨胀系数分别为 $1.0 \\times 10^{-5} /$ 度和 $2.0 \\times 10^{5}$ /度. 当温度升高到 $120^{\\circ} \\mathrm{C}$ 时, 双金属片将自动弯成圆弧形,如图所示. 试求双金属片弯曲的曲率半径. (忽略加热时金属片厚度的变化. )\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n温度开关用厚度均为 $0.20 \\mathrm{~mm}$ 的钢片和青铜片作感温元件; 在温度为 $20^{\\circ} \\mathrm{C}$ 时, 将它们紧贴, 两端焊接在一起, 成为等长的平直双金属片. 若钢和青铜的线膨胀系数分别为 $1.0 \\times 10^{-5} /$ 度和 $2.0 \\times 10^{5}$ /度. 当温度升高到 $120^{\\circ} \\mathrm{C}$ 时, 双金属片将自动弯成圆弧形,如图所示. 试求双金属片弯曲的曲率半径. (忽略加热时金属片厚度的变化. )\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_734a14c1a8de5e19f8ecg-03.jpg?height=142&width=140&top_left_y=1925&top_left_x=1575" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_414", "problem": "An experimental vessel having a glass viewing window $10 \\mathrm{~cm} \\times 10 \\mathrm{~cm}$ is evacuated by a moderately efficient vacuum pump. What is the approximate force on the window caused by the pressure difference across it?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nAn experimental vessel having a glass viewing window $10 \\mathrm{~cm} \\times 10 \\mathrm{~cm}$ is evacuated by a moderately efficient vacuum pump. What is the approximate force on the window caused by the pressure difference across it?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{kN}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{kN}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_112", "problem": "A small hard solid sphere of mass $m$ and negligible radius is connected to a thin rod of length $L$ and mass $2 m$. A second small hard solid sphere, of mass $M$ and negligible radius, is fired perpendicularly at the rod at a distance $h$ above the sphere attached to the rod, and sticks to it.\n\n[figure1]\nIn order for the rod not to rotate after the collision, the second sphere should have a mass $M$ given by\nA: $M=m$\nB: $M=1.5 m$\nC: $M=2 m$\nD: $M=3 m$\nE: Any mass $M$ will work. \n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA small hard solid sphere of mass $m$ and negligible radius is connected to a thin rod of length $L$ and mass $2 m$. A second small hard solid sphere, of mass $M$ and negligible radius, is fired perpendicularly at the rod at a distance $h$ above the sphere attached to the rod, and sticks to it.\n\n[figure1]\nIn order for the rod not to rotate after the collision, the second sphere should have a mass $M$ given by\n\nA: $M=m$\nB: $M=1.5 m$\nC: $M=2 m$\nD: $M=3 m$\nE: Any mass $M$ will work. \n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f726c6cf4a23f08e0214g-06.jpg?height=355&width=501&top_left_y=446&top_left_x=801" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1737", "problem": "某同学在原地进行单手运球训练中发现, 让篮球从静止开始下落并自由反弹, 弹起的最大高度比原来低 $20 \\mathrm{~cm}$ 。为了让球每次都能弹回到原来的高度, 当球回到最高点时, 向下拍打一次球, 每分钟拍打 100 次, 篮球质量为 $600 \\mathrm{~g}$ 。取重力加速度大小为 $10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。不计空气阻力和拍球瞬间的能量损失, 则该同学每次拍打小球需做功为— $\\mathrm{J}$, 拍打小球的平均功率为— $\\mathrm{W}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n某同学在原地进行单手运球训练中发现, 让篮球从静止开始下落并自由反弹, 弹起的最大高度比原来低 $20 \\mathrm{~cm}$ 。为了让球每次都能弹回到原来的高度, 当球回到最高点时, 向下拍打一次球, 每分钟拍打 100 次, 篮球质量为 $600 \\mathrm{~g}$ 。取重力加速度大小为 $10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。不计空气阻力和拍球瞬间的能量损失, 则该同学每次拍打小球需做功为— $\\mathrm{J}$, 拍打小球的平均功率为— $\\mathrm{W}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[该同学每次拍打小球需做功, 该同学拍打小球的平均功率]\n它们的单位依次是[$\\mathrm{J}$, $\\mathrm{W}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{J}$", "$\\mathrm{W}$" ], "answer_sequence": [ "该同学每次拍打小球需做功", "该同学拍打小球的平均功率" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_755", "problem": "A leak \n\nA hollow insulated cylinder of height $2 \\mathrm{H}$ and volume $2 \\mathrm{~V}$ is closed from below by an insulating piston. The cylinder is divided into two initially identical chambers by an insulating diaphragm of mass $m$. The diaphragm rests on a circular ledge and a gasket between them provides tight contact. Both chambers are filled with gaseous helium at pressure $p$ and temperature $T$. A force is applied to the piston, so that it moves upwards slowly.\n\n[figure1]\n\nFind the volume of the lower chamber $V_{0}$ when the gas starts to leak between the chambers", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA leak \n\nA hollow insulated cylinder of height $2 \\mathrm{H}$ and volume $2 \\mathrm{~V}$ is closed from below by an insulating piston. The cylinder is divided into two initially identical chambers by an insulating diaphragm of mass $m$. The diaphragm rests on a circular ledge and a gasket between them provides tight contact. Both chambers are filled with gaseous helium at pressure $p$ and temperature $T$. A force is applied to the piston, so that it moves upwards slowly.\n\n[figure1]\n\nFind the volume of the lower chamber $V_{0}$ when the gas starts to leak between the chambers\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_5a5b6f5fd43c588c9548g-1.jpg?height=862&width=520&top_left_y=588&top_left_x=800" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_639", "problem": "A unicyclist of total height $h$ goes around a circular track of radius $R$ while leaning inward at an angle $\\theta$ to the vertical. The acceleration due to gravity is $g$.\n\nNow model the unicyclist as a uniform rod of length $h$, where $h$ is less than $R$ but not negligible. This refined model introduces a correction to the previous result. What is the new expression for the angular velocity $\\omega$ ? Assume that the rod remains in the plane formed by the vertical and radial directions, and that $R$ is measured from the center of the circle to the point of contact at the ground.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA unicyclist of total height $h$ goes around a circular track of radius $R$ while leaning inward at an angle $\\theta$ to the vertical. The acceleration due to gravity is $g$.\n\nNow model the unicyclist as a uniform rod of length $h$, where $h$ is less than $R$ but not negligible. This refined model introduces a correction to the previous result. What is the new expression for the angular velocity $\\omega$ ? Assume that the rod remains in the plane formed by the vertical and radial directions, and that $R$ is measured from the center of the circle to the point of contact at the ground.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_675", "problem": "Deuterium is the heavy stable isotope of hydrogen, having a proton and neutron in its nucleus (protons and neutrons have approximately the same mass). Heavy water is a form of water that contains Deuterium and Oxygen. A heavy water nuclear reactor has heavy water between the nuclear fuel rods. Suppose that a neutron from the fuel rod has a head-on elastic collision with a Dueterium nucleus.\n\nWhat percentage of the initial kinetic energy is transferred to the deuteron?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nDeuterium is the heavy stable isotope of hydrogen, having a proton and neutron in its nucleus (protons and neutrons have approximately the same mass). Heavy water is a form of water that contains Deuterium and Oxygen. A heavy water nuclear reactor has heavy water between the nuclear fuel rods. Suppose that a neutron from the fuel rod has a head-on elastic collision with a Dueterium nucleus.\n\nWhat percentage of the initial kinetic energy is transferred to the deuteron?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_94", "problem": "A rock that has a mass of $12 \\mathrm{~kg}$ is sliding on a rough, horizontal surface. It has $24 \\mathrm{~J}$ of kinetic energy and the friction force on it is a constant $0.50 \\mathrm{~N}$. What distance will it slide before coming to rest?\nA: $2.0 \\mathrm{~m}$\nB: $12 \\mathrm{~m}$\nC: $24 \\mathrm{~m}$\nD: $36 \\mathrm{~m}$\nE: $48 \\mathrm{~m}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA rock that has a mass of $12 \\mathrm{~kg}$ is sliding on a rough, horizontal surface. It has $24 \\mathrm{~J}$ of kinetic energy and the friction force on it is a constant $0.50 \\mathrm{~N}$. What distance will it slide before coming to rest?\n\nA: $2.0 \\mathrm{~m}$\nB: $12 \\mathrm{~m}$\nC: $24 \\mathrm{~m}$\nD: $36 \\mathrm{~m}$\nE: $48 \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_663", "problem": "A diver shines a beam of light from a laser at $80^{\\circ}$ to the surface as shown below. How high above the water surface will the laser beam hit the vertical wall placed 2 $\\mathrm{m}$ away from the point $\\mathrm{A}$ (the place where the laser beam hits the surface of the water)?\n\n[figure1]\nA: About $15 \\mathrm{~m}$\nB: About $8 \\mathrm{~m}$\nC: About $2 \\mathrm{~m}$\nD: About $0.2 \\mathrm{~m}$\nE: It will never go above the water\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA diver shines a beam of light from a laser at $80^{\\circ}$ to the surface as shown below. How high above the water surface will the laser beam hit the vertical wall placed 2 $\\mathrm{m}$ away from the point $\\mathrm{A}$ (the place where the laser beam hits the surface of the water)?\n\n[figure1]\n\nA: About $15 \\mathrm{~m}$\nB: About $8 \\mathrm{~m}$\nC: About $2 \\mathrm{~m}$\nD: About $0.2 \\mathrm{~m}$\nE: It will never go above the water\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22e26a14ee6fdd9254b6g-04.jpg?height=385&width=648&top_left_y=2168&top_left_x=1172" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_860", "problem": "Initially the external field is turned off. Then the field magnitude is increased from zero to $E_{0}$ very slowly so that the electric field can be considered effectively time-independent in this question. The instantaneous value of the external field is denoted by $\\vec{E}=E \\hat{u}$, Find the instantaneous power absorbed by the atom from the external field in $0.75 \\mathrm{pt}$ terms of $\\vec{E}$ and $\\dot{\\vec{p}}$, where $\\vec{p}$ is the rate change of the induced dipole moment.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nInitially the external field is turned off. Then the field magnitude is increased from zero to $E_{0}$ very slowly so that the electric field can be considered effectively time-independent in this question. The instantaneous value of the external field is denoted by $\\vec{E}=E \\hat{u}$, Find the instantaneous power absorbed by the atom from the external field in $0.75 \\mathrm{pt}$ terms of $\\vec{E}$ and $\\dot{\\vec{p}}$, where $\\vec{p}$ is the rate change of the induced dipole moment.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_149", "problem": "A block of mass $m$ is launched horizontally onto a curved wedge of mass $M$ at a velocity $v$. What is the maximum height reached by the block after it shoots off the vertical segment of the wedge? Assume all surfaces are frictionless; both the block and the curved wedge are free to move. The curved wedge does not tilt or topple.\n\n[figure1]\nA: $\\frac{v^{2}}{2 g}$\nB: $\\left(\\frac{m}{m+M}\\right)^{2} \\cdot \\frac{v^{2}}{2 g}$\nC: $\\left(\\frac{M}{m+M}\\right)^{2} \\cdot \\frac{v^{2}}{2 g}$\nD: $\\frac{m}{m+M} \\cdot \\frac{v^{2}}{2 g}$\nE: $\\frac{M}{m+M} \\cdot \\frac{v^{2}}{2 g} $ \n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA block of mass $m$ is launched horizontally onto a curved wedge of mass $M$ at a velocity $v$. What is the maximum height reached by the block after it shoots off the vertical segment of the wedge? Assume all surfaces are frictionless; both the block and the curved wedge are free to move. The curved wedge does not tilt or topple.\n\n[figure1]\n\nA: $\\frac{v^{2}}{2 g}$\nB: $\\left(\\frac{m}{m+M}\\right)^{2} \\cdot \\frac{v^{2}}{2 g}$\nC: $\\left(\\frac{M}{m+M}\\right)^{2} \\cdot \\frac{v^{2}}{2 g}$\nD: $\\frac{m}{m+M} \\cdot \\frac{v^{2}}{2 g}$\nE: $\\frac{M}{m+M} \\cdot \\frac{v^{2}}{2 g} $ \n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-03.jpg?height=309&width=786&top_left_y=434&top_left_x=667" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_986", "problem": "Let us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nAt some radius $r_{3} \\ll r_{0}$, the gas becomes dense enough to be opaque to the heat radiation. Calculate the amount of heat $Q$ radiated away during the collapse from the radius $r_{0}$ down to $r_{3}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nLet us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nAt some radius $r_{3} \\ll r_{0}$, the gas becomes dense enough to be opaque to the heat radiation. Calculate the amount of heat $Q$ radiated away during the collapse from the radius $r_{0}$ down to $r_{3}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_734", "problem": "Tom is riding a hot air balloon by himself. There is a rope ladder hanging from the top of the balloon and reaching into the basket. At a moment when the balloon is stationary in the air, Tom begins climbing the ladder with speed $s$ relative to the ladder. Suppose Tom has mass $m$, the balloon has mass $M$. What is the speed of the balloon relative to the ground if air resistance can be neglected?\nA: $s$\nB: $\\frac{m}{M} s$\nC: $\\frac{m}{M+m} s$\nD: $\\sqrt{\\frac{m}{M+m}} s$\nE: 0\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTom is riding a hot air balloon by himself. There is a rope ladder hanging from the top of the balloon and reaching into the basket. At a moment when the balloon is stationary in the air, Tom begins climbing the ladder with speed $s$ relative to the ladder. Suppose Tom has mass $m$, the balloon has mass $M$. What is the speed of the balloon relative to the ground if air resistance can be neglected?\n\nA: $s$\nB: $\\frac{m}{M} s$\nC: $\\frac{m}{M+m} s$\nD: $\\sqrt{\\frac{m}{M+m}} s$\nE: 0\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_769", "problem": "A large truck breaks down out on the road and receives assistance from a small compact car as shown in the figure below.\n\n[figure1]\n\nThe car driver attempts to push the truck with the car. Unfortunately, the truck driver has left the brakes on the truck, and neither vehicle moves. Why does the truck not move?\nA: pushing force of the truck on the car, but in the opposite direction.\nB: Because the pushing force of the car on the truck is less than the pushing force of the truck on the car.\nC: Because the frictional force of the ground on the truck is equal to the frictional force of the truck on the ground, but in the opposite direction.\nD: Because the frictional force of the ground on the truck is greater than the frictional force of the truck on the ground.\nE: Because the pushing force of the car on the truck is equal to the frictional force of the ground on the truck, but in the opposite direction.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA large truck breaks down out on the road and receives assistance from a small compact car as shown in the figure below.\n\n[figure1]\n\nThe car driver attempts to push the truck with the car. Unfortunately, the truck driver has left the brakes on the truck, and neither vehicle moves. Why does the truck not move?\n\nA: pushing force of the truck on the car, but in the opposite direction.\nB: Because the pushing force of the car on the truck is less than the pushing force of the truck on the car.\nC: Because the frictional force of the ground on the truck is equal to the frictional force of the truck on the ground, but in the opposite direction.\nD: Because the frictional force of the ground on the truck is greater than the frictional force of the truck on the ground.\nE: Because the pushing force of the car on the truck is equal to the frictional force of the ground on the truck, but in the opposite direction.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9bac30101a7c40be9bg-02.jpg?height=166&width=960&top_left_y=1456&top_left_x=591" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_185", "problem": "A uniform block of mass $10 \\mathrm{~kg}$ is released at rest from the top of an incline of length $10 \\mathrm{~m}$ and inclination $30^{\\circ}$. The coefficients of static and kinetic friction between the incline and the block are $\\mu_{s}=0.15$ and $\\mu_{k}=0.1$. The end of the incline is connected to a frictionless horizontal surface. After a long time, how much energy is dissipated due to friction?\nA: $75 \\mathrm{~J}$\nB: $87 \\mathrm{~J} $\nC: $130 \\mathrm{~J}$\nD: $147 \\mathrm{~J}$\nE: $500 \\mathrm{~J}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA uniform block of mass $10 \\mathrm{~kg}$ is released at rest from the top of an incline of length $10 \\mathrm{~m}$ and inclination $30^{\\circ}$. The coefficients of static and kinetic friction between the incline and the block are $\\mu_{s}=0.15$ and $\\mu_{k}=0.1$. The end of the incline is connected to a frictionless horizontal surface. After a long time, how much energy is dissipated due to friction?\n\nA: $75 \\mathrm{~J}$\nB: $87 \\mathrm{~J} $\nC: $130 \\mathrm{~J}$\nD: $147 \\mathrm{~J}$\nE: $500 \\mathrm{~J}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1564", "problem": "如示意图所示, 一垂直放置的高为 $15.0 \\mathrm{~cm}$ 的圆柱形中空玻璃容器, 其底部玻璃较厚, 底部顶点 $\\mathrm{A}$ 点到容器底平面中心 $\\mathrm{B}$ 点的距离为 $8.0 \\mathrm{~cm}$, 底部上沿为一凸起的球冠, 球心 $\\mathrm{C}$ 点在 $\\mathrm{A}$ 点正下方, 球的半径为 $1.75 \\mathrm{~cm}$ 。已知空气和容器玻璃的折射率分别是 $n_{0}=1.0$ 和 $n_{1}=1.56$ 。只考虑近轴光线成像。已知: 当 $\\lambda<<1$ 时, $\\sin \\lambda \\approx \\lambda$ 。\n\n[图1]当容器内未装任何液体时, 求从 B 点发出的光线通过平凸玻璃柱，在玻璃柱对称轴上所成的像的位置，并判断像的虚实", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如示意图所示, 一垂直放置的高为 $15.0 \\mathrm{~cm}$ 的圆柱形中空玻璃容器, 其底部玻璃较厚, 底部顶点 $\\mathrm{A}$ 点到容器底平面中心 $\\mathrm{B}$ 点的距离为 $8.0 \\mathrm{~cm}$, 底部上沿为一凸起的球冠, 球心 $\\mathrm{C}$ 点在 $\\mathrm{A}$ 点正下方, 球的半径为 $1.75 \\mathrm{~cm}$ 。已知空气和容器玻璃的折射率分别是 $n_{0}=1.0$ 和 $n_{1}=1.56$ 。只考虑近轴光线成像。已知: 当 $\\lambda<<1$ 时, $\\sin \\lambda \\approx \\lambda$ 。\n\n[图1]\n\n问题:\n当容器内未装任何液体时, 求从 B 点发出的光线通过平凸玻璃柱，在玻璃柱对称轴上所成的像的位置，并判断像的虚实\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[像的虚实, 所成的像的位置]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-06.jpg?height=574&width=394&top_left_y=1572&top_left_x=1368", "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-13.jpg?height=589&width=1367&top_left_y=231&top_left_x=353" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "像的虚实", "所成的像的位置" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_172", "problem": "A mass is attached to one end of a rigid rod, while the other end of the rod is attached to a fixed horizontal axle. Initially the mass hangs at the end of the rod and the rod is vertical. The mass is given an initial kinetic energy $K$. If $K$ is very small, the mass behaves like a pendulum, performing small-angle oscillations with period $T_{0}$. As $K$ is increased, the period of the motion for the mass\nA: remains the same.\nB: increases, approaching a finite constant.\nC: decreases, approaching a finite non-zero constant.\nD: decreases, approaching zero.\nE: initially increases, then decreases.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA mass is attached to one end of a rigid rod, while the other end of the rod is attached to a fixed horizontal axle. Initially the mass hangs at the end of the rod and the rod is vertical. The mass is given an initial kinetic energy $K$. If $K$ is very small, the mass behaves like a pendulum, performing small-angle oscillations with period $T_{0}$. As $K$ is increased, the period of the motion for the mass\n\nA: remains the same.\nB: increases, approaching a finite constant.\nC: decreases, approaching a finite non-zero constant.\nD: decreases, approaching zero.\nE: initially increases, then decreases.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_492", "problem": "Radiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe Poynting-Robertson effect acts as another mechanism for cleaning out the solar system.\n\nBecause the particle is moving, the radiation force is not directed directly away from the sun. Find the torque $\\tau$ on the particle because of radiation pressure. You may assume that $v \\ll c$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nRadiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe Poynting-Robertson effect acts as another mechanism for cleaning out the solar system.\n\nBecause the particle is moving, the radiation force is not directed directly away from the sun. Find the torque $\\tau$ on the particle because of radiation pressure. You may assume that $v \\ll c$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_186", "problem": "A mass $m$ hangs from a massless spring connected to the roof of a box of mass $M$. When the box is held stationary, the mass-spring system oscillates vertically with angular frequency $\\omega$. If the box is dropped falls freely under gravity, how will the angular frequency change?\nA: $\\omega$ will be unchanged\nB: $\\omega$ will increase \nC: $\\omega$ will decrease\nD: Oscillations are impossible under these conditions\nE: $\\omega$ will increase or decrease depending on the values of $M$ and $m$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA mass $m$ hangs from a massless spring connected to the roof of a box of mass $M$. When the box is held stationary, the mass-spring system oscillates vertically with angular frequency $\\omega$. If the box is dropped falls freely under gravity, how will the angular frequency change?\n\nA: $\\omega$ will be unchanged\nB: $\\omega$ will increase \nC: $\\omega$ will decrease\nD: Oscillations are impossible under these conditions\nE: $\\omega$ will increase or decrease depending on the values of $M$ and $m$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1377", "problem": "图 a 是基于全内反射原理制备的折射率阶跃型光纤及其耦合光路示意图. 光纤内芯直径 $50 \\mu \\mathrm{m}$,折射率 $\\mathrm{n}_{1}=1.46$; 它的玻璃外包层的外径为 $125 \\mu \\mathrm{m}$, 折射率 $\\mathrm{n}_{2}=1.45$. 氦氛激光器输出一圆柱形平行光束, 为了将该激光束有效地耦合进入光纤传输, 可以在光纤前端放置一微球透镜进行聚焦和耦合, 微球透镜的直径 $\\mathrm{D}=3.00 \\mathrm{~mm}$, 折射率 $\\mathrm{n}=1.50$. 已知激光束中心轴通过微球透镜中心, 且与光纤对称轴重合. 空气折射率 $\\mathrm{n}_{0}=1.00$.为了使光线能在光纤内长距离传输, 在光纤端面处光线的最大入射角应为多大?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n图 a 是基于全内反射原理制备的折射率阶跃型光纤及其耦合光路示意图. 光纤内芯直径 $50 \\mu \\mathrm{m}$,折射率 $\\mathrm{n}_{1}=1.46$; 它的玻璃外包层的外径为 $125 \\mu \\mathrm{m}$, 折射率 $\\mathrm{n}_{2}=1.45$. 氦氛激光器输出一圆柱形平行光束, 为了将该激光束有效地耦合进入光纤传输, 可以在光纤前端放置一微球透镜进行聚焦和耦合, 微球透镜的直径 $\\mathrm{D}=3.00 \\mathrm{~mm}$, 折射率 $\\mathrm{n}=1.50$. 已知激光束中心轴通过微球透镜中心, 且与光纤对称轴重合. 空气折射率 $\\mathrm{n}_{0}=1.00$.\n\n问题:\n为了使光线能在光纤内长距离传输, 在光纤端面处光线的最大入射角应为多大?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$^{\\circ}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-25.jpg?height=200&width=708&top_left_y=1025&top_left_x=685" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$^{\\circ}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_344", "problem": "Consider a ball of monoatomic ideal gas at temperature $T$ which keeps a spherically symmetric mech anically stable stationary shape due to its own gravity field. Let the total mass of the gas be $\\boldsymbol{M}_{0}$, and its molar mass $\\boldsymbol{\\mu}$. We shall de scribe the radial mass distribution in terms of the total mass $M=M(r)$ inside a sphere of radius $r$, and the pressure $p=p(r)$ as a function of distance $r$ from the centre of the spherical gas cloud.\n\nConsider now a similar fully ionized plasma cloud. Assume that plasma is a macroscopically neutral mixture of electrons and protons. What is the proportional ity factor between the total gravitational energy and total thermal energy for the plasma cloud?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nConsider a ball of monoatomic ideal gas at temperature $T$ which keeps a spherically symmetric mech anically stable stationary shape due to its own gravity field. Let the total mass of the gas be $\\boldsymbol{M}_{0}$, and its molar mass $\\boldsymbol{\\mu}$. We shall de scribe the radial mass distribution in terms of the total mass $M=M(r)$ inside a sphere of radius $r$, and the pressure $p=p(r)$ as a function of distance $r$ from the centre of the spherical gas cloud.\n\nConsider now a similar fully ionized plasma cloud. Assume that plasma is a macroscopically neutral mixture of electrons and protons. What is the proportional ity factor between the total gravitational energy and total thermal energy for the plasma cloud?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_488", "problem": "Part A\n\nA \"Gilbert\" dipole consists of a pair of magnetic monopoles each with a magnitude $q_{m}$ but opposite magnetic charges separated by a distance $d$, where $d$ is small. In this case, assume that $-q_{m}$ is located at $z=0$ and $+q_{m}$ is located at $z=d$.\n\n[figure1]\n\nAssume that magnetic monopoles behave like electric monopoles according to a coulomb-like force\n\nand the magnetic field obeys\n\n$$\nF=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m 1} q_{m 2}}{r^{2}}\n$$\n\n$$\nB=F / q_{m} .\n$$\n\nBy the second expression, $q_{m}$ must be measured in Newtons per Tesla. But since Tesla are also Newtons per Ampere per meter, then $q_{m}$ is also measured in Ampere meters.\n\nAdding the two terms,\n\n$$\nB(z)=-\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{z^{2}}+\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{(z+d)^{2}}\n$$\n\nEvaluate this expression in the limit as $d \\rightarrow 0$, assuming that the product $q_{m} d=p_{m}$ is kept constant, keeping only the lowest non-zero term.\n\nSimplifying our previous expression,\n\n$$\nB(z)=\\frac{\\mu_{0}}{4 \\pi} q_{m} d\\left(\\frac{2+d / z}{z(z+d)^{2}}\\right) \\text {. }\n$$\n\nThus in the limit $d \\rightarrow 0$ we have\n\n$$\nB(z)=\\frac{\\mu_{0}}{2 \\pi} \\frac{q_{m} d}{z^{3}}=\\frac{\\mu_{0}}{2 \\pi} \\frac{p_{m}}{z^{3}}\n$$An \"Ampre\" dipole is a magnetic dipole produced by a current loop $I$ around a circle of radius $r$, where $r$ is small. Assume the that the $z$ axis is the axis of rotational symmetry for the circular loop, and the loop lies in the $x y$ plane at $z=0$.\n\n[figure2]\n\nLet $k I r^{\\gamma}$ have dimensions equal to that of the quantity $p_{m}$ defined above in Part A, where $k$ and $\\gamma$ are dimensionless constants. Determine the value of $\\gamma$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nPart A\n\nA \"Gilbert\" dipole consists of a pair of magnetic monopoles each with a magnitude $q_{m}$ but opposite magnetic charges separated by a distance $d$, where $d$ is small. In this case, assume that $-q_{m}$ is located at $z=0$ and $+q_{m}$ is located at $z=d$.\n\n[figure1]\n\nAssume that magnetic monopoles behave like electric monopoles according to a coulomb-like force\n\nand the magnetic field obeys\n\n$$\nF=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m 1} q_{m 2}}{r^{2}}\n$$\n\n$$\nB=F / q_{m} .\n$$\n\nBy the second expression, $q_{m}$ must be measured in Newtons per Tesla. But since Tesla are also Newtons per Ampere per meter, then $q_{m}$ is also measured in Ampere meters.\n\nAdding the two terms,\n\n$$\nB(z)=-\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{z^{2}}+\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{(z+d)^{2}}\n$$\n\nEvaluate this expression in the limit as $d \\rightarrow 0$, assuming that the product $q_{m} d=p_{m}$ is kept constant, keeping only the lowest non-zero term.\n\nSimplifying our previous expression,\n\n$$\nB(z)=\\frac{\\mu_{0}}{4 \\pi} q_{m} d\\left(\\frac{2+d / z}{z(z+d)^{2}}\\right) \\text {. }\n$$\n\nThus in the limit $d \\rightarrow 0$ we have\n\n$$\nB(z)=\\frac{\\mu_{0}}{2 \\pi} \\frac{q_{m} d}{z^{3}}=\\frac{\\mu_{0}}{2 \\pi} \\frac{p_{m}}{z^{3}}\n$$\n\nproblem:\nAn \"Ampre\" dipole is a magnetic dipole produced by a current loop $I$ around a circle of radius $r$, where $r$ is small. Assume the that the $z$ axis is the axis of rotational symmetry for the circular loop, and the loop lies in the $x y$ plane at $z=0$.\n\n[figure2]\n\nLet $k I r^{\\gamma}$ have dimensions equal to that of the quantity $p_{m}$ defined above in Part A, where $k$ and $\\gamma$ are dimensionless constants. Determine the value of $\\gamma$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-18.jpg?height=147&width=653&top_left_y=569&top_left_x=777", "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-19.jpg?height=296&width=653&top_left_y=421&top_left_x=779" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_746", "problem": "Three balls \n\nThree small identical balls (denoted as A, B, and C) of mass m each are connected with two massless rods of length $l$ so that one of the rods connects the balls $\\mathrm{A}$ and $\\mathrm{B}$, and the other rod connects the balls $\\mathrm{B}$ and $\\mathrm{C}$. The connection at the ball $\\mathrm{B}$ is hinged, and the angle between the rods can change effortlessly. The system rests in weightlessness so that all the balls lie on one line. The ball $\\mathrm{A}$ is given instantaneously a velocity perpendicular to the rods. Find the minimal distance $\\mathrm{d}$ between the balls $\\mathrm{A}$ and $\\mathrm{C}$ during the subsequent motion of the system. Any friction is to be neglected.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThree balls \n\nThree small identical balls (denoted as A, B, and C) of mass m each are connected with two massless rods of length $l$ so that one of the rods connects the balls $\\mathrm{A}$ and $\\mathrm{B}$, and the other rod connects the balls $\\mathrm{B}$ and $\\mathrm{C}$. The connection at the ball $\\mathrm{B}$ is hinged, and the angle between the rods can change effortlessly. The system rests in weightlessness so that all the balls lie on one line. The ball $\\mathrm{A}$ is given instantaneously a velocity perpendicular to the rods. Find the minimal distance $\\mathrm{d}$ between the balls $\\mathrm{A}$ and $\\mathrm{C}$ during the subsequent motion of the system. Any friction is to be neglected.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_111", "problem": "A uniform solid right prism whose cross section is an isosceles right triangle with height $h$ and width $w=2 h$ is placed on an incline that has a variable angle with the horizontal $\\theta$. What is the minimum coefficient of static friction so that the prism topples before it begins sliding as $\\theta$ is slowly increased from zero?\n\n[figure1]\nA: 0.71\nB: 1.41\nC: 1.50\nD: 1.73\nE: 3.00 \n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA uniform solid right prism whose cross section is an isosceles right triangle with height $h$ and width $w=2 h$ is placed on an incline that has a variable angle with the horizontal $\\theta$. What is the minimum coefficient of static friction so that the prism topples before it begins sliding as $\\theta$ is slowly increased from zero?\n\n[figure1]\n\nA: 0.71\nB: 1.41\nC: 1.50\nD: 1.73\nE: 3.00 \n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-22.jpg?height=244&width=399&top_left_y=488&top_left_x=863" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_427", "problem": "In the future, as the miniaturisation of microprocessors continues, an obvious limit to the technology is when the size of one active unit in a device approaches the size of an atom. We will explore, speculatively, an aspect of passing a current into and out of the structure of graphene.\n\nThe following Figures represent part of an infinite graphene layer for which we wish to find the measured resistance between a pair of atoms located at points $\\mathbf{A}$ and $\\mathbf{B}$. Each straight line (bond) between two nodes represents a resistance $R$.\n\nFigure: The single layer hexagonal structure of graphene.\n(a)\n[figure1]\n\n(b)\n[figure2]\n\n(c)\n[figure3]\n\nConsidering for the moment Fig. 6(a) in which an external wire at potential $V$ is attached to $\\mathbf{A}$, so that a current $I$ flows into $\\mathbf{A}$, and then flows out into the wider network, eventually being dissipated to $\\infty$ where the potential is zero. State the values of the three currents flowing away from $\\mathbf{A}$ in terms of $I$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn the future, as the miniaturisation of microprocessors continues, an obvious limit to the technology is when the size of one active unit in a device approaches the size of an atom. We will explore, speculatively, an aspect of passing a current into and out of the structure of graphene.\n\nThe following Figures represent part of an infinite graphene layer for which we wish to find the measured resistance between a pair of atoms located at points $\\mathbf{A}$ and $\\mathbf{B}$. Each straight line (bond) between two nodes represents a resistance $R$.\n\nFigure: The single layer hexagonal structure of graphene.\n(a)\n[figure1]\n\n(b)\n[figure2]\n\n(c)\n[figure3]\n\nConsidering for the moment Fig. 6(a) in which an external wire at potential $V$ is attached to $\\mathbf{A}$, so that a current $I$ flows into $\\mathbf{A}$, and then flows out into the wider network, eventually being dissipated to $\\infty$ where the potential is zero. State the values of the three currents flowing away from $\\mathbf{A}$ in terms of $I$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-7.jpg?height=360&width=366&top_left_y=188&top_left_x=434", "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-7.jpg?height=363&width=366&top_left_y=187&top_left_x=868", "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-7.jpg?height=365&width=349&top_left_y=183&top_left_x=1276" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_163", "problem": "A steel ball bearing bounces vertically on a steel plate. If the speed of the ball just before a bounce is $v_{i}$, the speed of the ball immediately afterward is $v_{f}=\\alpha v_{i}$, with $\\alpha<1$. Which one of the following graphs best shows the time between successive bounces, $\\tau$, as a function of time?\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA steel ball bearing bounces vertically on a steel plate. If the speed of the ball just before a bounce is $v_{i}$, the speed of the ball immediately afterward is $v_{f}=\\alpha v_{i}$, with $\\alpha<1$. Which one of the following graphs best shows the time between successive bounces, $\\tau$, as a function of time?\n\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-10.jpg?height=448&width=548&top_left_y=459&top_left_x=317", "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-10.jpg?height=449&width=550&top_left_y=1044&top_left_x=316", "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-10.jpg?height=450&width=547&top_left_y=455&top_left_x=1128", "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-10.jpg?height=453&width=550&top_left_y=1042&top_left_x=1129", "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-10.jpg?height=442&width=548&top_left_y=1633&top_left_x=317" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_63", "problem": "Two quantities are found to be inversely proportional to one another. When analyzing this relationship, a graph between $y$ and $\\frac{1}{x}$ is plotted. This graph would show:\nA: a straight line bisecting the $y$-axis.\nB: a straight line through the origin.\nC: a parabolic curve.\nD: a hyperbolic curve.\nE: a straight line bisecting the $\\frac{1}{x}$-axis.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo quantities are found to be inversely proportional to one another. When analyzing this relationship, a graph between $y$ and $\\frac{1}{x}$ is plotted. This graph would show:\n\nA: a straight line bisecting the $y$-axis.\nB: a straight line through the origin.\nC: a parabolic curve.\nD: a hyperbolic curve.\nE: a straight line bisecting the $\\frac{1}{x}$-axis.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c737821a79d87eb20c9cg-3.jpg?height=224&width=528&top_left_y=1924&top_left_x=1188" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1555", "problem": "某根水平固定的滑竿上有 $n(n \\geq 3)$ 个质量均为 $m$ 的相同滑扣（即可以滑动的圆环），每相邻的两个滑扣（极薄)之间有不可伸长的柔软轻质细线相连, 细线长度均为 $L$, 滑扣在滑竿上滑行的阻力恒为滑扣对滑竿正压力的 $\\mu$ 倍。开始时所有滑扣可近似地看成挨在一起（但未相互挤压）; 当给第 1 个滑扣一个初速度使其在滑等上开始滑行 (平动) ; 在滑扣滑行的过程中, 前、后滑扣之间的细线拉紧后都以共同的速度向前滑行, 但最后一个 (即第 $n$ 个) 滑扣固定在滑竿边缘。已知从第 1 个滑扣开始的 $(n-1)$ 个滑扣都依次拉紧, 继续滑行距离 $l(0 \\leq lF_{B}>F_{C}$\nB: $F_{A}>F_{C}>F_{B}$\nC: $F_{B}>F_{C}>F_{A}$\nD: $F_{C}>F_{A}>F_{B} $ \nE: $F_{A}=F_{B}=F_{C}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nFlasks A, B, and $\\mathrm{C}$ each have a circular base with a radius of $2 \\mathrm{~cm}$. An equal volume of water is poured into each flask, and none overflow. Rank the force of water $F$ on the base of the flask from greatest to least.\n\n[figure1]\n\nA: $F_{A}>F_{B}>F_{C}$\nB: $F_{A}>F_{C}>F_{B}$\nC: $F_{B}>F_{C}>F_{A}$\nD: $F_{C}>F_{A}>F_{B} $ \nE: $F_{A}=F_{B}=F_{C}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f726c6cf4a23f08e0214g-05.jpg?height=223&width=607&top_left_y=390&top_left_x=759" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1707", "problem": "$\\alpha$ 粒子和 $\\beta$ 粒子都沿垂直于磁场方向射入同一均匀磁场中, 发现这两种粒子沿相同半径的圆轨道运动。若 $\\alpha$ 粒子的质量为 $m_{1} 、 \\beta$ 粒子的是质量为 $m_{2}$, 则 $\\alpha$ 粒子与 $\\beta$ 粒子的动能之比（用 $m_{1}$ 和 $m_{2}$ 表示）是[ ]\nA: $\\frac{m_{2}}{m_{1}}$\nB: $\\frac{m_{1}}{m_{2}}$\nC: $\\frac{m_{1}}{4 m_{2}}$\nD: $\\frac{4 m_{2}}{m_{1}}$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n$\\alpha$ 粒子和 $\\beta$ 粒子都沿垂直于磁场方向射入同一均匀磁场中, 发现这两种粒子沿相同半径的圆轨道运动。若 $\\alpha$ 粒子的质量为 $m_{1} 、 \\beta$ 粒子的是质量为 $m_{2}$, 则 $\\alpha$ 粒子与 $\\beta$ 粒子的动能之比（用 $m_{1}$ 和 $m_{2}$ 表示）是[ ]\n\nA: $\\frac{m_{2}}{m_{1}}$\nB: $\\frac{m_{1}}{m_{2}}$\nC: $\\frac{m_{1}}{4 m_{2}}$\nD: $\\frac{4 m_{2}}{m_{1}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1279", "problem": "某同学用电荷量计（能测出一段时间内通过导体横截面的电荷量）测量地磁场强度, 完成了如下实验：如图，将面积为 $S$ 、电阻为 $R$ 的矩形导线框 abcd 沿图示方位水平放置于地面上某处, 将其从图示位置绕东西轴转 $180^{\\circ}$, 测得通过线框的电荷量为 $Q_{1}$; 将其从图示位置绕东西轴转 $90^{\\circ}$, 测得通过线框的电荷量为 $Q_{2}$ 。该处地磁场的磁感应强度大小为\n\n[图1]\nA: $\\frac{R}{S} \\sqrt{\\frac{Q_{1}^{2}}{4}+Q_{2}^{2}}$\nB: $\\frac{R}{S} \\sqrt{Q_{1}^{2}+Q_{2}^{2}}$\nC: $\\frac{R}{S} \\sqrt{\\frac{Q_{1}^{2}}{2}+Q_{1} Q_{2}+Q_{2}^{2}}$\nD: $\\frac{R}{S} \\sqrt{Q_{1}^{2}+Q_{1} Q_{2}+Q_{2}^{2}}$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n某同学用电荷量计（能测出一段时间内通过导体横截面的电荷量）测量地磁场强度, 完成了如下实验：如图，将面积为 $S$ 、电阻为 $R$ 的矩形导线框 abcd 沿图示方位水平放置于地面上某处, 将其从图示位置绕东西轴转 $180^{\\circ}$, 测得通过线框的电荷量为 $Q_{1}$; 将其从图示位置绕东西轴转 $90^{\\circ}$, 测得通过线框的电荷量为 $Q_{2}$ 。该处地磁场的磁感应强度大小为\n\n[图1]\n\nA: $\\frac{R}{S} \\sqrt{\\frac{Q_{1}^{2}}{4}+Q_{2}^{2}}$\nB: $\\frac{R}{S} \\sqrt{Q_{1}^{2}+Q_{2}^{2}}$\nC: $\\frac{R}{S} \\sqrt{\\frac{Q_{1}^{2}}{2}+Q_{1} Q_{2}+Q_{2}^{2}}$\nD: $\\frac{R}{S} \\sqrt{Q_{1}^{2}+Q_{1} Q_{2}+Q_{2}^{2}}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-02.jpg?height=280&width=454&top_left_y=1690&top_left_x=1212" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1630", "problem": "电子偶素原子 (Ps ) 是由电子 $\\mathrm{e}^{-}$与正电子 $\\mathrm{e}^{+}$(电子的反粒子, 其质量与电子的相同,电荷与电子的大小相等、符号相反) 组成的量子束缚体系, 其能级可类比氢原子能级得出。根据玻尔氢原子理论, 电子绕质子的圆周运动轨道角动量的取值是量子化的, 即为 $\\hbar$ 的整数倍。考虑到质子质量是有限的, 氢原子量子化条件应修正为: 电子与质子质心系中相对其质心的总轨道角动量取值为 $\\hbar$ 的整数倍。这一量子化条件可直接推广到其它两体束缚体系, 如电子偶素等。以下计算结果均保留四位有效数字。\n\n电子偶素基态原子不稳定, 可很快发生湮没而生成两个光子:\n\n$$\n\\mathrm{Ps} \\rightarrow \\gamma_{1}+\\gamma_{2}\n$$\n\n当基态电子偶素原子相对于实验室参照系以远小于光速的某速度运动时发生湮没, 在相对于该速度方向的偏角为 $\\theta_{1}$ 的方向上观测到生成的一个光子 $\\gamma_{1}$, 同时在相对于光子 $\\gamma_{1}$ 速度反方向的偏角为 $\\Delta \\theta(\\Delta \\theta<<1)$ 的方向上观测到\n\n[图1]\n另一个光子 $\\gamma_{2}$, 如图所示。\n\n已知: 氢原子基态能量 $E_{n=1}^{\\mathrm{H}}=-13.60 \\mathrm{eV}$, 电子质量 $m_{\\mathrm{e}}=0.5110 \\mathrm{MeV} / c^{2}$, 质子与电子的质量之比为 1836 。当基态电子偶素原子静止时, 求两个光子 $\\gamma_{1} 、 \\gamma_{2}$ 各自的能量 $E_{1} 、 E_{2}$ 与单个自由电子静能之差。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n电子偶素原子 (Ps ) 是由电子 $\\mathrm{e}^{-}$与正电子 $\\mathrm{e}^{+}$(电子的反粒子, 其质量与电子的相同,电荷与电子的大小相等、符号相反) 组成的量子束缚体系, 其能级可类比氢原子能级得出。根据玻尔氢原子理论, 电子绕质子的圆周运动轨道角动量的取值是量子化的, 即为 $\\hbar$ 的整数倍。考虑到质子质量是有限的, 氢原子量子化条件应修正为: 电子与质子质心系中相对其质心的总轨道角动量取值为 $\\hbar$ 的整数倍。这一量子化条件可直接推广到其它两体束缚体系, 如电子偶素等。以下计算结果均保留四位有效数字。\n\n电子偶素基态原子不稳定, 可很快发生湮没而生成两个光子:\n\n$$\n\\mathrm{Ps} \\rightarrow \\gamma_{1}+\\gamma_{2}\n$$\n\n当基态电子偶素原子相对于实验室参照系以远小于光速的某速度运动时发生湮没, 在相对于该速度方向的偏角为 $\\theta_{1}$ 的方向上观测到生成的一个光子 $\\gamma_{1}$, 同时在相对于光子 $\\gamma_{1}$ 速度反方向的偏角为 $\\Delta \\theta(\\Delta \\theta<<1)$ 的方向上观测到\n\n[图1]\n另一个光子 $\\gamma_{2}$, 如图所示。\n\n已知: 氢原子基态能量 $E_{n=1}^{\\mathrm{H}}=-13.60 \\mathrm{eV}$, 电子质量 $m_{\\mathrm{e}}=0.5110 \\mathrm{MeV} / c^{2}$, 质子与电子的质量之比为 1836 。\n\n问题:\n当基态电子偶素原子静止时, 求两个光子 $\\gamma_{1} 、 \\gamma_{2}$ 各自的能量 $E_{1} 、 E_{2}$ 与单个自由电子静能之差。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{eV}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-06.jpg?height=437&width=437&top_left_y=775&top_left_x=1278" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{eV}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_885", "problem": "When modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nFind the capacitance per unit length, $C_{x}$, of the co-axial cable. You may wish to $0.3 \\mathrm{pt}$ consider a length $\\Delta x$ of the cable.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nWhen modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nFind the capacitance per unit length, $C_{x}$, of the co-axial cable. You may wish to $0.3 \\mathrm{pt}$ consider a length $\\Delta x$ of the cable.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-1.jpg?height=358&width=844&top_left_y=1688&top_left_x=617" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_32", "problem": "A curious physics student drops a rock from a cliff that is $5.0 \\mathrm{~m}$ above a pond. The rock hits the water with velocity $v$ and then sinks to the bottom with the constant velocity $v$. It reaches the bottom of the pond $5.0 \\mathrm{~s}$ after it is dropped. Find the depth of the lake\nA: $10 \\mathrm{~m}$\nB: $20 \\mathrm{~m}$\nC: $30 \\mathrm{~m}$\nD: $40 \\mathrm{~m}$\nE: $50 \\mathrm{~m}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA curious physics student drops a rock from a cliff that is $5.0 \\mathrm{~m}$ above a pond. The rock hits the water with velocity $v$ and then sinks to the bottom with the constant velocity $v$. It reaches the bottom of the pond $5.0 \\mathrm{~s}$ after it is dropped. Find the depth of the lake\n\nA: $10 \\mathrm{~m}$\nB: $20 \\mathrm{~m}$\nC: $30 \\mathrm{~m}$\nD: $40 \\mathrm{~m}$\nE: $50 \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_805", "problem": "Trish is moving boxes of photocopy paper on a trolley. The top of the trolley is flat, and a box sits on it as shown. Trish pushes the trolley, accelerating it to the left, as shown. Which force, if any, is causing the box to accelerate? Ignore air resistance.\n\n[figure1]\nA: Applied force\nB: Friction force\nC: Gravitational force\nD: Normal force\nE: None - no force is required\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTrish is moving boxes of photocopy paper on a trolley. The top of the trolley is flat, and a box sits on it as shown. Trish pushes the trolley, accelerating it to the left, as shown. Which force, if any, is causing the box to accelerate? Ignore air resistance.\n\n[figure1]\n\nA: Applied force\nB: Friction force\nC: Gravitational force\nD: Normal force\nE: None - no force is required\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_fa5ea5309158b0c5d169g-06.jpg?height=480&width=642&top_left_y=405&top_left_x=707" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_160", "problem": "Three identical masses are connected with identical rigid rods and pivoted at point $A$. If the lowest mass receives a small horizontal push to the left, it oscillates with period $T_{1}$. If it instead receives a small push into the page, it oscillates with period $T_{2}$. The ratio $T_{1} / T_{2}$ is\n\n[figure1]\nA: $1 / 2$\nB: 1\nC: $\\sqrt{3} $ \nD: $2 \\sqrt{2}$\nE: $2 \\sqrt{5}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThree identical masses are connected with identical rigid rods and pivoted at point $A$. If the lowest mass receives a small horizontal push to the left, it oscillates with period $T_{1}$. If it instead receives a small push into the page, it oscillates with period $T_{2}$. The ratio $T_{1} / T_{2}$ is\n\n[figure1]\n\nA: $1 / 2$\nB: 1\nC: $\\sqrt{3} $ \nD: $2 \\sqrt{2}$\nE: $2 \\sqrt{5}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-12.jpg?height=298&width=464&top_left_y=409&top_left_x=836" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_352", "problem": "The power radiated in gravitational waves by an orbiting binary system is given by $P\\left(r, m_{1}, m_{2}\\right)=\\frac{32}{5} \\frac{G^{4}}{c^{5}} \\frac{\\left(m_{1} m_{2}\\right)^{2}\\left(m_{1}+m_{2}\\right)}{r^{5}}$ where $r$ is the distance between the centers of the two orbiting masses $m_{1}$ and $m_{2}$. It is known that the most compact object is a black hole. The size of a black hole is defined by its Schwarzschild radius $r_{s}=\\frac{2 G m}{c^{2}}$, where $m$ is the mass of\n\n Estimate the maximum distance $z=z(\\varepsilon, f)$ from the Earth to the gravitational wave source as a function of the strain $\\varepsilon$ and frequency $f$. Use the models derived previously in this problem.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe power radiated in gravitational waves by an orbiting binary system is given by $P\\left(r, m_{1}, m_{2}\\right)=\\frac{32}{5} \\frac{G^{4}}{c^{5}} \\frac{\\left(m_{1} m_{2}\\right)^{2}\\left(m_{1}+m_{2}\\right)}{r^{5}}$ where $r$ is the distance between the centers of the two orbiting masses $m_{1}$ and $m_{2}$. It is known that the most compact object is a black hole. The size of a black hole is defined by its Schwarzschild radius $r_{s}=\\frac{2 G m}{c^{2}}$, where $m$ is the mass of\n\n Estimate the maximum distance $z=z(\\varepsilon, f)$ from the Earth to the gravitational wave source as a function of the strain $\\varepsilon$ and frequency $f$. Use the models derived previously in this problem.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_266", "problem": "Consider the circuit shown. Each of the fixed resistors has a value of $10 \\Omega$.\n\nA current of 0.6 A flows through resistor R3.\n\n[figure1]\n\nThe total current flowing through the battery is:\nA: $\\quad 0.6 \\mathrm{~A}$\nB: $\\quad 0.9 \\mathrm{~A}$\nC: $\\quad 1.2 \\mathrm{~A}$\nD: $\\quad 1.8 \\mathrm{~A}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nConsider the circuit shown. Each of the fixed resistors has a value of $10 \\Omega$.\n\nA current of 0.6 A flows through resistor R3.\n\n[figure1]\n\nThe total current flowing through the battery is:\n\nA: $\\quad 0.6 \\mathrm{~A}$\nB: $\\quad 0.9 \\mathrm{~A}$\nC: $\\quad 1.2 \\mathrm{~A}$\nD: $\\quad 1.8 \\mathrm{~A}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ae10da91e53998b18280g-04.jpg?height=534&width=642&top_left_y=567&top_left_x=707" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_419", "problem": "This question is about estimations.\n\nEspecially at the start of research or when solving a complex problem, it is useful to have an estimate of the sort of outcome to expect; making approximate calculations is a useful skill. Use the suggestions given and any other estimated values of your own to answer the following:\n\na) What is the equivalent mass of a joule?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThis question is about estimations.\n\nEspecially at the start of research or when solving a complex problem, it is useful to have an estimate of the sort of outcome to expect; making approximate calculations is a useful skill. Use the suggestions given and any other estimated values of your own to answer the following:\n\na) What is the equivalent mass of a joule?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of kg, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "kg" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1541", "problem": "如图, 一质量分布均匀、半径为 $r$ 的刚性薄圆环落到粗糙的水平地面前的瞬间,圆环质心速度 $v_{0}$ 与坚直方向成 $\\theta\\left(\\frac{\\pi}{2}<\\theta<\\frac{3 \\pi}{2}\\right)$ 角, 并同时以\n\n[图1]\n$\\omega_{0}$ ( $\\omega_{0}$ 的正方向如图中箭头所示）绕通过其质心 $\\mathrm{O}$ 、且垂直环面的轴转动。已知圆环仅在其所在的坚直平面内运动, 在弹起前刚好与地面无相对滑动, 圆环与地面碰撞的恢复系数为 $k$,重力加速度大小为 $g$ 。忽略空气阻力。若让 $\\theta$ 角可变, 求圆环第二次落地点到首次落地点之间的水平距离 $s$ 随 $\\theta$ 变化的函数关系式、 $s$ 的最大值以及 $s$ 取最大值时 $r$ 、 $v_{0}$ 和 $\\omega_{0}$ 应满足的条件", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一质量分布均匀、半径为 $r$ 的刚性薄圆环落到粗糙的水平地面前的瞬间,圆环质心速度 $v_{0}$ 与坚直方向成 $\\theta\\left(\\frac{\\pi}{2}<\\theta<\\frac{3 \\pi}{2}\\right)$ 角, 并同时以\n\n[图1]\n$\\omega_{0}$ ( $\\omega_{0}$ 的正方向如图中箭头所示）绕通过其质心 $\\mathrm{O}$ 、且垂直环面的轴转动。已知圆环仅在其所在的坚直平面内运动, 在弹起前刚好与地面无相对滑动, 圆环与地面碰撞的恢复系数为 $k$,重力加速度大小为 $g$ 。忽略空气阻力。\n\n问题:\n若让 $\\theta$ 角可变, 求圆环第二次落地点到首次落地点之间的水平距离 $s$ 随 $\\theta$ 变化的函数关系式、 $s$ 的最大值以及 $s$ 取最大值时 $r$ 、 $v_{0}$ 和 $\\omega_{0}$ 应满足的条件\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[水平距离 $s$ 随 $\\theta$ 变化的函数关系式, $s$ 的最大值, $s$ 取最大值时 $r$ 、 $v_{0}$ 和 $\\omega_{0}$ 应满足的条件]\n它们的答案类型依次是[数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e3a9fdbbef225ad3aefbg-01.jpg?height=314&width=363&top_left_y=2576&top_left_x=1326" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "水平距离 $s$ 随 $\\theta$ 变化的函数关系式", "$s$ 的最大值", " $s$ 取最大值时 $r$ 、 $v_{0}$ 和 $\\omega_{0}$ 应满足的条件" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_471", "problem": "Consider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nLet $E_{0}$ be the minimum energy of the bouncing neutron and $f$ be the frequency of the bounce. Determine an order of magnitude estimate for the ratio $E / f$. It only needs to be accurate to within an order of magnitude or so, but you do need to show work!", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nLet $E_{0}$ be the minimum energy of the bouncing neutron and $f$ be the frequency of the bounce. Determine an order of magnitude estimate for the ratio $E / f$. It only needs to be accurate to within an order of magnitude or so, but you do need to show work!\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_489", "problem": "In this problem we consider a simplified model of the electromagnetic radiation inside a cubical box of side length $L$. In this model, the electric field has spatial dependence\n\n$$\nE(x, y, z)=E_{0} \\sin \\left(k_{x} x\\right) \\sin \\left(k_{y} y\\right) \\sin \\left(k_{z} z\\right)\n$$\n\nwhere one corner of the box lies at the origin and the box is aligned with the $x, y$, and $z$ axes. Let $h$ be Planck's constant, $k_{B}$ be Boltzmann's constant, and $c$ be the speed of light.\n\nThe electric field must be zero everywhere at the sides of the box. What condition does this impose on $k_{x}, k_{y}$, and $k_{z}$ ? (Assume that any of these may be negative, and include cases where one or more of the $k_{i}$ is zero, even though this causes $E$ to be zero.)", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nIn this problem we consider a simplified model of the electromagnetic radiation inside a cubical box of side length $L$. In this model, the electric field has spatial dependence\n\n$$\nE(x, y, z)=E_{0} \\sin \\left(k_{x} x\\right) \\sin \\left(k_{y} y\\right) \\sin \\left(k_{z} z\\right)\n$$\n\nwhere one corner of the box lies at the origin and the box is aligned with the $x, y$, and $z$ axes. Let $h$ be Planck's constant, $k_{B}$ be Boltzmann's constant, and $c$ be the speed of light.\n\nThe electric field must be zero everywhere at the sides of the box. What condition does this impose on $k_{x}, k_{y}$, and $k_{z}$ ? (Assume that any of these may be negative, and include cases where one or more of the $k_{i}$ is zero, even though this causes $E$ to be zero.)\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [condition imposes on $k_{x}, condition imposes on $k_{y}, condition imposes on $k_{z}].\nTheir answer types are, in order, [equation, equation, equation].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "condition imposes on $k_{x}", "condition imposes on $k_{y}", "condition imposes on $k_{z}" ], "type_sequence": [ "EQ", "EQ", "EQ" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_36", "problem": "Imagine that you're standing on the surface of a planet that shrinks to $1 / 10$ of its original diameter with no change in mass. Your weight on the shrunken planet would be\nA: $1 / 100$ as much.\nB: $1 / 10$ times as much.\nC: 10 times as much.\nD: 100 times as much.\nE: 1000 times as much.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nImagine that you're standing on the surface of a planet that shrinks to $1 / 10$ of its original diameter with no change in mass. Your weight on the shrunken planet would be\n\nA: $1 / 100$ as much.\nB: $1 / 10$ times as much.\nC: 10 times as much.\nD: 100 times as much.\nE: 1000 times as much.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_848", "problem": "[figure1]\n\nFig. 1 A monopole $q_{\\mathrm{m}}$ appears at a distance $h$ from a conducting thin film of thickness $d$. The origin of the coordinates is located on the upper surface.\n\nWe first focus on the initial response of the conducting thin film when at time $t=0$ a north monopole $q_{\\mathrm{m}}$ appears suddenly at the position $\\vec{r}_{\\mathrm{mp}}=h \\hat{z}(h>0)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nObtain the initial induced electric current density $\\vec{j}(\\vec{\\rho})$ in the conducting thin film $\\quad 0.6 \\mathrm{pt}$ at $t_{0}=0$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n[figure1]\n\nFig. 1 A monopole $q_{\\mathrm{m}}$ appears at a distance $h$ from a conducting thin film of thickness $d$. The origin of the coordinates is located on the upper surface.\n\nWe first focus on the initial response of the conducting thin film when at time $t=0$ a north monopole $q_{\\mathrm{m}}$ appears suddenly at the position $\\vec{r}_{\\mathrm{mp}}=h \\hat{z}(h>0)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nObtain the initial induced electric current density $\\vec{j}(\\vec{\\rho})$ in the conducting thin film $\\quad 0.6 \\mathrm{pt}$ at $t_{0}=0$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d32b3b2f89cebe6f1c2ag-2.jpg?height=642&width=1244&top_left_y=296&top_left_x=194", "https://cdn.mathpix.com/cropped/2024_03_14_300bd3a4734333693ab9g-1.jpg?height=142&width=987&top_left_y=1707&top_left_x=488" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1704", "problem": "用波长为 $633 \\mathrm{~nm}$ 的激光水平照射坚直圆珠笔中的小弹簧, 在距离弹簧 $4.2 \\mathrm{~m}$ 的光屏 (与激光水平照射方向垂直) 上形成衍射图像, 如图 a 所示。其右图与 1952 年拍摄的首张 DNA 分子双螺旋结构 X 射线衍射图像（图 b）十分相似。\n\n[图1]\n\n图 b\n\n说明: 由光学原理可知, 弹簧上两段互成角度的细铁丝的衍射、干涉图像与两条成同样角度、相同宽度的狭缝的衍射、干涉图像一致。图 b 是用波长为 $0.15 \\mathrm{~nm}$ 的平行 X 射线照射 DNA 分子样品后, 在距离样品 $9.0 \\mathrm{~cm}$ 的照相底片上拍摄的。假设 DNA 分子与底片平行，且均与 $\\mathrm{X}$ 射线照射方向垂直。根据图 $\\mathrm{b}$ 中给出的尺寸信息, 试估算 DNA 螺旋结构的半径 $R^{\\prime}$ 和螺距 $p^{\\prime}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n用波长为 $633 \\mathrm{~nm}$ 的激光水平照射坚直圆珠笔中的小弹簧, 在距离弹簧 $4.2 \\mathrm{~m}$ 的光屏 (与激光水平照射方向垂直) 上形成衍射图像, 如图 a 所示。其右图与 1952 年拍摄的首张 DNA 分子双螺旋结构 X 射线衍射图像（图 b）十分相似。\n\n[图1]\n\n图 b\n\n说明: 由光学原理可知, 弹簧上两段互成角度的细铁丝的衍射、干涉图像与两条成同样角度、相同宽度的狭缝的衍射、干涉图像一致。\n\n问题:\n图 b 是用波长为 $0.15 \\mathrm{~nm}$ 的平行 X 射线照射 DNA 分子样品后, 在距离样品 $9.0 \\mathrm{~cm}$ 的照相底片上拍摄的。假设 DNA 分子与底片平行，且均与 $\\mathrm{X}$ 射线照射方向垂直。根据图 $\\mathrm{b}$ 中给出的尺寸信息, 试估算 DNA 螺旋结构的半径 $R^{\\prime}$ 和螺距 $p^{\\prime}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[估算 DNA 螺旋结构的半径 $R^{\\prime}$, 估算 DNA 螺旋结构的螺距 $p^{\\prime}$ ]\n它们的单位依次是[$\\mathrm{~nm}$, $\\mathrm{~nm}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-04.jpg?height=960&width=1445&top_left_y=1810&top_left_x=297" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{~nm}$", "$\\mathrm{~nm}$" ], "answer_sequence": [ "估算 DNA 螺旋结构的半径 $R^{\\prime}$", "估算 DNA 螺旋结构的螺距 $p^{\\prime}$ " ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1017", "problem": "# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nAfter the bubble is formed for a while, it will be in thermal equilibrium with the surrounding. This bubble in still air will naturally fall towards the ground.\n\nCalculate the numerical value for $u$ using $\\eta=1.8 \\times 10^{-5} \\mathrm{kgm}^{-1} \\mathrm{~s}^{-1}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a range interval.\n\nproblem:\n# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nAfter the bubble is formed for a while, it will be in thermal equilibrium with the surrounding. This bubble in still air will naturally fall towards the ground.\n\nCalculate the numerical value for $u$ using $\\eta=1.8 \\times 10^{-5} \\mathrm{kgm}^{-1} \\mathrm{~s}^{-1}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~m} / \\mathrm{s}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an interval without any units, e.g. ANSWER=(1,2] \\cup[7,+\\infty)", "figure_urls": null, "answer": null, "solution": null, "answer_type": "IN", "unit": [ "\\mathrm{~m} / \\mathrm{s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1668", "problem": "Hyperloop 是一款利用胶囊状的运输车在水平管道中的快速运动来实现超高速运输的系统（见图 a）。它采用了 “气垫” 技术和 “直线电动机” 原理。\n\n“气垫” 技术是将内部高压气体从水平放置的运输车下半部的\n\n[图1]\n\n图a 细孔快速喷出（见图 b），以至于整个运输车被托离管壁非常小的距离,\n\n[图2]\n从而可忽略摩擦。运输车横截面是半径为 $R$ 的圆, 运输车下半部壁上均匀分布有沿径向的大量细孔, 单位面积内细孔个数为 $n$ ( $n>>1)$, 单个细孔面积为 $s$ 。运输车长度为 $l$, 质量为 $M$ 。气体的流动可认为遵从伯努利方程, 且温度不变, 细孔出口处气体的压强为较低的环境压强 $P_{\\text {low }}$ 。\n\n如图 c，在水平管道中固定有两条平行的水平光滑供电导轨（粗实线）, 运输车上固定有与导轨垂直的两根导线 (细实线) ; 导轨横截面为圆形, 半径为 $r_{\\mathrm{d}}$, 电阻率为 $\\rho_{\\mathrm{d}}$, 两导轨轴线间距为 $2\\left(D+r_{\\mathrm{d}}\\right)$; 两根导线的粗细可忽略, 间距为 $D$; 每根导线电阻是长度为 $D$ 的导轨电阻的 2 倍。两导线和导轨轴线均处于同一水平面内。导轨、导线电接触良好, 且所有接触电阻均可忽略。\n\n[图3]\n\n图c\n\n[图4]\n\n图d\n\n[图5]\n\n图e如图 d 所示, 当运输车进站时, 运输车以速度 $v_{0}$ 沿水平光滑导轨滑进匀强磁场区域,磁场边界与导线平行, 磁感应强度大小为 $B$, 方向垂直于导轨-导线平面向下。当两根导线全都进入磁场后, 求运输车滑动速度的大小。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\nHyperloop 是一款利用胶囊状的运输车在水平管道中的快速运动来实现超高速运输的系统（见图 a）。它采用了 “气垫” 技术和 “直线电动机” 原理。\n\n“气垫” 技术是将内部高压气体从水平放置的运输车下半部的\n\n[图1]\n\n图a 细孔快速喷出（见图 b），以至于整个运输车被托离管壁非常小的距离,\n\n[图2]\n从而可忽略摩擦。运输车横截面是半径为 $R$ 的圆, 运输车下半部壁上均匀分布有沿径向的大量细孔, 单位面积内细孔个数为 $n$ ( $n>>1)$, 单个细孔面积为 $s$ 。运输车长度为 $l$, 质量为 $M$ 。气体的流动可认为遵从伯努利方程, 且温度不变, 细孔出口处气体的压强为较低的环境压强 $P_{\\text {low }}$ 。\n\n如图 c，在水平管道中固定有两条平行的水平光滑供电导轨（粗实线）, 运输车上固定有与导轨垂直的两根导线 (细实线) ; 导轨横截面为圆形, 半径为 $r_{\\mathrm{d}}$, 电阻率为 $\\rho_{\\mathrm{d}}$, 两导轨轴线间距为 $2\\left(D+r_{\\mathrm{d}}\\right)$; 两根导线的粗细可忽略, 间距为 $D$; 每根导线电阻是长度为 $D$ 的导轨电阻的 2 倍。两导线和导轨轴线均处于同一水平面内。导轨、导线电接触良好, 且所有接触电阻均可忽略。\n\n[图3]\n\n图c\n\n[图4]\n\n图d\n\n[图5]\n\n图e\n\n问题:\n如图 d 所示, 当运输车进站时, 运输车以速度 $v_{0}$ 沿水平光滑导轨滑进匀强磁场区域,磁场边界与导线平行, 磁感应强度大小为 $B$, 方向垂直于导轨-导线平面向下。当两根导线全都进入磁场后, 求运输车滑动速度的大小。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-03.jpg?height=226&width=351&top_left_y=2194&top_left_x=1412", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-03.jpg?height=374&width=905&top_left_y=2486&top_left_x=884", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-04.jpg?height=268&width=420&top_left_y=914&top_left_x=430", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-04.jpg?height=263&width=440&top_left_y=919&top_left_x=842", "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-04.jpg?height=263&width=380&top_left_y=919&top_left_x=1272" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_806", "problem": "Jason is going to make a blouse for his mother for Mother's day. The pattern requires $2.5 \\mathrm{~m}^{2}$ of fabric so Jason asks for $1.7 \\mathrm{~m}$ of fabric that is $1.5 \\mathrm{~m}$ wide. When he gets home the measures this piece to be $1.50 \\mathrm{~m}$ wide but only $1.66 \\mathrm{~m}$ long. His measuring tape has an uncertainty of $1 \\mathrm{~cm}$. What are of fabric does Jason have?\nA: $\\quad 2.49 \\mathrm{~m}^{2} \\pm 0.01 \\mathrm{~m}^{2}$\nB: $2.49 \\mathrm{~m}^{2} \\pm 0.02 \\mathrm{~m}^{2}$\nC: $2.49 \\mathrm{~m}^{2} \\pm 0.03 \\mathrm{~m}^{2}$\nD: $2.55 \\mathrm{~m}^{2} \\pm 0.01 \\mathrm{~m}^{2}$\nE: $\\quad 2.55 \\mathrm{~m}^{2} \\pm 0.02 \\mathrm{~m}^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nJason is going to make a blouse for his mother for Mother's day. The pattern requires $2.5 \\mathrm{~m}^{2}$ of fabric so Jason asks for $1.7 \\mathrm{~m}$ of fabric that is $1.5 \\mathrm{~m}$ wide. When he gets home the measures this piece to be $1.50 \\mathrm{~m}$ wide but only $1.66 \\mathrm{~m}$ long. His measuring tape has an uncertainty of $1 \\mathrm{~cm}$. What are of fabric does Jason have?\n\nA: $\\quad 2.49 \\mathrm{~m}^{2} \\pm 0.01 \\mathrm{~m}^{2}$\nB: $2.49 \\mathrm{~m}^{2} \\pm 0.02 \\mathrm{~m}^{2}$\nC: $2.49 \\mathrm{~m}^{2} \\pm 0.03 \\mathrm{~m}^{2}$\nD: $2.55 \\mathrm{~m}^{2} \\pm 0.01 \\mathrm{~m}^{2}$\nE: $\\quad 2.55 \\mathrm{~m}^{2} \\pm 0.02 \\mathrm{~m}^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_78", "problem": "Two light waves, initially emitted in phase, will interfere constructively with maximum amplitude if the path-length difference between them is:\nA: 1.5 wavelengths\nB: one wavelength\nC: one-half wavelength\nD: one-quarter wavelength\nE: one-eighth wavelength\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo light waves, initially emitted in phase, will interfere constructively with maximum amplitude if the path-length difference between them is:\n\nA: 1.5 wavelengths\nB: one wavelength\nC: one-half wavelength\nD: one-quarter wavelength\nE: one-eighth wavelength\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1319", "problem": "如图, 一固定的坚直长导线载有恒定电流 $I$, 其旁边有一正方形导线框, 导线框可围绕过对边中心的坚直轴 $\\mathrm{O}_{1} \\mathrm{O}_{2}$ 转动, 转轴到长直导线的距离为 $b$ 。已知导线框的边长为 $2 a$ $(a0$ 时, 线框平面恰好逆时针转至水平, 此时断开 $\\mathrm{P} 、 \\mathrm{Q}$ 与外电路的连接, 此后线框如何运动？求 $\\mathrm{P} 、 \\mathrm{Q}$ 间电压 $V_{\\mathrm{PQ}}$ 随时间变化的关系式;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 在磁感应强度大小为 $B$ 、方向坚直向上的匀强磁场中, 有一均质刚性导电的正方形线框 abcd, 线框质量为 $m$ ，边长为 $l$, 总电阻为 $R$ 。线框可绕通过 ad 边和 bc 边中点的光滑轴 $\\mathrm{OO}^{\\prime}$ 转动。 $\\mathrm{P} 、 \\mathrm{Q}$ 点是线框引线的两端, $\\mathrm{OO}^{\\prime}$ 轴和 X 轴位于同一水平面内, 且相互垂直。不考虑线框自感。\n\n[图1]\n\n问题:\n$t=t_{0}>0$ 时, 线框平面恰好逆时针转至水平, 此时断开 $\\mathrm{P} 、 \\mathrm{Q}$ 与外电路的连接, 此后线框如何运动？求 $\\mathrm{P} 、 \\mathrm{Q}$ 间电压 $V_{\\mathrm{PQ}}$ 随时间变化的关系式;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$t=t_{0}>0$ 时, 线框平面恰好逆时针转至水平, 此时断开 $\\mathrm{P} 、 \\mathrm{Q}$ 与外电路的连接, 此后线框如何运动？, 求 $\\mathrm{P} 、 \\mathrm{Q}$ 间电压 $V_{\\mathrm{PQ}}$ 随时间变化的关系式;]\n它们的答案类型依次是[元组, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_a47de6806e8da0a0f86dg-02.jpg?height=440&width=674&top_left_y=1322&top_left_x=1222" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$t=t_{0}>0$ 时, 线框平面恰好逆时针转至水平, 此时断开 $\\mathrm{P} 、 \\mathrm{Q}$ 与外电路的连接, 此后线框如何运动？", "求 $\\mathrm{P} 、 \\mathrm{Q}$ 间电压 $V_{\\mathrm{PQ}}$ 随时间变化的关系式;" ], "type_sequence": [ "TUP", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_57", "problem": "A Boeing 737 jet is landing with a speed of $69 \\mathrm{~m} / \\mathrm{s}$. The jet touches down and has $750 \\mathrm{~m}$ of runway in which to reduce its speed to $6.1 \\mathrm{~m} / \\mathrm{s}$. What is the average acceleration, assumed to be uniform, of the plane during landing?\nA: $-0.32 \\frac{\\mathrm{m}}{\\mathrm{s}^{2}}$\nB: $-3.2 \\frac{\\mathrm{m}}{\\mathrm{s}^{2}}$\nC: $-6.3 \\frac{\\mathrm{m}}{\\mathrm{s}^{2}}$\nD: $-32 \\frac{m}{s^{2}}$\nE: $-63 \\frac{m}{s^{2}}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA Boeing 737 jet is landing with a speed of $69 \\mathrm{~m} / \\mathrm{s}$. The jet touches down and has $750 \\mathrm{~m}$ of runway in which to reduce its speed to $6.1 \\mathrm{~m} / \\mathrm{s}$. What is the average acceleration, assumed to be uniform, of the plane during landing?\n\nA: $-0.32 \\frac{\\mathrm{m}}{\\mathrm{s}^{2}}$\nB: $-3.2 \\frac{\\mathrm{m}}{\\mathrm{s}^{2}}$\nC: $-6.3 \\frac{\\mathrm{m}}{\\mathrm{s}^{2}}$\nD: $-32 \\frac{m}{s^{2}}$\nE: $-63 \\frac{m}{s^{2}}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1705", "problem": "折射率为 $n=1.50$ 、半径为 $R$ 的透明半圆柱体放在空气中, 其垂直于柱体轴线的横截面如图所示, 图中 $O$ 点为横截面与轴线的交点, 光仅允许从半圆柱体的平面 $A B$ 进入, 一束足够宽的平行单色光沿垂直于圆柱轴的方向以入射角 $\\mathrm{i}$ 射至 $A B$ 整个平面上,其中有一部分入射光束能通过半圆柱体从圆柱面射出, 这部分光束在入射到 $A B$ 面上时沿 $y$ 轴方向的长度用 $d$ 表示。本题不考虑在光线在透明圆柱体内经一次或多次反射后再射出柱体的复杂情形。\n\n[图1]当平行入射光的入射角 $\\mathrm{i}$ 从 $0^{\\circ}$ 到 $90^{\\circ}$ 变化时, 试求 $d$ 的最大值 $d_{\\max }$ 和最小值 $d_{\\min }$.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n折射率为 $n=1.50$ 、半径为 $R$ 的透明半圆柱体放在空气中, 其垂直于柱体轴线的横截面如图所示, 图中 $O$ 点为横截面与轴线的交点, 光仅允许从半圆柱体的平面 $A B$ 进入, 一束足够宽的平行单色光沿垂直于圆柱轴的方向以入射角 $\\mathrm{i}$ 射至 $A B$ 整个平面上,其中有一部分入射光束能通过半圆柱体从圆柱面射出, 这部分光束在入射到 $A B$ 面上时沿 $y$ 轴方向的长度用 $d$ 表示。本题不考虑在光线在透明圆柱体内经一次或多次反射后再射出柱体的复杂情形。\n\n[图1]\n\n问题:\n当平行入射光的入射角 $\\mathrm{i}$ 从 $0^{\\circ}$ 到 $90^{\\circ}$ 变化时, 试求 $d$ 的最大值 $d_{\\max }$ 和最小值 $d_{\\min }$.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$d_{\\max }$, $d_{\\min }$]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_1004b08dedac85274c96g-06.jpg?height=468&width=312&top_left_y=283&top_left_x=1249", "https://cdn.mathpix.com/cropped/2024_03_31_1004b08dedac85274c96g-16.jpg?height=437&width=377&top_left_y=1455&top_left_x=1162" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$d_{\\max }$", "$d_{\\min }$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_954", "problem": "A linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths $\\ell_{1}, \\ell_{2}, \\ell_{3}, \\ldots$ along the beam and with a fixed small separation $d$ between each tube. The tubes are connected to a high voltage, constant radio frequency $\\mathrm{AC}$ supply where the peak voltage of the $\\mathrm{AC}$ is $V_{0}$. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge $e$ and mass $m_{e}$ is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it \"drifts\" through the tube. It passes through a large potential difference between the tubes and, if the charged particle's motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10\n\n[figure1]\n\nFigure 9\n\n[figure2]\n\nFigure 10\n\n\nPreliminary: a resistor has a potential difference of $5 \\mathrm{~V}$ across it and a single electron flows through it. What is the thermal energy generated?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths $\\ell_{1}, \\ell_{2}, \\ell_{3}, \\ldots$ along the beam and with a fixed small separation $d$ between each tube. The tubes are connected to a high voltage, constant radio frequency $\\mathrm{AC}$ supply where the peak voltage of the $\\mathrm{AC}$ is $V_{0}$. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge $e$ and mass $m_{e}$ is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it \"drifts\" through the tube. It passes through a large potential difference between the tubes and, if the charged particle's motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10\n\n[figure1]\n\nFigure 9\n\n[figure2]\n\nFigure 10\n\n\nPreliminary: a resistor has a potential difference of $5 \\mathrm{~V}$ across it and a single electron flows through it. What is the thermal energy generated?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of J, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-10.jpg?height=300&width=454&top_left_y=818&top_left_x=296", "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-10.jpg?height=359&width=1128&top_left_y=757&top_left_x=795" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "J" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1674", "problem": "嫦娥 1 号奔月卫星与长征 3 号火箭分离后, 进入绕地运行的椭圆轨道, 近地点离地面高 $H_{n}=2.05 \\times 10^{2} \\mathrm{~km}$, 远地点离地面高 $H_{f}=5.0930 \\times 10^{4} \\mathrm{~km}$, 周期约为 16 小时, 称为 16 小时轨道 (如图中曲线 1 所示)。随后, 为了使卫星离地越来越远, 星载发动机先在远地点点火, 使卫星进入新轨道 (如图中曲线 2 所示), 以抬高近地点。后来又连续三次在抬高以后的近地点点火, 使卫星加速和变轨, 抬高远地点, 相继进入 24 小时轨道、 48 小时轨道和地月转移轨道 (分别如图中曲线 3、4、5 所示)。已知卫星质量 $m=2.350 \\times 10^{3} \\mathrm{~kg}$, 地球半径 $R=6.378 \\times 10^{3} \\mathrm{~km}$, 地面重力加速度 $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$, 月球半径 $r=1.738 \\times 10^{3} \\mathrm{~km}$ 。在 16 小时轨道的远地点点火时, 假设卫星所受推力的方向与卫星速度方向相同, 而且点火时间很短, 可以认为椭圆轨道长轴方向不变。设推力大小 $\\mathrm{F}=490 \\mathrm{~N}$, 要把近地点抬高到 $600 \\mathrm{~km}$ ，问点火时间应持绕多长?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n嫦娥 1 号奔月卫星与长征 3 号火箭分离后, 进入绕地运行的椭圆轨道, 近地点离地面高 $H_{n}=2.05 \\times 10^{2} \\mathrm{~km}$, 远地点离地面高 $H_{f}=5.0930 \\times 10^{4} \\mathrm{~km}$, 周期约为 16 小时, 称为 16 小时轨道 (如图中曲线 1 所示)。随后, 为了使卫星离地越来越远, 星载发动机先在远地点点火, 使卫星进入新轨道 (如图中曲线 2 所示), 以抬高近地点。后来又连续三次在抬高以后的近地点点火, 使卫星加速和变轨, 抬高远地点, 相继进入 24 小时轨道、 48 小时轨道和地月转移轨道 (分别如图中曲线 3、4、5 所示)。已知卫星质量 $m=2.350 \\times 10^{3} \\mathrm{~kg}$, 地球半径 $R=6.378 \\times 10^{3} \\mathrm{~km}$, 地面重力加速度 $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$, 月球半径 $r=1.738 \\times 10^{3} \\mathrm{~km}$ 。\n\n问题:\n在 16 小时轨道的远地点点火时, 假设卫星所受推力的方向与卫星速度方向相同, 而且点火时间很短, 可以认为椭圆轨道长轴方向不变。设推力大小 $\\mathrm{F}=490 \\mathrm{~N}$, 要把近地点抬高到 $600 \\mathrm{~km}$ ，问点火时间应持绕多长?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以分为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "分" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_59", "problem": "A block of wood initially at rest slides down an inclined plane. Neglecting friction, the kinetic energy of the block at the bottom of the plane is:\nA: all converted into heat.\nB: less than its kinetic energy at the top of the plane.\nC: dependent on the materials of which the block is made.\nD: dependent on the materials of which the inclined plane is made.\nE: equal to its potential energy (with respect to the bottom of the plane) when it was at the top of the plane.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA block of wood initially at rest slides down an inclined plane. Neglecting friction, the kinetic energy of the block at the bottom of the plane is:\n\nA: all converted into heat.\nB: less than its kinetic energy at the top of the plane.\nC: dependent on the materials of which the block is made.\nD: dependent on the materials of which the inclined plane is made.\nE: equal to its potential energy (with respect to the bottom of the plane) when it was at the top of the plane.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_606", "problem": "In parts this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinitely long wire wire on the $z$ axis is composed of positive charges with linear charge density $\\lambda$ which are at rest, and negative charges with linear charge density $-\\lambda$ moving with speed $v$ in the $z$ direction.\n\nDetermine the magnetic field $\\tilde{\\mathbf{B}}$ (magnitude) at points outside the wire.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn parts this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinitely long wire wire on the $z$ axis is composed of positive charges with linear charge density $\\lambda$ which are at rest, and negative charges with linear charge density $-\\lambda$ moving with speed $v$ in the $z$ direction.\n\nDetermine the magnetic field $\\tilde{\\mathbf{B}}$ (magnitude) at points outside the wire.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1474", "problem": "平行板电容器两极板分别位于 $z= \\pm \\frac{d}{2}$ 的平面内,电容器起初未被充电. 整个装置处于均匀磁场中, 磁感应强度大小为 $B$, 方向沿 $x$ 轴负方向, 如图所示.\n\n[图1]现在让介电常数为 $\\varepsilon$ 的电中性液体 (绝缘体) 在平行板电容器两极板之间匀速流动, 流速大小为 $v$, 方向沿 $y$ 轴正方向. 在相对液体静止的参考系 (即相对于电容器运动的参考系) $S^{\\prime}$ 中, 由于液体处在第1问所述的电场 $\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$ 中, 其正负电荷会因电场力作用而发生相对移动（即所谓极化效应）, 使得液体中出现附加的静电感应电场, 因而液体中总电场强度不再是 $\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$, 而是 $\\frac{\\varepsilon_{0}}{\\varepsilon}\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$, 这里 $\\varepsilon_{0}$ 是真空的介电常数. 这将导致在电容器参考系 $S$ 中电场不再为零. 试求电容器参考系 $S$ 中电场的强度以及电容器上、下极板之间的电势差. （结果用 $\\varepsilon_{0} 、 \\varepsilon 、 v 、 B$ 或 (和) $d$ 表出. )", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n平行板电容器两极板分别位于 $z= \\pm \\frac{d}{2}$ 的平面内,电容器起初未被充电. 整个装置处于均匀磁场中, 磁感应强度大小为 $B$, 方向沿 $x$ 轴负方向, 如图所示.\n\n[图1]\n\n问题:\n现在让介电常数为 $\\varepsilon$ 的电中性液体 (绝缘体) 在平行板电容器两极板之间匀速流动, 流速大小为 $v$, 方向沿 $y$ 轴正方向. 在相对液体静止的参考系 (即相对于电容器运动的参考系) $S^{\\prime}$ 中, 由于液体处在第1问所述的电场 $\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$ 中, 其正负电荷会因电场力作用而发生相对移动（即所谓极化效应）, 使得液体中出现附加的静电感应电场, 因而液体中总电场强度不再是 $\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$, 而是 $\\frac{\\varepsilon_{0}}{\\varepsilon}\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$, 这里 $\\varepsilon_{0}$ 是真空的介电常数. 这将导致在电容器参考系 $S$ 中电场不再为零. 试求电容器参考系 $S$ 中电场的强度以及电容器上、下极板之间的电势差. （结果用 $\\varepsilon_{0} 、 \\varepsilon 、 v 、 B$ 或 (和) $d$ 表出. )\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_734a14c1a8de5e19f8ecg-03.jpg?height=428&width=657&top_left_y=357&top_left_x=1405" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_570", "problem": "Suppose that, after some amount of work is done by the ion pumps, the charges on the outer and inner surfaces are $Q$ and $-Q$, respectively.\n\nOne charge layer by itself creates an electric field $E_{1}=Q /(2 \\epsilon A)$ in each direction. So the force between the two sides of the membrane $F_{E}=Q E_{1}=Q^{2} /(2 \\epsilon A)$.\n\nThis electric force is balanced by the spring force $F_{s}=k x$, where $x=d_{0}-d$. Equating these two forces and solving for $d$ gives gives\n\n$$\nd=d_{0}-\\frac{Q^{2}}{2 \\epsilon A k} .\n$$\n\nThe electric field inside the membrane (as produced by both the left and right plates) is $E=Q /\\left(\\epsilon_{0} \\kappa A\\right)$. So the voltage between them is\n\n$$\nV=E d=\\frac{Q}{\\epsilon A} d\n$$\n\nInserting the expression for $Q$ from part (a) gives\n\n$$\nV=\\frac{Q}{\\epsilon A}\\left(d_{0}-\\frac{Q^{2}}{2 \\epsilon A k}\\right)\n$$\n\nThis equation implies that as the charge $Q$ is increased, the voltage first increases and then decreases again.\n\nThe voltage $V$ first increases and then decreases as a function of $Q$, which implies that there is a maximum voltage $V_{\\max }$ to which the membrane can be charged. This voltage can be found by taking the derivative $d V / d Q$ and setting it equal to zero. This procedure gives\n\n$$\nV_{\\max }=\\sqrt{\\frac{k d_{0}^{3}}{\\epsilon A}}\\left(\\frac{2}{3}\\right)^{3 / 2} .\n$$\n\nThe corresponding charge at the maximum voltage is given by\n\n$$\nQ_{\\mathrm{V} \\max }^{2}=\\frac{2}{3} \\epsilon A k d_{0}\n$$\n\nFor the ion pumps to turn off, we must have $V_{\\max }>V_{\\mathrm{th}}$. Otherwise the pumps will continue to move charge across the membrane until it collapses. Setting $V_{\\max }>V_{\\text {th }}$ and solving for $k$ gives\n\n$$\nk>\\left(\\frac{3}{2}\\right)^{3} \\frac{V_{\\mathrm{th}}^{2} \\epsilon A}{d_{0}^{3}} .\n$$[figure1]\n\nThe wall of a neuron is made from an elastic membrane, which resists compression in the same way as a spring. It has an effective spring constant $k$ and an equilibrium thickness $d_{0}$. Assume that the membrane has a very large area $A$ and negligible curvature.\n\nThe neuron has \"ion pumps\" that can move ions across the membrane. In the resulting charged state, positive and negative ionic charge is arranged uniformly along the outer and inner surfaces of the membrane, respectively. The permittivity of the membrane is $\\epsilon$.\n\nSuppose that the ion pumps are first turned on in the uncharged state, and the membrane is charged very slowly (quasistatically). The pumps will only turn off when the voltage difference across the membrane becomes larger than a particular value $V_{\\text {th }}$.\n\nHow much work is done by the ion pumps in each of the following situations? Express your answers in terms of $k$ and $d_{0}$.\n\n($k$ is infinitesimally larger than the value derived before.)", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nSuppose that, after some amount of work is done by the ion pumps, the charges on the outer and inner surfaces are $Q$ and $-Q$, respectively.\n\nOne charge layer by itself creates an electric field $E_{1}=Q /(2 \\epsilon A)$ in each direction. So the force between the two sides of the membrane $F_{E}=Q E_{1}=Q^{2} /(2 \\epsilon A)$.\n\nThis electric force is balanced by the spring force $F_{s}=k x$, where $x=d_{0}-d$. Equating these two forces and solving for $d$ gives gives\n\n$$\nd=d_{0}-\\frac{Q^{2}}{2 \\epsilon A k} .\n$$\n\nThe electric field inside the membrane (as produced by both the left and right plates) is $E=Q /\\left(\\epsilon_{0} \\kappa A\\right)$. So the voltage between them is\n\n$$\nV=E d=\\frac{Q}{\\epsilon A} d\n$$\n\nInserting the expression for $Q$ from part (a) gives\n\n$$\nV=\\frac{Q}{\\epsilon A}\\left(d_{0}-\\frac{Q^{2}}{2 \\epsilon A k}\\right)\n$$\n\nThis equation implies that as the charge $Q$ is increased, the voltage first increases and then decreases again.\n\nThe voltage $V$ first increases and then decreases as a function of $Q$, which implies that there is a maximum voltage $V_{\\max }$ to which the membrane can be charged. This voltage can be found by taking the derivative $d V / d Q$ and setting it equal to zero. This procedure gives\n\n$$\nV_{\\max }=\\sqrt{\\frac{k d_{0}^{3}}{\\epsilon A}}\\left(\\frac{2}{3}\\right)^{3 / 2} .\n$$\n\nThe corresponding charge at the maximum voltage is given by\n\n$$\nQ_{\\mathrm{V} \\max }^{2}=\\frac{2}{3} \\epsilon A k d_{0}\n$$\n\nFor the ion pumps to turn off, we must have $V_{\\max }>V_{\\mathrm{th}}$. Otherwise the pumps will continue to move charge across the membrane until it collapses. Setting $V_{\\max }>V_{\\text {th }}$ and solving for $k$ gives\n\n$$\nk>\\left(\\frac{3}{2}\\right)^{3} \\frac{V_{\\mathrm{th}}^{2} \\epsilon A}{d_{0}^{3}} .\n$$\n\nproblem:\n[figure1]\n\nThe wall of a neuron is made from an elastic membrane, which resists compression in the same way as a spring. It has an effective spring constant $k$ and an equilibrium thickness $d_{0}$. Assume that the membrane has a very large area $A$ and negligible curvature.\n\nThe neuron has \"ion pumps\" that can move ions across the membrane. In the resulting charged state, positive and negative ionic charge is arranged uniformly along the outer and inner surfaces of the membrane, respectively. The permittivity of the membrane is $\\epsilon$.\n\nSuppose that the ion pumps are first turned on in the uncharged state, and the membrane is charged very slowly (quasistatically). The pumps will only turn off when the voltage difference across the membrane becomes larger than a particular value $V_{\\text {th }}$.\n\nHow much work is done by the ion pumps in each of the following situations? Express your answers in terms of $k$ and $d_{0}$.\n\n($k$ is infinitesimally larger than the value derived before.)\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_fed953f9e38b72bf8bd7g-14.jpg?height=358&width=927&top_left_y=469&top_left_x=596" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1461", "problem": "如图, 太空中有一由同心的内球和球壳构成的实验装置, 内球和球壳内表面之间为真空。内球半径为 $r=0.200 \\mathrm{~m}$, 温度保持恒定, 比辐射率为 $e=0.800$; 球壳的导热系数为 $\\kappa=1.00 \\times 10^{-2} \\mathrm{~J} \\cdot \\mathrm{m}^{-1} \\cdot \\mathrm{s}^{-1} \\cdot \\mathrm{K}^{-1}$, 内、外半径分别为 $R_{1}=0.900 \\mathrm{~m} 、 R_{2}=1.00 \\mathrm{~m}$ ，各表面的热表面可视为黑体; 该实验装置已处于热稳定状态, 此时球壳内表面比辐射率为 $E=0.800$ 。斯特藩常量为 $\\sigma=5.67 \\times 10^{-8} \\mathrm{~W} \\cdot \\mathrm{m}^{-2} \\cdot \\mathrm{K}^{-4}$, 宇宙微波背景辐射温度为 $T=2.73 \\mathrm{~K}$ 。若单位时间内由球壳内表面传递到球壳外表面的热量为 $Q=44.0 \\mathrm{~W}$, 求\n\n[图1]\n\n已知: 物体表面单位面积上的辐射功率与同温度下的黑体在该表面单位面积上的辐射功率之比称为比辐射率。当辐射照射到物体表面时, 物体表面单位面积吸收的辐射功率与照射到物体单位面积上的辐射功率之比称为吸收比。物体在某一温度下的吸收比等于其在同一温度下的比辐射率。当物体内某处在 $\\mathrm{z}$ 方向（热流方向）每单位距离温度的增量为 $\\frac{\\mathrm{d} T}{\\mathrm{~d} z}$ 时, 物体内该处单位时间在 $z$ 方向每单位面积流过的热量为 $-\\kappa \\frac{\\mathrm{d} T}{\\mathrm{~d} z}$, 此即傅里叶热传导定律。内球温度 $T_{0}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 太空中有一由同心的内球和球壳构成的实验装置, 内球和球壳内表面之间为真空。内球半径为 $r=0.200 \\mathrm{~m}$, 温度保持恒定, 比辐射率为 $e=0.800$; 球壳的导热系数为 $\\kappa=1.00 \\times 10^{-2} \\mathrm{~J} \\cdot \\mathrm{m}^{-1} \\cdot \\mathrm{s}^{-1} \\cdot \\mathrm{K}^{-1}$, 内、外半径分别为 $R_{1}=0.900 \\mathrm{~m} 、 R_{2}=1.00 \\mathrm{~m}$ ，各表面的热表面可视为黑体; 该实验装置已处于热稳定状态, 此时球壳内表面比辐射率为 $E=0.800$ 。斯特藩常量为 $\\sigma=5.67 \\times 10^{-8} \\mathrm{~W} \\cdot \\mathrm{m}^{-2} \\cdot \\mathrm{K}^{-4}$, 宇宙微波背景辐射温度为 $T=2.73 \\mathrm{~K}$ 。若单位时间内由球壳内表面传递到球壳外表面的热量为 $Q=44.0 \\mathrm{~W}$, 求\n\n[图1]\n\n已知: 物体表面单位面积上的辐射功率与同温度下的黑体在该表面单位面积上的辐射功率之比称为比辐射率。当辐射照射到物体表面时, 物体表面单位面积吸收的辐射功率与照射到物体单位面积上的辐射功率之比称为吸收比。物体在某一温度下的吸收比等于其在同一温度下的比辐射率。当物体内某处在 $\\mathrm{z}$ 方向（热流方向）每单位距离温度的增量为 $\\frac{\\mathrm{d} T}{\\mathrm{~d} z}$ 时, 物体内该处单位时间在 $z$ 方向每单位面积流过的热量为 $-\\kappa \\frac{\\mathrm{d} T}{\\mathrm{~d} z}$, 此即傅里叶热传导定律。\n\n问题:\n内球温度 $T_{0}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~K}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-03.jpg?height=286&width=286&top_left_y=1276&top_left_x=1479" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~K}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1498", "problem": "如图, 轨道型电磁发射器是由两条平行固定长直刚性金属导轨、高功率电源、接触导电性能良好的电枢和发射体等构成。电流从电流源输出, 经过导轨、电枢和另一条导轨构成闭合回路, 在空间中激发磁场。载流电枢在安培力作用下加速, 推动发射体前进。已知电枢质量为 $m_{s}$, 发射体质量为 $m_{a}$; 电枢每向前行进单位长度整个回路的电阻和电感的增加量分别为 $R_{r}^{\\prime}$ 和 $L_{r}^{\\prime}$; 电枢引入的电阻和电感分别为 $R_{s}$ 和 $L_{s}$; 回路连线引入的电阻和电感分别为 $R_{0}$ 和 $L_{0}$ 。导轨与电枢间摩擦以及空气阻力可忽略。\n\n[图1]设回路电流为恒流 (平顶脉冲电流) 、电枢和发射体的总质量为 $m_{s}+m_{a}=0.50 \\mathrm{~kg}$ 、导轨长度为 $x_{s m}=500 \\mathrm{~m}$ 、导轨上每单位长度电感增加量 $L_{r}^{\\prime}=1.0 \\mu \\mathrm{H} / \\mathrm{m}$, 若发射体开始时静止, 出口速度为 $v_{s m}=3.0 \\times 10^{3} \\mathrm{~m} / \\mathrm{s}$, 求回路电流 $I$ 和加速时间 $\\tau$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 轨道型电磁发射器是由两条平行固定长直刚性金属导轨、高功率电源、接触导电性能良好的电枢和发射体等构成。电流从电流源输出, 经过导轨、电枢和另一条导轨构成闭合回路, 在空间中激发磁场。载流电枢在安培力作用下加速, 推动发射体前进。已知电枢质量为 $m_{s}$, 发射体质量为 $m_{a}$; 电枢每向前行进单位长度整个回路的电阻和电感的增加量分别为 $R_{r}^{\\prime}$ 和 $L_{r}^{\\prime}$; 电枢引入的电阻和电感分别为 $R_{s}$ 和 $L_{s}$; 回路连线引入的电阻和电感分别为 $R_{0}$ 和 $L_{0}$ 。导轨与电枢间摩擦以及空气阻力可忽略。\n\n[图1]\n\n问题:\n设回路电流为恒流 (平顶脉冲电流) 、电枢和发射体的总质量为 $m_{s}+m_{a}=0.50 \\mathrm{~kg}$ 、导轨长度为 $x_{s m}=500 \\mathrm{~m}$ 、导轨上每单位长度电感增加量 $L_{r}^{\\prime}=1.0 \\mu \\mathrm{H} / \\mathrm{m}$, 若发射体开始时静止, 出口速度为 $v_{s m}=3.0 \\times 10^{3} \\mathrm{~m} / \\mathrm{s}$, 求回路电流 $I$ 和加速时间 $\\tau$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[the value of $I$, the value of $\\tau$]\n它们的单位依次是[A, s]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_07aa406e17d01fd01b36g-02.jpg?height=337&width=922&top_left_y=248&top_left_x=607" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "A", "s" ], "answer_sequence": [ "the value of $I$", "the value of $\\tau$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_881", "problem": "In this part we shall study the simplified model of the galaxy. You can ignore the velocities of the stars in the galaxy and assume the constant stellar concentration $n$. The characteristic size of the galaxy is $R$. The stellar concentration is small enough, so the stellar collisions are extemely rare and negligible. Let us consider a SBH with the mass $M \\gg m$ moving with the velocity $v$ through the galaxy. Surprisingly, the SBH experiences nonzero average force from the stars. This force slows the motion of the SBH and is called the force of dynamical friction for this reason. This part is devoted to the determination of this force.\n\n[figure1]\n\nFig. 1: The deflection of a star by the SBH with mass $M$. The impact parameter is $b$, the minimal distance between the star and the SBH is $r_{m}$.\n\nAs you obtained in the previous task, the expression for $F_{D F}$ includes the factor $\\log R / b_{1}$, which we will denote further as $\\log \\Lambda$. Calculate the value of $\\log \\Lambda$ for $M=10^{8} M_{S}, R=20 \\mathrm{kpc}=20 \\times 10^{3} \\mathrm{pc}$ and velocity $v=200 \\mathrm{~km} / \\mathrm{s}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nIn this part we shall study the simplified model of the galaxy. You can ignore the velocities of the stars in the galaxy and assume the constant stellar concentration $n$. The characteristic size of the galaxy is $R$. The stellar concentration is small enough, so the stellar collisions are extemely rare and negligible. Let us consider a SBH with the mass $M \\gg m$ moving with the velocity $v$ through the galaxy. Surprisingly, the SBH experiences nonzero average force from the stars. This force slows the motion of the SBH and is called the force of dynamical friction for this reason. This part is devoted to the determination of this force.\n\n[figure1]\n\nFig. 1: The deflection of a star by the SBH with mass $M$. The impact parameter is $b$, the minimal distance between the star and the SBH is $r_{m}$.\n\nAs you obtained in the previous task, the expression for $F_{D F}$ includes the factor $\\log R / b_{1}$, which we will denote further as $\\log \\Lambda$. Calculate the value of $\\log \\Lambda$ for $M=10^{8} M_{S}, R=20 \\mathrm{kpc}=20 \\times 10^{3} \\mathrm{pc}$ and velocity $v=200 \\mathrm{~km} / \\mathrm{s}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c97d93237c96d764622dg-2.jpg?height=391&width=1016&top_left_y=1318&top_left_x=523" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_588", "problem": "Suppose a domino stands upright on a table. It has height $h$, thickness $t$, width $w$ (as shown below), and mass $m$. The domino is free to rotate about its edges, but will not slide across the table.\n[figure1]\n\nSuppose we give the domino a sharp, horizontal impulsive push with total momentum $p$.\n\nWhat is the minimum value of $p$ to topple the domino?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nSuppose a domino stands upright on a table. It has height $h$, thickness $t$, width $w$ (as shown below), and mass $m$. The domino is free to rotate about its edges, but will not slide across the table.\n[figure1]\n\nSuppose we give the domino a sharp, horizontal impulsive push with total momentum $p$.\n\nWhat is the minimum value of $p$ to topple the domino?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-15.jpg?height=456&width=832&top_left_y=528&top_left_x=644" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_593", "problem": "Determine an expression for the magnitude of the magnetic field at a distance $r$ from the power line cable in terms of $I, r$, and fundamental constants.\n\nThe field is perpendicular to the wire and to the radius, and from Ampere's Law\n\n$$\n\\oint \\mathbf{B} \\cdot d \\mathbf{s}=\\mu_{0} I\n$$\n\nThe integral evaluates to $(2 \\pi r) B$, giving\n\n$$\nB=\\frac{\\mu_{0} I}{2 \\pi r}\n$$\n\nThis well-known result can also be written down without justification.An AC power line cable transmits electrical power using a sinusoidal waveform with frequency $60 \\mathrm{~Hz}$. The load receives an RMS voltage of $500 \\mathrm{kV}$ and requires $1000 \\mathrm{MW}$ of average power. For this problem, consider only the cable carrying current in one of the two directions, and ignore effects due to capacitance or inductance between the cable and with the ground.\n\nA local rancher thinks he might be able to extract electrical power from the cable using electromagnetic induction. The rancher constructs a rectangular loop of length $a$ and width $b1$.\n- The gas is then allowed to expand adiabatically (no heat is transferred to or from the gas) to pressure $P_{0}$\n- The gas is cooled at constant pressure back to the original state.\n\nThe adiabatic constant $\\gamma$ is defined in terms of the specific heat at constant pressure $C_{p}$ and the specific heat at constant volume $C_{v}$ by the ratio $\\gamma=C_{p} / C_{v}$.\n\nDetermine the efficiency of this cycle in terms of $\\alpha$ and the adiabatic constant $\\gamma$. As a reminder, efficiency is defined as the ratio of work out divided by heat in.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nAn ideal (but not necessarily perfect monatomic) gas undergoes the following cycle.\n\n- The gas starts at pressure $P_{0}$, volume $V_{0}$ and temperature $T_{0}$.\n- The gas is heated at constant volume to a pressure $\\alpha P_{0}$, where $\\alpha>1$.\n- The gas is then allowed to expand adiabatically (no heat is transferred to or from the gas) to pressure $P_{0}$\n- The gas is cooled at constant pressure back to the original state.\n\nThe adiabatic constant $\\gamma$ is defined in terms of the specific heat at constant pressure $C_{p}$ and the specific heat at constant volume $C_{v}$ by the ratio $\\gamma=C_{p} / C_{v}$.\n\nDetermine the efficiency of this cycle in terms of $\\alpha$ and the adiabatic constant $\\gamma$. As a reminder, efficiency is defined as the ratio of work out divided by heat in.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_157", "problem": "A disk of radius $r$ rolls uniformly without slipping around the inside of a fixed hoop of radius $R$. If the period of the disc's motion around the hoop is $T$, what is the instantaneous speed of the point on the disk opposite to the point of contact?\n\n[figure1]\nA: $2 \\pi(R+r) / T$\nB: $2 \\pi(R+2 r) / T$\nC: $4 \\pi(R-2 r) / T$\nD: $4 \\pi(R-r) / T $ \nE: $4 \\pi(R+r) / T$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA disk of radius $r$ rolls uniformly without slipping around the inside of a fixed hoop of radius $R$. If the period of the disc's motion around the hoop is $T$, what is the instantaneous speed of the point on the disk opposite to the point of contact?\n\n[figure1]\n\nA: $2 \\pi(R+r) / T$\nB: $2 \\pi(R+2 r) / T$\nC: $4 \\pi(R-2 r) / T$\nD: $4 \\pi(R-r) / T $ \nE: $4 \\pi(R+r) / T$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-06.jpg?height=369&width=361&top_left_y=1241&top_left_x=882" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_549", "problem": "When studying problems in special relativity it is often the invariant distance $\\Delta s$ between two events that is most important, where $\\Delta s$ is defined by\n\n$$\n(\\Delta s)^{2}=(c \\Delta t)^{2}-\\left[(\\Delta x)^{2}+(\\Delta y)^{2}+(\\Delta z)^{2}\\right]\n$$\n\nwhere $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$ is the speed of light.\n\nConsider the motion of a projectile launched with initial speed $v_{0}$ at angle of $\\theta_{0}$ above the horizontal. Assume that $g$, the acceleration of free fall, is constant for the motion of the projectile.\n\nThe radius of curvature of an object's trajectory can be estimated by assuming that the trajectory is part of a circle, determining the distance between the end points, and measuring the maximum height above the straight line that connects the endpoints. Assuming that we mean \"invariant distance\" as defined above, find the radius of curvature of the projectile's trajectory as a function of any or all of $\\theta_{0}, v_{0}, c$, and $g$. Assume that the projectile lands at the same level from which it was launched, and assume that the motion is not relativistic, so $v_{0} \\ll c$, and you can neglect terms with $v / c$ compared to terms without.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nWhen studying problems in special relativity it is often the invariant distance $\\Delta s$ between two events that is most important, where $\\Delta s$ is defined by\n\n$$\n(\\Delta s)^{2}=(c \\Delta t)^{2}-\\left[(\\Delta x)^{2}+(\\Delta y)^{2}+(\\Delta z)^{2}\\right]\n$$\n\nwhere $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$ is the speed of light.\n\nConsider the motion of a projectile launched with initial speed $v_{0}$ at angle of $\\theta_{0}$ above the horizontal. Assume that $g$, the acceleration of free fall, is constant for the motion of the projectile.\n\nThe radius of curvature of an object's trajectory can be estimated by assuming that the trajectory is part of a circle, determining the distance between the end points, and measuring the maximum height above the straight line that connects the endpoints. Assuming that we mean \"invariant distance\" as defined above, find the radius of curvature of the projectile's trajectory as a function of any or all of $\\theta_{0}, v_{0}, c$, and $g$. Assume that the projectile lands at the same level from which it was launched, and assume that the motion is not relativistic, so $v_{0} \\ll c$, and you can neglect terms with $v / c$ compared to terms without.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the distance between the end points, the maximum height above the straight line that connects the endpoints, the radius of curvature of the projectile's trajectory].\nTheir answer types are, in order, [expression, expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "the distance between the end points", "the maximum height above the straight line that connects the endpoints", "the radius of curvature of the projectile's trajectory" ], "type_sequence": [ "EX", "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_72", "problem": "A box of mass $m$ is pressed against (but is not attached to) an ideal spring of force constant $k$ and negligible mass. The spring is compressed a distance $x$. After it is released, the box slides up a frictionless incline as shown in the diagram at right and eventually stops. If we repeat this experiment with a box of mass $2 m$\n[figure1]\nA: the lighter box will go twice as high up the incline as the heavier box.\nB: just as it moves free of the spring, the lighter box will be moving twice as fast as the heavier box.\nC: both boxes will have the same speed just as they move free of the spring.\nD: both boxes will reach the same maximum height on the incline.\nE: just as it moves free of the spring, the heavier box will have twice as much kinetic energy as the lighter box.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA box of mass $m$ is pressed against (but is not attached to) an ideal spring of force constant $k$ and negligible mass. The spring is compressed a distance $x$. After it is released, the box slides up a frictionless incline as shown in the diagram at right and eventually stops. If we repeat this experiment with a box of mass $2 m$\n[figure1]\n\nA: the lighter box will go twice as high up the incline as the heavier box.\nB: just as it moves free of the spring, the lighter box will be moving twice as fast as the heavier box.\nC: both boxes will have the same speed just as they move free of the spring.\nD: both boxes will reach the same maximum height on the incline.\nE: just as it moves free of the spring, the heavier box will have twice as much kinetic energy as the lighter box.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ae7e25be7efc2df26f6eg-07.jpg?height=306&width=637&top_left_y=1625&top_left_x=1232" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_833", "problem": "In this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\n\nLet us solve a related problem. Let a star of mass $m$ transit by a point mass $M_{2} \\gg m$ being at rest. The minimal distance between the star and the point during the transit is $r_{m}$. The velocity of the star at large distance is $\\sigma$. Find the exact value of impact parameter $b$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\n\nLet us solve a related problem. Let a star of mass $m$ transit by a point mass $M_{2} \\gg m$ being at rest. The minimal distance between the star and the point during the transit is $r_{m}$. The velocity of the star at large distance is $\\sigma$. Find the exact value of impact parameter $b$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_281", "problem": "This question is about a familiar school practical to measure acceleration where a student investigates the acceleration of a toy car on a ramp.\n\nThe height of the ramp is changed and the resulting acceleration obtained from the measurements given below.\n\nThe experimental setup is shown in the diagram.\n\n[figure1]\n\nTo measure the acceleration, the student releases the car from rest at the top of the ramp and uses a stopwatch to time how long it takes for the car to reach the bottom of the ramp. The student records the following results:\n\n| Ramp height $/ \\mathrm{cm}$ | time $\\left(1^{\\text {st }}\\right.$ attempt $) / \\mathrm{s}$ | time $\\left(2^{\\text {nd }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(3^{\\text {rd }}\\right.$ attempt $) / \\mathrm{s}$ |\n| :--- | :--- | :--- | :--- |\n| 5 | 2.45 | 2.48 | 2.41 |\n\nCalculate the average speed of the toy car.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThis question is about a familiar school practical to measure acceleration where a student investigates the acceleration of a toy car on a ramp.\n\nThe height of the ramp is changed and the resulting acceleration obtained from the measurements given below.\n\nThe experimental setup is shown in the diagram.\n\n[figure1]\n\nTo measure the acceleration, the student releases the car from rest at the top of the ramp and uses a stopwatch to time how long it takes for the car to reach the bottom of the ramp. The student records the following results:\n\n| Ramp height $/ \\mathrm{cm}$ | time $\\left(1^{\\text {st }}\\right.$ attempt $) / \\mathrm{s}$ | time $\\left(2^{\\text {nd }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(3^{\\text {rd }}\\right.$ attempt $) / \\mathrm{s}$ |\n| :--- | :--- | :--- | :--- |\n| 5 | 2.45 | 2.48 | 2.41 |\n\nCalculate the average speed of the toy car.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of m/s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ae10da91e53998b18280g-09.jpg?height=260&width=1579&top_left_y=767&top_left_x=244" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m/s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_758", "problem": "Staircase \n\nThe equilibrium shape of bodies in zero-gravity is determined by the minimum of their surface energy. Thus, for example, the equilibrium shape of a water droplet turns out to be spherical: the sphere has the smallest surface area among bodies of the same volume. At low temperature, the equilibrium shape of crystals may have flat facets. The parts of the crystal surface that have a small angle $\\phi$ with the facet are in fact staircases of rare steps on this facet. The height of such steps is equal to the period of the crystal lattice $h$.\n[figure1]\n\nEquilibrium surface profile $y(x)$ of a certain crystal and the corresponding microscopic staircase are shown schematically in the figure, where $\\mathrm{n}$ denotes the step number, counting from $x=0$. The profile shape at $x>0$ can be approximated as $y(x)=-\\left(\\frac{x}{\\lambda}\\right)^{\\frac{3}{2}} h$, where $\\lambda=45 \\mu \\mathrm{m}$ and $h=0.3 \\mathrm{~nm}$.\n\nThe interaction energy $E$ of two steps depend on the distance $d$ between them as\n\n$$\nE(d)=\\mu d^{\\nu}\n$$\n\nwhere $\\mu$ is a constant. Assume that only adjacent steps interact. \n\nFind the numerical value for the exponent $\\nu$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nStaircase \n\nThe equilibrium shape of bodies in zero-gravity is determined by the minimum of their surface energy. Thus, for example, the equilibrium shape of a water droplet turns out to be spherical: the sphere has the smallest surface area among bodies of the same volume. At low temperature, the equilibrium shape of crystals may have flat facets. The parts of the crystal surface that have a small angle $\\phi$ with the facet are in fact staircases of rare steps on this facet. The height of such steps is equal to the period of the crystal lattice $h$.\n[figure1]\n\nEquilibrium surface profile $y(x)$ of a certain crystal and the corresponding microscopic staircase are shown schematically in the figure, where $\\mathrm{n}$ denotes the step number, counting from $x=0$. The profile shape at $x>0$ can be approximated as $y(x)=-\\left(\\frac{x}{\\lambda}\\right)^{\\frac{3}{2}} h$, where $\\lambda=45 \\mu \\mathrm{m}$ and $h=0.3 \\mathrm{~nm}$.\n\nThe interaction energy $E$ of two steps depend on the distance $d$ between them as\n\n$$\nE(d)=\\mu d^{\\nu}\n$$\n\nwhere $\\mu$ is a constant. Assume that only adjacent steps interact. \n\nFind the numerical value for the exponent $\\nu$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2878f1ea9876cd74ba9g-1.jpg?height=598&width=1418&top_left_y=684&top_left_x=344" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_517", "problem": "Suppose that, after some amount of work is done by the ion pumps, the charges on the outer and inner surfaces are $Q$ and $-Q$, respectively.\n\nOne charge layer by itself creates an electric field $E_{1}=Q /(2 \\epsilon A)$ in each direction. So the force between the two sides of the membrane $F_{E}=Q E_{1}=Q^{2} /(2 \\epsilon A)$.\n\nThis electric force is balanced by the spring force $F_{s}=k x$, where $x=d_{0}-d$. Equating these two forces and solving for $d$ gives gives\n\n$$\nd=d_{0}-\\frac{Q^{2}}{2 \\epsilon A k} .\n$$\n\nThe electric field inside the membrane (as produced by both the left and right plates) is $E=Q /\\left(\\epsilon_{0} \\kappa A\\right)$. So the voltage between them is\n\n$$\nV=E d=\\frac{Q}{\\epsilon A} d\n$$\n\nInserting the expression for $Q$ from part (a) gives\n\n$$\nV=\\frac{Q}{\\epsilon A}\\left(d_{0}-\\frac{Q^{2}}{2 \\epsilon A k}\\right)\n$$\n\nThis equation implies that as the charge $Q$ is increased, the voltage first increases and then decreases again.\n\nThe voltage $V$ first increases and then decreases as a function of $Q$, which implies that there is a maximum voltage $V_{\\max }$ to which the membrane can be charged. This voltage can be found by taking the derivative $d V / d Q$ and setting it equal to zero. This procedure gives\n\n$$\nV_{\\max }=\\sqrt{\\frac{k d_{0}^{3}}{\\epsilon A}}\\left(\\frac{2}{3}\\right)^{3 / 2} .\n$$\n\nThe corresponding charge at the maximum voltage is given by\n\n$$\nQ_{\\mathrm{V} \\max }^{2}=\\frac{2}{3} \\epsilon A k d_{0}\n$$\n\nFor the ion pumps to turn off, we must have $V_{\\max }>V_{\\mathrm{th}}$. Otherwise the pumps will continue to move charge across the membrane until it collapses. Setting $V_{\\max }>V_{\\text {th }}$ and solving for $k$ gives\n\n$$\nk>\\left(\\frac{3}{2}\\right)^{3} \\frac{V_{\\mathrm{th}}^{2} \\epsilon A}{d_{0}^{3}} .\n$$[figure1]\n\nThe wall of a neuron is made from an elastic membrane, which resists compression in the same way as a spring. It has an effective spring constant $k$ and an equilibrium thickness $d_{0}$. Assume that the membrane has a very large area $A$ and negligible curvature.\n\nThe neuron has \"ion pumps\" that can move ions across the membrane. In the resulting charged state, positive and negative ionic charge is arranged uniformly along the outer and inner surfaces of the membrane, respectively. The permittivity of the membrane is $\\epsilon$.\n\nSuppose that the ion pumps are first turned on in the uncharged state, and the membrane is charged very slowly (quasistatically). The pumps will only turn off when the voltage difference across the membrane becomes larger than a particular value $V_{\\text {th }}$.\n\nHow much work is done by the ion pumps in each of the following situations? Express your answers in terms of $k$ and $d_{0}$.\n\n($k$ is infinitesimally smaller than the value derived before.)", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nSuppose that, after some amount of work is done by the ion pumps, the charges on the outer and inner surfaces are $Q$ and $-Q$, respectively.\n\nOne charge layer by itself creates an electric field $E_{1}=Q /(2 \\epsilon A)$ in each direction. So the force between the two sides of the membrane $F_{E}=Q E_{1}=Q^{2} /(2 \\epsilon A)$.\n\nThis electric force is balanced by the spring force $F_{s}=k x$, where $x=d_{0}-d$. Equating these two forces and solving for $d$ gives gives\n\n$$\nd=d_{0}-\\frac{Q^{2}}{2 \\epsilon A k} .\n$$\n\nThe electric field inside the membrane (as produced by both the left and right plates) is $E=Q /\\left(\\epsilon_{0} \\kappa A\\right)$. So the voltage between them is\n\n$$\nV=E d=\\frac{Q}{\\epsilon A} d\n$$\n\nInserting the expression for $Q$ from part (a) gives\n\n$$\nV=\\frac{Q}{\\epsilon A}\\left(d_{0}-\\frac{Q^{2}}{2 \\epsilon A k}\\right)\n$$\n\nThis equation implies that as the charge $Q$ is increased, the voltage first increases and then decreases again.\n\nThe voltage $V$ first increases and then decreases as a function of $Q$, which implies that there is a maximum voltage $V_{\\max }$ to which the membrane can be charged. This voltage can be found by taking the derivative $d V / d Q$ and setting it equal to zero. This procedure gives\n\n$$\nV_{\\max }=\\sqrt{\\frac{k d_{0}^{3}}{\\epsilon A}}\\left(\\frac{2}{3}\\right)^{3 / 2} .\n$$\n\nThe corresponding charge at the maximum voltage is given by\n\n$$\nQ_{\\mathrm{V} \\max }^{2}=\\frac{2}{3} \\epsilon A k d_{0}\n$$\n\nFor the ion pumps to turn off, we must have $V_{\\max }>V_{\\mathrm{th}}$. Otherwise the pumps will continue to move charge across the membrane until it collapses. Setting $V_{\\max }>V_{\\text {th }}$ and solving for $k$ gives\n\n$$\nk>\\left(\\frac{3}{2}\\right)^{3} \\frac{V_{\\mathrm{th}}^{2} \\epsilon A}{d_{0}^{3}} .\n$$\n\nproblem:\n[figure1]\n\nThe wall of a neuron is made from an elastic membrane, which resists compression in the same way as a spring. It has an effective spring constant $k$ and an equilibrium thickness $d_{0}$. Assume that the membrane has a very large area $A$ and negligible curvature.\n\nThe neuron has \"ion pumps\" that can move ions across the membrane. In the resulting charged state, positive and negative ionic charge is arranged uniformly along the outer and inner surfaces of the membrane, respectively. The permittivity of the membrane is $\\epsilon$.\n\nSuppose that the ion pumps are first turned on in the uncharged state, and the membrane is charged very slowly (quasistatically). The pumps will only turn off when the voltage difference across the membrane becomes larger than a particular value $V_{\\text {th }}$.\n\nHow much work is done by the ion pumps in each of the following situations? Express your answers in terms of $k$ and $d_{0}$.\n\n($k$ is infinitesimally smaller than the value derived before.)\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_fed953f9e38b72bf8bd7g-14.jpg?height=358&width=927&top_left_y=469&top_left_x=596" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_785", "problem": "A block of mass $5 \\mathrm{~kg}$ lies at rest on a horizontal surface. An upwards force of $20 \\mathrm{~N}$ is applied to the block, as shown. Assuming $g=10 \\mathrm{~ms}^{-2}$, what is the weight of the block?\n[figure1]\nA: $3 \\mathrm{~kg}$\nB: $5 \\mathrm{~kg}$\nC: $30 \\mathrm{~N}$\nD: $50 \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA block of mass $5 \\mathrm{~kg}$ lies at rest on a horizontal surface. An upwards force of $20 \\mathrm{~N}$ is applied to the block, as shown. Assuming $g=10 \\mathrm{~ms}^{-2}$, what is the weight of the block?\n[figure1]\n\nA: $3 \\mathrm{~kg}$\nB: $5 \\mathrm{~kg}$\nC: $30 \\mathrm{~N}$\nD: $50 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9bac30101a7c40be9bg-05.jpg?height=317&width=416&top_left_y=441&top_left_x=266" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_420", "problem": "The figure shows a load of mass, $m$, supported by a simple pulley system with a tension $T$ in the cord. \n\n[figure1]\nFigure: Two light pulleys and a light cord.\n\nWhat length of cord must be extracted at the free end, $\\mathbf{X}$, to raise the mass through a distance $h$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe figure shows a load of mass, $m$, supported by a simple pulley system with a tension $T$ in the cord. \n\n[figure1]\nFigure: Two light pulleys and a light cord.\n\nWhat length of cord must be extracted at the free end, $\\mathbf{X}$, to raise the mass through a distance $h$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-4.jpg?height=497&width=391&top_left_y=742&top_left_x=838" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_297", "problem": "A physics experiment a golf ball and a squash ball are both launched horizontally in such a way that they both have the same momentum.\n\nThe mass of the golf ball is $46 \\mathrm{~g}$ and the mass of the squash ball is $23 \\mathrm{~g}$.\n\nThe ratio $\\frac{\\text { kinetic energy golf ball }}{\\text { kinetic energy squash ball }}$ is:\nA: $1: 4$\nB: $1: 2$\nC: 2:1\nD: $4: 1$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA physics experiment a golf ball and a squash ball are both launched horizontally in such a way that they both have the same momentum.\n\nThe mass of the golf ball is $46 \\mathrm{~g}$ and the mass of the squash ball is $23 \\mathrm{~g}$.\n\nThe ratio $\\frac{\\text { kinetic energy golf ball }}{\\text { kinetic energy squash ball }}$ is:\n\nA: $1: 4$\nB: $1: 2$\nC: 2:1\nD: $4: 1$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_171", "problem": "Two planets $A$ and $B$ have masses $m_{A}=2 m_{B}$. They orbit a star in circular orbits of radius $r_{A}=3 r_{B}$. Let $E_{i}$ and $L_{i}$ be the kinetic energy and the magnitude of the angular momentum of planet $i$, respectively. Which of the following is true?\nA: $E_{A}>E_{B}$ and $L_{A}>L_{B}$\nB: $E_{A}>E_{B}$ and $L_{A}L_{B} $ \nD: $E_{A}L_{B}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo planets $A$ and $B$ have masses $m_{A}=2 m_{B}$. They orbit a star in circular orbits of radius $r_{A}=3 r_{B}$. Let $E_{i}$ and $L_{i}$ be the kinetic energy and the magnitude of the angular momentum of planet $i$, respectively. Which of the following is true?\n\nA: $E_{A}>E_{B}$ and $L_{A}>L_{B}$\nB: $E_{A}>E_{B}$ and $L_{A}L_{B} $ \nD: $E_{A}L_{B}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_960", "problem": "The idea of centre of mass is an important concept and is more appreciated with examples of its use.\n\nA $34 \\mathrm{~cm}$ long uniform straight rod lies on a smooth horizontal surface and it is seen to be spinning round whilst also moving across the surface (translating). At one particular moment in time it is observed that the velocities of the ends of the rod are normal to the rod and have values, $2.6 \\mathrm{~m} \\mathrm{~s}^{-1}$ and $4.2 \\mathrm{~m} \\mathrm{~s}^{-1}$ as illustrated in Fig. 2.\n\n[figure1]\n\nFigure 2: A uniform rod which is rotating and translating across a smooth horizontal surface.\n\nAt what speed would you need to fly over the rod as an observer to see only its rotational motion?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe idea of centre of mass is an important concept and is more appreciated with examples of its use.\n\nA $34 \\mathrm{~cm}$ long uniform straight rod lies on a smooth horizontal surface and it is seen to be spinning round whilst also moving across the surface (translating). At one particular moment in time it is observed that the velocities of the ends of the rod are normal to the rod and have values, $2.6 \\mathrm{~m} \\mathrm{~s}^{-1}$ and $4.2 \\mathrm{~m} \\mathrm{~s}^{-1}$ as illustrated in Fig. 2.\n\n[figure1]\n\nFigure 2: A uniform rod which is rotating and translating across a smooth horizontal surface.\n\nAt what speed would you need to fly over the rod as an observer to see only its rotational motion?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~m} / \\mathrm{s}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_9de0d5715d0b2f377364g-04.jpg?height=560&width=355&top_left_y=1496&top_left_x=859" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m} / \\mathrm{s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1513", "problem": "平行板电容器两极板分别位于 $z= \\pm \\frac{d}{2}$ 的平面内,电容器起初未被充电. 整个装置处于均匀磁场中, 磁感应强度大小为 $B$, 方向沿 $x$ 轴负方向, 如图所示.\n\n[图1]在电容器参考系 $S$ 中只存在磁场; 而在以沿 $y$ 轴正方向的恒定速度 $(0, v, 0)$ （这里 $(0, v, 0)$ 表示为沿 $x 、 y 、 z$ 轴正方向的速度分量分别为 $0 、 v 、 0$, 以下类似) 相对于电容器运动的参考系 $S^{\\prime}$ 中, 可能既有电场 $\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$ 又有磁场 $\\left(B_{x}^{\\prime}, B_{y}^{\\prime}, B_{z}^{\\prime}\\right)$. 试在非相对论情形下, 从伽利略速度变换, 求出在参考系 $S^{\\prime}$ 中电场 $\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$ 和磁场 $\\left(B_{x}^{\\prime}, B_{y}^{\\prime}, B_{z}^{\\prime}\\right)$ 的表达式. 已知电荷量和作用在物体上的合力在伽利略变换下不变.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n平行板电容器两极板分别位于 $z= \\pm \\frac{d}{2}$ 的平面内,电容器起初未被充电. 整个装置处于均匀磁场中, 磁感应强度大小为 $B$, 方向沿 $x$ 轴负方向, 如图所示.\n\n[图1]\n\n问题:\n在电容器参考系 $S$ 中只存在磁场; 而在以沿 $y$ 轴正方向的恒定速度 $(0, v, 0)$ （这里 $(0, v, 0)$ 表示为沿 $x 、 y 、 z$ 轴正方向的速度分量分别为 $0 、 v 、 0$, 以下类似) 相对于电容器运动的参考系 $S^{\\prime}$ 中, 可能既有电场 $\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$ 又有磁场 $\\left(B_{x}^{\\prime}, B_{y}^{\\prime}, B_{z}^{\\prime}\\right)$. 试在非相对论情形下, 从伽利略速度变换, 求出在参考系 $S^{\\prime}$ 中电场 $\\left(E_{x}^{\\prime}, E_{y}^{\\prime}, E_{z}^{\\prime}\\right)$ 和磁场 $\\left(B_{x}^{\\prime}, B_{y}^{\\prime}, B_{z}^{\\prime}\\right)$ 的表达式. 已知电荷量和作用在物体上的合力在伽利略变换下不变.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[电场, 磁场]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_734a14c1a8de5e19f8ecg-03.jpg?height=428&width=657&top_left_y=357&top_left_x=1405" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "电场", "磁场" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_311", "problem": "Particles with mass $m$ and charge $q$ are launched from the origin with speed $v$, parallel to the $x$-axis. There is a screen at $x=l$\n\nNext, the electric field is turned off, a homogeneous $z$-directional magnetic field in region $l>x>0$ is swithced on, and the second particle is launched. Knowing that particle's speed is just big enough to reach the screen, find the magnetic fied strength $B$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nParticles with mass $m$ and charge $q$ are launched from the origin with speed $v$, parallel to the $x$-axis. There is a screen at $x=l$\n\nNext, the electric field is turned off, a homogeneous $z$-directional magnetic field in region $l>x>0$ is swithced on, and the second particle is launched. Knowing that particle's speed is just big enough to reach the screen, find the magnetic fied strength $B$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_19", "problem": "Two bodies of equal mass are moving in circular paths at equal speed. The first body moves in a circle whose radius is twice that of the second. The ratio of the radial acceleration of the first body to that of the second is:\nA: 1 to 4\nB: 1 to 3\nC: 1 to 2\nD: 1 to 1\nE: 1 to 0.5\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo bodies of equal mass are moving in circular paths at equal speed. The first body moves in a circle whose radius is twice that of the second. The ratio of the radial acceleration of the first body to that of the second is:\n\nA: 1 to 4\nB: 1 to 3\nC: 1 to 2\nD: 1 to 1\nE: 1 to 0.5\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1424", "problem": "一线光源, 已知它发出的光包含三种不同频率的可见光, 若要使它通过三棱镜分光, 最后能在屏上看到这三种不同频率的光谱线, 则除了光源、三棱镜和屏外, 必需的器件至少还应有?\n\n其中一个的位置应在?和?之间，\n\n另一个的位置应在?和?之间。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n一线光源, 已知它发出的光包含三种不同频率的可见光, 若要使它通过三棱镜分光, 最后能在屏上看到这三种不同频率的光谱线, 则除了光源、三棱镜和屏外, 必需的器件至少还应有?\n\n其中一个的位置应在?和?之间，\n\n另一个的位置应在?和?之间。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[必需的器件至少还应有?, 其中一个的位置应在?和?之间, 另一个的位置应在?和?之间]\n它们的答案类型依次是[数值, 元组, 元组]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "必需的器件至少还应有?", "其中一个的位置应在?和?之间", "另一个的位置应在?和?之间" ], "type_sequence": [ "NV", "TUP", "TUP" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_282", "problem": "This question is about a familiar school practical to measure acceleration where a student investigates the acceleration of a toy car on a ramp.\n\nThe height of the ramp is changed and the resulting acceleration obtained from the measurements given below.\n\nThe experimental setup is shown in the diagram.\n\n[figure1]\n\nTo measure the acceleration, the student releases the car from rest at the top of the ramp and uses a stopwatch to time how long it takes for the car to reach the bottom of the ramp. The student records the following results:\n\n| Ramp height $/ \\mathrm{cm}$ | time $\\left(1^{\\text {st }}\\right.$ attempt $) / \\mathrm{s}$ | time $\\left(2^{\\text {nd }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(3^{\\text {rd }}\\right.$ attempt $) / \\mathrm{s}$ |\n| :--- | :--- | :--- | :--- |\n| 5 | 2.45 | 2.48 | 2.41 |\n\nThe student's teacher states \"the final speed of the car is twice the average speed\"\n\nThe teacher states \"theory suggests that the acceleration of the car is directly proportional to the height of the ramp\"\n\nThe student takes further readings of time for different heights of the ramp\n\n| Ramp height $/ \\mathrm{cm}$ | time $\\left(1^{\\text {st }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(2^{\\text {nd }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(3^{\\text {rd }}\\right.$ attempt) $/ \\mathrm{s}$ |\n| :--- | :--- | :--- | :--- |\n| 9 | 1.84 | 1.82 | 1.82 |\n| 13 | 1.50 | 1.52 | 1.56 |\n\nTheory shows that the relationship between ramp height $(h)$ and acceleration $(a)$ is given by the equation $a=\\frac{g \\times h}{L}$ where $g$ is the acceleration due to gravity and $L$ is the length of the ramp.\n\nUse the student's data to calculate a value for the acceleration due to gravity.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThis question is about a familiar school practical to measure acceleration where a student investigates the acceleration of a toy car on a ramp.\n\nThe height of the ramp is changed and the resulting acceleration obtained from the measurements given below.\n\nThe experimental setup is shown in the diagram.\n\n[figure1]\n\nTo measure the acceleration, the student releases the car from rest at the top of the ramp and uses a stopwatch to time how long it takes for the car to reach the bottom of the ramp. The student records the following results:\n\n| Ramp height $/ \\mathrm{cm}$ | time $\\left(1^{\\text {st }}\\right.$ attempt $) / \\mathrm{s}$ | time $\\left(2^{\\text {nd }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(3^{\\text {rd }}\\right.$ attempt $) / \\mathrm{s}$ |\n| :--- | :--- | :--- | :--- |\n| 5 | 2.45 | 2.48 | 2.41 |\n\nThe student's teacher states \"the final speed of the car is twice the average speed\"\n\nThe teacher states \"theory suggests that the acceleration of the car is directly proportional to the height of the ramp\"\n\nThe student takes further readings of time for different heights of the ramp\n\n| Ramp height $/ \\mathrm{cm}$ | time $\\left(1^{\\text {st }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(2^{\\text {nd }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(3^{\\text {rd }}\\right.$ attempt) $/ \\mathrm{s}$ |\n| :--- | :--- | :--- | :--- |\n| 9 | 1.84 | 1.82 | 1.82 |\n| 13 | 1.50 | 1.52 | 1.56 |\n\nTheory shows that the relationship between ramp height $(h)$ and acceleration $(a)$ is given by the equation $a=\\frac{g \\times h}{L}$ where $g$ is the acceleration due to gravity and $L$ is the length of the ramp.\n\nUse the student's data to calculate a value for the acceleration due to gravity.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{m} / \\mathrm{s}^2$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ae10da91e53998b18280g-09.jpg?height=260&width=1579&top_left_y=767&top_left_x=244" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{m} / \\mathrm{s}^2$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1460", "problem": "超导体的一个重要应用是绕制强磁场磁体, 其使用的超导线材属于第二类超导体.如果将这类超导体置于磁感应强度为 $B_{a}$ 的外磁场中, 其磁力线将以磁通量子 (或称为磁通漩浴线) 的形式穿透超导体, 从而在超导体中形成正三角形的磁通格子, 如图 1 所示. 所谓的磁通量子, 如图 2 所示, 其中心是半径为 $\\xi$ 的正常态 (电阻不为零) 区域, 而其周围处于超导态 (电阻为零), 存在超导电流, 所携带的磁通量为 $\\phi_{0}=\\frac{h}{2 e}=2.07 \\times 10^{-15} \\mathrm{~Wb}$ (磁通量的最小单位)\n\n[图1]\n\n图 1\n\n[图2]\n\n图3随着 $B_{a}$ 的增大, 磁通漩涡线密度不断增加, 当 $B_{a}$ 达到某一临界值 $B_{c 2}$ 时, 整块超导体都变为正常态，假设磁通漩浴线芯的半径为 $\\xi=5 \\times 10^{-9} \\mathrm{~m}$, 求所对应的临界磁场 $B_{c 2}$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n超导体的一个重要应用是绕制强磁场磁体, 其使用的超导线材属于第二类超导体.如果将这类超导体置于磁感应强度为 $B_{a}$ 的外磁场中, 其磁力线将以磁通量子 (或称为磁通漩浴线) 的形式穿透超导体, 从而在超导体中形成正三角形的磁通格子, 如图 1 所示. 所谓的磁通量子, 如图 2 所示, 其中心是半径为 $\\xi$ 的正常态 (电阻不为零) 区域, 而其周围处于超导态 (电阻为零), 存在超导电流, 所携带的磁通量为 $\\phi_{0}=\\frac{h}{2 e}=2.07 \\times 10^{-15} \\mathrm{~Wb}$ (磁通量的最小单位)\n\n[图1]\n\n图 1\n\n[图2]\n\n图3\n\n问题:\n随着 $B_{a}$ 的增大, 磁通漩涡线密度不断增加, 当 $B_{a}$ 达到某一临界值 $B_{c 2}$ 时, 整块超导体都变为正常态，假设磁通漩浴线芯的半径为 $\\xi=5 \\times 10^{-9} \\mathrm{~m}$, 求所对应的临界磁场 $B_{c 2}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以\\mathrm{~T}为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-05.jpg?height=400&width=471&top_left_y=1433&top_left_x=304", "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-05.jpg?height=483&width=701&top_left_y=1366&top_left_x=863" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~T}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_899", "problem": "$$\nk_{x}=\\frac{e_{0} V_{a} \\pi}{\\hbar v_{z} \\ln \\frac{b}{a}}\n$$The two-slit electron interference experiment was first performed by Möllenstedt et al, MerliMissiroli and Pozzi in 1974 and Tonomura et al in 1989. In the two-slit electron interference experiment, a monochromatic electron point source emits particles at $S$ that first passes through an electron \"biprism\" before impinging on an observational plane; $S_{1}$ and $S_{2}$ are virtual sources at distance $d$. In the diagram, the filament is pointing into the page. Note that it is a very thin filament (not drawn to scale in the diagram).\n\n[figure1]\n\nThe electron \"biprism\" consists of a grounded cylindrical wire mesh with a fine filament $F$ at the center. The distance between the source and the \"biprism\" is $\\ell$, and the distance between the distance between the \"biprism\" and the screen is $L$.\n\nIn Tonomura et al experiment,\n\n$$\n\\begin{aligned}\n& v_{z}=c / 2 \\\\\n& V_{a}=10 \\mathrm{~V}, \\\\\n& V_{0}=50 \\mathrm{kV} \\text {, } \\\\\n& a=0.5 \\mu \\mathrm{m}, \\\\\n& b=5 \\mathrm{~mm}, \\\\\n& \\ell \\quad=25 \\mathrm{~cm}, \\\\\n& L=1.5 \\mathrm{~m}, \\\\\n& h=6.6 \\times 10^{-34} \\mathrm{Js} \\text {, } \\\\\n& \\text { electron charge, } e=1.6 \\times 10^{-19} \\mathrm{C} \\text {, } \\\\\n& \\text { mass of electron, } \\mathrm{m}_{0}=9.1 \\times 10^{-31} \\mathrm{~kg} \\text {, } \\\\\n& \\text { and the speed of light in vacuo, } c=3 \\times 10^{8} \\mathrm{~ms}^{-1}\n\\end{aligned}\n$$\n\ncalculate the value of $k_{x}$,", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\n$$\nk_{x}=\\frac{e_{0} V_{a} \\pi}{\\hbar v_{z} \\ln \\frac{b}{a}}\n$$\n\nproblem:\nThe two-slit electron interference experiment was first performed by Möllenstedt et al, MerliMissiroli and Pozzi in 1974 and Tonomura et al in 1989. In the two-slit electron interference experiment, a monochromatic electron point source emits particles at $S$ that first passes through an electron \"biprism\" before impinging on an observational plane; $S_{1}$ and $S_{2}$ are virtual sources at distance $d$. In the diagram, the filament is pointing into the page. Note that it is a very thin filament (not drawn to scale in the diagram).\n\n[figure1]\n\nThe electron \"biprism\" consists of a grounded cylindrical wire mesh with a fine filament $F$ at the center. The distance between the source and the \"biprism\" is $\\ell$, and the distance between the distance between the \"biprism\" and the screen is $L$.\n\nIn Tonomura et al experiment,\n\n$$\n\\begin{aligned}\n& v_{z}=c / 2 \\\\\n& V_{a}=10 \\mathrm{~V}, \\\\\n& V_{0}=50 \\mathrm{kV} \\text {, } \\\\\n& a=0.5 \\mu \\mathrm{m}, \\\\\n& b=5 \\mathrm{~mm}, \\\\\n& \\ell \\quad=25 \\mathrm{~cm}, \\\\\n& L=1.5 \\mathrm{~m}, \\\\\n& h=6.6 \\times 10^{-34} \\mathrm{Js} \\text {, } \\\\\n& \\text { electron charge, } e=1.6 \\times 10^{-19} \\mathrm{C} \\text {, } \\\\\n& \\text { mass of electron, } \\mathrm{m}_{0}=9.1 \\times 10^{-31} \\mathrm{~kg} \\text {, } \\\\\n& \\text { and the speed of light in vacuo, } c=3 \\times 10^{8} \\mathrm{~ms}^{-1}\n\\end{aligned}\n$$\n\ncalculate the value of $k_{x}$,\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $m^{-1}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_4f547aee877827e020bbg-1.jpg?height=1319&width=1091&top_left_y=1082&top_left_x=471" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$m^{-1}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_366", "problem": "Particles with mass $m$ and charge $q$ are launched from the origin with speed $v$, parallel to the $x$-axis. There is a screen at $x=l$\n\nNext, the electric field is turned off, a homogeneous $z$-directional magnetic field in region $l>x>0$ is swithced on, and the second particle is launched. Knowing that particle's speed is just big enough to reach the screen, find the magnetic fied strength $B$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nParticles with mass $m$ and charge $q$ are launched from the origin with speed $v$, parallel to the $x$-axis. There is a screen at $x=l$\n\nNext, the electric field is turned off, a homogeneous $z$-directional magnetic field in region $l>x>0$ is swithced on, and the second particle is launched. Knowing that particle's speed is just big enough to reach the screen, find the magnetic fied strength $B$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_854", "problem": "$$\nB_{z}^{\\prime}(\\rho, z ; t)=f\\left(\\rho, z+v_{0} t\\right)\n$$[figure1]\n\nFig. 1 A monopole $q_{\\mathrm{m}}$ appears at a distance $h$ from a conducting thin film of thickness $d$. The origin of the coordinates is located on the upper surface.\n\nWe first focus on the initial response of the conducting thin film when at time $t=0$ a north monopole $q_{\\mathrm{m}}$ appears suddenly at the position $\\vec{r}_{\\mathrm{mp}}=h \\hat{z}(h>0)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nFor $t>0$, the total magnetic field $\\vec{B}$ becomes $\\vec{B}(\\vec{\\rho}, z ; t)=\\vec{B}_{\\mathrm{mp}}(\\vec{\\rho}, z)+\\vec{B}^{\\prime}(\\vec{\\rho}, z ; t)$, by superposition, with $\\vec{B}^{\\prime}(\\vec{\\rho}, z ; t)$ due to the induced electric current in the thin film. You are required below to obtain an equation for $B_{z}^{\\prime}(\\rho, z ; t)$ near the $z=0$ thin film surface. The time-evolution behavior of $B_{z}^{\\prime}$ would reveal a moving image-monopole picture for the description of the $\\vec{B}^{\\prime}$ field near $z \\approx 0$ in $t>0$.\n\nThe equation for $B_{z}^{\\prime}$ inside the thin film is given below,\n\n$$\n\\frac{\\partial^{2} B_{z}^{\\prime}(\\rho, z ; t)}{\\partial z^{2}}=\\mu_{0} \\sigma \\frac{\\partial B_{z}^{\\prime}(\\rho, z ; t)}{\\partial t} .\n$$\n\nThis equation has been obtained from imposing inside the thin film the Maxwell equation and the Ohmic behavior of the conducting thin film ( $\\vec{j}=\\sigma \\vec{E}$, where $\\sigma$ is the electrical conductivity) while neglecting the displacement-current effect. Term being neglected on the left-hand side of Eq.(2) is $\\frac{1}{\\rho} \\frac{\\partial}{\\partial \\rho}\\left(\\rho \\frac{\\partial B_{z}^{\\prime}}{\\partial \\rho}\\right)$, based on the $h \\gg d$ condition.\n\nShow that your solution in $\\mathbf{A} .6$ reveals a moving image-monopole picture for the magnetic field $B_{z}^{\\prime}(\\rho, z \\approx 0 ; t)$, with a downwardly moving velocity. Find the speed $v_{0}$ of the image monopole in terms of known parameters from the problem text.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\n$$\nB_{z}^{\\prime}(\\rho, z ; t)=f\\left(\\rho, z+v_{0} t\\right)\n$$\n\nproblem:\n[figure1]\n\nFig. 1 A monopole $q_{\\mathrm{m}}$ appears at a distance $h$ from a conducting thin film of thickness $d$. The origin of the coordinates is located on the upper surface.\n\nWe first focus on the initial response of the conducting thin film when at time $t=0$ a north monopole $q_{\\mathrm{m}}$ appears suddenly at the position $\\vec{r}_{\\mathrm{mp}}=h \\hat{z}(h>0)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nFor $t>0$, the total magnetic field $\\vec{B}$ becomes $\\vec{B}(\\vec{\\rho}, z ; t)=\\vec{B}_{\\mathrm{mp}}(\\vec{\\rho}, z)+\\vec{B}^{\\prime}(\\vec{\\rho}, z ; t)$, by superposition, with $\\vec{B}^{\\prime}(\\vec{\\rho}, z ; t)$ due to the induced electric current in the thin film. You are required below to obtain an equation for $B_{z}^{\\prime}(\\rho, z ; t)$ near the $z=0$ thin film surface. The time-evolution behavior of $B_{z}^{\\prime}$ would reveal a moving image-monopole picture for the description of the $\\vec{B}^{\\prime}$ field near $z \\approx 0$ in $t>0$.\n\nThe equation for $B_{z}^{\\prime}$ inside the thin film is given below,\n\n$$\n\\frac{\\partial^{2} B_{z}^{\\prime}(\\rho, z ; t)}{\\partial z^{2}}=\\mu_{0} \\sigma \\frac{\\partial B_{z}^{\\prime}(\\rho, z ; t)}{\\partial t} .\n$$\n\nThis equation has been obtained from imposing inside the thin film the Maxwell equation and the Ohmic behavior of the conducting thin film ( $\\vec{j}=\\sigma \\vec{E}$, where $\\sigma$ is the electrical conductivity) while neglecting the displacement-current effect. Term being neglected on the left-hand side of Eq.(2) is $\\frac{1}{\\rho} \\frac{\\partial}{\\partial \\rho}\\left(\\rho \\frac{\\partial B_{z}^{\\prime}}{\\partial \\rho}\\right)$, based on the $h \\gg d$ condition.\n\nShow that your solution in $\\mathbf{A} .6$ reveals a moving image-monopole picture for the magnetic field $B_{z}^{\\prime}(\\rho, z \\approx 0 ; t)$, with a downwardly moving velocity. Find the speed $v_{0}$ of the image monopole in terms of known parameters from the problem text.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d32b3b2f89cebe6f1c2ag-2.jpg?height=642&width=1244&top_left_y=296&top_left_x=194" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_189", "problem": "A very long cylinder of dust is spinning about its axis with angular velocity $\\omega$ at steady state. Let $r$ be the distance from the axis. If the dust is only held together by gravity, the density of the dust is proportional to:\nA: $r^{-2}$\nB: $r^{-1}$\nC: the density does not depend on $r . $ \nD: $r$\nE: $r^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA very long cylinder of dust is spinning about its axis with angular velocity $\\omega$ at steady state. Let $r$ be the distance from the axis. If the dust is only held together by gravity, the density of the dust is proportional to:\n\nA: $r^{-2}$\nB: $r^{-1}$\nC: the density does not depend on $r . $ \nD: $r$\nE: $r^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_563", "problem": "In parts this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinitely long wire wire on the $z$ axis is composed of positive charges with linear charge density $\\lambda$ which are at rest, and negative charges with linear charge density $-\\lambda$ moving with speed $v$ in the $z$ direction.\n\nDetermine the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) at points outside the wire.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn parts this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinitely long wire wire on the $z$ axis is composed of positive charges with linear charge density $\\lambda$ which are at rest, and negative charges with linear charge density $-\\lambda$ moving with speed $v$ in the $z$ direction.\n\nDetermine the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) at points outside the wire.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1565", "problem": "如图, 两个质量均为 $m$ 的小球 A 和 B（均可视为质点）固定在中心位于 C、长为 $2 l$ 的刚性轻质细杆的两端, 构成一质点系.在坚直面内建立 $O x y$ 坐标, $O x$ 方向沿水平向右, $O y$ 方向坚直向上.初始时质点系中心 $C$ 位于原点 $O$, 并以初速度 $v_{0}$ 坚直上抛, 上抛过程中, $A 、 C 、 B$ 三点连线\n\n[图1]\n始终水平. 风速大小恒定为 $u$ 、方向沿 $x$ 轴正向, 小球在运动中所受空气阻力 $f$ 的大小与相对于空气运动速度 $v$ 的大小成正比, 方向相反, 即 $\\vec{f}=-k \\vec{v}, k$ 为正的常量. 当 $\\mathrm{C}$ 点升至最高点时, 恰好有一沿 $y$ 轴正向运动、质量为 $m_{1}$ 、速度大小为 $u_{1}$ 的小石块 (视为质点) 与小球 $\\mathrm{A}$ 发生坚直方向的碰撞, 设碰撞是完全弹性的, 时间极短. 此后 $\\mathrm{C}$ 点回落到上抛开始时的同一水平高度, 此时它在 $O x$ 方向上的位置记为 $s$, 将从上抛到落回的整个过程所用时间记为 $T$, 质点系旋转的圈数记为 $n$. 求质点系转动的初始角速度 $\\omega_{0}$, 及其回落到 $s$ 点时角速度 $\\omega_{s}$ 与 $n$ 的关系", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 两个质量均为 $m$ 的小球 A 和 B（均可视为质点）固定在中心位于 C、长为 $2 l$ 的刚性轻质细杆的两端, 构成一质点系.在坚直面内建立 $O x y$ 坐标, $O x$ 方向沿水平向右, $O y$ 方向坚直向上.初始时质点系中心 $C$ 位于原点 $O$, 并以初速度 $v_{0}$ 坚直上抛, 上抛过程中, $A 、 C 、 B$ 三点连线\n\n[图1]\n始终水平. 风速大小恒定为 $u$ 、方向沿 $x$ 轴正向, 小球在运动中所受空气阻力 $f$ 的大小与相对于空气运动速度 $v$ 的大小成正比, 方向相反, 即 $\\vec{f}=-k \\vec{v}, k$ 为正的常量. 当 $\\mathrm{C}$ 点升至最高点时, 恰好有一沿 $y$ 轴正向运动、质量为 $m_{1}$ 、速度大小为 $u_{1}$ 的小石块 (视为质点) 与小球 $\\mathrm{A}$ 发生坚直方向的碰撞, 设碰撞是完全弹性的, 时间极短. 此后 $\\mathrm{C}$ 点回落到上抛开始时的同一水平高度, 此时它在 $O x$ 方向上的位置记为 $s$, 将从上抛到落回的整个过程所用时间记为 $T$, 质点系旋转的圈数记为 $n$. 求质点系\n\n问题:\n转动的初始角速度 $\\omega_{0}$, 及其回落到 $s$ 点时角速度 $\\omega_{s}$ 与 $n$ 的关系\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-06.jpg?height=260&width=280&top_left_y=1395&top_left_x=1502", "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-26.jpg?height=120&width=220&top_left_y=2558&top_left_x=1609", "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-27.jpg?height=297&width=348&top_left_y=1096&top_left_x=1408", "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-28.jpg?height=257&width=237&top_left_y=2182&top_left_x=1595" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1329", "problem": "质量为 $M$ 的绝热薄壁容器处于远离其他星体的太空（可视为真空）中。在某惯性系中观察, 该容器的初始速度为零。容器的容积为 $V$, 容器中充有某种单原子分子理想气体, 气体的初始分子数、分子质量分别为 $N_{0} 、 m$, 气体的初始温度为 $T_{0} 。 t=0$ 时容器壁上出现面积为 $S$ 的一个小孔, 由于小孔漏气导致容器开始运动, 但容器没有转动。假设小孔较小, 容器中的气体在泄漏过程中始终处于平衡态。已知气体分子速度沿 $x$ 方向的分量 $v_{x}$ 的麦克斯韦分布函数为 $f\\left(v_{x}\\right)=\\sqrt{\\frac{m}{2 \\pi k T}} \\exp \\left(-\\frac{m v_{x}^{2}}{2 k T}\\right)$ ( $k$ 为玻尔兹曼常量)。在泄漏过程中, 求:\n\n已知积分公式: $\\int_{0}^{\\infty} x \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{2 A}, \\int_{0}^{\\infty} x^{2} \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{4} \\sqrt{\\frac{\\pi}{A^{3}}}, \\int_{0}^{\\infty} x^{3} \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{2 A^{2}}$$t$ 时刻容器中气体的温度;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n质量为 $M$ 的绝热薄壁容器处于远离其他星体的太空（可视为真空）中。在某惯性系中观察, 该容器的初始速度为零。容器的容积为 $V$, 容器中充有某种单原子分子理想气体, 气体的初始分子数、分子质量分别为 $N_{0} 、 m$, 气体的初始温度为 $T_{0} 。 t=0$ 时容器壁上出现面积为 $S$ 的一个小孔, 由于小孔漏气导致容器开始运动, 但容器没有转动。假设小孔较小, 容器中的气体在泄漏过程中始终处于平衡态。已知气体分子速度沿 $x$ 方向的分量 $v_{x}$ 的麦克斯韦分布函数为 $f\\left(v_{x}\\right)=\\sqrt{\\frac{m}{2 \\pi k T}} \\exp \\left(-\\frac{m v_{x}^{2}}{2 k T}\\right)$ ( $k$ 为玻尔兹曼常量)。在泄漏过程中, 求:\n\n已知积分公式: $\\int_{0}^{\\infty} x \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{2 A}, \\int_{0}^{\\infty} x^{2} \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{4} \\sqrt{\\frac{\\pi}{A^{3}}}, \\int_{0}^{\\infty} x^{3} \\mathrm{e}^{-A x^{2}} \\mathrm{~d} x=\\frac{1}{2 A^{2}}$\n\n问题:\n$t$ 时刻容器中气体的温度;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_478", "problem": "The flow of heat through a material can be described via the thermal conductivity $\\kappa$. If the two faces of a slab of material with thermal conductivity $\\kappa$, area $A$, and thickness $d$ are held at temperatures differing by $\\Delta T$, the thermal power $P$ transferred through the slab is\n\n$$\nP=\\frac{\\kappa A \\Delta T}{d}\n$$\n\nA large, flat lake in the upper Midwest has a uniform depth of 5.0 meters of water that is covered by a uniform layer of $1.0 \\mathrm{~cm}$ of ice. Cold air has moved into the region so that the upper surface of the ice is now maintained at a constant temperature of $-10{ }^{\\circ} \\mathrm{C}$ by the cold air (an infinitely large constant temperature heat sink). The bottom of the lake remains at a fixed $4.0^{\\circ} \\mathrm{C}$ because of contact with the earth (an infinitely large constant temperature heat source). It is reasonable to assume that heat flow is only in the vertical direction and that there is no convective motion in the water.\n\nAssuming the air stays at the same temperature for a long time, find the equilibrium thickness of the ice.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe flow of heat through a material can be described via the thermal conductivity $\\kappa$. If the two faces of a slab of material with thermal conductivity $\\kappa$, area $A$, and thickness $d$ are held at temperatures differing by $\\Delta T$, the thermal power $P$ transferred through the slab is\n\n$$\nP=\\frac{\\kappa A \\Delta T}{d}\n$$\n\nA large, flat lake in the upper Midwest has a uniform depth of 5.0 meters of water that is covered by a uniform layer of $1.0 \\mathrm{~cm}$ of ice. Cold air has moved into the region so that the upper surface of the ice is now maintained at a constant temperature of $-10{ }^{\\circ} \\mathrm{C}$ by the cold air (an infinitely large constant temperature heat sink). The bottom of the lake remains at a fixed $4.0^{\\circ} \\mathrm{C}$ because of contact with the earth (an infinitely large constant temperature heat source). It is reasonable to assume that heat flow is only in the vertical direction and that there is no convective motion in the water.\n\nAssuming the air stays at the same temperature for a long time, find the equilibrium thickness of the ice.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~m}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_422", "problem": "Clearly, any friction in the mechanism will lead to inefficiency: it is, however, instructive to examine the assumption that the pulley blocks have negligible mass.\n\nFigure shows a more realistic system, in which the lower half of the system (the pulley block) has a mass $m / k$, where $k$ is a numerical parameter.\n\n[figure1]\nFigure: Two pulley blocks and a light cord.\n\nWhat length of cord must be extracted at the free end, $\\mathbf{X}$, to raise the mass through a distance $h$?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nClearly, any friction in the mechanism will lead to inefficiency: it is, however, instructive to examine the assumption that the pulley blocks have negligible mass.\n\nFigure shows a more realistic system, in which the lower half of the system (the pulley block) has a mass $m / k$, where $k$ is a numerical parameter.\n\n[figure1]\nFigure: Two pulley blocks and a light cord.\n\nWhat length of cord must be extracted at the free end, $\\mathbf{X}$, to raise the mass through a distance $h$?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-5.jpg?height=902&width=460&top_left_y=323&top_left_x=1392" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1332", "problem": "平行板电容器极板 1 和 2 的面积均为 $S$, 水平固定放置,它们之间的距离为 $d$, 接入如图所示的电路中, 电源的电动势记为 $U$ 。不带电的导体薄平板 3 的质量为 $m$ 、尺寸与电容器极板相同。平板 3 平放在极板 2 的正上方, 且与极板 2 有良好的电接触。整个系统置于真空室内, 真空的介电常量为 $\\varepsilon_{0}$ 。闭合电键 $\\mathrm{K}$ 后, 平板 3 与极板 1 和 2 相继碰撞, 上下往复运动。假设导体板之间的电场均可视为匀强电场; 导线电阻和电源内阻足够小, 充放电时间可忽略不计; 平板 3 与极板 1 或 2 碰撞后立即在极短时间内达到静电平衡; 所有碰撞都是完全非弹性的。重力加速度大小为 $g$ 。\n\n[图1]\n\n已知积分公式\n\n$$\n\\int \\frac{\\mathrm{d} x}{\\sqrt{a x^{2}+b x}}=\\frac{1}{\\sqrt{a}} \\ln \\left(2 a x+b+2 \\sqrt{a} \\sqrt{a x^{2}+b x}\\right)+C \\text {, 其中 } a>0, C \\text { 为积分常数。 }\n$$求平板3运动的周期（用 $U$ 和题给条件表示）。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n平行板电容器极板 1 和 2 的面积均为 $S$, 水平固定放置,它们之间的距离为 $d$, 接入如图所示的电路中, 电源的电动势记为 $U$ 。不带电的导体薄平板 3 的质量为 $m$ 、尺寸与电容器极板相同。平板 3 平放在极板 2 的正上方, 且与极板 2 有良好的电接触。整个系统置于真空室内, 真空的介电常量为 $\\varepsilon_{0}$ 。闭合电键 $\\mathrm{K}$ 后, 平板 3 与极板 1 和 2 相继碰撞, 上下往复运动。假设导体板之间的电场均可视为匀强电场; 导线电阻和电源内阻足够小, 充放电时间可忽略不计; 平板 3 与极板 1 或 2 碰撞后立即在极短时间内达到静电平衡; 所有碰撞都是完全非弹性的。重力加速度大小为 $g$ 。\n\n[图1]\n\n已知积分公式\n\n$$\n\\int \\frac{\\mathrm{d} x}{\\sqrt{a x^{2}+b x}}=\\frac{1}{\\sqrt{a}} \\ln \\left(2 a x+b+2 \\sqrt{a} \\sqrt{a x^{2}+b x}\\right)+C \\text {, 其中 } a>0, C \\text { 为积分常数。 }\n$$\n\n问题:\n求平板3运动的周期（用 $U$ 和题给条件表示）。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_35bc41298eef336dfdafg-01.jpg?height=317&width=448&top_left_y=1966&top_left_x=1318", "https://cdn.mathpix.com/cropped/2024_03_31_35bc41298eef336dfdafg-12.jpg?height=245&width=368&top_left_y=1254&top_left_x=1412" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1387", "problem": "找到两块很大的金属平面, 如图 11 所示摆成 $\\theta_{0}=\\frac{\\pi}{6}$ 角, 角的顶点为 $\\mathrm{O}$ 点, 两块板之间在电压大小为 $\\mathrm{V}_{0}$ 的电源, 金属板和 $\\mathrm{O}$ 点比较靠近, 以至于在角内的电场线几乎为圆弧, $\\mathrm{A}$ 位于角内, $|O A|$ $=\\rho, \\mathrm{OA}$ 和下面的平面夹角为 $\\theta$ 。\n\n[图1]\n\n图 11计算 $\\mathrm{A}$ 的电场和电势大小;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n找到两块很大的金属平面, 如图 11 所示摆成 $\\theta_{0}=\\frac{\\pi}{6}$ 角, 角的顶点为 $\\mathrm{O}$ 点, 两块板之间在电压大小为 $\\mathrm{V}_{0}$ 的电源, 金属板和 $\\mathrm{O}$ 点比较靠近, 以至于在角内的电场线几乎为圆弧, $\\mathrm{A}$ 位于角内, $|O A|$ $=\\rho, \\mathrm{OA}$ 和下面的平面夹角为 $\\theta$ 。\n\n[图1]\n\n图 11\n\n问题:\n计算 $\\mathrm{A}$ 的电场和电势大小;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[计算 $\\mathrm{A}$ 的电场强度, 计算 $\\mathrm{A}$ 的电势大小]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-06.jpg?height=377&width=576&top_left_y=1513&top_left_x=777" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "计算 $\\mathrm{A}$ 的电场强度", "计算 $\\mathrm{A}$ 的电势大小" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_88", "problem": "On November 16, 2022, the Artemis 1 rocket lifted off from Launch Pad 39 at Cape Canaveral, FL for its trip to the moon. To lift off, the engines supplied 8.8 million pounds $\\left(4.0 \\times 10^{7} \\mathrm{~N}\\right)$ of thrust. Which of the following does NOT express this thrust force?\nA: $4.0 \\times 10^{10} \\mathrm{mN}$\nB: $4.0 \\times 10^{4} \\mathrm{kN}$\nC: $4.0 \\times 10^{1} \\mathrm{MN}$\nD: $4.0 \\times 10^{-3} \\mathrm{GN}$\nE: $4.0 \\times 10^{-5} \\mathrm{TN}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOn November 16, 2022, the Artemis 1 rocket lifted off from Launch Pad 39 at Cape Canaveral, FL for its trip to the moon. To lift off, the engines supplied 8.8 million pounds $\\left(4.0 \\times 10^{7} \\mathrm{~N}\\right)$ of thrust. Which of the following does NOT express this thrust force?\n\nA: $4.0 \\times 10^{10} \\mathrm{mN}$\nB: $4.0 \\times 10^{4} \\mathrm{kN}$\nC: $4.0 \\times 10^{1} \\mathrm{MN}$\nD: $4.0 \\times 10^{-3} \\mathrm{GN}$\nE: $4.0 \\times 10^{-5} \\mathrm{TN}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_179", "problem": "A mass of $M=100 \\mathrm{~g}$ is attached to the end of a string of length $R=2 \\mathrm{~m}$. A person swings the mass overhead such that their hand traverses a circle of radius $r=3 \\mathrm{~cm}$ at angular velocity $\\omega=10 \\mathrm{rad} / \\mathrm{s}$, ahead of the mass $M$ by an angle of $\\pi / 2$. Estimate the force of air resistance on the object.\n\n[figure1]\nA: $0.02 \\mathrm{~N}$\nB: $0.03 \\mathrm{~N}$\nC: $0.2 \\mathrm{~N}$\nD: $0.3 \\mathrm{~N} $\nE: $2 \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA mass of $M=100 \\mathrm{~g}$ is attached to the end of a string of length $R=2 \\mathrm{~m}$. A person swings the mass overhead such that their hand traverses a circle of radius $r=3 \\mathrm{~cm}$ at angular velocity $\\omega=10 \\mathrm{rad} / \\mathrm{s}$, ahead of the mass $M$ by an angle of $\\pi / 2$. Estimate the force of air resistance on the object.\n\n[figure1]\n\nA: $0.02 \\mathrm{~N}$\nB: $0.03 \\mathrm{~N}$\nC: $0.2 \\mathrm{~N}$\nD: $0.3 \\mathrm{~N} $\nE: $2 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-12.jpg?height=383&width=387&top_left_y=394&top_left_x=869" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_15", "problem": "The James Webb Space Telescope was launched on December 25, 2021 and arrived at its destination, LaGrange Point 2, on January 24, 2022. This point is approximately 1,500,000 km from the Earth and it is where the telescope will orbit the sun.How long will it take the radio signal sent by the James Webb Space Telescope to reach the Earth?\nA: $0.005 \\mathrm{~s}$\nB: $0.05 \\mathrm{~s}$\nC: $5 \\mathrm{~s}$\nD: $50 \\mathrm{~s}$\nE: $5000 \\mathrm{~s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe James Webb Space Telescope was launched on December 25, 2021 and arrived at its destination, LaGrange Point 2, on January 24, 2022. This point is approximately 1,500,000 km from the Earth and it is where the telescope will orbit the sun.How long will it take the radio signal sent by the James Webb Space Telescope to reach the Earth?\n\nA: $0.005 \\mathrm{~s}$\nB: $0.05 \\mathrm{~s}$\nC: $5 \\mathrm{~s}$\nD: $50 \\mathrm{~s}$\nE: $5000 \\mathrm{~s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1163", "problem": "# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nFind the numerical value of $\\frac{\\rho_{i} T_{i}}{\\rho_{a} T_{a}}-1$ using $\\gamma=0.0250 \\mathrm{Nm}^{-1}, R_{0}=1.00 \\mathrm{~cm}$, and $P_{a}=1.013 \\times 10^{5} \\mathrm{Nm}^{-2}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\n# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nFind the numerical value of $\\frac{\\rho_{i} T_{i}}{\\rho_{a} T_{a}}-1$ using $\\gamma=0.0250 \\mathrm{Nm}^{-1}, R_{0}=1.00 \\mathrm{~cm}$, and $P_{a}=1.013 \\times 10^{5} \\mathrm{Nm}^{-2}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_341", "problem": "ii) (2 points) According to Fermat's principle, light ray travels from one point to another along the path of shortest travel time. Suppose that in a certain medium, a light ray can propagate from $A$ to $B$ along the brachistochrone curve shown in the figure above. Find the refractive index $n=n(x, y)$ as a func tion of the coordinates $x$ and $y$ for this medium if $n(L, H)=1$\n\niii) (2 points) Show that the path of a light ray traveling in a medium with a variable refractive index $n(x, y) \\equiv n(y)$ satisfies the differential equation $\\frac{\\mathrm{d} y}{\\mathrm{~d} x}=\\sqrt{C \\cdot n(y)^{2}-1}$, where $C$ is a constant determined by boundary condi tions.Consider points $A$ and $B$ separated by height $H$ in the vertical direction and distance $L$ in the horizontal direction, placed in a gravitational field $g$ as shown in the figure below. A point mass can slide along a rail of fixed shape frictionlessly (including taking $90^{\\circ}$-turns) from $A$ to $B$. The brachistochrone curve is the curve minimizing the total travel time.[figure1]\n\nSolving the equations derived in parts ii) and iii), one may show that the brachistochrone curve is actually a segment of a cycloid. A cycloid is the curve traced by a fixed point on the rim of a circular wheel as it rolls along a straight line without slipping. For the special case $\\frac{L}{H}=\\frac{\\pi}{2}$ find the minimum ravel time $t_{\\min }$ between $A$ and $B$", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nii) (2 points) According to Fermat's principle, light ray travels from one point to another along the path of shortest travel time. Suppose that in a certain medium, a light ray can propagate from $A$ to $B$ along the brachistochrone curve shown in the figure above. Find the refractive index $n=n(x, y)$ as a func tion of the coordinates $x$ and $y$ for this medium if $n(L, H)=1$\n\niii) (2 points) Show that the path of a light ray traveling in a medium with a variable refractive index $n(x, y) \\equiv n(y)$ satisfies the differential equation $\\frac{\\mathrm{d} y}{\\mathrm{~d} x}=\\sqrt{C \\cdot n(y)^{2}-1}$, where $C$ is a constant determined by boundary condi tions.\n\nproblem:\nConsider points $A$ and $B$ separated by height $H$ in the vertical direction and distance $L$ in the horizontal direction, placed in a gravitational field $g$ as shown in the figure below. A point mass can slide along a rail of fixed shape frictionlessly (including taking $90^{\\circ}$-turns) from $A$ to $B$. The brachistochrone curve is the curve minimizing the total travel time.[figure1]\n\nSolving the equations derived in parts ii) and iii), one may show that the brachistochrone curve is actually a segment of a cycloid. A cycloid is the curve traced by a fixed point on the rim of a circular wheel as it rolls along a straight line without slipping. For the special case $\\frac{L}{H}=\\frac{\\pi}{2}$ find the minimum ravel time $t_{\\min }$ between $A$ and $B$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_706aca6df357b4c9a255g-2.jpg?height=336&width=664&top_left_y=436&top_left_x=56" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1449", "problem": "爱因斯坦等效原理可表述为: 在有引力作用的情况下的物理规律和没有引力但有适当加速度的参考系中的物理规律是相同的。作为一个例子，考察下面两种情况：引力红移现象的第一个实验验证是在地球表面附近利用穆斯堡尔探测器完成的, 穆斯堡尔探测器能以极高的精度分辨伽马光子的能量。按第（1）（i）问，在地球表面附近，A 处放置一个静止的伽马辐射源, 辐射的伽马光子的频率为 $v_{0} ; \\mathrm{B}$ 处放置一个穆斯堡尔探测器, 假设该探测器在相对于自身静止的参考系中仅能探测到频率为 $v_{0}$ 的伽马光子。为了探测到从 A 处发射的伽马光子，该穆斯堡尔探测器需要某一坚直向下的运动速度。1960-1964 年期间，庞德、雷布卡和斯奈德利用美国哈佛大学杰弗逊物理实验室的高塔多次做了这个实验, 实验中 $L=22.6 \\mathrm{~m}$ 。试问： $\\mathrm{A}$ 处发射的伽马光子被探测到时, 该穆斯堡尔探测器的运动速度为多大? 已知地球表面重力加速度 $g=9.81 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$, 真空中的光速 $c=3.00 \\times 10^{8} \\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n爱因斯坦等效原理可表述为: 在有引力作用的情况下的物理规律和没有引力但有适当加速度的参考系中的物理规律是相同的。作为一个例子，考察下面两种情况：\n\n问题:\n引力红移现象的第一个实验验证是在地球表面附近利用穆斯堡尔探测器完成的, 穆斯堡尔探测器能以极高的精度分辨伽马光子的能量。按第（1）（i）问，在地球表面附近，A 处放置一个静止的伽马辐射源, 辐射的伽马光子的频率为 $v_{0} ; \\mathrm{B}$ 处放置一个穆斯堡尔探测器, 假设该探测器在相对于自身静止的参考系中仅能探测到频率为 $v_{0}$ 的伽马光子。为了探测到从 A 处发射的伽马光子，该穆斯堡尔探测器需要某一坚直向下的运动速度。1960-1964 年期间，庞德、雷布卡和斯奈德利用美国哈佛大学杰弗逊物理实验室的高塔多次做了这个实验, 实验中 $L=22.6 \\mathrm{~m}$ 。试问： $\\mathrm{A}$ 处发射的伽马光子被探测到时, 该穆斯堡尔探测器的运动速度为多大? 已知地球表面重力加速度 $g=9.81 \\mathrm{~m} \\cdot \\mathrm{s}^{-2}$, 真空中的光速 $c=3.00 \\times 10^{8} \\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m} \\cdot \\mathrm{s}^{-1}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1097", "problem": "## Kelvin water dropper\n\nThe following facts about the surface tension may turn out to be useful for this problem. For the molecules of a liquid, the positions at the liquid-air interface are less favourable as compared with the positions in the bulk of the liquid. This interface is described by the so-called surface energy, $U=\\sigma S$, where $S$ is the surface area of the interface and $\\sigma$ is the surface tension coefficient of the liquid. Moreover, two fragments of the liquid surface pull each other with a force $F=\\sigma l$, where $l$ is the length of a straight line separating the fragments.\n\n[figure1]\n\nA long metallic pipe with internal diameter $d$ is pointing directly downwards. Water is slowly dripping from a nozzle at its lower end, see fig. Water can be considered to be electrically conducting; its surface tension is $\\sigma$ and its density is $\\rho$. A droplet of radius $r$ hangs below the nozzle. The radius grows slowly in time until the droplet separates from the nozzle due to the free fall acceleration $g$. Always assume that $d \\ll r$.\n\nAn apparatus called the \"Kelvin water dropper\" consists of two pipes, each identical to the one described in Part A, connected via a T-junction, see fig. The ends of both pipes are at the centres of two cylindrical electrodes (with height $L$ and diameter $D$ with $L \\gg D \\gg r$ ). For both tubes, the dripping rate is $n$ droplets per unit time. Droplets fall from height $H$ into conductive bowls underneath the nozzles, cross-connected to the electrodes as shown in the diagram. The electrodes are connected via a capacitance $C$. There is no net charge on the system of bowls and electrodes. Note that the top water container is earthed as shown. The first droplet to fall will have some microscopic charge which will cause an imbalance between the two sides and a small charge separation across the capacitor.\n\n[figure2]\n\nThe dropper's functioning can be hindered by the effect shown in Part A-iii. In addition, a limit $U_{\\max }$ to the achievable potential between the electrodes is set by the electrostatic push between a droplet and the bowl beneath it. Find $U_{\\max }$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n## Kelvin water dropper\n\nThe following facts about the surface tension may turn out to be useful for this problem. For the molecules of a liquid, the positions at the liquid-air interface are less favourable as compared with the positions in the bulk of the liquid. This interface is described by the so-called surface energy, $U=\\sigma S$, where $S$ is the surface area of the interface and $\\sigma$ is the surface tension coefficient of the liquid. Moreover, two fragments of the liquid surface pull each other with a force $F=\\sigma l$, where $l$ is the length of a straight line separating the fragments.\n\n[figure1]\n\nA long metallic pipe with internal diameter $d$ is pointing directly downwards. Water is slowly dripping from a nozzle at its lower end, see fig. Water can be considered to be electrically conducting; its surface tension is $\\sigma$ and its density is $\\rho$. A droplet of radius $r$ hangs below the nozzle. The radius grows slowly in time until the droplet separates from the nozzle due to the free fall acceleration $g$. Always assume that $d \\ll r$.\n\nAn apparatus called the \"Kelvin water dropper\" consists of two pipes, each identical to the one described in Part A, connected via a T-junction, see fig. The ends of both pipes are at the centres of two cylindrical electrodes (with height $L$ and diameter $D$ with $L \\gg D \\gg r$ ). For both tubes, the dripping rate is $n$ droplets per unit time. Droplets fall from height $H$ into conductive bowls underneath the nozzles, cross-connected to the electrodes as shown in the diagram. The electrodes are connected via a capacitance $C$. There is no net charge on the system of bowls and electrodes. Note that the top water container is earthed as shown. The first droplet to fall will have some microscopic charge which will cause an imbalance between the two sides and a small charge separation across the capacitor.\n\n[figure2]\n\nThe dropper's functioning can be hindered by the effect shown in Part A-iii. In addition, a limit $U_{\\max }$ to the achievable potential between the electrodes is set by the electrostatic push between a droplet and the bowl beneath it. Find $U_{\\max }$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_6b45dae650949cf89919g-1.jpg?height=657&width=406&top_left_y=939&top_left_x=334", "https://cdn.mathpix.com/cropped/2024_03_14_6b45dae650949cf89919g-1.jpg?height=754&width=557&top_left_y=1165&top_left_x=1229" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_95", "problem": "Three blocks ( $m_{1}=1 \\mathrm{~kg}, m_{2}=2 \\mathrm{~kg}$, $m_{3}=3 \\mathrm{~kg}$ ) connected by cords are pulled by a constant force, $F$, of $18 \\mathrm{~N}$\n\n[figure1]\non a frictionless horizontal table. $T_{2}$ is the tension in the rope between $m_{2}$ and $m_{3}$. What is $T_{2}$ ?\nA: $3 \\mathrm{~N}$\nB: $6 \\mathrm{~N}$\nC: $9 \\mathrm{~N}$\nD: $12 \\mathrm{~N}$\nE: $15 \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThree blocks ( $m_{1}=1 \\mathrm{~kg}, m_{2}=2 \\mathrm{~kg}$, $m_{3}=3 \\mathrm{~kg}$ ) connected by cords are pulled by a constant force, $F$, of $18 \\mathrm{~N}$\n\n[figure1]\non a frictionless horizontal table. $T_{2}$ is the tension in the rope between $m_{2}$ and $m_{3}$. What is $T_{2}$ ?\n\nA: $3 \\mathrm{~N}$\nB: $6 \\mathrm{~N}$\nC: $9 \\mathrm{~N}$\nD: $12 \\mathrm{~N}$\nE: $15 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_bd21c863b16c0f40a895g-05.jpg?height=120&width=748&top_left_y=1336&top_left_x=1076" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_596", "problem": "When studying problems in special relativity it is often the invariant distance $\\Delta s$ between two events that is most important, where $\\Delta s$ is defined by\n\n$$\n(\\Delta s)^{2}=(c \\Delta t)^{2}-\\left[(\\Delta x)^{2}+(\\Delta y)^{2}+(\\Delta z)^{2}\\right]\n$$\n\nwhere $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$ is the speed of light.\n\nConsider the motion of a projectile launched with initial speed $v_{0}$ at angle of $\\theta_{0}$ above the horizontal. Assume that $g$, the acceleration of free fall, is constant for the motion of the projectile.\n\nDerive an expression for the invariant distance of the projectile as a function of time $t$ as measured from the launch, assuming that it is launched at $t=0$. Express your answer as a function of any or all of $\\theta_{0}, v_{0}, c, g$, and $t$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nWhen studying problems in special relativity it is often the invariant distance $\\Delta s$ between two events that is most important, where $\\Delta s$ is defined by\n\n$$\n(\\Delta s)^{2}=(c \\Delta t)^{2}-\\left[(\\Delta x)^{2}+(\\Delta y)^{2}+(\\Delta z)^{2}\\right]\n$$\n\nwhere $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$ is the speed of light.\n\nConsider the motion of a projectile launched with initial speed $v_{0}$ at angle of $\\theta_{0}$ above the horizontal. Assume that $g$, the acceleration of free fall, is constant for the motion of the projectile.\n\nDerive an expression for the invariant distance of the projectile as a function of time $t$ as measured from the launch, assuming that it is launched at $t=0$. Express your answer as a function of any or all of $\\theta_{0}, v_{0}, c, g$, and $t$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_652", "problem": "A ball is dropped from the top of a building with no initial speed. After $1 \\mathrm{~s}$, a second ball is thrown downward with an initial vertical speed of $26 \\mathrm{~m} / \\mathrm{s}$. What is the minimum height of the building if the second ball hits the ground earlier than the first ball?\nA: $0.45 \\mathrm{~m}$\nB: $4.91 \\mathrm{~m}$\nC: $6.93 \\mathrm{~m}$\nD: $8.33 \\mathrm{~m}$\nE: $12.65 \\mathrm{~m}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA ball is dropped from the top of a building with no initial speed. After $1 \\mathrm{~s}$, a second ball is thrown downward with an initial vertical speed of $26 \\mathrm{~m} / \\mathrm{s}$. What is the minimum height of the building if the second ball hits the ground earlier than the first ball?\n\nA: $0.45 \\mathrm{~m}$\nB: $4.91 \\mathrm{~m}$\nC: $6.93 \\mathrm{~m}$\nD: $8.33 \\mathrm{~m}$\nE: $12.65 \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1713", "problem": "如图所示, 有一辆左右对称的光滑小车, 质量为 $M$, 放在光滑立平面上, 不考虑轮子质量, 重力加速度为 $g$, 将一个质点 $m=\\sqrt{2} M$ 的小球如图放置, 初态质点和小车都静止, 然后自由释放,\n小球下降 $r$ 之后进入半径为 $r$ 的圆弧, 经过圆心角为 $\\theta=\\frac{3 \\pi}{4}$ 后腾空一段距离 $l$ 后恰好对小车沿切线进入右侧圆弧, 最终上升到右侧与初态相同高度点。\n\n[图1]为了使得质点恰好进入右侧圆弧, $l$ 应当为多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图所示, 有一辆左右对称的光滑小车, 质量为 $M$, 放在光滑立平面上, 不考虑轮子质量, 重力加速度为 $g$, 将一个质点 $m=\\sqrt{2} M$ 的小球如图放置, 初态质点和小车都静止, 然后自由释放,\n小球下降 $r$ 之后进入半径为 $r$ 的圆弧, 经过圆心角为 $\\theta=\\frac{3 \\pi}{4}$ 后腾空一段距离 $l$ 后恰好对小车沿切线进入右侧圆弧, 最终上升到右侧与初态相同高度点。\n\n[图1]\n\n问题:\n为了使得质点恰好进入右侧圆弧, $l$ 应当为多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-07.jpg?height=363&width=697&top_left_y=401&top_left_x=722" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1600", "problem": "双星系统是一类重要的天文观测对象。假设某两星体均可视为质点, 其质量分别为 $M$ 和 $m$,一起围绕它们的质心做圆周运动,\n\n构成一双星系统, 观测到该系统的转动周期为 $T_{0}$ 。在某一时刻, $M$ 星突然发生爆炸而失去质量 $\\Delta M$ 。假设爆炸是瞬时的、相对于 $M$ 星是各向同性的，因而爆炸后 $M$ 星的残余体 $M^{\\prime}$ $\\left(M^{\\prime}=M-\\Delta M\\right)$ 星的瞬间速度与爆炸前瞬间 $M$ 星的速度相同,且爆炸过程和抛射物质 $\\Delta M$ 都对 $m$ 星没有影响。已知引力常量为 $G$ ，不考虑相对论效应。\n\n[图1]求爆炸前 $M$ 星和 $m$ 星之间的距离 $r_{0}$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n双星系统是一类重要的天文观测对象。假设某两星体均可视为质点, 其质量分别为 $M$ 和 $m$,一起围绕它们的质心做圆周运动,\n\n构成一双星系统, 观测到该系统的转动周期为 $T_{0}$ 。在某一时刻, $M$ 星突然发生爆炸而失去质量 $\\Delta M$ 。假设爆炸是瞬时的、相对于 $M$ 星是各向同性的，因而爆炸后 $M$ 星的残余体 $M^{\\prime}$ $\\left(M^{\\prime}=M-\\Delta M\\right)$ 星的瞬间速度与爆炸前瞬间 $M$ 星的速度相同,且爆炸过程和抛射物质 $\\Delta M$ 都对 $m$ 星没有影响。已知引力常量为 $G$ ，不考虑相对论效应。\n\n[图1]\n\n问题:\n求爆炸前 $M$ 星和 $m$ 星之间的距离 $r_{0}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_a47de6806e8da0a0f86dg-01.jpg?height=537&width=419&top_left_y=1171&top_left_x=1361" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1706", "problem": "真空中平行板电容器两极板的面积均为 $S$, 相距 $d$, 上、下极\n\n板所带电量分别为 $Q$ 和 $-Q(Q>0)$ 。现将一厚度为 $t$ 、面积为 $S / 2$\n\n（宽度和原来的极板相同, 长度是原来极板的一半）的金属片在上极板的正下方平行插入电容器, 将电容器分成如图所示的 $1 、 2 、 3$ 三部分。不考虑边缘效应。静电力常量为 $k$ 。试求\n\n\n[图1]电容器中 $1 、 2 、 3$ 三部分的电场强度;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n真空中平行板电容器两极板的面积均为 $S$, 相距 $d$, 上、下极\n\n板所带电量分别为 $Q$ 和 $-Q(Q>0)$ 。现将一厚度为 $t$ 、面积为 $S / 2$\n\n（宽度和原来的极板相同, 长度是原来极板的一半）的金属片在上极板的正下方平行插入电容器, 将电容器分成如图所示的 $1 、 2 、 3$ 三部分。不考虑边缘效应。静电力常量为 $k$ 。试求\n\n\n[图1]\n\n问题:\n电容器中 $1 、 2 、 3$ 三部分的电场强度;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$1$部分电场强度, $2$部分电场强度, $3$部分电场强度]\n它们的答案类型依次是[表达式, 表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://i.postimg.cc/c4YQHfdj/2016-CPho-Q13.png" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "$1$部分电场强度", "$2$部分电场强度", "$3$部分电场强度" ], "type_sequence": [ "EX", "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_905", "problem": "The two-slit electron interference experiment was first performed by Möllenstedt et al, MerliMissiroli and Pozzi in 1974 and Tonomura et al in 1989. In the two-slit electron interference experiment, a monochromatic electron point source emits particles at $S$ that first passes through an electron \"biprism\" before impinging on an observational plane; $S_{1}$ and $S_{2}$ are virtual sources at distance $d$. In the diagram, the filament is pointing into the page. Note that it is a very thin filament (not drawn to scale in the diagram).\n\n[figure1]\n\nThe electron \"biprism\" consists of a grounded cylindrical wire mesh with a fine filament $F$ at the center. The distance between the source and the \"biprism\" is $\\ell$, and the distance between the distance between the \"biprism\" and the screen is $L$.\n\nIn Tonomura et al experiment,\n\n$$\n\\begin{aligned}\n& v_{z}=c / 2 \\\\\n& V_{a}=10 \\mathrm{~V}, \\\\\n& V_{0}=50 \\mathrm{kV} \\text {, } \\\\\n& a=0.5 \\mu \\mathrm{m}, \\\\\n& b=5 \\mathrm{~mm}, \\\\\n& \\ell \\quad=25 \\mathrm{~cm}, \\\\\n& L=1.5 \\mathrm{~m}, \\\\\n& h=6.6 \\times 10^{-34} \\mathrm{Js} \\text {, } \\\\\n& \\text { electron charge, } e=1.6 \\times 10^{-19} \\mathrm{C} \\text {, } \\\\\n& \\text { mass of electron, } \\mathrm{m}_{0}=9.1 \\times 10^{-31} \\mathrm{~kg} \\text {, } \\\\\n& \\text { and the speed of light in vacuo, } c=3 \\times 10^{8} \\mathrm{~ms}^{-1}\n\\end{aligned}\n$$\n\ndetermine the fringe separation of the interference pattern on the screen,", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe two-slit electron interference experiment was first performed by Möllenstedt et al, MerliMissiroli and Pozzi in 1974 and Tonomura et al in 1989. In the two-slit electron interference experiment, a monochromatic electron point source emits particles at $S$ that first passes through an electron \"biprism\" before impinging on an observational plane; $S_{1}$ and $S_{2}$ are virtual sources at distance $d$. In the diagram, the filament is pointing into the page. Note that it is a very thin filament (not drawn to scale in the diagram).\n\n[figure1]\n\nThe electron \"biprism\" consists of a grounded cylindrical wire mesh with a fine filament $F$ at the center. The distance between the source and the \"biprism\" is $\\ell$, and the distance between the distance between the \"biprism\" and the screen is $L$.\n\nIn Tonomura et al experiment,\n\n$$\n\\begin{aligned}\n& v_{z}=c / 2 \\\\\n& V_{a}=10 \\mathrm{~V}, \\\\\n& V_{0}=50 \\mathrm{kV} \\text {, } \\\\\n& a=0.5 \\mu \\mathrm{m}, \\\\\n& b=5 \\mathrm{~mm}, \\\\\n& \\ell \\quad=25 \\mathrm{~cm}, \\\\\n& L=1.5 \\mathrm{~m}, \\\\\n& h=6.6 \\times 10^{-34} \\mathrm{Js} \\text {, } \\\\\n& \\text { electron charge, } e=1.6 \\times 10^{-19} \\mathrm{C} \\text {, } \\\\\n& \\text { mass of electron, } \\mathrm{m}_{0}=9.1 \\times 10^{-31} \\mathrm{~kg} \\text {, } \\\\\n& \\text { and the speed of light in vacuo, } c=3 \\times 10^{8} \\mathrm{~ms}^{-1}\n\\end{aligned}\n$$\n\ndetermine the fringe separation of the interference pattern on the screen,\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\AA$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_4f547aee877827e020bbg-1.jpg?height=1319&width=1091&top_left_y=1082&top_left_x=471" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\AA$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_243", "problem": "Earth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nThe modulus of Earths magnetic dipole moment is $M_{E}=8.05 \\times 10^{22} \\mathrm{~A} \\mathrm{~m}^{2}$. Earths magnetic field can be expressed in spherical coordinates as a function of the latitude $\\lambda$ and the distance $r$ to the center of the planet:\n\n$$\n\\vec{B}=\\frac{\\mu_{0}}{4 \\pi} \\frac{M_{E}}{r^{3}}\\left(-2 \\sin \\lambda \\vec{u}_{r}+\\cos \\lambda \\vec{u}_{\\lambda}\\right)\n$$\n\nwhere $u_{r}$ and $u_{\\lambda}$ are unitary vectors pointing radial and polar directions respectively, with azimuthal symmetry. However, for our purposes it will be useful to express the field modulus value along one of the field lines. These lines follow the equation $r=r_{E q} \\cos ^{2} \\lambda$ where $r_{E q}$ is the distance from the line to the center of the Earth at Equator. Moreover, the distance $r_{E q}$ can also be expressed as a function of the parameter $L=\\frac{r_{E q}}{R_{E}}$. With this notation, we can identify a field line with the parameter $L$.\n\nDetermine the modulus of the magnetic field $B$ along a field line as a function of the variables $\\lambda$ and $L$. The magnetic field in the surface of the Earth at the Equator is $B_{E}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nEarth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nThe modulus of Earths magnetic dipole moment is $M_{E}=8.05 \\times 10^{22} \\mathrm{~A} \\mathrm{~m}^{2}$. Earths magnetic field can be expressed in spherical coordinates as a function of the latitude $\\lambda$ and the distance $r$ to the center of the planet:\n\n$$\n\\vec{B}=\\frac{\\mu_{0}}{4 \\pi} \\frac{M_{E}}{r^{3}}\\left(-2 \\sin \\lambda \\vec{u}_{r}+\\cos \\lambda \\vec{u}_{\\lambda}\\right)\n$$\n\nwhere $u_{r}$ and $u_{\\lambda}$ are unitary vectors pointing radial and polar directions respectively, with azimuthal symmetry. However, for our purposes it will be useful to express the field modulus value along one of the field lines. These lines follow the equation $r=r_{E q} \\cos ^{2} \\lambda$ where $r_{E q}$ is the distance from the line to the center of the Earth at Equator. Moreover, the distance $r_{E q}$ can also be expressed as a function of the parameter $L=\\frac{r_{E q}}{R_{E}}$. With this notation, we can identify a field line with the parameter $L$.\n\nDetermine the modulus of the magnetic field $B$ along a field line as a function of the variables $\\lambda$ and $L$. The magnetic field in the surface of the Earth at the Equator is $B_{E}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=496&width=705&top_left_y=774&top_left_x=713", "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=220&width=504&top_left_y=1690&top_left_x=802" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1552", "problem": "下述实验或现象中，能够说明光具有粒子性的是\nA: 光的双缝干涉实验\nB: 黑体辐射\nC: 光电效应\nD: 康普顿效应\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n下述实验或现象中，能够说明光具有粒子性的是\n\nA: 光的双缝干涉实验\nB: 黑体辐射\nC: 光电效应\nD: 康普顿效应\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_558", "problem": "A large block of mass $m_{b}$ is located on a horizontal frictionless surface. A second block of mass $m_{t}$ is located on top of the first block; the coefficient of friction (both static and kinetic) between the two blocks is given by $\\mu$. All surfaces are horizontal; all motion is effectively one dimensional. A spring with spring constant $k$ is connected to the top block only; the spring obeys Hooke's Law equally in both extension and compression. Assume that the top block never falls off of the bottom block; you may assume that the bottom block is very, very long. The top block is moved a distance $A$ away from the equilibrium position and then released from rest.\n\n[figure1]\n\nDepending on the value of $A$, the motion can be divided into two types: motion that experiences no frictional energy losses and motion that does. Find the value $A_{c}$ that divides the two motion types. Write your answer in terms of any or all of $\\mu$, the acceleration of gravity $g$, the masses $m_{t}$ and $m_{b}$, and the spring constant $k$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA large block of mass $m_{b}$ is located on a horizontal frictionless surface. A second block of mass $m_{t}$ is located on top of the first block; the coefficient of friction (both static and kinetic) between the two blocks is given by $\\mu$. All surfaces are horizontal; all motion is effectively one dimensional. A spring with spring constant $k$ is connected to the top block only; the spring obeys Hooke's Law equally in both extension and compression. Assume that the top block never falls off of the bottom block; you may assume that the bottom block is very, very long. The top block is moved a distance $A$ away from the equilibrium position and then released from rest.\n\n[figure1]\n\nDepending on the value of $A$, the motion can be divided into two types: motion that experiences no frictional energy losses and motion that does. Find the value $A_{c}$ that divides the two motion types. Write your answer in terms of any or all of $\\mu$, the acceleration of gravity $g$, the masses $m_{t}$ and $m_{b}$, and the spring constant $k$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-07.jpg?height=204&width=1160&top_left_y=684&top_left_x=474" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_399", "problem": "Jaan Kalda. A comet's orbit intersects the Earth's orbit (which can be assumed to be circular of radius $R_{0}=$ $1.5 \\times 10^{8} \\mathrm{~km}$ ) at an angle $\\alpha=45^{\\circ}$. The comet's and the Earth's orbits lie on the same plane.\n\nFind the distance $R_{\\min }$ of the comet's perihelion $P$ from the Sun (i.e. its shortest distance). The distance $R_{\\max }$ of the comet's aphelion $A$ from the Sun (i.e. its longest distance) can be assumed to be much larger than $R_{0}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nJaan Kalda. A comet's orbit intersects the Earth's orbit (which can be assumed to be circular of radius $R_{0}=$ $1.5 \\times 10^{8} \\mathrm{~km}$ ) at an angle $\\alpha=45^{\\circ}$. The comet's and the Earth's orbits lie on the same plane.\n\nFind the distance $R_{\\min }$ of the comet's perihelion $P$ from the Sun (i.e. its shortest distance). The distance $R_{\\max }$ of the comet's aphelion $A$ from the Sun (i.e. its longest distance) can be assumed to be much larger than $R_{0}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_128", "problem": "The power output from a certain experimental car design to be shaped like a cube is proportional to the mass $m$ of the car. The force of air friction on the car is proportional to $A v^{2}$, where $v$ is the speed of the car and $A$ the cross sectional area. On a level surface the car has a maximum speed $v_{\\max }$. Assuming that all versions of this design have the same density, then which of the following is true?\nA: $v_{\\max } \\propto m^{1 / 9} \\mathrm{}$\nB: $v_{\\max } \\propto m^{1 / 7}$\nC: $v_{\\max } \\propto m^{1 / 3}$\nD: $v_{\\max } \\propto m^{2 / 3}$\nE: $v_{\\max } \\propto m^{3 / 4}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe power output from a certain experimental car design to be shaped like a cube is proportional to the mass $m$ of the car. The force of air friction on the car is proportional to $A v^{2}$, where $v$ is the speed of the car and $A$ the cross sectional area. On a level surface the car has a maximum speed $v_{\\max }$. Assuming that all versions of this design have the same density, then which of the following is true?\n\nA: $v_{\\max } \\propto m^{1 / 9} \\mathrm{}$\nB: $v_{\\max } \\propto m^{1 / 7}$\nC: $v_{\\max } \\propto m^{1 / 3}$\nD: $v_{\\max } \\propto m^{2 / 3}$\nE: $v_{\\max } \\propto m^{3 / 4}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_482", "problem": "Part (i)\n\nBy symmetry, the fields above and below the sheet are equal in magnitude and directed away from the sheet. By Gauss's Law, using a cylinder of base area $A$,\n\n$$\n2 E A=\\frac{\\sigma A}{\\epsilon_{0}} \\Rightarrow E=\\frac{\\sigma}{2 \\epsilon_{0}}\n$$\n\npointing directly away from the sheet in the $z$ direction, or\n\n$$\n\\mathbf{E}=\\frac{\\sigma}{2 \\epsilon} \\times \\begin{cases}\\hat{\\mathbf{z}} & \\text { above the sheet } \\\\ -\\hat{\\mathbf{z}} & \\text { below the sheet. }\\end{cases}\n$$In this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinite uniform sheet has a surface charge density $\\sigma$ and has an infinitesimal thickness. The sheet lies in the $x y$ plane.\n\nAssuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ (perpendicular to the sheet), determine the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) above and below the sheet.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\nHere is some context information for this question, which might assist you in solving it:\nPart (i)\n\nBy symmetry, the fields above and below the sheet are equal in magnitude and directed away from the sheet. By Gauss's Law, using a cylinder of base area $A$,\n\n$$\n2 E A=\\frac{\\sigma A}{\\epsilon_{0}} \\Rightarrow E=\\frac{\\sigma}{2 \\epsilon_{0}}\n$$\n\npointing directly away from the sheet in the $z$ direction, or\n\n$$\n\\mathbf{E}=\\frac{\\sigma}{2 \\epsilon} \\times \\begin{cases}\\hat{\\mathbf{z}} & \\text { above the sheet } \\\\ -\\hat{\\mathbf{z}} & \\text { below the sheet. }\\end{cases}\n$$\n\nproblem:\nIn this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinite uniform sheet has a surface charge density $\\sigma$ and has an infinitesimal thickness. The sheet lies in the $x y$ plane.\n\nAssuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{z}}$ (perpendicular to the sheet), determine the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) above and below the sheet.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) above the sheet, the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) below the sheet].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) above the sheet", "the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) below the sheet" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_506", "problem": "Determine an expression for the magnitude of the magnetic field at a distance $r$ from the power line cable in terms of $I, r$, and fundamental constants.\n\nThe field is perpendicular to the wire and to the radius, and from Ampere's Law\n\n$$\n\\oint \\mathbf{B} \\cdot d \\mathbf{s}=\\mu_{0} I\n$$\n\nThe integral evaluates to $(2 \\pi r) B$, giving\n\n$$\nB=\\frac{\\mu_{0} I}{2 \\pi r}\n$$\n\nThis well-known result can also be written down without justification.\n\nAssuming the loop is placed in this way, determine an expression for the emf induced in the loop (as a function of time) in terms of any or all of $I_{0}, h, a, b, N, \\omega, t$, and fundamental constants.\n\nFaraday's law states\n\n$$\n\\mathcal{E}=N \\frac{d \\Phi_{B}}{d t}\n$$\n\nwhere we have dropped the sign, which is not important, and $\\Phi_{B}$ is the magnetic flux through a single loop. The flux is\n\n$$\n\\Phi_{B}=\\int \\mathbf{B} \\cdot d \\mathbf{A}=b \\int_{h-a}^{h} B(r) d r\n$$\n\nwhere we have divided the loop into strips of radial width $d r$ and length $b$. Plugging in the result of part (i),\n\n$$\n\\Phi_{B}=b \\int_{h-a}^{h} \\frac{\\mu_{0} I}{2 \\pi r} d r=\\frac{\\mu_{0} I b}{2 \\pi} \\log \\frac{h}{h-a}\n$$\n\nThe time dependence comes only from the current,\n\n$$\n\\frac{d I}{d t}=\\omega I_{0} \\cos \\omega t\n$$\n\nTherefore, we have\n\n$$\n\\mathcal{E}=\\frac{N \\mu_{0} b}{2 \\pi} \\log \\frac{h}{h-a} I_{0} \\omega \\cos \\omega t\n$$An AC power line cable transmits electrical power using a sinusoidal waveform with frequency $60 \\mathrm{~Hz}$. The load receives an RMS voltage of $500 \\mathrm{kV}$ and requires $1000 \\mathrm{MW}$ of average power. For this problem, consider only the cable carrying current in one of the two directions, and ignore effects due to capacitance or inductance between the cable and with the ground.\n\nA local rancher thinks he might be able to extract electrical power from the cable using electromagnetic induction. The rancher constructs a rectangular loop of length $a$ and width $b10 \\mathrm{~J}$\nD: Remains unchanged\nE: More information is needed\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA sample of an ideal gas is being held at a constant pressure. What happens to the internal energy of the gas if $10 \\mathrm{~J}$ of heat energy are transferred to the gas?\n\nA: Increases by $10 \\mathrm{~J}$\nB: Increases by $<10 \\mathrm{~J}$\nC: Increases by $>10 \\mathrm{~J}$\nD: Remains unchanged\nE: More information is needed\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_196", "problem": "An object is thrown directly downward from the top of a 180 meter tall building. It takes 1.0 seconds for the object to fall the last 60 meters. With what initial downward speed was the object thrown from the roof?\nA: $15 \\mathrm{~m} / \\mathrm{s}$\nB: $25 \\mathrm{~m} / \\mathrm{s} $ \nC: $35 \\mathrm{~m} / \\mathrm{s}$\nD: $55 \\mathrm{~m} / \\mathrm{s}$\nE: insufficient information.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn object is thrown directly downward from the top of a 180 meter tall building. It takes 1.0 seconds for the object to fall the last 60 meters. With what initial downward speed was the object thrown from the roof?\n\nA: $15 \\mathrm{~m} / \\mathrm{s}$\nB: $25 \\mathrm{~m} / \\mathrm{s} $ \nC: $35 \\mathrm{~m} / \\mathrm{s}$\nD: $55 \\mathrm{~m} / \\mathrm{s}$\nE: insufficient information.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1412", "problem": "$1 \\mathrm{~mol}$ 的理想气体经历一循环过程 1-2-3-1,如 $p-T$ 图示所示. 过程 1-2 是等压过程, 过程 3-1 是通过 $p-T$ 图原点的直线上的一段, 描述过程 2-3 的方程为\n\n$$\nc_{1} p^{2}+c_{2} p=T\n$$\n\n式中 $c_{1}$ 和 $c_{2}$ 都是待定的常量, $p$ 和 $T$ 分别是气体的压强和绝对温度. 已知, 气体在状态 1 的压强、绝对温度分别为 $p_{1}$\n\n[图1]\n和 $T_{1}$, 气体在状态 2 的绝对温度以及在状态 3 的压强和绝对温度分别为 $T_{2}$ 以及 $p_{3}$ 和 $T_{3}$. 气体常量 $R$ 也是已知的.求常量 $c_{1}$ 和 $c_{2}$ 的值", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n$1 \\mathrm{~mol}$ 的理想气体经历一循环过程 1-2-3-1,如 $p-T$ 图示所示. 过程 1-2 是等压过程, 过程 3-1 是通过 $p-T$ 图原点的直线上的一段, 描述过程 2-3 的方程为\n\n$$\nc_{1} p^{2}+c_{2} p=T\n$$\n\n式中 $c_{1}$ 和 $c_{2}$ 都是待定的常量, $p$ 和 $T$ 分别是气体的压强和绝对温度. 已知, 气体在状态 1 的压强、绝对温度分别为 $p_{1}$\n\n[图1]\n和 $T_{1}$, 气体在状态 2 的绝对温度以及在状态 3 的压强和绝对温度分别为 $T_{2}$ 以及 $p_{3}$ 和 $T_{3}$. 气体常量 $R$ 也是已知的.\n\n问题:\n求常量 $c_{1}$ 和 $c_{2}$ 的值\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$c_{1}$ , $c_{2}$ ]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_58b1fc45927d60138a23g-06.jpg?height=385&width=503&top_left_y=624&top_left_x=1299" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$c_{1}$ ", "$c_{2}$ " ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1728", "problem": "如图, 质量分别为 $m_{\\mathrm{a}} 、 m_{\\mathrm{b}}$ 的小球 $\\mathrm{a} 、 \\mathrm{~b}$ 放置在光滑绝缘\n\n水平面上, 两球之间用一原长为 $l_{0}$ 、劲度系数为 $k_{0}$ 的绝缘轻\n\n(b)-minn-a 弹簧连接。\n\n[图1]若让两小球带等量同号电荷, 系统平衡时弹簧长度为 $L_{0}$ 。记静电力常量为 $K$ 。求小球所带电荷量和两球与弹簧构成的系统做微振动的频率（极化电荷的影响可忽略）。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 质量分别为 $m_{\\mathrm{a}} 、 m_{\\mathrm{b}}$ 的小球 $\\mathrm{a} 、 \\mathrm{~b}$ 放置在光滑绝缘\n\n水平面上, 两球之间用一原长为 $l_{0}$ 、劲度系数为 $k_{0}$ 的绝缘轻\n\n(b)-minn-a 弹簧连接。\n\n[图1]\n\n问题:\n若让两小球带等量同号电荷, 系统平衡时弹簧长度为 $L_{0}$ 。记静电力常量为 $K$ 。求小球所带电荷量和两球与弹簧构成的系统做微振动的频率（极化电荷的影响可忽略）。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://i.postimg.cc/Ssd3bpmp/2017-CPho-Q1.png", "https://cdn.mathpix.com/cropped/2024_03_31_a47de6806e8da0a0f86dg-07.jpg?height=174&width=739&top_left_y=1415&top_left_x=1024" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_9", "problem": "A spark is produced between two insulated surfaces, maintained at a constant potential difference of $5.0 \\times 10^{6} \\mathrm{~V}$. The energy output is $10^{-5} \\mathrm{~J}$. How many electrons have been transferred in the spark?\nA: $1.25 \\times 10^{7}$\nB: $1.20 \\times 10^{7}$\nC: $1.35 \\times 10^{7}$\nD: $1.25 \\times 10^{6}$\nE: $1.25 \\times 10^{8}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA spark is produced between two insulated surfaces, maintained at a constant potential difference of $5.0 \\times 10^{6} \\mathrm{~V}$. The energy output is $10^{-5} \\mathrm{~J}$. How many electrons have been transferred in the spark?\n\nA: $1.25 \\times 10^{7}$\nB: $1.20 \\times 10^{7}$\nC: $1.35 \\times 10^{7}$\nD: $1.25 \\times 10^{6}$\nE: $1.25 \\times 10^{8}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_229", "problem": "Let the two-ball body be placed on the flat horizontal surface as shown in Figure 2. The smaller ball is right under the larger one such that the line connecting the centers of the two balls is strictly perpendicular to the surface. It is obvious that such an equilibrium position is unstable and an insignificant random deflection will set the two-ball body into a motion due to gravity whose acceleration is $g=9.80 \\mathrm{~m} / \\mathrm{s}^{2}$.\n\n[figure1]\n\nFigure 2: Initial position of the two-ball body for Part B.\n\nAssume that the friction between the lower ball and the surface is so strong that there is no slipping at all times. Find the velocities of balls' centers at the time moment right before the larger ball hits the ground.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nLet the two-ball body be placed on the flat horizontal surface as shown in Figure 2. The smaller ball is right under the larger one such that the line connecting the centers of the two balls is strictly perpendicular to the surface. It is obvious that such an equilibrium position is unstable and an insignificant random deflection will set the two-ball body into a motion due to gravity whose acceleration is $g=9.80 \\mathrm{~m} / \\mathrm{s}^{2}$.\n\n[figure1]\n\nFigure 2: Initial position of the two-ball body for Part B.\n\nAssume that the friction between the lower ball and the surface is so strong that there is no slipping at all times. Find the velocities of balls' centers at the time moment right before the larger ball hits the ground.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [$v_1$, $v_2$].\nTheir units are, in order, [$\\frac{m}{s}$, $\\frac{m}{s}$], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_57b03088b694c15dae09g-2.jpg?height=740&width=609&top_left_y=343&top_left_x=758" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\frac{m}{s}$", "$\\frac{m}{s}$" ], "answer_sequence": [ "$v_1$", "$v_2$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_429", "problem": "[figure1]\nFigure: A table of stopping distances taken from the Highway Code. \n\nWhat is the kinetic energy of a car of mass $1000 \\mathrm{~kg}$ travelling at $30 \\mathrm{~m} \\mathrm{~s}^{-1}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\n[figure1]\nFigure: A table of stopping distances taken from the Highway Code. \n\nWhat is the kinetic energy of a car of mass $1000 \\mathrm{~kg}$ travelling at $30 \\mathrm{~m} \\mathrm{~s}^{-1}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of kJ, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-2.jpg?height=428&width=1579&top_left_y=2102&top_left_x=244" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "kJ" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_823", "problem": "The half maximum occurs at:\n\n$$\n\\delta=2 m \\pi \\pm \\frac{\\varepsilon}{2} \\text { (b1) }\n$$\n\nGiven that $T=0.5$, thus:\n\n$$\n\\begin{gathered}\nF \\sin ^{2} \\frac{\\delta}{2}=1 \\quad(\\mathrm{~b} 2) \\\\\n\\varepsilon=\\frac{4}{\\sqrt{F}}=\\frac{2(1-R)}{\\sqrt{R}}=\\frac{2(1-0.9)}{\\sqrt{0.9}}=0.21 \\mathrm{rad}\n\\end{gathered}\n$$Figure 1 shows a Fabry-Perot (F-P) etalon, in which air pressure is tunable. The F-P etalon consists of two glass plates with high-reflectivity inner surfaces. The two plates form a cavity in which light can be reflected back and forth. The outer surfaces of the plates are generally not parallel to the inner ones and do not affect the back-and-forth reflection. The air density in the etalon can be controlled. Light from a Sodium lamp is collimated by the lens L1 and then passes through the F-P etalon. The transmitivity of the etalon is given by $T=\\frac{1}{1+F \\sin ^{2}(\\delta / 2)}$, where $F=\\frac{4 R}{(1-R)^{2}}, \\mathrm{R}$ is the reflectivity of the inner surfaces, $\\delta=\\frac{4 \\pi n t \\cos \\theta}{\\lambda}$ is the phase shift of two neighboring rays, $\\mathrm{n}$ is the refractive index of the gas, $\\mathrm{t}$ is the spacing of inner surfaces, $\\theta$ is the incident angle, and $\\lambda$ is the light wavelength.\n\n[figure1]\n\nFigure 1\n\nThe Sodium lamp emits D1 $(\\lambda=589.6 \\mathrm{~nm})$ and D2 $(589 \\mathrm{~nm})$ spectral lines and is located in a tunable uniform magnetic field. For simplicity, an optical filter F1 is assumed to only allow the D1 line to pass through. The D1 line is then collimated to the F-P etalon by the lens L1. Circular interference fringes will be present on the focal plane of the lens L2 with a focal length $\\mathrm{f}=30 \\mathrm{~cm}$. Different fringes have the different incident angle $\\theta$. A microscope is used to observe the fringes. We take the reflectivity $\\mathrm{R}=90 \\%$ and the inner-surface spacing $\\mathrm{t}=1 \\mathrm{~cm}$.\n\nSome physical constants: $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}, e=1.6 \\times 10^{-19} \\mathrm{C}, m_{e}=9.1 \\times 10^{-31} \\mathrm{~kg}, c=3.0 \\times 10^{8} \\mathrm{~ms}^{-1}$.\n\n[figure2]\n\nFigure 2\nAs shown in Fig. 2, the width $\\varepsilon$ of the spectral line is defined as the full width of half maximum (FWHM) of light transmitivity T regarding the phase shift $\\delta$. The resolution of the F-P etalon is defined as follows: for two wavelengths $\\lambda$ and $\\lambda+\\Delta \\lambda$, when the central phase difference $\\Delta \\delta$ of both spectral lines is larger than $\\varepsilon$, they are thought to be resolvable; then the etalon resolution is $\\lambda / \\Delta \\lambda$ when $\\Delta \\delta=\\varepsilon$. For the vacuum case, the D1 line ( $\\lambda=589.6 \\mathrm{~nm}$ ), and because of the incident angle $\\theta \\approx 0$, take $\\cos \\theta \\approx 1.0$, please calculate the resolution $\\lambda / \\Delta \\lambda$ of the etalon.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe half maximum occurs at:\n\n$$\n\\delta=2 m \\pi \\pm \\frac{\\varepsilon}{2} \\text { (b1) }\n$$\n\nGiven that $T=0.5$, thus:\n\n$$\n\\begin{gathered}\nF \\sin ^{2} \\frac{\\delta}{2}=1 \\quad(\\mathrm{~b} 2) \\\\\n\\varepsilon=\\frac{4}{\\sqrt{F}}=\\frac{2(1-R)}{\\sqrt{R}}=\\frac{2(1-0.9)}{\\sqrt{0.9}}=0.21 \\mathrm{rad}\n\\end{gathered}\n$$\n\nproblem:\nFigure 1 shows a Fabry-Perot (F-P) etalon, in which air pressure is tunable. The F-P etalon consists of two glass plates with high-reflectivity inner surfaces. The two plates form a cavity in which light can be reflected back and forth. The outer surfaces of the plates are generally not parallel to the inner ones and do not affect the back-and-forth reflection. The air density in the etalon can be controlled. Light from a Sodium lamp is collimated by the lens L1 and then passes through the F-P etalon. The transmitivity of the etalon is given by $T=\\frac{1}{1+F \\sin ^{2}(\\delta / 2)}$, where $F=\\frac{4 R}{(1-R)^{2}}, \\mathrm{R}$ is the reflectivity of the inner surfaces, $\\delta=\\frac{4 \\pi n t \\cos \\theta}{\\lambda}$ is the phase shift of two neighboring rays, $\\mathrm{n}$ is the refractive index of the gas, $\\mathrm{t}$ is the spacing of inner surfaces, $\\theta$ is the incident angle, and $\\lambda$ is the light wavelength.\n\n[figure1]\n\nFigure 1\n\nThe Sodium lamp emits D1 $(\\lambda=589.6 \\mathrm{~nm})$ and D2 $(589 \\mathrm{~nm})$ spectral lines and is located in a tunable uniform magnetic field. For simplicity, an optical filter F1 is assumed to only allow the D1 line to pass through. The D1 line is then collimated to the F-P etalon by the lens L1. Circular interference fringes will be present on the focal plane of the lens L2 with a focal length $\\mathrm{f}=30 \\mathrm{~cm}$. Different fringes have the different incident angle $\\theta$. A microscope is used to observe the fringes. We take the reflectivity $\\mathrm{R}=90 \\%$ and the inner-surface spacing $\\mathrm{t}=1 \\mathrm{~cm}$.\n\nSome physical constants: $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}, e=1.6 \\times 10^{-19} \\mathrm{C}, m_{e}=9.1 \\times 10^{-31} \\mathrm{~kg}, c=3.0 \\times 10^{8} \\mathrm{~ms}^{-1}$.\n\n[figure2]\n\nFigure 2\nAs shown in Fig. 2, the width $\\varepsilon$ of the spectral line is defined as the full width of half maximum (FWHM) of light transmitivity T regarding the phase shift $\\delta$. The resolution of the F-P etalon is defined as follows: for two wavelengths $\\lambda$ and $\\lambda+\\Delta \\lambda$, when the central phase difference $\\Delta \\delta$ of both spectral lines is larger than $\\varepsilon$, they are thought to be resolvable; then the etalon resolution is $\\lambda / \\Delta \\lambda$ when $\\Delta \\delta=\\varepsilon$. For the vacuum case, the D1 line ( $\\lambda=589.6 \\mathrm{~nm}$ ), and because of the incident angle $\\theta \\approx 0$, take $\\cos \\theta \\approx 1.0$, please calculate the resolution $\\lambda / \\Delta \\lambda$ of the etalon.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_42bd302d6bc3274ac3b2g-1.jpg?height=691&width=1133&top_left_y=1236&top_left_x=473", "https://cdn.mathpix.com/cropped/2024_03_14_42bd302d6bc3274ac3b2g-2.jpg?height=571&width=868&top_left_y=1279&top_left_x=614" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_706", "problem": "On a dry day you comb your hair. Afterwards you notice that bringing the comb close to small bits of paper attracts the paper, causing it to jump off the table towards the comb. The correct explanation for this phenomena involves:\nA: The comb is positively charged after combing, and the paper is negatively charged.\nB: The comb is negatively charged after combing, and the paper is positively charged.\nC: The comb is charged and the paper has no net charge.\nD: The comb has no net charge and the paper has no net charge.\nE: The comb has no net charge, the paper has positive charge.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOn a dry day you comb your hair. Afterwards you notice that bringing the comb close to small bits of paper attracts the paper, causing it to jump off the table towards the comb. The correct explanation for this phenomena involves:\n\nA: The comb is positively charged after combing, and the paper is negatively charged.\nB: The comb is negatively charged after combing, and the paper is positively charged.\nC: The comb is charged and the paper has no net charge.\nD: The comb has no net charge and the paper has no net charge.\nE: The comb has no net charge, the paper has positive charge.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1408", "problem": "光子被电子散射时, 如果初态电子具有足够的动能, 以至于在散射过程中有能量从电子转移到光子, 则该散射被称为逆康普顿散射. 当低能光子与高能电子发生对头碰撞时, 就会出现逆康普顿散射. 已知电子静止质量为 $m_{e}$, 真空中的光速为 $c$. 若能量为 $E_{e}$ 的电子与能量为 $E_{\\gamma}$ 的光子相向对碰如果入射光子能量为 $2.00 \\mathrm{eV}$, 电子能量为 $1.00 \\square 10^{\\circ} \\mathrm{eV}$, 求散射后光子的能量. 已知 $m_{e}=0.511 \\square 10^{6} \\mathrm{eV} / c^{2}$. 计算中有必要时可利用近似: 如果 $|x|<<1$, 有 $\\sqrt{1 \\square x} \\square 1 \\square \\frac{1}{2} x$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n光子被电子散射时, 如果初态电子具有足够的动能, 以至于在散射过程中有能量从电子转移到光子, 则该散射被称为逆康普顿散射. 当低能光子与高能电子发生对头碰撞时, 就会出现逆康普顿散射. 已知电子静止质量为 $m_{e}$, 真空中的光速为 $c$. 若能量为 $E_{e}$ 的电子与能量为 $E_{\\gamma}$ 的光子相向对碰\n\n问题:\n如果入射光子能量为 $2.00 \\mathrm{eV}$, 电子能量为 $1.00 \\square 10^{\\circ} \\mathrm{eV}$, 求散射后光子的能量. 已知 $m_{e}=0.511 \\square 10^{6} \\mathrm{eV} / c^{2}$. 计算中有必要时可利用近似: 如果 $|x|<<1$, 有 $\\sqrt{1 \\square x} \\square 1 \\square \\frac{1}{2} x$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以 \\mathrm{eV}为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ " \\mathrm{eV}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_645", "problem": "Electromagnetic radiation with intensity (power per unit area) $I$ exerts a pressure on a surface perpendicular to the radiation given by $I / c$. The sun has a radiation intensity of $1.38 \\mathrm{~kW} / \\mathrm{m}^{2}$ from where you will be launching. You are building a solar sail spaceship - a spaceship that uses a large thin sheet to use light pressure to accelerate. The material you are building your sail out of has a mass per unit area of $0.1 \\mathrm{~g} / \\mathrm{m}^{2}$, and you have a payload of $25 \\mathrm{~g}$. Approximately how large does your sail have to be to accelerate your ship at $2 \\mathrm{~cm} / \\mathrm{s}^{2}$ ?\nA: $10 \\mathrm{~m}^{2}$\nB: $20 \\mathrm{~m}^{2}$\nC: $100 \\mathrm{~m}^{2}$\nD: $200 \\mathrm{~m}^{2}$\nE: $250 \\mathrm{~m}^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nElectromagnetic radiation with intensity (power per unit area) $I$ exerts a pressure on a surface perpendicular to the radiation given by $I / c$. The sun has a radiation intensity of $1.38 \\mathrm{~kW} / \\mathrm{m}^{2}$ from where you will be launching. You are building a solar sail spaceship - a spaceship that uses a large thin sheet to use light pressure to accelerate. The material you are building your sail out of has a mass per unit area of $0.1 \\mathrm{~g} / \\mathrm{m}^{2}$, and you have a payload of $25 \\mathrm{~g}$. Approximately how large does your sail have to be to accelerate your ship at $2 \\mathrm{~cm} / \\mathrm{s}^{2}$ ?\n\nA: $10 \\mathrm{~m}^{2}$\nB: $20 \\mathrm{~m}^{2}$\nC: $100 \\mathrm{~m}^{2}$\nD: $200 \\mathrm{~m}^{2}$\nE: $250 \\mathrm{~m}^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_290", "problem": "A Physics trolley starts from rest and has a constant acceleration. The velocity - time graph is linear as shown.\n\nThe corresponding graph of velocity against displacement is:\n\n[figure1]\n\n[figure2]\n\nGraph A\n\n[figure3]\n\nGraph C\n\n[figure4]\n\nGraph B\n\n[figure5]\n\nGraph D\nA: A\nB: B\nC: C\nD: D\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA Physics trolley starts from rest and has a constant acceleration. The velocity - time graph is linear as shown.\n\nThe corresponding graph of velocity against displacement is:\n\n[figure1]\n\n[figure2]\n\nGraph A\n\n[figure3]\n\nGraph C\n\n[figure4]\n\nGraph B\n\n[figure5]\n\nGraph D\n\nA: A\nB: B\nC: C\nD: D\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_236da82643280c9e45c1g-07.jpg?height=309&width=397&top_left_y=1256&top_left_x=1406", "https://cdn.mathpix.com/cropped/2024_03_06_236da82643280c9e45c1g-07.jpg?height=363&width=462&top_left_y=1663&top_left_x=246", "https://cdn.mathpix.com/cropped/2024_03_06_236da82643280c9e45c1g-07.jpg?height=348&width=457&top_left_y=2210&top_left_x=248", "https://cdn.mathpix.com/cropped/2024_03_06_236da82643280c9e45c1g-07.jpg?height=363&width=465&top_left_y=1663&top_left_x=864", "https://cdn.mathpix.com/cropped/2024_03_06_236da82643280c9e45c1g-07.jpg?height=351&width=465&top_left_y=2206&top_left_x=864" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_839", "problem": "In a uniform cylindrical pipe of length $L$, water is flowing steadily along the $+x$ direction with horizontal velocity $v_{0}$, density $\\rho_{0}$, and pressure $P_{0}$. As shown in Fig. 1 , the pipe is connected to a reservoir at a depth $h$ and opens into the atmosphere at pressure $P_{\\mathrm{a}}$.\n\nSuppose the flow-control valve $T$ at the end of the pipe is then shut instantly so that the oncoming liquid element next to the valve suffers both a pressure change $\\Delta P_{\\mathrm{s}} \\equiv P_{1}-P_{0}$ and a velocity change $\\Delta v=v_{1}-v_{0}$ with $v_{1} \\leq 0$. This causes a longitudinal wave of excess pressure $\\Delta P_{\\mathrm{s}}$ to travel upstream in the $-x$ direction with a speed of propagation $c$.\n\n[figure1]\n\nFig. 1: Steady flow in a uniform pipe.\nThe excess pressure $\\Delta P_{\\mathrm{s}}$ is related to the velocity change $\\Delta v$ by $\\Delta P_{\\mathrm{s}}=\\alpha \\rho_{0} c \\Delta v$. $1.6 \\mathrm{pt}$ The speed of propagation $c$ is given by $c=\\beta+\\sqrt{\\gamma B / \\rho_{0}}$. Find $\\alpha, \\beta$, and $\\gamma$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nIn a uniform cylindrical pipe of length $L$, water is flowing steadily along the $+x$ direction with horizontal velocity $v_{0}$, density $\\rho_{0}$, and pressure $P_{0}$. As shown in Fig. 1 , the pipe is connected to a reservoir at a depth $h$ and opens into the atmosphere at pressure $P_{\\mathrm{a}}$.\n\nSuppose the flow-control valve $T$ at the end of the pipe is then shut instantly so that the oncoming liquid element next to the valve suffers both a pressure change $\\Delta P_{\\mathrm{s}} \\equiv P_{1}-P_{0}$ and a velocity change $\\Delta v=v_{1}-v_{0}$ with $v_{1} \\leq 0$. This causes a longitudinal wave of excess pressure $\\Delta P_{\\mathrm{s}}$ to travel upstream in the $-x$ direction with a speed of propagation $c$.\n\n[figure1]\n\nFig. 1: Steady flow in a uniform pipe.\nThe excess pressure $\\Delta P_{\\mathrm{s}}$ is related to the velocity change $\\Delta v$ by $\\Delta P_{\\mathrm{s}}=\\alpha \\rho_{0} c \\Delta v$. $1.6 \\mathrm{pt}$ The speed of propagation $c$ is given by $c=\\beta+\\sqrt{\\gamma B / \\rho_{0}}$. Find $\\alpha, \\beta$, and $\\gamma$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the value of $ \\alpha$, the value of $ \\gamma$, the value of $ \\beta$].\nTheir answer types are, in order, [expression, expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_04e7339923135da91bb4g-1.jpg?height=542&width=1445&top_left_y=1825&top_left_x=243" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "the value of $ \\alpha$", "the value of $ \\gamma$", "the value of $ \\beta$" ], "type_sequence": [ "EX", "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1659", "problem": "如图, 物块 $A 、 C$ 置于光滑水平桌面上, 通过轻质滑轮和细绳悬挂物块 $\\mathrm{B}$, 物块 $\\mathrm{A} 、 \\mathrm{~B}$ 的质量均为 $2 \\mathrm{~kg}$ ，物块 $\\mathrm{C}$ 的质量为 $1 \\mathrm{~kg}$ ，重力加速度大小为 $10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。\n\n[图1]若三个物块同时由静止释放, 则物块 $A 、 B$ 和 C 加速度之比为", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 物块 $A 、 C$ 置于光滑水平桌面上, 通过轻质滑轮和细绳悬挂物块 $\\mathrm{B}$, 物块 $\\mathrm{A} 、 \\mathrm{~B}$ 的质量均为 $2 \\mathrm{~kg}$ ，物块 $\\mathrm{C}$ 的质量为 $1 \\mathrm{~kg}$ ，重力加速度大小为 $10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。\n\n[图1]\n\n问题:\n若三个物块同时由静止释放, 则物块 $A 、 B$ 和 C 加速度之比为\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-03.jpg?height=314&width=674&top_left_y=1119&top_left_x=1156" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1643", "problem": "用国家标准一级螺旋测微器 (直标度尺最小分度为 $0.5 \\mathrm{~mm}$, 丝杆螺距为 $0.5 \\mathrm{~mm}$,套管上分为 50 格刻度) 测量小球直径. \n\n测微器的初读数如图 (a) 所示, 其值为 $\\mathrm{mm}$,\n\n测量时如图（b）所示，\n\n其值为 $\\mathrm{mm}$, \n\n测得小球直径 $d=$ $\\mathrm{mm}$.\n\n[图1]\n\n图（a）\n\n[图2]\n\n图（b）", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n用国家标准一级螺旋测微器 (直标度尺最小分度为 $0.5 \\mathrm{~mm}$, 丝杆螺距为 $0.5 \\mathrm{~mm}$,套管上分为 50 格刻度) 测量小球直径. \n\n测微器的初读数如图 (a) 所示, 其值为 $\\mathrm{mm}$,\n\n测量时如图（b）所示，\n\n其值为 $\\mathrm{mm}$, \n\n测得小球直径 $d=$ $\\mathrm{mm}$.\n\n[图1]\n\n图（a）\n\n[图2]\n\n图（b）\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[图 (a) 所示其值, 图 (b) 所示其值, 小球直径]\n它们的答案类型依次是[数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_58b1fc45927d60138a23g-02.jpg?height=342&width=351&top_left_y=1571&top_left_x=590", "https://cdn.mathpix.com/cropped/2024_03_31_58b1fc45927d60138a23g-02.jpg?height=342&width=351&top_left_y=1571&top_left_x=1115" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "图 (a) 所示其值", "图 (b) 所示其值", "小球直径" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1393", "problem": "某金属材料发生光电效应的最大波长为 $\\lambda_{0}$, 将此材料制成一半径为 $R$ 的圆球, 并用绝缘线悬挂于真空室内。若以波长为 $\\lambda\\left(\\lambda<\\lambda_{0}\\right)$ 的单色光持续照射此金属球, 该金属球发生光电效应所产生光电子的最大初动能为? \n\n\n此金属球可带的电荷量最多为?\n\n(设无穷远处电势为零, 真空中半径为 $r$ 带电量为 $q$ 的导体球的电势为 $U=k \\frac{q}{r}$ 。)", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n某金属材料发生光电效应的最大波长为 $\\lambda_{0}$, 将此材料制成一半径为 $R$ 的圆球, 并用绝缘线悬挂于真空室内。若以波长为 $\\lambda\\left(\\lambda<\\lambda_{0}\\right)$ 的单色光持续照射此金属球, 该金属球发生光电效应所产生光电子的最大初动能为? \n\n\n此金属球可带的电荷量最多为?\n\n(设无穷远处电势为零, 真空中半径为 $r$ 带电量为 $q$ 的导体球的电势为 $U=k \\frac{q}{r}$ 。)\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[金属球可带的电荷量最多为, 金属球发生光电效应所产生光电子的最大初动能为]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "金属球可带的电荷量最多为", "金属球发生光电效应所产生光电子的最大初动能为" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_797", "problem": "Here are some definitions of weight given by students:\n\nI. The gravitational force exerted on an object.\n\nII. The normal-contact force exerted by a supporting surface on an object.\n\nIII. The normal-contact force exerted by an object on a supporting surface.\n\nWhich of the following is/are the correct definition? Select one:\nA: I only\nB: II and III\nC: III only\nD: I and III\nE: I, II and III\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nHere are some definitions of weight given by students:\n\nI. The gravitational force exerted on an object.\n\nII. The normal-contact force exerted by a supporting surface on an object.\n\nIII. The normal-contact force exerted by an object on a supporting surface.\n\nWhich of the following is/are the correct definition? Select one:\n\nA: I only\nB: II and III\nC: III only\nD: I and III\nE: I, II and III\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_178", "problem": "in the time for a car to complete a race. The car begins with an initial speed $v_{0}$ and maintains a constant acceleration $a$ throughout the race. Both $v_{0}$ and $a$ have independent uncertainties of $10 \\%$. Which contributes greater uncertainty to your estimate of the time for the car to complete the race?\nA: The uncertainty in $v_{0}$ for sufficiently short races and the uncertainty in $a$ for sufficiently long races. \nB: The uncertainty in $v_{0}$ for sufficiently long races and the uncertainty in $a$ for sufficiently short races.\nC: The uncertainty in $v_{0}$.\nD: The uncertainty in $a$.\nE: They contribute equally in the uncertainty.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nin the time for a car to complete a race. The car begins with an initial speed $v_{0}$ and maintains a constant acceleration $a$ throughout the race. Both $v_{0}$ and $a$ have independent uncertainties of $10 \\%$. Which contributes greater uncertainty to your estimate of the time for the car to complete the race?\n\nA: The uncertainty in $v_{0}$ for sufficiently short races and the uncertainty in $a$ for sufficiently long races. \nB: The uncertainty in $v_{0}$ for sufficiently long races and the uncertainty in $a$ for sufficiently short races.\nC: The uncertainty in $v_{0}$.\nD: The uncertainty in $a$.\nE: They contribute equally in the uncertainty.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1382", "problem": "一电流表, 其内阻 $R_{\\mathrm{g}}=10.0 \\Omega$, 如果将它与一阻值 $R_{0}=49990 \\Omega$ 的定值电阻串联,便可成为一量程 $U_{0}=50 \\mathrm{~V}$ 的电压表. 现把此电流表改装成一块双量程的电压表, 两个量程分别为 $U_{01}=5 \\mathrm{~V}$ 和 $U_{02}=10 \\mathrm{~V}$. 当用此电压表的 $5 \\mathrm{~V}$ 挡去测一直流电源两端的电压时, 电压表的示数为 $4.50 \\mathrm{~V}$; 当用此电压表的 $10 \\mathrm{~V}$ 挡去测量该电源两端的电压时, 电压表的示数为 $4.80 \\mathrm{~V}$. 问此电源的电动势为多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n一电流表, 其内阻 $R_{\\mathrm{g}}=10.0 \\Omega$, 如果将它与一阻值 $R_{0}=49990 \\Omega$ 的定值电阻串联,便可成为一量程 $U_{0}=50 \\mathrm{~V}$ 的电压表. 现把此电流表改装成一块双量程的电压表, 两个量程分别为 $U_{01}=5 \\mathrm{~V}$ 和 $U_{02}=10 \\mathrm{~V}$. 当用此电压表的 $5 \\mathrm{~V}$ 挡去测一直流电源两端的电压时, 电压表的示数为 $4.50 \\mathrm{~V}$; 当用此电压表的 $10 \\mathrm{~V}$ 挡去测量该电源两端的电压时, 电压表的示数为 $4.80 \\mathrm{~V}$. 问此电源的电动势为多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$V$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-10.jpg?height=200&width=671&top_left_y=1679&top_left_x=561" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$V$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_230", "problem": "Let the two-ball body be placed on an inclined plane set at an angle $\\alpha=30^{\\circ}$ against the horizontal. At the initial time moment the line connecting balls touching points with the surface is exactly parallel to the lower edge of the inclined plane as shown in Figure 3. In this Part assume that the friction between the balls and the surface is so strong that there is no slipping at all times. The two-ball body is released.\n\n[figure1]\n\nFigure 3: Initial position of the two-ball body for Parts C1-C2.\n\n[figure2]\n\nFigure 4: Initial equilibrium position of the two-ball body for Part C3.\n\nNow assume that the two-ball body is at rest on the inclined plane such that the line connecting balls' touching points with the surface is exactly perpendicular to the lower edge of the inclined plane as shown in Figure 4.\n\nFind the angular frequency of small oscillations of the two-ball body around the equilibrium position shown in Figure 4.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nLet the two-ball body be placed on an inclined plane set at an angle $\\alpha=30^{\\circ}$ against the horizontal. At the initial time moment the line connecting balls touching points with the surface is exactly parallel to the lower edge of the inclined plane as shown in Figure 3. In this Part assume that the friction between the balls and the surface is so strong that there is no slipping at all times. The two-ball body is released.\n\n[figure1]\n\nFigure 3: Initial position of the two-ball body for Parts C1-C2.\n\n[figure2]\n\nFigure 4: Initial equilibrium position of the two-ball body for Part C3.\n\nNow assume that the two-ball body is at rest on the inclined plane such that the line connecting balls' touching points with the surface is exactly perpendicular to the lower edge of the inclined plane as shown in Figure 4.\n\nFind the angular frequency of small oscillations of the two-ball body around the equilibrium position shown in Figure 4.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $s^{-1}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_57b03088b694c15dae09g-3.jpg?height=412&width=998&top_left_y=347&top_left_x=561", "https://cdn.mathpix.com/cropped/2024_03_06_57b03088b694c15dae09g-3.jpg?height=414&width=1005&top_left_y=926&top_left_x=560" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$s^{-1}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_573", "problem": "Beloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nThe protons are fired at a target consisting of Fluorine atoms $(Z=9)$. What is the distance of closest approach to the center of the Fluorine nuclei for the protons? You can assume that the Fluorine does not move.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nBeloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nThe protons are fired at a target consisting of Fluorine atoms $(Z=9)$. What is the distance of closest approach to the center of the Fluorine nuclei for the protons? You can assume that the Fluorine does not move.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~m}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=468&width=1592&top_left_y=438&top_left_x=259", "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=493&width=1268&top_left_y=1339&top_left_x=426" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_17", "problem": "A satellite with a mass of $2500 \\mathrm{~kg}$ is in orbit around Earth. If a second satellite with a mass $5000 \\mathrm{~kg}$, is to be placed at the same orbital distance from Earth, the second satellite must have a speed which is:\nA: half the speed of the lighter satellite.\nB: twice the speed of the lighter satellite.\nC: the same speed as the lighter satellite.\nD: four times the speed of the lighter satellite.\nE: dependent upon the location from which it is launched.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA satellite with a mass of $2500 \\mathrm{~kg}$ is in orbit around Earth. If a second satellite with a mass $5000 \\mathrm{~kg}$, is to be placed at the same orbital distance from Earth, the second satellite must have a speed which is:\n\nA: half the speed of the lighter satellite.\nB: twice the speed of the lighter satellite.\nC: the same speed as the lighter satellite.\nD: four times the speed of the lighter satellite.\nE: dependent upon the location from which it is launched.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_504", "problem": "The square of the standard deviation is the variance, so\n\n$$\n(\\Delta r)^{2}=\\left\\langle r^{2}\\right\\rangle-\\langle r\\rangle^{2}\n$$\n\nComputing the average value of $r^{2}$ is mathematically the exact same thing as computing the moment of inertia of a uniform disk; we have\n\n$$\n\\left\\langle r^{2}\\right\\rangle=\\frac{1}{\\pi R^{2}} \\int_{0}^{R} r^{2}(2 \\pi r d r)=\\frac{1}{2} R^{2}\n$$\n\nSimilarly, the average value of $r$ is\n\n$$\n\\langle r\\rangle=\\frac{1}{\\pi R^{2}} \\int_{0}^{R} r(2 \\pi r d r)=\\frac{2}{3} R\n$$\n\nThen we have\n\n$$\n\\Delta r=\\frac{R}{\\sqrt{18}}\n$$\n\nWe have\n\n$$\n\\Delta p_{r} \\approx p \\Delta \\theta=\\frac{h}{\\lambda} \\Delta \\theta\n$$\n\nApplying the uncertainty principle, we have\n\n$$\n\\Delta \\theta \\geq \\frac{\\sqrt{18} \\lambda}{4 \\pi R}\n$$\n\nSince the factors of $\\hbar$ canceled out, this is really a classical calculation; one could get a similar result by considering classical diffraction from a circular aperture.In this problem, use a particle-like model of photons: they propagate in straight lines and obey the law of reflection, but are subject to the quantum uncertainty principle. You may use small-angle approximations throughout the problem.\n\nA photon with wavelength $\\lambda$ has traveled from a distant star to a telescope mirror, which has a circular cross-section with radius $R$ and a focal length $f \\gg R$. The path of the photon is nearly aligned to the axis of the mirror, but has some slight uncertainty $\\Delta \\theta$. The photon reflects off the mirror and travels to a detector, where it is absorbed by a particular pixel on a charge-coupled device (CCD).\n\nSuppose the telescope mirror is manufactured so that photons coming in parallel to each other are focused to the same pixel on the CCD, regardless of where they hit the mirror. Then all small cross-sectional areas of the mirror are equally likely to include the point of reflection for a photon.\n\nSuppose we want to build a telescope that can tell with high probability whether a photon it detected from Alpha Centauri A came the left half or right half of the star. Approximately how large would a telescope have to be to achieve this? Alpha Centauri A is approximately $4 \\times 10^{16} \\mathrm{~m}$ from Earth and has a radius approximately $7 \\times 10^{8} \\mathrm{~m}$. Assume visible light with $\\lambda=500 \\mathrm{~nm}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\nThe square of the standard deviation is the variance, so\n\n$$\n(\\Delta r)^{2}=\\left\\langle r^{2}\\right\\rangle-\\langle r\\rangle^{2}\n$$\n\nComputing the average value of $r^{2}$ is mathematically the exact same thing as computing the moment of inertia of a uniform disk; we have\n\n$$\n\\left\\langle r^{2}\\right\\rangle=\\frac{1}{\\pi R^{2}} \\int_{0}^{R} r^{2}(2 \\pi r d r)=\\frac{1}{2} R^{2}\n$$\n\nSimilarly, the average value of $r$ is\n\n$$\n\\langle r\\rangle=\\frac{1}{\\pi R^{2}} \\int_{0}^{R} r(2 \\pi r d r)=\\frac{2}{3} R\n$$\n\nThen we have\n\n$$\n\\Delta r=\\frac{R}{\\sqrt{18}}\n$$\n\nWe have\n\n$$\n\\Delta p_{r} \\approx p \\Delta \\theta=\\frac{h}{\\lambda} \\Delta \\theta\n$$\n\nApplying the uncertainty principle, we have\n\n$$\n\\Delta \\theta \\geq \\frac{\\sqrt{18} \\lambda}{4 \\pi R}\n$$\n\nSince the factors of $\\hbar$ canceled out, this is really a classical calculation; one could get a similar result by considering classical diffraction from a circular aperture.\n\nproblem:\nIn this problem, use a particle-like model of photons: they propagate in straight lines and obey the law of reflection, but are subject to the quantum uncertainty principle. You may use small-angle approximations throughout the problem.\n\nA photon with wavelength $\\lambda$ has traveled from a distant star to a telescope mirror, which has a circular cross-section with radius $R$ and a focal length $f \\gg R$. The path of the photon is nearly aligned to the axis of the mirror, but has some slight uncertainty $\\Delta \\theta$. The photon reflects off the mirror and travels to a detector, where it is absorbed by a particular pixel on a charge-coupled device (CCD).\n\nSuppose the telescope mirror is manufactured so that photons coming in parallel to each other are focused to the same pixel on the CCD, regardless of where they hit the mirror. Then all small cross-sectional areas of the mirror are equally likely to include the point of reflection for a photon.\n\nSuppose we want to build a telescope that can tell with high probability whether a photon it detected from Alpha Centauri A came the left half or right half of the star. Approximately how large would a telescope have to be to achieve this? Alpha Centauri A is approximately $4 \\times 10^{16} \\mathrm{~m}$ from Earth and has a radius approximately $7 \\times 10^{8} \\mathrm{~m}$. Assume visible light with $\\lambda=500 \\mathrm{~nm}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~m}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~m}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_379", "problem": "Jaan Kalda. A comet's orbit intersects the Earth's orbit (which can be assumed to be circular of radius $R_{0}=$ $1.5 \\times 10^{8} \\mathrm{~km}$ ) at an angle $\\alpha=45^{\\circ}$. The comet's and the Earth's orbits lie on the same plane.\n\nFor how many days $t$ will the the black hole. comet's distance to the Sun be less that $R_{0}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nJaan Kalda. A comet's orbit intersects the Earth's orbit (which can be assumed to be circular of radius $R_{0}=$ $1.5 \\times 10^{8} \\mathrm{~km}$ ) at an angle $\\alpha=45^{\\circ}$. The comet's and the Earth's orbits lie on the same plane.\n\nFor how many days $t$ will the the black hole. comet's distance to the Sun be less that $R_{0}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_83cc90b265723e4db811g-1.jpg?height=250&width=663&top_left_y=628&top_left_x=2171" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_409", "problem": "Consider the circuit shown in the figure.\n\n[figure1]\n\nThe voltages between the nodes $D$, $E$, and $F$ are know to have the following values: $V_{D E}=7 \\mathrm{~V}, V_{D F}=15 \\mathrm{~V}$, and $V_{E F}=20 \\mathrm{~V}$; what is the value of the input voltage $V_{0}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nConsider the circuit shown in the figure.\n\n[figure1]\n\nThe voltages between the nodes $D$, $E$, and $F$ are know to have the following values: $V_{D E}=7 \\mathrm{~V}, V_{D F}=15 \\mathrm{~V}$, and $V_{E F}=20 \\mathrm{~V}$; what is the value of the input voltage $V_{0}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~V}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_57fd2cb729ee4cd65883g-2.jpg?height=619&width=623&top_left_y=988&top_left_x=807" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~V}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_62", "problem": "The James Webb Space Telescope was launched on December 25, 2021 and arrived at its destination, LaGrange Point 2, on January 24, 2022. This point is approximately 1,500,000 km from the Earth and it is where the telescope will orbit the sun.\n\n11. What was James Webb's former occupation?\nA: Astronomer\nB: Telescope designer/engineer\nC: Silicon valley entrepreneur and financier of the telescope project\nD: Space shuttle astronaut/pilot\nE: NASA Administrator\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe James Webb Space Telescope was launched on December 25, 2021 and arrived at its destination, LaGrange Point 2, on January 24, 2022. This point is approximately 1,500,000 km from the Earth and it is where the telescope will orbit the sun.\n\n11. What was James Webb's former occupation?\n\nA: Astronomer\nB: Telescope designer/engineer\nC: Silicon valley entrepreneur and financier of the telescope project\nD: Space shuttle astronaut/pilot\nE: NASA Administrator\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1210", "problem": "Although atomic nuclei are quantum objects, a number of phenomenological laws for their basic properties (like radius or binding energy) can be deduced from simple assumptions: (i) nuclei are built from nucleons (i.e. protons and neutrons); (ii) strong nuclear interaction holding these nucleons together has a very short range (it acts only between neighboring nucleons); (iii) the number of protons $(Z)$ in a given nucleus is approximately equal to the number of neutrons $(N)$, i.e. $Z \\approx N \\approx A / 2$, where $A$ is the total number of nucleons $(A \\gg 1)$.\n\n## Fission of heavy nuclei\n\nFission is a nuclear process in which a nucleus splits into smaller parts (lighter nuclei). Suppose that a nucleus with $A$ nucleons splits into only two equal parts as depicted in Fig. 2.\n\n[figure1]\n\nAssume that $d=2 R(A / 2)$ and evaluate the expression for $E_{\\text {kin }}$ obtained in part a) for $A=$ 100, 150, 200 and 250 (express the results in units of MeV). Estimate the values of $A$ for which fission is possible in the model described above?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nAlthough atomic nuclei are quantum objects, a number of phenomenological laws for their basic properties (like radius or binding energy) can be deduced from simple assumptions: (i) nuclei are built from nucleons (i.e. protons and neutrons); (ii) strong nuclear interaction holding these nucleons together has a very short range (it acts only between neighboring nucleons); (iii) the number of protons $(Z)$ in a given nucleus is approximately equal to the number of neutrons $(N)$, i.e. $Z \\approx N \\approx A / 2$, where $A$ is the total number of nucleons $(A \\gg 1)$.\n\n## Fission of heavy nuclei\n\nFission is a nuclear process in which a nucleus splits into smaller parts (lighter nuclei). Suppose that a nucleus with $A$ nucleons splits into only two equal parts as depicted in Fig. 2.\n\n[figure1]\n\nAssume that $d=2 R(A / 2)$ and evaluate the expression for $E_{\\text {kin }}$ obtained in part a) for $A=$ 100, 150, 200 and 250 (express the results in units of MeV). Estimate the values of $A$ for which fission is possible in the model described above?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_656c8e5e21a256bf8ee1g-2.jpg?height=594&width=1510&top_left_y=2036&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1291", "problem": "两根质量均匀分布的杆 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 质量均为 $m$ ，长均为 $l, \\mathrm{~A}$ 端被光滑较接到一固定点 (即 $\\mathrm{AB}$ 杆可在坚直平面内绕 $\\mathrm{A}$ 点无摩擦转动）。开始时 $\\mathrm{C}$ 点有外力保持两杆静止, $\\mathrm{A} 、 \\mathrm{C}$ 在同一水平线 $\\mathrm{AD}$ 上, $\\mathrm{A}$ 、 $B 、 C$ 三点都在同一坚直平面内, $\\angle A B C=60^{\\circ}$ 。某时刻撤去外力后两杆始终在坚直平面内运动。\n\n若两杆在 B 点固结在一起, 求当 $\\mathrm{AB}$ 杆运动到与水平线 $\\mathrm{AD}$ 的夹角为 $\\theta$ 时, $\\mathrm{AB}$ 杆绕 $\\mathrm{A}$ 点转动的角速度。\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n两根质量均匀分布的杆 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 质量均为 $m$ ，长均为 $l, \\mathrm{~A}$ 端被光滑较接到一固定点 (即 $\\mathrm{AB}$ 杆可在坚直平面内绕 $\\mathrm{A}$ 点无摩擦转动）。开始时 $\\mathrm{C}$ 点有外力保持两杆静止, $\\mathrm{A} 、 \\mathrm{C}$ 在同一水平线 $\\mathrm{AD}$ 上, $\\mathrm{A}$ 、 $B 、 C$ 三点都在同一坚直平面内, $\\angle A B C=60^{\\circ}$ 。某时刻撤去外力后两杆始终在坚直平面内运动。\n\n若两杆在 B 点固结在一起, 求\n\n问题:\n当 $\\mathrm{AB}$ 杆运动到与水平线 $\\mathrm{AD}$ 的夹角为 $\\theta$ 时, $\\mathrm{AB}$ 杆绕 $\\mathrm{A}$ 点转动的角速度。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_49158ed36459a540f197g-04.jpg?height=549&width=688&top_left_y=248&top_left_x=1164" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1416", "problem": "如图所示, 匝数为 $N_{1}$ 的原线圈和匝数为 $N_{2}$ 的副线圈绕在同一闭合铁芯上, 副线圈两端与电阻 $R$ 相连, 原线圈两端与平行金属导轨相连。两轨之间的距离为 $L$, 其电阻可不计。在虚线的左侧, 存在方向与导轨所在平面垂直的匀强磁场, 磁场的磁感应强度大小为 $B, p q$ 是一质量为 $m$ 电阻为 $r$ 与导轨垂直放置的金属杆, 它可在导轨上沿与导轨平行的方向无摩擦地滑动。假设在任何同一时刻通过线圈每一匝的磁通量相同, 两个线圈的电阻、铁芯中包括浴流在内的各种损耗都忽略不计, 且变压器中的电磁场完全限制在变压器铁芯中。现于 $t=0$ 时刻开始施一外力, 使杆从静止出发以恒定的加速度 $a$ 向左运动。不考虑连接导线的自感。若已知在某时刻 $t$ 时原线圈电流的大小为 $I_{1}$,\n\n[图1]求此时刻外力的功率", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图所示, 匝数为 $N_{1}$ 的原线圈和匝数为 $N_{2}$ 的副线圈绕在同一闭合铁芯上, 副线圈两端与电阻 $R$ 相连, 原线圈两端与平行金属导轨相连。两轨之间的距离为 $L$, 其电阻可不计。在虚线的左侧, 存在方向与导轨所在平面垂直的匀强磁场, 磁场的磁感应强度大小为 $B, p q$ 是一质量为 $m$ 电阻为 $r$ 与导轨垂直放置的金属杆, 它可在导轨上沿与导轨平行的方向无摩擦地滑动。假设在任何同一时刻通过线圈每一匝的磁通量相同, 两个线圈的电阻、铁芯中包括浴流在内的各种损耗都忽略不计, 且变压器中的电磁场完全限制在变压器铁芯中。现于 $t=0$ 时刻开始施一外力, 使杆从静止出发以恒定的加速度 $a$ 向左运动。不考虑连接导线的自感。若已知在某时刻 $t$ 时原线圈电流的大小为 $I_{1}$,\n\n[图1]\n\n问题:\n求此时刻外力的功率\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_93e0584db50294961a50g-05.jpg?height=213&width=691&top_left_y=1799&top_left_x=532" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_721", "problem": "Cherie is doing an experiment; she needs to heat a small sample to $700 \\mathrm{~K}$. There is only one oven available with a maximum temperature of $400 \\mathrm{~K}$. Could she heat the sample to $700 \\mathrm{~K}$ by using a large lens to focus the radiation from the oven onto the sample?\nA: Yes, if the volume of the oven is at least $7 / 4$ the volume of the sample.\nB: Yes, if the sample is placed at the focal point of the lens.\nC: No, because it would violate the conservation of energy.\nD: Yes, if the area of the front of the oven is at least 7/4 the area of the front of the sample.\nE: No, because it would violate the second law of thermodynamics.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nCherie is doing an experiment; she needs to heat a small sample to $700 \\mathrm{~K}$. There is only one oven available with a maximum temperature of $400 \\mathrm{~K}$. Could she heat the sample to $700 \\mathrm{~K}$ by using a large lens to focus the radiation from the oven onto the sample?\n\nA: Yes, if the volume of the oven is at least $7 / 4$ the volume of the sample.\nB: Yes, if the sample is placed at the focal point of the lens.\nC: No, because it would violate the conservation of energy.\nD: Yes, if the area of the front of the oven is at least 7/4 the area of the front of the sample.\nE: No, because it would violate the second law of thermodynamics.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1469", "problem": "太空中有一飞行器靠其自身动力维持在地球赤道的正上方 $L=a R_{e}$ 处, 相对于赤道上的一地面物资供应站保持静止. 这里, $R_{e}$ 为地球的半径, $a$ 为常数, $a>a_{m}$, 而\n\n$$\n\\alpha_{m}=\\left(\\frac{G M_{e}}{\\omega_{e}^{2} R_{e}^{3}}\\right)^{1 / 3}-1\n$$\n\n$M_{e}$ 和 $W_{e}$ 分别为地球的质量和自转角速度, $G$ 为引力常数. 设想从供应站到飞行器有一根用于运送物资的刚性、管壁匀质、质量为 $m_{p}$ 的坚直输送管, 输送管下端固定在地面上, 并设法保持输送管与地面始终垂直. 推送物资时, 把物资放进输送管下端内的平底托盘上, 沿管壁向上推进, 并保持托盘运行速度不致过大. 忽略托盘与管壁之间的摩擦力, 考虑地球的自转, 但不考虑地球的公转. 设某次所推送物资和托盘的总质量为 $m$.当飞行器离地面的高度 (记为 $L_{0}$ ) 为多少时, 在把物资送到飞行器的过程中, 地球引力和惯性离心力所做功的和为零?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n太空中有一飞行器靠其自身动力维持在地球赤道的正上方 $L=a R_{e}$ 处, 相对于赤道上的一地面物资供应站保持静止. 这里, $R_{e}$ 为地球的半径, $a$ 为常数, $a>a_{m}$, 而\n\n$$\n\\alpha_{m}=\\left(\\frac{G M_{e}}{\\omega_{e}^{2} R_{e}^{3}}\\right)^{1 / 3}-1\n$$\n\n$M_{e}$ 和 $W_{e}$ 分别为地球的质量和自转角速度, $G$ 为引力常数. 设想从供应站到飞行器有一根用于运送物资的刚性、管壁匀质、质量为 $m_{p}$ 的坚直输送管, 输送管下端固定在地面上, 并设法保持输送管与地面始终垂直. 推送物资时, 把物资放进输送管下端内的平底托盘上, 沿管壁向上推进, 并保持托盘运行速度不致过大. 忽略托盘与管壁之间的摩擦力, 考虑地球的自转, 但不考虑地球的公转. 设某次所推送物资和托盘的总质量为 $m$.\n\n问题:\n当飞行器离地面的高度 (记为 $L_{0}$ ) 为多少时, 在把物资送到飞行器的过程中, 地球引力和惯性离心力所做功的和为零?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1519", "problem": "电子偶素原子 (Ps ) 是由电子 $\\mathrm{e}^{-}$与正电子 $\\mathrm{e}^{+}$(电子的反粒子, 其质量与电子的相同,电荷与电子的大小相等、符号相反) 组成的量子束缚体系, 其能级可类比氢原子能级得出。根据玻尔氢原子理论, 电子绕质子的圆周运动轨道角动量的取值是量子化的, 即为 $\\hbar$ 的整数倍。考虑到质子质量是有限的, 氢原子量子化条件应修正为: 电子与质子质心系中相对其质心的总轨道角动量取值为 $\\hbar$ 的整数倍。这一量子化条件可直接推广到其它两体束缚体系, 如电子偶素等。以下计算结果均保留四位有效数字。\n\n电子偶素基态原子不稳定, 可很快发生湮没而生成两个光子:\n\n$$\n\\mathrm{Ps} \\rightarrow \\gamma_{1}+\\gamma_{2}\n$$\n\n当基态电子偶素原子相对于实验室参照系以远小于光速的某速度运动时发生湮没, 在相对于该速度方向的偏角为 $\\theta_{1}$ 的方向上观测到生成的一个光子 $\\gamma_{1}$, 同时在相对于光子 $\\gamma_{1}$ 速度反方向的偏角为 $\\Delta \\theta(\\Delta \\theta<<1)$ 的方向上观测到\n\n[图1]\n另一个光子 $\\gamma_{2}$, 如图所示。\n\n已知: 氢原子基态能量 $E_{n=1}^{\\mathrm{H}}=-13.60 \\mathrm{eV}$, 电子质量 $m_{\\mathrm{e}}=0.5110 \\mathrm{MeV} / c^{2}$, 质子与电子的质量之比为 1836 。求基态电子偶素原子速度 $v_{0}$ 的大小、两个光子 $\\gamma_{1}$ 和 $\\gamma_{2}$ 的能量 $E_{1}$ 和 $E_{2}$ 的表达式;并给出当 $\\theta_{1}=\\frac{\\pi}{3} 、 \\Delta \\theta=3.464 \\times 10^{-3}$ 时 $v_{0} 、 E_{1}$ 和 $E_{2}$ 的值。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n电子偶素原子 (Ps ) 是由电子 $\\mathrm{e}^{-}$与正电子 $\\mathrm{e}^{+}$(电子的反粒子, 其质量与电子的相同,电荷与电子的大小相等、符号相反) 组成的量子束缚体系, 其能级可类比氢原子能级得出。根据玻尔氢原子理论, 电子绕质子的圆周运动轨道角动量的取值是量子化的, 即为 $\\hbar$ 的整数倍。考虑到质子质量是有限的, 氢原子量子化条件应修正为: 电子与质子质心系中相对其质心的总轨道角动量取值为 $\\hbar$ 的整数倍。这一量子化条件可直接推广到其它两体束缚体系, 如电子偶素等。以下计算结果均保留四位有效数字。\n\n电子偶素基态原子不稳定, 可很快发生湮没而生成两个光子:\n\n$$\n\\mathrm{Ps} \\rightarrow \\gamma_{1}+\\gamma_{2}\n$$\n\n当基态电子偶素原子相对于实验室参照系以远小于光速的某速度运动时发生湮没, 在相对于该速度方向的偏角为 $\\theta_{1}$ 的方向上观测到生成的一个光子 $\\gamma_{1}$, 同时在相对于光子 $\\gamma_{1}$ 速度反方向的偏角为 $\\Delta \\theta(\\Delta \\theta<<1)$ 的方向上观测到\n\n[图1]\n另一个光子 $\\gamma_{2}$, 如图所示。\n\n已知: 氢原子基态能量 $E_{n=1}^{\\mathrm{H}}=-13.60 \\mathrm{eV}$, 电子质量 $m_{\\mathrm{e}}=0.5110 \\mathrm{MeV} / c^{2}$, 质子与电子的质量之比为 1836 。\n\n问题:\n求基态电子偶素原子速度 $v_{0}$ 的大小、两个光子 $\\gamma_{1}$ 和 $\\gamma_{2}$ 的能量 $E_{1}$ 和 $E_{2}$ 的表达式;并给出当 $\\theta_{1}=\\frac{\\pi}{3} 、 \\Delta \\theta=3.464 \\times 10^{-3}$ 时 $v_{0} 、 E_{1}$ 和 $E_{2}$ 的值。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$v_{0}$, $E_{1}$, $E_{2}$]\n它们的单位依次是[$\\mathrm{~m} / \\mathrm{s}$, $\\mathrm{MeV}$, $\\mathrm{MeV}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-06.jpg?height=437&width=437&top_left_y=775&top_left_x=1278" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{~m} / \\mathrm{s}$", "$\\mathrm{MeV}$", "$\\mathrm{MeV}$" ], "answer_sequence": [ "$v_{0}$", "$E_{1}$", "$E_{2}$" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_812", "problem": "## Introduction\n\nActive galactic nuclei (AGN) are supermassive black holes which form the centres of galaxies, and emit large amounts of energy in radiation and particle flows. One feature of many AGN are jetted outflows, which can be observed through radio emission, and sometimes also in other parts of the electromagnetic spectrum, including $x$-rays. These jets are large flows of plasma at relativistic speeds, over lengths of order $10^{20} \\mathrm{~m}$, which is tens of thousands of light years. The $\\mathrm{x}$-ray emission from jets is usually dominated by synchrotron emission from relativistic electrons gyrating in the magnetic field of the jet.\n\n[figure1]\n\nFigure 1: X-ray image of the jet from the Centaurus A AGN. Darker regions represent regions of higher intensity $x$-rays. Brighter regions within the fainter jet are called knots. (Snios et al., 2019)\n\nA simple model of the flow of jets assumes that the flow is steady and directed radially away from the central AGN, so approximately one dimensional, and that the plasma in the jet is in pressure equilibrium with its surroundings. There is assumed to be a constant rate per volume of mass injected into the jet from stars which lose their outer layers as they move through their life cycle.\n\nThe jet is described in terms of the coordinate representing distance from the AGN, $s$, and the opening radius $r$ of the conical jet. These distances are measured in parsecs, where $1 \\mathrm{pc}=3.086 \\times 10^{16} \\mathrm{~m}$. The speed of the jet flow is assumed to be directed radially away from the central AGN, and be a function of $s$ only. The plasma in the jet is comprised of electrons, protons, and some heavier ionised nuclei. The average energy carried by each particle in the jet, in the reference frame of the bulk flow of the jet (which we will call the jet frame), is $\\epsilon_{\\mathrm{av}}=\\mu_{\\mathrm{pp}} c^{2}+h$, where the term $h$ includes all thermal kinetic energy and potential energies in terms of the pressure $P$ and $n$ is the number density of the plasma.\n\nAs the stars, which the jet flows past, move through their life cycles they can lose part of their atmosphere. This results in a uniform rate of injection of mass per unit volume $\\alpha$ into the jet, and the injected particles are assumed to be at rest relative to the AGN.\n\nThis model can be applied to the Centaurus A jet. Centaurus A is one of the nearest AGN, so it is possible to observe its jet at relatively high spatial resolution. The total power carried by the jet is estimated to be $P_{\\mathrm{j}}=1 \\times 10^{36} \\mathrm{~J} \\cdot \\mathrm{s}^{-1}$. See below for a diagram of a simple geometrical description of the Centaurus A jet, including measurements of some jet parameters. $s_{1}$ is the coordinate of the start of the jet, and $s_{2}$ the coordinate of the end of the jet. In Centuarus A the average mass per particle is $\\mu_{\\mathrm{pp}}=0.59 m_{\\mathrm{p}}$ and $h=\\frac{13}{4} P / n$. The pressure in the plasma surrounding the jet is $P(s)=5.7 \\times 10^{-12}\\left(\\frac{s}{s_{0}}\\right)^{-1.5} \\mathrm{~Pa}$, where $s_{0}=1 \\mathrm{kpc}$.\n\n[figure2]\n\nFigure 2: The Centaurus A jet, showing the geometry compared to the active galactic nucleus (AGN).\n\nThe jet is described by the following parameters, all of which depend on the distance $s$ from the AGN:\n\n- the opening radius of the jet $r(s)$ in the AGN frame\n- the cross sectional area of the jet $A(s)$ in the AGN frame\n- the speed of the jet $v(s)$ in the AGN frame\n- the lorentz gamma factor of the jet $\\gamma(s)$ in the AGN frame\n- the number density $n(s)$ in the frame of the jet\n\nFind the number density of particles, $n^{\\prime}(s)$, in the frame of the AGN, in terms of the proper number density, $n(s)$ and other jet parameters. The proper number density is the number density in the frame which is locally co-moving with the jet plasma outflow, which we will call the jet frame.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\n## Introduction\n\nActive galactic nuclei (AGN) are supermassive black holes which form the centres of galaxies, and emit large amounts of energy in radiation and particle flows. One feature of many AGN are jetted outflows, which can be observed through radio emission, and sometimes also in other parts of the electromagnetic spectrum, including $x$-rays. These jets are large flows of plasma at relativistic speeds, over lengths of order $10^{20} \\mathrm{~m}$, which is tens of thousands of light years. The $\\mathrm{x}$-ray emission from jets is usually dominated by synchrotron emission from relativistic electrons gyrating in the magnetic field of the jet.\n\n[figure1]\n\nFigure 1: X-ray image of the jet from the Centaurus A AGN. Darker regions represent regions of higher intensity $x$-rays. Brighter regions within the fainter jet are called knots. (Snios et al., 2019)\n\nA simple model of the flow of jets assumes that the flow is steady and directed radially away from the central AGN, so approximately one dimensional, and that the plasma in the jet is in pressure equilibrium with its surroundings. There is assumed to be a constant rate per volume of mass injected into the jet from stars which lose their outer layers as they move through their life cycle.\n\nThe jet is described in terms of the coordinate representing distance from the AGN, $s$, and the opening radius $r$ of the conical jet. These distances are measured in parsecs, where $1 \\mathrm{pc}=3.086 \\times 10^{16} \\mathrm{~m}$. The speed of the jet flow is assumed to be directed radially away from the central AGN, and be a function of $s$ only. The plasma in the jet is comprised of electrons, protons, and some heavier ionised nuclei. The average energy carried by each particle in the jet, in the reference frame of the bulk flow of the jet (which we will call the jet frame), is $\\epsilon_{\\mathrm{av}}=\\mu_{\\mathrm{pp}} c^{2}+h$, where the term $h$ includes all thermal kinetic energy and potential energies in terms of the pressure $P$ and $n$ is the number density of the plasma.\n\nAs the stars, which the jet flows past, move through their life cycles they can lose part of their atmosphere. This results in a uniform rate of injection of mass per unit volume $\\alpha$ into the jet, and the injected particles are assumed to be at rest relative to the AGN.\n\nThis model can be applied to the Centaurus A jet. Centaurus A is one of the nearest AGN, so it is possible to observe its jet at relatively high spatial resolution. The total power carried by the jet is estimated to be $P_{\\mathrm{j}}=1 \\times 10^{36} \\mathrm{~J} \\cdot \\mathrm{s}^{-1}$. See below for a diagram of a simple geometrical description of the Centaurus A jet, including measurements of some jet parameters. $s_{1}$ is the coordinate of the start of the jet, and $s_{2}$ the coordinate of the end of the jet. In Centuarus A the average mass per particle is $\\mu_{\\mathrm{pp}}=0.59 m_{\\mathrm{p}}$ and $h=\\frac{13}{4} P / n$. The pressure in the plasma surrounding the jet is $P(s)=5.7 \\times 10^{-12}\\left(\\frac{s}{s_{0}}\\right)^{-1.5} \\mathrm{~Pa}$, where $s_{0}=1 \\mathrm{kpc}$.\n\n[figure2]\n\nFigure 2: The Centaurus A jet, showing the geometry compared to the active galactic nucleus (AGN).\n\nThe jet is described by the following parameters, all of which depend on the distance $s$ from the AGN:\n\n- the opening radius of the jet $r(s)$ in the AGN frame\n- the cross sectional area of the jet $A(s)$ in the AGN frame\n- the speed of the jet $v(s)$ in the AGN frame\n- the lorentz gamma factor of the jet $\\gamma(s)$ in the AGN frame\n- the number density $n(s)$ in the frame of the jet\n\nFind the number density of particles, $n^{\\prime}(s)$, in the frame of the AGN, in terms of the proper number density, $n(s)$ and other jet parameters. The proper number density is the number density in the frame which is locally co-moving with the jet plasma outflow, which we will call the jet frame.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2416d49d47cb88c0a72bg-1.jpg?height=651&width=805&top_left_y=1045&top_left_x=631", "https://cdn.mathpix.com/cropped/2024_03_14_2416d49d47cb88c0a72bg-2.jpg?height=654&width=1356&top_left_y=821&top_left_x=356" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1634", "problem": "卫星的运动可有地面的观测来决定, 而知道了卫星的运动, 又可以用空间的飞行体或地面上物体的运动, 这都涉及到时间和空间坐标的测定, 为简化分析和计算, 不考虑地球的自转和公转, 把它作惯性系。\n\n根据狭义相对论, 运动的钟比静止的钟慢, 钟在引力场中慢。现在来考虑在上述测量中\n相对论的这两种效应。已知天上卫星的钟与地面观测站的钟零点已对准, 假设卫星在离地面 $h=2.00 \\times 10^{4} \\mathrm{~m}$ 的圆形轨道上运行, 地球半径 $\\mathrm{R}$ 、光速 $\\mathrm{c}$ 和地球表面重力加速度 $\\mathrm{g}$ 取小题 2 中给的值。根据狭义相对论, 试估算地上的钟经过 24 小时后, 它的示数与卫星上的钟的示数差多少? 设在处理这一问题时可以把匀速直线运动的时钟走慢的公式用于匀速圆周运动。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n卫星的运动可有地面的观测来决定, 而知道了卫星的运动, 又可以用空间的飞行体或地面上物体的运动, 这都涉及到时间和空间坐标的测定, 为简化分析和计算, 不考虑地球的自转和公转, 把它作惯性系。\n\n根据狭义相对论, 运动的钟比静止的钟慢, 钟在引力场中慢。现在来考虑在上述测量中\n相对论的这两种效应。已知天上卫星的钟与地面观测站的钟零点已对准, 假设卫星在离地面 $h=2.00 \\times 10^{4} \\mathrm{~m}$ 的圆形轨道上运行, 地球半径 $\\mathrm{R}$ 、光速 $\\mathrm{c}$ 和地球表面重力加速度 $\\mathrm{g}$ 取小题 2 中给的值。\n\n问题:\n根据狭义相对论, 试估算地上的钟经过 24 小时后, 它的示数与卫星上的钟的示数差多少? 设在处理这一问题时可以把匀速直线运动的时钟走慢的公式用于匀速圆周运动。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$$\\mu s$$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$$\\mu s$$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_778", "problem": "A book lies at rest on a table. The table is at rest on the surface of the Earth. The Newton's Third Law reaction force to the gravitational force of the Earth on the book is:\nA: the gravitational force exerted by the Earth on the book.\nB: the normal force exerted by the table on the book.\nC: the gravitational force exerted by the table on the book.\nD: the normal force exerted by the Earth on the table.\nE: the gravitational force exerted by the book on the Earth.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA book lies at rest on a table. The table is at rest on the surface of the Earth. The Newton's Third Law reaction force to the gravitational force of the Earth on the book is:\n\nA: the gravitational force exerted by the Earth on the book.\nB: the normal force exerted by the table on the book.\nC: the gravitational force exerted by the table on the book.\nD: the normal force exerted by the Earth on the table.\nE: the gravitational force exerted by the book on the Earth.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_91", "problem": "A small rocket with a mass of $5000 \\mathrm{~kg}$ is to be launched vertically. What rate of ejection of gas, with an exhaust speed $1000 \\mathrm{~m} / \\mathrm{s}$, is necessary to provide the thrust to impart an initial upward acceleration of $2 g$ ?\nA: $50 \\mathrm{~kg} / \\mathrm{s}$\nB: $100 \\mathrm{~kg} / \\mathrm{s}$\nC: $150 \\mathrm{~kg} / \\mathrm{s}$\nD: $200 \\mathrm{~kg} / \\mathrm{s}$\nE: $250 \\mathrm{~kg} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA small rocket with a mass of $5000 \\mathrm{~kg}$ is to be launched vertically. What rate of ejection of gas, with an exhaust speed $1000 \\mathrm{~m} / \\mathrm{s}$, is necessary to provide the thrust to impart an initial upward acceleration of $2 g$ ?\n\nA: $50 \\mathrm{~kg} / \\mathrm{s}$\nB: $100 \\mathrm{~kg} / \\mathrm{s}$\nC: $150 \\mathrm{~kg} / \\mathrm{s}$\nD: $200 \\mathrm{~kg} / \\mathrm{s}$\nE: $250 \\mathrm{~kg} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1604", "problem": "嫦娥 1 号奔月卫星与长征 3 号火箭分离后, 进入绕地运行的椭圆轨道, 近地点离地面高 $H_{n}=2.05 \\times 10^{2} \\mathrm{~km}$, 远地点离地面高 $H_{f}=5.0930 \\times 10^{4} \\mathrm{~km}$, 周期约为 16 小时, 称为 16 小时轨道 (如图中曲线 1 所示)。随后, 为了使卫星离地越来越远, 星载发动机先在远地点点火, 使卫星进入新轨道 (如图中曲线 2 所示), 以抬高近地点。后来又连续三次在抬高以后的近地点点火, 使卫星加速和变轨, 抬高远地点, 相继进入 24 小时轨道、 48 小时轨道和地月转移轨道 (分别如图中曲线 3、4、5 所示)。已知卫星质量 $m=2.350 \\times 10^{3} \\mathrm{~kg}$, 地球半径 $R=6.378 \\times 10^{3} \\mathrm{~km}$, 地面重力加速度 $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$, 月球半径 $r=1.738 \\times 10^{3} \\mathrm{~km}$ 。卫星最后进入绕月圆形轨道, 距月面高度 $\\mathrm{H}_{\\mathrm{m}}$ 约为 $200 \\mathrm{~km}$, 周期 $\\mathrm{T}_{\\mathrm{m}}=127$ 分钟, 试据此估算月球质量与地球质量之比值。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n嫦娥 1 号奔月卫星与长征 3 号火箭分离后, 进入绕地运行的椭圆轨道, 近地点离地面高 $H_{n}=2.05 \\times 10^{2} \\mathrm{~km}$, 远地点离地面高 $H_{f}=5.0930 \\times 10^{4} \\mathrm{~km}$, 周期约为 16 小时, 称为 16 小时轨道 (如图中曲线 1 所示)。随后, 为了使卫星离地越来越远, 星载发动机先在远地点点火, 使卫星进入新轨道 (如图中曲线 2 所示), 以抬高近地点。后来又连续三次在抬高以后的近地点点火, 使卫星加速和变轨, 抬高远地点, 相继进入 24 小时轨道、 48 小时轨道和地月转移轨道 (分别如图中曲线 3、4、5 所示)。已知卫星质量 $m=2.350 \\times 10^{3} \\mathrm{~kg}$, 地球半径 $R=6.378 \\times 10^{3} \\mathrm{~km}$, 地面重力加速度 $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$, 月球半径 $r=1.738 \\times 10^{3} \\mathrm{~km}$ 。\n\n问题:\n卫星最后进入绕月圆形轨道, 距月面高度 $\\mathrm{H}_{\\mathrm{m}}$ 约为 $200 \\mathrm{~km}$, 周期 $\\mathrm{T}_{\\mathrm{m}}=127$ 分钟, 试据此估算月球质量与地球质量之比值。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_546", "problem": "Part A\n\nA \"Gilbert\" dipole consists of a pair of magnetic monopoles each with a magnitude $q_{m}$ but opposite magnetic charges separated by a distance $d$, where $d$ is small. In this case, assume that $-q_{m}$ is located at $z=0$ and $+q_{m}$ is located at $z=d$.\n\n[figure1]\n\nAssume that magnetic monopoles behave like electric monopoles according to a coulomb-like force\n\nand the magnetic field obeys\n\n$$\nF=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m 1} q_{m 2}}{r^{2}}\n$$\n\n$$\nB=F / q_{m} .\n$$\n\nBy the second expression, $q_{m}$ must be measured in Newtons per Tesla. But since Tesla are also Newtons per Ampere per meter, then $q_{m}$ is also measured in Ampere meters.\n\nAdding the two terms,\n\n$$\nB(z)=-\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{z^{2}}+\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{(z+d)^{2}}\n$$\n\nEvaluate this expression in the limit as $d \\rightarrow 0$, assuming that the product $q_{m} d=p_{m}$ is kept constant, keeping only the lowest non-zero term.\n\nSimplifying our previous expression,\n\n$$\nB(z)=\\frac{\\mu_{0}}{4 \\pi} q_{m} d\\left(\\frac{2+d / z}{z(z+d)^{2}}\\right) \\text {. }\n$$\n\nThus in the limit $d \\rightarrow 0$ we have\n\n$$\nB(z)=\\frac{\\mu_{0}}{2 \\pi} \\frac{q_{m} d}{z^{3}}=\\frac{\\mu_{0}}{2 \\pi} \\frac{p_{m}}{z^{3}}\n$$\n\nPart B\n\nAn \"Ampre\" dipole is a magnetic dipole produced by a current loop $I$ around a circle of radius $r$, where $r$ is small. Assume the that the $z$ axis is the axis of rotational symmetry for the circular loop, and the loop lies in the $x y$ plane at $z=0$.\n\n[figure2]\n\nApplying the Biot-Savart law, with $\\mathbf{s}$ the vector from the point on the loop to the point on the $z$ axis,\n\n$$\nB(z)=\\frac{\\mu_{0} I}{4 \\pi} \\oint \\frac{d \\mathbf{l} \\times \\mathbf{s}}{s^{3}}=\\frac{\\mu_{0} I}{4 \\pi} \\frac{2 \\pi r}{r^{2}+z^{2}} \\sin \\theta\n$$\n\nwhere $\\theta$ is the angle between the point on the loop and the center of the loop as measured by the point on the $z$ axis, so\n\n$$\n\\sin \\theta=\\frac{r}{\\sqrt{r^{2}+z^{2}}}\n$$\n\nThen we have\n\n$$\nB(z)=\\frac{\\mu_{0} I}{4 \\pi} \\frac{2 \\pi r^{2}}{\\left(r^{2}+z^{2}\\right)^{3 / 2}}\n$$\n\nLet $k I r^{\\gamma}$ have dimensions equal to that of the quantity $p_{m}$ defined above in Part A.\n\nEvaluate the expression in Part bi in the limit as $r \\rightarrow 0$, assuming that the product $k I r^{\\gamma}=p_{m}^{\\prime}$ is kept constant, keeping only the lowest non-zero term.\n\n$$\nB(z)=\\frac{\\mu_{0} I}{4 \\pi} \\frac{2 \\pi r^{2}}{\\left(r^{2}+z^{2}\\right)^{3 / 2}} \\approx \\frac{\\mu_{0} I}{2 \\pi} \\frac{\\pi r^{2}}{z^{3}}=\\frac{\\mu_{0}}{2 \\pi} \\frac{\\pi}{k} \\frac{p_{m}^{\\prime}}{z^{3}}\n$$Now we try to compare the two approaches if we model a physical magnet as being composed of densely packed microscopic dipoles.\n\n[figure3]\n\nA cylinder of this uniform magnetic material has a radius $R$ and a length $L$. It is composed of $N$ magnetic dipoles that could be either all Ampre type or all Gilbert type. $N$ is a very large number. The axis of rotation of the cylinder and all of the dipoles are all aligned with the $z$ axis and all point in the same direction as defined above so that the magnetic field outside the cylinder is the same in either dipole case as you previously determined. Below is a picture of the two dipole models; they are cubes of side $d<\\lambda_{1}\\right)$ 的单色可见光照射同一个杨氏双缝干涉实验装置。观察到波长为 $\\lambda_{1}$ 的光的干涉条纹的 $1 、 2$ 级亮纹之间原本是暗纹的位置出现了波长为 $\\lambda_{2}$ 的光的干涉条纹的 1 级亮纹, 则两种光的波长之比 $\\lambda_{2}: \\lambda_{1}=$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n先用波长为 $\\lambda_{1}$ 的单色可见光照射杨氏双缝干涉实验装置; 再加上波长为 $\\lambda_{2}\\left(\\lambda_{2}>\\lambda_{1}\\right)$ 的单色可见光照射同一个杨氏双缝干涉实验装置。观察到波长为 $\\lambda_{1}$ 的光的干涉条纹的 $1 、 2$ 级亮纹之间原本是暗纹的位置出现了波长为 $\\lambda_{2}$ 的光的干涉条纹的 1 级亮纹, 则两种光的波长之比 $\\lambda_{2}: \\lambda_{1}=$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_464", "problem": "In this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinite uniform sheet has a surface charge density $\\sigma$ and has an infinitesimal thickness. The sheet lies in the $x y$ plane.\n\nAssuming the sheet is at rest, determine the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) above and below the sheet.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nIn this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinite uniform sheet has a surface charge density $\\sigma$ and has an infinitesimal thickness. The sheet lies in the $x y$ plane.\n\nAssuming the sheet is at rest, determine the electric field $\\tilde{\\mathbf{E}}$ (magnitude and direction) above and below the sheet.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the electric field $\\tilde{\\mathbf{E}}$ above the sheet, the electric field $\\tilde{\\mathbf{E}}$ below the sheet].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "the electric field $\\tilde{\\mathbf{E}}$ above the sheet", "the electric field $\\tilde{\\mathbf{E}}$ below the sheet" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1593", "problem": "Z-籍缩作为惯性约束核聚变的一种可能方式, 近年来受到特别重视, 其原理如图所示. 图中, 长 $20 \\mathrm{~mm}$ 、直径为 $5 \\mu \\mathrm{m}$ 的铇丝组成的两个共轴的圆柱面阵列, 殠间通以超强电流,钨丝阵列在安培力的作用下以极大的加速度向内运动, 即所谓自笣缩效应; 铇丝的巨大动量转移到处于阵列中心的直径为豪米量级的氛氛靶球上, 可以使靯球压缩后达到高温高密度状态, 实现核聚变. 设内圈有 $N$ 根钨丝 (可视为长直导线) 均匀地分布在半径为 $r$ 的圆周上, 通有总电流 $I_{\\text {内 }}=2 \\times 10^{7} \\mathrm{~A}$; 外圈有 $M$ 根钨丝, 均匀地分布在半径为 $R$ 的圆周上, 每根钨丝所通过的电流同内圈铇丝. 已知通有电流 $i$ 的长直导线在距其 $r$ 处产生的磁感应强度大小为 $k_{m} \\frac{i}{r}$, 式中比例常量 $k_{\\mathrm{m}}=2 \\times 10^{-7} \\mathrm{~T} \\cdot \\mathrm{m} / \\mathrm{A}=2 \\times 10^{-7} \\mathrm{~N} / \\mathrm{A}^{2}$.\n\n[图1]若不考虑外圈钨丝, 内聪钨丝阵列熔化后形成了圆柱面, 且篗缩为半径 $r=0.25 \\mathrm{~cm}$ 的圆柱面时, 求柱面上单位面积所受到的安培力, 这相当于多少个大气压?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\nZ-籍缩作为惯性约束核聚变的一种可能方式, 近年来受到特别重视, 其原理如图所示. 图中, 长 $20 \\mathrm{~mm}$ 、直径为 $5 \\mu \\mathrm{m}$ 的铇丝组成的两个共轴的圆柱面阵列, 殠间通以超强电流,钨丝阵列在安培力的作用下以极大的加速度向内运动, 即所谓自笣缩效应; 铇丝的巨大动量转移到处于阵列中心的直径为豪米量级的氛氛靶球上, 可以使靯球压缩后达到高温高密度状态, 实现核聚变. 设内圈有 $N$ 根钨丝 (可视为长直导线) 均匀地分布在半径为 $r$ 的圆周上, 通有总电流 $I_{\\text {内 }}=2 \\times 10^{7} \\mathrm{~A}$; 外圈有 $M$ 根钨丝, 均匀地分布在半径为 $R$ 的圆周上, 每根钨丝所通过的电流同内圈铇丝. 已知通有电流 $i$ 的长直导线在距其 $r$ 处产生的磁感应强度大小为 $k_{m} \\frac{i}{r}$, 式中比例常量 $k_{\\mathrm{m}}=2 \\times 10^{-7} \\mathrm{~T} \\cdot \\mathrm{m} / \\mathrm{A}=2 \\times 10^{-7} \\mathrm{~N} / \\mathrm{A}^{2}$.\n\n[图1]\n\n问题:\n若不考虑外圈钨丝, 内聪钨丝阵列熔化后形成了圆柱面, 且篗缩为半径 $r=0.25 \\mathrm{~cm}$ 的圆柱面时, 求柱面上单位面积所受到的安培力, 这相当于多少个大气压?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以\\mathrm{~atm}为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ea729d4659bcaa2c4b91g-04.jpg?height=677&width=1188&top_left_y=632&top_left_x=454", "https://cdn.mathpix.com/cropped/2024_03_31_ea729d4659bcaa2c4b91g-17.jpg?height=395&width=371&top_left_y=1847&top_left_x=1368" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~atm}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1492", "problem": "爱因斯坦引力理论预言物质分布的变化会导致时空几何结构的波动一一引力波。为简明起见, 考虑沿 $z$ 轴传播的平面引力波。对于任意给定的 $z$, 在 $x-y$ 二维空间中两个无限邻近点 $(x, y)$ 和 $(x+\\mathrm{d} x, y+\\mathrm{d} y)$ 之间距离 $\\mathrm{d} r$ 的表达式为\n\n$$\n\\mathrm{d} r=\\sqrt{\\left(1+f_{1}\\right)(\\mathrm{d} x)^{2}+f_{2}(\\mathrm{~d} x \\mathrm{~d} y+\\mathrm{d} y \\mathrm{~d} x)+\\left(1-f_{1}\\right)(\\mathrm{d} y)^{2}}\n$$\n\n引力波体现为 $f_{1}$ 和 $f_{2}$ 的变化 (波动)。\n\n假设一列平面引力波传来时, $f_{1}$ 和 $f_{2}$ 可表示为\n\n$$\nf_{1}=A \\sin \\left[\\omega\\left(t-\\frac{Z}{c}\\right)\\right], \\quad f_{2}=0 ; \\quad 0l)$ 处.若碰前滑块 $\\mathrm{A}$ 的速度为 $v_{0}$, 求碰撞过程中轴受到的作用力的冲量", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一长为 $2 l$ 的轻质刚性细杆位于水平的光滑桌面上, 杆的两端分别固定一质量为 $m$ 的小物块 $\\mathrm{D}$ 和一质量为 $\\alpha m$ ( $\\alpha$ 为常数) 的小物块 $\\mathrm{B}$, 杆可绕通过小物块 B 所在端的坚直固定转轴无摩擦地转动. 一质量为 $m$ 的小环 $\\mathrm{C}$ 套在细杆上 (C 与杆密接), 可沿杆滑动, 环 $\\mathrm{C}$ 与杆之间的摩擦可忽略. 一轻质弹簧原长为 $l$, 劲度系数为 $k$, 两端分别与小环 $\\mathrm{C}$和物块 $\\mathrm{B}$ 相连. 一质量为 $m$ 的小滑块 $\\mathrm{A}$ 在桌面上以垂直于杆的速度飞向物块 $\\mathrm{D}$, 并与之发生完全弹性正碰, 碰撞时间极短. 碰撞 时滑块 $\\mathrm{C}$ 恰好静止在距轴为 $r(r>l)$ 处.\n\n问题:\n若碰前滑块 $\\mathrm{A}$ 的速度为 $v_{0}$, 求碰撞过程中轴受到的作用力的冲量\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_92", "problem": "Three charges are located at the vertices of a right triangle as shown at right. What is the magnitude of the resultant electric field at the midpoint, $M$, on a line between $A$ and $C$ ?\n[figure1]\nA: $1.44 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\nB: $1.84 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\nC: $1.14 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\nD: $2.64 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\nE: $1.04 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThree charges are located at the vertices of a right triangle as shown at right. What is the magnitude of the resultant electric field at the midpoint, $M$, on a line between $A$ and $C$ ?\n[figure1]\n\nA: $1.44 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\nB: $1.84 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\nC: $1.14 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\nD: $2.64 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\nE: $1.04 \\times 10^{7} \\mathrm{~N} / \\mathrm{C}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ae7e25be7efc2df26f6eg-08.jpg?height=382&width=463&top_left_y=611&top_left_x=1338" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1512", "problem": "横截面积为 $S$ 和 $2 S$ 的两圆柱形容器按图示方式连接成一气缸, 每个圆筒中各置有一活塞, 两活塞间的距离为 $l$, 用硬杆相连, 形成“工”形活塞, 它把整个气缸分隔成三个气室, 其\n\n[图1]\n中 I、III 室密闭摩尔数分别为 $v$ 和 $2 v$ 的同种理想气体, 两个气室內都有电加热器; II 室的缸壁上开有一小孔, 与大气相通; $1 \\mathrm{~mol}$ 该种气体内能为 $C T$ ( $C$ 是气体摩尔热容量,\n$T$ 是气体的绝对温度）。当三个气室中气体的温度均为 $T_{1}$ 时, “工”形活塞在气缸中恰好在图所示的位置处于平衡状态, 这时 I 室内气柱长亦为 $l$, II 室内空气的摩尔数为 $\\frac{3}{2} v_{0}$ 。已知大气压不变, 气缸壁和活塞都是绝热的, 不计活塞与气缸之间的摩擦。现通过电热器对 I、III 两室中的气体缓慢加热, 直至 I 室内气体的温度升为其初始状态温度的 2 倍时, 活塞左移距离 $d$ 。已知理想气体常量为 $R$ 。求III 室内气体初态气柱的长度;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n横截面积为 $S$ 和 $2 S$ 的两圆柱形容器按图示方式连接成一气缸, 每个圆筒中各置有一活塞, 两活塞间的距离为 $l$, 用硬杆相连, 形成“工”形活塞, 它把整个气缸分隔成三个气室, 其\n\n[图1]\n中 I、III 室密闭摩尔数分别为 $v$ 和 $2 v$ 的同种理想气体, 两个气室內都有电加热器; II 室的缸壁上开有一小孔, 与大气相通; $1 \\mathrm{~mol}$ 该种气体内能为 $C T$ ( $C$ 是气体摩尔热容量,\n$T$ 是气体的绝对温度）。当三个气室中气体的温度均为 $T_{1}$ 时, “工”形活塞在气缸中恰好在图所示的位置处于平衡状态, 这时 I 室内气柱长亦为 $l$, II 室内空气的摩尔数为 $\\frac{3}{2} v_{0}$ 。已知大气压不变, 气缸壁和活塞都是绝热的, 不计活塞与气缸之间的摩擦。现通过电热器对 I、III 两室中的气体缓慢加热, 直至 I 室内气体的温度升为其初始状态温度的 2 倍时, 活塞左移距离 $d$ 。已知理想气体常量为 $R$ 。求\n\n问题:\nIII 室内气体初态气柱的长度;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_d716ce15f03757bb482eg-03.jpg?height=160&width=372&top_left_y=2307&top_left_x=1316" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1473", "problem": "如图, 导热性能良好的气缸 A 和 B 高度均为 $h$ (已除开活塞的厚度）, 横截面积不同, 坚直浸没在温度为 $T_{0}$ 的恒温槽内, 它们的底部由一细管连通（细管容积可忽略）。两气缸内各有一个活塞, 质量\n\n[图1]\n分别为 $m_{\\mathrm{A}}=2 m$ 和 $m_{\\mathrm{B}}=m$, 活塞与气缸之间无摩擦, 两活塞的下方为理想气体, 上方为真空。当两活塞下方气体处于平衡状态时, 两活塞底面相对于气缸底的高度均为 $\\frac{h}{2}$ 。现保持恒温槽温度不变, 在两活塞上面同时各缓慢加上同样大小的压力, 让压力从零缓慢增加, 直至其大小等于 $2 m g$ ( $g$ 为重力加速度) 为止, 并一直保持两活塞上的压力不变; 系统再次达到平衡后, 缓慢升高恒温慒的温度, 对气体加热, 直至气缸 B 中活塞底面恰好回到高度为 $\\frac{h}{2}$ 处。求气缸內气体的最后的温度", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 导热性能良好的气缸 A 和 B 高度均为 $h$ (已除开活塞的厚度）, 横截面积不同, 坚直浸没在温度为 $T_{0}$ 的恒温槽内, 它们的底部由一细管连通（细管容积可忽略）。两气缸内各有一个活塞, 质量\n\n[图1]\n分别为 $m_{\\mathrm{A}}=2 m$ 和 $m_{\\mathrm{B}}=m$, 活塞与气缸之间无摩擦, 两活塞的下方为理想气体, 上方为真空。当两活塞下方气体处于平衡状态时, 两活塞底面相对于气缸底的高度均为 $\\frac{h}{2}$ 。现保持恒温槽温度不变, 在两活塞上面同时各缓慢加上同样大小的压力, 让压力从零缓慢增加, 直至其大小等于 $2 m g$ ( $g$ 为重力加速度) 为止, 并一直保持两活塞上的压力不变; 系统再次达到平衡后, 缓慢升高恒温慒的温度, 对气体加热, 直至气缸 B 中活塞底面恰好回到高度为 $\\frac{h}{2}$ 处。求\n\n问题:\n气缸內气体的最后的温度\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-06.jpg?height=242&width=343&top_left_y=530&top_left_x=1413" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1560", "problem": "如图，介质薄膜波导由三层均匀介质组中间层 1 为波导薄膜, 其折射率为 $n_{1}$, 光波在其中传底层 0 为祄底, 其折射率为 $n_{0}$; 上层 2 为覆盖层, 折为 $n_{2} ; n_{1}>n_{0} \\geq n_{2}$ 。光在薄膜层 1 里来回反射, 沿锯齿波导延伸方向传播。图中, $\\theta_{i j}$ 是光波在介质 $j$ 表面上射角, $\\theta_{\\mathrm{t} j}$ 是光波在介质 $j$ 表面上的折射角。\n\n[图1]已知波导薄膜的厚度为 $d$, 求能够在薄膜波导中传输的光波在该介质中的最长波长 $\\lambda_{\\text {max }}$ 。已知: 两介质 $j$ 与 $k$ 的交界面上的反射系数（即反射光的电场强度与入射光的电场强度之比）为\n\n$$\nr_{j k}=\\frac{n_{j} \\cos \\theta_{\\mathrm{i} j}-n_{k} \\cos \\theta_{\\mathrm{tk}}}{n_{j} \\cos \\theta_{\\mathrm{i} j}+n_{k} \\cos \\theta_{\\mathrm{t} k}}=\\left|r_{j k}\\right| e^{-i \\varphi_{j k}}\n$$\n\n式中， $\\theta_{\\mathrm{ij}}$ 和 $\\theta_{\\mathrm{t} j}$ 是分别是光波在介质 $j$ 的表面上的入射角和折射角，余类推; 正弦函数和余弦函数在复数域中可定义为\n\n$$\n\\sin \\theta=\\frac{e^{i \\theta}-e^{-i \\theta}}{2 i}, \\quad \\cos \\theta=\\frac{e^{i \\theta}+e^{-i \\theta}}{2}\n$$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图，介质薄膜波导由三层均匀介质组中间层 1 为波导薄膜, 其折射率为 $n_{1}$, 光波在其中传底层 0 为祄底, 其折射率为 $n_{0}$; 上层 2 为覆盖层, 折为 $n_{2} ; n_{1}>n_{0} \\geq n_{2}$ 。光在薄膜层 1 里来回反射, 沿锯齿波导延伸方向传播。图中, $\\theta_{i j}$ 是光波在介质 $j$ 表面上射角, $\\theta_{\\mathrm{t} j}$ 是光波在介质 $j$ 表面上的折射角。\n\n[图1]\n\n问题:\n已知波导薄膜的厚度为 $d$, 求能够在薄膜波导中传输的光波在该介质中的最长波长 $\\lambda_{\\text {max }}$ 。已知: 两介质 $j$ 与 $k$ 的交界面上的反射系数（即反射光的电场强度与入射光的电场强度之比）为\n\n$$\nr_{j k}=\\frac{n_{j} \\cos \\theta_{\\mathrm{i} j}-n_{k} \\cos \\theta_{\\mathrm{tk}}}{n_{j} \\cos \\theta_{\\mathrm{i} j}+n_{k} \\cos \\theta_{\\mathrm{t} k}}=\\left|r_{j k}\\right| e^{-i \\varphi_{j k}}\n$$\n\n式中， $\\theta_{\\mathrm{ij}}$ 和 $\\theta_{\\mathrm{t} j}$ 是分别是光波在介质 $j$ 的表面上的入射角和折射角，余类推; 正弦函数和余弦函数在复数域中可定义为\n\n$$\n\\sin \\theta=\\frac{e^{i \\theta}-e^{-i \\theta}}{2 i}, \\quad \\cos \\theta=\\frac{e^{i \\theta}+e^{-i \\theta}}{2}\n$$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e3a9fdbbef225ad3aefbg-04.jpg?height=540&width=731&top_left_y=381&top_left_x=1162", "https://cdn.mathpix.com/cropped/2024_03_31_e3a9fdbbef225ad3aefbg-22.jpg?height=214&width=354&top_left_y=2172&top_left_x=768" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_811", "problem": "The moving image-monopole concept developed in A. 7 for $B_{z}^{\\prime}$ near $z \\approx 0$ can be assumed to hold also for the $\\vec{B}^{\\prime}$ field in the $z \\geq 0$ region. This assumption is good as long as the time evolution is sufficiently slow in the conducting thin film response.\n\n[figure1]\n\nFig. 2 A monopole $q_{\\mathrm{m}}$ moves with a constant velocity $\\vec{v}$ and a constant height $h$ from the conducting thin film. As shown are its coordinates at $t=0$.\n\nA monopole $q_{\\mathrm{m}}$ (Fig. 2) is caused to move in a constant velocity $v \\hat{x}$, with $v \\ll c$, and a constant height, at $z=h$, motion up to the present moment $(t=0)$. Its present coordinates $(x, y)$ are $(0,0)$. Our focus is on the magnetic potential $\\Phi_{+}$due to all image monopoles generated by this moving monopole along its trajectory.\n\nBy splitting $q_{m}$ 's trajectory into discrete time steps (a very small time step $\\tau$ ), we replace the motion of the $q_{m}$ by a hopping at the beginning moment of each time step. The hopping is represented by a simultaneous removal and creation of the monopoles. The position of the created monopole coincides with a point on its trajectory right at the beginning moment of this time step. Thus the position of the removed monopole coincides with its trajectory position at the beginning moment of the previous time step. This is achieved by a simultaneous sudden appearance of two magnetic monopoles: $q_{\\mathrm{m}}$ and $-q_{\\mathrm{m}}$ at, respectively, the trajectory positions corresponding to the beginning moments of this and the previous time step. The two positions are separated by a hopping distance $\\Delta x=v \\tau$. This time-step approach facilitates the determination of all the image magnetic monopoles, and their positions, that are generated in all the time steps.\n\n[figure2]\n\nFig. 3 A dipole with an upward-pointing magnetic dipole moment $\\vec{m}$ moves with a constant $\\vec{v}$ and a constant height $h$ from the conducting thin film. As shown are its coordinates at $t=0$.\n\nNow consider a point-like moving magnetic dipole as shown in Fig. 3. The dipole, with a dipole moment $\\vec{m}=m \\hat{z}$, is caused to move in a constant velocity $v \\hat{x}$, and a constant height $(z=h)$ motion up to the present moment $(t=0)$, where its present coordinates are at $(0,0)$. The point-like dipole can be represented by two slightly displaced monopoles as has been mentioned in the Introduction section. The location of the magnetic dipole is chosen to be that of the north monopole, and $\\vec{m}$ is assumed kept fixed.\n\nFind the force $\\vec{F}$ acting upon the point-like magnetic dipole by the conducting $1.5 \\mathrm{pt}$ thin film at $t=0$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nThe moving image-monopole concept developed in A. 7 for $B_{z}^{\\prime}$ near $z \\approx 0$ can be assumed to hold also for the $\\vec{B}^{\\prime}$ field in the $z \\geq 0$ region. This assumption is good as long as the time evolution is sufficiently slow in the conducting thin film response.\n\n[figure1]\n\nFig. 2 A monopole $q_{\\mathrm{m}}$ moves with a constant velocity $\\vec{v}$ and a constant height $h$ from the conducting thin film. As shown are its coordinates at $t=0$.\n\nA monopole $q_{\\mathrm{m}}$ (Fig. 2) is caused to move in a constant velocity $v \\hat{x}$, with $v \\ll c$, and a constant height, at $z=h$, motion up to the present moment $(t=0)$. Its present coordinates $(x, y)$ are $(0,0)$. Our focus is on the magnetic potential $\\Phi_{+}$due to all image monopoles generated by this moving monopole along its trajectory.\n\nBy splitting $q_{m}$ 's trajectory into discrete time steps (a very small time step $\\tau$ ), we replace the motion of the $q_{m}$ by a hopping at the beginning moment of each time step. The hopping is represented by a simultaneous removal and creation of the monopoles. The position of the created monopole coincides with a point on its trajectory right at the beginning moment of this time step. Thus the position of the removed monopole coincides with its trajectory position at the beginning moment of the previous time step. This is achieved by a simultaneous sudden appearance of two magnetic monopoles: $q_{\\mathrm{m}}$ and $-q_{\\mathrm{m}}$ at, respectively, the trajectory positions corresponding to the beginning moments of this and the previous time step. The two positions are separated by a hopping distance $\\Delta x=v \\tau$. This time-step approach facilitates the determination of all the image magnetic monopoles, and their positions, that are generated in all the time steps.\n\n[figure2]\n\nFig. 3 A dipole with an upward-pointing magnetic dipole moment $\\vec{m}$ moves with a constant $\\vec{v}$ and a constant height $h$ from the conducting thin film. As shown are its coordinates at $t=0$.\n\nNow consider a point-like moving magnetic dipole as shown in Fig. 3. The dipole, with a dipole moment $\\vec{m}=m \\hat{z}$, is caused to move in a constant velocity $v \\hat{x}$, and a constant height $(z=h)$ motion up to the present moment $(t=0)$, where its present coordinates are at $(0,0)$. The point-like dipole can be represented by two slightly displaced monopoles as has been mentioned in the Introduction section. The location of the magnetic dipole is chosen to be that of the north monopole, and $\\vec{m}$ is assumed kept fixed.\n\nFind the force $\\vec{F}$ acting upon the point-like magnetic dipole by the conducting $1.5 \\mathrm{pt}$ thin film at $t=0$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d32b3b2f89cebe6f1c2ag-3.jpg?height=323&width=763&top_left_y=2123&top_left_x=658", "https://cdn.mathpix.com/cropped/2024_03_14_d32b3b2f89cebe6f1c2ag-4.jpg?height=388&width=805&top_left_y=1639&top_left_x=631", "https://cdn.mathpix.com/cropped/2024_03_14_300bd3a4734333693ab9g-3.jpg?height=382&width=724&top_left_y=1975&top_left_x=690" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_930", "problem": "The two-slit electron interference experiment was first performed by Möllenstedt et al, MerliMissiroli and Pozzi in 1974 and Tonomura et al in 1989. In the two-slit electron interference experiment, a monochromatic electron point source emits particles at $S$ that first passes through an electron \"biprism\" before impinging on an observational plane; $S_{1}$ and $S_{2}$ are virtual sources at distance $d$. In the diagram, the filament is pointing into the page. Note that it is a very thin filament (not drawn to scale in the diagram).\n\n[figure1]\n\nThe electron \"biprism\" consists of a grounded cylindrical wire mesh with a fine filament $F$ at the center. The distance between the source and the \"biprism\" is $\\ell$, and the distance between the distance between the \"biprism\" and the screen is $L$.\n\nAn incoming electron plane wave with wave vector $k_{z}$ is deflected by the \"biprism\" due to the $x$-component of the force exerted on the electron. Determine $k_{x}$ the $x$-component of the wave vector due to the \"biprism\" in terms of the electron charge, $e, v_{z}, V_{a}, k_{z}, a$ and $b$, where $e$ and $v_{z}$ are the charge and the $z$-component of the velocity of the electrons $\\left(k_{x} \\ll k_{z}\\right)$. Note that $\\vec{k}=\\frac{2 \\pi \\vec{p}}{h}$ where $h$ is the Planck constant.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nThe two-slit electron interference experiment was first performed by Möllenstedt et al, MerliMissiroli and Pozzi in 1974 and Tonomura et al in 1989. In the two-slit electron interference experiment, a monochromatic electron point source emits particles at $S$ that first passes through an electron \"biprism\" before impinging on an observational plane; $S_{1}$ and $S_{2}$ are virtual sources at distance $d$. In the diagram, the filament is pointing into the page. Note that it is a very thin filament (not drawn to scale in the diagram).\n\n[figure1]\n\nThe electron \"biprism\" consists of a grounded cylindrical wire mesh with a fine filament $F$ at the center. The distance between the source and the \"biprism\" is $\\ell$, and the distance between the distance between the \"biprism\" and the screen is $L$.\n\nAn incoming electron plane wave with wave vector $k_{z}$ is deflected by the \"biprism\" due to the $x$-component of the force exerted on the electron. Determine $k_{x}$ the $x$-component of the wave vector due to the \"biprism\" in terms of the electron charge, $e, v_{z}, V_{a}, k_{z}, a$ and $b$, where $e$ and $v_{z}$ are the charge and the $z$-component of the velocity of the electrons $\\left(k_{x} \\ll k_{z}\\right)$. Note that $\\vec{k}=\\frac{2 \\pi \\vec{p}}{h}$ where $h$ is the Planck constant.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_4f547aee877827e020bbg-1.jpg?height=1319&width=1091&top_left_y=1082&top_left_x=471" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1616", "problem": "如图, 两根内径相同的绝缘细管 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 连接成倒 $\\mathrm{V}$ 字形, 坚直放置,连接点 $\\mathrm{B}$ 处可视为一段很短的圆弧; 两管长度均为 $l=2.0 \\mathrm{~m}$, 倾角 $\\alpha=37^{\\circ}$,处于方向坚直向下的匀强电场中, 场强大小 $E=10000 \\mathrm{~V} / \\mathrm{m}$ 。一质量 $m=1.0 \\times 10^{-4} \\mathrm{~kg}$ 、带电量 $-q=-2.0 \\times 10^{-7} \\mathrm{C}$ 的小球 (小球直径比细管内径稍小,可视为质点), 从 $\\mathrm{A}$ 点由静止开始在管内运动, 小球与 $\\mathrm{AB}$ 管壁间的动摩擦因数为 $\\mu_{1}=0.50$,小球与 $\\mathrm{BC}$ 管壁间的动摩擦因数为 $\\mu_{2}=0.25$ 。小球在运动过程中带电量保持不变。已知重力加速度大小 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}, \\sin 37^{\\circ}=\\frac{3}{5}$ 。求\n\n[图1]小球分别在 $\\mathrm{AB}$ 管和 $\\mathrm{BC}$ 管中运动直至静止的总路程 $S_{\\mathrm{AB}}$ 和 $S_{\\mathrm{BC}}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 两根内径相同的绝缘细管 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 连接成倒 $\\mathrm{V}$ 字形, 坚直放置,连接点 $\\mathrm{B}$ 处可视为一段很短的圆弧; 两管长度均为 $l=2.0 \\mathrm{~m}$, 倾角 $\\alpha=37^{\\circ}$,处于方向坚直向下的匀强电场中, 场强大小 $E=10000 \\mathrm{~V} / \\mathrm{m}$ 。一质量 $m=1.0 \\times 10^{-4} \\mathrm{~kg}$ 、带电量 $-q=-2.0 \\times 10^{-7} \\mathrm{C}$ 的小球 (小球直径比细管内径稍小,可视为质点), 从 $\\mathrm{A}$ 点由静止开始在管内运动, 小球与 $\\mathrm{AB}$ 管壁间的动摩擦因数为 $\\mu_{1}=0.50$,小球与 $\\mathrm{BC}$ 管壁间的动摩擦因数为 $\\mu_{2}=0.25$ 。小球在运动过程中带电量保持不变。已知重力加速度大小 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}, \\sin 37^{\\circ}=\\frac{3}{5}$ 。求\n\n[图1]\n\n问题:\n小球分别在 $\\mathrm{AB}$ 管和 $\\mathrm{BC}$ 管中运动直至静止的总路程 $S_{\\mathrm{AB}}$ 和 $S_{\\mathrm{BC}}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[小球在 $\\mathrm{AB}$ 管中运动直至静止的总路程 $S_{\\mathrm{AB}}$, 小球在 $\\mathrm{BC}$ 管中运动直至静止的总路程 $S_{\\mathrm{BC}}$]\n它们的单位依次是[$\\mathrm{~m}$, $\\mathrm{~m}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-04.jpg?height=197&width=414&top_left_y=1735&top_left_x=1569" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{~m}$", "$\\mathrm{~m}$" ], "answer_sequence": [ "小球在 $\\mathrm{AB}$ 管中运动直至静止的总路程 $S_{\\mathrm{AB}}$", "小球在 $\\mathrm{BC}$ 管中运动直至静止的总路程 $S_{\\mathrm{BC}}$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_53", "problem": "Two simple pendula are $60 \\mathrm{~cm}$ and $63 \\mathrm{~cm}$ in length. They hang vertically, one in front of the other. If they are set in motion simultaneously, find the time taken for one to gain a complete cycle of oscillation on the other.\nA: $15.4 \\mathrm{~s}$\nB: $15.7 \\mathrm{~s}$\nC: $31.5 \\mathrm{~s}$\nD: $62.4 \\mathrm{~s}$\nE: $79.7 \\mathrm{~s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo simple pendula are $60 \\mathrm{~cm}$ and $63 \\mathrm{~cm}$ in length. They hang vertically, one in front of the other. If they are set in motion simultaneously, find the time taken for one to gain a complete cycle of oscillation on the other.\n\nA: $15.4 \\mathrm{~s}$\nB: $15.7 \\mathrm{~s}$\nC: $31.5 \\mathrm{~s}$\nD: $62.4 \\mathrm{~s}$\nE: $79.7 \\mathrm{~s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_697", "problem": "In the previous question, at $t=t_{1}$, at what rate does the string wind around the cylinder (i.e. at what speed does L1 get shorted)?\nA: $\\frac{L_{1}}{R} V_{1}$\nB: $\\frac{L_{1}+R}{R} V_{1}$\nC: $\\frac{R}{L_{1}} V_{1}$\nD: $\\frac{R^{2}}{L_{1}^{2}} V_{1}$\nE: $\\frac{L_{1}^{2}}{R^{2}} V_{1}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn the previous question, at $t=t_{1}$, at what rate does the string wind around the cylinder (i.e. at what speed does L1 get shorted)?\n\nA: $\\frac{L_{1}}{R} V_{1}$\nB: $\\frac{L_{1}+R}{R} V_{1}$\nC: $\\frac{R}{L_{1}} V_{1}$\nD: $\\frac{R^{2}}{L_{1}^{2}} V_{1}$\nE: $\\frac{L_{1}^{2}}{R^{2}} V_{1}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_581", "problem": "In general, whenever an electric and a magnetic field are at an angle to each other, energy is transferred; for example, this principle is the reason electromagnetic radiation transfers energy. The power transferred per unit area is given by the Poynting vector:\n\n$$\n\\vec{S}=\\frac{1}{\\mu_{0}} \\vec{E} \\times \\vec{B}\n$$\n\nIn each part of this problem, the last subpart asks you to verify that the rate of energy transfer agrees with the formula for the Poynting vector. Therefore, you should not use the formula for the Poynting vector before the last subpart!\n\nA parallel plate capacitor consists of two discs of radius $R$ separated by a distance $d \\ll R$. The capacitor carries charge $Q$, and is being charged by a small, constant current $I$.\n\nWhat is the power $P$ delivered to the capacitor?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn general, whenever an electric and a magnetic field are at an angle to each other, energy is transferred; for example, this principle is the reason electromagnetic radiation transfers energy. The power transferred per unit area is given by the Poynting vector:\n\n$$\n\\vec{S}=\\frac{1}{\\mu_{0}} \\vec{E} \\times \\vec{B}\n$$\n\nIn each part of this problem, the last subpart asks you to verify that the rate of energy transfer agrees with the formula for the Poynting vector. Therefore, you should not use the formula for the Poynting vector before the last subpart!\n\nA parallel plate capacitor consists of two discs of radius $R$ separated by a distance $d \\ll R$. The capacitor carries charge $Q$, and is being charged by a small, constant current $I$.\n\nWhat is the power $P$ delivered to the capacitor?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_359", "problem": "[figure1]\n\nA circuit is made out of a battery, a switch, res istors and capacitors as shown in the image. The resistors all have a resistance of $R$, the capacitor all have a capacitance of $C$ and the battery has a voltage of $U$. The point $\\mathrm{A}$ is connected to the ground and so it has a potential of $0 \\mathrm{~V}$. In the be ginning the switch is open and all the capacitor have no charge.\nWhat is the potential at point $D$ after we have closed the switch and waited for all the potentials to stabilize?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\n[figure1]\n\nA circuit is made out of a battery, a switch, res istors and capacitors as shown in the image. The resistors all have a resistance of $R$, the capacitor all have a capacitance of $C$ and the battery has a voltage of $U$. The point $\\mathrm{A}$ is connected to the ground and so it has a potential of $0 \\mathrm{~V}$. In the be ginning the switch is open and all the capacitor have no charge.\nWhat is the potential at point $D$ after we have closed the switch and waited for all the potentials to stabilize?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of U, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e6c1582ce7f1c05fa0a6g-1.jpg?height=657&width=642&top_left_y=257&top_left_x=794", "https://cdn.mathpix.com/cropped/2024_03_14_83cc90b265723e4db811g-2.jpg?height=291&width=664&top_left_y=1682&top_left_x=772", "https://cdn.mathpix.com/cropped/2024_03_14_83cc90b265723e4db811g-2.jpg?height=347&width=504&top_left_y=192&top_left_x=1553" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "U" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_449", "problem": "A car travelling at approximately $30 \\mathrm{~m} \\mathrm{~s}^{-1}$ in the country is required by law to halve its speed on entering a built-up area. What fraction of its kinetic energy is lost in doing this?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA car travelling at approximately $30 \\mathrm{~m} \\mathrm{~s}^{-1}$ in the country is required by law to halve its speed on entering a built-up area. What fraction of its kinetic energy is lost in doing this?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1688", "problem": "${ }_{92}^{238} \\mathrm{U}$ (铀核) 衰变为 ${ }_{88}^{222} \\mathrm{Rn}$ (氡核) 要经过\nA: 8 次 $\\alpha$ 衰变, 16 次 $\\beta$ 衰变\nB: 3 次 $\\alpha$ 衰变, 4 次 $\\beta$ 衰变\nC: 4 次 $\\alpha$ 衰变, 16 次 $\\beta$ 衰变\nD: 4 次 $\\alpha$ 衰变, 4 次 $\\beta$ 衰变\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n${ }_{92}^{238} \\mathrm{U}$ (铀核) 衰变为 ${ }_{88}^{222} \\mathrm{Rn}$ (氡核) 要经过\n\nA: 8 次 $\\alpha$ 衰变, 16 次 $\\beta$ 衰变\nB: 3 次 $\\alpha$ 衰变, 4 次 $\\beta$ 衰变\nC: 4 次 $\\alpha$ 衰变, 16 次 $\\beta$ 衰变\nD: 4 次 $\\alpha$ 衰变, 4 次 $\\beta$ 衰变\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1391", "problem": "天文观测表明, 远处的星系均离我们而去. 著名的哈勃定律指出, 星系离开我们的速度大小 $v=H D$, 其中 $D$ 为星系与我们之间的距离, 该距离通常以百方秒差距 ( $\\mathrm{Mpc}$ ) 为单位: $H$ 为哈勃常数, 最新的测量结果为 $H=67.80 \\mathrm{~km} /(\\mathrm{s} \\cdot \\mathrm{Mpc})$. 当星系离开我们远去时, 它发出的光谱线的波长会变长 (称为红移). 红移量 $z$ 被定义为 $z=\\frac{\\lambda^{\\prime}-\\lambda}{\\lambda}$, 其中 $\\lambda^{\\prime}$ 是我们观测到的星系中某恒星发出的谱线的波长, 而 $\\lambda$ 是实验室中测得的同种原子发出的相应的谱线的波长, 该红移可用多普勒效应解释. 绝大部分星系的红移量 $z$ 远小于 1 , 即星系退行的速度远小于光速. 在一次天文观测中发现从天鹰座的一个星系中射来的氢原子光谱中有两条谱线, 它们的频率 $v^{\\prime}$ 分别为 $4.549 \\times 10^{14} \\mathrm{~Hz}$ 和 $6.141 \\times 10^{14} \\mathrm{~Hz}$. 由于这两条谱线处于可见光频率区间，可假设它们属于氢原子的巴尔末系, 即为由 $n>2$ 的能级向 $k=2$ 的能级跃迁而产生的光谱. (已知氢原子的基态能量 $E_{0}=-13.60 \\mathrm{eV}$, 真空中光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 普朗克常量 $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 电子电荷量 $e=1.602 \\times 10^{-19} \\mathrm{C}$ )求该星系发出的光谱线的红移量 $z$ 和该星系远离我们的速度大小 $v$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n天文观测表明, 远处的星系均离我们而去. 著名的哈勃定律指出, 星系离开我们的速度大小 $v=H D$, 其中 $D$ 为星系与我们之间的距离, 该距离通常以百方秒差距 ( $\\mathrm{Mpc}$ ) 为单位: $H$ 为哈勃常数, 最新的测量结果为 $H=67.80 \\mathrm{~km} /(\\mathrm{s} \\cdot \\mathrm{Mpc})$. 当星系离开我们远去时, 它发出的光谱线的波长会变长 (称为红移). 红移量 $z$ 被定义为 $z=\\frac{\\lambda^{\\prime}-\\lambda}{\\lambda}$, 其中 $\\lambda^{\\prime}$ 是我们观测到的星系中某恒星发出的谱线的波长, 而 $\\lambda$ 是实验室中测得的同种原子发出的相应的谱线的波长, 该红移可用多普勒效应解释. 绝大部分星系的红移量 $z$ 远小于 1 , 即星系退行的速度远小于光速. 在一次天文观测中发现从天鹰座的一个星系中射来的氢原子光谱中有两条谱线, 它们的频率 $v^{\\prime}$ 分别为 $4.549 \\times 10^{14} \\mathrm{~Hz}$ 和 $6.141 \\times 10^{14} \\mathrm{~Hz}$. 由于这两条谱线处于可见光频率区间，可假设它们属于氢原子的巴尔末系, 即为由 $n>2$ 的能级向 $k=2$ 的能级跃迁而产生的光谱. (已知氢原子的基态能量 $E_{0}=-13.60 \\mathrm{eV}$, 真空中光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 普朗克常量 $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 电子电荷量 $e=1.602 \\times 10^{-19} \\mathrm{C}$ )\n\n问题:\n求该星系发出的光谱线的红移量 $z$ 和该星系远离我们的速度大小 $v$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以\\mathrm{~m} / \\mathrm{s}为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~m} / \\mathrm{s}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1666", "problem": "为训练宇航员能在失重状态下工作和生活, 需要创造一种失重的环境. 在地球表面附近, 当飞机模拟某些在重力作用下的运动时, 就可以在飞机座船内实现短时间的完全失重状态. 现要求一架飞机在 $v_{1}=500 \\mathrm{~m} / \\mathrm{s}$ 时进入失重状态的试验, 在速率为 $v_{2}=1000 \\mathrm{~m} / \\mathrm{s}$ 时退出失重状态试验. 重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$\n\n在上述给定的速率要求下, 该飞机需要模拟何重运动, 方可在一定范围内任意选择失重的时间的长短? 试定量讨论影响失重时间长短的因素.飞机模拟这种运动时, 可选择的失重状态的时间范围是多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个区间。\n这里是一些可能会帮助你解决问题的先验信息提示：\n为训练宇航员能在失重状态下工作和生活, 需要创造一种失重的环境. 在地球表面附近, 当飞机模拟某些在重力作用下的运动时, 就可以在飞机座船内实现短时间的完全失重状态. 现要求一架飞机在 $v_{1}=500 \\mathrm{~m} / \\mathrm{s}$ 时进入失重状态的试验, 在速率为 $v_{2}=1000 \\mathrm{~m} / \\mathrm{s}$ 时退出失重状态试验. 重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$\n\n在上述给定的速率要求下, 该飞机需要模拟何重运动, 方可在一定范围内任意选择失重的时间的长短? 试定量讨论影响失重时间长短的因素.\n\n问题:\n飞机模拟这种运动时, 可选择的失重状态的时间范围是多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以s为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含任何单位的区间，例如ANSWER=(1,2] \\cup[7,+\\infty)", "figure_urls": null, "answer": null, "solution": null, "answer_type": "IN", "unit": [ "s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_774", "problem": "The cockatoo and the sparrow collide head-on in mid-air. During the collision, the magnitude of the force exerted by the cockatoo on the sparrow is:\n\nSelect one:\nA: Four times the magnitude of the force exerted by the sparrow on the cockatoo\nB: Twice the magnitude of the force exerted by the sparrow on the cockatoo\nC: The same as the magnitude of the force exerted by the sparrow on the cockatoo\nD: Half the magnitude of the force exerted by the sparrow on the cockatoo\nE: One quarter the magnitude of the force exerted by the sparrow on the cockatoo\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe cockatoo and the sparrow collide head-on in mid-air. During the collision, the magnitude of the force exerted by the cockatoo on the sparrow is:\n\nSelect one:\n\nA: Four times the magnitude of the force exerted by the sparrow on the cockatoo\nB: Twice the magnitude of the force exerted by the sparrow on the cockatoo\nC: The same as the magnitude of the force exerted by the sparrow on the cockatoo\nD: Half the magnitude of the force exerted by the sparrow on the cockatoo\nE: One quarter the magnitude of the force exerted by the sparrow on the cockatoo\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_870", "problem": "Free vortices move about in space with the flow 2 . In other words each element of the filament moves with the velocity $\\vec{v}$ of the fluid at the position of that element.[^1]\n\nAs an example, consider a pair of counter-rotating straight vortices placed initially at distance $r_{0}$ from each other, see Fig. 3. Each vortex produces velocity $v_{0}=\\kappa / r_{0}$ at the axis of another. As a result, the vortex pair moves rectilinearly with constant speed $v_{0}=\\kappa / r_{0}$ so that the distance between them remains unchanged.\n\n[figure1]\n\nFig. 3: Parallel vortex filaments with opposite circulations.\n\nFind the distance $\\mathrm{AB}(t)$ between the vortices $\\mathrm{A}$ and $\\mathrm{B}$ at time $t$. Treat $\\mathrm{AB}(0)$ as given.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nFree vortices move about in space with the flow 2 . In other words each element of the filament moves with the velocity $\\vec{v}$ of the fluid at the position of that element.[^1]\n\nAs an example, consider a pair of counter-rotating straight vortices placed initially at distance $r_{0}$ from each other, see Fig. 3. Each vortex produces velocity $v_{0}=\\kappa / r_{0}$ at the axis of another. As a result, the vortex pair moves rectilinearly with constant speed $v_{0}=\\kappa / r_{0}$ so that the distance between them remains unchanged.\n\n[figure1]\n\nFig. 3: Parallel vortex filaments with opposite circulations.\n\nFind the distance $\\mathrm{AB}(t)$ between the vortices $\\mathrm{A}$ and $\\mathrm{B}$ at time $t$. Treat $\\mathrm{AB}(0)$ as given.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_61fc31149c0f627b45f3g-3.jpg?height=369&width=531&top_left_y=775&top_left_x=768" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_135", "problem": "A rod moves freely between the horizontal floor and the slanted wall. When the end in contact with the floor is moving at $v$, what is the speed of the end in contact with the wall?\n\n[figure1]\nA: $v \\frac{\\sin \\theta}{\\cos (\\alpha-\\theta)}$.\nB: $v \\frac{\\sin (\\alpha-\\theta)}{\\cos (\\alpha+\\theta)}$.\nC: $v \\frac{\\cos (\\alpha-\\theta)}{\\sin (\\alpha+\\theta)}$.\nD: $v \\frac{\\cos \\theta}{\\cos (\\alpha-\\theta)} \\cdot $ \nE: $v \\frac{\\sin \\theta}{\\cos (\\alpha+\\theta)}$.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA rod moves freely between the horizontal floor and the slanted wall. When the end in contact with the floor is moving at $v$, what is the speed of the end in contact with the wall?\n\n[figure1]\n\nA: $v \\frac{\\sin \\theta}{\\cos (\\alpha-\\theta)}$.\nB: $v \\frac{\\sin (\\alpha-\\theta)}{\\cos (\\alpha+\\theta)}$.\nC: $v \\frac{\\cos (\\alpha-\\theta)}{\\sin (\\alpha+\\theta)}$.\nD: $v \\frac{\\cos \\theta}{\\cos (\\alpha-\\theta)} \\cdot $ \nE: $v \\frac{\\sin \\theta}{\\cos (\\alpha+\\theta)}$.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f726c6cf4a23f08e0214g-07.jpg?height=265&width=783&top_left_y=1399&top_left_x=671" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1623", "problem": "有两块无限大的均匀带电平面, 一块带正电, 一块带负电, 单位面积所带电荷量的数值相等. 现把两带电平面正交放置如图所示. 图中直线 $A_{1} B_{1}$ 和 $A_{2} B_{2}$ 分别为带正电的平面和带负电的平面与纸面正交的交线, $O$ 为两交线的交点.\n\n[图1]若每个带电平面单独产生的电场是 $E_{0}=1.0 \\mathrm{~V} / \\mathrm{m}$, 则求出(i)中相邻两等势面间的距离 $\\mathrm{d}=$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n有两块无限大的均匀带电平面, 一块带正电, 一块带负电, 单位面积所带电荷量的数值相等. 现把两带电平面正交放置如图所示. 图中直线 $A_{1} B_{1}$ 和 $A_{2} B_{2}$ 分别为带正电的平面和带负电的平面与纸面正交的交线, $O$ 为两交线的交点.\n\n[图1]\n\n问题:\n若每个带电平面单独产生的电场是 $E_{0}=1.0 \\mathrm{~V} / \\mathrm{m}$, 则求出(i)中相邻两等势面间的距离 $\\mathrm{d}=$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以m为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-04.jpg?height=509&width=537&top_left_y=802&top_left_x=654" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_403", "problem": "A wire loop is hovering in outer space (weightless vacuum) with its plane parallel to the $x y$-plane. In $x<0$ there is a homogeneous magnetic field parallel to the $z$-axis. The rigid rectangular loop is $l=10 \\mathrm{~cm}$ wide and $h=30 \\mathrm{~cm}$ long. The loop is made of copper wire with a circular cross section (radius $r=1.0 \\mathrm{~mm}$ ). At $t=0 \\mathrm{~s}$ the external magnetic field starts to decrease at a rate of $0.025 \\mathrm{~T} / \\mathrm{s}$.\n\nWe can try to increase the acceleration in many ways. How does the result in i) change What if we use the thicker copper wire to make a twice bigger loop $(r=2.0 \\mathrm{~mm}, l=$ $20 \\mathrm{~cm}, h=60 \\mathrm{~cm})$ and immerse it $24 \\mathrm{~cm}$ into the external field?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA wire loop is hovering in outer space (weightless vacuum) with its plane parallel to the $x y$-plane. In $x<0$ there is a homogeneous magnetic field parallel to the $z$-axis. The rigid rectangular loop is $l=10 \\mathrm{~cm}$ wide and $h=30 \\mathrm{~cm}$ long. The loop is made of copper wire with a circular cross section (radius $r=1.0 \\mathrm{~mm}$ ). At $t=0 \\mathrm{~s}$ the external magnetic field starts to decrease at a rate of $0.025 \\mathrm{~T} / \\mathrm{s}$.\n\nWe can try to increase the acceleration in many ways. How does the result in i) change What if we use the thicker copper wire to make a twice bigger loop $(r=2.0 \\mathrm{~mm}, l=$ $20 \\mathrm{~cm}, h=60 \\mathrm{~cm})$ and immerse it $24 \\mathrm{~cm}$ into the external field?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $m^2/s$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$m^2/s$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_20", "problem": "On October 4, the 2022 Nobel Prize in Physics was awarded to three individuals, Alain Aspect, John F. Clauser, and Anton Zeilinger, for experiments with entangled photons. In how many other categories are Nobel Prizes awarded?\nA: One\nB: Two\nC: Three\nD: Four\nE: Five\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nOn October 4, the 2022 Nobel Prize in Physics was awarded to three individuals, Alain Aspect, John F. Clauser, and Anton Zeilinger, for experiments with entangled photons. In how many other categories are Nobel Prizes awarded?\n\nA: One\nB: Two\nC: Three\nD: Four\nE: Five\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1375", "problem": "在固体材料中, 考虑相互作用后, 可以利用“准粒子”的概念研究材料的物理性质。准粒子的动能与动量之间的关系可能与真实粒子的不同。当外加电场或磁场时, 准粒子的运动往往可以用经典力学的方法来处理。在某种二维界面结构中, 存在电量为 $q$ 、有效质量为 $m$ 的准粒子, 它只能在 $x$ - $y$ 平面内运动, 其动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}+\\alpha p$,其中 $\\alpha$ 为正的常量。对于质量为 $m$ 的真实的自由粒子, 动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}$,试从动能定理出发，推导该粒子运动的速度 $v$ 与动量 $\\boldsymbol{p}$ 之间的关系式;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n在固体材料中, 考虑相互作用后, 可以利用“准粒子”的概念研究材料的物理性质。准粒子的动能与动量之间的关系可能与真实粒子的不同。当外加电场或磁场时, 准粒子的运动往往可以用经典力学的方法来处理。在某种二维界面结构中, 存在电量为 $q$ 、有效质量为 $m$ 的准粒子, 它只能在 $x$ - $y$ 平面内运动, 其动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}+\\alpha p$,其中 $\\alpha$ 为正的常量。\n\n问题:\n对于质量为 $m$ 的真实的自由粒子, 动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}$,试从动能定理出发，推导该粒子运动的速度 $v$ 与动量 $\\boldsymbol{p}$ 之间的关系式;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_925", "problem": "Let us first consider a space elevator, which is a cylindrical wire with a uniform cross section $A$ and is homogeneous with density $\\rho$. It is a cylinder positioned vertically at the equator. Its height is greater than the height of the geostationary satellite orbit, so that the stress (force per unit area) on the bottom of the cylinder is zero. The cylinder is in tension along its entire length, with the stress adjusting itself so that each element of the cylinder is in equilibrium under the action of the gravitational, centrifugal, and tension forces.\n\nFind the expression for maximum stress of the cylinder in terms of $\\rho, R_{G}, R$ and the gravitational acceleration $g$. If the cylinder is made of steel whose density is $7900 \\mathrm{~kg} / \\mathrm{m}^{3}$, tensile strength is $5.0 \\mathrm{GPa}$, evaluate the ratio between the maximum stress and the tensile strength of steel. Tensile strength is the maximum stress a material can withstand.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nLet us first consider a space elevator, which is a cylindrical wire with a uniform cross section $A$ and is homogeneous with density $\\rho$. It is a cylinder positioned vertically at the equator. Its height is greater than the height of the geostationary satellite orbit, so that the stress (force per unit area) on the bottom of the cylinder is zero. The cylinder is in tension along its entire length, with the stress adjusting itself so that each element of the cylinder is in equilibrium under the action of the gravitational, centrifugal, and tension forces.\n\nFind the expression for maximum stress of the cylinder in terms of $\\rho, R_{G}, R$ and the gravitational acceleration $g$. If the cylinder is made of steel whose density is $7900 \\mathrm{~kg} / \\mathrm{m}^{3}$, tensile strength is $5.0 \\mathrm{GPa}$, evaluate the ratio between the maximum stress and the tensile strength of steel. Tensile strength is the maximum stress a material can withstand.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1579", "problem": "如图, $1 \\mathrm{~mol}$ 单原子理想气体构成的系统分别经历循环过程 $a b c d a$ 和 $a b c^{\\prime} a$ 。已知理想气体在任一缓慢变化过程中, 压强 $p$ 和体积 $V$ 满足函数关系 $p=f(V)$ 。\n\n[图1]计算系统经 $b c^{\\prime}$ 直线变化过程中的摩尔热容", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, $1 \\mathrm{~mol}$ 单原子理想气体构成的系统分别经历循环过程 $a b c d a$ 和 $a b c^{\\prime} a$ 。已知理想气体在任一缓慢变化过程中, 压强 $p$ 和体积 $V$ 满足函数关系 $p=f(V)$ 。\n\n[图1]\n\n问题:\n计算系统经 $b c^{\\prime}$ 直线变化过程中的摩尔热容\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e3a9fdbbef225ad3aefbg-03.jpg?height=982&width=785&top_left_y=1682&top_left_x=1092" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_197", "problem": "A mass on a frictionless table is attached to the midpoint of an originally unstretched spring fixed at the ends. If the mass is displaced a distance $A$ parallel to the table surface but perpendicular to the spring, it exhibits oscillations. The period $T$ of the oscillations\nA: does not depend on $A$.\nB: increases as $A$ increases, approaching a fixed value.\nC: decreases as $A$ increases, approaching a fixed value. \nD: is approximately constant for small values of $A$, then increases without bound.\nE: is approximately constant for small values of $A$, then decreases without bound.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA mass on a frictionless table is attached to the midpoint of an originally unstretched spring fixed at the ends. If the mass is displaced a distance $A$ parallel to the table surface but perpendicular to the spring, it exhibits oscillations. The period $T$ of the oscillations\n\nA: does not depend on $A$.\nB: increases as $A$ increases, approaching a fixed value.\nC: decreases as $A$ increases, approaching a fixed value. \nD: is approximately constant for small values of $A$, then increases without bound.\nE: is approximately constant for small values of $A$, then decreases without bound.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_722", "problem": "Ann and Betty are good at throwing and catching balls. They can always throw a ball for the other to catch it with only negligible motion. This time they want to have more fun and play the game on skateboards as shown. Initially they are both at rest. Ann first throws the ball to Betty at speed u. After catching the ball, Betty throws the ball back to Ann at the same speed. After Ann catches the ball, Betty is moving faster than Ann by how much? Assuming\n\nmass of the ball $=\\mathrm{m}$\n\nmass of Ann and the skateboard = mass of Betty and the skateboard $=\\mathrm{M}$\n\n[figure1]\nA: $\\frac{M m+2 m^{2}}{M(M+m)} u$\nB: $\\frac{2 m^{2}}{M(M+m)} u$\nC: $\\frac{2 m}{M+m} u$\nD: $-\\frac{2 m}{M+m} u$\nE: $\\frac{m}{M} u$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAnn and Betty are good at throwing and catching balls. They can always throw a ball for the other to catch it with only negligible motion. This time they want to have more fun and play the game on skateboards as shown. Initially they are both at rest. Ann first throws the ball to Betty at speed u. After catching the ball, Betty throws the ball back to Ann at the same speed. After Ann catches the ball, Betty is moving faster than Ann by how much? Assuming\n\nmass of the ball $=\\mathrm{m}$\n\nmass of Ann and the skateboard = mass of Betty and the skateboard $=\\mathrm{M}$\n\n[figure1]\n\nA: $\\frac{M m+2 m^{2}}{M(M+m)} u$\nB: $\\frac{2 m^{2}}{M(M+m)} u$\nC: $\\frac{2 m}{M+m} u$\nD: $-\\frac{2 m}{M+m} u$\nE: $\\frac{m}{M} u$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22e26a14ee6fdd9254b6g-03.jpg?height=295&width=632&top_left_y=736&top_left_x=302" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1655", "problem": "如图, 两劲度系数均为 $k$ 的同样的轻弹性绳的上端固定在一水平面上, 下端连在一起悬挂一质量为 $m$ 的小物块。平衡时, 轻弹性\n\n[图1]\n绳与水平面的夹角为 $\\alpha_{0}$, 弹性绳长度为 $l_{0}$ 。现将小物块向下拉一段微小的距离后从静止释放。若 $k=0.50 \\mathrm{~N} / \\mathrm{m}, m=50 \\mathrm{~g}, \\alpha_{0}=30^{\\circ}, l_{0}=2.0 \\mathrm{~m}$ ，求小物块振动的周期 $T$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 两劲度系数均为 $k$ 的同样的轻弹性绳的上端固定在一水平面上, 下端连在一起悬挂一质量为 $m$ 的小物块。平衡时, 轻弹性\n\n[图1]\n绳与水平面的夹角为 $\\alpha_{0}$, 弹性绳长度为 $l_{0}$ 。现将小物块向下拉一段微小的距离后从静止释放。\n\n问题:\n若 $k=0.50 \\mathrm{~N} / \\mathrm{m}, m=50 \\mathrm{~g}, \\alpha_{0}=30^{\\circ}, l_{0}=2.0 \\mathrm{~m}$ ，求小物块振动的周期 $T$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~s}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_d716ce15f03757bb482eg-04.jpg?height=266&width=537&top_left_y=2494&top_left_x=1408" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_136", "problem": "A uniform rod of length $l$ lies on a frictionless horizontal surface. One end of the rod is attached to a pivot. An un-stretched spring of length $L \\gg l$ lies on the surface perpendicular to the rod; one end of the spring is attached to the movable end of the rod, and the other end is attached to a fixed post. When the rod is rotated slightly about the pivot, it oscillates at frequency $f$.\n\n[figure1]\n The spring attachment is moved back to the end of the rod; the post is moved so that it is in line with the rod and the pivot and the spring is unstretched. The post is then moved away from the pivot by an additional amount $l$. What is the new frequency of oscillation?\n\n[figure2]\nA: $f / 3$\nB: $f / \\sqrt{3}$\nC: $f $ \nD: $\\sqrt{3} f$\nE: $3 f$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA uniform rod of length $l$ lies on a frictionless horizontal surface. One end of the rod is attached to a pivot. An un-stretched spring of length $L \\gg l$ lies on the surface perpendicular to the rod; one end of the spring is attached to the movable end of the rod, and the other end is attached to a fixed post. When the rod is rotated slightly about the pivot, it oscillates at frequency $f$.\n\n[figure1]\n The spring attachment is moved back to the end of the rod; the post is moved so that it is in line with the rod and the pivot and the spring is unstretched. The post is then moved away from the pivot by an additional amount $l$. What is the new frequency of oscillation?\n\n[figure2]\n\nA: $f / 3$\nB: $f / \\sqrt{3}$\nC: $f $ \nD: $\\sqrt{3} f$\nE: $3 f$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-14.jpg?height=301&width=632&top_left_y=511&top_left_x=730", "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-16.jpg?height=181&width=781&top_left_y=430&top_left_x=669" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1582", "problem": "超导体的一个重要应用是绕制强磁场磁体, 其使用的超导线材属于第二类超导体.如果将这类超导体置于磁感应强度为 $B_{a}$ 的外磁场中, 其磁力线将以磁通量子 (或称为磁通漩浴线) 的形式穿透超导体, 从而在超导体中形成正三角形的磁通格子, 如图 1 所示. 所谓的磁通量子, 如图 2 所示, 其中心是半径为 $\\xi$ 的正常态 (电阻不为零) 区域, 而其周围处于超导态 (电阻为零), 存在超导电流, 所携带的磁通量为 $\\phi_{0}=\\frac{h}{2 e}=2.07 \\times 10^{-15} \\mathrm{~Wb}$ (磁通量的最小单位)\n\n[图1]\n\n图 1\n\n[图2]\n\n图3若 $B_{a}=5 \\times 10^{-2} \\mathrm{~T}$, 求此时磁通涡旋线之间距离 $a$.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n超导体的一个重要应用是绕制强磁场磁体, 其使用的超导线材属于第二类超导体.如果将这类超导体置于磁感应强度为 $B_{a}$ 的外磁场中, 其磁力线将以磁通量子 (或称为磁通漩浴线) 的形式穿透超导体, 从而在超导体中形成正三角形的磁通格子, 如图 1 所示. 所谓的磁通量子, 如图 2 所示, 其中心是半径为 $\\xi$ 的正常态 (电阻不为零) 区域, 而其周围处于超导态 (电阻为零), 存在超导电流, 所携带的磁通量为 $\\phi_{0}=\\frac{h}{2 e}=2.07 \\times 10^{-15} \\mathrm{~Wb}$ (磁通量的最小单位)\n\n[图1]\n\n图 1\n\n[图2]\n\n图3\n\n问题:\n若 $B_{a}=5 \\times 10^{-2} \\mathrm{~T}$, 求此时磁通涡旋线之间距离 $a$.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以\\mathrm{~m}为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-05.jpg?height=400&width=471&top_left_y=1433&top_left_x=304", "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-05.jpg?height=483&width=701&top_left_y=1366&top_left_x=863" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~m}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_180", "problem": "A car is turning left along a circular track of radius $r$ at a constant speed $v$. A cylindrical beaker is placed vertically inside the car. The beaker has a small hole on its right side. If the water's highest point in the beaker is a height $h$ above the hole, at what instantaneous speed does water escape the hole, from a passenger's perspective?\nA: $\\sqrt{2 g h} $ \nB: $\\sqrt{\\left(v^{2} / r\\right) h}$\nC: $\\sqrt{g h}$\nD: $\\sqrt{h \\sqrt{\\left(v^{2} / r\\right)^{2}+g^{2}}}$\nE: $\\sqrt{2 h \\sqrt{\\left(v^{2} / r\\right)^{2}+g^{2}}}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA car is turning left along a circular track of radius $r$ at a constant speed $v$. A cylindrical beaker is placed vertically inside the car. The beaker has a small hole on its right side. If the water's highest point in the beaker is a height $h$ above the hole, at what instantaneous speed does water escape the hole, from a passenger's perspective?\n\nA: $\\sqrt{2 g h} $ \nB: $\\sqrt{\\left(v^{2} / r\\right) h}$\nC: $\\sqrt{g h}$\nD: $\\sqrt{h \\sqrt{\\left(v^{2} / r\\right)^{2}+g^{2}}}$\nE: $\\sqrt{2 h \\sqrt{\\left(v^{2} / r\\right)^{2}+g^{2}}}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1331", "problem": "太阳是我们赖以生存的恒星.它的主要成分是氢元素, 在自身引力的作用下收缩而导致升温, 当温度高到一定程度时, 中性原子将电离成质子和电子组成的等离子体, 并在其核心区域达到约 $1.05 \\times 107 \\mathrm{~K}$ 的高温和 $1.6 \\times 105 \\mathrm{~kg} / \\mathrm{m}^{3}$ 以上的高密度, 产生热核聚变而放出巨大能量,从而抗衡自身的引力收缩达到平衡, 而成为恒星.太阳内部主要核反应过程为\n\n${ }^{1} \\mathrm{H}+{ }^{1} \\mathrm{H} \\rightarrow \\mathrm{D}+\\mathrm{e}^{+}+\\mathrm{V}_{\\mathrm{e}}$\n\n$\\mathrm{D}+{ }^{1} \\mathrm{H} \\rightarrow{ }^{3} \\mathrm{He}+x$ (II)\n\n${ }^{3} \\mathrm{He}+{ }^{3} \\mathrm{He} \\rightarrow{ }^{4} \\mathrm{He}+{ }^{1} \\mathrm{H}+{ }^{1} \\mathrm{H}$\n\n其中第一个反应的概率由弱相互作用主导, 概率很低这恰好可以使得能量缓慢释放. 反应产物正电子 $e^{+}$会与电子 $e^{-}$湮灭为 $\\gamma$ 射线, 即\n\n$\\mathrm{e}^{+}+\\mathrm{e}^{-} \\rightarrow \\gamma^{+\\gamma} \\quad$ (IV )\n已知: 质子 $\\left({ }^{1} \\mathrm{H}\\right)$ 、氛 (D) 、氦-3 $\\left({ }^{3} \\mathrm{He}\\right)$ 和电子的质量分别为 $938.27 、 1875.61 、 2808.38 、 3727.36$和 $0.51\\left(\\mathrm{MeV} / \\mathrm{c}^{2}\\right.$ ) (误差为 $0.01 \\mathrm{MeV} / \\mathrm{c}^{2}$ ), $c$ 为真空中的光速, 中微子 $\\mathrm{v}_{\\mathrm{e}}$ 的质量小于 $3 \\mathrm{eV} / \\mathrm{c}^{2}$. 普朗克常量 $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}, c=3.0 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 玻尔兹曼常量 $\\mathrm{k}=1.381 \\times 10^{-23} \\mathrm{~J} / \\mathrm{K}$. 电子电量 $e=1.602 \\times 10^{-19} \\mathrm{C}$.试用理想气体模型估算处于热平衡状态的各种粒子的平均动能及太阳核心区的压强 （请分别用 $\\mathrm{eV}$ 和 atm 为单位）", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n太阳是我们赖以生存的恒星.它的主要成分是氢元素, 在自身引力的作用下收缩而导致升温, 当温度高到一定程度时, 中性原子将电离成质子和电子组成的等离子体, 并在其核心区域达到约 $1.05 \\times 107 \\mathrm{~K}$ 的高温和 $1.6 \\times 105 \\mathrm{~kg} / \\mathrm{m}^{3}$ 以上的高密度, 产生热核聚变而放出巨大能量,从而抗衡自身的引力收缩达到平衡, 而成为恒星.太阳内部主要核反应过程为\n\n${ }^{1} \\mathrm{H}+{ }^{1} \\mathrm{H} \\rightarrow \\mathrm{D}+\\mathrm{e}^{+}+\\mathrm{V}_{\\mathrm{e}}$\n\n$\\mathrm{D}+{ }^{1} \\mathrm{H} \\rightarrow{ }^{3} \\mathrm{He}+x$ (II)\n\n${ }^{3} \\mathrm{He}+{ }^{3} \\mathrm{He} \\rightarrow{ }^{4} \\mathrm{He}+{ }^{1} \\mathrm{H}+{ }^{1} \\mathrm{H}$\n\n其中第一个反应的概率由弱相互作用主导, 概率很低这恰好可以使得能量缓慢释放. 反应产物正电子 $e^{+}$会与电子 $e^{-}$湮灭为 $\\gamma$ 射线, 即\n\n$\\mathrm{e}^{+}+\\mathrm{e}^{-} \\rightarrow \\gamma^{+\\gamma} \\quad$ (IV )\n已知: 质子 $\\left({ }^{1} \\mathrm{H}\\right)$ 、氛 (D) 、氦-3 $\\left({ }^{3} \\mathrm{He}\\right)$ 和电子的质量分别为 $938.27 、 1875.61 、 2808.38 、 3727.36$和 $0.51\\left(\\mathrm{MeV} / \\mathrm{c}^{2}\\right.$ ) (误差为 $0.01 \\mathrm{MeV} / \\mathrm{c}^{2}$ ), $c$ 为真空中的光速, 中微子 $\\mathrm{v}_{\\mathrm{e}}$ 的质量小于 $3 \\mathrm{eV} / \\mathrm{c}^{2}$. 普朗克常量 $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}, c=3.0 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 玻尔兹曼常量 $\\mathrm{k}=1.381 \\times 10^{-23} \\mathrm{~J} / \\mathrm{K}$. 电子电量 $e=1.602 \\times 10^{-19} \\mathrm{C}$.\n\n问题:\n试用理想气体模型估算处于热平衡状态的各种粒子的平均动能及太阳核心区的压强 （请分别用 $\\mathrm{eV}$ 和 atm 为单位）\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[平均动能, 平均动能]\n它们的单位依次是[KeV, atm]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "KeV", "atm" ], "answer_sequence": [ "平均动能", "平均动能" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_442", "problem": "The figure shows a load of mass, $m$, supported by a simple pulley system with a tension $T$ in the cord. What is the value of the tension?\n\n[figure1]\nFigure: Two light pulleys and a light cord.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe figure shows a load of mass, $m$, supported by a simple pulley system with a tension $T$ in the cord. What is the value of the tension?\n\n[figure1]\nFigure: Two light pulleys and a light cord.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-4.jpg?height=497&width=391&top_left_y=742&top_left_x=838" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_116", "problem": "In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_{1}=400 \\mathrm{~g}, M_{2}=200 \\mathrm{~g}$, and $M_{4}=500 \\mathrm{~g}$. What is the value of $M_{3}$ for the system to be in static equilibrium?\n\n[figure1]\nA: $300 \\mathrm{~g}$\nB: $400 \\mathrm{~g}$\nC: $500 \\mathrm{~g}$\nD: $600 \\mathrm{~g}$\nE: $700 \\mathrm{~g} $\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIn the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_{1}=400 \\mathrm{~g}, M_{2}=200 \\mathrm{~g}$, and $M_{4}=500 \\mathrm{~g}$. What is the value of $M_{3}$ for the system to be in static equilibrium?\n\n[figure1]\n\nA: $300 \\mathrm{~g}$\nB: $400 \\mathrm{~g}$\nC: $500 \\mathrm{~g}$\nD: $600 \\mathrm{~g}$\nE: $700 \\mathrm{~g} $\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f726c6cf4a23f08e0214g-04.jpg?height=626&width=982&top_left_y=405&top_left_x=577" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1553", "problem": "质量为 $m_{\\mathrm{A}}$ 的 $\\mathrm{A}$ 球, 以某一速度沿光滑水平面向静止的 $\\mathrm{B}$ 球运动, 并与 $\\mathrm{B}$ 球发生弹性正碰. 假设 $\\mathrm{B}$ 球的质量 $m_{\\mathrm{B}}$ 可选取为不同的值, 则\nA: 当 $m_{\\mathrm{B}}=m_{\\mathrm{A}}$ 时, 碰后 $\\mathrm{B}$ 球的速度最大\nB: 当 $m_{\\mathrm{B}}=m_{\\mathrm{A}}$ 时, 碰后 $\\mathrm{B}$ 球的动能最大\nC: 在保持 $m_{\\mathrm{B}}>m_{\\mathrm{A}}$ 的条件下, $m_{\\mathrm{B}}$ 越小, 碰后 $\\mathrm{B}$ 球的速度越大\nD: 在保持 $m_{\\mathrm{B}}m_{\\mathrm{A}}$ 的条件下, $m_{\\mathrm{B}}$ 越小, 碰后 $\\mathrm{B}$ 球的速度越大\nD: 在保持 $m_{\\mathrm{B}}b$ centered about the origin. Since the shell is neutral, the enclosed charge is $q$, so by spherical symmetry\n\n$$\nE(r)=\\frac{q}{4 \\pi \\epsilon_{0} r^{2}}\n$$\n\noutside the shell. Just outside the shell, the field is $q / 4 \\pi \\epsilon_{0} b^{2}$.\n\nSince the shell is conducting, the electrostatic field is zero inside it. By Gauss's law, this is achieved by having a charge of $-q$ on the inner surface $r=a$ and a charge of $q$ on the outer surface $r=b$, both uniformly distributed.\n\nFor $rb$ centered about the origin. Since the shell is neutral, the enclosed charge is $q$, so by spherical symmetry\n\n$$\nE(r)=\\frac{q}{4 \\pi \\epsilon_{0} r^{2}}\n$$\n\noutside the shell. Just outside the shell, the field is $q / 4 \\pi \\epsilon_{0} b^{2}$.\n\nSince the shell is conducting, the electrostatic field is zero inside it. By Gauss's law, this is achieved by having a charge of $-q$ on the inner surface $r=a$ and a charge of $q$ on the outer surface $r=b$, both uniformly distributed.\n\nFor $r>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nFind the expression for the force $\\vec{f}$ acting on the ion due to the polarised atom. Show that this force is attractive regardless of the sign of the charge of the ion.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n## To Commemorate the Centenary of Rutherford's Atomic Nucleus: the Scattering of an Ion by a Neutral Atom\n\n[figure1]\n\nAn ion of mass $m$, charge $Q$, is moving with an initial non-relativistic speed $v_{0}$ from a great distance towards the vicinity of a neutral atom of mass $M>>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nFind the expression for the force $\\vec{f}$ acting on the ion due to the polarised atom. Show that this force is attractive regardless of the sign of the charge of the ion.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_51c4dc0e7c52a1226310g-1.jpg?height=462&width=1495&top_left_y=701&top_left_x=315" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1368", "problem": "质子数与中子数互换的核互为镜像核, 例如 ${ }^{3} \\mathrm{He}$ 是 ${ }^{3} \\mathrm{H}$ 的镜像核, 同样 ${ }^{3} \\mathrm{H}$ 是 ${ }^{3} \\mathrm{He}$的镜像核。已知 ${ }^{3} \\mathrm{H}$ 和 ${ }^{3} \\mathrm{He}$ 原子的质量分别是 $m_{3_{n}}=3.016050 \\mathrm{u}$ 和 $m_{3_{k}}=3.016029 \\mathrm{u}$, 中子和质子质量分别是 $m_{n}=1.008665 \\mathrm{u}$ 和 $m_{p}=1.007825 \\mathrm{u}, 1 u=\\frac{931.5}{c^{2}} \\mathrm{MeV}$, 式中 $\\mathrm{c}$ 为光速, 静电力常量 $k=\\frac{1.44}{e^{2}} \\mathrm{MeV} \\cdot \\mathrm{fm}$, 式中 $\\mathrm{e}$ 为电子的电荷量。试计算 ${ }^{3} \\mathrm{H}$ 和 ${ }^{3} \\mathrm{He}$ 的结合能之差为多少 $\\mathrm{MeV}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n质子数与中子数互换的核互为镜像核, 例如 ${ }^{3} \\mathrm{He}$ 是 ${ }^{3} \\mathrm{H}$ 的镜像核, 同样 ${ }^{3} \\mathrm{H}$ 是 ${ }^{3} \\mathrm{He}$的镜像核。已知 ${ }^{3} \\mathrm{H}$ 和 ${ }^{3} \\mathrm{He}$ 原子的质量分别是 $m_{3_{n}}=3.016050 \\mathrm{u}$ 和 $m_{3_{k}}=3.016029 \\mathrm{u}$, 中子和质子质量分别是 $m_{n}=1.008665 \\mathrm{u}$ 和 $m_{p}=1.007825 \\mathrm{u}, 1 u=\\frac{931.5}{c^{2}} \\mathrm{MeV}$, 式中 $\\mathrm{c}$ 为光速, 静电力常量 $k=\\frac{1.44}{e^{2}} \\mathrm{MeV} \\cdot \\mathrm{fm}$, 式中 $\\mathrm{e}$ 为电子的电荷量。\n\n问题:\n试计算 ${ }^{3} \\mathrm{H}$ 和 ${ }^{3} \\mathrm{He}$ 的结合能之差为多少 $\\mathrm{MeV}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$$ \\mathrm{MeV} $$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$$ \\mathrm{MeV} $$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1641", "problem": "2016 年 9 月，G20 峰会在杭州隆重召开，其会议厅的装饰设计既展示出中国建筑的节能环保理念，又体现了浙江的竹文化特色。图 a 给出了其部分墙面采用的微孔竹板装饰的局部放大照片，该装饰同时又实现了对声波的共振吸收. 竹板上有一系列不同面积、周期性排列的长方形微孔，声波进入微孔后导致微孔中的空气柱做简谐振动. 单个微孔和竹板后的空气层, 可简化成一个亥姆霍兹共振器, 如图 b 所示. 假设微孔深度均为 1 、单个微孔后的空气腔体体积均为 $V_{0}$ 、微孔横截面积记为 $\\mathrm{S}$. 声波在空气层中传播可视为绝热过程, 声波传播速度 $v_{s}$ 与空气密度 $\\rho$ 及体积弹性模量 $\\kappa$ 的关系为\n\n$v_{s}=\\sqrt{\\frac{\\kappa}{\\rho}}$\n\n其中 $\\kappa$ 是气体压强的增加量 $\\Delta p$ 与其体积 $\\mathrm{V}$ 相对变化量之比\n\n$\\kappa=-\\frac{\\Delta p}{\\Delta V / V}=-V \\frac{\\Delta p}{\\Delta V}$\n\n已知标准状态 $\\left(273 \\mathrm{~K}, \\quad l \\mathrm{~atm}=1.01 \\times 10^{5} \\mathrm{~Pa}\\right)$ 下空气 (可视为理想气体) 的摩尔质量 $\\mathrm{M}_{\\mathrm{mol}}=29.0 \\mathrm{~g} / \\mathrm{mol}$, 热容比\n$\\gamma=\\frac{7}{5}$, 气体普适常量 $\\mathrm{R}=8.31 \\mathrm{~J} /(\\mathrm{K} \\cdot \\mathrm{mol})$.\n\n[图1]\n\n图 a. 微孔竹板墙照片 (局部)\n\n[图2]\n\n图 b. 微孔竹板装置简化图及对应的亥㛂霍兹共振器模型求标准状态下空气的密度和声波在空气中的传播速度 $v_{s}$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n2016 年 9 月，G20 峰会在杭州隆重召开，其会议厅的装饰设计既展示出中国建筑的节能环保理念，又体现了浙江的竹文化特色。图 a 给出了其部分墙面采用的微孔竹板装饰的局部放大照片，该装饰同时又实现了对声波的共振吸收. 竹板上有一系列不同面积、周期性排列的长方形微孔，声波进入微孔后导致微孔中的空气柱做简谐振动. 单个微孔和竹板后的空气层, 可简化成一个亥姆霍兹共振器, 如图 b 所示. 假设微孔深度均为 1 、单个微孔后的空气腔体体积均为 $V_{0}$ 、微孔横截面积记为 $\\mathrm{S}$. 声波在空气层中传播可视为绝热过程, 声波传播速度 $v_{s}$ 与空气密度 $\\rho$ 及体积弹性模量 $\\kappa$ 的关系为\n\n$v_{s}=\\sqrt{\\frac{\\kappa}{\\rho}}$\n\n其中 $\\kappa$ 是气体压强的增加量 $\\Delta p$ 与其体积 $\\mathrm{V}$ 相对变化量之比\n\n$\\kappa=-\\frac{\\Delta p}{\\Delta V / V}=-V \\frac{\\Delta p}{\\Delta V}$\n\n已知标准状态 $\\left(273 \\mathrm{~K}, \\quad l \\mathrm{~atm}=1.01 \\times 10^{5} \\mathrm{~Pa}\\right)$ 下空气 (可视为理想气体) 的摩尔质量 $\\mathrm{M}_{\\mathrm{mol}}=29.0 \\mathrm{~g} / \\mathrm{mol}$, 热容比\n$\\gamma=\\frac{7}{5}$, 气体普适常量 $\\mathrm{R}=8.31 \\mathrm{~J} /(\\mathrm{K} \\cdot \\mathrm{mol})$.\n\n[图1]\n\n图 a. 微孔竹板墙照片 (局部)\n\n[图2]\n\n图 b. 微孔竹板装置简化图及对应的亥㛂霍兹共振器模型\n\n问题:\n求标准状态下空气的密度和声波在空气中的传播速度 $v_{s}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-04.jpg?height=340&width=408&top_left_y=635&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-04.jpg?height=325&width=762&top_left_y=637&top_left_x=841" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_308", "problem": "Stopping distance is the sum of thinking distance and braking distance.\n\nWhich of the following changes gives the longest stopping distance of a vehicle being driven fast along a straight road and coming to a stop with a constant deceleration using just the vehicle brakes?\nA: Doubling the initial speed of the vehicle\nB: Doubling the mass of the vehicle\nC: Doubling the reaction time of the driver\nD: Halving the braking force of the vehicle (for example due to road conditions)\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nStopping distance is the sum of thinking distance and braking distance.\n\nWhich of the following changes gives the longest stopping distance of a vehicle being driven fast along a straight road and coming to a stop with a constant deceleration using just the vehicle brakes?\n\nA: Doubling the initial speed of the vehicle\nB: Doubling the mass of the vehicle\nC: Doubling the reaction time of the driver\nD: Halving the braking force of the vehicle (for example due to road conditions)\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_836", "problem": "When the cloud temperature $T$ is much smaller than equivalent temperature $T_{0}$, the optical potential can be well approximated by a cylindrically symmetric harmonic potential $U_{\\text {dip }}(\\rho, z)=-U_{\\text {depth }}+\\frac{1}{2} m \\Omega_{\\rho}^{2} \\rho^{2}+$ $\\frac{1}{2} m \\Omega_{z}^{2} z^{2}$, where $m$ is the mass of a sodium atom and $\\Omega_{\\rho}, \\Omega_{z}$ are oscillation frequencies in the corresponding directions.\n\nFind the expression for $\\Omega_{\\rho}, \\Omega_{z}$ in terms of $T_{0}, m, D_{0}, z_{R}$ and $k_{B}$. Here $k_{B}$ is the $0.5 \\mathrm{pt}$ Boltzmann constant.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nWhen the cloud temperature $T$ is much smaller than equivalent temperature $T_{0}$, the optical potential can be well approximated by a cylindrically symmetric harmonic potential $U_{\\text {dip }}(\\rho, z)=-U_{\\text {depth }}+\\frac{1}{2} m \\Omega_{\\rho}^{2} \\rho^{2}+$ $\\frac{1}{2} m \\Omega_{z}^{2} z^{2}$, where $m$ is the mass of a sodium atom and $\\Omega_{\\rho}, \\Omega_{z}$ are oscillation frequencies in the corresponding directions.\n\nFind the expression for $\\Omega_{\\rho}, \\Omega_{z}$ in terms of $T_{0}, m, D_{0}, z_{R}$ and $k_{B}$. Here $k_{B}$ is the $0.5 \\mathrm{pt}$ Boltzmann constant.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [$\\Omega_{z}$ , $\\Omega_{\\rho}$ ].\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$\\Omega_{z}$ ", "$\\Omega_{\\rho}$ " ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_156", "problem": "A hoop of radius $r$ is launched to the right at initial speed $v_{0}$ at ground-level. As it is launched, it is also spun counterclockwise at angular velocity $3 v_{0} / r$. The coefficient of kinetic friction between the ground and the hoop is $\\mu_{k}$.\n\nHow long does it take the hoop to return to its starting position?\nA: $\\frac{v_{0}}{2 \\mu_{k} g}$\nB: $\\frac{2 v_{0}}{\\mu_{k} g} $\nC: $\\frac{\\mu_{k} v_{0}}{2 g}$\nD: $\\frac{\\mu_{k} v_{0}}{g}$\nE: $\\frac{2 \\mu_{k} v_{0}}{g}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA hoop of radius $r$ is launched to the right at initial speed $v_{0}$ at ground-level. As it is launched, it is also spun counterclockwise at angular velocity $3 v_{0} / r$. The coefficient of kinetic friction between the ground and the hoop is $\\mu_{k}$.\n\nHow long does it take the hoop to return to its starting position?\n\nA: $\\frac{v_{0}}{2 \\mu_{k} g}$\nB: $\\frac{2 v_{0}}{\\mu_{k} g} $\nC: $\\frac{\\mu_{k} v_{0}}{2 g}$\nD: $\\frac{\\mu_{k} v_{0}}{g}$\nE: $\\frac{2 \\mu_{k} v_{0}}{g}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1001", "problem": "Although atomic nuclei are quantum objects, a number of phenomenological laws for their basic properties (like radius or binding energy) can be deduced from simple assumptions: (i) nuclei are built from nucleons (i.e. protons and neutrons); (ii) strong nuclear interaction holding these nucleons together has a very short range (it acts only between neighboring nucleons); (iii) the number of protons $(Z)$ in a given nucleus is approximately equal to the number of neutrons $(N)$, i.e. $Z \\approx N \\approx A / 2$, where $A$ is the total number of nucleons $(A \\gg 1)$.\n\n## Binding energy of atomic nuclei - volume and surface terms\n\nBinding energy of a nucleus is the energy required to disassemble it into separate nucleons and it essentially comes from the attractive nuclear force of each nucleon with its neighbors. If a given nucleon is not on the surface of the nucleus, it contributes to the total binding energy with $a_{V}=15.8$ $\\mathrm{MeV}\\left(1 \\mathrm{MeV}=1.602 \\cdot 10^{-13} \\mathrm{~J}\\right)$. The contribution of one surface nucleon to the binding energy is approximately $a_{V} / 2$. Express the binding energy $E_{b}$ of a nucleus with $A$ nucleons in terms of $A, a_{V}$, and $f$, and by including the surface correction.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nAlthough atomic nuclei are quantum objects, a number of phenomenological laws for their basic properties (like radius or binding energy) can be deduced from simple assumptions: (i) nuclei are built from nucleons (i.e. protons and neutrons); (ii) strong nuclear interaction holding these nucleons together has a very short range (it acts only between neighboring nucleons); (iii) the number of protons $(Z)$ in a given nucleus is approximately equal to the number of neutrons $(N)$, i.e. $Z \\approx N \\approx A / 2$, where $A$ is the total number of nucleons $(A \\gg 1)$.\n\n## Binding energy of atomic nuclei - volume and surface terms\n\nBinding energy of a nucleus is the energy required to disassemble it into separate nucleons and it essentially comes from the attractive nuclear force of each nucleon with its neighbors. If a given nucleon is not on the surface of the nucleus, it contributes to the total binding energy with $a_{V}=15.8$ $\\mathrm{MeV}\\left(1 \\mathrm{MeV}=1.602 \\cdot 10^{-13} \\mathrm{~J}\\right)$. The contribution of one surface nucleon to the binding energy is approximately $a_{V} / 2$. Express the binding energy $E_{b}$ of a nucleus with $A$ nucleons in terms of $A, a_{V}$, and $f$, and by including the surface correction.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of MeV, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without any units and equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": [ "MeV" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_842", "problem": "A single electron transistor (SET) consists of a quantum dot, which is a small isolated conductor where electrons can be localised, and of several electrodes in its vicinity. The gate electrode couples capacitatively to the quantum dot, while the two other electrodes --- the source and the drain --- are connected via tunnel junctions, through which electrons can tunnel due to quantum mechanics. A simplified circuit diagram for an SET is shown in the figure.\n[figure1]\n\nCircuit diagram representation of an SET. QD is the quantum dot, $\\mathrm{S}$ is the source, $\\mathrm{D}$ is the drain and $\\mathrm{G}$ is the gate.\n\nThe capacitance of the gate is $C_{g}$ and the capacitance of the tunnel junctions is $C_{t} \\ll C_{g}$. Consider $C_{g}$ to be the total capacitance of the quantum dot. In this part of the problem, the source and the drain are held at zero potential, and the voltage on the gate electrode is fixed at $V_{g}$.\n\nConsider a state of the SET in which the quantum dot contains $n$ electrons.\n\nTunnelling of electrons onto or off the dot limits the lifetime of their energy states. This tunnelling can be modelled using an effective resistance of the tunnel junction with the characteristic tunnelling time equal to the characteristic time for charging or discharging the quantum dot through the junction.\nEstimate the tunnelling time for a quantum dot in terms of capacitance $C_{t}$ and effective resistance $R_{t}$ of the tunnel junction.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA single electron transistor (SET) consists of a quantum dot, which is a small isolated conductor where electrons can be localised, and of several electrodes in its vicinity. The gate electrode couples capacitatively to the quantum dot, while the two other electrodes --- the source and the drain --- are connected via tunnel junctions, through which electrons can tunnel due to quantum mechanics. A simplified circuit diagram for an SET is shown in the figure.\n[figure1]\n\nCircuit diagram representation of an SET. QD is the quantum dot, $\\mathrm{S}$ is the source, $\\mathrm{D}$ is the drain and $\\mathrm{G}$ is the gate.\n\nThe capacitance of the gate is $C_{g}$ and the capacitance of the tunnel junctions is $C_{t} \\ll C_{g}$. Consider $C_{g}$ to be the total capacitance of the quantum dot. In this part of the problem, the source and the drain are held at zero potential, and the voltage on the gate electrode is fixed at $V_{g}$.\n\nConsider a state of the SET in which the quantum dot contains $n$ electrons.\n\nTunnelling of electrons onto or off the dot limits the lifetime of their energy states. This tunnelling can be modelled using an effective resistance of the tunnel junction with the characteristic tunnelling time equal to the characteristic time for charging or discharging the quantum dot through the junction.\nEstimate the tunnelling time for a quantum dot in terms of capacitance $C_{t}$ and effective resistance $R_{t}$ of the tunnel junction.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-3.jpg?height=582&width=868&top_left_y=2076&top_left_x=594" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1730", "problem": "可能用到的物理常量和公式:\n\n真空中的光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$;\n\n已知地球表面的重力加速度的大小为 $g$;\n\n已知普朗克常量为 $h, \\hbar=\\frac{h}{2 \\pi}$;\n\n$\\int \\frac{1}{1-x^{2}} \\mathrm{~d} x=\\frac{1}{2} \\ln \\frac{1+x}{1-x}+C,|x|<1$ 。\n\n山西大同某煤矿相对于秦皇岛的高度为 $h_{\\mathrm{c}}$ 。质量为 $m_{\\mathrm{t}}$ 的火车载有质量为 $m_{\\mathrm{c}}$ 的煤, 从大同沿大秦线铁路行驶路程 $l$ 后到达秦皇岛, 卸载后空车返回。从大同到秦皇岛的过程中, 火车和煤总势能的一部分克服铁轨和空气阻力做功, 其余部分由发电机转换成电能, 平均转换效率为 $\\eta_{1}$, 电能被全部储存于蓄电池中以用于返程。空车在返程中由储存的电能驱动电动机克服重力和阻力做功, 存储电能转化为对外做功的平均转换效率为 $\\eta_{2}$ 。假设大秦线轨道上火车平均每运行单位距离克服阻力需要做的功与运行时（火车或火车和煤）总重量成正比,比例系数为常数 $\\mu$, 火车由大同出发时携带的电能为零。\n\n去程火车发电机的输入能量是\n\n$$\nE_{1}=\\left(m_{\\mathrm{c}}+m_{\\mathrm{t}}\\right) g\\left(h_{\\mathrm{c}}-0\\right)-\\mu\\left(m_{\\mathrm{c}}+m_{\\mathrm{t}}\\right) g l\n$$已知火车在从大同到秦皇岛的铁轨上运行的平均速率为 $v$, 请给出发电机的平均输出功率 $P$ 与题给的其它物理量的关系。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n可能用到的物理常量和公式:\n\n真空中的光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$;\n\n已知地球表面的重力加速度的大小为 $g$;\n\n已知普朗克常量为 $h, \\hbar=\\frac{h}{2 \\pi}$;\n\n$\\int \\frac{1}{1-x^{2}} \\mathrm{~d} x=\\frac{1}{2} \\ln \\frac{1+x}{1-x}+C,|x|<1$ 。\n\n山西大同某煤矿相对于秦皇岛的高度为 $h_{\\mathrm{c}}$ 。质量为 $m_{\\mathrm{t}}$ 的火车载有质量为 $m_{\\mathrm{c}}$ 的煤, 从大同沿大秦线铁路行驶路程 $l$ 后到达秦皇岛, 卸载后空车返回。从大同到秦皇岛的过程中, 火车和煤总势能的一部分克服铁轨和空气阻力做功, 其余部分由发电机转换成电能, 平均转换效率为 $\\eta_{1}$, 电能被全部储存于蓄电池中以用于返程。空车在返程中由储存的电能驱动电动机克服重力和阻力做功, 存储电能转化为对外做功的平均转换效率为 $\\eta_{2}$ 。假设大秦线轨道上火车平均每运行单位距离克服阻力需要做的功与运行时（火车或火车和煤）总重量成正比,比例系数为常数 $\\mu$, 火车由大同出发时携带的电能为零。\n\n去程火车发电机的输入能量是\n\n$$\nE_{1}=\\left(m_{\\mathrm{c}}+m_{\\mathrm{t}}\\right) g\\left(h_{\\mathrm{c}}-0\\right)-\\mu\\left(m_{\\mathrm{c}}+m_{\\mathrm{t}}\\right) g l\n$$\n\n问题:\n已知火车在从大同到秦皇岛的铁轨上运行的平均速率为 $v$, 请给出发电机的平均输出功率 $P$ 与题给的其它物理量的关系。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_295", "problem": "The speed of sound in air depends on the temperature of the air. Sound travels faster in warmer air and slower in cooler air. During the day the ground is warmed by the sun and the air just above the ground is warmer than the air higher up. In this situation sound waves can curve upwards creating \"sound shadows\" where sounds are not heard by listeners on the ground.\n\nThis phenomenon is an example of:\nA: Dispersion\nB: Diffraction\nC: Refraction\nD: Reflection\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nThe speed of sound in air depends on the temperature of the air. Sound travels faster in warmer air and slower in cooler air. During the day the ground is warmed by the sun and the air just above the ground is warmer than the air higher up. In this situation sound waves can curve upwards creating \"sound shadows\" where sounds are not heard by listeners on the ground.\n\nThis phenomenon is an example of:\n\nA: Dispersion\nB: Diffraction\nC: Refraction\nD: Reflection\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_562", "problem": "Part (i)\n\nBy symmetry, the fields above and below the sheet are equal in magnitude and directed away from the sheet. By Gauss's Law, using a cylinder of base area $A$,\n\n$$\n2 E A=\\frac{\\sigma A}{\\epsilon_{0}} \\Rightarrow E=\\frac{\\sigma}{2 \\epsilon_{0}}\n$$\n\npointing directly away from the sheet in the $z$ direction, or\n\n$$\n\\mathbf{E}=\\frac{\\sigma}{2 \\epsilon} \\times \\begin{cases}\\hat{\\mathbf{z}} & \\text { above the sheet } \\\\ -\\hat{\\mathbf{z}} & \\text { below the sheet. }\\end{cases}\n$$In this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinite uniform sheet has a surface charge density $\\sigma$ and has an infinitesimal thickness. The sheet lies in the $x y$ plane.\n\nAssuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{x}}$, determine the magnetic field $\\tilde{\\mathbf{B}}$ (magnitude and direction) above and below the sheet.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\nHere is some context information for this question, which might assist you in solving it:\nPart (i)\n\nBy symmetry, the fields above and below the sheet are equal in magnitude and directed away from the sheet. By Gauss's Law, using a cylinder of base area $A$,\n\n$$\n2 E A=\\frac{\\sigma A}{\\epsilon_{0}} \\Rightarrow E=\\frac{\\sigma}{2 \\epsilon_{0}}\n$$\n\npointing directly away from the sheet in the $z$ direction, or\n\n$$\n\\mathbf{E}=\\frac{\\sigma}{2 \\epsilon} \\times \\begin{cases}\\hat{\\mathbf{z}} & \\text { above the sheet } \\\\ -\\hat{\\mathbf{z}} & \\text { below the sheet. }\\end{cases}\n$$\n\nproblem:\nIn this problem assume that velocities $v$ are much less than the speed of light $c$, and therefore ignore relativistic contraction of lengths or time dilation.\n\nAn infinite uniform sheet has a surface charge density $\\sigma$ and has an infinitesimal thickness. The sheet lies in the $x y$ plane.\n\nAssuming the sheet is moving with velocity $\\tilde{\\mathbf{v}}=v \\hat{\\mathbf{x}}$, determine the magnetic field $\\tilde{\\mathbf{B}}$ (magnitude and direction) above and below the sheet.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the magnetic field $\\tilde{\\mathbf{B}}$ above the sheet, the magnetic field $\\tilde{\\mathbf{B}}$ below the sheet].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "the magnetic field $\\tilde{\\mathbf{B}}$ above the sheet", "the magnetic field $\\tilde{\\mathbf{B}}$ below the sheet" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_153", "problem": "A packing crate with mass $m=115 \\mathrm{~kg}$ is slid up a $5.00 \\mathrm{~m}$ long ramp which makes an angle of $20.0^{\\circ}$ with respect to the horizontal by an applied force of $F=1.00 \\times 10^{3} \\mathrm{~N}$ directed parallel to the ramp's incline. A frictional force of magnitude $f=4.00 \\times 10^{2} \\mathrm{~N}$ resists the motion. If the crate starts from rest, what is its speed at the top of the ramp?\nA: $4.24 \\mathrm{~m} / \\mathrm{s} $ \nB: $5.11 \\mathrm{~m} / \\mathrm{s}$\nC: $7.22 \\mathrm{~m} / \\mathrm{s}$\nD: $8.26 \\mathrm{~m} / \\mathrm{s}$\nE: $9.33 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA packing crate with mass $m=115 \\mathrm{~kg}$ is slid up a $5.00 \\mathrm{~m}$ long ramp which makes an angle of $20.0^{\\circ}$ with respect to the horizontal by an applied force of $F=1.00 \\times 10^{3} \\mathrm{~N}$ directed parallel to the ramp's incline. A frictional force of magnitude $f=4.00 \\times 10^{2} \\mathrm{~N}$ resists the motion. If the crate starts from rest, what is its speed at the top of the ramp?\n\nA: $4.24 \\mathrm{~m} / \\mathrm{s} $ \nB: $5.11 \\mathrm{~m} / \\mathrm{s}$\nC: $7.22 \\mathrm{~m} / \\mathrm{s}$\nD: $8.26 \\mathrm{~m} / \\mathrm{s}$\nE: $9.33 \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_761", "problem": "What is the force of the pigeon on the turtle?\nA: $0.5 \\mathrm{~kg}$\nB: $5 \\mathrm{~N}$\nC: $55 \\mathrm{~N}$\nD: $0 \\mathrm{~N}$\nE: $5.5 \\mathrm{~kg}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the force of the pigeon on the turtle?\n\nA: $0.5 \\mathrm{~kg}$\nB: $5 \\mathrm{~N}$\nC: $55 \\mathrm{~N}$\nD: $0 \\mathrm{~N}$\nE: $5.5 \\mathrm{~kg}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_763", "problem": "Skye is collecting coins as a fundraiser for her ice hockey team. She has a shoebox full of 5c coins. To the nearest order of the magnitude, what is the mass of the shoebox with the coins in it?\nA: $\\quad 0.3 \\mathrm{~kg}$\nB: $3 \\mathrm{~kg}$\nC: $30 \\mathrm{~kg}$\nD: $300 \\mathrm{~kg}$\nE: $3000 \\mathrm{~kg}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nSkye is collecting coins as a fundraiser for her ice hockey team. She has a shoebox full of 5c coins. To the nearest order of the magnitude, what is the mass of the shoebox with the coins in it?\n\nA: $\\quad 0.3 \\mathrm{~kg}$\nB: $3 \\mathrm{~kg}$\nC: $30 \\mathrm{~kg}$\nD: $300 \\mathrm{~kg}$\nE: $3000 \\mathrm{~kg}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1495", "problem": "一质量为 $m$ 、长为 $L$ 的匀质细杆, 可绕过其一端的光滑水平轴 $O$ 在坚直平面内自由转动. 杆在水平状态由静止开始下摆,已知系统的动能等于系统的质量全部集中在质心时随质心一起运动的动能和系统在质心系 (随质心平动的参考系) 中的动能之和, 求常数 $k$ 的值", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一质量为 $m$ 、长为 $L$ 的匀质细杆, 可绕过其一端的光滑水平轴 $O$ 在坚直平面内自由转动. 杆在水平状态由静止开始下摆,\n\n问题:\n已知系统的动能等于系统的质量全部集中在质心时随质心一起运动的动能和系统在质心系 (随质心平动的参考系) 中的动能之和, 求常数 $k$ 的值\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1551", "problem": "电子偶素原子 (Ps ) 是由电子 $\\mathrm{e}^{-}$与正电子 $\\mathrm{e}^{+}$(电子的反粒子, 其质量与电子的相同,电荷与电子的大小相等、符号相反) 组成的量子束缚体系, 其能级可类比氢原子能级得出。根据玻尔氢原子理论, 电子绕质子的圆周运动轨道角动量的取值是量子化的, 即为 $\\hbar$ 的整数倍。考虑到质子质量是有限的, 氢原子量子化条件应修正为: 电子与质子质心系中相对其质心的总轨道角动量取值为 $\\hbar$ 的整数倍。这一量子化条件可直接推广到其它两体束缚体系, 如电子偶素等。以下计算结果均保留四位有效数字。当基态电子偶素原子相对于实验室参照系以与光速 $c$ 可比拟的速度 $v_{0}$ 运动时湮没生成两个光子, 求生成的两个光子 $\\gamma_{1}$ 和 $\\gamma_{2}$ 的能量 $E_{1}$ 和 $E_{2}$ 、以及光子 $\\gamma_{2}$ 的运动方向相对于 $v_{0}$ 的方向的偏角 $\\theta_{2}$ (如图所示) 与 $\\theta_{1}$ 之间的关系式; 并给出当 $v_{0}=\\frac{c}{2} 、 \\theta_{1}=\\frac{\\pi}{3}$ 时 $E_{1} 、 E_{2}$ 和 $\\theta_{2}$ 的值。\n\n已知: 氢原子基态能量 $E_{n=1}^{\\mathrm{H}}=-13.60 \\mathrm{eV}$, 电子质量 $m_{\\mathrm{e}}=0.5110 \\mathrm{MeV} / c^{2}$, 质子与电子的质量之比为 1836 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n电子偶素原子 (Ps ) 是由电子 $\\mathrm{e}^{-}$与正电子 $\\mathrm{e}^{+}$(电子的反粒子, 其质量与电子的相同,电荷与电子的大小相等、符号相反) 组成的量子束缚体系, 其能级可类比氢原子能级得出。根据玻尔氢原子理论, 电子绕质子的圆周运动轨道角动量的取值是量子化的, 即为 $\\hbar$ 的整数倍。考虑到质子质量是有限的, 氢原子量子化条件应修正为: 电子与质子质心系中相对其质心的总轨道角动量取值为 $\\hbar$ 的整数倍。这一量子化条件可直接推广到其它两体束缚体系, 如电子偶素等。以下计算结果均保留四位有效数字。\n\n问题:\n当基态电子偶素原子相对于实验室参照系以与光速 $c$ 可比拟的速度 $v_{0}$ 运动时湮没生成两个光子, 求生成的两个光子 $\\gamma_{1}$ 和 $\\gamma_{2}$ 的能量 $E_{1}$ 和 $E_{2}$ 、以及光子 $\\gamma_{2}$ 的运动方向相对于 $v_{0}$ 的方向的偏角 $\\theta_{2}$ (如图所示) 与 $\\theta_{1}$ 之间的关系式; 并给出当 $v_{0}=\\frac{c}{2} 、 \\theta_{1}=\\frac{\\pi}{3}$ 时 $E_{1} 、 E_{2}$ 和 $\\theta_{2}$ 的值。\n\n已知: 氢原子基态能量 $E_{n=1}^{\\mathrm{H}}=-13.60 \\mathrm{eV}$, 电子质量 $m_{\\mathrm{e}}=0.5110 \\mathrm{MeV} / c^{2}$, 质子与电子的质量之比为 1836 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$E_{1}$, $E_{2}$, $\\theta_{2}$]\n它们的单位依次是[$\\mathrm{MeV}$, $\\mathrm{MeV}$, None]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{MeV}$", "$\\mathrm{MeV}$", null ], "answer_sequence": [ "$E_{1}$", "$E_{2}$", "$\\theta_{2}$" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1162", "problem": "# A Three-body Problem and LISA \n\n[figure1]\n\nFIGURE 1 Coplanar orbits of three bodies.\n\nConsider the case $M=m$. If $\\mu$ is now given a small radial perturbation (along $\\mathrm{O} \\mu$ ), what is the angular frequency of oscillation of $\\mu$ about the unperturbed position in terms of $\\omega_{0}$ ? Assume that the angular momentum of $\\mu$ is conserved.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n# A Three-body Problem and LISA \n\n[figure1]\n\nFIGURE 1 Coplanar orbits of three bodies.\n\nConsider the case $M=m$. If $\\mu$ is now given a small radial perturbation (along $\\mathrm{O} \\mu$ ), what is the angular frequency of oscillation of $\\mu$ about the unperturbed position in terms of $\\omega_{0}$ ? Assume that the angular momentum of $\\mu$ is conserved.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_27986d152c1a1dded222g-1.jpg?height=918&width=919&top_left_y=506&top_left_x=619", "https://cdn.mathpix.com/cropped/2024_03_14_2e1e5a594468530ddb04g-4.jpg?height=379&width=398&top_left_y=2139&top_left_x=1449" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1657", "problem": "在一水平直线上相距 $18 \\mathrm{~m}$ 的 $\\mathrm{A} 、 \\mathrm{~B}$ 两点放置两个波源。这两个波源振动的方向相同、振幅相等、频率都是 $30 \\mathrm{~Hz}$, 且有相位差 $\\pi$ 。它们沿同一条直线在其两边的媒质中各激起简谐横波。波在媒质中的传播速度为 $360 \\mathrm{~m} / \\mathrm{s}$ 。这两列波在 $\\mathrm{A} 、 \\mathrm{~B}$ 两点所在直线上因干涉而振幅等于原来各自振幅的点有___个, 它们到 $\\mathrm{A}$ 点的距离依次为 ____m。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n在一水平直线上相距 $18 \\mathrm{~m}$ 的 $\\mathrm{A} 、 \\mathrm{~B}$ 两点放置两个波源。这两个波源振动的方向相同、振幅相等、频率都是 $30 \\mathrm{~Hz}$, 且有相位差 $\\pi$ 。它们沿同一条直线在其两边的媒质中各激起简谐横波。波在媒质中的传播速度为 $360 \\mathrm{~m} / \\mathrm{s}$ 。这两列波在 $\\mathrm{A} 、 \\mathrm{~B}$ 两点所在直线上因干涉而振幅等于原来各自振幅的点有___个, 它们到 $\\mathrm{A}$ 点的距离依次为 ____m。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[这两列波在 A 、 B A、 B 两点所在直线上因干涉而振幅等于原来各自振幅的点个数, 它们到 A 点的距离]\n它们的单位依次是[个, m]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "个", "m" ], "answer_sequence": [ "这两列波在 A 、 B A、 B 两点所在直线上因干涉而振幅等于原来各自振幅的点个数", "它们到 A 点的距离" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_127", "problem": "A small ball of mass $3 m$ is at rest on the ground. A second small ball of mass $m$ is positioned above the ground by a vertical massless rod of length $L$ that is also attached to the ball on the ground. The original orientation of the rod is directly vertical, and the top ball is given a small horizontal nudge. There is no friction; assume that everything happens in a single plane.\n\n21. Determine the horizontal displacement $x$ of the second ball just before it hits the ground.\nA: $x=\\frac{3}{4} L $ \nB: $x=\\frac{3}{5} L$\nC: $x=\\frac{1}{4} L$\nD: $x=\\frac{1}{3} L$\nE: $x=\\frac{2}{5} L$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA small ball of mass $3 m$ is at rest on the ground. A second small ball of mass $m$ is positioned above the ground by a vertical massless rod of length $L$ that is also attached to the ball on the ground. The original orientation of the rod is directly vertical, and the top ball is given a small horizontal nudge. There is no friction; assume that everything happens in a single plane.\n\n21. Determine the horizontal displacement $x$ of the second ball just before it hits the ground.\n\nA: $x=\\frac{3}{4} L $ \nB: $x=\\frac{3}{5} L$\nC: $x=\\frac{1}{4} L$\nD: $x=\\frac{1}{3} L$\nE: $x=\\frac{2}{5} L$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_950", "problem": "A filament lamp has a resistance which we can assume is proportional to its temperature in kelvin. A $50 \\mathrm{~W}$ bulb operates on $230 \\mathrm{~V}$ at a temperature of $2250 \\mathrm{~K}$. What is the resistance of the bulb at room temperature of $27^{\\circ} \\mathrm{C}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA filament lamp has a resistance which we can assume is proportional to its temperature in kelvin. A $50 \\mathrm{~W}$ bulb operates on $230 \\mathrm{~V}$ at a temperature of $2250 \\mathrm{~K}$. What is the resistance of the bulb at room temperature of $27^{\\circ} \\mathrm{C}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_709", "problem": "A metal rod of length $L$ is inclined on the rails in a uniform magnetic field of flux density $\\mathrm{B}$ as shown in the graph. If the rod moves with constant velocity v upward, what is the induced emf on the rod?\n\n[figure1]\nA: $\\frac{2 B L v}{\\sqrt{3}}$\nB: $\\frac{\\sqrt{3} B L v}{2}$\nC: $B L v$\nD: $\\frac{B L v}{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA metal rod of length $L$ is inclined on the rails in a uniform magnetic field of flux density $\\mathrm{B}$ as shown in the graph. If the rod moves with constant velocity v upward, what is the induced emf on the rod?\n\n[figure1]\n\nA: $\\frac{2 B L v}{\\sqrt{3}}$\nB: $\\frac{\\sqrt{3} B L v}{2}$\nC: $B L v$\nD: $\\frac{B L v}{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22e26a14ee6fdd9254b6g-03.jpg?height=415&width=328&top_left_y=362&top_left_x=1321" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_751", "problem": "Ice pellets \n\nAn interesting weather phenomenon can occur when the temperature profile in the atmosphere shows an inversion. The solid blue line in figure 1 shows such a temperature profile. The inversion occurs at heights between $1 \\mathrm{~km}$ and $2 \\mathrm{~km}$. Under these conditions snow falling through the atmosphere (partially) melts in the warmer layer and (partially) freezes again before reaching the ground in the form of \"ice pellets\".\n\n[figure1]\n\nFigura 1: Atmospheric temperature $T$ vs. height $h$ above the ground.\n\nAssume that a small, spherical ice droplet almost completely melts while falling through the atmospheric layer between $h_{A}$ and $h_{B}$ where the temperature is above freezing point.\n\nFind, as precisely as possible, the temperature of the droplet at ground level if there were no inversion and the temperature profile followed the dashed line below a height of $2 \\mathrm{~km}$.\n\nNeglect evaporation, condensation and size changes of the droplet. Assume that water and ice have very high thermal conductivity and that the density of the atmosphere is constant with height. Use $c_{\\text {water }}=4.2 \\mathrm{~kJ} \\mathrm{~kg}^{-1} \\mathrm{~K}^{-1}$\n\nfor the the specific heat of water and $c_{i c e}=2.1 \\mathrm{~kJ} \\mathrm{~kg}^{-1} \\mathrm{~K}^{-1}$ for that of ice. The specific latent heat for the melting of ice is $L=334 \\mathrm{~kJ} \\mathrm{~kg}^{-1}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question has more than one correct answer, you need to include them all.\n\nproblem:\nIce pellets \n\nAn interesting weather phenomenon can occur when the temperature profile in the atmosphere shows an inversion. The solid blue line in figure 1 shows such a temperature profile. The inversion occurs at heights between $1 \\mathrm{~km}$ and $2 \\mathrm{~km}$. Under these conditions snow falling through the atmosphere (partially) melts in the warmer layer and (partially) freezes again before reaching the ground in the form of \"ice pellets\".\n\n[figure1]\n\nFigura 1: Atmospheric temperature $T$ vs. height $h$ above the ground.\n\nAssume that a small, spherical ice droplet almost completely melts while falling through the atmospheric layer between $h_{A}$ and $h_{B}$ where the temperature is above freezing point.\n\nFind, as precisely as possible, the temperature of the droplet at ground level if there were no inversion and the temperature profile followed the dashed line below a height of $2 \\mathrm{~km}$.\n\nNeglect evaporation, condensation and size changes of the droplet. Assume that water and ice have very high thermal conductivity and that the density of the atmosphere is constant with height. Use $c_{\\text {water }}=4.2 \\mathrm{~kJ} \\mathrm{~kg}^{-1} \\mathrm{~K}^{-1}$\n\nfor the the specific heat of water and $c_{i c e}=2.1 \\mathrm{~kJ} \\mathrm{~kg}^{-1} \\mathrm{~K}^{-1}$ for that of ice. The specific latent heat for the melting of ice is $L=334 \\mathrm{~kJ} \\mathrm{~kg}^{-1}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nTheir units are, in order, [$^{\\circ} \\mathrm{C}$, $^{\\circ} \\mathrm{C}$], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_cfe12d29430fb70c15dbg-1.jpg?height=718&width=1005&top_left_y=595&top_left_x=560" ], "answer": null, "solution": null, "answer_type": "MA", "unit": [ "$^{\\circ} \\mathrm{C}$", "$^{\\circ} \\mathrm{C}$" ], "answer_sequence": null, "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_818", "problem": "An electric current $I$ (Figure 3) flows along a homogeneous conducting bar with length $L$, resistivity $\\rho$, thermal conductivity $k$. The two ends of the bar are located at coordinates $x=0$ and $x=L$ in the $O X$ axis. The temperature at $x=0$ is $T_{1}$, at $x=L$ is $T_{2}\\left(T_{1}>T_{2}\\right)$, both temperatures are kept constant.\n\n[figure1]\n\nFigure 3\n\nThe heat current $q(x)$ (the amount of heat transferred via perpendicular cross-section per unit time) flowing in the bar is described by the Fourier law\n\n$$\nq(x)=-k S \\frac{d T(x)}{d x}\n$$\n\nhere $k$ is thermal conductivity, and $S$ is the cross-sectional area of the bar.\n\n Find the temperature distribution $T(x)$ when $x$ varies along the bar at the steady state assuming no heat loss to the surroundings.\n\nHint: the equation $\\frac{d^{2} T(x)}{d x^{2}}=a$ has the solution $T(x)=\\frac{1}{2} a x^{2}+C_{1} x+C_{2}$, where $C_{1}$ and $C_{2}$ are derived from boundary conditions.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nAn electric current $I$ (Figure 3) flows along a homogeneous conducting bar with length $L$, resistivity $\\rho$, thermal conductivity $k$. The two ends of the bar are located at coordinates $x=0$ and $x=L$ in the $O X$ axis. The temperature at $x=0$ is $T_{1}$, at $x=L$ is $T_{2}\\left(T_{1}>T_{2}\\right)$, both temperatures are kept constant.\n\n[figure1]\n\nFigure 3\n\nThe heat current $q(x)$ (the amount of heat transferred via perpendicular cross-section per unit time) flowing in the bar is described by the Fourier law\n\n$$\nq(x)=-k S \\frac{d T(x)}{d x}\n$$\n\nhere $k$ is thermal conductivity, and $S$ is the cross-sectional area of the bar.\n\n Find the temperature distribution $T(x)$ when $x$ varies along the bar at the steady state assuming no heat loss to the surroundings.\n\nHint: the equation $\\frac{d^{2} T(x)}{d x^{2}}=a$ has the solution $T(x)=\\frac{1}{2} a x^{2}+C_{1} x+C_{2}$, where $C_{1}$ and $C_{2}$ are derived from boundary conditions.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_6e4e3efc8af7423242a0g-3.jpg?height=354&width=1311&top_left_y=925&top_left_x=361" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_265", "problem": "A student predicts that a steel ball dropped from rest from a height of $2 \\mathrm{~m}$ will hit the floor after 0.63 seconds. In their calculation they used an approximate value for the acceleration due to gravity of $\\mathrm{g}=10 \\mathrm{~m} / \\mathrm{s}^{2}$.\n\nThe accepted value for the acceleration due to gravity is $g=9.8 \\mathrm{~m} / \\mathrm{s}^{2}$.\n\nThis means their calculated time will be:\nA: unaffected\nB: too long by about $1 \\%$\nC: too short by about $1 \\%$\nD: too long by about 0.2 seconds\nE: too short by about 0.2 seconds\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA student predicts that a steel ball dropped from rest from a height of $2 \\mathrm{~m}$ will hit the floor after 0.63 seconds. In their calculation they used an approximate value for the acceleration due to gravity of $\\mathrm{g}=10 \\mathrm{~m} / \\mathrm{s}^{2}$.\n\nThe accepted value for the acceleration due to gravity is $g=9.8 \\mathrm{~m} / \\mathrm{s}^{2}$.\n\nThis means their calculated time will be:\n\nA: unaffected\nB: too long by about $1 \\%$\nC: too short by about $1 \\%$\nD: too long by about 0.2 seconds\nE: too short by about 0.2 seconds\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_959", "problem": "A spherical shaped party balloon can be filled by blowing air into it. We observe that it is difficult to start the balloon expanding, but it becomes easier once the rubber has stretched a little. It is easier to inflate the balloon a second time. This behaviour is illustrated by the graph of Fig. 9 and is described by the equation,\n\n$$\nP_{\\text {in }}-P_{\\text {out }}=\\frac{C}{r_{0}^{2} r}\\left[1-\\left(\\frac{r_{0}}{r}\\right)^{6}\\right]\n$$\n\nwhere $P_{\\text {in }}$ is the pressure inside the balloon, $P_{\\text {out }}$ is the external atmospheric pressure, $r_{0}$ is the uninflated radius of the balloon, $r$ is the radius of the balloon, and $C$ is a constant.\n\n\n By taking two regions on the graph, estimate the work done in blowing up the balloon to a radius of $6 \\mathrm{~cm}$. From the equation $\\mathrm{WD}=F \\Delta x$ we obtain $\\mathrm{WD}=P \\Delta V$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA spherical shaped party balloon can be filled by blowing air into it. We observe that it is difficult to start the balloon expanding, but it becomes easier once the rubber has stretched a little. It is easier to inflate the balloon a second time. This behaviour is illustrated by the graph of Fig. 9 and is described by the equation,\n\n$$\nP_{\\text {in }}-P_{\\text {out }}=\\frac{C}{r_{0}^{2} r}\\left[1-\\left(\\frac{r_{0}}{r}\\right)^{6}\\right]\n$$\n\nwhere $P_{\\text {in }}$ is the pressure inside the balloon, $P_{\\text {out }}$ is the external atmospheric pressure, $r_{0}$ is the uninflated radius of the balloon, $r$ is the radius of the balloon, and $C$ is a constant.\n\n\n By taking two regions on the graph, estimate the work done in blowing up the balloon to a radius of $6 \\mathrm{~cm}$. From the equation $\\mathrm{WD}=F \\Delta x$ we obtain $\\mathrm{WD}=P \\Delta V$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of J, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "J" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_406", "problem": "Consider a hypothetical interstellar travel with a photon-propelled spaceship of initial rest mass $M=1 \\times 10^{5} \\mathrm{~kg}$. The on-board fuel (antimatter) is annihilated with an equal mass of matter to create photons yielding a reactive force. The matter required for annihilation is collected from the very sparse plasma of the interstellar space (assume that the velocity of the interstellar plasma is zero in the Earth's frame of reference). The speed of light is $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$.\n\nThe frequency of the emitted photons is measured by an observer on Earth. What is the frequency of the last photons (emitted just before the engine is switched off), as measured on Earth, if the frequency in the the space ship frame remains constant and equal to $f_{0}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider a hypothetical interstellar travel with a photon-propelled spaceship of initial rest mass $M=1 \\times 10^{5} \\mathrm{~kg}$. The on-board fuel (antimatter) is annihilated with an equal mass of matter to create photons yielding a reactive force. The matter required for annihilation is collected from the very sparse plasma of the interstellar space (assume that the velocity of the interstellar plasma is zero in the Earth's frame of reference). The speed of light is $c=3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$.\n\nThe frequency of the emitted photons is measured by an observer on Earth. What is the frequency of the last photons (emitted just before the engine is switched off), as measured on Earth, if the frequency in the the space ship frame remains constant and equal to $f_{0}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1585", "problem": "如图所示, 一 $\\mathrm{U}$ 形光滑导轨串有一电阻 $R$, 放置在匀强的外磁场中, 导轨平面与磁场方向垂直. 一电阻可以忽略不计但有一定质量的金属杆 ab 跨接在导轨上, 可沿导轨方向平移. 现从静止开始对 $\\mathrm{ab}$ 杆施加向右的恒力 $F$, 若忽略杆和 U 形导轨的自感, 则在杆的运动过程中, 下列哪种说法是正确的?\n\n[图1]\nA: 外磁场对载流杆 $a b$ 作用力对 $a b$ 杆做功, 但外磁场的能量是不变的\nB: 外力 $\\mathrm{F}$ 的功总是等于电阻 $R$ 上消耗的功\nC: 外磁场对载流杆 $\\mathrm{ab}$ 作用力的功率与电阻 $R$ 上消耗的功率两者的大小是相等的\nD: 电阻 $R$ 上消耗的功率存在最大值\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 一 $\\mathrm{U}$ 形光滑导轨串有一电阻 $R$, 放置在匀强的外磁场中, 导轨平面与磁场方向垂直. 一电阻可以忽略不计但有一定质量的金属杆 ab 跨接在导轨上, 可沿导轨方向平移. 现从静止开始对 $\\mathrm{ab}$ 杆施加向右的恒力 $F$, 若忽略杆和 U 形导轨的自感, 则在杆的运动过程中, 下列哪种说法是正确的?\n\n[图1]\n\nA: 外磁场对载流杆 $a b$ 作用力对 $a b$ 杆做功, 但外磁场的能量是不变的\nB: 外力 $\\mathrm{F}$ 的功总是等于电阻 $R$ 上消耗的功\nC: 外磁场对载流杆 $\\mathrm{ab}$ 作用力的功率与电阻 $R$ 上消耗的功率两者的大小是相等的\nD: 电阻 $R$ 上消耗的功率存在最大值\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-01.jpg?height=297&width=408&top_left_y=1873&top_left_x=264" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_890", "problem": "The most outstanding fact in cosmology is that our universe is expanding. Space is continuously created as time lapses. The expansion of space indicates that, when the universe expands, the distance between objects in our universe also expands. It is convenient to use \"comoving\" coordinate system $\\vec{r}=(x, y, z)$ to label points in our expanding universe, in which the coordinate distance $\\Delta r=\\left|\\vec{r}_{2}-\\vec{r}_{1}\\right|=\\sqrt{\\left(x_{2}-x_{1}\\right)^{2}+\\left(y_{2}-y_{1}\\right)^{2}+\\left(z_{2}-z_{1}\\right)^{2}}$ between objects 1 and 2 does not change. (Here we assume no peculiar motion, i.e. no additional motion of those objects other than the motion following the expansion of the universe.) The situation is illustrated in the figure below (the figure has two space dimensions, but our universe actually has three space dimensions).\n\n[figure1]\n\nThe modern theory of cosmology is built upon Einstein's general relativity. However, under proper assumptions, a simplified understanding under the framework of Newton's theory of gravity is also possible. In the following questions, we shall work in the framework of Newton's gravity.\n\nTo measure the physical distance, a \"scale factor\" $a(t)$ is introduced such that the physical distance $\\Delta r_{\\mathrm{p}}$ between the comoving points $\\vec{r}_{1}$ and $\\vec{r}_{2}$ is \n$$\n\\Delta r_{\\mathrm{p}}=a(t) \\Delta r,\n$$\n\nThe expansion of the universe implies that $a(t)$ is an increasing function of time.\n\nOn large scales - scales much larger than galaxies and their clusters - our universe is approximately homogeneous and isotropic. So let us consider a toy model of our universe, which is filled with uniformly distributed particles. There are so many particles, such that we model them as a continuous fluid. Furthermore, we assume the number of particles is\nconserved.\n\nCurrently, our universe is dominated by non-relativistic matter, whose kinetic energy is negligible compared to its mass energy. Let $\\rho_{\\mathrm{m}}(t)$ be the physical energy density (i.e. energyper unit physical volume, which is dominated by mass energy for non-relativistic matter and the gravitational potential energy is not counted as part of the \"physical energy density\") of non-relativistic matter at time $t$. We use $t_{0}$ to denote the present time.\n\nBesides non-relativistic matter, there is also a small amount of radiation in our current universe, which is made of massless particles, for example, photons. The physical wavelength of massless particles increases with the universe expansion as $\\lambda_{\\mathrm{p}} \\propto a(t)$. Let the physical energy density of radiation be $\\rho_{\\mathrm{r}}(t)$.\n\nConsider a gas of non-interacting photons which has thermal equilibrium distribution. In this situation, the temperature of the photon depends on time as $T(t) \\propto[a(t)]^{\\gamma}$.\n\nCalculate the numerical value of $\\gamma$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe most outstanding fact in cosmology is that our universe is expanding. Space is continuously created as time lapses. The expansion of space indicates that, when the universe expands, the distance between objects in our universe also expands. It is convenient to use \"comoving\" coordinate system $\\vec{r}=(x, y, z)$ to label points in our expanding universe, in which the coordinate distance $\\Delta r=\\left|\\vec{r}_{2}-\\vec{r}_{1}\\right|=\\sqrt{\\left(x_{2}-x_{1}\\right)^{2}+\\left(y_{2}-y_{1}\\right)^{2}+\\left(z_{2}-z_{1}\\right)^{2}}$ between objects 1 and 2 does not change. (Here we assume no peculiar motion, i.e. no additional motion of those objects other than the motion following the expansion of the universe.) The situation is illustrated in the figure below (the figure has two space dimensions, but our universe actually has three space dimensions).\n\n[figure1]\n\nThe modern theory of cosmology is built upon Einstein's general relativity. However, under proper assumptions, a simplified understanding under the framework of Newton's theory of gravity is also possible. In the following questions, we shall work in the framework of Newton's gravity.\n\nTo measure the physical distance, a \"scale factor\" $a(t)$ is introduced such that the physical distance $\\Delta r_{\\mathrm{p}}$ between the comoving points $\\vec{r}_{1}$ and $\\vec{r}_{2}$ is \n$$\n\\Delta r_{\\mathrm{p}}=a(t) \\Delta r,\n$$\n\nThe expansion of the universe implies that $a(t)$ is an increasing function of time.\n\nOn large scales - scales much larger than galaxies and their clusters - our universe is approximately homogeneous and isotropic. So let us consider a toy model of our universe, which is filled with uniformly distributed particles. There are so many particles, such that we model them as a continuous fluid. Furthermore, we assume the number of particles is\nconserved.\n\nCurrently, our universe is dominated by non-relativistic matter, whose kinetic energy is negligible compared to its mass energy. Let $\\rho_{\\mathrm{m}}(t)$ be the physical energy density (i.e. energyper unit physical volume, which is dominated by mass energy for non-relativistic matter and the gravitational potential energy is not counted as part of the \"physical energy density\") of non-relativistic matter at time $t$. We use $t_{0}$ to denote the present time.\n\nBesides non-relativistic matter, there is also a small amount of radiation in our current universe, which is made of massless particles, for example, photons. The physical wavelength of massless particles increases with the universe expansion as $\\lambda_{\\mathrm{p}} \\propto a(t)$. Let the physical energy density of radiation be $\\rho_{\\mathrm{r}}(t)$.\n\nConsider a gas of non-interacting photons which has thermal equilibrium distribution. In this situation, the temperature of the photon depends on time as $T(t) \\propto[a(t)]^{\\gamma}$.\n\nCalculate the numerical value of $\\gamma$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_88fc0d3da9200bd5a832g-1.jpg?height=657&width=1455&top_left_y=1116&top_left_x=312" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1649", "problem": "真空中平行板电容器两极板的面积均为 $S$, 相距 $d$, 上、下极\n\n板所带电量分别为 $Q$ 和 $-Q(Q>0)$ 。现将一厚度为 $t$ 、面积为 $S / 2$\n\n（宽度和原来的极板相同, 长度是原来极板的一半）的金属片在上极板的正下方平行插入电容器, 将电容器分成如图所示的 $1 、 2 、 3$ 三部分。不考虑边缘效应。静电力常量为 $k$ 。试求\n\n\n[图1]插入金属片过程中外力所做的功。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n真空中平行板电容器两极板的面积均为 $S$, 相距 $d$, 上、下极\n\n板所带电量分别为 $Q$ 和 $-Q(Q>0)$ 。现将一厚度为 $t$ 、面积为 $S / 2$\n\n（宽度和原来的极板相同, 长度是原来极板的一半）的金属片在上极板的正下方平行插入电容器, 将电容器分成如图所示的 $1 、 2 、 3$ 三部分。不考虑边缘效应。静电力常量为 $k$ 。试求\n\n\n[图1]\n\n问题:\n插入金属片过程中外力所做的功。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://i.postimg.cc/c4YQHfdj/2016-CPho-Q13.png" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1716", "problem": "1914 年，弗兰克-赫兹用电子碰撞原子的方法使原子从低能级激发到高能级，从而证明了原子能级的存在。加速电子碰撞自由的氢原子, 使某氢原子从基态激发到激发态。该氢原子仅能发出一条可见光波长范围 ( 400nm 760nm ) 内的光谱线。仅考虑一维正碰。\n\n已知 $h c=1240 \\mathrm{~nm} \\cdot \\mathrm{eV}$, 其中 $h$ 为普朗克常量, $c$ 为真空中的光速; 质子质量近似为电子质量的 1836 倍, 氢原子在碰撞前的速度可忽略。求加速后电子动能 $E_{\\mathrm{k}}$ 的范围;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个区间。\n这里是一些可能会帮助你解决问题的先验信息提示：\n1914 年，弗兰克-赫兹用电子碰撞原子的方法使原子从低能级激发到高能级，从而证明了原子能级的存在。加速电子碰撞自由的氢原子, 使某氢原子从基态激发到激发态。该氢原子仅能发出一条可见光波长范围 ( 400nm 760nm ) 内的光谱线。仅考虑一维正碰。\n\n已知 $h c=1240 \\mathrm{~nm} \\cdot \\mathrm{eV}$, 其中 $h$ 为普朗克常量, $c$ 为真空中的光速; 质子质量近似为电子质量的 1836 倍, 氢原子在碰撞前的速度可忽略。\n\n问题:\n求加速后电子动能 $E_{\\mathrm{k}}$ 的范围;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{eV}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含任何单位的区间，例如ANSWER=(1,2] \\cup[7,+\\infty)", "figure_urls": null, "answer": null, "solution": null, "answer_type": "IN", "unit": [ "$\\mathrm{eV}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_119", "problem": "A trough half-filled with water is suspended from wires, as shown. The tension is initially the same in each wire.\n\n[figure1]\n\nA boat is placed in the trough directly under the left wire. It floats without touching the sides of the trough or overflowing the water. How does the tension in the wires change as a result?\nA: The tension is not affected in either wire.\nB: The tension increases equally in both wires. \nC: The tension increases in the left wire and decreases in the right wire.\nD: The tension increases in the left wire and stays the same in the right wire.\nE: The tension increases in the left wire and increases by a smaller amount in the right wire.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA trough half-filled with water is suspended from wires, as shown. The tension is initially the same in each wire.\n\n[figure1]\n\nA boat is placed in the trough directly under the left wire. It floats without touching the sides of the trough or overflowing the water. How does the tension in the wires change as a result?\n\nA: The tension is not affected in either wire.\nB: The tension increases equally in both wires. \nC: The tension increases in the left wire and decreases in the right wire.\nD: The tension increases in the left wire and stays the same in the right wire.\nE: The tension increases in the left wire and increases by a smaller amount in the right wire.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-05.jpg?height=325&width=618&top_left_y=1250&top_left_x=751" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1285", "problem": "从左至右在同一水平地面上依次有 3 个质点 $\\mathrm{a} 、 \\mathrm{~b} 、 \\mathrm{c}$, 且三者共线, $\\mathrm{a}$ 与 $\\mathrm{b}$ 相距 $l_{1}, \\mathrm{~b}$ 与 $\\mathrm{c}$ 相距 $l_{2}$ 。现同时将它们从其初始位置抛出。已知质点 $\\mathrm{b}$ 以初速度 $v_{0}$ 坚直上抛, 质点 $\\mathrm{c}$ 以某一初速度坚直上抛。设在这 3 个质点的运动过程中, $\\mathrm{a}$ 能碰到质点 $\\mathrm{b}$ 和 $\\mathrm{c}$; 并假设质点 $\\mathrm{a}$ 的质量远大于质点 $\\mathrm{b}$ 的质量, 且 $\\mathrm{a}$ 与 $\\mathrm{b}$ 碰撞时间极短。求质点 $\\mathrm{c}$ 的初速度 $v_{\\mathrm{c}}$ 和质点 $\\mathrm{a}$ 的初速度所满足的条件。所求的结果均用题中的已知量表示出来。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n从左至右在同一水平地面上依次有 3 个质点 $\\mathrm{a} 、 \\mathrm{~b} 、 \\mathrm{c}$, 且三者共线, $\\mathrm{a}$ 与 $\\mathrm{b}$ 相距 $l_{1}, \\mathrm{~b}$ 与 $\\mathrm{c}$ 相距 $l_{2}$ 。现同时将它们从其初始位置抛出。已知质点 $\\mathrm{b}$ 以初速度 $v_{0}$ 坚直上抛, 质点 $\\mathrm{c}$ 以某一初速度坚直上抛。设在这 3 个质点的运动过程中, $\\mathrm{a}$ 能碰到质点 $\\mathrm{b}$ 和 $\\mathrm{c}$; 并假设质点 $\\mathrm{a}$ 的质量远大于质点 $\\mathrm{b}$ 的质量, 且 $\\mathrm{a}$ 与 $\\mathrm{b}$ 碰撞时间极短。求质点 $\\mathrm{c}$ 的初速度 $v_{\\mathrm{c}}$ 和质点 $\\mathrm{a}$ 的初速度所满足的条件。所求的结果均用题中的已知量表示出来。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[质点 $\\mathrm{c}$ 的初速度 $v_{\\mathrm{c}}$, 质点 $\\mathrm{a}$ 的水平初速度 $v_{\\mathrm{a}}$, 质点 $\\mathrm{a}$ 的垂直初速度 $v_{\\mathrm{a}}$]\n它们的答案类型依次是[数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "质点 $\\mathrm{c}$ 的初速度 $v_{\\mathrm{c}}$", "质点 $\\mathrm{a}$ 的水平初速度 $v_{\\mathrm{a}}$", "质点 $\\mathrm{a}$ 的垂直初速度 $v_{\\mathrm{a}}$" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_262", "problem": "A battery and two identical bulbs are connected in parallel.\n\nThe current through each bulb is $2 \\mathrm{~A}$ and the potential difference across each bulb is $12 \\mathrm{~V}$.\n\nThe battery voltage and current through the battery are:\n\n| | Battery voltage / V | Current through
battery / A |\n| :--- | :---: | :---: |\n| A. | 6 | 2 |\n| B. | 6 | 4 |\n| C. | 12 | 2 |\n| D. | 12 | 4 |\n| E. | 24 | 2 |\n| F. | 24 | 4 |\n\n[figure1]\nA: A\nB: B\nC: C\nD: D\nE: E\nF: F\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA battery and two identical bulbs are connected in parallel.\n\nThe current through each bulb is $2 \\mathrm{~A}$ and the potential difference across each bulb is $12 \\mathrm{~V}$.\n\nThe battery voltage and current through the battery are:\n\n| | Battery voltage / V | Current through
battery / A |\n| :--- | :---: | :---: |\n| A. | 6 | 2 |\n| B. | 6 | 4 |\n| C. | 12 | 2 |\n| D. | 12 | 4 |\n| E. | 24 | 2 |\n| F. | 24 | 4 |\n\n[figure1]\n\nA: A\nB: B\nC: C\nD: D\nE: E\nF: F\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D, E, F].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_236da82643280c9e45c1g-04.jpg?height=899&width=417&top_left_y=1378&top_left_x=1408" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_222", "problem": "A particle of mass $m$ moving at speed $v_{0}$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$.\n If the collision is perfectly elastic, what is the maximum possible fractional momentum transfer, $f_{\\max }$ ?\nA: $0\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nFind the rate of change of the angular momentum about the $z$-axis.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nA Tippe top is a special kind of top that can spontaneously invert once it has been set spinning. One can model a Tippe top as a sphere of radius $R$ that is truncated, with a stem added. It has rotational symmetry about an axis through the stem, which is at angle $\\theta$ from the vertical. As shown in Figure 1(a), its centre of mass $C$ is offset from its geometric centre $O$ by $\\alpha R$ along its symmetry axis. The Tippe top makes contact with the surface it rests on at point $A$; we assume this surface is planar, and refer to it as the floor. Given certain geometrical constraints and if spun fast enough initially, the Tippe top will tip so that the stem points increasingly downwards, until it starts to spin on in its stem, and eventually comes to a stop.\n[figure1]\n\nFigure 1. Views of the Tippe top (a) from the side and (b) from above\n\nLet $x y z$ be the rotating reference frame defined such that $\\hat{\\mathbf{z}}$ is stationary and upwards, and the top's symmetry axis is within the $x z$-plane. Two views of the Tippe top are shown in Figure 1: from the side, and from above. As shown in Figure 1(b), the top's symmetry axis is aligned with the $x$-axis when viewed from above.\n\nFigure 2 shows the top's motion at several phases after it is started spinning:\n\n(a) phase I: immediately after it is initially set spinning, with $\\theta \\sim 0$\n\n(b) phase II: soon after, having tipped to angle $0<\\theta<\\frac{\\pi}{2}$\n\n(c) phase III: when the stem first touches the floor, with $\\theta>\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nFind the rate of change of the angular momentum about the $z$-axis.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-1.jpg?height=522&width=1332&top_left_y=1121&top_left_x=360", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=578&width=1778&top_left_y=316&top_left_x=184", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=417&width=545&top_left_y=1296&top_left_x=527", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=431&width=397&top_left_y=1298&top_left_x=1189" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_677", "problem": "A light beam with two monochromatic components $\\mathrm{X}$ and $\\mathrm{Y}$ passes into a glass block and splits into two monochromatic beams when it comes out of the glass block. If the frequency of component $\\mathrm{X}$ is smaller than that of $Y$, which of the following ray diagrams is correct?\n\na)\n\n[figure1]\n\nb)\n\n[figure2]\n\nc)\n\n[figure3]\n\nd)\n\n[figure4]\nA: A\nB: B\nC: C\nD: D\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA light beam with two monochromatic components $\\mathrm{X}$ and $\\mathrm{Y}$ passes into a glass block and splits into two monochromatic beams when it comes out of the glass block. If the frequency of component $\\mathrm{X}$ is smaller than that of $Y$, which of the following ray diagrams is correct?\n\na)\n\n[figure1]\n\nb)\n\n[figure2]\n\nc)\n\n[figure3]\n\nd)\n\n[figure4]\n\nA: A\nB: B\nC: C\nD: D\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_a0d0ab960474d1b0609cg-06.jpg?height=459&width=502&top_left_y=833&top_left_x=1142", "https://cdn.mathpix.com/cropped/2024_03_06_a0d0ab960474d1b0609cg-06.jpg?height=461&width=499&top_left_y=1420&top_left_x=1149", "https://cdn.mathpix.com/cropped/2024_03_06_a0d0ab960474d1b0609cg-06.jpg?height=485&width=487&top_left_y=2013&top_left_x=1147", "https://cdn.mathpix.com/cropped/2024_03_06_a0d0ab960474d1b0609cg-07.jpg?height=483&width=483&top_left_y=146&top_left_x=257" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_800", "problem": "Beatrice wants to plot a straight line graph and find the value of $k$ from the gradient of the graph. To do this, she should:\nA: Plot $T$ vs $m$, and then the gradient is $\\frac{1}{k}$.\nB: Plot $T$ vs $m$, and then the gradient is $\\frac{2 \\pi}{\\sqrt{k}}$.\nC: Plot $T^{2}$ vs $m$, and then the gradient is $\\frac{2 \\pi}{\\sqrt{k}}$.\nD: Plot $T^{2}$ vs $m$, and then the gradient is $\\frac{2 \\pi}{k}$.\nE: Plot $T^{2}$ vs $m$, and then the gradient is $\\frac{4 \\pi^{2}}{k}$.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nBeatrice wants to plot a straight line graph and find the value of $k$ from the gradient of the graph. To do this, she should:\n\nA: Plot $T$ vs $m$, and then the gradient is $\\frac{1}{k}$.\nB: Plot $T$ vs $m$, and then the gradient is $\\frac{2 \\pi}{\\sqrt{k}}$.\nC: Plot $T^{2}$ vs $m$, and then the gradient is $\\frac{2 \\pi}{\\sqrt{k}}$.\nD: Plot $T^{2}$ vs $m$, and then the gradient is $\\frac{2 \\pi}{k}$.\nE: Plot $T^{2}$ vs $m$, and then the gradient is $\\frac{4 \\pi^{2}}{k}$.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1502", "problem": "质量为 $m_{1}$ 的小滑块, 沿一倾角为 $\\theta$ 的光滑斜面滑下, 斜面质量为 $m_{2}$, 置于光滑的水平桌面上。设重力加速度为 $g$, 斜面在水平桌面上运动的加速度的大小为\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n质量为 $m_{1}$ 的小滑块, 沿一倾角为 $\\theta$ 的光滑斜面滑下, 斜面质量为 $m_{2}$, 置于光滑的水平桌面上。设重力加速度为 $g$, 斜面在水平桌面上运动的加速度的大小为\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_93e0584db50294961a50g-02.jpg?height=176&width=415&top_left_y=1537&top_left_x=1306" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1514", "problem": "在 “利用电流传感器（相当于理想电流表）测定干电池电动势和内阻” 的实验中, 某同学利用两个电流传感器和定值电阻 $R_{0}=2000 \\Omega$ 以及滑动变阻器, 设计了如图 a 所示的电路,进行实验。该同学测出的实验数据如下表所示\n\n| | 1 | 2 | 3 | 4 | 5 |\n| :---: | :---: | :---: | :---: | :---: | :---: |\n| $I_{1}(\\mathrm{~mA})$ | 1.35 | 1.30 | 1.20 | 1.10 | 1.05 |\n| $I_{2}(\\mathrm{~A})$ | 0.30 | 0.40 | 0.60 | 0.80 | 0.90 |\n\n表中 $I_{1}$ 和 $I_{2}$ 分别是通过电流传感器 1 和 2 的电流。该电流的值通过数据采集器输入到计算机, 数据采集器和计算机对原电路的影响可忽略。\n\n[图1]\n\n图a\n\n[图2]\n\n图 b由 $I_{1} \\sim I_{2}$ 图线得出, 被测电池的电动势为 $V$, 内阻为 $\\Omega$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n在 “利用电流传感器（相当于理想电流表）测定干电池电动势和内阻” 的实验中, 某同学利用两个电流传感器和定值电阻 $R_{0}=2000 \\Omega$ 以及滑动变阻器, 设计了如图 a 所示的电路,进行实验。该同学测出的实验数据如下表所示\n\n| | 1 | 2 | 3 | 4 | 5 |\n| :---: | :---: | :---: | :---: | :---: | :---: |\n| $I_{1}(\\mathrm{~mA})$ | 1.35 | 1.30 | 1.20 | 1.10 | 1.05 |\n| $I_{2}(\\mathrm{~A})$ | 0.30 | 0.40 | 0.60 | 0.80 | 0.90 |\n\n表中 $I_{1}$ 和 $I_{2}$ 分别是通过电流传感器 1 和 2 的电流。该电流的值通过数据采集器输入到计算机, 数据采集器和计算机对原电路的影响可忽略。\n\n[图1]\n\n图a\n\n[图2]\n\n图 b\n\n问题:\n由 $I_{1} \\sim I_{2}$ 图线得出, 被测电池的电动势为 $V$, 内阻为 $\\Omega$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[the value of $V$, the value of $\\Omega$]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-04.jpg?height=371&width=691&top_left_y=634&top_left_x=311", "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-04.jpg?height=451&width=557&top_left_y=591&top_left_x=1155" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "the value of $V$", "the value of $\\Omega$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_715", "problem": "A child sitting on an inflatable mattress in the swimming pool had a $20 \\mathrm{~kg}$ steel weight with him. When he dropped it to the bottom of the pool the water level in the pool:\n\n[figure1]\nA: Went down\nB: Went up\nC: Did not change\nD: Depends on the mass of the child and mattress\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA child sitting on an inflatable mattress in the swimming pool had a $20 \\mathrm{~kg}$ steel weight with him. When he dropped it to the bottom of the pool the water level in the pool:\n\n[figure1]\n\nA: Went down\nB: Went up\nC: Did not change\nD: Depends on the mass of the child and mattress\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22e26a14ee6fdd9254b6g-05.jpg?height=494&width=585&top_left_y=333&top_left_x=1212" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1367", "problem": "如图, 质量分别为 $m_{\\mathrm{a}} 、 m_{\\mathrm{b}}$ 的小球 $\\mathrm{a} 、 \\mathrm{~b}$ 放置在光滑绝缘\n\n水平面上, 两球之间用一原长为 $l_{0}$ 、劲度系数为 $k_{0}$ 的绝缘轻\n\n(b)-minn-a 弹簧连接。\n\n[图1]$t=0$ 时, 弹簧处于原长, 小球 $\\mathrm{a}$ 有一沿两球连线向右的初速度 $v_{0}$, 小球 $\\mathrm{b}$ 静止。若运动过程中弹簧始终处于弹性形变范围内, 求两球在任一时刻 $t(t>0)$ 的速度。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 质量分别为 $m_{\\mathrm{a}} 、 m_{\\mathrm{b}}$ 的小球 $\\mathrm{a} 、 \\mathrm{~b}$ 放置在光滑绝缘\n\n水平面上, 两球之间用一原长为 $l_{0}$ 、劲度系数为 $k_{0}$ 的绝缘轻\n\n(b)-minn-a 弹簧连接。\n\n[图1]\n\n问题:\n$t=0$ 时, 弹簧处于原长, 小球 $\\mathrm{a}$ 有一沿两球连线向右的初速度 $v_{0}$, 小球 $\\mathrm{b}$ 静止。若运动过程中弹簧始终处于弹性形变范围内, 求两球在任一时刻 $t(t>0)$ 的速度。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[a球在任一时刻 $t(t>0)$ 的速度, b球在任一时刻 $t(t>0)$ 的速度]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://i.postimg.cc/Ssd3bpmp/2017-CPho-Q1.png", "https://cdn.mathpix.com/cropped/2024_03_31_a47de6806e8da0a0f86dg-06.jpg?height=220&width=691&top_left_y=1295&top_left_x=1065" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "a球在任一时刻 $t(t>0)$ 的速度", "b球在任一时刻 $t(t>0)$ 的速度" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_651", "problem": "The number of field lines crossing a given area is given by the electric flux. For a closed surface (like a cube or sphere) the total electric flux is given by Gauss's law:\n\nElectric flux through a closed surface $=Q / \\epsilon_{0}$,\n\nwhere $Q$ is the net charge inside the closed surface. A point charge $Q=24 q$ is placed somewhere inside a cube. The electric flux through the bottom surface of the cube is measured to be $\\phi=\\frac{3.98 q}{\\varepsilon_{0}}$. What is the approximate location of the charge?\nA: Slightly off center.\nB: Slightly off center, towards the top surface.\nC: Slightly off center, towards the bottom surface.\nD: Far from the center, closer to the bottom surface.\nE: At the center of the cube.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe number of field lines crossing a given area is given by the electric flux. For a closed surface (like a cube or sphere) the total electric flux is given by Gauss's law:\n\nElectric flux through a closed surface $=Q / \\epsilon_{0}$,\n\nwhere $Q$ is the net charge inside the closed surface. A point charge $Q=24 q$ is placed somewhere inside a cube. The electric flux through the bottom surface of the cube is measured to be $\\phi=\\frac{3.98 q}{\\varepsilon_{0}}$. What is the approximate location of the charge?\n\nA: Slightly off center.\nB: Slightly off center, towards the top surface.\nC: Slightly off center, towards the bottom surface.\nD: Far from the center, closer to the bottom surface.\nE: At the center of the cube.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_264", "problem": "An electric kettle rated at $2.5 \\mathrm{~kW}$ and designed for use in the UK at a mains voltage of $230 \\mathrm{~V}$ takes just under 3 minutes to bring $1.25 \\mathrm{~L}$ of cold water to the boil when used in the UK.\n\nAssume the resistance of the heating element remains constant.\n\nApproximately how long would the same kettle take to boil the same quantity of cold water when used in America where the domestic mains voltage is only $110 \\mathrm{~V}$ ?\nA: $\\quad 1 \\frac{1}{2}$ minutes\nB: 3 minutes\nC: $\\quad 6 \\frac{1}{2}$ minutes\nD: 13 minutes\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nAn electric kettle rated at $2.5 \\mathrm{~kW}$ and designed for use in the UK at a mains voltage of $230 \\mathrm{~V}$ takes just under 3 minutes to bring $1.25 \\mathrm{~L}$ of cold water to the boil when used in the UK.\n\nAssume the resistance of the heating element remains constant.\n\nApproximately how long would the same kettle take to boil the same quantity of cold water when used in America where the domestic mains voltage is only $110 \\mathrm{~V}$ ?\n\nA: $\\quad 1 \\frac{1}{2}$ minutes\nB: 3 minutes\nC: $\\quad 6 \\frac{1}{2}$ minutes\nD: 13 minutes\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_502", "problem": "Scientists have recently detected a new star, the MAR-Kappa. The star is almost a perfect blackbody, and its measured light spectrum is shown below.\n\n[figure1]\n\nThe total measured light intensity from MAR-Kappa is $I=1.12 \\times 10^{-8} \\mathrm{~W} / \\mathrm{m}^{2}$. The mass of MAR-Kappa is estimated to be $3.5 \\times 10^{30} \\mathrm{~kg}$. It is stationary relative to the sun. You may find the Stefan-Boltzmann law useful, which states the power emitted by a blackbody with area $A$ is $\\sigma A T^{4}$.\n\nThe spectrum of wavelengths $\\lambda$ emitted from a blackbody only depends on $h, c, k_{B}, \\lambda$, and $T$. \n\nThe \"lines\" in the spectrum result from atoms in the star absorbing specific wavelengths of the emitted light. One contribution to the width of the spectral lines is the Doppler shift associated with the thermal motion of the atoms in the star. The spectral line at $\\lambda=389 \\mathrm{~nm}$ is due to helium. \n\nOver the course of a year, MAR-Kappa appears to oscillate between two positions in the background night sky, which are an angular distance of $1.6 \\times 10^{-6} \\mathrm{rad}$ apart. How far away is MAR-Kappa? Assume that MAR-Kappa lies in the same plane as the Earth's orbit, which is circular with radius $1.5 \\times 10^{11} \\mathrm{~m}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nScientists have recently detected a new star, the MAR-Kappa. The star is almost a perfect blackbody, and its measured light spectrum is shown below.\n\n[figure1]\n\nThe total measured light intensity from MAR-Kappa is $I=1.12 \\times 10^{-8} \\mathrm{~W} / \\mathrm{m}^{2}$. The mass of MAR-Kappa is estimated to be $3.5 \\times 10^{30} \\mathrm{~kg}$. It is stationary relative to the sun. You may find the Stefan-Boltzmann law useful, which states the power emitted by a blackbody with area $A$ is $\\sigma A T^{4}$.\n\nThe spectrum of wavelengths $\\lambda$ emitted from a blackbody only depends on $h, c, k_{B}, \\lambda$, and $T$. \n\nThe \"lines\" in the spectrum result from atoms in the star absorbing specific wavelengths of the emitted light. One contribution to the width of the spectral lines is the Doppler shift associated with the thermal motion of the atoms in the star. The spectral line at $\\lambda=389 \\mathrm{~nm}$ is due to helium. \n\nOver the course of a year, MAR-Kappa appears to oscillate between two positions in the background night sky, which are an angular distance of $1.6 \\times 10^{-6} \\mathrm{rad}$ apart. How far away is MAR-Kappa? Assume that MAR-Kappa lies in the same plane as the Earth's orbit, which is circular with radius $1.5 \\times 10^{11} \\mathrm{~m}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~m}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_fed953f9e38b72bf8bd7g-17.jpg?height=640&width=1051&top_left_y=507&top_left_x=537", "https://cdn.mathpix.com/cropped/2024_03_06_fed953f9e38b72bf8bd7g-18.jpg?height=152&width=1352&top_left_y=970&top_left_x=408" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1", "problem": "A $4.0 \\mathrm{~kg}$ mass at the end of a spring moves with simple harmonic motion on a horizontal frictionless table with a period of $2.0 \\mathrm{~s}$ and an amplitude of $2.0 \\mathrm{~m}$. Determine the maximum force exerted on the spring.\nA: $\\quad 25.1 \\mathrm{~N}$\nB: $158 \\mathrm{~N}$\nC: $39.5 \\mathrm{~N}$\nD: $63.0 \\mathrm{~N}$\nE: $79.0 \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA $4.0 \\mathrm{~kg}$ mass at the end of a spring moves with simple harmonic motion on a horizontal frictionless table with a period of $2.0 \\mathrm{~s}$ and an amplitude of $2.0 \\mathrm{~m}$. Determine the maximum force exerted on the spring.\n\nA: $\\quad 25.1 \\mathrm{~N}$\nB: $158 \\mathrm{~N}$\nC: $39.5 \\mathrm{~N}$\nD: $63.0 \\mathrm{~N}$\nE: $79.0 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1283", "problem": "具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：已知一个以角速度为 $\\omega$ 旋转的球形颗粒 (半径为 $r$ ) 在粘滞系数为 $\\eta$ 的液体中受到的粘滞摩擦力矩大小为 $M_{\\mathrm{f}}=8 \\pi \\eta r^{3} \\omega$ 。已知水的粘滞系数 $\\eta=8.00 \\times 10^{-4} \\mathrm{~Pa} \\cdot \\mathrm{s}$, 求金纳米球颗粒在此光束照射下达到稳定旋转时的转速 (转数/秒) $f$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：\n\n问题:\n已知一个以角速度为 $\\omega$ 旋转的球形颗粒 (半径为 $r$ ) 在粘滞系数为 $\\eta$ 的液体中受到的粘滞摩擦力矩大小为 $M_{\\mathrm{f}}=8 \\pi \\eta r^{3} \\omega$ 。已知水的粘滞系数 $\\eta=8.00 \\times 10^{-4} \\mathrm{~Pa} \\cdot \\mathrm{s}$, 求金纳米球颗粒在此光束照射下达到稳定旋转时的转速 (转数/秒) $f$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~s}^{-1}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-02.jpg?height=450&width=450&top_left_y=1883&top_left_x=1292" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~s}^{-1}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_572", "problem": "A particle of mass $m$ moves under a force similar to that of an ideal spring, except that the force repels the particle from the origin:\n\n$$\nF=+m \\alpha^{2} x\n$$\n\nIn simple harmonic motion, the position of the particle as a function of time can be written\n\n$$\nx(t)=A \\cos \\omega t+B \\sin \\omega t\n$$\n\nLikewise, in the present case we have\n\n$$\nx(t)=A f_{1}(t)+B f_{2}(t)\n$$\n\nfor some appropriate functions $f_{1}$ and $f_{2}$.\n\nA second, identical particle begins at position $x(0)=0$ with velocity $v(0)=v_{0}$. The second particle becomes closer and closer to the first particle as time goes on. What is $v_{0}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA particle of mass $m$ moves under a force similar to that of an ideal spring, except that the force repels the particle from the origin:\n\n$$\nF=+m \\alpha^{2} x\n$$\n\nIn simple harmonic motion, the position of the particle as a function of time can be written\n\n$$\nx(t)=A \\cos \\omega t+B \\sin \\omega t\n$$\n\nLikewise, in the present case we have\n\n$$\nx(t)=A f_{1}(t)+B f_{2}(t)\n$$\n\nfor some appropriate functions $f_{1}$ and $f_{2}$.\n\nA second, identical particle begins at position $x(0)=0$ with velocity $v(0)=v_{0}$. The second particle becomes closer and closer to the first particle as time goes on. What is $v_{0}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_662", "problem": "A radioactive substance decays, and an emitted particle passes through a uniform magnetic field pointing into page, as shown. Which labeling is correct?\n\n[figure1]\nA: $1=$ alpha; $2=$ beta; $3=$ gamma\nB: $1=$ alpha; $3=$ beta; $2=$ gamma\nC: 2=alpha; $1=$ beta; $3=$ gamma\nD: 2=alpha; 3=beta; 1=gamma\nE: $3=$ alpha; $1=$ beta; $2=$ gamma\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA radioactive substance decays, and an emitted particle passes through a uniform magnetic field pointing into page, as shown. Which labeling is correct?\n\n[figure1]\n\nA: $1=$ alpha; $2=$ beta; $3=$ gamma\nB: $1=$ alpha; $3=$ beta; $2=$ gamma\nC: 2=alpha; $1=$ beta; $3=$ gamma\nD: 2=alpha; 3=beta; 1=gamma\nE: $3=$ alpha; $1=$ beta; $2=$ gamma\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-06.jpg?height=534&width=572&top_left_y=1191&top_left_x=1210" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_407", "problem": "Tools: an empty 1-litre bottle, a small (about $100 \\mathrm{ml}$ ) cup of known volume (or other tool for measuring water volumes), a smartphone with installed \"Physics Toolbox Sensor Suite\" or \"Physics Toolbox Pro\" (mark on your paper, which version did you use).\n\nIf you blow near the bottle's mouth, a whistling sound can be generated: a gentle to moderately strong) air flow needs to pass the bottle's mouth perpendicularly to the bottle's axis. Your task is to study the de- pendence of the frequency $f$ of the generated sound as a function of the volume $V$ occupied by water inside the bottle.\n\nEither based on theoretical consideration or on the data analysis, suggest a functional dependence of $f$ on $V$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nTools: an empty 1-litre bottle, a small (about $100 \\mathrm{ml}$ ) cup of known volume (or other tool for measuring water volumes), a smartphone with installed \"Physics Toolbox Sensor Suite\" or \"Physics Toolbox Pro\" (mark on your paper, which version did you use).\n\nIf you blow near the bottle's mouth, a whistling sound can be generated: a gentle to moderately strong) air flow needs to pass the bottle's mouth perpendicularly to the bottle's axis. Your task is to study the de- pendence of the frequency $f$ of the generated sound as a function of the volume $V$ occupied by water inside the bottle.\n\nEither based on theoretical consideration or on the data analysis, suggest a functional dependence of $f$ on $V$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_952", "problem": "In the U-tube half filled with water of Fig. 3, the tube has a cross-sectional area $A$. The U-tube has a horizontal acceleration, $a$, to the right and in the plane of the U-tube. This will cause a height difference $h$ in the levels of the water.\n\n[figure1]\n\nFigure 3: U-tube containing water.\n\nThe arms of the tube are a distance $L$ apart, By considering the forces on a thin disc of water in the tube or otherwise, deduce an equation relating $h$ to $a, g$ and $L$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn the U-tube half filled with water of Fig. 3, the tube has a cross-sectional area $A$. The U-tube has a horizontal acceleration, $a$, to the right and in the plane of the U-tube. This will cause a height difference $h$ in the levels of the water.\n\n[figure1]\n\nFigure 3: U-tube containing water.\n\nThe arms of the tube are a distance $L$ apart, By considering the forces on a thin disc of water in the tube or otherwise, deduce an equation relating $h$ to $a, g$ and $L$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-04.jpg?height=528&width=551&top_left_y=844&top_left_x=1141", "https://cdn.mathpix.com/cropped/2024_03_06_cffb3a26a947b3b523f1g-3.jpg?height=360&width=483&top_left_y=1322&top_left_x=909" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1439", "problem": "如图, 某圆形薄片材料置于 $x O y$ 水平面上, 圆心位于坐标原点 $O ; x O y$ 平面上方存在大小为 $E$ 、沿 $z$ 轴负向的匀强电场, 以该圆形材料为底的圆柱体区域内存在大小为 $B$ 、沿 $z$ 轴正向的匀强磁场, 圆柱体区域外无磁场。从原点 $O$ 向 $x O y$ 平面上方的各方向均匀发射电荷量为 $q$ 、质量为 $m$ 、速度大小为 $v$ 的带正电荷的粒子。粒子所受重力的影响可忽略, 不考虑粒子间的相互作用。\n\n[图1]\n\n若在粒子每次与材料表面碰撞后的瞬间, 速度坚直分量反向, 水平分量方向不变, 坚直方向的速度大小和水平方向的速度大小均按同比例减小, 以至于动能减小 $10 \\%$ 。\n\n已知 $\\int d u \\sqrt{1+u^{2}}=\\frac{1}{2} u \\sqrt{1+u^{2}}+\\frac{1}{2} \\ln \\left(u+\\sqrt{1+u^{2}}\\right)+C, C$ 为积分常数。求在粒子射出直至它第一次与材料表面发生碰撞的过程中, 粒子在 $x O y$ 平面上的投影点走过路程的最大值;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 某圆形薄片材料置于 $x O y$ 水平面上, 圆心位于坐标原点 $O ; x O y$ 平面上方存在大小为 $E$ 、沿 $z$ 轴负向的匀强电场, 以该圆形材料为底的圆柱体区域内存在大小为 $B$ 、沿 $z$ 轴正向的匀强磁场, 圆柱体区域外无磁场。从原点 $O$ 向 $x O y$ 平面上方的各方向均匀发射电荷量为 $q$ 、质量为 $m$ 、速度大小为 $v$ 的带正电荷的粒子。粒子所受重力的影响可忽略, 不考虑粒子间的相互作用。\n\n[图1]\n\n若在粒子每次与材料表面碰撞后的瞬间, 速度坚直分量反向, 水平分量方向不变, 坚直方向的速度大小和水平方向的速度大小均按同比例减小, 以至于动能减小 $10 \\%$ 。\n\n已知 $\\int d u \\sqrt{1+u^{2}}=\\frac{1}{2} u \\sqrt{1+u^{2}}+\\frac{1}{2} \\ln \\left(u+\\sqrt{1+u^{2}}\\right)+C, C$ 为积分常数。\n\n问题:\n求在粒子射出直至它第一次与材料表面发生碰撞的过程中, 粒子在 $x O y$ 平面上的投影点走过路程的最大值;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_a47de6806e8da0a0f86dg-02.jpg?height=462&width=648&top_left_y=2459&top_left_x=1275" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1233", "problem": "Gaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\n## Manzanares prototype\n\nThe prototype chimney built in Manzanares, Spain, had a height of $195 \\mathrm{~m}$, and a radius $5 \\mathrm{~m}$. The collector is circular with diameter of $244 \\mathrm{~m}$. The specific heat of the air under typical operational conditions of the prototype solar chimney is $1012 \\mathrm{~J} / \\mathrm{kg} \\mathrm{K}$, the density of the hot air is about $0.9 \\mathrm{~kg} / \\mathrm{m}^{3}$, and the typical temperature of the atmosphere $T_{\\text {Air }}=295 \\mathrm{~K}$. In Manzanares, the solar power per unit of horizontal surface is typically $150 \\mathrm{~W} / \\mathrm{m}^{2}$ during a sunny day.\n\nHow large is the rise in the air temperature as it enters the chimney (warm air) from the surrounding (cold air)? Write the general formula and evaluate it for the prototype chimney.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nGaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\n## Manzanares prototype\n\nThe prototype chimney built in Manzanares, Spain, had a height of $195 \\mathrm{~m}$, and a radius $5 \\mathrm{~m}$. The collector is circular with diameter of $244 \\mathrm{~m}$. The specific heat of the air under typical operational conditions of the prototype solar chimney is $1012 \\mathrm{~J} / \\mathrm{kg} \\mathrm{K}$, the density of the hot air is about $0.9 \\mathrm{~kg} / \\mathrm{m}^{3}$, and the typical temperature of the atmosphere $T_{\\text {Air }}=295 \\mathrm{~K}$. In Manzanares, the solar power per unit of horizontal surface is typically $150 \\mathrm{~W} / \\mathrm{m}^{2}$ during a sunny day.\n\nHow large is the rise in the air temperature as it enters the chimney (warm air) from the surrounding (cold air)? Write the general formula and evaluate it for the prototype chimney.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of K, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_fd946cfac82ef740b1dag-1.jpg?height=977&width=1644&top_left_y=1453&top_left_x=206" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "K" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1021", "problem": "# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nThe bubble is initially formed with warmer air inside. Find the minimum numerical value of $T_{i}$ such that the bubble can float in still air. Use $T_{a}=300 \\mathrm{~K}, \\rho_{s}=1000 \\mathrm{kgm}^{-3}$, $\\rho_{a}=1.30 \\mathrm{kgm}^{-3}, t=100 \\mathrm{~nm}$ and $g=9.80 \\mathrm{~ms}^{-2}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\n# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nThe bubble is initially formed with warmer air inside. Find the minimum numerical value of $T_{i}$ such that the bubble can float in still air. Use $T_{a}=300 \\mathrm{~K}, \\rho_{s}=1000 \\mathrm{kgm}^{-3}$, $\\rho_{a}=1.30 \\mathrm{kgm}^{-3}, t=100 \\mathrm{~nm}$ and $g=9.80 \\mathrm{~ms}^{-2}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of K, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "K" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_16", "problem": "A force applied to a rocket gives it an upward acceleration equal to 2 times the acceleration of gravity. The magnitude of the force is equal to:\nA: One-half the weight of the rocket\nB: The weight of the rocket\nC: Two times the weight of the rocket\nD: Three times the weight of the rocket\nE: Four times the weight of the rocket\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA force applied to a rocket gives it an upward acceleration equal to 2 times the acceleration of gravity. The magnitude of the force is equal to:\n\nA: One-half the weight of the rocket\nB: The weight of the rocket\nC: Two times the weight of the rocket\nD: Three times the weight of the rocket\nE: Four times the weight of the rocket\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1295", "problem": "如图所示, $A$ 为放在水平光滑桌面上的长方形物块,在它上面放有物块 $B$ 和 $C, A 、 B 、 C$ 的质量分别为 $m 、 5 m 、 m$ 。\n$B 、 C$ 与 $A$ 之间的静摩擦系数和滑动摩擦系数皆为 0.10 。 $\\mathrm{K}$ 为轻滑轮, 绕过轻滑轮连接 $B$ 和 $C$ 的轻细绳都处于水平放置。现用沿水平方向的恒定外力 $F$ 拉滑轮, 使 $A$ 的加速度等于 $0.20 \\mathrm{~g}$, $g$ 为重力加速度。在这种情况时, $B 、 A$ 之间沿水平方向的作用力大小等于 ,$C 、 A$ 之间沿水平方向的作用力大小等于?\n\n外力 $F$ 的大小等于?\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n如图所示, $A$ 为放在水平光滑桌面上的长方形物块,在它上面放有物块 $B$ 和 $C, A 、 B 、 C$ 的质量分别为 $m 、 5 m 、 m$ 。\n$B 、 C$ 与 $A$ 之间的静摩擦系数和滑动摩擦系数皆为 0.10 。 $\\mathrm{K}$ 为轻滑轮, 绕过轻滑轮连接 $B$ 和 $C$ 的轻细绳都处于水平放置。现用沿水平方向的恒定外力 $F$ 拉滑轮, 使 $A$ 的加速度等于 $0.20 \\mathrm{~g}$, $g$ 为重力加速度。在这种情况时, $B 、 A$ 之间沿水平方向的作用力大小等于 ,$C 、 A$ 之间沿水平方向的作用力大小等于?\n\n外力 $F$ 的大小等于?\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$B 、 A$ 之间沿水平方向的作用力大小等于, $C 、 A$ 之间沿水平方向的作用力大小等于]\n它们的单位依次是[mg, mg]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_93e0584db50294961a50g-03.jpg?height=140&width=503&top_left_y=226&top_left_x=1202" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "mg", "mg" ], "answer_sequence": [ "$B 、 A$ 之间沿水平方向的作用力大小等于", "$C 、 A$ 之间沿水平方向的作用力大小等于" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_358", "problem": "A space station at a geostationary orbit has a form of a cylinder of length $L=100 \\mathrm{~km}$ and radius $R=1 \\mathrm{~km}$ is filled with air (molar mass $M=29 \\mathrm{~g} / \\mathrm{mol}$ ) at the atmospheric pressure and temperatur $T=295 \\mathrm{~K}$ and the cylindrical walls serve as ground for the people living inside. It rotates around its axis so as to create normal gravity $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$ at the \"ground\"\n\n\nA spherical balloon of radius $r=3 \\mathrm{~m}$ is filled with helium (molar mass $M^{\\prime}=$ $4 \\mathrm{~g} / \\mathrm{mol}$ ), and is used to lift a weight of unknown mass $m$. The weight is fixed to the ball with a light rope of length $L=20 \\mathrm{~m}$, and the system rises until coming to a stop at the height $H=500 \\mathrm{~m}$ from the \"ground\". Determ ine the value of the mass $m$.\n\nA rope of linear mass density $\\lambda=1 \\mathrm{~kg} / \\mathrm{m}$ is fixed to the \"ground\" at two diametrically opposite points of the cylinder (so that the distance between the endpoints of the rop is $2 R$ ). Let $A, B$, and $C$ denote the two end points and the middle point of the rope, re spectively.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA space station at a geostationary orbit has a form of a cylinder of length $L=100 \\mathrm{~km}$ and radius $R=1 \\mathrm{~km}$ is filled with air (molar mass $M=29 \\mathrm{~g} / \\mathrm{mol}$ ) at the atmospheric pressure and temperatur $T=295 \\mathrm{~K}$ and the cylindrical walls serve as ground for the people living inside. It rotates around its axis so as to create normal gravity $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$ at the \"ground\"\n\n\nA spherical balloon of radius $r=3 \\mathrm{~m}$ is filled with helium (molar mass $M^{\\prime}=$ $4 \\mathrm{~g} / \\mathrm{mol}$ ), and is used to lift a weight of unknown mass $m$. The weight is fixed to the ball with a light rope of length $L=20 \\mathrm{~m}$, and the system rises until coming to a stop at the height $H=500 \\mathrm{~m}$ from the \"ground\". Determ ine the value of the mass $m$.\n\nA rope of linear mass density $\\lambda=1 \\mathrm{~kg} / \\mathrm{m}$ is fixed to the \"ground\" at two diametrically opposite points of the cylinder (so that the distance between the endpoints of the rop is $2 R$ ). Let $A, B$, and $C$ denote the two end points and the middle point of the rope, re spectively.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~kg}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~kg}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_543", "problem": "Connect an ideal voltmeter between $A$ and $B$.\n\nAn ideal voltmeter has infinite resistance, so no current flows between A and B. By symmetry, the same current must flow down each leg, so the current in each leg is $I_{s} / 2$.\n\nAssume the potential at the bottom is zero. The potential at $\\mathrm{A}$ is the same as the junction to the left of A, so\n\n$$\nV_{A}=\\frac{I_{s}}{2} 2 R=I_{s} R\n$$\n\nThe potential at B is found the same way,\n\n$$\nV_{B}=\\frac{I_{s}}{2} 4 R=2 I_{s} R\n$$\n\nThe difference is\n\n$$\nV_{A}-V_{B}=-I_{s} R\n$$\n\nThe sign is not important for scoring purposes.\n\nConnect instead an ideal ammeter between $A$ and $B$.\n\nAn ideal ammeter has zero resistance, so we just need to find the current through the effective $6 R$ resistor that connects the two vertical branches. This current will flow to the left.\n\nBy symmetry, the current through each vertical resistance of $2 R$ must be the same, as well as the currents through each vertical resistance of $4 R$. This gives the system of equations\n\n$$\n\\begin{aligned}\nI_{s} & =I_{2}+I_{4} \\\\\nI_{2} & =I_{6}+I_{4} \\\\\nI_{4}(4 R) & =I_{2}(2 R)+I_{6}(6 R)\n\\end{aligned}\n$$\n\nCopyright (C)2015 American Association of Physics Teachers\n\nEliminating $I_{2}$ gives\n\n$$\n\\begin{aligned}\nI_{s} & =I_{6}+2 I_{4} \\\\\n4 I_{4} & =2\\left(I_{6}+I_{4}\\right)+6 I_{6}\n\\end{aligned}\n$$\n\nFinally, eliminating $I_{4}$ gives $I_{4}=4 I_{6}$ and\n\n$$\nI_{6}=\\frac{1}{9} I_{s}\n$$Consider the circuit shown below. $I_{s}$ is a constant current source, meaning that no matter what device is connected between points $\\mathrm{A}$ and $\\mathrm{B}$, the current provided by the constant current source is the same.\n\n[figure1]\n\nIt turns out that it is possible to replace the above circuit with a new circuit as follows:\n\n[figure2]\n\nFrom the point of view of any passive resistance that is connected between A and B the circuits are identical. You don't need to prove this statement, but you do need to find $I_{t}$ and $R_{t}$ in terms of any or all of $R$ and $I_{s}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nConnect an ideal voltmeter between $A$ and $B$.\n\nAn ideal voltmeter has infinite resistance, so no current flows between A and B. By symmetry, the same current must flow down each leg, so the current in each leg is $I_{s} / 2$.\n\nAssume the potential at the bottom is zero. The potential at $\\mathrm{A}$ is the same as the junction to the left of A, so\n\n$$\nV_{A}=\\frac{I_{s}}{2} 2 R=I_{s} R\n$$\n\nThe potential at B is found the same way,\n\n$$\nV_{B}=\\frac{I_{s}}{2} 4 R=2 I_{s} R\n$$\n\nThe difference is\n\n$$\nV_{A}-V_{B}=-I_{s} R\n$$\n\nThe sign is not important for scoring purposes.\n\nConnect instead an ideal ammeter between $A$ and $B$.\n\nAn ideal ammeter has zero resistance, so we just need to find the current through the effective $6 R$ resistor that connects the two vertical branches. This current will flow to the left.\n\nBy symmetry, the current through each vertical resistance of $2 R$ must be the same, as well as the currents through each vertical resistance of $4 R$. This gives the system of equations\n\n$$\n\\begin{aligned}\nI_{s} & =I_{2}+I_{4} \\\\\nI_{2} & =I_{6}+I_{4} \\\\\nI_{4}(4 R) & =I_{2}(2 R)+I_{6}(6 R)\n\\end{aligned}\n$$\n\nCopyright (C)2015 American Association of Physics Teachers\n\nEliminating $I_{2}$ gives\n\n$$\n\\begin{aligned}\nI_{s} & =I_{6}+2 I_{4} \\\\\n4 I_{4} & =2\\left(I_{6}+I_{4}\\right)+6 I_{6}\n\\end{aligned}\n$$\n\nFinally, eliminating $I_{4}$ gives $I_{4}=4 I_{6}$ and\n\n$$\nI_{6}=\\frac{1}{9} I_{s}\n$$\n\nproblem:\nConsider the circuit shown below. $I_{s}$ is a constant current source, meaning that no matter what device is connected between points $\\mathrm{A}$ and $\\mathrm{B}$, the current provided by the constant current source is the same.\n\n[figure1]\n\nIt turns out that it is possible to replace the above circuit with a new circuit as follows:\n\n[figure2]\n\nFrom the point of view of any passive resistance that is connected between A and B the circuits are identical. You don't need to prove this statement, but you do need to find $I_{t}$ and $R_{t}$ in terms of any or all of $R$ and $I_{s}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-05.jpg?height=347&width=838&top_left_y=474&top_left_x=619", "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-06.jpg?height=358&width=754&top_left_y=892&top_left_x=729" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_796", "problem": "Ali and Beatrice are doing an experiment to measure the spring constant, $k$, of a spring. The spring constant is a measure of how stiff or stretchy a spring is. Ali is going to use a static method, using the equation $k=\\frac{F_{\\text {applied }}}{s}=\\frac{m g}{s}$, where $m$ is the mass of a weight attached to the hanging spring, and $s$ is the distance that the spring stretches by.\n\nWhich method will give Ali the most accurate and precise result for $k$ ?\nA: Hang one weight on the spring, and measure how far the spring stretches. Put this pair of values into the equation.\nB: Hang two different weights on the spring (one at a time), and measure how far each weight stretches the spring. Put these two pairs of values into the equation and average the results.\nC: Hang at least three different weights on the spring (one at a time), and measure how far each weight stretches the spring. Put these pairs of values into the equation and average the results.\nD: Hang at least three different weights on the spring (one at a time), and measure how far each weight stretches the spring. Plot a graph of $m$ vs $s$ and find the value of $k$ from the gradient of a line of best fit for the data.\nE: Hang at least three different weights on the spring (one at a time), and measure how far each weight stretches the spring. Plot a graph of $m$ vs $s$ and find the value of $k$ from the gradient of a line of best fit to the data that also passes through the origin.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAli and Beatrice are doing an experiment to measure the spring constant, $k$, of a spring. The spring constant is a measure of how stiff or stretchy a spring is. Ali is going to use a static method, using the equation $k=\\frac{F_{\\text {applied }}}{s}=\\frac{m g}{s}$, where $m$ is the mass of a weight attached to the hanging spring, and $s$ is the distance that the spring stretches by.\n\nWhich method will give Ali the most accurate and precise result for $k$ ?\n\nA: Hang one weight on the spring, and measure how far the spring stretches. Put this pair of values into the equation.\nB: Hang two different weights on the spring (one at a time), and measure how far each weight stretches the spring. Put these two pairs of values into the equation and average the results.\nC: Hang at least three different weights on the spring (one at a time), and measure how far each weight stretches the spring. Put these pairs of values into the equation and average the results.\nD: Hang at least three different weights on the spring (one at a time), and measure how far each weight stretches the spring. Plot a graph of $m$ vs $s$ and find the value of $k$ from the gradient of a line of best fit for the data.\nE: Hang at least three different weights on the spring (one at a time), and measure how far each weight stretches the spring. Plot a graph of $m$ vs $s$ and find the value of $k$ from the gradient of a line of best fit to the data that also passes through the origin.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_643", "problem": "Consider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nAn estimate for the regime where quantum effects become important can be found by simply considering when the deBroglie wavelength of the particle is on the same order as the height of a bounce. Assuming that the deBroglie wavelength is defined by the maximum momentum of the bouncing particle, determine the value of the energy $E_{q}$ where quantum effects become important. Write your answer in terms of some or all of $g, m$, and Planck's constant $h$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nAn estimate for the regime where quantum effects become important can be found by simply considering when the deBroglie wavelength of the particle is on the same order as the height of a bounce. Assuming that the deBroglie wavelength is defined by the maximum momentum of the bouncing particle, determine the value of the energy $E_{q}$ where quantum effects become important. Write your answer in terms of some or all of $g, m$, and Planck's constant $h$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1610", "problem": "某电磁轨道炮的简化模型如图 a 所示, 两圆柱形固定导轨相互平行, 其对称轴所在平面与水平面的夹角为 $\\theta$, 两导轨的长均为 $\\mathrm{L}$ 、半径均为 $\\mathrm{b}$ 、每单位长度的电阻均为 $\\lambda$, 两导轨之间的最近距离为 $\\mathrm{d}(\\mathrm{d}$很小). 一弹丸质量为 $\\mathrm{m}$ ( $\\mathrm{m}$ 较小) 的金属弹丸 (可视为薄片) 置于两导轨之间, 弹丸直径为 $\\mathrm{d}$ 、电阻为 $\\mathrm{R}$, 与导轨保持良好接触. 两导轨下端横截面共面, 下端 (通过两根与相应导轨同轴的、较长的硬导线) 与一电流为 I 的理想恒流源 (恒流源内部的能量损耗可不计) 相连, 不考虑空气阻力和摩擦阻力, 重力加速度大小图 a. 某电磁轨道炮的简化模型为 $g$, 真空磁导率为 $\\mu_{0}$. 考虑一弹丸自导轨下端从静止开始被磁场加速直至射出的过程.\n\n[图1]\n\n图 a. 某电磁轨道炮的简化模型在 $\\theta=0^{\\circ}$ 的条件下, 若导轨和弹丸的电阻均可忽略, 求弹丸出射时的动能与理想恒流源所做的功之比.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某电磁轨道炮的简化模型如图 a 所示, 两圆柱形固定导轨相互平行, 其对称轴所在平面与水平面的夹角为 $\\theta$, 两导轨的长均为 $\\mathrm{L}$ 、半径均为 $\\mathrm{b}$ 、每单位长度的电阻均为 $\\lambda$, 两导轨之间的最近距离为 $\\mathrm{d}(\\mathrm{d}$很小). 一弹丸质量为 $\\mathrm{m}$ ( $\\mathrm{m}$ 较小) 的金属弹丸 (可视为薄片) 置于两导轨之间, 弹丸直径为 $\\mathrm{d}$ 、电阻为 $\\mathrm{R}$, 与导轨保持良好接触. 两导轨下端横截面共面, 下端 (通过两根与相应导轨同轴的、较长的硬导线) 与一电流为 I 的理想恒流源 (恒流源内部的能量损耗可不计) 相连, 不考虑空气阻力和摩擦阻力, 重力加速度大小图 a. 某电磁轨道炮的简化模型为 $g$, 真空磁导率为 $\\mu_{0}$. 考虑一弹丸自导轨下端从静止开始被磁场加速直至射出的过程.\n\n[图1]\n\n图 a. 某电磁轨道炮的简化模型\n\n问题:\n在 $\\theta=0^{\\circ}$ 的条件下, 若导轨和弹丸的电阻均可忽略, 求弹丸出射时的动能与理想恒流源所做的功之比.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以%为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-03.jpg?height=539&width=808&top_left_y=250&top_left_x=521" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "%" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1418", "problem": "2014 年 3 月 8 日凌晨 2 点 40 分, 马来西亚航空公司一架波音 777-200 飞机与管制中心失去联系。2014 年 3 月 24 日晚, 初步确定失事地点位于南纬 $31^{\\circ} 52^{\\prime}$ 、东经 $115^{\\circ} 52^{\\prime}$ 的澳大利亚西南城市珀斯附近的海域。有一颗绕地球做匀速圆周运动的卫星, 每天上午同一时刻在该区域正上方对海面拍照, 则\nA: 该卫星一定是地球同步卫星\nB: 该卫星轨道平面与南纬 $31^{\\circ} 52^{\\prime}$ 所确定的平面共面\nC: 该卫星运行周期一定是地球自转周期的整数倍\nD: 地球自转周期一定是该卫星运行周期的整数倍\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n2014 年 3 月 8 日凌晨 2 点 40 分, 马来西亚航空公司一架波音 777-200 飞机与管制中心失去联系。2014 年 3 月 24 日晚, 初步确定失事地点位于南纬 $31^{\\circ} 52^{\\prime}$ 、东经 $115^{\\circ} 52^{\\prime}$ 的澳大利亚西南城市珀斯附近的海域。有一颗绕地球做匀速圆周运动的卫星, 每天上午同一时刻在该区域正上方对海面拍照, 则\n\nA: 该卫星一定是地球同步卫星\nB: 该卫星轨道平面与南纬 $31^{\\circ} 52^{\\prime}$ 所确定的平面共面\nC: 该卫星运行周期一定是地球自转周期的整数倍\nD: 地球自转周期一定是该卫星运行周期的整数倍\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1327", "problem": "在固体材料中, 考虑相互作用后, 可以利用“准粒子”的概念研究材料的物理性质。准粒子的动能与动量之间的关系可能与真实粒子的不同。当外加电场或磁场时, 准粒子的运动往往可以用经典力学的方法来处理。在某种二维界面结构中, 存在电量为 $q$ 、有效质量为 $m$ 的准粒子, 它只能在 $x$ - $y$ 平面内运动, 其动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}+\\alpha p$,其中 $\\alpha$ 为正的常量。将该二维界面结构置于匀强磁场中, 磁场沿 $z$ 轴正方向, 磁感应强度大小为 $B$, 求动能为 $K$ 的准粒子做匀速率圆周运动的半径、周期和角动量的大小;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n在固体材料中, 考虑相互作用后, 可以利用“准粒子”的概念研究材料的物理性质。准粒子的动能与动量之间的关系可能与真实粒子的不同。当外加电场或磁场时, 准粒子的运动往往可以用经典力学的方法来处理。在某种二维界面结构中, 存在电量为 $q$ 、有效质量为 $m$ 的准粒子, 它只能在 $x$ - $y$ 平面内运动, 其动能 $K$ 与动量大小 $p$ 之间的关系可表示为 $K=\\frac{p^{2}}{2 m}+\\alpha p$,其中 $\\alpha$ 为正的常量。\n\n问题:\n将该二维界面结构置于匀强磁场中, 磁场沿 $z$ 轴正方向, 磁感应强度大小为 $B$, 求动能为 $K$ 的准粒子做匀速率圆周运动的半径、周期和角动量的大小;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[求动能为 $K$ 的准粒子做匀速率圆周运动的半径, 求动能为 $K$ 的准粒子做匀速率圆周运动的周期, 求动能为 $K$ 的准粒子做匀速率圆周运动的角动量]\n它们的答案类型依次是[表达式, 表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "求动能为 $K$ 的准粒子做匀速率圆周运动的半径", "求动能为 $K$ 的准粒子做匀速率圆周运动的周期", "求动能为 $K$ 的准粒子做匀速率圆周运动的角动量" ], "type_sequence": [ "EX", "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1485", "problem": "如图, 一劲度系数为 $k$ 的轻弹簧左端固定, 右端连一质量为 $m$ 的小球; 弹簧水平, 它处于自然状态时小球位于坐标原点 $O$; 小球可在水平地面上滑动, 它与地面之间的动摩擦因数为 $\\mu$ 。初始时小球速度为零, 将此时弹簧相对于其原长的伸长记为 $-A_{0}\\left(A_{0}>0\\right.$,但 $A_{0}$ 并不是已知量）。重力加速度大小为 $g$, 假设最大静摩擦力等于滑动摩擦力。\n\n[图1]如果小球完成第一次向右运动至原点右边后, 至多只能向左运动, 求小球最终静止的位置, 和此种情形下 $A_{0}$ 应满足的条件;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一劲度系数为 $k$ 的轻弹簧左端固定, 右端连一质量为 $m$ 的小球; 弹簧水平, 它处于自然状态时小球位于坐标原点 $O$; 小球可在水平地面上滑动, 它与地面之间的动摩擦因数为 $\\mu$ 。初始时小球速度为零, 将此时弹簧相对于其原长的伸长记为 $-A_{0}\\left(A_{0}>0\\right.$,但 $A_{0}$ 并不是已知量）。重力加速度大小为 $g$, 假设最大静摩擦力等于滑动摩擦力。\n\n[图1]\n\n问题:\n如果小球完成第一次向右运动至原点右边后, 至多只能向左运动, 求小球最终静止的位置, 和此种情形下 $A_{0}$ 应满足的条件;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[小球最终静止的位置, 此种情形下 $A_{0}$ 应满足的条件]\n它们的答案类型依次是[方程, 方程]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-01.jpg?height=137&width=511&top_left_y=1362&top_left_x=1229" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "小球最终静止的位置", "此种情形下 $A_{0}$ 应满足的条件" ], "type_sequence": [ "EQ", "EQ" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_438", "problem": "This question explores issues around a commonplace domestic electrical device.\n\nModern sets of decorative lights, based on LED technology, are popular, efficient and safe. However, their forerunners had a poor reputation on safety grounds.\n\nFigure shows the principles of a set of 'fairy lights' used to decorate an indoor tree in, say, the 1960s. It consists of 20 identical $12 \\mathrm{~V}$ incandescent light bulbs (now illegal on energy grounds!) connected in series.\n\n[figure1]\n\nFigure: Decorative tree lights circuit of the 1960s.\n\nDifficulties arise if a bulb fails (i.e. the filament breaks, effectively giving the bulb infinite resistance).\n\nClearly, a householder could substitute a new bulb for each one in turn until the faulty bulb has been discovered and replaced. This is very laborious so the householder decides to try use a voltmeter to discover the faulty bulb. (DO NOT actually do any such thing as the method adopted is a physics exercise to illustrate the dangers involved: probing electrical apparatus, especially when it is live, in this way can be FATAL)\n\nLet us assume bulb $\\boldsymbol{6}$ (between points $\\mathbf{X}$ and $\\mathbf{Y}$ ) is faulty.\n\nThe householder decides to set the switch to 'on'. What p.d. will they now measure between $\\mathbf{W}$ and $\\mathbf{X}$ ? And between $\\mathbf{W}$ and $\\mathbf{Y}$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nThis question explores issues around a commonplace domestic electrical device.\n\nModern sets of decorative lights, based on LED technology, are popular, efficient and safe. However, their forerunners had a poor reputation on safety grounds.\n\nFigure shows the principles of a set of 'fairy lights' used to decorate an indoor tree in, say, the 1960s. It consists of 20 identical $12 \\mathrm{~V}$ incandescent light bulbs (now illegal on energy grounds!) connected in series.\n\n[figure1]\n\nFigure: Decorative tree lights circuit of the 1960s.\n\nDifficulties arise if a bulb fails (i.e. the filament breaks, effectively giving the bulb infinite resistance).\n\nClearly, a householder could substitute a new bulb for each one in turn until the faulty bulb has been discovered and replaced. This is very laborious so the householder decides to try use a voltmeter to discover the faulty bulb. (DO NOT actually do any such thing as the method adopted is a physics exercise to illustrate the dangers involved: probing electrical apparatus, especially when it is live, in this way can be FATAL)\n\nLet us assume bulb $\\boldsymbol{6}$ (between points $\\mathbf{X}$ and $\\mathbf{Y}$ ) is faulty.\n\nThe householder decides to set the switch to 'on'. What p.d. will they now measure between $\\mathbf{W}$ and $\\mathbf{X}$ ? And between $\\mathbf{W}$ and $\\mathbf{Y}$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [p.d. between $W$ and $Y$, p.d. between $W$ and $X$].\nTheir units are, in order, [$V$, $V$], but units shouldn't be included in your concluded answer.\nTheir answer types are, in order, [numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-6.jpg?height=403&width=1328&top_left_y=672&top_left_x=361" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$V$", "$V$" ], "answer_sequence": [ "p.d. between $W$ and $Y$", "p.d. between $W$ and $X$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1378", "problem": "某电磁轨道炮的简化模型如图 a 所示, 两圆柱形固定导轨相互平行, 其对称轴所在平面与水平面的夹角为 $\\theta$, 两导轨的长均为 $\\mathrm{L}$ 、半径均为 $\\mathrm{b}$ 、每单位长度的电阻均为 $\\lambda$, 两导轨之间的最近距离为 $\\mathrm{d}(\\mathrm{d}$很小). 一弹丸质量为 $\\mathrm{m}$ ( $\\mathrm{m}$ 较小) 的金属弹丸 (可视为薄片) 置于两导轨之间, 弹丸直径为 $\\mathrm{d}$ 、电阻为 $\\mathrm{R}$, 与导轨保持良好接触. 两导轨下端横截面共面, 下端 (通过两根与相应导轨同轴的、较长的硬导线) 与一电流为 I 的理想恒流源 (恒流源内部的能量损耗可不计) 相连, 不考虑空气阻力和摩擦阻力, 重力加速度大小图 a. 某电磁轨道炮的简化模型为 $g$, 真空磁导率为 $\\mu_{0}$. 考虑一弹丸自导轨下端从静止开始被磁场加速直至射出的过程.\n\n[图1]\n\n图 a. 某电磁轨道炮的简化模型求在弹丸加速过程中任意时刻、以及弹丸出射时刻理想恒流源两端的电压;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某电磁轨道炮的简化模型如图 a 所示, 两圆柱形固定导轨相互平行, 其对称轴所在平面与水平面的夹角为 $\\theta$, 两导轨的长均为 $\\mathrm{L}$ 、半径均为 $\\mathrm{b}$ 、每单位长度的电阻均为 $\\lambda$, 两导轨之间的最近距离为 $\\mathrm{d}(\\mathrm{d}$很小). 一弹丸质量为 $\\mathrm{m}$ ( $\\mathrm{m}$ 较小) 的金属弹丸 (可视为薄片) 置于两导轨之间, 弹丸直径为 $\\mathrm{d}$ 、电阻为 $\\mathrm{R}$, 与导轨保持良好接触. 两导轨下端横截面共面, 下端 (通过两根与相应导轨同轴的、较长的硬导线) 与一电流为 I 的理想恒流源 (恒流源内部的能量损耗可不计) 相连, 不考虑空气阻力和摩擦阻力, 重力加速度大小图 a. 某电磁轨道炮的简化模型为 $g$, 真空磁导率为 $\\mu_{0}$. 考虑一弹丸自导轨下端从静止开始被磁场加速直至射出的过程.\n\n[图1]\n\n图 a. 某电磁轨道炮的简化模型\n\n问题:\n求在弹丸加速过程中任意时刻、以及弹丸出射时刻理想恒流源两端的电压;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-03.jpg?height=539&width=808&top_left_y=250&top_left_x=521" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_201", "problem": "The density of the Earth increases gradually from around $3 \\mathrm{~g} / \\mathrm{cm}^{3}$ at the crust to about $13 \\mathrm{~g} / \\mathrm{cm}^{3}$ at the core. Which one of these plots could show local gravitational acceleration as a function of distance from Earth's center?\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe density of the Earth increases gradually from around $3 \\mathrm{~g} / \\mathrm{cm}^{3}$ at the crust to about $13 \\mathrm{~g} / \\mathrm{cm}^{3}$ at the core. Which one of these plots could show local gravitational acceleration as a function of distance from Earth's center?\n\nA: [figure1]\nB: [figure2]\nC: [figure3]\nD: [figure4]\nE: [figure5]\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-04.jpg?height=391&width=377&top_left_y=409&top_left_x=321", "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-04.jpg?height=385&width=379&top_left_y=1003&top_left_x=320", "https://cdn.mathpix.com/cropped/2024_03_06_56b704dda4b4f71f3893g-03.jpg?height=412&width=379&top_left_y=2122&top_left_x=320", "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-04.jpg?height=396&width=385&top_left_y=404&top_left_x=1138", "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-04.jpg?height=382&width=377&top_left_y=1002&top_left_x=1140" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_862", "problem": "The moving image-monopole concept developed in A. 7 for $B_{z}^{\\prime}$ near $z \\approx 0$ can be assumed to hold also for the $\\vec{B}^{\\prime}$ field in the $z \\geq 0$ region. This assumption is good as long as the time evolution is sufficiently slow in the conducting thin film response.\n\n[figure1]\n\nFig. 2 A monopole $q_{\\mathrm{m}}$ moves with a constant velocity $\\vec{v}$ and a constant height $h$ from the conducting thin film. As shown are its coordinates at $t=0$.\n\nA monopole $q_{\\mathrm{m}}$ (Fig. 2) is caused to move in a constant velocity $v \\hat{x}$, with $v \\ll c$, and a constant height, at $z=h$, motion up to the present moment $(t=0)$. Its present coordinates $(x, y)$ are $(0,0)$. Our focus is on the magnetic potential $\\Phi_{+}$due to all image monopoles generated by this moving monopole along its trajectory.\n\nBy splitting $q_{m}$ 's trajectory into discrete time steps (a very small time step $\\tau$ ), we replace the motion of the $q_{m}$ by a hopping at the beginning moment of each time step. The hopping is represented by a simultaneous removal and creation of the monopoles. The position of the created monopole coincides with a point on its trajectory right at the beginning moment of this time step. Thus the position of the removed monopole coincides with its trajectory position at the beginning moment of the previous time step. This is achieved by a simultaneous sudden appearance of two magnetic monopoles: $q_{\\mathrm{m}}$ and $-q_{\\mathrm{m}}$ at, respectively, the trajectory positions corresponding to the beginning moments of this and the previous time step. The two positions are separated by a hopping distance $\\Delta x=v \\tau$. This time-step approach facilitates the determination of all the image magnetic monopoles, and their positions, that are generated in all the time steps.\n\nWrite down the present $(t=0)$ positions of all the image monopoles of the\n\n$0.8 \\mathrm{pt}$ types $q_{\\mathrm{m}}$ and $-q_{\\mathrm{m}}$. The beginning moments of the time steps are at $t=-n \\tau$, where $n \\geq 0$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question has more than one correct answer, you need to include them all.\n\nproblem:\nThe moving image-monopole concept developed in A. 7 for $B_{z}^{\\prime}$ near $z \\approx 0$ can be assumed to hold also for the $\\vec{B}^{\\prime}$ field in the $z \\geq 0$ region. This assumption is good as long as the time evolution is sufficiently slow in the conducting thin film response.\n\n[figure1]\n\nFig. 2 A monopole $q_{\\mathrm{m}}$ moves with a constant velocity $\\vec{v}$ and a constant height $h$ from the conducting thin film. As shown are its coordinates at $t=0$.\n\nA monopole $q_{\\mathrm{m}}$ (Fig. 2) is caused to move in a constant velocity $v \\hat{x}$, with $v \\ll c$, and a constant height, at $z=h$, motion up to the present moment $(t=0)$. Its present coordinates $(x, y)$ are $(0,0)$. Our focus is on the magnetic potential $\\Phi_{+}$due to all image monopoles generated by this moving monopole along its trajectory.\n\nBy splitting $q_{m}$ 's trajectory into discrete time steps (a very small time step $\\tau$ ), we replace the motion of the $q_{m}$ by a hopping at the beginning moment of each time step. The hopping is represented by a simultaneous removal and creation of the monopoles. The position of the created monopole coincides with a point on its trajectory right at the beginning moment of this time step. Thus the position of the removed monopole coincides with its trajectory position at the beginning moment of the previous time step. This is achieved by a simultaneous sudden appearance of two magnetic monopoles: $q_{\\mathrm{m}}$ and $-q_{\\mathrm{m}}$ at, respectively, the trajectory positions corresponding to the beginning moments of this and the previous time step. The two positions are separated by a hopping distance $\\Delta x=v \\tau$. This time-step approach facilitates the determination of all the image magnetic monopoles, and their positions, that are generated in all the time steps.\n\nWrite down the present $(t=0)$ positions of all the image monopoles of the\n\n$0.8 \\mathrm{pt}$ types $q_{\\mathrm{m}}$ and $-q_{\\mathrm{m}}$. The beginning moments of the time steps are at $t=-n \\tau$, where $n \\geq 0$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d32b3b2f89cebe6f1c2ag-3.jpg?height=323&width=763&top_left_y=2123&top_left_x=658" ], "answer": null, "solution": null, "answer_type": "MA", "unit": [ null, null ], "answer_sequence": null, "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_224", "problem": "Thermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nParticles in the exobase with enough outwards velocity will escape gravitational attraction.\n\nAssuming a Maxwellian distribution, determine the probability that a hydrogen atom has a velocity greater than the escape velocity in the exobase.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nParticles in the exobase with enough outwards velocity will escape gravitational attraction.\n\nAssuming a Maxwellian distribution, determine the probability that a hydrogen atom has a velocity greater than the escape velocity in the exobase.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_61a2ff399c33d9b3cd3bg-1.jpg?height=968&width=1044&top_left_y=1240&top_left_x=302" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1176", "problem": "Although atomic nuclei are quantum objects, a number of phenomenological laws for their basic properties (like radius or binding energy) can be deduced from simple assumptions: (i) nuclei are built from nucleons (i.e. protons and neutrons); (ii) strong nuclear interaction holding these nucleons together has a very short range (it acts only between neighboring nucleons); (iii) the number of protons $(Z)$ in a given nucleus is approximately equal to the number of neutrons $(N)$, i.e. $Z \\approx N \\approx A / 2$, where $A$ is the total number of nucleons $(A \\gg 1)$.\n\n## Transfer reactions\n\n[figure1]\n\nIn modern physics, the energetics of nuclei and their reactions is described in terms of masses. For example, if a nucleus (with zero velocity) is in an excited state with energy $E_{\\text {exc }}$ above the ground state, its mass is $m=m_{0}+E_{e x c} / c^{2}$, where $m_{0}$ is its mass in the ground state at rest. The nuclear reaction ${ }^{16} \\mathrm{O}+{ }^{54} \\mathrm{Fe} \\rightarrow{ }^{12} \\mathrm{C}+{ }^{58} \\mathrm{Ni}$ is an example of the so-called \"transfer reactions\", in which a part of one nucleus (\"cluster\") is transferred to the other (see Fig. 3). In our example the transferred part is a ${ }^{4} \\mathrm{He}$-cluster ( $\\alpha$-particle). The transfer reactions occur with maximum probability if the velocity of the projectile-like reaction product (in our case: ${ }^{12} \\mathrm{C}$ ) is equal both in magnitude and direction to the velocity of projectile (in our case: ${ }^{16} \\mathrm{O}$ ). The target ${ }^{54} \\mathrm{Fe}$ is initially at rest. In the reaction, ${ }^{58} \\mathrm{Ni}$ is excited into one of its higher-lying states. Find the excitation energy of that state (and express it units of $\\mathrm{MeV}$ ) if the kinetic energy of the projectile ${ }^{16} \\mathrm{O}$ is $50 \\mathrm{MeV}$. The speed of light is $\\mathrm{c}=3 \\cdot 10^{8} \\mathrm{~m} / \\mathrm{s}$.\n\n| 1. | $\\mathrm{M}\\left({ }^{16} \\mathrm{O}\\right)$ | 15.99491 a.m.u. |\n| ---: | ---: | :--- |\n| 2. | $\\mathrm{M}\\left({ }^{54} \\mathrm{Fe}\\right)$ | 53.93962 a.m.u. |\n| 3. | $\\mathrm{M}\\left({ }^{12} \\mathrm{C}\\right)$ | 12.00000 a.m.u. |\n| 4. | $\\mathrm{M}\\left({ }^{58} \\mathrm{Ni}\\right)$ | 57.93535 a.m.u. |\n| Table 1. The rest masses of the reactants in their ground states. 1 a.m.u. $=1.6605 \\cdot 10^{-27} \\mathrm{~kg}$ | | |", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nAlthough atomic nuclei are quantum objects, a number of phenomenological laws for their basic properties (like radius or binding energy) can be deduced from simple assumptions: (i) nuclei are built from nucleons (i.e. protons and neutrons); (ii) strong nuclear interaction holding these nucleons together has a very short range (it acts only between neighboring nucleons); (iii) the number of protons $(Z)$ in a given nucleus is approximately equal to the number of neutrons $(N)$, i.e. $Z \\approx N \\approx A / 2$, where $A$ is the total number of nucleons $(A \\gg 1)$.\n\n## Transfer reactions\n\n[figure1]\n\nIn modern physics, the energetics of nuclei and their reactions is described in terms of masses. For example, if a nucleus (with zero velocity) is in an excited state with energy $E_{\\text {exc }}$ above the ground state, its mass is $m=m_{0}+E_{e x c} / c^{2}$, where $m_{0}$ is its mass in the ground state at rest. The nuclear reaction ${ }^{16} \\mathrm{O}+{ }^{54} \\mathrm{Fe} \\rightarrow{ }^{12} \\mathrm{C}+{ }^{58} \\mathrm{Ni}$ is an example of the so-called \"transfer reactions\", in which a part of one nucleus (\"cluster\") is transferred to the other (see Fig. 3). In our example the transferred part is a ${ }^{4} \\mathrm{He}$-cluster ( $\\alpha$-particle). The transfer reactions occur with maximum probability if the velocity of the projectile-like reaction product (in our case: ${ }^{12} \\mathrm{C}$ ) is equal both in magnitude and direction to the velocity of projectile (in our case: ${ }^{16} \\mathrm{O}$ ). The target ${ }^{54} \\mathrm{Fe}$ is initially at rest. In the reaction, ${ }^{58} \\mathrm{Ni}$ is excited into one of its higher-lying states. Find the excitation energy of that state (and express it units of $\\mathrm{MeV}$ ) if the kinetic energy of the projectile ${ }^{16} \\mathrm{O}$ is $50 \\mathrm{MeV}$. The speed of light is $\\mathrm{c}=3 \\cdot 10^{8} \\mathrm{~m} / \\mathrm{s}$.\n\n| 1. | $\\mathrm{M}\\left({ }^{16} \\mathrm{O}\\right)$ | 15.99491 a.m.u. |\n| ---: | ---: | :--- |\n| 2. | $\\mathrm{M}\\left({ }^{54} \\mathrm{Fe}\\right)$ | 53.93962 a.m.u. |\n| 3. | $\\mathrm{M}\\left({ }^{12} \\mathrm{C}\\right)$ | 12.00000 a.m.u. |\n| 4. | $\\mathrm{M}\\left({ }^{58} \\mathrm{Ni}\\right)$ | 57.93535 a.m.u. |\n| Table 1. The rest masses of the reactants in their ground states. 1 a.m.u. $=1.6605 \\cdot 10^{-27} \\mathrm{~kg}$ | | |\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of MeV, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_656c8e5e21a256bf8ee1g-3.jpg?height=716&width=1450&top_left_y=1815&top_left_x=306" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "MeV" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1647", "problem": "如图, 一小角度单摆的轻质摆杆的长度 $\\mathrm{AB}=L$, 地球半经 $\\mathrm{OC}=R$,单摆的悬点到地面的距离 $\\mathrm{AC}=L$ 。已知地球质量为 $M$, 引力常量为 $G$ 。当 $L<>R$ 时，单摆做简谐运动的周期为___。悬点相对于地球不动, 不考虑地球自转。\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n如图, 一小角度单摆的轻质摆杆的长度 $\\mathrm{AB}=L$, 地球半经 $\\mathrm{OC}=R$,单摆的悬点到地面的距离 $\\mathrm{AC}=L$ 。已知地球质量为 $M$, 引力常量为 $G$ 。当 $L<>R$ 时，单摆做简谐运动的周期为___。悬点相对于地球不动, 不考虑地球自转。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[当 $L<>R$ 时，单摆做简谐运动的周期]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_17b1131fe8d911867aa0g-03.jpg?height=574&width=217&top_left_y=741&top_left_x=1545" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "当 $L<>R$ 时，单摆做简谐运动的周期" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_312", "problem": "Consider a spacecraft shaped as a homogeneous pipe, which is closed at both ends. The spacecraft is rotating around its center of mass with angular velocity $\\omega$, around an axis perpendicular to the pipe, in order to simulate gravity. The spacecraft is filled with air of molar mass $\\mu$, which has pressure $p_{0}$ at the rota- tion axis. The diameter of the spacecraft is much smaller than its length. Added during the competition: the temperature is $T$.\n\nCalculate the pressure $p$ as a function of the distance $r$ from the rotation axis.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nConsider a spacecraft shaped as a homogeneous pipe, which is closed at both ends. The spacecraft is rotating around its center of mass with angular velocity $\\omega$, around an axis perpendicular to the pipe, in order to simulate gravity. The spacecraft is filled with air of molar mass $\\mu$, which has pressure $p_{0}$ at the rota- tion axis. The diameter of the spacecraft is much smaller than its length. Added during the competition: the temperature is $T$.\n\nCalculate the pressure $p$ as a function of the distance $r$ from the rotation axis.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1211", "problem": "Let us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nEventually we cannot neglect the effect of the pressure on the dynamics of the gas and the collapse stops at $r=r_{4}$ (with $r_{4} \\ll r_{3}$ ). However, the radiation loss can still be neglected and the temperature is not yet high enough to ignite nuclear fusion. The pressure of such a protostar is not uniform anymore, but rough estimates with inaccurate numerical prefactors can still be done. Estimate the final radius $r_{4}$ and the respective temperature $T_{4}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nLet us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nEventually we cannot neglect the effect of the pressure on the dynamics of the gas and the collapse stops at $r=r_{4}$ (with $r_{4} \\ll r_{3}$ ). However, the radiation loss can still be neglected and the temperature is not yet high enough to ignite nuclear fusion. The pressure of such a protostar is not uniform anymore, but rough estimates with inaccurate numerical prefactors can still be done. Estimate the final radius $r_{4}$ and the respective temperature $T_{4}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [$r_{4}$, $T_{4}$].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$r_{4}$", "$T_{4}$" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_241", "problem": "Thermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nExobase is defined as the height above which a radially outward moving particle will suffer less than one backscattering collision on average. This means that the mean free path has to be equal to the scale height, which is defined as the height where the atmospheres density is $\\frac{1}{e}$ lower than on Earths surface $\\left(R_{E}=6.37 \\times 10^{6} \\mathrm{~m}\\right)$. The mean free path $\\lambda$ is the average distance covered by a moving particle in a gas (that we consider to be ideal) between two consecutive collisions and this can be expressed by the following equality:\n\n$$\n\\lambda(h)=\\frac{1}{\\sigma n_{V}(h)}\n$$\n\nwhere $\\sigma$ is the effective cross sectional area for the collision hydrogen atom-atmosphere $\\sigma=2 \\times$ $10^{-19} m^{2}$ and $n_{V}$ is the number of molecules per unit volume. Atmospheres density decreases with exponentially from altitude $250 \\mathrm{~km}$ :\n\n$$\nP(h)=P_{R e f} \\exp \\left(-\\frac{\\left(h-h_{R e f}\\right)}{H}\\right),\n$$\n\nwhere we know that at an altitude of $250 \\mathrm{~km}$, the pressure is $21 \\mu \\mathrm{Pa} . H$ is the scale height, and its value is $H=60 \\mathrm{~km}$.\n\nDetermine the exobase height $h_{E B}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThermal atmospheric escape is a process in which small gas molecules reach speeds high enough to escape the gravitational field of the Earth and reach outer space. Particles in a gas collide with each other due to their thermal energy. These collisions constantly accelerate and decelerate particles, varying their speed continuously. Paticles might be accelerated until they reach the socalled escape velocity and leave the atmosphere. This process, known as Jeans escape, is critical for the formation and maintenance or the evaporation of the atmosphere of a planet. It is believed that it played an important role in the loss of water from Venus and Mars atmospheres, due to their lower escape velocity.\n\nIn this problem we will quantify the scattering (collisions between gas particles) and rate of loss of hydrogen in Earths atmosphere. The modulus and direction velocity distribution of the molecules of a gas of mass $m$ and at a temperature $T$ is given by the Maxwellian distribution:\n\n$$\nf(v) d^{3} v=\\left(\\frac{m}{2 \\pi k T}\\right)^{\\frac{3}{2}} \\exp \\left(-\\frac{m v^{2}}{2 k T}\\right) v^{2} \\sin (\\theta) d v d \\theta d \\varphi\n$$\n\nwhere $d^{3} v$ is the velocity differential. In spherical coordinates it is expressed as $d^{3} v=v^{2} \\sin (\\theta) d v d \\theta d \\varphi$.\n\n[figure1]\n\nFigure 1: Schematic illustration of the spherical coordinate\n\nThus at any temperature there can always be some molecules whose velocity is greater than the escape velocity. A molecule located in the lower part of the atmosphere would not, in general, be able to escape to outer space even though its velocity is greater than the limit velocity because it would soon collide with other molecules, losing a big part of its energy. In order to escape, these molecules need to be at a certain height such that density is so low that their probability of colliding is negligible. The region in the atmosphere where this condition is satisfied is called exosphere and its lower boundary, which separates the dense zone from the exosphere, is called exobase. The temperature at the exobase is roughly $1000 \\mathrm{~K}$.\n\nExobase is defined as the height above which a radially outward moving particle will suffer less than one backscattering collision on average. This means that the mean free path has to be equal to the scale height, which is defined as the height where the atmospheres density is $\\frac{1}{e}$ lower than on Earths surface $\\left(R_{E}=6.37 \\times 10^{6} \\mathrm{~m}\\right)$. The mean free path $\\lambda$ is the average distance covered by a moving particle in a gas (that we consider to be ideal) between two consecutive collisions and this can be expressed by the following equality:\n\n$$\n\\lambda(h)=\\frac{1}{\\sigma n_{V}(h)}\n$$\n\nwhere $\\sigma$ is the effective cross sectional area for the collision hydrogen atom-atmosphere $\\sigma=2 \\times$ $10^{-19} m^{2}$ and $n_{V}$ is the number of molecules per unit volume. Atmospheres density decreases with exponentially from altitude $250 \\mathrm{~km}$ :\n\n$$\nP(h)=P_{R e f} \\exp \\left(-\\frac{\\left(h-h_{R e f}\\right)}{H}\\right),\n$$\n\nwhere we know that at an altitude of $250 \\mathrm{~km}$, the pressure is $21 \\mu \\mathrm{Pa} . H$ is the scale height, and its value is $H=60 \\mathrm{~km}$.\n\nDetermine the exobase height $h_{E B}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of km, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_61a2ff399c33d9b3cd3bg-1.jpg?height=968&width=1044&top_left_y=1240&top_left_x=302" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "km" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1407", "problem": "图示两条虚线之间为一光学元件所在处, $A B$ 为其主光轴, $P$ 是一点光源, 其傍轴光线通过此光学元件成像于 $Q$ 点。该光学元件可能是 [ ]\n\n[图1]\nA: 薄凸透镜\nB: 薄凹透镜\nC: 凸球面镜\nD: 凹球面镜\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n图示两条虚线之间为一光学元件所在处, $A B$ 为其主光轴, $P$ 是一点光源, 其傍轴光线通过此光学元件成像于 $Q$ 点。该光学元件可能是 [ ]\n\n[图1]\n\nA: 薄凸透镜\nB: 薄凹透镜\nC: 凸球面镜\nD: 凹球面镜\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_93e0584db50294961a50g-02.jpg?height=218&width=600&top_left_y=706&top_left_x=1060" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_875", "problem": "A Tippe top is a special kind of top that can spontaneously invert once it has been set spinning. One can model a Tippe top as a sphere of radius $R$ that is truncated, with a stem added. It has rotational symmetry about an axis through the stem, which is at angle $\\theta$ from the vertical. As shown in Figure 1(a), its centre of mass $C$ is offset from its geometric centre $O$ by $\\alpha R$ along its symmetry axis. The Tippe top makes contact with the surface it rests on at point $A$; we assume this surface is planar, and refer to it as the floor. Given certain geometrical constraints and if spun fast enough initially, the Tippe top will tip so that the stem points increasingly downwards, until it starts to spin on in its stem, and eventually comes to a stop.\n[figure1]\n\nFigure 1. Views of the Tippe top (a) from the side and (b) from above\n\nLet $x y z$ be the rotating reference frame defined such that $\\hat{\\mathbf{z}}$ is stationary and upwards, and the top's symmetry axis is within the $x z$-plane. Two views of the Tippe top are shown in Figure 1: from the side, and from above. As shown in Figure 1(b), the top's symmetry axis is aligned with the $x$-axis when viewed from above.\n\nFigure 2 shows the top's motion at several phases after it is started spinning:\n\n(a) phase I: immediately after it is initially set spinning, with $\\theta \\sim 0$\n\n(b) phase II: soon after, having tipped to angle $0<\\theta<\\frac{\\pi}{2}$\n\n(c) phase III: when the stem first touches the floor, with $\\theta>\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nFind the total energy of a spinning Tippe top, in terms of time derivatives of the Euler angles, $v_{x}$, and $v_{y}$. For partial marks, you may leave your answer in terms of $\\dot{\\mathbf{s}}=\\frac{d \\mathbf{s}}{d t}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA Tippe top is a special kind of top that can spontaneously invert once it has been set spinning. One can model a Tippe top as a sphere of radius $R$ that is truncated, with a stem added. It has rotational symmetry about an axis through the stem, which is at angle $\\theta$ from the vertical. As shown in Figure 1(a), its centre of mass $C$ is offset from its geometric centre $O$ by $\\alpha R$ along its symmetry axis. The Tippe top makes contact with the surface it rests on at point $A$; we assume this surface is planar, and refer to it as the floor. Given certain geometrical constraints and if spun fast enough initially, the Tippe top will tip so that the stem points increasingly downwards, until it starts to spin on in its stem, and eventually comes to a stop.\n[figure1]\n\nFigure 1. Views of the Tippe top (a) from the side and (b) from above\n\nLet $x y z$ be the rotating reference frame defined such that $\\hat{\\mathbf{z}}$ is stationary and upwards, and the top's symmetry axis is within the $x z$-plane. Two views of the Tippe top are shown in Figure 1: from the side, and from above. As shown in Figure 1(b), the top's symmetry axis is aligned with the $x$-axis when viewed from above.\n\nFigure 2 shows the top's motion at several phases after it is started spinning:\n\n(a) phase I: immediately after it is initially set spinning, with $\\theta \\sim 0$\n\n(b) phase II: soon after, having tipped to angle $0<\\theta<\\frac{\\pi}{2}$\n\n(c) phase III: when the stem first touches the floor, with $\\theta>\\frac{\\pi}{2}$\n\n(d) phase IV: after inversion, when the top is spinning on its stem, with $\\theta \\sim \\pi$\n\n(e) phase V: in its final state, at rest on its stem $\\theta=\\pi$.\n\nTheory\n[figure2]\n\nFigure 2. Phases I to $\\mathbf{V}$ of the Tippe top's motion, shown in the $x z$-plane\n\nLet $X Y Z$ be the inertial frame, where the surface the top is on is wholly in the $X Y$-plane. The frame $x y z$ is defined as above, and reached from $X Y Z$ via rotation around the $Z$ axis by $\\phi$. The transformation from the $X Y Z$ frame to frame $x y z$ is shown in Figure 3(a). In particular, $\\hat{\\mathbf{z}}=\\hat{\\mathbf{Z}}$.\n\n(a)\n\n[figure3]\n\n(b)\n\n[figure4]\n\nFigure 3. Transformations between reference frames: (a) to $x y z$ from $X Y Z$, and (b) to 123 from $x y z$\n\nAny rotational motion in 3-dimensional space can be described by the three Euler angles $(\\theta, \\phi, \\psi)$. The transformations between the inertial frame $X Y Z$, the intermediate frame $x y z$, and the top's frame 123 can be understood in terms of these Euler angles.\n\nIn our description of the Tippe top's motion, the angles $\\theta$ and $\\phi$ are the standard zenith and azimuthal angles respectively, in spherical polar coordinates. In the $X Y Z$ frame they are defined as follows: $\\theta$ is the angle of the top's symmetry axis from the vertical $Z$-axis, representing how far from vertical its stem is, while $\\phi$ represents the top's angular position about the $Z$-axis, and is defined as the angle between the $X Z$-plane and the plane through points $O, A, C$ (i.e. the vertical projection of the top's symmetry axis).\n\nThe third Euler angle $\\psi$ describes the rotation of the top about its own symmetry axis, i.e. its 'spin', which has angular velocity $\\dot{\\psi}$.\n\nThe reference frame of the spinning top is defined as a new rotating frame 123, which is reached by rotating $x y z$ by $\\theta$ around $\\hat{\\mathbf{y}}$ : 'tilting' the $\\hat{\\mathbf{z}}$-axis down by $\\theta$ to meet the top's symmetry axis $\\hat{\\mathbf{3}}$. The transformation from the $x y z$ frame to the 123 frame is shown in Figure 3(b). In particular, $\\hat{\\mathbf{2}}=\\hat{\\mathbf{y}}$.\n\nNOTE: For a reference frame $\\widetilde{\\mathbf{K}}$ rotating in inertial frame $\\mathbf{K}$ with angular velocity $\\omega$, the time derivatives of a vector $\\mathbf{A}$ within both frames $\\mathbf{K}$ and $\\widetilde{\\mathbf{K}}$ are related via:\n\n$$\n\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\mathbf{K}}=\\left(\\frac{\\partial \\mathbf{A}}{\\partial t}\\right)_{\\widetilde{\\mathbf{K}}}+\\boldsymbol{\\omega} \\times \\mathbf{A}\n$$\n\nThe motion that a Tippe top undergoes is complex, involving the time evolution of the three Euler angles, as well the translational velocities (or positions) and the motion of the top's symmetry axis. All of these parameters are coupled. To solve for the motion of a Tippe top, one would use standard tools including Newton's laws to prepare the system of equations, then program a computer to solve them numerically via simulation.\n\nIn this question, you will perform the first part of this process, investigating the physics of the Tippe top to set up the system of equations.\n\nFriction between the Tippe top and the surface it is moving on drives the motion of the Tippe top. Assume that the top remains in contact with the floor at point $A$, until such time as the stem contacts the floor. It is in motion at point $A$ with velocity $\\mathbf{v}_{A}$ relative to the floor. The frictional coefficient $\\mu_{k}$ between the top and floor is kinetic, with $\\left|\\mathbf{F}_{\\mathbf{f}}\\right|=\\mu_{k} N$, where $\\mathbf{F}_{f}=F_{f, x} \\widehat{\\mathbf{x}}+F_{f, y} \\hat{\\mathbf{y}}$ is the frictional force, and $N$ is the magnitude of the normal force. Assume that the top is initially set spinning only, i.e. there is no translational impulse given to the top.\n\nLet the mass of the Tippe top be $m$. Its moments of inertia are: $I_{3}$ about the axis of symmetry is, and $I_{1}=I_{2}$ about the mutually perpendicular principal axes. Let $\\mathbf{s}$ be the position vector of the centre of mass, and $\\mathbf{a}=\\overrightarrow{C A}$ be the vector from the centre of mass to the point of contact.\n\nUnless otherwise specified, give your answers in the $x y z$ reference frame for full marks. All torques and angular momentum are considered about the centre of mass $C$, unless otherwise specified. You may give your answers in terms of $N$. Except for part A.8, you need only consider the top where $\\theta<\\frac{\\pi}{2}$, and the stem is not in contact with the floor.\n \nFind the total energy of a spinning Tippe top, in terms of time derivatives of the Euler angles, $v_{x}$, and $v_{y}$. For partial marks, you may leave your answer in terms of $\\dot{\\mathbf{s}}=\\frac{d \\mathbf{s}}{d t}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-1.jpg?height=522&width=1332&top_left_y=1121&top_left_x=360", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=578&width=1778&top_left_y=316&top_left_x=184", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=417&width=545&top_left_y=1296&top_left_x=527", "https://cdn.mathpix.com/cropped/2024_03_14_691ce25cdf5feefac5b9g-2.jpg?height=431&width=397&top_left_y=1298&top_left_x=1189" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_623", "problem": "A newly discovered subatomic particle, the $S$ meson, has a mass $M$. When at rest, it lives for exactly $\\tau=3 \\times 10^{-8}$ seconds before decaying into two identical particles called $P$ mesons (peons?) that each have a mass of $\\alpha M$.\n\nIn a reference frame where the $\\mathrm{S}$ meson is at rest, determine the momentum of each $\\mathrm{P}$ meson particle in terms of $M, \\alpha$, the speed of light $c$, and any numerical constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA newly discovered subatomic particle, the $S$ meson, has a mass $M$. When at rest, it lives for exactly $\\tau=3 \\times 10^{-8}$ seconds before decaying into two identical particles called $P$ mesons (peons?) that each have a mass of $\\alpha M$.\n\nIn a reference frame where the $\\mathrm{S}$ meson is at rest, determine the momentum of each $\\mathrm{P}$ meson particle in terms of $M, \\alpha$, the speed of light $c$, and any numerical constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1697", "problem": "通常电容器两极板间有多层电介质, 并有漏电现象,为了探究其规律性, 采用如图所示的简单模型. 电容器的两极板面积均为 $A$, 其间充有两层电介质 1 和 2 , 第 1 层电介质的介电常\n\n| $\\varepsilon_{1} \\sigma_{1}$ |\n| :---: |\n| $\\varepsilon_{2} \\sigma_{2}$ |\n\n数、电导率 (即电阻率的倒数) 和厚度分别为 $\\varepsilon_{1} 、 \\sigma_{1}$ 和 $d_{1}$, 第 2 层电介质的则为 $\\varepsilon_{2} 、 \\sigma_{2}$ 和 $d_{2}$. 现在两极板加一直流电压 $U$, 电容器处于稳定状态.计算净电荷量在两介质交界面处的值", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n通常电容器两极板间有多层电介质, 并有漏电现象,为了探究其规律性, 采用如图所示的简单模型. 电容器的两极板面积均为 $A$, 其间充有两层电介质 1 和 2 , 第 1 层电介质的介电常\n\n| $\\varepsilon_{1} \\sigma_{1}$ |\n| :---: |\n| $\\varepsilon_{2} \\sigma_{2}$ |\n\n数、电导率 (即电阻率的倒数) 和厚度分别为 $\\varepsilon_{1} 、 \\sigma_{1}$ 和 $d_{1}$, 第 2 层电介质的则为 $\\varepsilon_{2} 、 \\sigma_{2}$ 和 $d_{2}$. 现在两极板加一直流电压 $U$, 电容器处于稳定状态.\n\n问题:\n计算净电荷量在两介质交界面处的值\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_529", "problem": "Radiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe force of radiation on a spherical particle of radius $r$ is given by\n\n$$\nF=P Q \\pi r^{2}\n$$\n\nwhere $P$ is the radiation pressure and $Q$ is a dimensionless quality factor that depends on the relative size of the particle $r$ and the wavelength of light $\\lambda$. Throughout this problem assume that the sun emits a single wavelength $\\lambda_{\\max }$; unless told otherwise, leave your answers in terms of symbolic variables.\n\nAssuming that the particle has a density $\\rho$, derive an expression for the ratio $\\frac{F_{\\text {radiation }}}{F_{\\text {gravity }}}$ in terms of $L_{\\odot}$, mass of sun $M_{\\odot}, \\rho$, particle radius $r$, and quality factor $Q$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nRadiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe force of radiation on a spherical particle of radius $r$ is given by\n\n$$\nF=P Q \\pi r^{2}\n$$\n\nwhere $P$ is the radiation pressure and $Q$ is a dimensionless quality factor that depends on the relative size of the particle $r$ and the wavelength of light $\\lambda$. Throughout this problem assume that the sun emits a single wavelength $\\lambda_{\\max }$; unless told otherwise, leave your answers in terms of symbolic variables.\n\nAssuming that the particle has a density $\\rho$, derive an expression for the ratio $\\frac{F_{\\text {radiation }}}{F_{\\text {gravity }}}$ in terms of $L_{\\odot}$, mass of sun $M_{\\odot}, \\rho$, particle radius $r$, and quality factor $Q$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_453", "problem": "The figure shows a more complex system, known as a block and tackle, consisting of two light pulley blocks and a light cord.\n\n[figure1]\nFigure: Two light pulley blocks and a light cord.\n\nWhat is the value of the tension in the cord?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe figure shows a more complex system, known as a block and tackle, consisting of two light pulley blocks and a light cord.\n\n[figure1]\nFigure: Two light pulley blocks and a light cord.\n\nWhat is the value of the tension in the cord?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-4.jpg?height=1074&width=514&top_left_y=1593&top_left_x=1356" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1734", "problem": "电子感应加速器利用变化的磁场来加速电子。电子绕平均半径为 $R$ 的环形轨道（轨道位于真空管道内）运动, 磁感应强度方向与环形轨道平面垂直。电子被感应电场加速, 感应电场的方向与环形轨道相切。电子电荷量为 $e$ 。\n\n[图1]设电子做圆周运动的环形轨道上的磁感应强度大小的增加率为 $\\frac{\\Delta B}{\\Delta t}$, 求在环形轨道切线方向感应电场作用在电子上力", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n电子感应加速器利用变化的磁场来加速电子。电子绕平均半径为 $R$ 的环形轨道（轨道位于真空管道内）运动, 磁感应强度方向与环形轨道平面垂直。电子被感应电场加速, 感应电场的方向与环形轨道相切。电子电荷量为 $e$ 。\n\n[图1]\n\n问题:\n设电子做圆周运动的环形轨道上的磁感应强度大小的增加率为 $\\frac{\\Delta B}{\\Delta t}$, 求在环形轨道切线方向感应电场作用在电子上力\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-05.jpg?height=371&width=417&top_left_y=1893&top_left_x=1362" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_703", "problem": "The expansion of a rod when heated is described by the equation $L(T)=L_{0}(1+\\alpha \\Delta T)$ where $L_{0}$ is the length of the rod at an initial temperature and $\\Delta T$ is the change in temperature, and $\\alpha$ is a coefficient that depends on the type of material used. Now suppose we have a ring with $\\alpha=10^{-5} /{ }^{\\circ} \\mathrm{C}$ and a sphere with $\\alpha=0.5 \\times 10^{-5} /{ }^{\\circ} \\mathrm{C}$. The sphere is of radius $10.005 \\mathrm{~cm}$, the ring has an inner radius of $10.000 \\mathrm{~cm}$, both when measured at $20^{\\circ} \\mathrm{C}$. At what temperature does the sphere just fit through the ring?\nA: $100^{\\circ} \\mathrm{C}$\nB: $120^{\\circ} \\mathrm{C}$\nC: $70^{\\circ} \\mathrm{C}$\nD: $360^{\\circ} \\mathrm{C}$\nE: $210^{\\circ} \\mathrm{C}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe expansion of a rod when heated is described by the equation $L(T)=L_{0}(1+\\alpha \\Delta T)$ where $L_{0}$ is the length of the rod at an initial temperature and $\\Delta T$ is the change in temperature, and $\\alpha$ is a coefficient that depends on the type of material used. Now suppose we have a ring with $\\alpha=10^{-5} /{ }^{\\circ} \\mathrm{C}$ and a sphere with $\\alpha=0.5 \\times 10^{-5} /{ }^{\\circ} \\mathrm{C}$. The sphere is of radius $10.005 \\mathrm{~cm}$, the ring has an inner radius of $10.000 \\mathrm{~cm}$, both when measured at $20^{\\circ} \\mathrm{C}$. At what temperature does the sphere just fit through the ring?\n\nA: $100^{\\circ} \\mathrm{C}$\nB: $120^{\\circ} \\mathrm{C}$\nC: $70^{\\circ} \\mathrm{C}$\nD: $360^{\\circ} \\mathrm{C}$\nE: $210^{\\circ} \\mathrm{C}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_883", "problem": "In this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\n\nLet us solve a related problem. Let a star of mass $m$ transit by a point mass $M_{2} \\gg m$ being at rest. The minimal distance between the star and the point during the transit is $r_{m}$. The velocity of the star at large distance is $\\sigma$. Find the exact value of impact parameter $b$.\n\nIf a star approaches the SBH binary for a distance about $a$, it participates in a complex 3-body interaction with the binary that almost always results in a star being shot out with the velocity about $v_{\\text {bin }}$ (the velocity of the star at the large distance after interaction). We will call such a strong interaction a collision of a star with the SBH binary. Acceleration and the shot of the star after the collision is called \"gravitational slingshot\".\nEstimate the characteristic time $\\Delta t$ between two successive collisions of the SBH binary with stars. Take into account that $\\sigma \\ll v_{\\text {bin }}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\n\nLet us solve a related problem. Let a star of mass $m$ transit by a point mass $M_{2} \\gg m$ being at rest. The minimal distance between the star and the point during the transit is $r_{m}$. The velocity of the star at large distance is $\\sigma$. Find the exact value of impact parameter $b$.\n\nIf a star approaches the SBH binary for a distance about $a$, it participates in a complex 3-body interaction with the binary that almost always results in a star being shot out with the velocity about $v_{\\text {bin }}$ (the velocity of the star at the large distance after interaction). We will call such a strong interaction a collision of a star with the SBH binary. Acceleration and the shot of the star after the collision is called \"gravitational slingshot\".\nEstimate the characteristic time $\\Delta t$ between two successive collisions of the SBH binary with stars. Take into account that $\\sigma \\ll v_{\\text {bin }}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_750", "problem": "A leak \n\nA hollow insulated cylinder of height $2 \\mathrm{H}$ and volume $2 \\mathrm{~V}$ is closed from below by an insulating piston. The cylinder is divided into two initially identical chambers by an insulating diaphragm of mass $m$. The diaphragm rests on a circular ledge and a gasket between them provides tight contact. Both chambers are filled with gaseous helium at pressure $p$ and temperature $T$. A force is applied to the piston, so that it moves upwards slowly.\n\n[figure1]\n\nFind the temperature $T_{1}$ in the upper chamber when the piston touches the diaphragm.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA leak \n\nA hollow insulated cylinder of height $2 \\mathrm{H}$ and volume $2 \\mathrm{~V}$ is closed from below by an insulating piston. The cylinder is divided into two initially identical chambers by an insulating diaphragm of mass $m$. The diaphragm rests on a circular ledge and a gasket between them provides tight contact. Both chambers are filled with gaseous helium at pressure $p$ and temperature $T$. A force is applied to the piston, so that it moves upwards slowly.\n\n[figure1]\n\nFind the temperature $T_{1}$ in the upper chamber when the piston touches the diaphragm.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_5a5b6f5fd43c588c9548g-1.jpg?height=862&width=520&top_left_y=588&top_left_x=800" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_676", "problem": "What is the speed of a particle observed to have a momentum of $5 \\mathrm{MeV} / \\mathrm{c}$ and a total relativistic energy of $10 \\mathrm{MeV}$ ?\nA: c\nB: $0.75 \\mathrm{c}$\nC: $0.5 \\mathrm{c}$\nD: $\\mathrm{c} / \\sqrt{3}$\nE: $0.25 \\mathrm{c}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhat is the speed of a particle observed to have a momentum of $5 \\mathrm{MeV} / \\mathrm{c}$ and a total relativistic energy of $10 \\mathrm{MeV}$ ?\n\nA: c\nB: $0.75 \\mathrm{c}$\nC: $0.5 \\mathrm{c}$\nD: $\\mathrm{c} / \\sqrt{3}$\nE: $0.25 \\mathrm{c}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1143", "problem": "# A Three-body Problem and LISA \n\n[figure1]\n\nFIGURE 1 Coplanar orbits of three bodies.\n\nThe Laser Interferometry Space Antenna (LISA) is a group of three identical spacecrafts for detecting low frequency gravitational waves. Each of the spacecrafts is placed at the corners of an equilateral triangle as shown in Figure 2 and Figure 3. The sides (or 'arms') are about 5.0 million kilometres long. The LISA constellation is in an earth-like orbit around the Sun trailing the Earth by $20^{\\circ}$. Each of them moves on a slightly inclined individual orbit around the Sun. Effectively, the three spacecrafts appear to roll about their common centre one revolution per year.\n\nThey are continuously transmitting and receiving laser signals between each other. Overall, they detect the gravitational waves by measuring tiny changes in the arm lengths using interferometric means. A collision of massive objects, such as blackholes, in nearby galaxies is an example of the sources of gravitational waves.\n\n[figure2]\n\nFIGURE 2 Illustration of the LISA orbit. The three spacecraft roll about their centre of mass with a period of 1 year. Initially, they trail the Earth by $20^{\\circ}$. (Picture from D.A. Shaddock, \"An Overview of the Laser Interferometer Space Antenna\", Publications of the Astronomical Society of Australia, 2009, 26, pp.128-132.).\n\n[figure3]\n\nFIGURE 3 Enlarged view of the three spacecrafts trailing the Earth. A, B and $\\mathrm{C}$ are the three spacecrafts at the corners of the equilateral triangle.\n\nIn the plane containing the three spacecrafts, what is the relative speed of one spacecraft with respect to another?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\n# A Three-body Problem and LISA \n\n[figure1]\n\nFIGURE 1 Coplanar orbits of three bodies.\n\nThe Laser Interferometry Space Antenna (LISA) is a group of three identical spacecrafts for detecting low frequency gravitational waves. Each of the spacecrafts is placed at the corners of an equilateral triangle as shown in Figure 2 and Figure 3. The sides (or 'arms') are about 5.0 million kilometres long. The LISA constellation is in an earth-like orbit around the Sun trailing the Earth by $20^{\\circ}$. Each of them moves on a slightly inclined individual orbit around the Sun. Effectively, the three spacecrafts appear to roll about their common centre one revolution per year.\n\nThey are continuously transmitting and receiving laser signals between each other. Overall, they detect the gravitational waves by measuring tiny changes in the arm lengths using interferometric means. A collision of massive objects, such as blackholes, in nearby galaxies is an example of the sources of gravitational waves.\n\n[figure2]\n\nFIGURE 2 Illustration of the LISA orbit. The three spacecraft roll about their centre of mass with a period of 1 year. Initially, they trail the Earth by $20^{\\circ}$. (Picture from D.A. Shaddock, \"An Overview of the Laser Interferometer Space Antenna\", Publications of the Astronomical Society of Australia, 2009, 26, pp.128-132.).\n\n[figure3]\n\nFIGURE 3 Enlarged view of the three spacecrafts trailing the Earth. A, B and $\\mathrm{C}$ are the three spacecrafts at the corners of the equilateral triangle.\n\nIn the plane containing the three spacecrafts, what is the relative speed of one spacecraft with respect to another?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{m} / \\mathrm{s}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_27986d152c1a1dded222g-1.jpg?height=918&width=919&top_left_y=506&top_left_x=619", "https://cdn.mathpix.com/cropped/2024_03_14_27986d152c1a1dded222g-2.jpg?height=843&width=1629&top_left_y=1210&top_left_x=281", "https://cdn.mathpix.com/cropped/2024_03_14_27986d152c1a1dded222g-3.jpg?height=396&width=1396&top_left_y=339&top_left_x=386", "https://cdn.mathpix.com/cropped/2024_03_14_2e1e5a594468530ddb04g-7.jpg?height=889&width=1263&top_left_y=596&top_left_x=274" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{m} / \\mathrm{s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1302", "problem": "由玻尔理论可知, 当氢原子中的核外电子由一个轨道跃迁到另一轨道时, 有可能\nA: 发射出光子, 电子的动能减少, 原子的势能减少\nB: 发射出光子, 电子的动能增加, 原子的势能减少\nC: 吸收光子, 电子的动能减少, 原子的势能增加\nD: 吸收光子，电子的动能增加，原子的势能减少\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n由玻尔理论可知, 当氢原子中的核外电子由一个轨道跃迁到另一轨道时, 有可能\n\nA: 发射出光子, 电子的动能减少, 原子的势能减少\nB: 发射出光子, 电子的动能增加, 原子的势能减少\nC: 吸收光子, 电子的动能减少, 原子的势能增加\nD: 吸收光子，电子的动能增加，原子的势能减少\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_109", "problem": "Consider a flat uniform square of mass $M$ and side length $L$. Cut a circle out of the square that has a diameter equal to the length of the side of the square, with the same center as the square. Determine the moment of inertia of the remaining shape about an axis through the center and perpendicular to the plane of the square.\nA: $\\left(\\frac{1}{6}-\\frac{\\pi}{32}\\right) M L^{2} $ \nB: $\\left(\\frac{1}{12}-\\frac{\\pi}{64}\\right) M L^{2}$\nC: $\\left(\\frac{\\pi}{24}-\\frac{1}{3 \\pi}\\right) M L^{2}$\nD: $\\left(\\frac{1}{2 \\pi}-\\frac{1}{16}\\right) M L^{2}$\nE: $\\left(\\frac{1}{2 \\pi}-\\frac{1}{8}\\right) M L^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nConsider a flat uniform square of mass $M$ and side length $L$. Cut a circle out of the square that has a diameter equal to the length of the side of the square, with the same center as the square. Determine the moment of inertia of the remaining shape about an axis through the center and perpendicular to the plane of the square.\n\nA: $\\left(\\frac{1}{6}-\\frac{\\pi}{32}\\right) M L^{2} $ \nB: $\\left(\\frac{1}{12}-\\frac{\\pi}{64}\\right) M L^{2}$\nC: $\\left(\\frac{\\pi}{24}-\\frac{1}{3 \\pi}\\right) M L^{2}$\nD: $\\left(\\frac{1}{2 \\pi}-\\frac{1}{16}\\right) M L^{2}$\nE: $\\left(\\frac{1}{2 \\pi}-\\frac{1}{8}\\right) M L^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_773", "problem": "An elevator is moving upwards a constant speed. Ignoring any friction, which statement is correct?\nA: The kinetic energy of the elevator is constant.\nB: The gravitational potential energy of the Earth-Elevator system is constant.\nC: The mechanical energy of the Earth-Elevator system is constant.\nD: a and $\\mathrm{c}$ are both correct, but $\\mathrm{b}$ is not correct.\nE: $\\mathrm{a}, \\mathrm{b}$ and $\\mathrm{c}$ are all correct.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn elevator is moving upwards a constant speed. Ignoring any friction, which statement is correct?\n\nA: The kinetic energy of the elevator is constant.\nB: The gravitational potential energy of the Earth-Elevator system is constant.\nC: The mechanical energy of the Earth-Elevator system is constant.\nD: a and $\\mathrm{c}$ are both correct, but $\\mathrm{b}$ is not correct.\nE: $\\mathrm{a}, \\mathrm{b}$ and $\\mathrm{c}$ are all correct.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1483", "problem": "农用平板车的简化模型如图 $\\mathrm{a}$ 所示, 两车轮的半径均为 $\\mathrm{r}$ (忽略内外半径差), 质量均为 $\\mathrm{m}$ (车轮辐条的质量可忽略), 两轮可 $2 \\mathrm{~m}$ 绕过其中心的光滑细车轴转动 (轴 $\\mathrm{m}$ 的质量可忽略); 车平板长为 1 、质把手量为 $2 \\mathrm{~m}$, 平板的质心恰好位于车轮的轴上; 两车把手 (可视为细直杆) 1 的长均为 21 、质量均为 $\\mathrm{m}$, 且把手\n前端与平板对齐, 平板、把手和车轴固连成一个整体, 车轮、平板和把手各自的质量分布都是均匀的. 重力加速度大小为 $\\mathrm{g}$.在把手与水平地面碰撞前的瞬间立即撤去卡住两车轮的装置, 同时将车轮和轴锁死, 在碰后的瞬间立即解锁, 假设碰撞时间较短 (但不为零), 碰后把手末端在坚直方向不反弹. 已知把手与地面、车轮与地面之间的滑动摩擦系数均为 $\\mu$ (最大静摩擦力等于滑动摩擦力). 求在车轮从开始运动直至静止的过程中, 车轴移动的距离.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n农用平板车的简化模型如图 $\\mathrm{a}$ 所示, 两车轮的半径均为 $\\mathrm{r}$ (忽略内外半径差), 质量均为 $\\mathrm{m}$ (车轮辐条的质量可忽略), 两轮可 $2 \\mathrm{~m}$ 绕过其中心的光滑细车轴转动 (轴 $\\mathrm{m}$ 的质量可忽略); 车平板长为 1 、质把手量为 $2 \\mathrm{~m}$, 平板的质心恰好位于车轮的轴上; 两车把手 (可视为细直杆) 1 的长均为 21 、质量均为 $\\mathrm{m}$, 且把手\n前端与平板对齐, 平板、把手和车轴固连成一个整体, 车轮、平板和把手各自的质量分布都是均匀的. 重力加速度大小为 $\\mathrm{g}$.\n\n问题:\n在把手与水平地面碰撞前的瞬间立即撤去卡住两车轮的装置, 同时将车轮和轴锁死, 在碰后的瞬间立即解锁, 假设碰撞时间较短 (但不为零), 碰后把手末端在坚直方向不反弹. 已知把手与地面、车轮与地面之间的滑动摩擦系数均为 $\\mu$ (最大静摩擦力等于滑动摩擦力). 求在车轮从开始运动直至静止的过程中, 车轴移动的距离.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-14.jpg?height=465&width=443&top_left_y=1640&top_left_x=315", "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-14.jpg?height=357&width=665&top_left_y=1757&top_left_x=929", "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-16.jpg?height=442&width=462&top_left_y=1800&top_left_x=317", "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-16.jpg?height=368&width=711&top_left_y=1820&top_left_x=952" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1700", "problem": "牛顿曾观察到一束细日光射到有灰尘的反射镜上面会产生干涉条纹。为了分析这一现象背后的物理, 考虑如图所示的简单实验。一平板玻璃的折射率为 $n$, 厚度为 $t$, 下表面涂有水银反射层,上表面撒有滑石粉（灰尘粒子）。观察者 $\\mathrm{O}$ 和单色点光源 $\\mathrm{L}$ （光线的波长为 $\\lambda$ ）的连线垂直于镜面 (垂足为 $\\mathrm{N}$ ), $\\mathrm{LN}=a, \\mathrm{ON}=b$ 。反射镜面上的某灰尘粒子 $\\mathrm{P}$ 与直线 $\\mathrm{ON}$ 的距离为 $r$ $(b>a>>r>t)$ 。观察者可以观察到明暗相间的环形条纹。求第 $m$ 个亮环到 $\\mathrm{N}$ 点的距离;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n牛顿曾观察到一束细日光射到有灰尘的反射镜上面会产生干涉条纹。为了分析这一现象背后的物理, 考虑如图所示的简单实验。一平板玻璃的折射率为 $n$, 厚度为 $t$, 下表面涂有水银反射层,上表面撒有滑石粉（灰尘粒子）。观察者 $\\mathrm{O}$ 和单色点光源 $\\mathrm{L}$ （光线的波长为 $\\lambda$ ）的连线垂直于镜面 (垂足为 $\\mathrm{N}$ ), $\\mathrm{LN}=a, \\mathrm{ON}=b$ 。反射镜面上的某灰尘粒子 $\\mathrm{P}$ 与直线 $\\mathrm{ON}$ 的距离为 $r$ $(b>a>>r>t)$ 。观察者可以观察到明暗相间的环形条纹。\n\n问题:\n求第 $m$ 个亮环到 $\\mathrm{N}$ 点的距离;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_17b1131fe8d911867aa0g-11.jpg?height=629&width=714&top_left_y=2144&top_left_x=1068" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1419", "problem": "某电动汽车自重 $2.0 \\mathrm{t}$, 其电池额定容量为 $50 \\mathrm{kWh}$ 。车行驶时受到的阻力约为车重的十分之一。电池瞬时功率最高可达 $90 \\mathrm{~kW}$, 理论续航里程为 $400 \\mathrm{~km}$ 。国家电网的充电桩可在电池额定容量的 $30 \\% \\sim 80 \\%$ 范围内应用快充技术 $(500 \\mathrm{~V}, 50 \\mathrm{~A})$ 充电, 而便携充电器 $(220 \\mathrm{~V}, 16 \\mathrm{~A})$ 可将电池容量从零充至 100\\%; 不计充电电源的内阻。当汽车电池剩余电量为其额定值的 30\\%时, 下列说法正确的是\nA: 汽车至少还能行驶 $130 \\mathrm{~km}$\nB: 用国家电网充电桩将电池容量充至其额定值的 $80 \\%$, 理论上需要 $40 \\mathrm{~min}$\nC: 用便携充电器将电池电量充至其额定值的 $80 \\%$, 理论上需要 $7 \\mathrm{~h}$ 以上\nD: 此电动汽车的最高行驶速度可超过 $130 \\mathrm{~km} / \\mathrm{h}$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n某电动汽车自重 $2.0 \\mathrm{t}$, 其电池额定容量为 $50 \\mathrm{kWh}$ 。车行驶时受到的阻力约为车重的十分之一。电池瞬时功率最高可达 $90 \\mathrm{~kW}$, 理论续航里程为 $400 \\mathrm{~km}$ 。国家电网的充电桩可在电池额定容量的 $30 \\% \\sim 80 \\%$ 范围内应用快充技术 $(500 \\mathrm{~V}, 50 \\mathrm{~A})$ 充电, 而便携充电器 $(220 \\mathrm{~V}, 16 \\mathrm{~A})$ 可将电池容量从零充至 100\\%; 不计充电电源的内阻。当汽车电池剩余电量为其额定值的 30\\%时, 下列说法正确的是\n\nA: 汽车至少还能行驶 $130 \\mathrm{~km}$\nB: 用国家电网充电桩将电池容量充至其额定值的 $80 \\%$, 理论上需要 $40 \\mathrm{~min}$\nC: 用便携充电器将电池电量充至其额定值的 $80 \\%$, 理论上需要 $7 \\mathrm{~h}$ 以上\nD: 此电动汽车的最高行驶速度可超过 $130 \\mathrm{~km} / \\mathrm{h}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1352", "problem": "盤状星云脉冲星的鎘射脉冲周期是 $0.033 \\mathrm{~s}$ 。假设它是由均匀分布的物质构成的球体, 脉冲周期是它的旋转周期, 万有引力是唯一能阻止它离心分解的力, 已知万有引力常量 $G=6.67 \\times 10^{-11} \\mathrm{~m}^{3} \\cdot \\mathrm{kg}^{-1} \\cdot \\mathrm{s}^{-2}$, 由于脉冲星表面的物质未分离, 故可估算出此脉冲星密度的下限是", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n盤状星云脉冲星的鎘射脉冲周期是 $0.033 \\mathrm{~s}$ 。假设它是由均匀分布的物质构成的球体, 脉冲周期是它的旋转周期, 万有引力是唯一能阻止它离心分解的力, 已知万有引力常量 $G=6.67 \\times 10^{-11} \\mathrm{~m}^{3} \\cdot \\mathrm{kg}^{-1} \\cdot \\mathrm{s}^{-2}$, 由于脉冲星表面的物质未分离, 故可估算出此脉冲星密度的下限是\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{kg} \\cdot \\mathrm{m}^{-3}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{kg} \\cdot \\mathrm{m}^{-3}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_782", "problem": "An elevator is moving upwards with constant upwards acceleration. At some point in the elevator's motion, a bolt breaks loose and drops from the ceiling. What is the motion of the bolt as seen by an observer standing inside the elevator? Ignore air resistance.\nA: The bolt immediately moves downwards, at constant speed.\nB: The bolt initially moves upwards, then slows, reverses direction and moves downwards.\nC: The bolt immediately moves downwards, with acceleration less than $g$.\nD: The bolt immediately moves downwards, with acceleration equal to $g$.\nE: The bolt immediately moves downwards, with acceleration greater than $g$.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn elevator is moving upwards with constant upwards acceleration. At some point in the elevator's motion, a bolt breaks loose and drops from the ceiling. What is the motion of the bolt as seen by an observer standing inside the elevator? Ignore air resistance.\n\nA: The bolt immediately moves downwards, at constant speed.\nB: The bolt initially moves upwards, then slows, reverses direction and moves downwards.\nC: The bolt immediately moves downwards, with acceleration less than $g$.\nD: The bolt immediately moves downwards, with acceleration equal to $g$.\nE: The bolt immediately moves downwards, with acceleration greater than $g$.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_193", "problem": "Consider two identical masses that interact only by gravitational attraction to each other. If one mass is fixed in place and the other is released from rest, then the two masses collide in time $T$. If both masses are released from rest, they collide in time\nA: $T / 4$\nB: $T /(2 \\sqrt{2})$\nC: $T / 2$\nD: $T / \\sqrt{2} $\nE: $T$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nConsider two identical masses that interact only by gravitational attraction to each other. If one mass is fixed in place and the other is released from rest, then the two masses collide in time $T$. If both masses are released from rest, they collide in time\n\nA: $T / 4$\nB: $T /(2 \\sqrt{2})$\nC: $T / 2$\nD: $T / \\sqrt{2} $\nE: $T$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1227", "problem": "# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nAfter the bubble is formed for a while, it will be in thermal equilibrium with the surrounding. This bubble in still air will naturally fall towards the ground.\n\nThe above calculations suggest that the terms involving the surface tension $\\gamma$ add very little to the accuracy of the result. In all of the questions below, you can neglect the surface tension terms.\n\nAssume that the total charge is not too large (i.e. $\\frac{q^{2}}{\\varepsilon_{0} R_{0}^{4}}<R$. Determine the period of the orbit $T$ in terms of any or all of $r, R, Q, e$, and any necessary fundamental constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA spherical region of space of radius $R$ has a uniform charge density and total charge $+Q$. An electron of charge $-e$ is free to move inside or outside the sphere, under the influence of the charge density alone. For this first part ignore radiation effects.\n\nConsider a circular orbit for the electron where $r>R$. Determine the period of the orbit $T$ in terms of any or all of $r, R, Q, e$, and any necessary fundamental constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_141", "problem": "An upright rod of length $\\ell$ is launched into the air with vertical velocity $v_{y}$. It is given enough angular momentum so that the rod rotates by an angle of $2 \\pi$ before landing. Find the initial horizontal velocity of the bottom of the rod.\nA: $\\frac{2 \\ell g}{v_{y}^{2}}$\nB: $\\frac{\\pi \\ell g}{2 v_{y}} $ \nC: $\\sqrt{\\ell g}$\nD: $\\frac{2 v_{y}}{\\ell g}$\nE: $\\frac{2 v_{y}^{2}}{\\pi \\sqrt{\\ell g}}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn upright rod of length $\\ell$ is launched into the air with vertical velocity $v_{y}$. It is given enough angular momentum so that the rod rotates by an angle of $2 \\pi$ before landing. Find the initial horizontal velocity of the bottom of the rod.\n\nA: $\\frac{2 \\ell g}{v_{y}^{2}}$\nB: $\\frac{\\pi \\ell g}{2 v_{y}} $ \nC: $\\sqrt{\\ell g}$\nD: $\\frac{2 v_{y}}{\\ell g}$\nE: $\\frac{2 v_{y}^{2}}{\\pi \\sqrt{\\ell g}}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_494", "problem": "In general, whenever an electric and a magnetic field are at an angle to each other, energy is transferred; for example, this principle is the reason electromagnetic radiation transfers energy. The power transferred per unit area is given by the Poynting vector:\n\n$$\n\\vec{S}=\\frac{1}{\\mu_{0}} \\vec{E} \\times \\vec{B}\n$$\n\nIn each part of this problem, the last subpart asks you to verify that the rate of energy transfer agrees with the formula for the Poynting vector. Therefore, you should not use the formula for the Poynting vector before the last subpart!\n\nA long solenoid of radius $R$ has $\\mathcal{N}$ turns of wire per unit length. The solenoid carries current $I$, and this current is increased at a small, constant rate $\\frac{d I}{d t}$.\n\nWhat is the electric field $E$ just inside the surface of the solenoid? Draw its direction on a diagram.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn general, whenever an electric and a magnetic field are at an angle to each other, energy is transferred; for example, this principle is the reason electromagnetic radiation transfers energy. The power transferred per unit area is given by the Poynting vector:\n\n$$\n\\vec{S}=\\frac{1}{\\mu_{0}} \\vec{E} \\times \\vec{B}\n$$\n\nIn each part of this problem, the last subpart asks you to verify that the rate of energy transfer agrees with the formula for the Poynting vector. Therefore, you should not use the formula for the Poynting vector before the last subpart!\n\nA long solenoid of radius $R$ has $\\mathcal{N}$ turns of wire per unit length. The solenoid carries current $I$, and this current is increased at a small, constant rate $\\frac{d I}{d t}$.\n\nWhat is the electric field $E$ just inside the surface of the solenoid? Draw its direction on a diagram.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1615", "problem": "如图, 两劲度系数均为 $k$ 的同样的轻弹性绳的上端固定在一水平面上, 下端连在一起悬挂一质量为 $m$ 的小物块。平衡时, 轻弹性\n\n[图1]\n绳与水平面的夹角为 $\\alpha_{0}$, 弹性绳长度为 $l_{0}$ 。现将小物块向下拉一段微小的距离后从静止释放。当小物块向下拉一段微小的距离 $0.010 \\mathrm{~m}$ 时, 写出该小物块相对于平衡位置的偏离随时间变化的方程。\n\n已知: 当 $x<<1$ 时, $\\frac{1}{1+x} \\approx 1-x, \\sqrt{1+x} \\approx 1+\\frac{1}{2} x$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个方程。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 两劲度系数均为 $k$ 的同样的轻弹性绳的上端固定在一水平面上, 下端连在一起悬挂一质量为 $m$ 的小物块。平衡时, 轻弹性\n\n[图1]\n绳与水平面的夹角为 $\\alpha_{0}$, 弹性绳长度为 $l_{0}$ 。现将小物块向下拉一段微小的距离后从静止释放。\n\n问题:\n当小物块向下拉一段微小的距离 $0.010 \\mathrm{~m}$ 时, 写出该小物块相对于平衡位置的偏离随时间变化的方程。\n\n已知: 当 $x<<1$ 时, $\\frac{1}{1+x} \\approx 1-x, \\sqrt{1+x} \\approx 1+\\frac{1}{2} x$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个方程，例如ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_d716ce15f03757bb482eg-04.jpg?height=266&width=537&top_left_y=2494&top_left_x=1408" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_113", "problem": "Two springs of spring constants $k_{1}$ and $k_{2}$, respectively, are connected in series and stretched, as shown below. What is the ratio of their potential energies, $U_{1} / U_{2}$ ?\n\n[figure1]\nA: 1\nB: $k_{1} / k_{2}$\nC: $k_{2} / k_{1} $ \nD: $\\left(k_{1} / k_{2}\\right)^{2}$\nE: $\\left(k_{2} / k_{1}\\right)^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo springs of spring constants $k_{1}$ and $k_{2}$, respectively, are connected in series and stretched, as shown below. What is the ratio of their potential energies, $U_{1} / U_{2}$ ?\n\n[figure1]\n\nA: 1\nB: $k_{1} / k_{2}$\nC: $k_{2} / k_{1} $ \nD: $\\left(k_{1} / k_{2}\\right)^{2}$\nE: $\\left(k_{2} / k_{1}\\right)^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f8112dc52e2354b9eb24g-03.jpg?height=120&width=797&top_left_y=360&top_left_x=661" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1355", "problem": "嫦娥 1 号奔月卫星与长征 3 号火箭分离后, 进入绕地运行的椭圆轨道, 近地点离地面高 $H_{n}=2.05 \\times 10^{2} \\mathrm{~km}$, 远地点离地面高 $H_{f}=5.0930 \\times 10^{4} \\mathrm{~km}$, 周期约为 16 小时, 称为 16 小时轨道 (如图中曲线 1 所示)。随后, 为了使卫星离地越来越远, 星载发动机先在远地点点火, 使卫星进入新轨道 (如图中曲线 2 所示), 以抬高近地点。后来又连续三次在抬高以后的近地点点火, 使卫星加速和变轨, 抬高远地点, 相继进入 24 小时轨道、 48 小时轨道和地月转移轨道 (分别如图中曲线 3、4、5 所示)。已知卫星质量 $m=2.350 \\times 10^{3} \\mathrm{~kg}$, 地球半径 $R=6.378 \\times 10^{3} \\mathrm{~km}$, 地面重力加速度 $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$, 月球半径 $r=1.738 \\times 10^{3} \\mathrm{~km}$ 。试计算 16 小时轨道的半长轴 $\\mathrm{a}$ 和半短轴 $\\mathrm{b}$ 的长度, 以及椭圆偏心率 $\\mathrm{e}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n嫦娥 1 号奔月卫星与长征 3 号火箭分离后, 进入绕地运行的椭圆轨道, 近地点离地面高 $H_{n}=2.05 \\times 10^{2} \\mathrm{~km}$, 远地点离地面高 $H_{f}=5.0930 \\times 10^{4} \\mathrm{~km}$, 周期约为 16 小时, 称为 16 小时轨道 (如图中曲线 1 所示)。随后, 为了使卫星离地越来越远, 星载发动机先在远地点点火, 使卫星进入新轨道 (如图中曲线 2 所示), 以抬高近地点。后来又连续三次在抬高以后的近地点点火, 使卫星加速和变轨, 抬高远地点, 相继进入 24 小时轨道、 48 小时轨道和地月转移轨道 (分别如图中曲线 3、4、5 所示)。已知卫星质量 $m=2.350 \\times 10^{3} \\mathrm{~kg}$, 地球半径 $R=6.378 \\times 10^{3} \\mathrm{~km}$, 地面重力加速度 $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$, 月球半径 $r=1.738 \\times 10^{3} \\mathrm{~km}$ 。\n\n问题:\n试计算 16 小时轨道的半长轴 $\\mathrm{a}$ 和半短轴 $\\mathrm{b}$ 的长度, 以及椭圆偏心率 $\\mathrm{e}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[the value of a, the value of b, the value of e]\n它们的单位依次是[km, km, None]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "km", "km", null ], "answer_sequence": [ "the value of a", "the value of b", "the value of e" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_855", "problem": "In the jets from AGN, we have populations of highly energetic electrons in regions with strong magnetic fields. This creates the conditions for the emission of high fluxes of synchrotron radiation. The electrons are often so highly energetic, that they can be described as ultra relativistic with $\\gamma \\gg 1$.\n \nAs the electron is accelerated due to the magnetic field it emits electromagnetic radiation. In a frame at which the electron is momentarily at rest, there is no preferred direction for the emission of the radiation. Half is emitted in the forward direction, and half in the backward direction. However, in the frame of the observer, for an electron moving at an ultra relativistic speed, with $\\gamma \\gg 1$, the radiation is concentrated in a forward cone with $\\theta \\lesssim 1 / \\gamma$ (so the total angle of cone is $2 / \\gamma$ ). As the electron is gyrating around the magnetic field, any observer will only see pulses of radiation as the forward cone sweeps through the line of sight.\n[figure1]\n\nFigure 3: The diagram on the left shows the distribution of power in radiation from an electron accelerating up the page in the frame at which the electron in momentarily at rest. The diagram on the right shows the distribution of power in radiation for the same electron in the observer's frame, where most radiation is emitted in the forward cone. In the observers frame, the direction of the electron's acceleration is shown by a vector labelled $\\mathbf{a}$ and the direction of its velocity is shown by a vector labelled $\\mathbf{v}$.\n\nThe total synchrotron power emitted is\n\n$$\nP_{\\mathrm{s}}=\\frac{1}{6 \\pi \\varepsilon_{0}}\\left(\\frac{q^{4} B^{2} \\sin ^{2} \\phi}{m^{4} c^{5}}\\right) E^{2}\n$$\n Estimate the time, $\\tau$, for an electron of energy $E$ to lose its energy through synchrotron cooling.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn the jets from AGN, we have populations of highly energetic electrons in regions with strong magnetic fields. This creates the conditions for the emission of high fluxes of synchrotron radiation. The electrons are often so highly energetic, that they can be described as ultra relativistic with $\\gamma \\gg 1$.\n \nAs the electron is accelerated due to the magnetic field it emits electromagnetic radiation. In a frame at which the electron is momentarily at rest, there is no preferred direction for the emission of the radiation. Half is emitted in the forward direction, and half in the backward direction. However, in the frame of the observer, for an electron moving at an ultra relativistic speed, with $\\gamma \\gg 1$, the radiation is concentrated in a forward cone with $\\theta \\lesssim 1 / \\gamma$ (so the total angle of cone is $2 / \\gamma$ ). As the electron is gyrating around the magnetic field, any observer will only see pulses of radiation as the forward cone sweeps through the line of sight.\n[figure1]\n\nFigure 3: The diagram on the left shows the distribution of power in radiation from an electron accelerating up the page in the frame at which the electron in momentarily at rest. The diagram on the right shows the distribution of power in radiation for the same electron in the observer's frame, where most radiation is emitted in the forward cone. In the observers frame, the direction of the electron's acceleration is shown by a vector labelled $\\mathbf{a}$ and the direction of its velocity is shown by a vector labelled $\\mathbf{v}$.\n\nThe total synchrotron power emitted is\n\n$$\nP_{\\mathrm{s}}=\\frac{1}{6 \\pi \\varepsilon_{0}}\\left(\\frac{q^{4} B^{2} \\sin ^{2} \\phi}{m^{4} c^{5}}\\right) E^{2}\n$$\n Estimate the time, $\\tau$, for an electron of energy $E$ to lose its energy through synchrotron cooling.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_2416d49d47cb88c0a72bg-4.jpg?height=166&width=842&top_left_y=1502&top_left_x=618" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_454", "problem": "The resultant amplitude of the waves from the slits arriving at the screen at a maxima (a bright fringe), where the incoming waves arrive in phase, is 2A.\n\nThe intensity at the maxima is 4I.\n\nThe intensity at the minimum is 0.This question relates to ways in which light energy may be concentrated in an interference pattern.\n\nThe figure shows wave fronts at normal incidence on a Young's double slit arrangement, illuminating them so that they both radiate in phase, each with an amplitude $A$ at the screen. One such slit alone will cause an intensity of illumination of $I$, where $I \\propto A^{2}$, in the central region of the screen where the Young's fringes pattern forms.\n\n[figure1]\nFigure: Young's double slits.\n\nWhat is the average intensity across the whole pattern?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe resultant amplitude of the waves from the slits arriving at the screen at a maxima (a bright fringe), where the incoming waves arrive in phase, is 2A.\n\nThe intensity at the maxima is 4I.\n\nThe intensity at the minimum is 0.\n\nproblem:\nThis question relates to ways in which light energy may be concentrated in an interference pattern.\n\nThe figure shows wave fronts at normal incidence on a Young's double slit arrangement, illuminating them so that they both radiate in phase, each with an amplitude $A$ at the screen. One such slit alone will cause an intensity of illumination of $I$, where $I \\propto A^{2}$, in the central region of the screen where the Young's fringes pattern forms.\n\n[figure1]\nFigure: Young's double slits.\n\nWhat is the average intensity across the whole pattern?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-3.jpg?height=522&width=859&top_left_y=584&top_left_x=610" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_654", "problem": "A $1 \\mathrm{~kg}$ object slides $3.6 \\mathrm{~m}$ down a ramp with a $35^{\\circ}$ slope. It has an initial speed of $2 \\mathrm{~m} / \\mathrm{s}$ and a final speed of $1.06 \\mathrm{~m} / \\mathrm{s}$ when it reaches the bottom. What was the work done by kinetic friction $\\left(\\mu_{k}=0.3\\right)$ when the object has slid to the bottom of the ramp?\nA: $10.6 \\mathrm{Nm}$\nB: $21.7 \\mathrm{Nm}$\nC: $0 \\mathrm{Nm}$\nD: $19.6 \\mathrm{Nm}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA $1 \\mathrm{~kg}$ object slides $3.6 \\mathrm{~m}$ down a ramp with a $35^{\\circ}$ slope. It has an initial speed of $2 \\mathrm{~m} / \\mathrm{s}$ and a final speed of $1.06 \\mathrm{~m} / \\mathrm{s}$ when it reaches the bottom. What was the work done by kinetic friction $\\left(\\mu_{k}=0.3\\right)$ when the object has slid to the bottom of the ramp?\n\nA: $10.6 \\mathrm{Nm}$\nB: $21.7 \\mathrm{Nm}$\nC: $0 \\mathrm{Nm}$\nD: $19.6 \\mathrm{Nm}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1545", "problem": "某秋天清晨, 气温为 $4.0^{\\circ} \\mathrm{C}$,一加水员到实验园区给一内径为 $2.00 \\mathrm{~m}$ 、高为 $2.00 \\mathrm{~m}$ 的圆柱形不锈钢蒸馏水罐加水。罐体导热良好。罐外有一内径为 $4.00 \\mathrm{~cm}$的透明圆柱形观察柱，底部与罐相连（连接处很短）,顶部与大气相通，如图所示。加完水后，加水员在水面上覆盖一层轻质防蒸发膜（不溶于水，与罐壁无摩擦），并密闭了罐顶的加水口。此时加水员通过观察柱上的刻度看到罐内水高为 $1.00 \\mathrm{~m}$ 。从清晨到中午，气温缓慢升至 $24.0^{\\circ} \\mathrm{C}$ ，问此时\n\n[图1]\n观察柱内水位为多少？假设中间无人用水，水的蒸发及罐和观察柱体积随温度的变化可忽略。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某秋天清晨, 气温为 $4.0^{\\circ} \\mathrm{C}$,一加水员到实验园区给一内径为 $2.00 \\mathrm{~m}$ 、高为 $2.00 \\mathrm{~m}$ 的圆柱形不锈钢蒸馏水罐加水。罐体导热良好。罐外有一内径为 $4.00 \\mathrm{~cm}$的透明圆柱形观察柱，底部与罐相连（连接处很短）,顶部与大气相通，如图所示。加完水后，加水员在水面上覆盖一层轻质防蒸发膜（不溶于水，与罐壁无摩擦），并密闭了罐顶的加水口。此时加水员通过观察柱上的刻度看到罐内水高为 $1.00 \\mathrm{~m}$ 。\n\n问题:\n从清晨到中午，气温缓慢升至 $24.0^{\\circ} \\mathrm{C}$ ，问此时\n\n[图1]\n观察柱内水位为多少？假设中间无人用水，水的蒸发及罐和观察柱体积随温度的变化可忽略。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~m}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_49158ed36459a540f197g-01.jpg?height=588&width=671&top_left_y=1822&top_left_x=1161" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_384", "problem": "Tools At least three pairs of large color less medical semi-transparent latex rubber gloves; a roll of transparent and strong office tape; a pair of sharp scissors; at least four sheets of A4 or larger size graphing paper three rulers; a flexible measuring tape with a length of at least one meter; extra-fine poin universal surface marker. Rubber gloves can be cut as needed into pieces. The pieces of gloves can be fixed to your working table either directly using the tape and/or with the help of a ruler (to achieve a firmer fixing).\n\nLatex is a highly stretchable elastic ma terial for which it can be assumed that its volume remains constant during stretching up to the breaking point.\n\nFor each of the tasks, sketch your experimental setup and explain the steps you made to obtain the best possible precision, and tab ulate the directly measured data\n\nDetermine and plot the stressstrain relationship for latex bands. Stress is defined as the tension force per cross sectional area. Express the stress $\\sigma$ in relat ive units, normalised to the maximal stress at the breaking point.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nTools At least three pairs of large color less medical semi-transparent latex rubber gloves; a roll of transparent and strong office tape; a pair of sharp scissors; at least four sheets of A4 or larger size graphing paper three rulers; a flexible measuring tape with a length of at least one meter; extra-fine poin universal surface marker. Rubber gloves can be cut as needed into pieces. The pieces of gloves can be fixed to your working table either directly using the tape and/or with the help of a ruler (to achieve a firmer fixing).\n\nLatex is a highly stretchable elastic ma terial for which it can be assumed that its volume remains constant during stretching up to the breaking point.\n\nFor each of the tasks, sketch your experimental setup and explain the steps you made to obtain the best possible precision, and tab ulate the directly measured data\n\nDetermine and plot the stressstrain relationship for latex bands. Stress is defined as the tension force per cross sectional area. Express the stress $\\sigma$ in relat ive units, normalised to the maximal stress at the breaking point.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_918", "problem": "In the following we will study the case where the neutral atoms are placed in an external laser field that varies in time and space as $\\vec{E}(\\vec{r}, t)=\\hat{u} . E_{0}(\\vec{r}) \\cos \\omega t$. The induced dipole moments $\\vec{p}$ will oscillate with the driving laser field frequency $\\omega$. It is well known that an oscillating dipole itself emits electromagnetic radiation. By doing so, electron receives some recoil momentum that causes an electromagnetic friction resulting in a phase shift between the applied electric field and the induced dipole moment. Therefore, the induced dipole moment takes the form $\\vec{p}(\\vec{r}, t)=\\hat{u} E_{0}(\\vec{r}) \\alpha(\\omega) \\cos [\\omega t+\\varphi(\\omega)]$. Here, both the polarizability $\\alpha$ and the phase shift $\\varphi$ depend on the driving frequency $\\omega$. Due to the oscillating nature, all physical quantities of our interest reveal themselves only via the corresponding time-averaged values over a period $2 \\pi / \\omega$ of the laser field. The time-averaged value of a periodically varying quantity is defined as $\\langle f(t)\\rangle=\\frac{\\omega}{2 \\pi} \\int_{0}^{2 \\pi / \\omega} f(t) d t$. Hereafter, the notation $\\langle\\ldots\\rangle$ means time-average of the enclosed quantity.\n\nLaser intensity $I(\\vec{r})$ is related to amplitude of the laser electric field $E_{0}$ as $I(\\vec{r})=\\frac{\\varepsilon_{0} c E_{0}^{2}(\\vec{r})}{2}$, where $\\epsilon_{0}$ is the permittivity of free space and $c$ is the speed of light.\n\n Find the induced dipole potential energy $U_{\\text {dip }}(\\vec{r})=\\left\\langle U_{\\text {induced }}(\\vec{r}, t)\\right\\rangle$ in term of $1.0 \\mathrm{pt}$ $\\alpha, \\varphi, \\varepsilon_{0}, c$, and $I(\\vec{r})$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn the following we will study the case where the neutral atoms are placed in an external laser field that varies in time and space as $\\vec{E}(\\vec{r}, t)=\\hat{u} . E_{0}(\\vec{r}) \\cos \\omega t$. The induced dipole moments $\\vec{p}$ will oscillate with the driving laser field frequency $\\omega$. It is well known that an oscillating dipole itself emits electromagnetic radiation. By doing so, electron receives some recoil momentum that causes an electromagnetic friction resulting in a phase shift between the applied electric field and the induced dipole moment. Therefore, the induced dipole moment takes the form $\\vec{p}(\\vec{r}, t)=\\hat{u} E_{0}(\\vec{r}) \\alpha(\\omega) \\cos [\\omega t+\\varphi(\\omega)]$. Here, both the polarizability $\\alpha$ and the phase shift $\\varphi$ depend on the driving frequency $\\omega$. Due to the oscillating nature, all physical quantities of our interest reveal themselves only via the corresponding time-averaged values over a period $2 \\pi / \\omega$ of the laser field. The time-averaged value of a periodically varying quantity is defined as $\\langle f(t)\\rangle=\\frac{\\omega}{2 \\pi} \\int_{0}^{2 \\pi / \\omega} f(t) d t$. Hereafter, the notation $\\langle\\ldots\\rangle$ means time-average of the enclosed quantity.\n\nLaser intensity $I(\\vec{r})$ is related to amplitude of the laser electric field $E_{0}$ as $I(\\vec{r})=\\frac{\\varepsilon_{0} c E_{0}^{2}(\\vec{r})}{2}$, where $\\epsilon_{0}$ is the permittivity of free space and $c$ is the speed of light.\n\n Find the induced dipole potential energy $U_{\\text {dip }}(\\vec{r})=\\left\\langle U_{\\text {induced }}(\\vec{r}, t)\\right\\rangle$ in term of $1.0 \\mathrm{pt}$ $\\alpha, \\varphi, \\varepsilon_{0}, c$, and $I(\\vec{r})$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1619", "problem": "如俯视图, 在水平面内有两个分别以 $\\mathrm{O}$ 点与 $\\mathrm{O}_{1}$ 点为圆心的导电半圆弧内切于 $\\mathrm{M}$ 点, 半圆 $\\mathrm{O}$ 的半径为 $2 a$,半圆 $\\mathrm{O}_{1}$ 的半径为 $a$;两个半圆弧和圆 $\\mathrm{O}$ 的半径 $\\mathrm{ON}$ 围成的区域内充满垂直于水平面向下的匀强磁场 (未画出),磁感应强度大小为 $B$; 其余区域没有磁场。半径 $\\mathrm{OP}$ 为一均匀细金属棒, 以恒定的角速度 $\\omega$ 绕 $\\mathrm{O}$ 点顺时针旋转, 旋转过程中金属棒 $\\mathrm{OP}$ 与两个半圆弧均接触良好。已知金属棒 $\\mathrm{OP}$ 的电阻为 $R$, 两个半圆弧的电阻可忽略。开始时 $\\mathrm{P}$ 点与 $\\mathrm{M}$ 点重合。在 $t\\left(0 \\leq t \\leq \\frac{\\pi}{\\omega}\\right)$ 时刻, 半径 $\\mathrm{OP}$ 与半圆 $\\mathrm{O}_{1}$交于 $\\mathrm{Q}$ 点。求\n\n[图1]金属棒 OP 所受到的原磁场 $B$ 的作用力的大小。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如俯视图, 在水平面内有两个分别以 $\\mathrm{O}$ 点与 $\\mathrm{O}_{1}$ 点为圆心的导电半圆弧内切于 $\\mathrm{M}$ 点, 半圆 $\\mathrm{O}$ 的半径为 $2 a$,半圆 $\\mathrm{O}_{1}$ 的半径为 $a$;两个半圆弧和圆 $\\mathrm{O}$ 的半径 $\\mathrm{ON}$ 围成的区域内充满垂直于水平面向下的匀强磁场 (未画出),磁感应强度大小为 $B$; 其余区域没有磁场。半径 $\\mathrm{OP}$ 为一均匀细金属棒, 以恒定的角速度 $\\omega$ 绕 $\\mathrm{O}$ 点顺时针旋转, 旋转过程中金属棒 $\\mathrm{OP}$ 与两个半圆弧均接触良好。已知金属棒 $\\mathrm{OP}$ 的电阻为 $R$, 两个半圆弧的电阻可忽略。开始时 $\\mathrm{P}$ 点与 $\\mathrm{M}$ 点重合。在 $t\\left(0 \\leq t \\leq \\frac{\\pi}{\\omega}\\right)$ 时刻, 半径 $\\mathrm{OP}$ 与半圆 $\\mathrm{O}_{1}$交于 $\\mathrm{Q}$ 点。求\n\n[图1]\n\n问题:\n金属棒 OP 所受到的原磁场 $B$ 的作用力的大小。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-02.jpg?height=343&width=554&top_left_y=1279&top_left_x=1205" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_723", "problem": "Sodium has a crystal structure (right) which can be thought of as a repetition of a unit cell shown on the left. The different colors just represent different relative positions in the unit cell.\n[figure1]\n\nGiven that sodium's unit cell is a cube with edge $a=$ $0.428 \\mathrm{~nm}$ and its atomic weight is $23 \\mathrm{~g} / \\mathrm{mol}$, what is the density of sodium?\nA: $0.47 \\mathrm{~g} / \\mathrm{cm}^{3}$\nB: $0.94 \\mathrm{~g} / \\mathrm{cm}^{3}$\nC: $1.41 \\mathrm{~g} / \\mathrm{cm}^{3}$\nD: $4.23 \\mathrm{~g} / \\mathrm{cm}^{3}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nSodium has a crystal structure (right) which can be thought of as a repetition of a unit cell shown on the left. The different colors just represent different relative positions in the unit cell.\n[figure1]\n\nGiven that sodium's unit cell is a cube with edge $a=$ $0.428 \\mathrm{~nm}$ and its atomic weight is $23 \\mathrm{~g} / \\mathrm{mol}$, what is the density of sodium?\n\nA: $0.47 \\mathrm{~g} / \\mathrm{cm}^{3}$\nB: $0.94 \\mathrm{~g} / \\mathrm{cm}^{3}$\nC: $1.41 \\mathrm{~g} / \\mathrm{cm}^{3}$\nD: $4.23 \\mathrm{~g} / \\mathrm{cm}^{3}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_a0d0ab960474d1b0609cg-06.jpg?height=328&width=614&top_left_y=2015&top_left_x=298" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1164", "problem": "Gaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\nHow does the pressure of the gases vary along the height of the chimney?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nGaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\nHow does the pressure of the gases vary along the height of the chimney?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_fd946cfac82ef740b1dag-1.jpg?height=977&width=1644&top_left_y=1453&top_left_x=206" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_210", "problem": "A rectangular slab sits on a frictionless surface. A sphere sits on the slab. There is sufficient friction between the sphere and the slab such that the sphere will not slip relative to the slab. A force to the right is applied to the slab, with both the slab and the sphere initially at rest.\n\n[figure1]\n\nThe sphere will then:\nA: begin spinning clockwise while its center of mass accelerates to the right.\nB: begin spinning counterclockwise while its center of mass accelerates to the left.\nC: begin spinning clockwise while its center of mass accelerates to the left.\nD: begin spinning counterclockwise while its center of mass accelerates to the right.\nE: not spin, while its center of mass accelerates to the right.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA rectangular slab sits on a frictionless surface. A sphere sits on the slab. There is sufficient friction between the sphere and the slab such that the sphere will not slip relative to the slab. A force to the right is applied to the slab, with both the slab and the sphere initially at rest.\n\n[figure1]\n\nThe sphere will then:\n\nA: begin spinning clockwise while its center of mass accelerates to the right.\nB: begin spinning counterclockwise while its center of mass accelerates to the left.\nC: begin spinning clockwise while its center of mass accelerates to the left.\nD: begin spinning counterclockwise while its center of mass accelerates to the right.\nE: not spin, while its center of mass accelerates to the right.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_17b50ad575c0c7f654b7g-20.jpg?height=322&width=824&top_left_y=389&top_left_x=648" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_568", "problem": "The intensity of solar radiation at the surface of the sun is $\\sigma T_{s}^{4}$, so the intensity at the planet's orbit radius is\n\n$$\nI=\\sigma T_{s}^{4} \\frac{R_{s}^{2}}{R_{0}^{2}}\n$$\n\nThe area subtended by the planet is $\\pi R^{2}$, so\n\n$$\nP=\\pi \\sigma T_{s}^{4} \\frac{R^{2} R_{s}^{2}}{R_{0}^{2}}\n$$In this problem, we will investigate a simple thermodynamic model for the conversion of solar energy into wind. Consider a planet of radius $R$, and assume that it rotates so that the same side always faces the Sun. The bright side facing the Sun has a constant uniform temperature $T_{1}$, while the dark side has a constant uniform temperature $T_{2}$. The orbit radius of the planet is $R_{0}$, the Sun has temperature $T_{s}$, and the radius of the Sun is $R_{s}$. Assume that outer space has zero temperature, and treat all objects as ideal blackbodies.\n\nIn order to keep both $T_{1}$ and $T_{2}$ constant, heat must be continually transferred from the bright side to the dark side. By viewing the two hemispheres as the two reservoirs of a reversible heat engine, work can be performed from this temperature difference, which appears in the form of wind power. For simplicity, we assume all of this power is immediately captured and stored by windmills.\n\nFind the wind power $P_{w}$ in terms of $P$ and the temperature ratio $T_{2} / T_{1}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe intensity of solar radiation at the surface of the sun is $\\sigma T_{s}^{4}$, so the intensity at the planet's orbit radius is\n\n$$\nI=\\sigma T_{s}^{4} \\frac{R_{s}^{2}}{R_{0}^{2}}\n$$\n\nThe area subtended by the planet is $\\pi R^{2}$, so\n\n$$\nP=\\pi \\sigma T_{s}^{4} \\frac{R^{2} R_{s}^{2}}{R_{0}^{2}}\n$$\n\nproblem:\nIn this problem, we will investigate a simple thermodynamic model for the conversion of solar energy into wind. Consider a planet of radius $R$, and assume that it rotates so that the same side always faces the Sun. The bright side facing the Sun has a constant uniform temperature $T_{1}$, while the dark side has a constant uniform temperature $T_{2}$. The orbit radius of the planet is $R_{0}$, the Sun has temperature $T_{s}$, and the radius of the Sun is $R_{s}$. Assume that outer space has zero temperature, and treat all objects as ideal blackbodies.\n\nIn order to keep both $T_{1}$ and $T_{2}$ constant, heat must be continually transferred from the bright side to the dark side. By viewing the two hemispheres as the two reservoirs of a reversible heat engine, work can be performed from this temperature difference, which appears in the form of wind power. For simplicity, we assume all of this power is immediately captured and stored by windmills.\n\nFind the wind power $P_{w}$ in terms of $P$ and the temperature ratio $T_{2} / T_{1}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_225", "problem": "The characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nStart with a case, where a point small charged object with a charge $+Q$ is fastened to the table. The center of the dipole is fixed at the distance $L$ from the charged object (see Figure 1). The dipole consists of two identical small balls fastened to the tiny, rigid rod with a length $d, d \\ll L$, so that the moment of inertia can be ignored. Each of the balls has a mass $m$ and have charge $+q$ and $-q$. The dipole can rotate around its center in a plane parallel to the surface of the smooth table.\n\n[figure1]\n\nFigure 1: Schematic representation of the system used in section 1.1\n\nCalculate the period of the small oscillations $T$ of the dipole around its stable equilibrium axis in the electrostatic field of the charged object.\n\nNow, the dipole freely moves around the charged object, which is still fastened to the table. The dipole is launched with an initial velocity $\\mathbf{u}$, as shown in Figure 2. The parameters of the system are chosen in such a way that the period of oscillation of the dipole in the electrostatic field of the fixed object is large enough to assume that the dipole is always oriented along the line between the dipole and the charged object.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nStart with a case, where a point small charged object with a charge $+Q$ is fastened to the table. The center of the dipole is fixed at the distance $L$ from the charged object (see Figure 1). The dipole consists of two identical small balls fastened to the tiny, rigid rod with a length $d, d \\ll L$, so that the moment of inertia can be ignored. Each of the balls has a mass $m$ and have charge $+q$ and $-q$. The dipole can rotate around its center in a plane parallel to the surface of the smooth table.\n\n[figure1]\n\nFigure 1: Schematic representation of the system used in section 1.1\n\nCalculate the period of the small oscillations $T$ of the dipole around its stable equilibrium axis in the electrostatic field of the charged object.\n\nNow, the dipole freely moves around the charged object, which is still fastened to the table. The dipole is launched with an initial velocity $\\mathbf{u}$, as shown in Figure 2. The parameters of the system are chosen in such a way that the period of oscillation of the dipole in the electrostatic field of the fixed object is large enough to assume that the dipole is always oriented along the line between the dipole and the charged object.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ed4e92416bdbac30298dg-1.jpg?height=179&width=1171&top_left_y=1187&top_left_x=477" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_594", "problem": "Part A\n\nA \"Gilbert\" dipole consists of a pair of magnetic monopoles each with a magnitude $q_{m}$ but opposite magnetic charges separated by a distance $d$, where $d$ is small. In this case, assume that $-q_{m}$ is located at $z=0$ and $+q_{m}$ is located at $z=d$.\n\n[figure1]\n\nAssume that magnetic monopoles behave like electric monopoles according to a coulomb-like force\n\nand the magnetic field obeys\n\n$$\nF=\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m 1} q_{m 2}}{r^{2}}\n$$\n\n$$\nB=F / q_{m} .\n$$\n\nBy the second expression, $q_{m}$ must be measured in Newtons per Tesla. But since Tesla are also Newtons per Ampere per meter, then $q_{m}$ is also measured in Ampere meters.\n\nAdding the two terms,\n\n$$\nB(z)=-\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{z^{2}}+\\frac{\\mu_{0}}{4 \\pi} \\frac{q_{m}}{(z+d)^{2}}\n$$\n\nEvaluate this expression in the limit as $d \\rightarrow 0$, assuming that the product $q_{m} d=p_{m}$ is kept constant, keeping only the lowest non-zero term.\n\nSimplifying our previous expression,\n\n$$\nB(z)=\\frac{\\mu_{0}}{4 \\pi} q_{m} d\\left(\\frac{2+d / z}{z(z+d)^{2}}\\right) \\text {. }\n$$\n\nThus in the limit $d \\rightarrow 0$ we have\n\n$$\nB(z)=\\frac{\\mu_{0}}{2 \\pi} \\frac{q_{m} d}{z^{3}}=\\frac{\\mu_{0}}{2 \\pi} \\frac{p_{m}}{z^{3}}\n$$\n\nPart B\n\nAn \"Ampre\" dipole is a magnetic dipole produced by a current loop $I$ around a circle of radius $r$, where $r$ is small. Assume the that the $z$ axis is the axis of rotational symmetry for the circular loop, and the loop lies in the $x y$ plane at $z=0$.\n\n[figure2]\n\nApplying the Biot-Savart law, with $\\mathbf{s}$ the vector from the point on the loop to the point on the $z$ axis,\n\n$$\nB(z)=\\frac{\\mu_{0} I}{4 \\pi} \\oint \\frac{d \\mathbf{l} \\times \\mathbf{s}}{s^{3}}=\\frac{\\mu_{0} I}{4 \\pi} \\frac{2 \\pi r}{r^{2}+z^{2}} \\sin \\theta\n$$\n\nwhere $\\theta$ is the angle between the point on the loop and the center of the loop as measured by the point on the $z$ axis, so\n\n$$\n\\sin \\theta=\\frac{r}{\\sqrt{r^{2}+z^{2}}}\n$$\n\nThen we have\n\n$$\nB(z)=\\frac{\\mu_{0} I}{4 \\pi} \\frac{2 \\pi r^{2}}{\\left(r^{2}+z^{2}\\right)^{3 / 2}}\n$$\n\nLet $k I r^{\\gamma}$ have dimensions equal to that of the quantity $p_{m}$ defined above in Part A.\n\nEvaluate the expression in Part bi in the limit as $r \\rightarrow 0$, assuming that the product $k I r^{\\gamma}=p_{m}^{\\prime}$ is kept constant, keeping only the lowest non-zero term.\n\n$$\nB(z)=\\frac{\\mu_{0} I}{4 \\pi} \\frac{2 \\pi r^{2}}{\\left(r^{2}+z^{2}\\right)^{3 / 2}} \\approx \\frac{\\mu_{0} I}{2 \\pi} \\frac{\\pi r^{2}}{z^{3}}=\\frac{\\mu_{0}}{2 \\pi} \\frac{\\pi}{k} \\frac{p_{m}^{\\prime}}{z^{3}}\n$$Now we try to compare the two approaches if we model a physical magnet as being composed of densely packed microscopic dipoles.\n\n[figure3]\n\nA cylinder of this uniform magnetic material has a radius $R$ and a length $L$. It is composed of $N$ magnetic dipoles that could be either all Ampre type or all Gilbert type. $N$ is a very large number. The axis of rotation of the cylinder and all of the dipoles are all aligned with the $z$ axis and all point in the same direction as defined above so that the magnetic field outside the cylinder is the same in either dipole case as you previously determined. Below is a picture of the two dipole models; they are cubes of side $d<0$ and $z<0$, interact with each other).", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n## Magnetic straws\n\nConsider a cylindrical tube made of a superconducting material. The length of the tube is $l$ and the inner radius is $r$ with $l \\gg r$. The centre of the tube coincides with the origin, and its axis coincides with the $z$-axis.\n\n[figure1]\n\nThere is a magnetic flux $\\Phi$ through the central cross-section of the tube, $z=0, x^{2}+y^{2}0$ and $z<0$, interact with each other).\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_acce87fa3c24869ce68cg-2.jpg?height=816&width=417&top_left_y=777&top_left_x=320" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_661", "problem": "It takes a ray of light roughly 2.5 million years to travel from our galaxy to the Andromeda galaxy. In a new science fiction film, travelers aboard a spaceship reach the Andromeda galaxy by traveling at high speed. The travelers get about 15 years older during the journey. Which of the following statements is true?\n\n\n[figure1]\nA: This is impossible, although an observer on Earth would see the travelers aging slowly, in fact the observers age more quickly.\nB: This is impossible, the travelers would need to live for at least 2.5 million years to reach the Andromeda galaxy.\nC: This is a possible scenario. An observer in the Andromeda galaxy would measure the travel time to be over 2.5 million years, but time runs more slowly for the travelers.\nD: This is a possible scenario, but only if the travelers go faster than the speed of light.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIt takes a ray of light roughly 2.5 million years to travel from our galaxy to the Andromeda galaxy. In a new science fiction film, travelers aboard a spaceship reach the Andromeda galaxy by traveling at high speed. The travelers get about 15 years older during the journey. Which of the following statements is true?\n\n\n[figure1]\n\nA: This is impossible, although an observer on Earth would see the travelers aging slowly, in fact the observers age more quickly.\nB: This is impossible, the travelers would need to live for at least 2.5 million years to reach the Andromeda galaxy.\nC: This is a possible scenario. An observer in the Andromeda galaxy would measure the travel time to be over 2.5 million years, but time runs more slowly for the travelers.\nD: This is a possible scenario, but only if the travelers go faster than the speed of light.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-03.jpg?height=391&width=803&top_left_y=296&top_left_x=178" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_903", "problem": "When modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nFind the inductance per unit length, $L_{x}$, of the cable.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nWhen modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nFind the inductance per unit length, $L_{x}$, of the cable.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-1.jpg?height=358&width=844&top_left_y=1688&top_left_x=617" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1644", "problem": "如图, 一个上端固定、内半径为 $R_{1}$ 的玻璃圆筒底部浸没在大容器中的水面之下, 未与容器底接触; 将另一个外半径为 $R_{2}\\left(R_{2}\\right.$ 略小于 $\\left.R_{1}\\right)$ 的同质玻璃制成的圆筒（上端固定）放置其中，不接触容器底，保持两玻璃圆筒的中轴线重合且坚直。设水与玻璃之间的接触角为 $\\theta$ （即气-液界面在水的最高处的切线与固-液界面的夹角, 见局部放大图)。已知水的密度和表面张力系数分别为 $\\rho$和 $\\alpha$ ，地面重力加速度的大小为 $g$ 。\n\n[图1]试导出系统在水由于毛细作用上升的过程中释放的热量", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一个上端固定、内半径为 $R_{1}$ 的玻璃圆筒底部浸没在大容器中的水面之下, 未与容器底接触; 将另一个外半径为 $R_{2}\\left(R_{2}\\right.$ 略小于 $\\left.R_{1}\\right)$ 的同质玻璃制成的圆筒（上端固定）放置其中，不接触容器底，保持两玻璃圆筒的中轴线重合且坚直。设水与玻璃之间的接触角为 $\\theta$ （即气-液界面在水的最高处的切线与固-液界面的夹角, 见局部放大图)。已知水的密度和表面张力系数分别为 $\\rho$和 $\\alpha$ ，地面重力加速度的大小为 $g$ 。\n\n[图1]\n\n问题:\n试导出系统在水由于毛细作用上升的过程中释放的热量\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_07aa406e17d01fd01b36g-03.jpg?height=556&width=631&top_left_y=907&top_left_x=1112" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_371", "problem": "A drone is pulling a cuboid with a rope as shown in the sketch; the cuboid is sliding slowly, with a constant speed, on the hori zontal floor. The cuboid is made from an ho mogeneous material. You may take measure ments from the sketch (on a separate page) assuming that the dimensions and distances on it are correct within an unknown scale factor. In order to help you in case you don't have access to a printer, and need to read the problem texts directly from the com puter screen, some auxiliary dashed lines are shown in the diagram (which might or might not be useful).\n\nAssuming that the maximal flight height $z_{\\max }$ of a drone with no load is limited by the power of its motor, find $z_{\\max }$ if it is known that the power is just enough for the drone to lift a load equal to $50 \\%$ of its mass off the ground. You may neglect the effect of turbulence on drone's thrust.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA drone is pulling a cuboid with a rope as shown in the sketch; the cuboid is sliding slowly, with a constant speed, on the hori zontal floor. The cuboid is made from an ho mogeneous material. You may take measure ments from the sketch (on a separate page) assuming that the dimensions and distances on it are correct within an unknown scale factor. In order to help you in case you don't have access to a printer, and need to read the problem texts directly from the com puter screen, some auxiliary dashed lines are shown in the diagram (which might or might not be useful).\n\nAssuming that the maximal flight height $z_{\\max }$ of a drone with no load is limited by the power of its motor, find $z_{\\max }$ if it is known that the power is just enough for the drone to lift a load equal to $50 \\%$ of its mass off the ground. You may neglect the effect of turbulence on drone's thrust.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of \\mathrm{~km}, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "\\mathrm{~km}" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_270", "problem": "This question is about a familiar school practical to measure acceleration where a student investigates the acceleration of a toy car on a ramp.\n\nThe height of the ramp is changed and the resulting acceleration obtained from the measurements given below.\n\nThe experimental setup is shown in the diagram.\n\n[figure1]\n\nTo measure the acceleration, the student releases the car from rest at the top of the ramp and uses a stopwatch to time how long it takes for the car to reach the bottom of the ramp. The student records the following results:\n\n| Ramp height $/ \\mathrm{cm}$ | time $\\left(1^{\\text {st }}\\right.$ attempt $) / \\mathrm{s}$ | time $\\left(2^{\\text {nd }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(3^{\\text {rd }}\\right.$ attempt $) / \\mathrm{s}$ |\n| :--- | :--- | :--- | :--- |\n| 5 | 2.45 | 2.48 | 2.41 |\n\nThe student's teacher states \"the final speed of the car is twice the average speed\"\n\nCalculate the average speed of the toy car.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThis question is about a familiar school practical to measure acceleration where a student investigates the acceleration of a toy car on a ramp.\n\nThe height of the ramp is changed and the resulting acceleration obtained from the measurements given below.\n\nThe experimental setup is shown in the diagram.\n\n[figure1]\n\nTo measure the acceleration, the student releases the car from rest at the top of the ramp and uses a stopwatch to time how long it takes for the car to reach the bottom of the ramp. The student records the following results:\n\n| Ramp height $/ \\mathrm{cm}$ | time $\\left(1^{\\text {st }}\\right.$ attempt $) / \\mathrm{s}$ | time $\\left(2^{\\text {nd }}\\right.$ attempt) $/ \\mathrm{s}$ | time $\\left(3^{\\text {rd }}\\right.$ attempt $) / \\mathrm{s}$ |\n| :--- | :--- | :--- | :--- |\n| 5 | 2.45 | 2.48 | 2.41 |\n\nThe student's teacher states \"the final speed of the car is twice the average speed\"\n\nCalculate the average speed of the toy car.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of m/s, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ae10da91e53998b18280g-09.jpg?height=260&width=1579&top_left_y=767&top_left_x=244" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m/s" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_235", "problem": "Earth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nIf the mirror point lies not far from the surface of the Earth the proton collides with the particles of the atmosphere. This distance is small compared with Earth radius $\\left(R_{E}=6400 \\mathrm{~km}\\right)$, so we will assume that the particles will collide when the mirror point is in the surface of the Earth.\n\nCompute the length of the helical path of the proton in a whole cycle, taking into account the two movements we have considered in this problem.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nEarth is a very interesting magnetic system. Earth is frequently approximated as a huge magnetic dipole and therefore its magnetic field is not uniform. Due to this non-uniformity there are some zones in the magnetosphere in which charged particles get trapped. These zones are known as Van Allen belts and particles inside them have three main movements: gyration around each magnetic field line (Gyro motion), movement along the field line (Bounce motion), and rotation of lines around the magnetic axis of the Earth (ignored in this question).\n\n[figure1]\n\nDue to the first two movements, particles travel along a helical path around the field lines. A key parameter to define such movement is the pitch angle $\\alpha$, which is the ratio of the perpendicular and the parallel velocity components to the field line :\n\n$$\n\\tan \\alpha=\\frac{v_{\\perp}}{v_{\\|}}\n$$\n\n[figure2]\n\nIf the mirror point lies not far from the surface of the Earth the proton collides with the particles of the atmosphere. This distance is small compared with Earth radius $\\left(R_{E}=6400 \\mathrm{~km}\\right)$, so we will assume that the particles will collide when the mirror point is in the surface of the Earth.\n\nCompute the length of the helical path of the proton in a whole cycle, taking into account the two movements we have considered in this problem.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of m, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=496&width=705&top_left_y=774&top_left_x=713", "https://cdn.mathpix.com/cropped/2024_03_06_add3ec5e4b807bd424cfg-1.jpg?height=220&width=504&top_left_y=1690&top_left_x=802" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1426", "problem": "具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：若把入射激光束换成方波脉冲激光束, 脉冲宽度为 $T_{1}$ （此期间内光强仍为 $I$ ）, 脉冲之间的间歇时间为 $T_{2}$ 。取第一个脉冲的光开始照到颗粒的时刻为计时零点 $t_{0}=0$, 求第 $n$ 个完整脉冲周期 $\\left(T_{1}+T_{2}\\right)$ 后的瞬间颗粒的转速 $f_{n}$ 的表达式, 并给出转速极限 $f_{n \\rightarrow \\infty}$ 的表达式。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：\n\n问题:\n若把入射激光束换成方波脉冲激光束, 脉冲宽度为 $T_{1}$ （此期间内光强仍为 $I$ ）, 脉冲之间的间歇时间为 $T_{2}$ 。取第一个脉冲的光开始照到颗粒的时刻为计时零点 $t_{0}=0$, 求第 $n$ 个完整脉冲周期 $\\left(T_{1}+T_{2}\\right)$ 后的瞬间颗粒的转速 $f_{n}$ 的表达式, 并给出转速极限 $f_{n \\rightarrow \\infty}$ 的表达式。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[第 $n$ 个完整脉冲周期 $\\left(T_{1}+T_{2}\\right)$ 后的瞬间颗粒的转速 $f_{n}$ 的表达式, 转速极限 $f_{n \\rightarrow \\infty}$ 的表达式。]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-02.jpg?height=450&width=450&top_left_y=1883&top_left_x=1292" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "第 $n$ 个完整脉冲周期 $\\left(T_{1}+T_{2}\\right)$ 后的瞬间颗粒的转速 $f_{n}$ 的表达式", "转速极限 $f_{n \\rightarrow \\infty}$ 的表达式。" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1171", "problem": "A point charge $q$ is placed in the vicinity of a grounded metallic sphere of radius $R$ [see Fig. 1(a)], and consequently a surface charge distribution is induced on the sphere. To calculate the electric field and potential from the distribution of the surface charge is a formidable task. However, the calculation can be considerably simplified by using the so called method of images. In this method, the electric field and potential produced by the charge distributed on the sphere can be represented as an electric field and potential of a single point charge $q$ ' placed inside the sphere (you do not have to prove it). **Note: The electric field of this image charge $q$ ' reproduces the electric field and the potential only outside the sphere (including its surface).**\n\n[figure1]\n\nThe symmetry of the problem dictates that the charge $q$ ' should be placed on the line connecting the point charge $q$ and the center of the sphere [see Fig. 1(b)].\n\nFind the magnitude of force acting on charge $q$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA point charge $q$ is placed in the vicinity of a grounded metallic sphere of radius $R$ [see Fig. 1(a)], and consequently a surface charge distribution is induced on the sphere. To calculate the electric field and potential from the distribution of the surface charge is a formidable task. However, the calculation can be considerably simplified by using the so called method of images. In this method, the electric field and potential produced by the charge distributed on the sphere can be represented as an electric field and potential of a single point charge $q$ ' placed inside the sphere (you do not have to prove it). **Note: The electric field of this image charge $q$ ' reproduces the electric field and the potential only outside the sphere (including its surface).**\n\n[figure1]\n\nThe symmetry of the problem dictates that the charge $q$ ' should be placed on the line connecting the point charge $q$ and the center of the sphere [see Fig. 1(b)].\n\nFind the magnitude of force acting on charge $q$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_843c51f18af1a9802d4bg-1.jpg?height=774&width=1627&top_left_y=1046&top_left_x=220" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_556", "problem": "A beam of muons is maintained in a circular orbit by a uniform magnetic field. Neglect energy loss due to electromagnetic radiation.\n\nThe mass of the muon is $1.88 \\times 10^{-28} \\mathrm{~kg}$, its charge is $-1.602 \\times 10^{-19} \\mathrm{C}$, and its half-life is $1.523 \\mu \\mathrm{s}$.\n\nThe speed of the muons is much less than the speed of light. It is found that half of the muons decay during each full orbit. What is the magnitude of the magnetic field?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA beam of muons is maintained in a circular orbit by a uniform magnetic field. Neglect energy loss due to electromagnetic radiation.\n\nThe mass of the muon is $1.88 \\times 10^{-28} \\mathrm{~kg}$, its charge is $-1.602 \\times 10^{-19} \\mathrm{C}$, and its half-life is $1.523 \\mu \\mathrm{s}$.\n\nThe speed of the muons is much less than the speed of light. It is found that half of the muons decay during each full orbit. What is the magnitude of the magnetic field?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_8", "problem": "A $20,000 \\mathrm{~kg}$ truck is traveling at $25 \\mathrm{~km} / \\mathrm{hr}$. At what speed does a $1000 \\mathrm{~kg}$ car need to travel to have the same kinetic energy as the truck?\nA: $112 \\mathrm{~km} / \\mathrm{hr}$\nB: $132 \\mathrm{~km} / \\mathrm{hr}$\nC: $102 \\mathrm{~km} / \\mathrm{hr}$\nD: $79.0 \\mathrm{~km} / \\mathrm{hr}$\nE: $89.0 \\mathrm{~km} / \\mathrm{hr}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA $20,000 \\mathrm{~kg}$ truck is traveling at $25 \\mathrm{~km} / \\mathrm{hr}$. At what speed does a $1000 \\mathrm{~kg}$ car need to travel to have the same kinetic energy as the truck?\n\nA: $112 \\mathrm{~km} / \\mathrm{hr}$\nB: $132 \\mathrm{~km} / \\mathrm{hr}$\nC: $102 \\mathrm{~km} / \\mathrm{hr}$\nD: $79.0 \\mathrm{~km} / \\mathrm{hr}$\nE: $89.0 \\mathrm{~km} / \\mathrm{hr}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1660", "problem": "田径场上某同学将一铅球以初速度 $v_{0}$ 抛出, 该铅球抛出点的高度为 $H$ 。铅球在田径场上的落点与铅球抛出点的最大水平距离为___ , 对应的抛射角为重力加速度大小为 $g$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n田径场上某同学将一铅球以初速度 $v_{0}$ 抛出, 该铅球抛出点的高度为 $H$ 。铅球在田径场上的落点与铅球抛出点的最大水平距离为___ , 对应的抛射角为重力加速度大小为 $g$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_331", "problem": "The power radiated in gravitational waves by an orbiting binary system is given by $P\\left(r, m_{1}, m_{2}\\right)=\\frac{32}{5} \\frac{G^{4}}{c^{5}} \\frac{\\left(m_{1} m_{2}\\right)^{2}\\left(m_{1}+m_{2}\\right)}{r^{5}}$ where $r$ is the distance between the centers of the two orbiting masses $m_{1}$ and $m_{2}$. It is known that the most compact object is a black hole. The size of a black hole is defined by its Schwarzschild radius $r_{s}=\\frac{2 G m}{c^{2}}$, where $m$ is the mass of\n\nDerive the energy density $u=$ $u(\\varepsilon, E)$ in a uniformly stretched elastic band in terms of the strain $\\varepsilon$ and the Elastic (Young's) modulus $E$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe power radiated in gravitational waves by an orbiting binary system is given by $P\\left(r, m_{1}, m_{2}\\right)=\\frac{32}{5} \\frac{G^{4}}{c^{5}} \\frac{\\left(m_{1} m_{2}\\right)^{2}\\left(m_{1}+m_{2}\\right)}{r^{5}}$ where $r$ is the distance between the centers of the two orbiting masses $m_{1}$ and $m_{2}$. It is known that the most compact object is a black hole. The size of a black hole is defined by its Schwarzschild radius $r_{s}=\\frac{2 G m}{c^{2}}$, where $m$ is the mass of\n\nDerive the energy density $u=$ $u(\\varepsilon, E)$ in a uniformly stretched elastic band in terms of the strain $\\varepsilon$ and the Elastic (Young's) modulus $E$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_24", "problem": "A block with a mass of $1.5 \\mathrm{~kg}$ is at rest on a ramp as shown at right. The ramp is angled at $50^{\\circ}$ to the horizontal and has a rough surface with coefficients of static and kinetic frictions that are $\\mu_{\\mathrm{s}}=0.65$ and $\\mu_{\\mathrm{k}}=0.3$, respectively. The block is being held in place by a spring with $k=40 \\mathrm{~N} / \\mathrm{m}$. What is the minimum distance the spring must be stretched in order for the block to be at rest?\n\n\n[figure1]\nA: $0.043 \\mathrm{~m}$\nB: $0.053 \\mathrm{~m}$\nC: $0.13 \\mathrm{~m}$\nD: $0.17 \\mathrm{~m}$\nE: $0.21 \\mathrm{~m}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA block with a mass of $1.5 \\mathrm{~kg}$ is at rest on a ramp as shown at right. The ramp is angled at $50^{\\circ}$ to the horizontal and has a rough surface with coefficients of static and kinetic frictions that are $\\mu_{\\mathrm{s}}=0.65$ and $\\mu_{\\mathrm{k}}=0.3$, respectively. The block is being held in place by a spring with $k=40 \\mathrm{~N} / \\mathrm{m}$. What is the minimum distance the spring must be stretched in order for the block to be at rest?\n\n\n[figure1]\n\nA: $0.043 \\mathrm{~m}$\nB: $0.053 \\mathrm{~m}$\nC: $0.13 \\mathrm{~m}$\nD: $0.17 \\mathrm{~m}$\nE: $0.21 \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ae7e25be7efc2df26f6eg-09.jpg?height=390&width=491&top_left_y=301&top_left_x=1359" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1222", "problem": "## Air flow around a wing\n\nFor this part of the problem, the following information may be useful. For a flow of liquid or gas in a tube along a streamline, $p+\\rho g h+\\frac{1}{2} \\rho v^{2}=$ const., assuming that the velocity $v$ is much less than the speed of sound. Here $\\rho$ is the density, $h$ is the height, $g$ is free fall acceleration and $p$ is hydrostatic pressure. Streamlines are defined as the trajectories of fluid particles (assuming that the flow pattern is stationary). Note that the term $\\frac{1}{2} \\rho v^{2}$ is called the dynamic pressure.\n\nIn the fig. shown below, a cross-section of an aircraft wing is depicted together with streamlines of the air flow around the wing, as seen in the wing's reference frame. Assume that (a) the air flow is purely two-dimensional (i.e. that the velocity vectors of air lie in the plane of the figure); (b) the streamline pattern is independent of the aircraft speed; (c) there is no wind; (d) the dynamic pressure is much smaller than the atmospheric pressure, $p_{0}=1.0 \\times 10^{5} \\mathrm{~Pa}$.\n\nYou can use a ruler to take measurements from the fig. on the answer sheet.\n\n[figure1]\n\nIf the aircraft's ground speed is $v_{0}=100 \\mathrm{~m} / \\mathrm{s}$, what is the speed of the air, $v_{P}$, at the point $P$ (marked in the fig.) with respect to the ground?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\n## Air flow around a wing\n\nFor this part of the problem, the following information may be useful. For a flow of liquid or gas in a tube along a streamline, $p+\\rho g h+\\frac{1}{2} \\rho v^{2}=$ const., assuming that the velocity $v$ is much less than the speed of sound. Here $\\rho$ is the density, $h$ is the height, $g$ is free fall acceleration and $p$ is hydrostatic pressure. Streamlines are defined as the trajectories of fluid particles (assuming that the flow pattern is stationary). Note that the term $\\frac{1}{2} \\rho v^{2}$ is called the dynamic pressure.\n\nIn the fig. shown below, a cross-section of an aircraft wing is depicted together with streamlines of the air flow around the wing, as seen in the wing's reference frame. Assume that (a) the air flow is purely two-dimensional (i.e. that the velocity vectors of air lie in the plane of the figure); (b) the streamline pattern is independent of the aircraft speed; (c) there is no wind; (d) the dynamic pressure is much smaller than the atmospheric pressure, $p_{0}=1.0 \\times 10^{5} \\mathrm{~Pa}$.\n\nYou can use a ruler to take measurements from the fig. on the answer sheet.\n\n[figure1]\n\nIf the aircraft's ground speed is $v_{0}=100 \\mathrm{~m} / \\mathrm{s}$, what is the speed of the air, $v_{P}$, at the point $P$ (marked in the fig.) with respect to the ground?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~m} / \\mathrm{s}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_acce87fa3c24869ce68cg-1.jpg?height=260&width=948&top_left_y=1269&top_left_x=1039" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m} / \\mathrm{s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_742", "problem": "The graph below is showing the wavefronts of a plane wave with frequency $3 \\mathrm{~Hz}$ travelling at speed $v=6 \\mathrm{~m} / \\mathrm{s}$ hitting a board at an angle $\\pi / 6$. What is the phase difference between the waves at two points $1 \\mathrm{~m}$ apart along the board?\n\n[figure1]\nA: $\\frac{\\pi}{2}$\nB: $\\frac{\\pi}{3}$\nC: $\\frac{\\pi}{4}$\nD: $\\frac{\\pi}{6}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe graph below is showing the wavefronts of a plane wave with frequency $3 \\mathrm{~Hz}$ travelling at speed $v=6 \\mathrm{~m} / \\mathrm{s}$ hitting a board at an angle $\\pi / 6$. What is the phase difference between the waves at two points $1 \\mathrm{~m}$ apart along the board?\n\n[figure1]\n\nA: $\\frac{\\pi}{2}$\nB: $\\frac{\\pi}{3}$\nC: $\\frac{\\pi}{4}$\nD: $\\frac{\\pi}{6}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22e26a14ee6fdd9254b6g-03.jpg?height=280&width=781&top_left_y=1367&top_left_x=1103" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1129", "problem": "A point charge $q$ is placed in the vicinity of a grounded metallic sphere of radius $R$ [see Fig. 1(a)], and consequently a surface charge distribution is induced on the sphere. To calculate the electric field and potential from the distribution of the surface charge is a formidable task. However, the calculation can be considerably simplified by using the so called method of images. In this method, the electric field and potential produced by the charge distributed on the sphere can be represented as an electric field and potential of a single point charge $q$ ' placed inside the sphere (you do not have to prove it). **Note: The electric field of this image charge $q$ ' reproduces the electric field and the potential only outside the sphere (including its surface).**\n\n[figure1]\n\nConsider a point charge $q$ placed at a distance $d$ from the center of a grounded metallic sphere of radius $R$. We are interested in how the grounded metallic sphere affects the electric field at point $A$ on the opposite side of the sphere (see Fig. 2). Point $A$ is on the line connecting charge $q$ and the center of the sphere; its distance from the point charge $q$ is $r$.\n\nIn which limit of $d$ does the grounded metallic sphere screen the field of the charge $q$ completely, such that the electric field at point $A$ is exactly zero?\n\n[figure2]\n\nFig 2. The electric field at point $A$ is partially screened by the grounded sphere.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA point charge $q$ is placed in the vicinity of a grounded metallic sphere of radius $R$ [see Fig. 1(a)], and consequently a surface charge distribution is induced on the sphere. To calculate the electric field and potential from the distribution of the surface charge is a formidable task. However, the calculation can be considerably simplified by using the so called method of images. In this method, the electric field and potential produced by the charge distributed on the sphere can be represented as an electric field and potential of a single point charge $q$ ' placed inside the sphere (you do not have to prove it). **Note: The electric field of this image charge $q$ ' reproduces the electric field and the potential only outside the sphere (including its surface).**\n\n[figure1]\n\nConsider a point charge $q$ placed at a distance $d$ from the center of a grounded metallic sphere of radius $R$. We are interested in how the grounded metallic sphere affects the electric field at point $A$ on the opposite side of the sphere (see Fig. 2). Point $A$ is on the line connecting charge $q$ and the center of the sphere; its distance from the point charge $q$ is $r$.\n\nIn which limit of $d$ does the grounded metallic sphere screen the field of the charge $q$ completely, such that the electric field at point $A$ is exactly zero?\n\n[figure2]\n\nFig 2. The electric field at point $A$ is partially screened by the grounded sphere.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_843c51f18af1a9802d4bg-1.jpg?height=774&width=1627&top_left_y=1046&top_left_x=220", "https://cdn.mathpix.com/cropped/2024_03_14_843c51f18af1a9802d4bg-2.jpg?height=428&width=893&top_left_y=545&top_left_x=587" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_724", "problem": "A closed 5-litre cylinder containing $0.25 \\mathrm{~g}$ of a substance in solid and liquid forms was slowly heated using a constant power. A graph of the temperature as a function of time is shown below. The heat capacity of the liquid is $2.43 \\mathrm{~J} /(\\mathrm{g} \\cdot \\mathrm{K})$, the latent heat of melting is $105 \\mathrm{~J} / \\mathrm{g}$, and the molar density is $46 \\mathrm{~g} / \\mathrm{mol}$.\n\n[figure1]\n\nWhat is the latent heat of evaporation of the substance?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA closed 5-litre cylinder containing $0.25 \\mathrm{~g}$ of a substance in solid and liquid forms was slowly heated using a constant power. A graph of the temperature as a function of time is shown below. The heat capacity of the liquid is $2.43 \\mathrm{~J} /(\\mathrm{g} \\cdot \\mathrm{K})$, the latent heat of melting is $105 \\mathrm{~J} / \\mathrm{g}$, and the molar density is $46 \\mathrm{~g} / \\mathrm{mol}$.\n\n[figure1]\n\nWhat is the latent heat of evaporation of the substance?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of J/g, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-08.jpg?height=1071&width=1612&top_left_y=825&top_left_x=251" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "J/g" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_195", "problem": "A spring stretched to double its unstretched length has a potential energy $U_{0}$. If the spring is cut in half, and each half spring is stretched to double its unstretched length, then the total potential energy stored in the two half springs will be\nA: $4 U_{0}$\nB: $2 U_{0}$\nC: $U_{0} $ \nD: $U_{0} / 2$\nE: $U_{0} / 4$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA spring stretched to double its unstretched length has a potential energy $U_{0}$. If the spring is cut in half, and each half spring is stretched to double its unstretched length, then the total potential energy stored in the two half springs will be\n\nA: $4 U_{0}$\nB: $2 U_{0}$\nC: $U_{0} $ \nD: $U_{0} / 2$\nE: $U_{0} / 4$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_205", "problem": "A uniform thin circular rubber band of mass $M$ and spring constant $k$ has an original radius $R$. Now it is tossed into the air. Assume it remains circular when stabilized in air and rotates at angular speed $\\omega$ about its center uniformly. Which of the following gives the new radius of the rubber band?\nA: $(2 \\pi k R) /\\left(2 \\pi k-M \\omega^{2}\\right)$\nB: $(4 \\pi k R) /\\left(4 \\pi k-M \\omega^{2}\\right)$\nC: $\\left(8 \\pi^{2} k R\\right) /\\left(8 \\pi^{2} k-M \\omega^{2}\\right)$\nD: $\\left(4 \\pi^{2} k R\\right) /\\left(4 \\pi^{2} k-M \\omega^{2}\\right) $ \nE: $(4 \\pi k R) /\\left(2 \\pi k-M \\omega^{2}\\right)$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA uniform thin circular rubber band of mass $M$ and spring constant $k$ has an original radius $R$. Now it is tossed into the air. Assume it remains circular when stabilized in air and rotates at angular speed $\\omega$ about its center uniformly. Which of the following gives the new radius of the rubber band?\n\nA: $(2 \\pi k R) /\\left(2 \\pi k-M \\omega^{2}\\right)$\nB: $(4 \\pi k R) /\\left(4 \\pi k-M \\omega^{2}\\right)$\nC: $\\left(8 \\pi^{2} k R\\right) /\\left(8 \\pi^{2} k-M \\omega^{2}\\right)$\nD: $\\left(4 \\pi^{2} k R\\right) /\\left(4 \\pi^{2} k-M \\omega^{2}\\right) $ \nE: $(4 \\pi k R) /\\left(2 \\pi k-M \\omega^{2}\\right)$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_206", "problem": "A massless rope passes over a frictionless pulley. Particles of mass $M$ and $M+m$ are suspended from the two different ends of the rope. If $m=0$, the tension $T$ in the pulley rope is $M g$. If instead the value $m$ increases to infinity, the value of the tension\nA: stays constant\nB: decreases, approaching a nonzero constant\nC: decreases, approaching zero\nD: increases, approaching a finite constant \nE: increases to infinity\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA massless rope passes over a frictionless pulley. Particles of mass $M$ and $M+m$ are suspended from the two different ends of the rope. If $m=0$, the tension $T$ in the pulley rope is $M g$. If instead the value $m$ increases to infinity, the value of the tension\n\nA: stays constant\nB: decreases, approaching a nonzero constant\nC: decreases, approaching zero\nD: increases, approaching a finite constant \nE: increases to infinity\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_592", "problem": "Beloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nAssuming the dome is charged to $500 \\mathrm{kV}$, determine the strength of the electric field at the surface of the dome.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nBeloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nAssuming the dome is charged to $500 \\mathrm{kV}$, determine the strength of the electric field at the surface of the dome.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~V} / \\mathrm{m}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=468&width=1592&top_left_y=438&top_left_x=259", "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=493&width=1268&top_left_y=1339&top_left_x=426" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~V} / \\mathrm{m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1663", "problem": "一颗人造地球通讯卫星 (同步卫星) 对地球的张角能覆盖赤道上空东经 $\\theta_{0}-\\Delta \\theta$ 到东经 $\\theta_{0}+\\Delta \\theta$ 之间的区域。已知地球半径为 $R_{0}$, 地球表面处的重力加速度大小为 $g$, 地球自转周期为 $T 。 \\Delta \\theta$ 的值等于\nA: $\\arcsin \\left(\\frac{4 \\pi^{2} R_{0}}{T^{2} g}\\right)^{1 / 3}$\nB: $2 \\arcsin \\left(\\frac{4 \\pi^{2} R_{0}}{T^{2} g}\\right)^{1 / 3}$\nC: $\\arccos \\left(\\frac{4 \\pi^{2} R_{0}}{T^{2} g}\\right)^{1 / 3}$\nD: $2 \\arccos \\left(\\frac{4 \\pi^{2} R_{0}}{T^{2} g}\\right)^{1 / 3}$\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n一颗人造地球通讯卫星 (同步卫星) 对地球的张角能覆盖赤道上空东经 $\\theta_{0}-\\Delta \\theta$ 到东经 $\\theta_{0}+\\Delta \\theta$ 之间的区域。已知地球半径为 $R_{0}$, 地球表面处的重力加速度大小为 $g$, 地球自转周期为 $T 。 \\Delta \\theta$ 的值等于\n\nA: $\\arcsin \\left(\\frac{4 \\pi^{2} R_{0}}{T^{2} g}\\right)^{1 / 3}$\nB: $2 \\arcsin \\left(\\frac{4 \\pi^{2} R_{0}}{T^{2} g}\\right)^{1 / 3}$\nC: $\\arccos \\left(\\frac{4 \\pi^{2} R_{0}}{T^{2} g}\\right)^{1 / 3}$\nD: $2 \\arccos \\left(\\frac{4 \\pi^{2} R_{0}}{T^{2} g}\\right)^{1 / 3}$\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_824", "problem": "Calculation in the previous part shows that in order to build the space elevator, it is neccessary to have light materials with very high tensile strength. Carbon nanotubes are materials that meet such requirements because of strong chemical bondings between very light atoms. Two natural polymorphs of carbon are diamond and graphite. In diamond every carbon atom is surrounded by four nearest neighbor (NN) atoms to form a tetrahedron. Graphite has a layer structure. In each layer, carbon atoms are arranged in a hexagonal plane lattice with three NNs. Although diamond is known as the hardest materials, covalent bondings between carbon atoms in hexagonal layers of graphite is stronger than those between carbon atoms in diamond tetrahedra. Graphite is much softer than diamond because of the van der Waals bonding between carbon atoms of different layers, which is much weaker than covalent bonding.\n\n[figure1]\n\nFigure 2. Graphite structure\n(a)\n[figure2]\n\nFigure 3. Graphene (a) and carbon nanotube (b).\n\nA monatomic layer in graphite is called graphene and has monoatomic thickness. Isolated graphene sheet is not stable and has a tendency to roll up to form carbon spheres or carbon nanotubes. The hexagonal crystal lattice of graphene is depicted in Fig. 4. The distance between two NN carbon atoms is $a=0.142 \\mathrm{~nm}$ and the distance between two closest parallel bondings is $b=0.246 \\mathrm{~nm}$. Because the covalent bondings between carbon atoms in graphene are very strong, mechanical properties of carbon nanotubes are very special. They have an extremely large Young's modulus and tensile strength, as well as a very light density. Young's modulus is defined as the ratio of the stress along an axis to the strain (ratio of deformation over initial length) along that axis in the range of stress in which Hooke's law holds.\n\n[figure3]\n\nFigure 4. Graphene.\n\n## Theory\n\n[figure4]\n\nFigure 5. An illustration of a carbon nanotube with 9 carbon-carbon parallel bondings. Note: In this problem, there are 27 carbon-carbon parallel bondings. (1) parallel bond; (2) slanted bond; (3) tube axis.\n\nNow we examine some mechanical properties of a carbon nanotube having 27 carbon-carbon bondings parallel to the tube axis (for an illustration, see Figure 5). The bonding between two carbon atoms can be described by the Morse potential $V(x)=V_{0}\\left(e^{-4 \\frac{x}{a}}-2 e^{-2 \\frac{x}{a}}\\right)$. Here $a=0.142 \\mathrm{~nm}$ is the equilibrium distance between two $\\mathrm{NN}$ carbon atoms, $V_{0}=4.93 \\mathrm{eV}$ is the bonding energy, and $x$ is the displacement of the atom from the equilibrium position. Hereafter, we approximate the Morse potential by a quadratic potential $V(x)=P+Q x^{2}$. All non-nearest-neighbor interactions are neglected. In this approximation, one can propose that carbon atoms are bonded through \"springs\" with the spring constant $k$. Changes in angles between bonds are neglected.\n\n Calculate the value of the maximum extension $x_{\\max }$ of the spring.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nCalculation in the previous part shows that in order to build the space elevator, it is neccessary to have light materials with very high tensile strength. Carbon nanotubes are materials that meet such requirements because of strong chemical bondings between very light atoms. Two natural polymorphs of carbon are diamond and graphite. In diamond every carbon atom is surrounded by four nearest neighbor (NN) atoms to form a tetrahedron. Graphite has a layer structure. In each layer, carbon atoms are arranged in a hexagonal plane lattice with three NNs. Although diamond is known as the hardest materials, covalent bondings between carbon atoms in hexagonal layers of graphite is stronger than those between carbon atoms in diamond tetrahedra. Graphite is much softer than diamond because of the van der Waals bonding between carbon atoms of different layers, which is much weaker than covalent bonding.\n\n[figure1]\n\nFigure 2. Graphite structure\n(a)\n[figure2]\n\nFigure 3. Graphene (a) and carbon nanotube (b).\n\nA monatomic layer in graphite is called graphene and has monoatomic thickness. Isolated graphene sheet is not stable and has a tendency to roll up to form carbon spheres or carbon nanotubes. The hexagonal crystal lattice of graphene is depicted in Fig. 4. The distance between two NN carbon atoms is $a=0.142 \\mathrm{~nm}$ and the distance between two closest parallel bondings is $b=0.246 \\mathrm{~nm}$. Because the covalent bondings between carbon atoms in graphene are very strong, mechanical properties of carbon nanotubes are very special. They have an extremely large Young's modulus and tensile strength, as well as a very light density. Young's modulus is defined as the ratio of the stress along an axis to the strain (ratio of deformation over initial length) along that axis in the range of stress in which Hooke's law holds.\n\n[figure3]\n\nFigure 4. Graphene.\n\n## Theory\n\n[figure4]\n\nFigure 5. An illustration of a carbon nanotube with 9 carbon-carbon parallel bondings. Note: In this problem, there are 27 carbon-carbon parallel bondings. (1) parallel bond; (2) slanted bond; (3) tube axis.\n\nNow we examine some mechanical properties of a carbon nanotube having 27 carbon-carbon bondings parallel to the tube axis (for an illustration, see Figure 5). The bonding between two carbon atoms can be described by the Morse potential $V(x)=V_{0}\\left(e^{-4 \\frac{x}{a}}-2 e^{-2 \\frac{x}{a}}\\right)$. Here $a=0.142 \\mathrm{~nm}$ is the equilibrium distance between two $\\mathrm{NN}$ carbon atoms, $V_{0}=4.93 \\mathrm{eV}$ is the bonding energy, and $x$ is the displacement of the atom from the equilibrium position. Hereafter, we approximate the Morse potential by a quadratic potential $V(x)=P+Q x^{2}$. All non-nearest-neighbor interactions are neglected. In this approximation, one can propose that carbon atoms are bonded through \"springs\" with the spring constant $k$. Changes in angles between bonds are neglected.\n\n Calculate the value of the maximum extension $x_{\\max }$ of the spring.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of nm, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-2.jpg?height=549&width=525&top_left_y=1850&top_left_x=731", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-3.jpg?height=430&width=900&top_left_y=710&top_left_x=518", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-3.jpg?height=451&width=691&top_left_y=1856&top_left_x=660", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-4.jpg?height=944&width=1144&top_left_y=172&top_left_x=184" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "nm" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_118", "problem": "A cylindrical space station produces 'artificial gravity' by rotating with angular frequency $\\omega$. Consider working in the reference frame rotating with the space station. In this frame, an astronaut is initially at rest standing on the floor, facing in the direction that the space station is rotating. The astronaut jumps up vertically relative to the floor of the space station, with an initial speed less than that of the speed of the floor. Just after leaving the floor, the motion of the astronaut, relative to the space station floor,\n\n[figure1]\nA: always has a component of acceleration directed toward the floor, and they land at the same point they jumped from.\nB: always has a component of acceleration directed toward the floor, and they land in front of the point they jumped from. \nC: always has a component of acceleration directed toward the floor, and they land behind the point they jumped from.\nD: has a component of acceleration directed away from the floor, and they land behind the point they jumped from.\nE: has a zero acceleration relative to the floor, and the astronaut never reaches the floor again.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA cylindrical space station produces 'artificial gravity' by rotating with angular frequency $\\omega$. Consider working in the reference frame rotating with the space station. In this frame, an astronaut is initially at rest standing on the floor, facing in the direction that the space station is rotating. The astronaut jumps up vertically relative to the floor of the space station, with an initial speed less than that of the speed of the floor. Just after leaving the floor, the motion of the astronaut, relative to the space station floor,\n\n[figure1]\n\nA: always has a component of acceleration directed toward the floor, and they land at the same point they jumped from.\nB: always has a component of acceleration directed toward the floor, and they land in front of the point they jumped from. \nC: always has a component of acceleration directed toward the floor, and they land behind the point they jumped from.\nD: has a component of acceleration directed away from the floor, and they land behind the point they jumped from.\nE: has a zero acceleration relative to the floor, and the astronaut never reaches the floor again.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_6e7e781e6599f99f638bg-13.jpg?height=279&width=533&top_left_y=500&top_left_x=796" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1306", "problem": "爱因斯坦引力理论预言物质分布的变化会导致时空几何结构的波动一一引力波。为简明起见, 考虑沿 $z$ 轴传播的平面引力波。对于任意给定的 $z$, 在 $x-y$ 二维空间中两个无限邻近点 $(x, y)$ 和 $(x+\\mathrm{d} x, y+\\mathrm{d} y)$ 之间距离 $\\mathrm{d} r$ 的表达式为\n\n$$\n\\mathrm{d} r=\\sqrt{\\left(1+f_{1}\\right)(\\mathrm{d} x)^{2}+f_{2}(\\mathrm{~d} x \\mathrm{~d} y+\\mathrm{d} y \\mathrm{~d} x)+\\left(1-f_{1}\\right)(\\mathrm{d} y)^{2}}\n$$\n\n引力波体现为 $f_{1}$ 和 $f_{2}$ 的变化 (波动)。假定引力波的波源为双星系统。设该双星系统两星体质量均为 $M$ 。取双星系统的质心为坐标原点 $O^{\\prime}$, 双星系统在 $x^{\\prime}-y^{\\prime}$ 二维空间中旋转。已知在特定条件下, $f_{1}$ 和 $f_{2}$ 可近似表示为\n\n$$\nf_{1}=\\frac{8 \\pi G}{c^{4}} \\frac{2}{l} \\frac{\\mathrm{d}^{2}}{\\mathrm{~d} t^{2}}\\left[I_{1}\\left(t-\\frac{l}{c}\\right)\\right], \\quad f_{2}=\\frac{8 \\pi G}{c^{4}} \\frac{2}{l} \\frac{\\mathrm{d}^{2}}{\\mathrm{~d} t^{2}}\\left[I_{2}\\left(t-\\frac{l}{c}\\right)\\right]\n$$\n\n式中 $G$ 为引力常量, $l$ 为在 $z^{\\prime}$ 轴上引力波探测点到 $O^{\\prime}$ 点的距离 (远大于双星系统中两星体之间的距离），而\n\n$$\nI_{1}(t)=\\left[x_{1}^{\\prime 2}(t)+x_{2}^{\\prime 2}(t)\\right] M, \\quad I_{2}(t)=x_{1}^{\\prime}(t) y_{2}^{\\prime}(t) M\n$$\n\n$\\left(x_{1}^{\\prime}(t), y_{1}^{\\prime}(t)\\right)$ 、 $\\left(x_{2}^{\\prime}(t), y_{2}^{\\prime}(t)\\right)$ 为两星体在其质心系中的坐标（注: 在这样的定义下, $d r$ 表达中的系数应替换为 $f_{1} \\rightarrow f_{1}, f_{2} \\rightarrow 2 f_{2}$ ) 。已知该引力波的频率为 $\\omega$, 假设引力波辐射对双星系统轨道运动的影响可忽略, 求 $f_{1}$ 和 $f_{2}$ 在探测点处的振幅。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n爱因斯坦引力理论预言物质分布的变化会导致时空几何结构的波动一一引力波。为简明起见, 考虑沿 $z$ 轴传播的平面引力波。对于任意给定的 $z$, 在 $x-y$ 二维空间中两个无限邻近点 $(x, y)$ 和 $(x+\\mathrm{d} x, y+\\mathrm{d} y)$ 之间距离 $\\mathrm{d} r$ 的表达式为\n\n$$\n\\mathrm{d} r=\\sqrt{\\left(1+f_{1}\\right)(\\mathrm{d} x)^{2}+f_{2}(\\mathrm{~d} x \\mathrm{~d} y+\\mathrm{d} y \\mathrm{~d} x)+\\left(1-f_{1}\\right)(\\mathrm{d} y)^{2}}\n$$\n\n引力波体现为 $f_{1}$ 和 $f_{2}$ 的变化 (波动)。\n\n问题:\n假定引力波的波源为双星系统。设该双星系统两星体质量均为 $M$ 。取双星系统的质心为坐标原点 $O^{\\prime}$, 双星系统在 $x^{\\prime}-y^{\\prime}$ 二维空间中旋转。已知在特定条件下, $f_{1}$ 和 $f_{2}$ 可近似表示为\n\n$$\nf_{1}=\\frac{8 \\pi G}{c^{4}} \\frac{2}{l} \\frac{\\mathrm{d}^{2}}{\\mathrm{~d} t^{2}}\\left[I_{1}\\left(t-\\frac{l}{c}\\right)\\right], \\quad f_{2}=\\frac{8 \\pi G}{c^{4}} \\frac{2}{l} \\frac{\\mathrm{d}^{2}}{\\mathrm{~d} t^{2}}\\left[I_{2}\\left(t-\\frac{l}{c}\\right)\\right]\n$$\n\n式中 $G$ 为引力常量, $l$ 为在 $z^{\\prime}$ 轴上引力波探测点到 $O^{\\prime}$ 点的距离 (远大于双星系统中两星体之间的距离），而\n\n$$\nI_{1}(t)=\\left[x_{1}^{\\prime 2}(t)+x_{2}^{\\prime 2}(t)\\right] M, \\quad I_{2}(t)=x_{1}^{\\prime}(t) y_{2}^{\\prime}(t) M\n$$\n\n$\\left(x_{1}^{\\prime}(t), y_{1}^{\\prime}(t)\\right)$ 、 $\\left(x_{2}^{\\prime}(t), y_{2}^{\\prime}(t)\\right)$ 为两星体在其质心系中的坐标（注: 在这样的定义下, $d r$ 表达中的系数应替换为 $f_{1} \\rightarrow f_{1}, f_{2} \\rightarrow 2 f_{2}$ ) 。已知该引力波的频率为 $\\omega$, 假设引力波辐射对双星系统轨道运动的影响可忽略, 求 $f_{1}$ 和 $f_{2}$ 在探测点处的振幅。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1729", "problem": "一座平顶房屋, 顶的面积 $S=40 \\mathrm{~m}^{2}$. 第一次连续下了 $\\mathrm{t}=24$ 小时的雨, 雨滴沿竖直方向以 $\\mathrm{v}=5.0 \\mathrm{~m} / \\mathrm{s}$ 的速度落到屋顶, 假定雨滴撞击屋顶的时间极短且不反弹, 并立即流走. 第二次气温在摄氏零下若干度, 而且是下冻雨, 也下了 24 小时, 全部冻雨落到屋顶便都结成冰并留在屋顶上, 测得冰层的厚度 $\\mathrm{d}=25 \\mathrm{~mm}$. 已知两次下雨的雨量相等, 冰的密度为 $9 \\times 10^{2} \\mathrm{~kg} / \\mathrm{m}^{3}$. 由以上数据可估算得第二次下的冻雨结成冰对屋顶的压力为 $N$, 第一次下雨过程中, 雨对屋顶的撞击使整个屋顶受到的压力为 $N$.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n一座平顶房屋, 顶的面积 $S=40 \\mathrm{~m}^{2}$. 第一次连续下了 $\\mathrm{t}=24$ 小时的雨, 雨滴沿竖直方向以 $\\mathrm{v}=5.0 \\mathrm{~m} / \\mathrm{s}$ 的速度落到屋顶, 假定雨滴撞击屋顶的时间极短且不反弹, 并立即流走. 第二次气温在摄氏零下若干度, 而且是下冻雨, 也下了 24 小时, 全部冻雨落到屋顶便都结成冰并留在屋顶上, 测得冰层的厚度 $\\mathrm{d}=25 \\mathrm{~mm}$. 已知两次下雨的雨量相等, 冰的密度为 $9 \\times 10^{2} \\mathrm{~kg} / \\mathrm{m}^{3}$. 由以上数据可估算得第二次下的冻雨结成冰对屋顶的压力为 $N$, 第一次下雨过程中, 雨对屋顶的撞击使整个屋顶受到的压力为 $N$.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[第二次下的冻雨结成冰对屋顶的压力, 第一次下雨过程中, 雨对屋顶的撞击使整个屋顶受到的压力]\n它们的单位依次是[$N$, $N$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$N$", "$N$" ], "answer_sequence": [ "第二次下的冻雨结成冰对屋顶的压力", "第一次下雨过程中, 雨对屋顶的撞击使整个屋顶受到的压力" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_139", "problem": "Kepler's Laws state that\n\nI. the orbits of planets are elliptical with one focus at the sun,\n\nII. a line connecting the sun and a planet sweeps out equal areas in equal times, and\n\nIII. the square the period of a planet's orbit is proportional to the cube of its semimajor axis.\n\nWhich of these laws would remain true if the force of gravity were proportional to $1 / r^{3}$ rather than $1 / r^{2}$ ?\nA: Only I.\nB: Only II. \nC: Only III.\nD: Both II and III.\nE: None of the above.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nKepler's Laws state that\n\nI. the orbits of planets are elliptical with one focus at the sun,\n\nII. a line connecting the sun and a planet sweeps out equal areas in equal times, and\n\nIII. the square the period of a planet's orbit is proportional to the cube of its semimajor axis.\n\nWhich of these laws would remain true if the force of gravity were proportional to $1 / r^{3}$ rather than $1 / r^{2}$ ?\n\nA: Only I.\nB: Only II. \nC: Only III.\nD: Both II and III.\nE: None of the above.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_648", "problem": "Relative to an inertial frame at the centre of the earth, approximately how fast is a person in Vancouver (latitude $\\left.49.3^{\\circ} \\mathrm{N}\\right)$ moving due to the Earth's rotation around its axis?\nA: $700 \\mathrm{~m} / \\mathrm{s}$\nB: $300 \\mathrm{~m} / \\mathrm{s}$\nC: $450 \\mathrm{~m} / \\mathrm{s}$\nD: $400 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nRelative to an inertial frame at the centre of the earth, approximately how fast is a person in Vancouver (latitude $\\left.49.3^{\\circ} \\mathrm{N}\\right)$ moving due to the Earth's rotation around its axis?\n\nA: $700 \\mathrm{~m} / \\mathrm{s}$\nB: $300 \\mathrm{~m} / \\mathrm{s}$\nC: $450 \\mathrm{~m} / \\mathrm{s}$\nD: $400 \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_207", "problem": "As shown in the figure, a ping-pong ball with mass $m$ with initial horizontal velocity $v$ and angular velocity $\\omega$ comes into contact with the ground. Friction is not negligible, so both the velocity and angular velocity of the ping-pong ball changes. What is the critical velocity $v_{c}$ such that the ping-pong will stop and remain stopped? Treat the ping-pong ball as a hollow sphere.\n\n[figure1]\nA: $v=\\frac{2}{3} R \\omega $ \nB: $v=\\frac{2}{5} R \\omega$\nC: $v=R \\omega$\nD: $v=\\frac{3}{5} R \\omega$\nE: $v=\\frac{5}{3} R \\omega$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAs shown in the figure, a ping-pong ball with mass $m$ with initial horizontal velocity $v$ and angular velocity $\\omega$ comes into contact with the ground. Friction is not negligible, so both the velocity and angular velocity of the ping-pong ball changes. What is the critical velocity $v_{c}$ such that the ping-pong will stop and remain stopped? Treat the ping-pong ball as a hollow sphere.\n\n[figure1]\n\nA: $v=\\frac{2}{3} R \\omega $ \nB: $v=\\frac{2}{5} R \\omega$\nC: $v=R \\omega$\nD: $v=\\frac{3}{5} R \\omega$\nE: $v=\\frac{5}{3} R \\omega$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_06de5165d0cf42eb6dbcg-18.jpg?height=409&width=531&top_left_y=473&top_left_x=794" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_86", "problem": "The distance between the electron and the proton in the hydrogen atom is about $0.53 \\times 10^{-10} \\mathrm{~m}$. By what factor is the electrical force between the electron and proton stronger than the gravitational force between them?\nA: $1.3 \\times 10^{39}$\nB: $2.3 \\times 10^{39}$\nC: $3.3 \\times 10^{39}$\nD: $4.3 \\times 10^{39}$\nE: $5.3 \\times 10^{39}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nThe distance between the electron and the proton in the hydrogen atom is about $0.53 \\times 10^{-10} \\mathrm{~m}$. By what factor is the electrical force between the electron and proton stronger than the gravitational force between them?\n\nA: $1.3 \\times 10^{39}$\nB: $2.3 \\times 10^{39}$\nC: $3.3 \\times 10^{39}$\nD: $4.3 \\times 10^{39}$\nE: $5.3 \\times 10^{39}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_436", "problem": "The figure shows a more complex system, known as a block and tackle, consisting of two light pulley blocks and a light cord.\n\n[figure1]\nFigure: Two light pulley blocks and a light cord.\n\nHow much work is done by the external agent pulling cord towards $\\mathbf{X}$?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe figure shows a more complex system, known as a block and tackle, consisting of two light pulley blocks and a light cord.\n\n[figure1]\nFigure: Two light pulley blocks and a light cord.\n\nHow much work is done by the external agent pulling cord towards $\\mathbf{X}$?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-4.jpg?height=1074&width=514&top_left_y=1593&top_left_x=1356" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_719", "problem": "Consider motion in only one dimension. An object with mass $m_{1}$ and initial speed $v_{1, i}$ in the direction of the positive $x$ axis collides with a stationary object of mass $m_{2}$. The first object bounces back, moving with speed $v_{1, f}$ in the direction of the negative $x$ axis. The second object has a final speed $v_{2, f}$. If the collision was perfectly elastic, which of the following must be true?\nA: $m_{1}=m_{2}$\nB: $m_{1}>m_{2}$\nC: Such a collision is impossible\nD: $m_{1}m_{2}$\nC: Such a collision is impossible\nD: $m_{1}\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nEstimate the time $t_{2}$ needed for the radius to shrink from $r_{0}$ to $r_{2}=0.95 r_{0}$. Neglect the change of the gravity field at the position of a falling gas particle.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nLet us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser than the gas) and the initial temperature of the gas is uniformly $T_{0}$. The gas may be assumed to be ideal. The average molar mass of the gas is $\\mu$ and its adiabatic index is $\\gamma>\\frac{4}{3}$. Assume that $G \\frac{m \\mu}{r_{0}} \\gg R T_{0}$, where $R$ is the gas constant and $G$ is the gravitational constant.\n\nEstimate the time $t_{2}$ needed for the radius to shrink from $r_{0}$ to $r_{2}=0.95 r_{0}$. Neglect the change of the gravity field at the position of a falling gas particle.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1381", "problem": "如示意图所示, 一垂直放置的高为 $15.0 \\mathrm{~cm}$ 的圆柱形中空玻璃容器, 其底部玻璃较厚, 底部顶点 $\\mathrm{A}$ 点到容器底平面中心 $\\mathrm{B}$ 点的距离为 $8.0 \\mathrm{~cm}$, 底部上沿为一凸起的球冠, 球心 $\\mathrm{C}$ 点在 $\\mathrm{A}$ 点正下方, 球的半径为 $1.75 \\mathrm{~cm}$ 。已知空气和容器玻璃的折射率分别是 $n_{0}=1.0$ 和 $n_{1}=1.56$ 。只考虑近轴光线成像。已知: 当 $\\lambda<<1$ 时, $\\sin \\lambda \\approx \\lambda$ 。\n\n[图1]当容器内装满折射率为 1.30 的液体时, 求从 $\\mathrm{B}$ 点发出的光线通过平凸玻璃柱的上表面折射后所成像点的位置, 并判断这个像的虚实", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如示意图所示, 一垂直放置的高为 $15.0 \\mathrm{~cm}$ 的圆柱形中空玻璃容器, 其底部玻璃较厚, 底部顶点 $\\mathrm{A}$ 点到容器底平面中心 $\\mathrm{B}$ 点的距离为 $8.0 \\mathrm{~cm}$, 底部上沿为一凸起的球冠, 球心 $\\mathrm{C}$ 点在 $\\mathrm{A}$ 点正下方, 球的半径为 $1.75 \\mathrm{~cm}$ 。已知空气和容器玻璃的折射率分别是 $n_{0}=1.0$ 和 $n_{1}=1.56$ 。只考虑近轴光线成像。已知: 当 $\\lambda<<1$ 时, $\\sin \\lambda \\approx \\lambda$ 。\n\n[图1]\n\n问题:\n当容器内装满折射率为 1.30 的液体时, 求从 $\\mathrm{B}$ 点发出的光线通过平凸玻璃柱的上表面折射后所成像点的位置, 并判断这个像的虚实\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[所成的像的位置, 像的虚实]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-06.jpg?height=574&width=394&top_left_y=1572&top_left_x=1368", "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-14.jpg?height=629&width=1250&top_left_y=665&top_left_x=403" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "所成的像的位置", "像的虚实" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_61", "problem": "An object of mass $m$ initially at rest experiences a uniform, non-zero acceleration. Which of the following statements is correct?\nA: The object's displacement is directly proportional to its time of travel.\nB: The object's velocity is directly proportional to its time of travel.\nC: The object's acceleration is directly proportional to its time of travel.\nD: The object's kinetic energy is directly proportional to its time of travel.\nE: The net force acting on the object is directly proportional to its time of travel.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn object of mass $m$ initially at rest experiences a uniform, non-zero acceleration. Which of the following statements is correct?\n\nA: The object's displacement is directly proportional to its time of travel.\nB: The object's velocity is directly proportional to its time of travel.\nC: The object's acceleration is directly proportional to its time of travel.\nD: The object's kinetic energy is directly proportional to its time of travel.\nE: The net force acting on the object is directly proportional to its time of travel.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_240", "problem": "The characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part, analyze a situation when an angular momentum is not conserved. The system is the same as in the previous part with the only difference that the dipole is fixed and the charged small object with a mass $2 m$ is moving around the dipole. The electrostatic field of the dipole is easier to describe in the polar system of coordinates, which is defined with the distance $r$ from the center of the dipole, and angle $\\theta$ counted counterclockwise, as shown in Figure 3.\n\n[figure1]\n\nFigure 3: The system analyzed in Part 2. (Direction of the vector $\\mathbf{E}_{\\mathbf{n}}$ and $\\mathbf{E}_{\\mathbf{t}}$ could be wrong)\n\nDetermine tangential component of the velocity $v_{\\tau 2}$ of the charged object as a function of coordinates $r$ and $\\theta$. Hint: $\\frac{d \\theta}{d t}=\\omega$ angular velocity", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nIn this part, analyze a situation when an angular momentum is not conserved. The system is the same as in the previous part with the only difference that the dipole is fixed and the charged small object with a mass $2 m$ is moving around the dipole. The electrostatic field of the dipole is easier to describe in the polar system of coordinates, which is defined with the distance $r$ from the center of the dipole, and angle $\\theta$ counted counterclockwise, as shown in Figure 3.\n\n[figure1]\n\nFigure 3: The system analyzed in Part 2. (Direction of the vector $\\mathbf{E}_{\\mathbf{n}}$ and $\\mathbf{E}_{\\mathbf{t}}$ could be wrong)\n\nDetermine tangential component of the velocity $v_{\\tau 2}$ of the charged object as a function of coordinates $r$ and $\\theta$. Hint: $\\frac{d \\theta}{d t}=\\omega$ angular velocity\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ed4e92416bdbac30298dg-2.jpg?height=477&width=1442&top_left_y=1694&top_left_x=336" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_508", "problem": "An AC power line cable transmits electrical power using a sinusoidal waveform with frequency $60 \\mathrm{~Hz}$. The load receives an RMS voltage of $500 \\mathrm{kV}$ and requires $1000 \\mathrm{MW}$ of average power. For this problem, consider only the cable carrying current in one of the two directions, and ignore effects due to capacitance or inductance between the cable and with the ground.\n\nSuppose that the load on the power line cable is a residential area that behaves like a pure resistor.\n\nWhat is the RMS current carried in the cable?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nAn AC power line cable transmits electrical power using a sinusoidal waveform with frequency $60 \\mathrm{~Hz}$. The load receives an RMS voltage of $500 \\mathrm{kV}$ and requires $1000 \\mathrm{MW}$ of average power. For this problem, consider only the cable carrying current in one of the two directions, and ignore effects due to capacitance or inductance between the cable and with the ground.\n\nSuppose that the load on the power line cable is a residential area that behaves like a pure resistor.\n\nWhat is the RMS current carried in the cable?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~A}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~A}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1309", "problem": "2016 年 2 月 11 日美国国家科学基金会宣布: 美国的 “激光干涉引力波天文台”（LIGO）的两台孪生引力波探测器首次直接探测到了引力波。该引力波是由 13 亿光年之外的两颗黑洞在合并的最后阶段产生的。初始质量分别为 29 倍太阳质量和 36 倍太阳质量的两颗黑洞, 合并成了一颗 62 倍太阳质量、高速旋转的黑洞; 亏损的质量以引力波的形式释放到宇宙空间。这亏损的质量为 ____ $\\mathrm{kg}$ ，相当于 ____ $\\mathrm{J}$ 的能量。已知太阳质量约为 $2.0 \\times 10^{30} \\mathrm{~kg}$, 光在真空中的速度为 $3.0 \\times 10^{8} \\mathrm{~m}$ 。\n7. 在一水平直线上相距 $18 \\mathrm{~m}$ 的 $\\mathrm{A} 、 \\mathrm{~B}$ 两点放置两个波源。这两个波源振动的方向相同、振幅相等、频率都是 $30 \\mathrm{~Hz}$, 且有相位差 $\\pi$ 。它们沿同一条直线在其两边的媒质中各激起简谐横波。波在媒质中的传播速度为 $360 \\mathrm{~m} / \\mathrm{s}$ 。这两列波在 $\\mathrm{A} 、 \\mathrm{~B}$ 两点所在直线上因干涉而振幅等于原来各自振幅的点有个, 它们到 $\\mathrm{A}$ 点的距离依次为 m。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n2016 年 2 月 11 日美国国家科学基金会宣布: 美国的 “激光干涉引力波天文台”（LIGO）的两台孪生引力波探测器首次直接探测到了引力波。该引力波是由 13 亿光年之外的两颗黑洞在合并的最后阶段产生的。初始质量分别为 29 倍太阳质量和 36 倍太阳质量的两颗黑洞, 合并成了一颗 62 倍太阳质量、高速旋转的黑洞; 亏损的质量以引力波的形式释放到宇宙空间。这亏损的质量为 ____ $\\mathrm{kg}$ ，相当于 ____ $\\mathrm{J}$ 的能量。已知太阳质量约为 $2.0 \\times 10^{30} \\mathrm{~kg}$, 光在真空中的速度为 $3.0 \\times 10^{8} \\mathrm{~m}$ 。\n7. 在一水平直线上相距 $18 \\mathrm{~m}$ 的 $\\mathrm{A} 、 \\mathrm{~B}$ 两点放置两个波源。这两个波源振动的方向相同、振幅相等、频率都是 $30 \\mathrm{~Hz}$, 且有相位差 $\\pi$ 。它们沿同一条直线在其两边的媒质中各激起简谐横波。波在媒质中的传播速度为 $360 \\mathrm{~m} / \\mathrm{s}$ 。这两列波在 $\\mathrm{A} 、 \\mathrm{~B}$ 两点所在直线上因干涉而振幅等于原来各自振幅的点有个, 它们到 $\\mathrm{A}$ 点的距离依次为 m。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[这亏损的质量为, 这亏损相当于能量]\n它们的单位依次是[kg, J]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "kg", "J" ], "answer_sequence": [ "这亏损的质量为", "这亏损相当于能量" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_799", "problem": "A block of mass $5 \\mathrm{~kg}$ lies at rest on a horizontal surface. An upwards force of $20 \\mathrm{~N}$ is applied to the block, as shown. Assuming $g=10 \\mathrm{~ms}^{-2}$, what is the weight of the block?\n\n[figure1]Assuming $g=10 \\mathrm{~m} \\mathrm{~s}^{-2}$, what is the magnitude of the normal force exerted by the surface on the block in Question 8?\nA: $20 \\mathrm{~N}$\nB: $30 \\mathrm{~N}$\nC: $50 \\mathrm{~N}$\nD: $70 \\mathrm{~N}$\nE: $5 \\mathrm{~kg}-20 \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\nHere is some context information for this question, which might assist you in solving it:\nA block of mass $5 \\mathrm{~kg}$ lies at rest on a horizontal surface. An upwards force of $20 \\mathrm{~N}$ is applied to the block, as shown. Assuming $g=10 \\mathrm{~ms}^{-2}$, what is the weight of the block?\n\n[figure1]\n\nproblem:\nAssuming $g=10 \\mathrm{~m} \\mathrm{~s}^{-2}$, what is the magnitude of the normal force exerted by the surface on the block in Question 8?\n\nA: $20 \\mathrm{~N}$\nB: $30 \\mathrm{~N}$\nC: $50 \\mathrm{~N}$\nD: $70 \\mathrm{~N}$\nE: $5 \\mathrm{~kg}-20 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9bac30101a7c40be9bg-05.jpg?height=317&width=416&top_left_y=441&top_left_x=266" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1142", "problem": "A point charge $q$ is placed in the vicinity of a grounded metallic sphere of radius $R$ [see Fig. 1(a)], and consequently a surface charge distribution is induced on the sphere. To calculate the electric field and potential from the distribution of the surface charge is a formidable task. However, the calculation can be considerably simplified by using the so called method of images. In this method, the electric field and potential produced by the charge distributed on the sphere can be represented as an electric field and potential of a single point charge $q$ ' placed inside the sphere (you do not have to prove it). **Note: The electric field of this image charge $q$ ' reproduces the electric field and the potential only outside the sphere (including its surface).**\n\n[figure1]\n\nA point charge $q$ with mass $m$ is suspended on a thread of length $L$ which is attached to a wall, in the vicinity of the grounded metallic sphere. In your considerations, ignore all electrostatic effects of the wall. The point charge makes a mathematical pendulum (see Fig. 3). The point at which the thread is attached to the wall is at a distancel from the center of the sphere. Assume that the effects of gravity are negligible.\n\n[figure2]\n\nFind the magnitude of the electric force acting on the point charge $q$ for a given angle $\\alpha$ and indicate the direction in a clear diagram.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA point charge $q$ is placed in the vicinity of a grounded metallic sphere of radius $R$ [see Fig. 1(a)], and consequently a surface charge distribution is induced on the sphere. To calculate the electric field and potential from the distribution of the surface charge is a formidable task. However, the calculation can be considerably simplified by using the so called method of images. In this method, the electric field and potential produced by the charge distributed on the sphere can be represented as an electric field and potential of a single point charge $q$ ' placed inside the sphere (you do not have to prove it). **Note: The electric field of this image charge $q$ ' reproduces the electric field and the potential only outside the sphere (including its surface).**\n\n[figure1]\n\nA point charge $q$ with mass $m$ is suspended on a thread of length $L$ which is attached to a wall, in the vicinity of the grounded metallic sphere. In your considerations, ignore all electrostatic effects of the wall. The point charge makes a mathematical pendulum (see Fig. 3). The point at which the thread is attached to the wall is at a distancel from the center of the sphere. Assume that the effects of gravity are negligible.\n\n[figure2]\n\nFind the magnitude of the electric force acting on the point charge $q$ for a given angle $\\alpha$ and indicate the direction in a clear diagram.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_843c51f18af1a9802d4bg-1.jpg?height=774&width=1627&top_left_y=1046&top_left_x=220", "https://i.postimg.cc/rmvjsmdm/image.png", "https://cdn.mathpix.com/cropped/2024_03_14_a3a9b2d64aba1218b8e1g-3.jpg?height=414&width=803&top_left_y=2023&top_left_x=615" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_880", "problem": "Calculation in the previous part shows that in order to build the space elevator, it is neccessary to have light materials with very high tensile strength. Carbon nanotubes are materials that meet such requirements because of strong chemical bondings between very light atoms. Two natural polymorphs of carbon are diamond and graphite. In diamond every carbon atom is surrounded by four nearest neighbor (NN) atoms to form a tetrahedron. Graphite has a layer structure. In each layer, carbon atoms are arranged in a hexagonal plane lattice with three NNs. Although diamond is known as the hardest materials, covalent bondings between carbon atoms in hexagonal layers of graphite is stronger than those between carbon atoms in diamond tetrahedra. Graphite is much softer than diamond because of the van der Waals bonding between carbon atoms of different layers, which is much weaker than covalent bonding.\n\n[figure1]\n\nFigure 2. Graphite structure\n(a)\n[figure2]\n\nFigure 3. Graphene (a) and carbon nanotube (b).\n\nA monatomic layer in graphite is called graphene and has monoatomic thickness. Isolated graphene sheet is not stable and has a tendency to roll up to form carbon spheres or carbon nanotubes. The hexagonal crystal lattice of graphene is depicted in Fig. 4. The distance between two NN carbon atoms is $a=0.142 \\mathrm{~nm}$ and the distance between two closest parallel bondings is $b=0.246 \\mathrm{~nm}$. Because the covalent bondings between carbon atoms in graphene are very strong, mechanical properties of carbon nanotubes are very special. They have an extremely large Young's modulus and tensile strength, as well as a very light density. Young's modulus is defined as the ratio of the stress along an axis to the strain (ratio of deformation over initial length) along that axis in the range of stress in which Hooke's law holds.\n\n[figure3]\n\nFigure 4. Graphene.\n\n## Theory\n\n[figure4]\n\nFigure 5. An illustration of a carbon nanotube with 9 carbon-carbon parallel bondings. Note: In this problem, there are 27 carbon-carbon parallel bondings. (1) parallel bond; (2) slanted bond; (3) tube axis.\n\nNow we examine some mechanical properties of a carbon nanotube having 27 carbon-carbon bondings parallel to the tube axis (for an illustration, see Figure 5). The bonding between two carbon atoms can be described by the Morse potential $V(x)=V_{0}\\left(e^{-4 \\frac{x}{a}}-2 e^{-2 \\frac{x}{a}}\\right)$. Here $a=0.142 \\mathrm{~nm}$ is the equilibrium distance between two $\\mathrm{NN}$ carbon atoms, $V_{0}=4.93 \\mathrm{eV}$ is the bonding energy, and $x$ is the displacement of the atom from the equilibrium position. Hereafter, we approximate the Morse potential by a quadratic potential $V(x)=P+Q x^{2}$. All non-nearest-neighbor interactions are neglected. In this approximation, one can propose that carbon atoms are bonded through \"springs\" with the spring constant $k$. Changes in angles between bonds are neglected.\n\nCalculate the value of the Young's modulus of the carbon nanotube.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nCalculation in the previous part shows that in order to build the space elevator, it is neccessary to have light materials with very high tensile strength. Carbon nanotubes are materials that meet such requirements because of strong chemical bondings between very light atoms. Two natural polymorphs of carbon are diamond and graphite. In diamond every carbon atom is surrounded by four nearest neighbor (NN) atoms to form a tetrahedron. Graphite has a layer structure. In each layer, carbon atoms are arranged in a hexagonal plane lattice with three NNs. Although diamond is known as the hardest materials, covalent bondings between carbon atoms in hexagonal layers of graphite is stronger than those between carbon atoms in diamond tetrahedra. Graphite is much softer than diamond because of the van der Waals bonding between carbon atoms of different layers, which is much weaker than covalent bonding.\n\n[figure1]\n\nFigure 2. Graphite structure\n(a)\n[figure2]\n\nFigure 3. Graphene (a) and carbon nanotube (b).\n\nA monatomic layer in graphite is called graphene and has monoatomic thickness. Isolated graphene sheet is not stable and has a tendency to roll up to form carbon spheres or carbon nanotubes. The hexagonal crystal lattice of graphene is depicted in Fig. 4. The distance between two NN carbon atoms is $a=0.142 \\mathrm{~nm}$ and the distance between two closest parallel bondings is $b=0.246 \\mathrm{~nm}$. Because the covalent bondings between carbon atoms in graphene are very strong, mechanical properties of carbon nanotubes are very special. They have an extremely large Young's modulus and tensile strength, as well as a very light density. Young's modulus is defined as the ratio of the stress along an axis to the strain (ratio of deformation over initial length) along that axis in the range of stress in which Hooke's law holds.\n\n[figure3]\n\nFigure 4. Graphene.\n\n## Theory\n\n[figure4]\n\nFigure 5. An illustration of a carbon nanotube with 9 carbon-carbon parallel bondings. Note: In this problem, there are 27 carbon-carbon parallel bondings. (1) parallel bond; (2) slanted bond; (3) tube axis.\n\nNow we examine some mechanical properties of a carbon nanotube having 27 carbon-carbon bondings parallel to the tube axis (for an illustration, see Figure 5). The bonding between two carbon atoms can be described by the Morse potential $V(x)=V_{0}\\left(e^{-4 \\frac{x}{a}}-2 e^{-2 \\frac{x}{a}}\\right)$. Here $a=0.142 \\mathrm{~nm}$ is the equilibrium distance between two $\\mathrm{NN}$ carbon atoms, $V_{0}=4.93 \\mathrm{eV}$ is the bonding energy, and $x$ is the displacement of the atom from the equilibrium position. Hereafter, we approximate the Morse potential by a quadratic potential $V(x)=P+Q x^{2}$. All non-nearest-neighbor interactions are neglected. In this approximation, one can propose that carbon atoms are bonded through \"springs\" with the spring constant $k$. Changes in angles between bonds are neglected.\n\nCalculate the value of the Young's modulus of the carbon nanotube.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{GPa}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-2.jpg?height=549&width=525&top_left_y=1850&top_left_x=731", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-3.jpg?height=430&width=900&top_left_y=710&top_left_x=518", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-3.jpg?height=451&width=691&top_left_y=1856&top_left_x=660", "https://cdn.mathpix.com/cropped/2024_03_14_9fac1528de308fa347bbg-4.jpg?height=944&width=1144&top_left_y=172&top_left_x=184" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{GPa}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1317", "problem": "如图所示, 以质量为 $m$ 半径为 $R$ 的由绝缘材料制成的薄球壳, 均匀带正电, 电荷量为 $\\mathrm{Q}$,球壳下面有与球壳固连的底座, 底座静止在光滑的水平面上。球壳内部有一劲度系数为 $\\eta$ 的轻弹簧 (质量不计）, 弹簧始终处于水平位置, 其一端与球壳内壁固连, 另一端恰好位于球心处, 球壳上开有一个小孔 $C$, 小孔位于过球心的水平线上。在此水平线上离球壳很远的 $O$ 处有一质量也为 $m$ 电荷量也为 $Q$ 的带正电的点电荷 $P$, 它以足够大的初速度 $v_{0}$ 沿水平的 $O C$ 方向开始运动。并知 $P$ 能通过小孔 $C$ 进入球壳内, 不考虑重力和底座的影响, 已知静电力常量为 $k$, 求 $P$ 刚进入 $C$ 孔出来所经历的时间。\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图所示, 以质量为 $m$ 半径为 $R$ 的由绝缘材料制成的薄球壳, 均匀带正电, 电荷量为 $\\mathrm{Q}$,球壳下面有与球壳固连的底座, 底座静止在光滑的水平面上。球壳内部有一劲度系数为 $\\eta$ 的轻弹簧 (质量不计）, 弹簧始终处于水平位置, 其一端与球壳内壁固连, 另一端恰好位于球心处, 球壳上开有一个小孔 $C$, 小孔位于过球心的水平线上。在此水平线上离球壳很远的 $O$ 处有一质量也为 $m$ 电荷量也为 $Q$ 的带正电的点电荷 $P$, 它以足够大的初速度 $v_{0}$ 沿水平的 $O C$ 方向开始运动。并知 $P$ 能通过小孔 $C$ 进入球壳内, 不考虑重力和底座的影响, 已知静电力常量为 $k$, 求 $P$ 刚进入 $C$ 孔出来所经历的时间。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_93e0584db50294961a50g-06.jpg?height=278&width=792&top_left_y=1008&top_left_x=601" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_523", "problem": "An ideal rocket when empty of fuel has a mass $m_{r}$ and will carry a mass of fuel $m_{f}$. The fuel burns and is ejected with an exhaust speed of $v_{e}$ relative to the rocket. The fuel burns at a constant mass rate for a total time $T_{b}$. Ignore gravity; assume the rocket is far from any other body.\n\nAssuming that the rocket starts from rest, determine the final speed of the rocket in terms of any or all of $m_{r}, m_{f}, v_{e}, T_{b}$, and any relevant fundamental constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nAn ideal rocket when empty of fuel has a mass $m_{r}$ and will carry a mass of fuel $m_{f}$. The fuel burns and is ejected with an exhaust speed of $v_{e}$ relative to the rocket. The fuel burns at a constant mass rate for a total time $T_{b}$. Ignore gravity; assume the rocket is far from any other body.\n\nAssuming that the rocket starts from rest, determine the final speed of the rocket in terms of any or all of $m_{r}, m_{f}, v_{e}, T_{b}$, and any relevant fundamental constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_600", "problem": "Two masses $m$ separated by a distance $l$ are given initial velocities $v_{0}$ as shown in the diagram. The masses interact only through universal gravitation.\n\n[figure1]\n\nUnder what conditions will the masses follow circular orbits of diameter $l$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nTwo masses $m$ separated by a distance $l$ are given initial velocities $v_{0}$ as shown in the diagram. The masses interact only through universal gravitation.\n\n[figure1]\n\nUnder what conditions will the masses follow circular orbits of diameter $l$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_05f4d0107dd82de1849ag-09.jpg?height=613&width=791&top_left_y=436&top_left_x=662" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1358", "problem": "如图 5 所示，我们以太阳为参考系，地球绕太阳的运动周期为 $T_{1}$, 以地球为参考系，月球绕地球的运动周期为 $T_{2}$, 则相邻两次月球一地球一太阳排列成几乎一条直线的时间差约为___。记地球绕太阳的轨道半径为 $r_{E}$, 月球绕地球的轨道半径为$r_{M}\\left(r_{E}>>r_{M}\\right)$, 以太阳为参考系, 月球运动过程,加速度的最小值和最大值的大小比例为___\n\n[图1]\n\n图 5", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n如图 5 所示，我们以太阳为参考系，地球绕太阳的运动周期为 $T_{1}$, 以地球为参考系，月球绕地球的运动周期为 $T_{2}$, 则相邻两次月球一地球一太阳排列成几乎一条直线的时间差约为___。记地球绕太阳的轨道半径为 $r_{E}$, 月球绕地球的轨道半径为$r_{M}\\left(r_{E}>>r_{M}\\right)$, 以太阳为参考系, 月球运动过程,加速度的最小值和最大值的大小比例为___\n\n[图1]\n\n图 5\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[相邻两次月球一地球一太阳排列成几乎一条直线的时间差, 以太阳为参考系, 月球运动过程,加速度的最小值和最大值的大小比例为]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-03.jpg?height=434&width=534&top_left_y=1551&top_left_x=790" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "相邻两次月球一地球一太阳排列成几乎一条直线的时间差", "以太阳为参考系, 月球运动过程,加速度的最小值和最大值的大小比例为" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_760", "problem": "Newton's Law of Gravitation, the force due to gravity of an object with mass M on a different object of mass $\\mathrm{m}$, is given as:\n\n$$\nF=\\frac{G M m}{r^{2}}\n$$\n\nWhere G is the gravitational constant, $6.6743 \\times 10^{-11} \\mathrm{~m}^{3} \\mathrm{~kg}^{-1} \\mathrm{~s}^{-2}$\n\nAssuming G, M and $\\mathrm{m}$ are constant, what combination of variables will give a straight-line relationship?\nA: $\\quad \\mathrm{Fvgr}^{2}$\nB: $\\quad \\mathrm{F} v \\mathrm{r}$\nC: $\\quad$ F vs $1 / \\mathrm{r}$\nD: $\\quad \\mathrm{F}$ vs $1 / \\mathrm{r}^{2}$\nE: None of the above\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nNewton's Law of Gravitation, the force due to gravity of an object with mass M on a different object of mass $\\mathrm{m}$, is given as:\n\n$$\nF=\\frac{G M m}{r^{2}}\n$$\n\nWhere G is the gravitational constant, $6.6743 \\times 10^{-11} \\mathrm{~m}^{3} \\mathrm{~kg}^{-1} \\mathrm{~s}^{-2}$\n\nAssuming G, M and $\\mathrm{m}$ are constant, what combination of variables will give a straight-line relationship?\n\nA: $\\quad \\mathrm{Fvgr}^{2}$\nB: $\\quad \\mathrm{F} v \\mathrm{r}$\nC: $\\quad$ F vs $1 / \\mathrm{r}$\nD: $\\quad \\mathrm{F}$ vs $1 / \\mathrm{r}^{2}$\nE: None of the above\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1374", "problem": "某种回旋加速器的设计方案如俯视图 a 所示, 图中粗黑线段为两个正对的极板，其间仅在带电粒子经过的过程中存在匀强电场, 两极板间电势差为 $U$ 。两个极板的板面中部各有一狭缝(沿 OP 方向的狭长区域），带电粒子可通过狭缝穿越极板（见图 b）；两细虚线间（除开两极板之间的区域）既无电场也无磁场；其它部分存在匀强磁场, 磁感应强度方向垂直于纸面。在离子源 $\\mathrm{S}$ 中产生的质量为 $\\mathrm{m}$ 、带电量为 $q(q>0)$ 的离子, 由静止开始被电场加速, 经狭缝中的 $\\mathrm{O}$ 点进入磁场区域, $\\mathrm{O}$ 点到极板右端的距离为 $D$, 到出射孔 $\\mathrm{P}$ 的距离为 $b D$ (常数 $b$ 为大于 2 的自然数）。已知磁感应强度大小在零到 $B_{\\text {max }}$ 之间可调, 离子从离子源上方的 $\\mathrm{O}$ 点射入磁场区域, 最终只能从出射孔 P 射出。假设如果离子打到器壁或离子源外壁便即被吸收。忽略相对论效应。求\n[图1]\n\n图a \n\n[图2]\n\n图 b出射离子能量的最大值。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某种回旋加速器的设计方案如俯视图 a 所示, 图中粗黑线段为两个正对的极板，其间仅在带电粒子经过的过程中存在匀强电场, 两极板间电势差为 $U$ 。两个极板的板面中部各有一狭缝(沿 OP 方向的狭长区域），带电粒子可通过狭缝穿越极板（见图 b）；两细虚线间（除开两极板之间的区域）既无电场也无磁场；其它部分存在匀强磁场, 磁感应强度方向垂直于纸面。在离子源 $\\mathrm{S}$ 中产生的质量为 $\\mathrm{m}$ 、带电量为 $q(q>0)$ 的离子, 由静止开始被电场加速, 经狭缝中的 $\\mathrm{O}$ 点进入磁场区域, $\\mathrm{O}$ 点到极板右端的距离为 $D$, 到出射孔 $\\mathrm{P}$ 的距离为 $b D$ (常数 $b$ 为大于 2 的自然数）。已知磁感应强度大小在零到 $B_{\\text {max }}$ 之间可调, 离子从离子源上方的 $\\mathrm{O}$ 点射入磁场区域, 最终只能从出射孔 P 射出。假设如果离子打到器壁或离子源外壁便即被吸收。忽略相对论效应。求\n[图1]\n\n图a \n\n[图2]\n\n图 b\n\n问题:\n出射离子能量的最大值。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-03.jpg?height=896&width=894&top_left_y=243&top_left_x=889", "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-03.jpg?height=331&width=371&top_left_y=1248&top_left_x=1345" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1496", "problem": "Ioffe-Pritchard 磁阱可用来束缚原子的运动, 其主要部分如图所示。四根均通有恒定电流 $I$ 的长直导线 1、2、3、4 都垂直于 $x-y$ 平面, 它们与 $x-y$ 平面的交点是边长为 $2 a$ 、中心在原点 $O$ 的正方形的顶点, 导线 1、2 所在平面与 $x$ 轴平行, 各导线中电流方向已在图中标出。整个装置置于匀强磁场 $\\boldsymbol{B}_{0}=B_{0} \\boldsymbol{k}(\\boldsymbol{k}$ 为 $z$ 轴正方向单位矢量）中。已知真空磁导率为 $\\mu_{0}$ 。\n\n[图1]在磁阱中运动的原子最容易从 $x-y$ 平面上什么位置逸出？求刚好能够逸出磁阶的原子的动能。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\nIoffe-Pritchard 磁阱可用来束缚原子的运动, 其主要部分如图所示。四根均通有恒定电流 $I$ 的长直导线 1、2、3、4 都垂直于 $x-y$ 平面, 它们与 $x-y$ 平面的交点是边长为 $2 a$ 、中心在原点 $O$ 的正方形的顶点, 导线 1、2 所在平面与 $x$ 轴平行, 各导线中电流方向已在图中标出。整个装置置于匀强磁场 $\\boldsymbol{B}_{0}=B_{0} \\boldsymbol{k}(\\boldsymbol{k}$ 为 $z$ 轴正方向单位矢量）中。已知真空磁导率为 $\\mu_{0}$ 。\n\n[图1]\n\n问题:\n在磁阱中运动的原子最容易从 $x-y$ 平面上什么位置逸出？求刚好能够逸出磁阶的原子的动能。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-02.jpg?height=349&width=343&top_left_y=1119&top_left_x=1413" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_574", "problem": "Consider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nA second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height $H$, we can write\n\n$$\n2 \\int_{0}^{H} p d x=\\left(n+\\frac{1}{2}\\right) h\n$$\n\nwhere $p$ is the momentum as a function of height $x$ above the ground, $n$ is a non-negative integer, and $h$ is Planck's constant.\n\nDetermine the allowed energies $E_{n}$ as a function of the integer $n$, and some or all of $g$, $m$, and Planck's constant $h$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nConsider a particle of mass $m$ that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is $E$ and the acceleration of free fall is $g$. Treat the particle as a point mass and assume the motion is non-relativistic.\n\nA second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height $H$, we can write\n\n$$\n2 \\int_{0}^{H} p d x=\\left(n+\\frac{1}{2}\\right) h\n$$\n\nwhere $p$ is the momentum as a function of height $x$ above the ground, $n$ is a non-negative integer, and $h$ is Planck's constant.\n\nDetermine the allowed energies $E_{n}$ as a function of the integer $n$, and some or all of $g$, $m$, and Planck's constant $h$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1732", "problem": "如图所示, 半径为 $R$ 、质量为 $m_{0}$ 的光滑均匀圆环, 套在光滑坚直细轴 $O O^{\\prime}$ 上, 可沿 $O O^{\\prime}$ 轴滑动或绕 $O O^{\\prime}$ 轴旋转. 圆环上串着两个质量均为 $m$ 的小球. 开始时让圆环以某一角速度绕 $O O$ 轴转动, 两小球自圆环顶端同时从静止开始释放.\n\n[图1]设开始时圆环绕 $O O^{\\prime}$ 轴转动的角速度为 $\\omega_{b}$, 在两小球从环顶下滑过程中, 应满足什么条件, 圆环才有可能沿 $O O$ 轴上滑?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图所示, 半径为 $R$ 、质量为 $m_{0}$ 的光滑均匀圆环, 套在光滑坚直细轴 $O O^{\\prime}$ 上, 可沿 $O O^{\\prime}$ 轴滑动或绕 $O O^{\\prime}$ 轴旋转. 圆环上串着两个质量均为 $m$ 的小球. 开始时让圆环以某一角速度绕 $O O$ 轴转动, 两小球自圆环顶端同时从静止开始释放.\n\n[图1]\n\n问题:\n设开始时圆环绕 $O O^{\\prime}$ 轴转动的角速度为 $\\omega_{b}$, 在两小球从环顶下滑过程中, 应满足什么条件, 圆环才有可能沿 $O O$ 轴上滑?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ea729d4659bcaa2c4b91g-02.jpg?height=414&width=368&top_left_y=821&top_left_x=1255", "https://cdn.mathpix.com/cropped/2024_03_31_ea729d4659bcaa2c4b91g-10.jpg?height=391&width=220&top_left_y=2129&top_left_x=1566", "https://cdn.mathpix.com/cropped/2024_03_31_ea729d4659bcaa2c4b91g-11.jpg?height=77&width=831&top_left_y=371&top_left_x=485" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_754", "problem": "Disk in gas \n\nConsider a thin flat disk of mass $M$ and face area $S$ at temperature $T_{1}$ resting initially in weightlessness in a gas of mass density $\\rho$ at temperature $T_{0}\\left(T_{1}=1000 T_{0}\\right)$. One of the faces of the disk is covered with a thermally insulating layer, the other face has a very good thermal contact with the surrounding gas: gas molecules of mass $m$ obtain the temperature of the disk during a single collision with the surface. Estimate the initial acceleration $a_{0}$ and maximal speed $v_{\\max }$ of the disk during its subsequent motion. Assume the heat capacity of the disk to be on the of order $N k_{B}$, where $N$ is the number of atoms in it, and $k_{B}$ is the Boltzmann constant, and molar masses of the gas and the disk's material to be of the same order. The mean free path length of molecules is much larger than the size of the disk. Neglect any edge effects occuring at the edge of the disk.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nDisk in gas \n\nConsider a thin flat disk of mass $M$ and face area $S$ at temperature $T_{1}$ resting initially in weightlessness in a gas of mass density $\\rho$ at temperature $T_{0}\\left(T_{1}=1000 T_{0}\\right)$. One of the faces of the disk is covered with a thermally insulating layer, the other face has a very good thermal contact with the surrounding gas: gas molecules of mass $m$ obtain the temperature of the disk during a single collision with the surface. Estimate the initial acceleration $a_{0}$ and maximal speed $v_{\\max }$ of the disk during its subsequent motion. Assume the heat capacity of the disk to be on the of order $N k_{B}$, where $N$ is the number of atoms in it, and $k_{B}$ is the Boltzmann constant, and molar masses of the gas and the disk's material to be of the same order. The mean free path length of molecules is much larger than the size of the disk. Neglect any edge effects occuring at the edge of the disk.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the initial acceleration $a_{0}$, the maximal speed $v_{\\max }$ of the disk during its subsequent motion].\nTheir answer types are, in order, [numerical value, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "the initial acceleration $a_{0}$", "the maximal speed $v_{\\max }$ of the disk during its subsequent motion" ], "type_sequence": [ "NV", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_628", "problem": "A particle is constrained to move on the inner surface of a frictionless parabolic bowl whose crosssection has equation $z=k r^{2}$. The particle begins at a height $z_{0}$ above the bottom of the bowl with a horizontal velocity $v_{0}$ along the surface of the bowl. The acceleration due to gravity is $g$.\n[figure1]\n\nFor a particular value of horizontal velocity $v_{0}$, which we will name $v_{h}$, the particle moves in a horizontal circle. What is $v_{h}$ in terms of $g, z_{0}$, and/or $k$ ?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA particle is constrained to move on the inner surface of a frictionless parabolic bowl whose crosssection has equation $z=k r^{2}$. The particle begins at a height $z_{0}$ above the bottom of the bowl with a horizontal velocity $v_{0}$ along the surface of the bowl. The acceleration due to gravity is $g$.\n[figure1]\n\nFor a particular value of horizontal velocity $v_{0}$, which we will name $v_{h}$, the particle moves in a horizontal circle. What is $v_{h}$ in terms of $g, z_{0}$, and/or $k$ ?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_3899026513eb55709c81g-13.jpg?height=392&width=1266&top_left_y=476&top_left_x=428" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_695", "problem": "Two metal balls of the same diameter but different mass are dropped simultaneously from a high cliff . Which of the statements below describes best what happens?\nA: The heavier one has the same acceleration from the start\nB: The lighter one has higher acceleration from the start\nC: They both have the same acceleration at the start but then the lighter one has higher acceleration.\nD: They both have the same acceleration at the start but then the heavier one has higher acceleration.\nE: They both have the acceleration all the way to the bottom\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo metal balls of the same diameter but different mass are dropped simultaneously from a high cliff . Which of the statements below describes best what happens?\n\nA: The heavier one has the same acceleration from the start\nB: The lighter one has higher acceleration from the start\nC: They both have the same acceleration at the start but then the lighter one has higher acceleration.\nD: They both have the same acceleration at the start but then the heavier one has higher acceleration.\nE: They both have the acceleration all the way to the bottom\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_583", "problem": "A vacuum system consists of a chamber of volume $V$ connected to a vacuum pump that is a cylinder with a piston that moves left and right. The minimum volume in the pump cylinder is $V_{0}$, and the maximum volume in the cylinder is $V_{0}+\\Delta V$. You should assume that $\\Delta V \\ll V$.\n\n[figure1]\n\nThe cylinder has two valves. The inlet valve opens when the pressure inside the cylinder is lower than the pressure in the chamber, but closes when the piston moves to the right. The outlet valve opens when the pressure inside the cylinder is greater than atmospheric pressure $P_{a}$, and closes when the piston moves to the left. A motor drives the piston to move back and forth. The piston moves at such a rate that heat is not conducted in or out of the gas contained in the cylinder during the pumping cycle. One complete cycle takes a time $\\Delta t$. You should assume that $\\Delta t$ is a very small quantity, but $\\Delta V / \\Delta t=R$ is finite. The gas in the chamber is ideal monatomic and remains at a fixed temperature of $T_{a}$.\n\nStart with assumption that $V_{0}=0$ and there are no leaks in the system.\n\nFor the remainder of this problem $00)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nFind the initial magnetic flux $\\Phi_{\\mathrm{B}}$ through surfaces at $z=0$, and at $z=-d$. $\\quad 0.4 \\mathrm{pt}$", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\nHere is some context information for this question, which might assist you in solving it:\n$\\vec{B} =\\frac{\\mu_{0} q_{\\mathrm{m}}}{4 \\pi}\\left[\\frac{(z-h) \\hat{z}+\\vec{\\rho}}{\\left[(z-h)^{2}+\\rho^{2}\\right]^{3 / 2}}+\\frac{(z+h) \\hat{z}+\\vec{\\rho}}{\\left[(z+h)^{2}+\\rho^{2}\\right]^{3 / 2}}\\right] $\n\nIn the $z \\leq-d$ region, the magnetic field $\\vec{B}=\\vec{B}^{\\prime}+\\vec{B}_{\\mathrm{mp}}$ at $t=t_{0}=0$ is given by\n\n$$\n\\vec{B}=0\n$$\n\nproblem:\n[figure1]\n\nFig. 1 A monopole $q_{\\mathrm{m}}$ appears at a distance $h$ from a conducting thin film of thickness $d$. The origin of the coordinates is located on the upper surface.\n\nWe first focus on the initial response of the conducting thin film when at time $t=0$ a north monopole $q_{\\mathrm{m}}$ appears suddenly at the position $\\vec{r}_{\\mathrm{mp}}=h \\hat{z}(h>0)$, as is shown in Fig. 1 . The monopole remains stationary in all later times $(t>0)$.\n\nOur interest here is the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ in regions $z \\geq 0$ and $z \\leq-d$, and the induced electric current density in the thin film. The total magnetic field $\\vec{B}=\\vec{B}_{\\mathrm{mp}}+\\vec{B}^{\\prime}$, where magnetic fields $\\vec{B}_{\\mathrm{mp}}$ and $\\vec{B}^{\\prime}$ are, respectively, due to the monopole and the induced current in the thin film. The initial $\\vec{B}(\\vec{\\rho}, z)$ we refer to is at the time $t_{0}$, which falls within the interval $h / c \\leq t_{0} \\ll \\tau_{\\mathrm{c}}$. Here $\\tau_{\\mathrm{c}}$ is a time constant characterizing the subsequent response of the thin film, and $c$ is the speed of light in vacuum. In this problem, we take the limit $h / c \\rightarrow 0$ and hence let $t_{0}=0$.\n\nThe calculation of the initial total magnetic field $\\vec{B}(\\vec{\\rho}, z)$ (at $t_{0}=0$ ) is facilitated by introducing an image monopole. For $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\geq 0$, the image monopole has a magnetic charge $q_{\\mathrm{m}}$ and is located at $z=-h$. On the other hand, for $\\vec{B}(\\vec{\\rho}, z)$ in the region $z \\leq-d$, the image monopole has a magnetic charge $-q_{\\mathrm{m}}$ and is located at $z=h$.\n\nFind the initial magnetic flux $\\Phi_{\\mathrm{B}}$ through surfaces at $z=0$, and at $z=-d$. $\\quad 0.4 \\mathrm{pt}$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d32b3b2f89cebe6f1c2ag-2.jpg?height=642&width=1244&top_left_y=296&top_left_x=194" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_589", "problem": "A light bulb has a solid cylindrical filament of length $L$ and radius $a$, and consumes power $P$. You are to design a new light bulb, using a cylindrical filament of the same material, operating at the same voltage, and emitting the same spectrum of light, which will consume power $n P$. What are the length and radius of the new filament? Assume that the temperature of the filament is approximately uniform across its cross-section; the filament doesn't emit light from the ends; and energy loss due to convection is minimal.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nA light bulb has a solid cylindrical filament of length $L$ and radius $a$, and consumes power $P$. You are to design a new light bulb, using a cylindrical filament of the same material, operating at the same voltage, and emitting the same spectrum of light, which will consume power $n P$. What are the length and radius of the new filament? Assume that the temperature of the filament is approximately uniform across its cross-section; the filament doesn't emit light from the ends; and energy loss due to convection is minimal.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the length of the new filament, the radius of the new filament].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "the length of the new filament", "the radius of the new filament" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1488", "problem": "如图, 在磁感应强度大小为 $B$ 、方向坚直向上的匀强磁场中, 有一均质刚性导电的正方形线框 abcd, 线框质量为 $m$ ，边长为 $l$, 总电阻为 $R$ 。线框可绕通过 ad 边和 bc 边中点的光滑轴 $\\mathrm{OO}^{\\prime}$ 转动。 $\\mathrm{P} 、 \\mathrm{Q}$ 点是线框引线的两端, $\\mathrm{OO}^{\\prime}$ 轴和 X 轴位于同一水平面内, 且相互垂直。不考虑线框自感。\n\n[图1]线框做上述运动一段时间后, 当其所在平面与 $\\mathrm{X}$ 轴夹角为 $\\theta_{1}\\left(\\frac{\\pi}{4} \\leq \\theta_{1} \\leq \\frac{3 \\pi}{4}\\right)$ 时, 将 $\\mathrm{P}$ 、 $\\mathrm{Q}$ 短路, 线框再转一小角度 $\\alpha$ 后停止, 求 $\\alpha$ 与 $\\theta_{1}$ 的关系式和 $\\alpha$ 的最小值。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 在磁感应强度大小为 $B$ 、方向坚直向上的匀强磁场中, 有一均质刚性导电的正方形线框 abcd, 线框质量为 $m$ ，边长为 $l$, 总电阻为 $R$ 。线框可绕通过 ad 边和 bc 边中点的光滑轴 $\\mathrm{OO}^{\\prime}$ 转动。 $\\mathrm{P} 、 \\mathrm{Q}$ 点是线框引线的两端, $\\mathrm{OO}^{\\prime}$ 轴和 X 轴位于同一水平面内, 且相互垂直。不考虑线框自感。\n\n[图1]\n\n问题:\n线框做上述运动一段时间后, 当其所在平面与 $\\mathrm{X}$ 轴夹角为 $\\theta_{1}\\left(\\frac{\\pi}{4} \\leq \\theta_{1} \\leq \\frac{3 \\pi}{4}\\right)$ 时, 将 $\\mathrm{P}$ 、 $\\mathrm{Q}$ 短路, 线框再转一小角度 $\\alpha$ 后停止, 求 $\\alpha$ 与 $\\theta_{1}$ 的关系式和 $\\alpha$ 的最小值。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[$\\alpha$ 与 $\\theta_{1}$ 的关系式, $\\alpha$ 的最小值]\n它们的答案类型依次是[方程, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_a47de6806e8da0a0f86dg-02.jpg?height=440&width=674&top_left_y=1322&top_left_x=1222" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$\\alpha$ 与 $\\theta_{1}$ 的关系式", "$\\alpha$ 的最小值" ], "type_sequence": [ "EQ", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1693", "problem": "1914 年，弗兰克-赫兹用电子碰撞原子的方法使原子从低能级激发到高能级，从而证明了原子能级的存在。加速电子碰撞自由的氢原子, 使某氢原子从基态激发到激发态。该氢原子仅能发出一条可见光波长范围 ( 400nm 760nm ) 内的光谱线。仅考虑一维正碰。\n\n已知 $h c=1240 \\mathrm{~nm} \\cdot \\mathrm{eV}$, 其中 $h$ 为普朗克常量, $c$ 为真空中的光速; 质子质量近似为电子质量的 1836 倍, 氢原子在碰撞前的速度可忽略。求该氢原子能发出的可见光的波长;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n1914 年，弗兰克-赫兹用电子碰撞原子的方法使原子从低能级激发到高能级，从而证明了原子能级的存在。加速电子碰撞自由的氢原子, 使某氢原子从基态激发到激发态。该氢原子仅能发出一条可见光波长范围 ( 400nm 760nm ) 内的光谱线。仅考虑一维正碰。\n\n已知 $h c=1240 \\mathrm{~nm} \\cdot \\mathrm{eV}$, 其中 $h$ 为普朗克常量, $c$ 为真空中的光速; 质子质量近似为电子质量的 1836 倍, 氢原子在碰撞前的速度可忽略。\n\n问题:\n求该氢原子能发出的可见光的波长;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~nm}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~nm}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_39", "problem": "For a particular conductor, the magnetic flux density and current are at right angles to one another. The component of force acting on the conductor is\nA: $\\mathrm{BIL} \\cos \\theta$\nB: BIL $\\sin \\theta$\nC: BIL $\\tan \\theta$\nD: $\\mathrm{BL} \\sin \\theta$ e.BL $\\cos \\theta$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nFor a particular conductor, the magnetic flux density and current are at right angles to one another. The component of force acting on the conductor is\n\nA: $\\mathrm{BIL} \\cos \\theta$\nB: BIL $\\sin \\theta$\nC: BIL $\\tan \\theta$\nD: $\\mathrm{BL} \\sin \\theta$ e.BL $\\cos \\theta$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_853", "problem": "he schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\\pm \\varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale.\n\n[figure1]\n\n\nCalculate the pressure $p_{B}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nhe schematic below shows the Hadley circulation in the Earth's tropical atmosphere around the spring equinox. Air rises from the equator and moves poleward in both hemispheres before descending in the subtropics at latitudes $\\pm \\varphi_{d}$ (where positive and negative latitudes refer to the northern and southern hemisphere respectively). The angular momentum about the Earth's spin axis is conserved for the upper branches of the circulation (enclosed by the dashed oval). Note that the schematic is not drawn to scale.\n\n[figure1]\n\n\nCalculate the pressure $p_{B}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of h P a, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e7b9ba69a6dd20c5cfd1g-1.jpg?height=1148&width=1151&top_left_y=1096&top_left_x=384" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "h P a" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_733", "problem": "A polarizer is a device that lets the component of light parallel to the polarizer's direction to go through and absorb the perpendicular direction. For example, a vertically oriented polarizer only allows the vertical component of the light to go through and the transmitted light is a vertically polarized light. Now we have three polarizers as shown below. The first polarizer is vertically oriented and the last one is horizontally oriented. The middle one is oriented at an angle $\\theta$ with respect to the vertical direction. When a randomly oriented light with intensity $\\mathrm{L}$ is passed into this series of polarizer, what is the intensity of the light after it passes through the third polarizer?\n\n[figure1]\nA: 0\nB: $\\mathrm{L} / 8$\nC: $\\mathrm{L} / 4 \\cos ^{2} \\theta$\nD: $\\mathrm{L} / 2 \\cos \\theta \\sin \\theta$\nE: $\\mathrm{L} / 2 \\cos ^{2} \\theta \\sin ^{2} \\theta$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA polarizer is a device that lets the component of light parallel to the polarizer's direction to go through and absorb the perpendicular direction. For example, a vertically oriented polarizer only allows the vertical component of the light to go through and the transmitted light is a vertically polarized light. Now we have three polarizers as shown below. The first polarizer is vertically oriented and the last one is horizontally oriented. The middle one is oriented at an angle $\\theta$ with respect to the vertical direction. When a randomly oriented light with intensity $\\mathrm{L}$ is passed into this series of polarizer, what is the intensity of the light after it passes through the third polarizer?\n\n[figure1]\n\nA: 0\nB: $\\mathrm{L} / 8$\nC: $\\mathrm{L} / 4 \\cos ^{2} \\theta$\nD: $\\mathrm{L} / 2 \\cos \\theta \\sin \\theta$\nE: $\\mathrm{L} / 2 \\cos ^{2} \\theta \\sin ^{2} \\theta$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22e26a14ee6fdd9254b6g-05.jpg?height=307&width=648&top_left_y=1771&top_left_x=1172" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_239", "problem": "Induction (or asynchronous) motors are the simplest and most reliable electric motors. They are powered by alternating current, and they do not contain commutators, slip rings or brushes. They consist of a stator and a rotor (see fig. 1). The stator is a fixed set of coils, which produces a rotating magnetic field in the plane perpendicular to the axis of the motor. The rotor is just a cage, i.e., a set of closed metallic loops attached to the axis of the motor. The rotating magnetic field produced by the stator induces electric current in the loops of the cage, which behave as magnetic dipoles, and interact with the external field of the stator. As a result, a torque is exerted on the rotor, and it starts rotating.\n\n[figure1]\n\nFigure 1: The structure of an induction motor\n\nIn a simplified model (see fig. 2) we assume that the magnetic induction vector $\\mathbf{B}$ produced by the stator is rotating in the $x-y$ plane at a constant angular velocity $\\Omega$, and it has a constant magnitude $\\mathbf{B}$. The axis of the rotor is in the $z$ direction. The rotor is assumed to be a flat coil of area $A$, winding number $N$, Ohmic resistance $R$ and self inductance $L$. The vector $\\mathbf{n}$ perpendicular to this coil is rotating also in the $x-y$ plane.\n\n[figure2]\n\nFigure 2: The simplified model, seen from the $z$ axis\n\nFirst we study the stationary operation of the motor, when its angular velocity $\\omega$ and torque $T$ are constant. As we shall see, when the motor is loaded, its angular velocity $\\omega$ gets smaller than $\\Omega$. It is convenient to characterize this shift by the slip\n\n$$\ns=\\frac{\\Omega-\\omega}{\\Omega}\n$$\n\nwhich is a dimensionless number between 0 and 1 .\n\nDetermine the efficiency $\\eta$ of the motor as a function of the slip $s$. (Assume that there is no energy loss due to friction, radiation.)", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nInduction (or asynchronous) motors are the simplest and most reliable electric motors. They are powered by alternating current, and they do not contain commutators, slip rings or brushes. They consist of a stator and a rotor (see fig. 1). The stator is a fixed set of coils, which produces a rotating magnetic field in the plane perpendicular to the axis of the motor. The rotor is just a cage, i.e., a set of closed metallic loops attached to the axis of the motor. The rotating magnetic field produced by the stator induces electric current in the loops of the cage, which behave as magnetic dipoles, and interact with the external field of the stator. As a result, a torque is exerted on the rotor, and it starts rotating.\n\n[figure1]\n\nFigure 1: The structure of an induction motor\n\nIn a simplified model (see fig. 2) we assume that the magnetic induction vector $\\mathbf{B}$ produced by the stator is rotating in the $x-y$ plane at a constant angular velocity $\\Omega$, and it has a constant magnitude $\\mathbf{B}$. The axis of the rotor is in the $z$ direction. The rotor is assumed to be a flat coil of area $A$, winding number $N$, Ohmic resistance $R$ and self inductance $L$. The vector $\\mathbf{n}$ perpendicular to this coil is rotating also in the $x-y$ plane.\n\n[figure2]\n\nFigure 2: The simplified model, seen from the $z$ axis\n\nFirst we study the stationary operation of the motor, when its angular velocity $\\omega$ and torque $T$ are constant. As we shall see, when the motor is loaded, its angular velocity $\\omega$ gets smaller than $\\Omega$. It is convenient to characterize this shift by the slip\n\n$$\ns=\\frac{\\Omega-\\omega}{\\Omega}\n$$\n\nwhich is a dimensionless number between 0 and 1 .\n\nDetermine the efficiency $\\eta$ of the motor as a function of the slip $s$. (Assume that there is no energy loss due to friction, radiation.)\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_84b5561c4961da250149g-1.jpg?height=417&width=849&top_left_y=892&top_left_x=627", "https://cdn.mathpix.com/cropped/2024_03_06_84b5561c4961da250149g-1.jpg?height=441&width=570&top_left_y=1796&top_left_x=772" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_668", "problem": "We are using X-rays (the wavelength of the order of $0.1 \\mathrm{~nm})$ to measure the spacing between the atoms or molecules in crystals. If we use a beam of neutrons for the same purpose their energy should be of the order of:\nA: $10^{-15} \\mathrm{~J}$\nB: $10^{-10} \\mathrm{~J}$\nC: $10^{-20} \\mathrm{~J}$\nD: $10^{-5} \\mathrm{~J}$\nE: $10^{-25} \\mathrm{~J}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWe are using X-rays (the wavelength of the order of $0.1 \\mathrm{~nm})$ to measure the spacing between the atoms or molecules in crystals. If we use a beam of neutrons for the same purpose their energy should be of the order of:\n\nA: $10^{-15} \\mathrm{~J}$\nB: $10^{-10} \\mathrm{~J}$\nC: $10^{-20} \\mathrm{~J}$\nD: $10^{-5} \\mathrm{~J}$\nE: $10^{-25} \\mathrm{~J}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_234", "problem": "Induction (or asynchronous) motors are the simplest and most reliable electric motors. They are powered by alternating current, and they do not contain commutators, slip rings or brushes. They consist of a stator and a rotor (see fig. 1). The stator is a fixed set of coils, which produces a rotating magnetic field in the plane perpendicular to the axis of the motor. The rotor is just a cage, i.e., a set of closed metallic loops attached to the axis of the motor. The rotating magnetic field produced by the stator induces electric current in the loops of the cage, which behave as magnetic dipoles, and interact with the external field of the stator. As a result, a torque is exerted on the rotor, and it starts rotating.\n\n[figure1]\n\nFigure 1: The structure of an induction motor\n\nIn a simplified model (see fig. 2) we assume that the magnetic induction vector $\\mathbf{B}$ produced by the stator is rotating in the $x-y$ plane at a constant angular velocity $\\Omega$, and it has a constant magnitude $\\mathbf{B}$. The axis of the rotor is in the $z$ direction. The rotor is assumed to be a flat coil of area $A$, winding number $N$, Ohmic resistance $R$ and self inductance $L$. The vector $\\mathbf{n}$ perpendicular to this coil is rotating also in the $x-y$ plane.\n\n[figure2]\n\nFigure 2: The simplified model, seen from the $z$ axis\n\n\nFirst we study the stationary operation of the motor, when its angular velocity $\\omega$ and torque $T$ are constant. As we shall see, when the motor is loaded, its angular velocity $\\omega$ gets smaller than $\\Omega$. It is convenient to characterize this shift by the slip\n\n$$\ns=\\frac{\\Omega-\\omega}{\\Omega}\n$$\n\nwhich is a dimensionless number between 0 and 1 .\n\nAssume that the load is small, so the slip $s \\ll 1$. Determine the time average of the torque $T$ exerted by the motor as a function of the slip $s$. (Use reasonable neglections.)", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nInduction (or asynchronous) motors are the simplest and most reliable electric motors. They are powered by alternating current, and they do not contain commutators, slip rings or brushes. They consist of a stator and a rotor (see fig. 1). The stator is a fixed set of coils, which produces a rotating magnetic field in the plane perpendicular to the axis of the motor. The rotor is just a cage, i.e., a set of closed metallic loops attached to the axis of the motor. The rotating magnetic field produced by the stator induces electric current in the loops of the cage, which behave as magnetic dipoles, and interact with the external field of the stator. As a result, a torque is exerted on the rotor, and it starts rotating.\n\n[figure1]\n\nFigure 1: The structure of an induction motor\n\nIn a simplified model (see fig. 2) we assume that the magnetic induction vector $\\mathbf{B}$ produced by the stator is rotating in the $x-y$ plane at a constant angular velocity $\\Omega$, and it has a constant magnitude $\\mathbf{B}$. The axis of the rotor is in the $z$ direction. The rotor is assumed to be a flat coil of area $A$, winding number $N$, Ohmic resistance $R$ and self inductance $L$. The vector $\\mathbf{n}$ perpendicular to this coil is rotating also in the $x-y$ plane.\n\n[figure2]\n\nFigure 2: The simplified model, seen from the $z$ axis\n\n\nFirst we study the stationary operation of the motor, when its angular velocity $\\omega$ and torque $T$ are constant. As we shall see, when the motor is loaded, its angular velocity $\\omega$ gets smaller than $\\Omega$. It is convenient to characterize this shift by the slip\n\n$$\ns=\\frac{\\Omega-\\omega}{\\Omega}\n$$\n\nwhich is a dimensionless number between 0 and 1 .\n\nAssume that the load is small, so the slip $s \\ll 1$. Determine the time average of the torque $T$ exerted by the motor as a function of the slip $s$. (Use reasonable neglections.)\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_84b5561c4961da250149g-1.jpg?height=417&width=849&top_left_y=892&top_left_x=627", "https://cdn.mathpix.com/cropped/2024_03_06_84b5561c4961da250149g-1.jpg?height=441&width=570&top_left_y=1796&top_left_x=772" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_396", "problem": "In this problem, we analyse the working principle of a speed camera. The transmitter of the speed camera emits an electomagnetic wave of frequency $f_{0}=24 \\mathrm{GHz}$ having waveform $\\cos \\left(2 \\pi f_{0} t\\right)$. The wave gets reflected from an approaching car moving with speed $v$. The reflected wave is recorded by the receiver of the speed camera.\n\nIn the speed camera, the received waveform is multiplied with the original emitted waveform. Express all the frequency components present in the multiplied signal.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nIn this problem, we analyse the working principle of a speed camera. The transmitter of the speed camera emits an electomagnetic wave of frequency $f_{0}=24 \\mathrm{GHz}$ having waveform $\\cos \\left(2 \\pi f_{0} t\\right)$. The wave gets reflected from an approaching car moving with speed $v$. The reflected wave is recorded by the receiver of the speed camera.\n\nIn the speed camera, the received waveform is multiplied with the original emitted waveform. Express all the frequency components present in the multiplied signal.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the value of $f_{\\text {high }}$, the value of $f_{\\text {low}}$].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "the value of $f_{\\text {high }}$", "the value of $f_{\\text {low}}$" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1636", "problem": "一斜䢃形透明介质䢃尖, 尖角为 $\\theta$, 高为 $h$. 今以尖角顶点为坐标原点, 建立坐标系如图(a)所示; 䢃尖斜面实际上是由一系列微小台阶组成的, 在图(a)中看来, 每一个小台阶的前侧面与 $\\mathrm{xz}$ 平面平行, 上表面与 $\\mathrm{yz}$ 平面平行. 䢃尖介质的折射率 $n$ 随 $x$ 而变化, $n(x)=1+b x$, 其中常数 $b>0$. 一束波长为 $\\lambda$ 的单色平行光沿 $x$ 轴正方向照射䢃尖; 䢃尖后放置一薄凸透镜, 在䢃尖与薄凸透镜之间放一档板, 在档板上刻有一系列与 $z$ 方向平行、沿 $y$ 方向排列的透光狭缝, 如图(b)所示. 入射光的波面 (即与平行入射光线垂直的平面)、䢃尖底面、档板平面都与 $x$ 轴垂直, 透镜主光轴为 $x$ 轴. 要求通过各狭缝的透射光彼此在透镜焦点处得到加强而形成亮纹. 已知第一条狭缝位于 $y=0$ 处; 物和像之间各光线的光程相等.\n\n[图1]\n\n[图2]求其余各狭缝的 $y$ 坐标", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n一斜䢃形透明介质䢃尖, 尖角为 $\\theta$, 高为 $h$. 今以尖角顶点为坐标原点, 建立坐标系如图(a)所示; 䢃尖斜面实际上是由一系列微小台阶组成的, 在图(a)中看来, 每一个小台阶的前侧面与 $\\mathrm{xz}$ 平面平行, 上表面与 $\\mathrm{yz}$ 平面平行. 䢃尖介质的折射率 $n$ 随 $x$ 而变化, $n(x)=1+b x$, 其中常数 $b>0$. 一束波长为 $\\lambda$ 的单色平行光沿 $x$ 轴正方向照射䢃尖; 䢃尖后放置一薄凸透镜, 在䢃尖与薄凸透镜之间放一档板, 在档板上刻有一系列与 $z$ 方向平行、沿 $y$ 方向排列的透光狭缝, 如图(b)所示. 入射光的波面 (即与平行入射光线垂直的平面)、䢃尖底面、档板平面都与 $x$ 轴垂直, 透镜主光轴为 $x$ 轴. 要求通过各狭缝的透射光彼此在透镜焦点处得到加强而形成亮纹. 已知第一条狭缝位于 $y=0$ 处; 物和像之间各光线的光程相等.\n\n[图1]\n\n[图2]\n\n问题:\n求其余各狭缝的 $y$ 坐标\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_734a14c1a8de5e19f8ecg-04.jpg?height=483&width=922&top_left_y=1232&top_left_x=196", "https://cdn.mathpix.com/cropped/2024_03_31_734a14c1a8de5e19f8ecg-04.jpg?height=836&width=711&top_left_y=1027&top_left_x=1346" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_725", "problem": "A closed 5-litre cylinder containing $0.25 \\mathrm{~g}$ of a substance in solid and liquid forms was slowly heated using a constant power. A graph of the temperature as a function of time is shown below. The heat capacity of the liquid is $2.43 \\mathrm{~J} /(\\mathrm{g} \\cdot \\mathrm{K})$, the latent heat of melting is $105 \\mathrm{~J} / \\mathrm{g}$, and the molar density is $46 \\mathrm{~g} / \\mathrm{mol}$.\n\n[figure1]\n\nWhat is the melting point of the substance?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA closed 5-litre cylinder containing $0.25 \\mathrm{~g}$ of a substance in solid and liquid forms was slowly heated using a constant power. A graph of the temperature as a function of time is shown below. The heat capacity of the liquid is $2.43 \\mathrm{~J} /(\\mathrm{g} \\cdot \\mathrm{K})$, the latent heat of melting is $105 \\mathrm{~J} / \\mathrm{g}$, and the molar density is $46 \\mathrm{~g} / \\mathrm{mol}$.\n\n[figure1]\n\nWhat is the melting point of the substance?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of ${ }^{\\circ} \\mathrm{C}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-08.jpg?height=1071&width=1612&top_left_y=825&top_left_x=251" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "${ }^{\\circ} \\mathrm{C}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_218", "problem": "A juggler juggles $N$ identical balls, catching and tossing one ball at a time. Assuming that the juggler requires a minimum time $T$ between ball tosses, the minimum possible power required for the juggler to continue juggling is proportional to\nA: $N^{0}$\nB: $N^{1}$\nC: $N^{2} $ \nD: $N^{3}$\nE: $N^{4}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA juggler juggles $N$ identical balls, catching and tossing one ball at a time. Assuming that the juggler requires a minimum time $T$ between ball tosses, the minimum possible power required for the juggler to continue juggling is proportional to\n\nA: $N^{0}$\nB: $N^{1}$\nC: $N^{2} $ \nD: $N^{3}$\nE: $N^{4}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1637", "problem": "如图，两个相同的正方形刚性细金属框 $\\mathrm{ABCD}$ 和 $\\mathrm{A}^{\\prime} \\mathrm{B}^{\\prime} \\mathrm{C}^{\\prime} \\mathrm{D}^{\\prime}$ 的质量均为 $m$, 边长均为 $a$,每边电阻均为 $R$; 两框部分地交叠在同一平面内, 两框交叠部分长为 $l$, 电接触良好。将整个系统置于恒定的匀强磁场中, 磁感应强度大小为 $B_{0}$, 方向垂直于框面（纸面）向纸面内。现将磁场突然撤去, 求流过框边重叠部分 $A^{\\prime} D$的横截面的总电荷量。不计摩擦、重力和框的电感。\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n如图，两个相同的正方形刚性细金属框 $\\mathrm{ABCD}$ 和 $\\mathrm{A}^{\\prime} \\mathrm{B}^{\\prime} \\mathrm{C}^{\\prime} \\mathrm{D}^{\\prime}$ 的质量均为 $m$, 边长均为 $a$,每边电阻均为 $R$; 两框部分地交叠在同一平面内, 两框交叠部分长为 $l$, 电接触良好。将整个系统置于恒定的匀强磁场中, 磁感应强度大小为 $B_{0}$, 方向垂直于框面（纸面）向纸面内。现将磁场突然撤去, 求流过框边重叠部分 $A^{\\prime} D$的横截面的总电荷量。不计摩擦、重力和框的电感。\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_17b1131fe8d911867aa0g-04.jpg?height=520&width=602&top_left_y=2013&top_left_x=1138", "https://cdn.mathpix.com/cropped/2024_03_31_17b1131fe8d911867aa0g-10.jpg?height=366&width=599&top_left_y=1913&top_left_x=1134" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_793", "problem": "A freely-falling body has a constant acceleration of $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$. This means that\nA: the body falls $9.8 \\mathrm{~m}$ every second\nB: the body falls $9.8 \\mathrm{~m}$ during the first second only.\nC: the speed of the body increases by $9.8 \\mathrm{~m} / \\mathrm{s}$ every second.\nD: the acceleration of the body increases by $9.8 \\mathrm{~m} / \\mathrm{s}$ every second.\nE: the acceleration of the body increases by $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$ every second.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA freely-falling body has a constant acceleration of $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$. This means that\n\nA: the body falls $9.8 \\mathrm{~m}$ every second\nB: the body falls $9.8 \\mathrm{~m}$ during the first second only.\nC: the speed of the body increases by $9.8 \\mathrm{~m} / \\mathrm{s}$ every second.\nD: the acceleration of the body increases by $9.8 \\mathrm{~m} / \\mathrm{s}$ every second.\nE: the acceleration of the body increases by $9.8 \\mathrm{~m} / \\mathrm{s}^{2}$ every second.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1505", "problem": "如图 $\\mathrm{a}$, 旅行车上有一个半径为 $\\mathrm{R}$ 的三脚圆登 (可视为刚性结构), 三个相同登脚的端点连线 (均水平) 构成边长为 $a$ 的等边三角形, 登子质心位于其轴上的 $\\mathrm{G}$ 点. 半径为 $\\mathrm{r}$ 的一圆筒形薄壁茶杯放在登面上,杯底中心位于登面中心 0 点处, 茶杯质量为 $\\mathrm{m}$ (远小于登子质量), 其中杯底质量为 $\\frac{m}{5}$ (杯壁和杯底各自的质量分布都是均匀的), 杯高为 $\\mathrm{H}$ (与杯高相比, 杯底厚度可忽略). 杯中盛有茶水, 茶水密度为 $\\rho$. 重力加速度大小为 $\\mathrm{g}$.\n\n[图1]\n\n图 a\n\n[图2]\n\n图 b为了使茶水杯所盛茶水尽可能多并保持足够稳定，杯中茶水的最佳高度是多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图 $\\mathrm{a}$, 旅行车上有一个半径为 $\\mathrm{R}$ 的三脚圆登 (可视为刚性结构), 三个相同登脚的端点连线 (均水平) 构成边长为 $a$ 的等边三角形, 登子质心位于其轴上的 $\\mathrm{G}$ 点. 半径为 $\\mathrm{r}$ 的一圆筒形薄壁茶杯放在登面上,杯底中心位于登面中心 0 点处, 茶杯质量为 $\\mathrm{m}$ (远小于登子质量), 其中杯底质量为 $\\frac{m}{5}$ (杯壁和杯底各自的质量分布都是均匀的), 杯高为 $\\mathrm{H}$ (与杯高相比, 杯底厚度可忽略). 杯中盛有茶水, 茶水密度为 $\\rho$. 重力加速度大小为 $\\mathrm{g}$.\n\n[图1]\n\n图 a\n\n[图2]\n\n图 b\n\n问题:\n为了使茶水杯所盛茶水尽可能多并保持足够稳定，杯中茶水的最佳高度是多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-01.jpg?height=591&width=423&top_left_y=1069&top_left_x=448", "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-01.jpg?height=597&width=420&top_left_y=1066&top_left_x=1160" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_680", "problem": "A person is knitting a sweater using a spherical ball of yarn with an initial radius of $15 \\mathrm{~cm}(a t \\mathrm{t}=0)$. The radius of the ball of yarn is $10 \\mathrm{~cm}$ after 5 hours $(t=5$ h) of knitting. What is the radius of the ball of yarn at time $t=7 \\mathrm{~h}$, assuming that the person is knitting at a constant rate?\nA: $8 \\mathrm{~cm}$\nB: $7.07 \\mathrm{~cm}$\nC: $6.16 \\mathrm{~cm}$\nD: $3.68 \\mathrm{~cm}$\nE: $0 \\mathrm{~cm}$ (the yarn would run out)\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA person is knitting a sweater using a spherical ball of yarn with an initial radius of $15 \\mathrm{~cm}(a t \\mathrm{t}=0)$. The radius of the ball of yarn is $10 \\mathrm{~cm}$ after 5 hours $(t=5$ h) of knitting. What is the radius of the ball of yarn at time $t=7 \\mathrm{~h}$, assuming that the person is knitting at a constant rate?\n\nA: $8 \\mathrm{~cm}$\nB: $7.07 \\mathrm{~cm}$\nC: $6.16 \\mathrm{~cm}$\nD: $3.68 \\mathrm{~cm}$\nE: $0 \\mathrm{~cm}$ (the yarn would run out)\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_900", "problem": "The distribution of electron energies in a jet from an AGN is typically a power law, of the form $f(\\epsilon)=\\kappa \\epsilon^{-p}$, where $f(\\epsilon) d \\epsilon$ is the number density of particles with energies between $\\epsilon$ and $\\epsilon+d \\epsilon$. The corresponding spectrum of synchrotron emission depends on the electron energy distribution, rather than the spectrum for an individual electron. This spectrum is\n\n$$\nj(\\nu) d \\nu \\propto B^{(1+p) / 2} \\nu^{(1-p) / 2} d \\nu .\n$$\n\nHere $j(\\nu) d \\nu$ is the energy per unit volume emitted as photons with frequencies between $\\nu$ and $\\nu+d \\nu$\n\nObservations of the Centaurus A jet, and other jets, show a knotty structure, with compact regions of brighter emission called knots. Observations of these knots at different times have shown both motion and brightness changes for some knots. Two possible mechanisms for the reductions in brightness are adiabatic expansion of the gas in the knot, and synchrotron cooling of electrons in the gas in knot.\n\nThe magnetic field in the plasma in the jets is assumed to be frozen in. Considering an arbitrary volume of plasma, the magnetic flux through the surface bounding it must remain constant, even as the volume containing the plasma changes shape and size.\n\nFind $f(\\epsilon)$, the distribution of electron energies after adiabatic expansion of a spherical knot to a volume $V$ on the distribution of electron energy densities, given that the knot of volume $V_{0}$ has an initial distribution of electrons $f_{0}(\\epsilon)=$ $\\kappa_{0} \\epsilon^{-p}$, where $f_{0}(\\epsilon) d \\epsilon$ is the number density of particles with energies between $\\epsilon$ and $\\epsilon+d \\epsilon$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nThe distribution of electron energies in a jet from an AGN is typically a power law, of the form $f(\\epsilon)=\\kappa \\epsilon^{-p}$, where $f(\\epsilon) d \\epsilon$ is the number density of particles with energies between $\\epsilon$ and $\\epsilon+d \\epsilon$. The corresponding spectrum of synchrotron emission depends on the electron energy distribution, rather than the spectrum for an individual electron. This spectrum is\n\n$$\nj(\\nu) d \\nu \\propto B^{(1+p) / 2} \\nu^{(1-p) / 2} d \\nu .\n$$\n\nHere $j(\\nu) d \\nu$ is the energy per unit volume emitted as photons with frequencies between $\\nu$ and $\\nu+d \\nu$\n\nObservations of the Centaurus A jet, and other jets, show a knotty structure, with compact regions of brighter emission called knots. Observations of these knots at different times have shown both motion and brightness changes for some knots. Two possible mechanisms for the reductions in brightness are adiabatic expansion of the gas in the knot, and synchrotron cooling of electrons in the gas in knot.\n\nThe magnetic field in the plasma in the jets is assumed to be frozen in. Considering an arbitrary volume of plasma, the magnetic flux through the surface bounding it must remain constant, even as the volume containing the plasma changes shape and size.\n\nFind $f(\\epsilon)$, the distribution of electron energies after adiabatic expansion of a spherical knot to a volume $V$ on the distribution of electron energy densities, given that the knot of volume $V_{0}$ has an initial distribution of electrons $f_{0}(\\epsilon)=$ $\\kappa_{0} \\epsilon^{-p}$, where $f_{0}(\\epsilon) d \\epsilon$ is the number density of particles with energies between $\\epsilon$ and $\\epsilon+d \\epsilon$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1328", "problem": "可能用到的物理常量和公式:\n\n真空中的光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$;\n\n已知地球表面的重力加速度的大小为 $g$;\n\n已知普朗克常量为 $h, \\hbar=\\frac{h}{2 \\pi}$;\n\n$\\int \\frac{1}{1-x^{2}} \\mathrm{~d} x=\\frac{1}{2} \\ln \\frac{1+x}{1-x}+C,|x|<1$ 。\n\n山西大同某煤矿相对于秦皇岛的高度为 $h_{\\mathrm{c}}$ 。质量为 $m_{\\mathrm{t}}$ 的火车载有质量为 $m_{\\mathrm{c}}$ 的煤, 从大同沿大秦线铁路行驶路程 $l$ 后到达秦皇岛, 卸载后空车返回。从大同到秦皇岛的过程中, 火车和煤总势能的一部分克服铁轨和空气阻力做功, 其余部分由发电机转换成电能, 平均转换效率为 $\\eta_{1}$, 电能被全部储存于蓄电池中以用于返程。空车在返程中由储存的电能驱动电动机克服重力和阻力做功, 存储电能转化为对外做功的平均转换效率为 $\\eta_{2}$ 。假设大秦线轨道上火车平均每运行单位距离克服阻力需要做的功与运行时（火车或火车和煤）总重量成正比,比例系数为常数 $\\mu$, 火车由大同出发时携带的电能为零。若空车返回大同时还有剩余的电能, 求该电能 $E$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n可能用到的物理常量和公式:\n\n真空中的光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$;\n\n已知地球表面的重力加速度的大小为 $g$;\n\n已知普朗克常量为 $h, \\hbar=\\frac{h}{2 \\pi}$;\n\n$\\int \\frac{1}{1-x^{2}} \\mathrm{~d} x=\\frac{1}{2} \\ln \\frac{1+x}{1-x}+C,|x|<1$ 。\n\n山西大同某煤矿相对于秦皇岛的高度为 $h_{\\mathrm{c}}$ 。质量为 $m_{\\mathrm{t}}$ 的火车载有质量为 $m_{\\mathrm{c}}$ 的煤, 从大同沿大秦线铁路行驶路程 $l$ 后到达秦皇岛, 卸载后空车返回。从大同到秦皇岛的过程中, 火车和煤总势能的一部分克服铁轨和空气阻力做功, 其余部分由发电机转换成电能, 平均转换效率为 $\\eta_{1}$, 电能被全部储存于蓄电池中以用于返程。空车在返程中由储存的电能驱动电动机克服重力和阻力做功, 存储电能转化为对外做功的平均转换效率为 $\\eta_{2}$ 。假设大秦线轨道上火车平均每运行单位距离克服阻力需要做的功与运行时（火车或火车和煤）总重量成正比,比例系数为常数 $\\mu$, 火车由大同出发时携带的电能为零。\n\n问题:\n若空车返回大同时还有剩余的电能, 求该电能 $E$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_981", "problem": "A ball, thrown with an initial speed $v_{0}$, moves in a homogeneous gravitational field in the $x-z$ plane, where the $x$-axis is horizontal, and the $z$-axis is vertical and antiparallel to the free fall acceleration $g$. Neglect the effect of air drag.\n\nBy adjusting the launching angle for a ball thrown with a fixed initial speed $v_{0}$ from the origin, targets can be hit within the region given by\n\n$$\nz \\leq z_{0}-k x^{2}\n$$\n\nYou can use this fact without proving it. Find the constants $z_{0}$ and $k$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nA ball, thrown with an initial speed $v_{0}$, moves in a homogeneous gravitational field in the $x-z$ plane, where the $x$-axis is horizontal, and the $z$-axis is vertical and antiparallel to the free fall acceleration $g$. Neglect the effect of air drag.\n\nBy adjusting the launching angle for a ball thrown with a fixed initial speed $v_{0}$ from the origin, targets can be hit within the region given by\n\n$$\nz \\leq z_{0}-k x^{2}\n$$\n\nYou can use this fact without proving it. Find the constants $z_{0}$ and $k$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [$z_{0}$, k].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$z_{0}$", "k" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1315", "problem": "塞曼发现了钠光 $\\mathrm{D}$ 线在磁场中分裂成三条, 洛仑兹根据经典电磁理论对此做出了解释,他们因此荣获 1902 年诺贝尔物理学奖。假定原子中的价电子 (质量为 $m$, 电荷量为 $-e, e>0$ )受到一指向原子中心的等效线性回复力 $-m \\omega_{0}^{2} \\boldsymbol{r}$ ( $\\boldsymbol{r}$ 为价电子相对于原子中心的位矢）作用,做固有圆频率为 $\\omega_{0}$ 的简谐振动, 发出圆频率为 $\\omega_{0}$ 的光。现将该原子置于沿 $z$ 轴正方向的匀\n强磁场中, 磁感应强度大小为 $B$ (为方便起见, 将 $B$ 参数化为 $B=\\frac{2 m}{e} \\omega_{\\mathrm{L}}$ )。\n\n已知: 在转动角速度为 $\\omega$ 的转动参考系中, 运动电子受到的惯性力除惯性离心力外还受到科里奥利力作用, 当电子相对于转动参考系运动速度为 $v^{\\prime}$ 时, 作用于电子的科里奥利力为 $\\boldsymbol{f}_{\\mathrm{c}}=-2 m \\boldsymbol{\\omega} \\times \\boldsymbol{v}^{\\prime}$ 。选一绕磁场方向匀角速转动的参考系, 使价电子在该参考系中做简谐振动, 导出该电子运动的动力学方程在直角坐标系中的分量形式并求出其解;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n塞曼发现了钠光 $\\mathrm{D}$ 线在磁场中分裂成三条, 洛仑兹根据经典电磁理论对此做出了解释,他们因此荣获 1902 年诺贝尔物理学奖。假定原子中的价电子 (质量为 $m$, 电荷量为 $-e, e>0$ )受到一指向原子中心的等效线性回复力 $-m \\omega_{0}^{2} \\boldsymbol{r}$ ( $\\boldsymbol{r}$ 为价电子相对于原子中心的位矢）作用,做固有圆频率为 $\\omega_{0}$ 的简谐振动, 发出圆频率为 $\\omega_{0}$ 的光。现将该原子置于沿 $z$ 轴正方向的匀\n强磁场中, 磁感应强度大小为 $B$ (为方便起见, 将 $B$ 参数化为 $B=\\frac{2 m}{e} \\omega_{\\mathrm{L}}$ )。\n\n已知: 在转动角速度为 $\\omega$ 的转动参考系中, 运动电子受到的惯性力除惯性离心力外还受到科里奥利力作用, 当电子相对于转动参考系运动速度为 $v^{\\prime}$ 时, 作用于电子的科里奥利力为 $\\boldsymbol{f}_{\\mathrm{c}}=-2 m \\boldsymbol{\\omega} \\times \\boldsymbol{v}^{\\prime}$ 。\n\n问题:\n选一绕磁场方向匀角速转动的参考系, 使价电子在该参考系中做简谐振动, 导出该电子运动的动力学方程在直角坐标系中的分量形式并求出其解;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[导出该电子运动的动力学方程在直角坐标系中的分量形式, 求解该电子运动的动力学方程在直角坐标系中的分量]\n它们的答案类型依次是[方程, 方程]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-19.jpg?height=348&width=459&top_left_y=2310&top_left_x=1301" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "导出该电子运动的动力学方程在直角坐标系中的分量形式", "求解该电子运动的动力学方程在直角坐标系中的分量" ], "type_sequence": [ "EQ", "EQ" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1586", "problem": "菲涅尔透镜又称同心圆阶梯透镜, 它是由很多个同轴环带套在一起构成的, 其迎光面是平面，折射面除中心是一个球冠外, 其它环带分别是属于不同球面的球台侧面, 其纵剖面如右图所示。这样的结构可以避免普通大口径球面透镜既厚又重的缺点。菲涅尔透镜的设计主要是确定每个环带的齿形 (即它所属球面的球半径和球心)，各环带都是一个独立的 (部分) 球面透镜, 它们的焦距不同, 但必须保证具有共同的焦点（即图中 F 点）。已知透镜材料的折射率为 $n$, 从透镜中心 $\\mathrm{O}$ (球冠的顶点) 到焦点 $\\mathrm{F}$ 的距离（焦距）为 $f$ (平行于光轴的平行光都能经环带折射后会聚到 $\\mathrm{F}$ 点), 相邻环带的间距为 $d$ ( $d$ 很小,可忽略同一带内的球面像差; $d$ 又不是非常小，可忽略衍射效应）。求\n\n[图1]该透镜的有效半径的最大值和有效环带的条数。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n菲涅尔透镜又称同心圆阶梯透镜, 它是由很多个同轴环带套在一起构成的, 其迎光面是平面，折射面除中心是一个球冠外, 其它环带分别是属于不同球面的球台侧面, 其纵剖面如右图所示。这样的结构可以避免普通大口径球面透镜既厚又重的缺点。菲涅尔透镜的设计主要是确定每个环带的齿形 (即它所属球面的球半径和球心)，各环带都是一个独立的 (部分) 球面透镜, 它们的焦距不同, 但必须保证具有共同的焦点（即图中 F 点）。已知透镜材料的折射率为 $n$, 从透镜中心 $\\mathrm{O}$ (球冠的顶点) 到焦点 $\\mathrm{F}$ 的距离（焦距）为 $f$ (平行于光轴的平行光都能经环带折射后会聚到 $\\mathrm{F}$ 点), 相邻环带的间距为 $d$ ( $d$ 很小,可忽略同一带内的球面像差; $d$ 又不是非常小，可忽略衍射效应）。求\n\n[图1]\n\n问题:\n该透镜的有效半径的最大值和有效环带的条数。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[该透镜的有效半径的最大值, 该透镜的有效环带的条数]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-04.jpg?height=317&width=322&top_left_y=1184&top_left_x=707" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "该透镜的有效半径的最大值", "该透镜的有效环带的条数" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_771", "problem": "A block of mass $5 \\mathrm{~kg}$ sits at rest on a horizontal surface. A downwards force of $20 \\mathrm{~N}$ is applied to the block, as shown.\n\n[figure1]\n\nWhat is the weight of the block? Select one:\nA: $5 \\mathrm{~kg}$\nB: $25 \\mathrm{~kg}$\nC: $25 \\mathrm{~N}$\nD: $50 \\mathrm{~N}$\nE: $\\quad 70 \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA block of mass $5 \\mathrm{~kg}$ sits at rest on a horizontal surface. A downwards force of $20 \\mathrm{~N}$ is applied to the block, as shown.\n\n[figure1]\n\nWhat is the weight of the block? Select one:\n\nA: $5 \\mathrm{~kg}$\nB: $25 \\mathrm{~kg}$\nC: $25 \\mathrm{~N}$\nD: $50 \\mathrm{~N}$\nE: $\\quad 70 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_b458d4c4b7b005fcc348g-04.jpg?height=372&width=556&top_left_y=445&top_left_x=316" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_49", "problem": "When descending mountain roads, large trucks pulling a heavy load can burn up the brakes. Once the brakes are no longer useful, the driver may need to guide the truck up a \"runaway truck lane\" on the side of the road. The runaway truck lane is directed uphill and often has a thick layer or sand or gravel or both on the surface. Which of the following is one of the reasons the truck will stop?\nA: An increase in kinetic energy\nB: A decrease in potential energy\nC: A decrease in fuel\nD: A transfer of energy to the gravel on the track of the runaway truck lane\nE: The change in temperature of the engine\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nWhen descending mountain roads, large trucks pulling a heavy load can burn up the brakes. Once the brakes are no longer useful, the driver may need to guide the truck up a \"runaway truck lane\" on the side of the road. The runaway truck lane is directed uphill and often has a thick layer or sand or gravel or both on the surface. Which of the following is one of the reasons the truck will stop?\n\nA: An increase in kinetic energy\nB: A decrease in potential energy\nC: A decrease in fuel\nD: A transfer of energy to the gravel on the track of the runaway truck lane\nE: The change in temperature of the engine\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1356", "problem": "图 1 所示器件由相互紧密接触的金属层 $\\mathrm{M}$ 薄绝缘层 $\\mathrm{I}$ 和金属层 $\\mathrm{M}$ 构成, 按照经典物理的观点, 在 $\\mathrm{I}$ 层绝缘性能理想的情况下, 电子不可能从一个金属层穿过绝缘层到达另一个绝\n\n[图1]\n\nM I M 缘层。但是按照量子物理的原理, 在一定的条件下, 这种渡越是可能的。习惯上将这一过程称之为“隧穿”, 它是电子具有波动性的结果。隧穿是单个电子的过程, 是独立的事件。通过绝缘层转移的电荷量只能是电子电荷量 $-e\\left(e=1.6 \\times 10^{-19} c\\right)$ 的整数倍, 因此也称为“单电子隧穿”, MIM 器件也称为“隧穿结”,或“单电子隧穿结”。\n\n本题涉及对单电子檤穿过程控制的库仑阻塞原理, 由于据此可望制成尺寸很小的单电子器件, 这是目前研究的很多、有应用前景的领域。假定 $V_{A B}=0.10 \\mathrm{mV}$, 是刚能发生的隧穿电压, 试估算电容 $C$ 的大小", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n图 1 所示器件由相互紧密接触的金属层 $\\mathrm{M}$ 薄绝缘层 $\\mathrm{I}$ 和金属层 $\\mathrm{M}$ 构成, 按照经典物理的观点, 在 $\\mathrm{I}$ 层绝缘性能理想的情况下, 电子不可能从一个金属层穿过绝缘层到达另一个绝\n\n[图1]\n\nM I M 缘层。但是按照量子物理的原理, 在一定的条件下, 这种渡越是可能的。习惯上将这一过程称之为“隧穿”, 它是电子具有波动性的结果。隧穿是单个电子的过程, 是独立的事件。通过绝缘层转移的电荷量只能是电子电荷量 $-e\\left(e=1.6 \\times 10^{-19} c\\right)$ 的整数倍, 因此也称为“单电子隧穿”, MIM 器件也称为“隧穿结”,或“单电子隧穿结”。\n\n本题涉及对单电子檤穿过程控制的库仑阻塞原理, 由于据此可望制成尺寸很小的单电子器件, 这是目前研究的很多、有应用前景的领域。\n\n问题:\n假定 $V_{A B}=0.10 \\mathrm{mV}$, 是刚能发生的隧穿电压, 试估算电容 $C$ 的大小\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以F为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_1004b08dedac85274c96g-04.jpg?height=126&width=386&top_left_y=337&top_left_x=1166" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "F" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_579", "problem": "Radiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe Poynting-Robertson effect acts as another mechanism for cleaning out the solar system.\n\nAssume that a particle is in a circular orbit around the sun. Find the speed of the particle $v$ in terms of $M_{\\odot}$, distance from sun $R$, and any other fundamental constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nRadiation pressure from the sun is responsible for cleaning out the inner solar system of small particles.\n\nThe Poynting-Robertson effect acts as another mechanism for cleaning out the solar system.\n\nAssume that a particle is in a circular orbit around the sun. Find the speed of the particle $v$ in terms of $M_{\\odot}$, distance from sun $R$, and any other fundamental constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_552", "problem": "An ideal (but not necessarily perfect monatomic) gas undergoes the following cycle.\n\n- The gas starts at pressure $P_{0}$, volume $V_{0}$ and temperature $T_{0}$.\n- The gas is heated at constant volume to a pressure $\\alpha P_{0}$, where $\\alpha>1$.\n- The gas is then allowed to expand adiabatically (no heat is transferred to or from the gas) to pressure $P_{0}$\n- The gas is cooled at constant pressure back to the original state.\n\nThe adiabatic constant $\\gamma$ is defined in terms of the specific heat at constant pressure $C_{p}$ and the specific heat at constant volume $C_{v}$ by the ratio $\\gamma=C_{p} / C_{v}$.\n\nWhat is $\\gamma$ for this gas?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nAn ideal (but not necessarily perfect monatomic) gas undergoes the following cycle.\n\n- The gas starts at pressure $P_{0}$, volume $V_{0}$ and temperature $T_{0}$.\n- The gas is heated at constant volume to a pressure $\\alpha P_{0}$, where $\\alpha>1$.\n- The gas is then allowed to expand adiabatically (no heat is transferred to or from the gas) to pressure $P_{0}$\n- The gas is cooled at constant pressure back to the original state.\n\nThe adiabatic constant $\\gamma$ is defined in terms of the specific heat at constant pressure $C_{p}$ and the specific heat at constant volume $C_{v}$ by the ratio $\\gamma=C_{p} / C_{v}$.\n\nWhat is $\\gamma$ for this gas?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_318", "problem": "A Zener diode is connected to a source of alternating current as shown in the figure. The current is sinus oidal $I=I_{0} \\cos \\omega t$ with a constant amplitude. The inductance $L$ of the inductor is such that $L \\omega I_{0} \\gg V_{1}, V_{2}$, where $V_{1}$ and $V_{2}$ are the break down voltages $\\left(V_{1}>V_{2}\\right)$. The current-voltage characteristic of the Zener diode is shown in the figure. In the following assume that a very long time has passed since the current source was first turned on.\n\n[figure1]\n\n Find the peak-to-peak amplitude of the current changes $\\Delta I$ in the inductor.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA Zener diode is connected to a source of alternating current as shown in the figure. The current is sinus oidal $I=I_{0} \\cos \\omega t$ with a constant amplitude. The inductance $L$ of the inductor is such that $L \\omega I_{0} \\gg V_{1}, V_{2}$, where $V_{1}$ and $V_{2}$ are the break down voltages $\\left(V_{1}>V_{2}\\right)$. The current-voltage characteristic of the Zener diode is shown in the figure. In the following assume that a very long time has passed since the current source was first turned on.\n\n[figure1]\n\n Find the peak-to-peak amplitude of the current changes $\\Delta I$ in the inductor.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_e6c1582ce7f1c05fa0a6g-2.jpg?height=424&width=694&top_left_y=754&top_left_x=82" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1431", "problem": "如图所示, 一电容器由固定在共同导电底座上的 $N+1$ 片对顶双扇形薄金属板和固定在可旋转的导电对称轴上的 $N$片对顶双扇形薄金属板组成, 所有顶点共轴, 轴线与所有板面垂直, 两组板面各自在垂直于轴线的平面上的投影重合, 板面扇形半径均为 $R$, 圆心角均为 $\\theta_{0}$ $\\left(\\frac{\\pi}{2} \\leq \\theta_{0}<\\pi\\right)$; 固定金属板和可\n\n[图1]\n旋转的金属板相间排列, 两相邻金属板之间距离均为 $s$. 此电容器的电容 $C$ 值与可旋转金属板的转角 $\\theta$ 有关. 已知静电力常量为 $k$.假设 $\\theta_{0}=\\frac{\\pi}{2}$, 考虑边缘效应后, 第 (1) 问中的 $C(\\theta)$ 可视为在其最大值和最小值之间光清变化的函数\n\n$$\nC(\\theta)=\\frac{1}{2}\\left(C_{\\max }+C_{\\min }\\right)+\\frac{1}{2}\\left(C_{\\max }-C_{\\min }\\right) \\cos 2 \\theta\n$$\n\n式中, $C_{\\max }$ 可由第 (1) 问的结果估算, 而 $C_{\\min }$ 是因边缘效应计入的, 它与 $C_{\\max }$ 的比值 $\\lambda$ 是已知的. 若转轴以角速度 $\\omega_{m}$ 匀速转动, 且 $\\theta=\\omega_{m} t$, 在极板间加一交流电压 $V=V_{0} \\cos \\omega t$. 试计算电容器在交流电压作用下能量在一个变化周期内的平均值, 并给出该平均值取最大值时所对应的 $\\omega_{m}$.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图所示, 一电容器由固定在共同导电底座上的 $N+1$ 片对顶双扇形薄金属板和固定在可旋转的导电对称轴上的 $N$片对顶双扇形薄金属板组成, 所有顶点共轴, 轴线与所有板面垂直, 两组板面各自在垂直于轴线的平面上的投影重合, 板面扇形半径均为 $R$, 圆心角均为 $\\theta_{0}$ $\\left(\\frac{\\pi}{2} \\leq \\theta_{0}<\\pi\\right)$; 固定金属板和可\n\n[图1]\n旋转的金属板相间排列, 两相邻金属板之间距离均为 $s$. 此电容器的电容 $C$ 值与可旋转金属板的转角 $\\theta$ 有关. 已知静电力常量为 $k$.\n\n问题:\n假设 $\\theta_{0}=\\frac{\\pi}{2}$, 考虑边缘效应后, 第 (1) 问中的 $C(\\theta)$ 可视为在其最大值和最小值之间光清变化的函数\n\n$$\nC(\\theta)=\\frac{1}{2}\\left(C_{\\max }+C_{\\min }\\right)+\\frac{1}{2}\\left(C_{\\max }-C_{\\min }\\right) \\cos 2 \\theta\n$$\n\n式中, $C_{\\max }$ 可由第 (1) 问的结果估算, 而 $C_{\\min }$ 是因边缘效应计入的, 它与 $C_{\\max }$ 的比值 $\\lambda$ 是已知的. 若转轴以角速度 $\\omega_{m}$ 匀速转动, 且 $\\theta=\\omega_{m} t$, 在极板间加一交流电压 $V=V_{0} \\cos \\omega t$. 试计算电容器在交流电压作用下能量在一个变化周期内的平均值, 并给出该平均值取最大值时所对应的 $\\omega_{m}$.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[电容器在交流电压作用下能量在一个变化周期内的平均值, $\\omega_{m}$]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ea729d4659bcaa2c4b91g-03.jpg?height=490&width=899&top_left_y=315&top_left_x=681" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "电容器在交流电压作用下能量在一个变化周期内的平均值", "$\\omega_{m}$" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1590", "problem": "有 3 种不同波长的光, 每种光同时发出、同时中断, 且光强都相同, 总的光强为 $I$, 脉冲宽度 (发光持续时间) 为 $\\tau$, 光脉冲的光强 $I$ 随时间 $t$ 的变化如图所示。该光脉冲正入射\n\n[图1]\n\n$L$\n\n到一长为 $L$ 的透明玻璃棒, 不考虑光在玻璃棒中的传输损失和端面的反射损失。在通过玻璃棒后光脉冲的光强 $I$ 随时间 $t$ 的变化最可能的图示是（虚线部分为入射前的波形）\nA: ![]([图2])\nB: ![]([图3])\nC: ![]([图4])\nD: ![]([图5])\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n有 3 种不同波长的光, 每种光同时发出、同时中断, 且光强都相同, 总的光强为 $I$, 脉冲宽度 (发光持续时间) 为 $\\tau$, 光脉冲的光强 $I$ 随时间 $t$ 的变化如图所示。该光脉冲正入射\n\n[图1]\n\n$L$\n\n到一长为 $L$ 的透明玻璃棒, 不考虑光在玻璃棒中的传输损失和端面的反射损失。在通过玻璃棒后光脉冲的光强 $I$ 随时间 $t$ 的变化最可能的图示是（虚线部分为入射前的波形）\n\nA: ![]([图2])\nB: ![]([图3])\nC: ![]([图4])\nD: ![]([图5])\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_d716ce15f03757bb482eg-01.jpg?height=314&width=1106&top_left_y=1859&top_left_x=560", "https://i.postimg.cc/Pr9pLV3M/2017-CPho-Q4-A.png", "https://i.postimg.cc/ZR5Ks6sR/2017-CPho-Q4-B.png", "https://i.postimg.cc/sD7Qh9kd/2017-CPho-Q4-C.png", "https://i.postimg.cc/v86DJtHg/2017-CPho-Q4-D.png" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_688", "problem": "A closed 5-litre cylinder containing $0.25 \\mathrm{~g}$ of a substance in solid and liquid forms was slowly heated using a constant power. A graph of the temperature as a function of time is shown below. The heat capacity of the liquid is $2.43 \\mathrm{~J} /(\\mathrm{g} \\cdot \\mathrm{K})$, the latent heat of melting is $105 \\mathrm{~J} / \\mathrm{g}$, and the molar density is $46 \\mathrm{~g} / \\mathrm{mol}$.\n\n[figure1]\n\nWhat is the boiling point of the substance?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA closed 5-litre cylinder containing $0.25 \\mathrm{~g}$ of a substance in solid and liquid forms was slowly heated using a constant power. A graph of the temperature as a function of time is shown below. The heat capacity of the liquid is $2.43 \\mathrm{~J} /(\\mathrm{g} \\cdot \\mathrm{K})$, the latent heat of melting is $105 \\mathrm{~J} / \\mathrm{g}$, and the molar density is $46 \\mathrm{~g} / \\mathrm{mol}$.\n\n[figure1]\n\nWhat is the boiling point of the substance?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of ${ }^{\\circ} \\mathrm{C}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_e69eaad2d8db15283465g-08.jpg?height=1071&width=1612&top_left_y=825&top_left_x=251" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "${ }^{\\circ} \\mathrm{C}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_342", "problem": "David stands at the bottom of an infinite stair case with both step width and height being equal to $d$. The corner of each step is slightly rounded. In the middle of each step, there is initially an upright domino of length $\\sqrt{5} d$ nd negligible thickness. Behind the base of ach domino, there is a small ridge that prevents it from sliding backward. David gives the first domino some initial angular velo city, and the dominoes start falling into each ther. All collisions are perfectly inelastic and there is no friction between two dom inoes. David notices that after a while, al dominoes have equal initial angular velocity $\\omega$. Find $\\omega$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nDavid stands at the bottom of an infinite stair case with both step width and height being equal to $d$. The corner of each step is slightly rounded. In the middle of each step, there is initially an upright domino of length $\\sqrt{5} d$ nd negligible thickness. Behind the base of ach domino, there is a small ridge that prevents it from sliding backward. David gives the first domino some initial angular velo city, and the dominoes start falling into each ther. All collisions are perfectly inelastic and there is no friction between two dom inoes. David notices that after a while, al dominoes have equal initial angular velocity $\\omega$. Find $\\omega$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_935", "problem": "A car tyre lasts typically $40000 \\mathrm{~km}$. Estimate the number of rotations it makes during its lifetime.\nA: $10^{5}$\nB: $10^{6}$\nC: $10^{7}$\nD: $10^{8}$\nE: $10^{9}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA car tyre lasts typically $40000 \\mathrm{~km}$. Estimate the number of rotations it makes during its lifetime.\n\nA: $10^{5}$\nB: $10^{6}$\nC: $10^{7}$\nD: $10^{8}$\nE: $10^{9}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_580", "problem": "A newly discovered subatomic particle, the $S$ meson, has a mass $M$. When at rest, it lives for exactly $\\tau=3 \\times 10^{-8}$ seconds before decaying into two identical particles called $P$ mesons (peons?) that each have a mass of $\\alpha M$.\n\nIn a reference frame where the $\\mathrm{S}$ meson is at rest, determine the velocity of each $\\mathrm{P}$ meson particle in terms of $M, \\alpha$, the speed of light $c$, and any numerical constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA newly discovered subatomic particle, the $S$ meson, has a mass $M$. When at rest, it lives for exactly $\\tau=3 \\times 10^{-8}$ seconds before decaying into two identical particles called $P$ mesons (peons?) that each have a mass of $\\alpha M$.\n\nIn a reference frame where the $\\mathrm{S}$ meson is at rest, determine the velocity of each $\\mathrm{P}$ meson particle in terms of $M, \\alpha$, the speed of light $c$, and any numerical constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1383", "problem": "具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：假设颗粒对光的吸收截面（颗粒吸收的光功率与入射光强之比）为 $\\sigma_{\\mathrm{abs}}=0.123 \\pi R^{2}$,求该束激光作用在颗粒上沿旋转对称轴的力矩的大小 $M$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n具有一定能量、动量的光子还具有角动量。圆偏振光的光子的角动量大小为 $\\hbar$ 。光子被物体吸收后, 光子的能量、动量和角动量就全部传给物体。物体吸收光子获得的角动量可以使物体转动。科学家利用这一原理, 在连续的圆偏振激光照射下, 实现了纳米颗粒的高速转动, 获得了迄今为止液体环境中转速最高的微尺度转子。\n\n如图, 一金纳米球颗粒放置在两片水平光滑玻璃平板之间，并整体（包括玻璃平板）浸在水中。一束圆偏振激光从上往下照射到金纳米颗粒上。已知该束入射激光在真空中的波长 $\\lambda=830 \\mathrm{~nm}$, 经显微物镜聚焦后（仍假设为平面波, 每个光子具有沿传播方向的角动量 $\\hbar$ ) 光斑直径 $d=1.20 \\times 10^{-6} \\mathrm{~m}$, 功率 $P=20.0 \\mathrm{~mW}$ 。金纳米球颗粒的半径 $R=100 \\mathrm{~nm}$, 金的密度 $\\rho=19.32 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$ 。忽略光在介质\n\n圆偏振激光\n[图1]\n界面上的反射以及玻璃、水对光的吸收等损失, 仅从金纳米颗粒吸收光子获得角动量驱动其转动的角度分析下列问题（计算结果取 3 位有效数字）：\n\n问题:\n假设颗粒对光的吸收截面（颗粒吸收的光功率与入射光强之比）为 $\\sigma_{\\mathrm{abs}}=0.123 \\pi R^{2}$,求该束激光作用在颗粒上沿旋转对称轴的力矩的大小 $M$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~N} \\cdot \\mathrm{m}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc7c55a7ed04447daac3g-02.jpg?height=450&width=450&top_left_y=1883&top_left_x=1292" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~N} \\cdot \\mathrm{m}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_383", "problem": "A drone is pulling a cuboid with a rope as shown in the sketch; the cuboid is sliding slowly, with a constant speed, on the hori zontal floor. The cuboid is made from an ho mogeneous material. You may take measure ments from the sketch (on a separate page) assuming that the dimensions and distances on it are correct within an unknown scale factor. In order to help you in case you don't have access to a printer, and need to read the problem texts directly from the com puter screen, some auxiliary dashed lines are shown in the diagram (which might or might not be useful).\n\nFind the coefficient of friction between the cuboid and the floor.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA drone is pulling a cuboid with a rope as shown in the sketch; the cuboid is sliding slowly, with a constant speed, on the hori zontal floor. The cuboid is made from an ho mogeneous material. You may take measure ments from the sketch (on a separate page) assuming that the dimensions and distances on it are correct within an unknown scale factor. In order to help you in case you don't have access to a printer, and need to read the problem texts directly from the com puter screen, some auxiliary dashed lines are shown in the diagram (which might or might not be useful).\n\nFind the coefficient of friction between the cuboid and the floor.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_683c031a1f6fc545249eg-4.jpg?height=571&width=668&top_left_y=289&top_left_x=770" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_840", "problem": "An electromagnetic wave can propagate in a transmission line in two opposite directions. For each direction of propagation, the characteristic impedance $Z_{0}$ can be used to relate the voltage $V_{0}$ and current $I_{0}$ amplitudes as in the Ohm's law, $Z_{0}=V_{0} / I_{0}$.\n\nConsider an interface between two transmission lines, with characteristic impedances $Z_{0}$ and $Z_{1}$. A schematic diagram of the circuit is shown below.\n\n[figure1]\n\nCircuit diagram of a transmission line of impedance $Z_{0}$ connected to a transmission line of impedance $Z_{1}$. The physical size of the interface is much smaller than the wavelength.\n\nWhen a signal $V_{\\mathrm{i}}$ sent into the transmission line with impedance $Z_{0}$ reaches the interface it is partially transmitted into the second transmission line, resulting in a signal $V_{\\mathrm{t}}$ in that line which propagates forward. Some of the signal may also be reflected, resulting in a backward propagating signal in the initial transmission line $V_{\\mathrm{r}}$.\n\nFind the reflectance of the interface $\\Gamma=V_{\\mathrm{r}} / V_{\\mathrm{i}}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nAn electromagnetic wave can propagate in a transmission line in two opposite directions. For each direction of propagation, the characteristic impedance $Z_{0}$ can be used to relate the voltage $V_{0}$ and current $I_{0}$ amplitudes as in the Ohm's law, $Z_{0}=V_{0} / I_{0}$.\n\nConsider an interface between two transmission lines, with characteristic impedances $Z_{0}$ and $Z_{1}$. A schematic diagram of the circuit is shown below.\n\n[figure1]\n\nCircuit diagram of a transmission line of impedance $Z_{0}$ connected to a transmission line of impedance $Z_{1}$. The physical size of the interface is much smaller than the wavelength.\n\nWhen a signal $V_{\\mathrm{i}}$ sent into the transmission line with impedance $Z_{0}$ reaches the interface it is partially transmitted into the second transmission line, resulting in a signal $V_{\\mathrm{t}}$ in that line which propagates forward. Some of the signal may also be reflected, resulting in a backward propagating signal in the initial transmission line $V_{\\mathrm{r}}$.\n\nFind the reflectance of the interface $\\Gamma=V_{\\mathrm{r}} / V_{\\mathrm{i}}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-3.jpg?height=166&width=782&top_left_y=842&top_left_x=637" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_67", "problem": "An older and heavier sister can balance on a seesaw with her younger and lighter brother. This can happen because of balanced\nA: forces.\nB: torques.\nC: momenta.\nD: energies.\nE: inertia.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn older and heavier sister can balance on a seesaw with her younger and lighter brother. This can happen because of balanced\n\nA: forces.\nB: torques.\nC: momenta.\nD: energies.\nE: inertia.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_446", "problem": "Clearly, any friction in the mechanism will lead to inefficiency: it is, however, instructive to examine the assumption that the pulley blocks have negligible mass.\n\nFigure shows a more realistic system, in which the lower half of the system (the pulley block) has a mass $m / k$, where $k$ is a numerical parameter.\n\n[figure1]\nFigure: Two pulley blocks and a light cord.\n\nWhat is the value of the tension in the cord?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nClearly, any friction in the mechanism will lead to inefficiency: it is, however, instructive to examine the assumption that the pulley blocks have negligible mass.\n\nFigure shows a more realistic system, in which the lower half of the system (the pulley block) has a mass $m / k$, where $k$ is a numerical parameter.\n\n[figure1]\nFigure: Two pulley blocks and a light cord.\n\nWhat is the value of the tension in the cord?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8ea586ad37c37c010606g-5.jpg?height=902&width=460&top_left_y=323&top_left_x=1392" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1463", "problem": "横截面积为 $S$ 和 $2 S$ 的两圆柱形容器按图示方式连接成一气缸, 每个圆筒中各置有一活塞, 两活塞间的距离为 $l$, 用硬杆相连, 形成“工”形活塞, 它把整个气缸分隔成三个气室, 其\n\n[图1]\n中 I、III 室密闭摩尔数分别为 $v$ 和 $2 v$ 的同种理想气体, 两个气室內都有电加热器; II 室的缸壁上开有一小孔, 与大气相通; $1 \\mathrm{~mol}$ 该种气体内能为 $C T$ ( $C$ 是气体摩尔热容量,\n$T$ 是气体的绝对温度）。当三个气室中气体的温度均为 $T_{1}$ 时, “工”形活塞在气缸中恰好在图所示的位置处于平衡状态, 这时 I 室内气柱长亦为 $l$, II 室内空气的摩尔数为 $\\frac{3}{2} v_{0}$ 。已知大气压不变, 气缸壁和活塞都是绝热的, 不计活塞与气缸之间的摩擦。现通过电热器对 I、III 两室中的气体缓慢加热, 直至 I 室内气体的温度升为其初始状态温度的 2 倍时, 活塞左移距离 $d$ 。已知理想气体常量为 $R$ 。求III 室内气体末态的温度;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n横截面积为 $S$ 和 $2 S$ 的两圆柱形容器按图示方式连接成一气缸, 每个圆筒中各置有一活塞, 两活塞间的距离为 $l$, 用硬杆相连, 形成“工”形活塞, 它把整个气缸分隔成三个气室, 其\n\n[图1]\n中 I、III 室密闭摩尔数分别为 $v$ 和 $2 v$ 的同种理想气体, 两个气室內都有电加热器; II 室的缸壁上开有一小孔, 与大气相通; $1 \\mathrm{~mol}$ 该种气体内能为 $C T$ ( $C$ 是气体摩尔热容量,\n$T$ 是气体的绝对温度）。当三个气室中气体的温度均为 $T_{1}$ 时, “工”形活塞在气缸中恰好在图所示的位置处于平衡状态, 这时 I 室内气柱长亦为 $l$, II 室内空气的摩尔数为 $\\frac{3}{2} v_{0}$ 。已知大气压不变, 气缸壁和活塞都是绝热的, 不计活塞与气缸之间的摩擦。现通过电热器对 I、III 两室中的气体缓慢加热, 直至 I 室内气体的温度升为其初始状态温度的 2 倍时, 活塞左移距离 $d$ 。已知理想气体常量为 $R$ 。求\n\n问题:\nIII 室内气体末态的温度;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_d716ce15f03757bb482eg-03.jpg?height=160&width=372&top_left_y=2307&top_left_x=1316" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1511", "problem": "在任意给定温度 $T$ (热力学温度) 下, 黑体热辐射功率最大的波长 $\\lambda_{\\text {max }}$ 满足 $\\lambda_{\\text {max }} T=b(b$为常量）; 黑体单位面积上的热辐射功率 $P=\\sigma T^{4}$ （ $\\sigma$ 为常量）。假定地球与太阳以及人体均可视为黑体。已知 $\\lambda_{\\max }^{\\text {(太阳 })}=5.0 \\times 10^{-7} \\mathrm{~m}$, 日地距离约为太阳半径的 200 倍, $\\lambda_{\\max }^{(\\text {(体) }}=9.3 \\times 10^{-6} \\mathrm{~m}$ 。由此可估算出, 太阳表面的温度约为 ___ ${ }^{\\circ} \\mathrm{C}$, 地球表面的平均温度约为 ___ ${ }^{\\circ} \\mathrm{C}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n在任意给定温度 $T$ (热力学温度) 下, 黑体热辐射功率最大的波长 $\\lambda_{\\text {max }}$ 满足 $\\lambda_{\\text {max }} T=b(b$为常量）; 黑体单位面积上的热辐射功率 $P=\\sigma T^{4}$ （ $\\sigma$ 为常量）。假定地球与太阳以及人体均可视为黑体。已知 $\\lambda_{\\max }^{\\text {(太阳 })}=5.0 \\times 10^{-7} \\mathrm{~m}$, 日地距离约为太阳半径的 200 倍, $\\lambda_{\\max }^{(\\text {(体) }}=9.3 \\times 10^{-6} \\mathrm{~m}$ 。由此可估算出, 太阳表面的温度约为 ___ ${ }^{\\circ} \\mathrm{C}$, 地球表面的平均温度约为 ___ ${ }^{\\circ} \\mathrm{C}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[太阳表面的温度约为, 地球表面的平均温度约为]\n它们的单位依次是[${ }^{\\circ} \\mathrm{C}$, ${ }^{\\circ} \\mathrm{C}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "${ }^{\\circ} \\mathrm{C}$", "${ }^{\\circ} \\mathrm{C}$" ], "answer_sequence": [ "太阳表面的温度约为", "地球表面的平均温度约为" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1409", "problem": "从地球上看太阳时, 对太阳直径的张角 $\\theta=0.53^{\\circ}$, 取地球表面上纬度为 $1^{\\circ}$ 的长度 $l=110 \\mathrm{~km}$, 地球表面处的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 地球公转的周期 $T=365$ 天。试仅用以上数据\n计算地球和太阳密度之比。假设太阳和地球都是质量均匀分布的球体。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n从地球上看太阳时, 对太阳直径的张角 $\\theta=0.53^{\\circ}$, 取地球表面上纬度为 $1^{\\circ}$ 的长度 $l=110 \\mathrm{~km}$, 地球表面处的重力加速度 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$, 地球公转的周期 $T=365$ 天。试仅用以上数据\n计算地球和太阳密度之比。假设太阳和地球都是质量均匀分布的球体。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_620", "problem": "The electric potential at the center of a cube with uniform charge density $\\rho$ and side length $a$ is\n\n$$\n\\Phi \\approx \\frac{0.1894 \\rho a^{2}}{\\epsilon_{0}}\n$$\n\nYou do not need to derive this. ${ }^{1}$\n\nFor the entirety of this problem, any computed numerical constants should be written to three significant figures.\n\nWhat is the electric potential at a corner of the same cube? Write your answer in terms of $\\rho, a, \\epsilon_{0}$, and any necessary numerical constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe electric potential at the center of a cube with uniform charge density $\\rho$ and side length $a$ is\n\n$$\n\\Phi \\approx \\frac{0.1894 \\rho a^{2}}{\\epsilon_{0}}\n$$\n\nYou do not need to derive this. ${ }^{1}$\n\nFor the entirety of this problem, any computed numerical constants should be written to three significant figures.\n\nWhat is the electric potential at a corner of the same cube? Write your answer in terms of $\\rho, a, \\epsilon_{0}$, and any necessary numerical constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_919", "problem": "The fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nIt turns out that binding an integer number of vortices $(n>1)$ with each electron generates a bigger surrounding whirlpool, hence pushes away all other electrons. Therefore, the system can considerably reduce its electrostatic\n\nCoulomb energy at the corresponding filling factor. Determine the scaling exponent $\\alpha$ of the amount of energy gain for each electron $\\Delta U(B) \\propto B^{\\alpha}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nIt turns out that binding an integer number of vortices $(n>1)$ with each electron generates a bigger surrounding whirlpool, hence pushes away all other electrons. Therefore, the system can considerably reduce its electrostatic\n\nCoulomb energy at the corresponding filling factor. Determine the scaling exponent $\\alpha$ of the amount of energy gain for each electron $\\Delta U(B) \\propto B^{\\alpha}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_273", "problem": "A thermistor is an electrical component. The resistance of a thermistor decreases as the temperature of the thermistor increases.\n\nWhich graph shows the resistance of a thermistor against current through the thermistor?\n\n[figure1]\n\nA\n\n[figure2]\n\nB\n\n[figure3]\n\nC\n\n[figure4]\n\nD\nA: A\nB: B\nC: C\nD: D\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA thermistor is an electrical component. The resistance of a thermistor decreases as the temperature of the thermistor increases.\n\nWhich graph shows the resistance of a thermistor against current through the thermistor?\n\n[figure1]\n\nA\n\n[figure2]\n\nB\n\n[figure3]\n\nC\n\n[figure4]\n\nD\n\nA: A\nB: B\nC: C\nD: D\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-04.jpg?height=383&width=354&top_left_y=568&top_left_x=246", "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-04.jpg?height=385&width=371&top_left_y=567&top_left_x=634", "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-04.jpg?height=374&width=371&top_left_y=567&top_left_x=1025", "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-04.jpg?height=385&width=374&top_left_y=567&top_left_x=1429" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1369", "problem": "为了提高风力发电的效率，我国目前正逐步采用变桨距（即调节风机叶片与风轮平面之间的夹角, 当风速小时使叶片的迎风面积增大，当风速超过一定限度时使叶片的迎风面积减小，以稳定其输出功率）控制风力发电机替代定桨距控制风力发电机。图 8a 所示中风力发电机每片叶片长度为 $54 \\mathrm{~m}$, 定浆距风机和变浆距风机的功率与风速的对应关系如图 8b 所示, 所处地域全天风速均为 $7.5 \\mathrm{~m} / \\mathrm{s}$, 空气密度为 $1.29 \\mathrm{~kg} / \\mathrm{m}^{3}$, 煤的燃烧值为 $2.9 \\times 10^{7} \\mathrm{~J} / \\mathrm{kg}$ 。每小时进入一台变桨距控制风力发电机的风的初始动能与完全燃烧 ___kg 煤所放出的热量相当, 变桨距控制风力发电机将风能转化成电能的效率为___\\%。\n\n[图1]\n\n图 8a", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n为了提高风力发电的效率，我国目前正逐步采用变桨距（即调节风机叶片与风轮平面之间的夹角, 当风速小时使叶片的迎风面积增大，当风速超过一定限度时使叶片的迎风面积减小，以稳定其输出功率）控制风力发电机替代定桨距控制风力发电机。图 8a 所示中风力发电机每片叶片长度为 $54 \\mathrm{~m}$, 定浆距风机和变浆距风机的功率与风速的对应关系如图 8b 所示, 所处地域全天风速均为 $7.5 \\mathrm{~m} / \\mathrm{s}$, 空气密度为 $1.29 \\mathrm{~kg} / \\mathrm{m}^{3}$, 煤的燃烧值为 $2.9 \\times 10^{7} \\mathrm{~J} / \\mathrm{kg}$ 。每小时进入一台变桨距控制风力发电机的风的初始动能与完全燃烧 ___kg 煤所放出的热量相当, 变桨距控制风力发电机将风能转化成电能的效率为___\\%。\n\n[图1]\n\n图 8a\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[每小时进入一台变桨距控制风力发电机的风的初始动能与完全燃烧多少煤所放出的热量相当, 变桨距控制风力发电机将风能转化成电能的效率]\n它们的单位依次是[kg, %]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-03.jpg?height=297&width=419&top_left_y=1893&top_left_x=1527" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "kg", "%" ], "answer_sequence": [ "每小时进入一台变桨距控制风力发电机的风的初始动能与完全燃烧多少煤所放出的热量相当", "变桨距控制风力发电机将风能转化成电能的效率" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_947", "problem": "A rod of mass $m_{1}$ is constrained to move vertically by a pair of guides, as shown in Fig. 4. The rod is in contact with a smooth wedge of mass $m_{2}$ and angle $\\theta$, which itself sits on a smooth horizontal surface. At time $t=0$ the rod is released and moves downwards, whilst the wedge accelerates to the right.\n\nFrom this, write down an expression for the speed of the rod, $u$, Using this and the ideas introduced earlier, write down an expression for the acceleration of the rod.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA rod of mass $m_{1}$ is constrained to move vertically by a pair of guides, as shown in Fig. 4. The rod is in contact with a smooth wedge of mass $m_{2}$ and angle $\\theta$, which itself sits on a smooth horizontal surface. At time $t=0$ the rod is released and moves downwards, whilst the wedge accelerates to the right.\n\nFrom this, write down an expression for the speed of the rod, $u$, Using this and the ideas introduced earlier, write down an expression for the acceleration of the rod.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1694", "problem": "在一条笔直的公路上依次设置三盛交通信号灯 $\\mathrm{L}_{1} 、 \\mathrm{~L}_{2}$ 和 $\\mathrm{L}_{3}, \\mathrm{~L}_{2}$ 与 $\\mathrm{L}_{1}$ 相距为 $80 \\mathrm{~m}$,\n\n$\\mathrm{L}_{3}$ 与 $\\mathrm{L}_{1}$ 相距为 $120 \\mathrm{~m}$. 每盛信号灯显示绿色的时间间隔都是 $20 \\mathrm{~s}$, 显示红色的时间间隔都是 $40 \\mathrm{~s}, \\mathrm{~L}_{1}$ 与 $\\mathrm{L}_{3}$ 同时显示绿色, $\\mathrm{L}_{2}$ 则在 $\\mathrm{L}_{1}$ 显示红色经历 $10 \\mathrm{~s}$ 时开始显示绿色. 规定车辆通过三搵信号灯经历的时间不得超过 $150 \\mathrm{~s}$. 若有一辆匀速向前行驶的汽车通过 $\\mathrm{L}_{1}$ 的时刻正好是 $\\mathrm{L}_{1}$ 刚开始显示绿色的时刻, 则此汽车能不停顿地通过三盛信号灯的最大速率是 $\\mathrm{m} / \\mathrm{s}$. 若一辆匀速向前行驶的自行车通过 $\\mathrm{L}_{1}$ 的时刻是 $\\mathrm{L}_{1}$ 显示绿色经历了 $10 \\mathrm{~s}$ 的时刻,则此自行车能不停顿地通过三盛信号灯的最小速率是 $\\mathrm{m} / \\mathrm{s}$.", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n\n问题:\n在一条笔直的公路上依次设置三盛交通信号灯 $\\mathrm{L}_{1} 、 \\mathrm{~L}_{2}$ 和 $\\mathrm{L}_{3}, \\mathrm{~L}_{2}$ 与 $\\mathrm{L}_{1}$ 相距为 $80 \\mathrm{~m}$,\n\n$\\mathrm{L}_{3}$ 与 $\\mathrm{L}_{1}$ 相距为 $120 \\mathrm{~m}$. 每盛信号灯显示绿色的时间间隔都是 $20 \\mathrm{~s}$, 显示红色的时间间隔都是 $40 \\mathrm{~s}, \\mathrm{~L}_{1}$ 与 $\\mathrm{L}_{3}$ 同时显示绿色, $\\mathrm{L}_{2}$ 则在 $\\mathrm{L}_{1}$ 显示红色经历 $10 \\mathrm{~s}$ 时开始显示绿色. 规定车辆通过三搵信号灯经历的时间不得超过 $150 \\mathrm{~s}$. 若有一辆匀速向前行驶的汽车通过 $\\mathrm{L}_{1}$ 的时刻正好是 $\\mathrm{L}_{1}$ 刚开始显示绿色的时刻, 则此汽车能不停顿地通过三盛信号灯的最大速率是 $\\mathrm{m} / \\mathrm{s}$. 若一辆匀速向前行驶的自行车通过 $\\mathrm{L}_{1}$ 的时刻是 $\\mathrm{L}_{1}$ 显示绿色经历了 $10 \\mathrm{~s}$ 的时刻,则此自行车能不停顿地通过三盛信号灯的最小速率是 $\\mathrm{m} / \\mathrm{s}$.\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{m} / \\mathrm{s}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{m} / \\mathrm{s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_859", "problem": "When modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nA lumped element model of the cable is constructed by considering the inductance and capacitance of short sections of the cable. The inductance is assumed to be a property of the inner core, and the capacitance links the core with the shielding. A diagram of the lumped element model is shown below.\n\n[figure2]\n\nCircuit diagram of lumped element model of coaxial cable.\nFind $b / a$ if the cable has impedance $Z_{0}=50 \\Omega$ and is made using a dielectric material with $\\varepsilon_{\\mathrm{r}}=4.0$ and $\\mu_{\\mathrm{r}}=1.0$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nWhen modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nA lumped element model of the cable is constructed by considering the inductance and capacitance of short sections of the cable. The inductance is assumed to be a property of the inner core, and the capacitance links the core with the shielding. A diagram of the lumped element model is shown below.\n\n[figure2]\n\nCircuit diagram of lumped element model of coaxial cable.\nFind $b / a$ if the cable has impedance $Z_{0}=50 \\Omega$ and is made using a dielectric material with $\\varepsilon_{\\mathrm{r}}=4.0$ and $\\mu_{\\mathrm{r}}=1.0$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-1.jpg?height=358&width=844&top_left_y=1688&top_left_x=617", "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-2.jpg?height=260&width=834&top_left_y=1041&top_left_x=608" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1580", "problem": "如图, $\\mathrm{MCN} 、 \\mathrm{PDQ}$ 为两条间距 $L=0.50 \\mathrm{~m}$ 、足够长的平行光滑导轨, MP 水平且与导轨垂直, 导轨与水平面的夹角 $\\theta=30^{\\circ}$, 导轨底端连接一电阻值 $R=2.0 \\Omega$ 的电阻; 导轨的 $\\mathrm{MC}$ 段与 $\\mathrm{PD}$ 段长度均为 $l=4.5 \\mathrm{~m}$, 电阻均为 $r=2.25 \\Omega$, 且电阻分布均匀, 导轨其余部分电阻不计; 一根质量 $m=0.20 \\mathrm{~kg}$ 、电阻可忽略的金属棒, 置于导轨底端,与导轨垂直并接触良好; 整个装置处于磁感应强度大小 $B=2.0 \\mathrm{~T}$ 、方向垂直导轨平面向上的匀强磁场中。现在对金属棒施加一沿导轨平面向上的拉力 $F$, 使棒从静止开始以大小为 $a=1.0 \\mathrm{~m} / \\mathrm{s}^{2}$ 的加速度沿导轨平面向上做匀加速直线运动。重力加速度大小 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。\n\n[图1]求在金属棒开始运动至其到达 $\\mathrm{CD}$ 位置的过程中拉力 $F$ 的最大值。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, $\\mathrm{MCN} 、 \\mathrm{PDQ}$ 为两条间距 $L=0.50 \\mathrm{~m}$ 、足够长的平行光滑导轨, MP 水平且与导轨垂直, 导轨与水平面的夹角 $\\theta=30^{\\circ}$, 导轨底端连接一电阻值 $R=2.0 \\Omega$ 的电阻; 导轨的 $\\mathrm{MC}$ 段与 $\\mathrm{PD}$ 段长度均为 $l=4.5 \\mathrm{~m}$, 电阻均为 $r=2.25 \\Omega$, 且电阻分布均匀, 导轨其余部分电阻不计; 一根质量 $m=0.20 \\mathrm{~kg}$ 、电阻可忽略的金属棒, 置于导轨底端,与导轨垂直并接触良好; 整个装置处于磁感应强度大小 $B=2.0 \\mathrm{~T}$ 、方向垂直导轨平面向上的匀强磁场中。现在对金属棒施加一沿导轨平面向上的拉力 $F$, 使棒从静止开始以大小为 $a=1.0 \\mathrm{~m} / \\mathrm{s}^{2}$ 的加速度沿导轨平面向上做匀加速直线运动。重力加速度大小 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}$ 。\n\n[图1]\n\n问题:\n求在金属棒开始运动至其到达 $\\mathrm{CD}$ 位置的过程中拉力 $F$ 的最大值。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~N}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-05.jpg?height=388&width=540&top_left_y=263&top_left_x=1449" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~N}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1185", "problem": "## To Commemorate the Centenary of Rutherford's Atomic Nucleus: the Scattering of an Ion by a Neutral Atom\n\n[figure1]\n\nAn ion of mass $m$, charge $Q$, is moving with an initial non-relativistic speed $v_{0}$ from a great distance towards the vicinity of a neutral atom of mass $M>>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nFind the expression for $r_{\\min }$, the distance of the closest approach, as shown in Figure 1.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\n## To Commemorate the Centenary of Rutherford's Atomic Nucleus: the Scattering of an Ion by a Neutral Atom\n\n[figure1]\n\nAn ion of mass $m$, charge $Q$, is moving with an initial non-relativistic speed $v_{0}$ from a great distance towards the vicinity of a neutral atom of mass $M>>m$ and of electrical polarisability $\\alpha$. The impact parameter is $b$ as shown in Figure 1.\n\nThe atom is instantaneously polarised by the electric field $\\vec{E}$ of the in-coming (approaching) ion. The resulting electric dipole moment of the atom is $\\vec{p}=\\alpha \\vec{E}$. Ignore any radiative losses in this problem.\n\nFind the expression for $r_{\\min }$, the distance of the closest approach, as shown in Figure 1.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_51c4dc0e7c52a1226310g-1.jpg?height=462&width=1495&top_left_y=701&top_left_x=315" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1333", "problem": "嫦娥 1 号奔月卫星与长征 3 号火箭分离后, 进入绕地运行的椭圆轨道, 近地点离地面高 $H_{n}=2.05 \\times 10^{2} \\mathrm{~km}$, 远地点离地面高 $H_{f}=5.0930 \\times 10^{4} \\mathrm{~km}$, 周期约为 16 小时, 称为 16 小时轨道 (如图中曲线 1 所示)。随后, 为了使卫星离地越来越远, 星载发动机先在远地点点火, 使卫星进入新轨道 (如图中曲线 2 所示), 以抬高近地点。后来又连续三次在抬高以后的近地点点火, 使卫星加速和变轨, 抬高远地点, 相继进入 24 小时轨道、 48 小时轨道和地月转移轨道 (分别如图中曲线 3、4、5 所示)。已知卫星质量 $m=2.350 \\times 10^{3} \\mathrm{~kg}$, 地球半径 $R=6.378 \\times 10^{3} \\mathrm{~km}$, 地面重力加速度 $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$, 月球半径 $r=1.738 \\times 10^{3} \\mathrm{~km}$ 。试根据题给数据计算卫星在 16 小时轨道的实际运行周期。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n嫦娥 1 号奔月卫星与长征 3 号火箭分离后, 进入绕地运行的椭圆轨道, 近地点离地面高 $H_{n}=2.05 \\times 10^{2} \\mathrm{~km}$, 远地点离地面高 $H_{f}=5.0930 \\times 10^{4} \\mathrm{~km}$, 周期约为 16 小时, 称为 16 小时轨道 (如图中曲线 1 所示)。随后, 为了使卫星离地越来越远, 星载发动机先在远地点点火, 使卫星进入新轨道 (如图中曲线 2 所示), 以抬高近地点。后来又连续三次在抬高以后的近地点点火, 使卫星加速和变轨, 抬高远地点, 相继进入 24 小时轨道、 48 小时轨道和地月转移轨道 (分别如图中曲线 3、4、5 所示)。已知卫星质量 $m=2.350 \\times 10^{3} \\mathrm{~kg}$, 地球半径 $R=6.378 \\times 10^{3} \\mathrm{~km}$, 地面重力加速度 $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$, 月球半径 $r=1.738 \\times 10^{3} \\mathrm{~km}$ 。\n\n问题:\n试根据题给数据计算卫星在 16 小时轨道的实际运行周期。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以分为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_40a5e1e69014b22d267bg-08.jpg?height=64&width=515&top_left_y=2441&top_left_x=679" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "分" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1417", "problem": "某种回旋加速器的设计方案如俯视图 a 所示, 图中粗黑线段为两个正对的极板，其间仅在带电粒子经过的过程中存在匀强电场, 两极板间电势差为 $U$ 。两个极板的板面中部各有一狭缝(沿 OP 方向的狭长区域），带电粒子可通过狭缝穿越极板（见图 b）；两细虚线间（除开两极板之间的区域）既无电场也无磁场；其它部分存在匀强磁场, 磁感应强度方向垂直于纸面。在离子源 $\\mathrm{S}$ 中产生的质量为 $\\mathrm{m}$ 、带电量为 $q(q>0)$ 的离子, 由静止开始被电场加速, 经狭缝中的 $\\mathrm{O}$ 点进入磁场区域, $\\mathrm{O}$ 点到极板右端的距离为 $D$, 到出射孔 $\\mathrm{P}$ 的距离为 $b D$ (常数 $b$ 为大于 2 的自然数）。已知磁感应强度大小在零到 $B_{\\text {max }}$ 之间可调, 离子从离子源上方的 $\\mathrm{O}$ 点射入磁场区域, 最终只能从出射孔 P 射出。假设如果离子打到器壁或离子源外壁便即被吸收。忽略相对论效应。求\n[图1]\n\n图a \n\n[图2]\n\n图 b可能的磁感应强度 $B$ 的最小值;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某种回旋加速器的设计方案如俯视图 a 所示, 图中粗黑线段为两个正对的极板，其间仅在带电粒子经过的过程中存在匀强电场, 两极板间电势差为 $U$ 。两个极板的板面中部各有一狭缝(沿 OP 方向的狭长区域），带电粒子可通过狭缝穿越极板（见图 b）；两细虚线间（除开两极板之间的区域）既无电场也无磁场；其它部分存在匀强磁场, 磁感应强度方向垂直于纸面。在离子源 $\\mathrm{S}$ 中产生的质量为 $\\mathrm{m}$ 、带电量为 $q(q>0)$ 的离子, 由静止开始被电场加速, 经狭缝中的 $\\mathrm{O}$ 点进入磁场区域, $\\mathrm{O}$ 点到极板右端的距离为 $D$, 到出射孔 $\\mathrm{P}$ 的距离为 $b D$ (常数 $b$ 为大于 2 的自然数）。已知磁感应强度大小在零到 $B_{\\text {max }}$ 之间可调, 离子从离子源上方的 $\\mathrm{O}$ 点射入磁场区域, 最终只能从出射孔 P 射出。假设如果离子打到器壁或离子源外壁便即被吸收。忽略相对论效应。求\n[图1]\n\n图a \n\n[图2]\n\n图 b\n\n问题:\n可能的磁感应强度 $B$ 的最小值;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-03.jpg?height=896&width=894&top_left_y=243&top_left_x=889", "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-03.jpg?height=331&width=371&top_left_y=1248&top_left_x=1345" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1537", "problem": "如图, 一劲度系数为 $k$ 的轻弹簧左端固定, 右端连一质量为 $m$ 的小球; 弹簧水平, 它处于自然状态时小球位于坐标原点 $O$; 小球可在水平地面上滑动, 它与地面之间的动摩擦因数为 $\\mu$ 。初始时小球速度为零, 将此时弹簧相对于其原长的伸长记为 $-A_{0}\\left(A_{0}>0\\right.$,但 $A_{0}$ 并不是已知量）。重力加速度大小为 $g$, 假设最大静摩擦力等于滑动摩擦力。\n\n[图1]如果小球至多只能向右运动, 求小球最终静止的位置, 和此种情形下 $A_{0}$ 应满足的条件;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 一劲度系数为 $k$ 的轻弹簧左端固定, 右端连一质量为 $m$ 的小球; 弹簧水平, 它处于自然状态时小球位于坐标原点 $O$; 小球可在水平地面上滑动, 它与地面之间的动摩擦因数为 $\\mu$ 。初始时小球速度为零, 将此时弹簧相对于其原长的伸长记为 $-A_{0}\\left(A_{0}>0\\right.$,但 $A_{0}$ 并不是已知量）。重力加速度大小为 $g$, 假设最大静摩擦力等于滑动摩擦力。\n\n[图1]\n\n问题:\n如果小球至多只能向右运动, 求小球最终静止的位置, 和此种情形下 $A_{0}$ 应满足的条件;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[小球最终静止的位置, 此种情形下 $A_{0}$ 应满足的条件]\n它们的答案类型依次是[方程, 方程]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-01.jpg?height=137&width=511&top_left_y=1362&top_left_x=1229" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "小球最终静止的位置", "此种情形下 $A_{0}$ 应满足的条件" ], "type_sequence": [ "EQ", "EQ" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_68", "problem": "Earth satellites such as the International Space Station orbit at altitudes that are mainly above Earth's atmosphere. A simple and accurate way to comprehend the orbit of these satellites is to view them as\nA: balanced between gravitational and centripetal forces.\nB: beyond the main pull of Earth's gravity.\nC: in mechanical equilibrium with a net force of zero.\nD: having sufficient tangential velocities to fall around rather than into Earth.\nE: All of the above.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nEarth satellites such as the International Space Station orbit at altitudes that are mainly above Earth's atmosphere. A simple and accurate way to comprehend the orbit of these satellites is to view them as\n\nA: balanced between gravitational and centripetal forces.\nB: beyond the main pull of Earth's gravity.\nC: in mechanical equilibrium with a net force of zero.\nD: having sufficient tangential velocities to fall around rather than into Earth.\nE: All of the above.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_595", "problem": "A student designs a simple integrated circuit device that has two inputs, $V_{a}$ and $V_{b}$, and two outputs, $V_{o}$ and $V_{g}$. The inputs are effectively connected internally to a single resistor with effectively infinite resistance. The outputs are effectively connected internally to a perfect source of $\\operatorname{emf} \\mathcal{E}$. The integrated circuit is configured so that $\\mathcal{E}=G\\left(V_{a}-V_{b}\\right)$, where $G$ is a very large number somewhere between $10^{7}$ and $10^{9}$. The circuits below are chosen so that the precise value of $G$ is unimportant. On the left is an internal schematic for the device; on the right is the symbol that is used in circuit diagrams.\n[figure1]\n\nThe key idea is that if $\\mathcal{E}$ is finite, then $V_{a} \\approx V_{b}$, since $G$ is so large. If we work exactly, then the answers will contain terms like $\\left(V_{b}-V_{a}\\right) / G$ which are negligible. Thus we can find the same answers by just setting $V_{a}=V_{b}$.\n\nConsider the following circuit. All four resistors have identical resistance $R$. Determine $V_{\\text {out }}$ in terms of any or all of $V_{1}, V_{2}$, and $R$.\n\n[figure2]", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA student designs a simple integrated circuit device that has two inputs, $V_{a}$ and $V_{b}$, and two outputs, $V_{o}$ and $V_{g}$. The inputs are effectively connected internally to a single resistor with effectively infinite resistance. The outputs are effectively connected internally to a perfect source of $\\operatorname{emf} \\mathcal{E}$. The integrated circuit is configured so that $\\mathcal{E}=G\\left(V_{a}-V_{b}\\right)$, where $G$ is a very large number somewhere between $10^{7}$ and $10^{9}$. The circuits below are chosen so that the precise value of $G$ is unimportant. On the left is an internal schematic for the device; on the right is the symbol that is used in circuit diagrams.\n[figure1]\n\nThe key idea is that if $\\mathcal{E}$ is finite, then $V_{a} \\approx V_{b}$, since $G$ is so large. If we work exactly, then the answers will contain terms like $\\left(V_{b}-V_{a}\\right) / G$ which are negligible. Thus we can find the same answers by just setting $V_{a}=V_{b}$.\n\nConsider the following circuit. All four resistors have identical resistance $R$. Determine $V_{\\text {out }}$ in terms of any or all of $V_{1}, V_{2}$, and $R$.\n\n[figure2]\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_33ea9fa74c8b34628eedg-05.jpg?height=158&width=676&top_left_y=666&top_left_x=754", "https://cdn.mathpix.com/cropped/2024_03_14_33ea9fa74c8b34628eedg-06.jpg?height=406&width=873&top_left_y=366&top_left_x=564" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_968", "problem": "A linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths $\\ell_{1}, \\ell_{2}, \\ell_{3}, \\ldots$ along the beam and with a fixed small separation $d$ between each tube. The tubes are connected to a high voltage, constant radio frequency $\\mathrm{AC}$ supply where the peak voltage of the $\\mathrm{AC}$ is $V_{0}$. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge $e$ and mass $m_{e}$ is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it \"drifts\" through the tube. It passes through a large potential difference between the tubes and, if the charged particle's motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10\n\n[figure1]\n\nFigure 9\n\n[figure2]\n\nFigure 10\n\nWhat is the maximum potential difference between adjacent tubes connected to the AC supply as shown?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths $\\ell_{1}, \\ell_{2}, \\ell_{3}, \\ldots$ along the beam and with a fixed small separation $d$ between each tube. The tubes are connected to a high voltage, constant radio frequency $\\mathrm{AC}$ supply where the peak voltage of the $\\mathrm{AC}$ is $V_{0}$. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge $e$ and mass $m_{e}$ is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it \"drifts\" through the tube. It passes through a large potential difference between the tubes and, if the charged particle's motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10\n\n[figure1]\n\nFigure 9\n\n[figure2]\n\nFigure 10\n\nWhat is the maximum potential difference between adjacent tubes connected to the AC supply as shown?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-10.jpg?height=300&width=454&top_left_y=818&top_left_x=296", "https://cdn.mathpix.com/cropped/2024_03_06_42e449efbde8c50a3134g-10.jpg?height=359&width=1128&top_left_y=757&top_left_x=795" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1373", "problem": "一种拉伸传感器的示意图如图 a 所示: 它由一半径为 $r_{2}$ 的圆柱形塑料棒和在上面紧密缠绕 $N(N>>1)$圈的一层细绳组成; 绳柔软绝缘, 半径为 $r_{1}$, 外表面均匀涂有厚度为 $t\\left(t<>1)$圈的一层细绳组成; 绳柔软绝缘, 半径为 $r_{1}$, 外表面均匀涂有厚度为 $t\\left(t<F_{b}$, 则 $\\mathrm{a}$ 对 $\\mathrm{b}$ 的作用力\n[图1]\nA: 必为推力\nB: 必为拉力\nC: 可能为推力, 也可能为拉力\nD: 可能为零\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图所示, 两块固连在一起的物块 $\\mathbf{a}$ 和 $\\mathbf{b}$, 质量分别为 $m_{\\mathrm{a}}$ 和 $m_{\\mathrm{b}}$, 放在水平的光滑桌面上. 现同时施给它们方向如图所示的推力 $F_{a}$ 和拉力 $F_{b}$, 已知 $F_{2}>F_{b}$, 则 $\\mathrm{a}$ 对 $\\mathrm{b}$ 的作用力\n[图1]\n\nA: 必为推力\nB: 必为拉力\nC: 可能为推力, 也可能为拉力\nD: 可能为零\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ff26c2613277b882b9edg-01.jpg?height=95&width=440&top_left_y=952&top_left_x=702" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_732", "problem": "If he directs the beam (from the previous question) to be $20^{\\circ}$ to the surface, how high above the water surface will the laser beam hit the vertical wall placed $2 \\mathrm{~m}$ away from the point A (the place where the laser beam hits the surface of the water)?\nA: About $15 \\mathrm{~m}$\nB: About $8 \\mathrm{~m}$\nC: About $2 \\mathrm{~m}$\nD: About $0.2 \\mathrm{~m}$\nE: It will never go above the water\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nIf he directs the beam (from the previous question) to be $20^{\\circ}$ to the surface, how high above the water surface will the laser beam hit the vertical wall placed $2 \\mathrm{~m}$ away from the point A (the place where the laser beam hits the surface of the water)?\n\nA: About $15 \\mathrm{~m}$\nB: About $8 \\mathrm{~m}$\nC: About $2 \\mathrm{~m}$\nD: About $0.2 \\mathrm{~m}$\nE: It will never go above the water\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1493", "problem": "如图所示, 一电容器由固定在共同导电底座上的 $N+1$ 片对顶双扇形薄金属板和固定在可旋转的导电对称轴上的 $N$片对顶双扇形薄金属板组成, 所有顶点共轴, 轴线与所有板面垂直, 两组板面各自在垂直于轴线的平面上的投影重合, 板面扇形半径均为 $R$, 圆心角均为 $\\theta_{0}$ $\\left(\\frac{\\pi}{2} \\leq \\theta_{0}<\\pi\\right)$; 固定金属板和可\n\n[图1]\n旋转的金属板相间排列, 两相邻金属板之间距离均为 $s$. 此电容器的电容 $C$ 值与可旋转金属板的转角 $\\theta$ 有关. 已知静电力常量为 $k$.当电容器电容接近最大时, 与电动势为 $E$ 的电源接通充电 (充电过程中保持可旋转金属板的转角不变), 稳定后断开电源, 求此时电容器极板所带电荷量和驱动可旋转金属板的力矩", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图所示, 一电容器由固定在共同导电底座上的 $N+1$ 片对顶双扇形薄金属板和固定在可旋转的导电对称轴上的 $N$片对顶双扇形薄金属板组成, 所有顶点共轴, 轴线与所有板面垂直, 两组板面各自在垂直于轴线的平面上的投影重合, 板面扇形半径均为 $R$, 圆心角均为 $\\theta_{0}$ $\\left(\\frac{\\pi}{2} \\leq \\theta_{0}<\\pi\\right)$; 固定金属板和可\n\n[图1]\n旋转的金属板相间排列, 两相邻金属板之间距离均为 $s$. 此电容器的电容 $C$ 值与可旋转金属板的转角 $\\theta$ 有关. 已知静电力常量为 $k$.\n\n问题:\n当电容器电容接近最大时, 与电动势为 $E$ 的电源接通充电 (充电过程中保持可旋转金属板的转角不变), 稳定后断开电源, 求此时电容器极板所带电荷量和驱动可旋转金属板的力矩\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ea729d4659bcaa2c4b91g-03.jpg?height=490&width=899&top_left_y=315&top_left_x=681" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_27", "problem": "A tiny ball with a mass of $0.6 \\mathrm{~g}$ carries a charge of magnitude $8 \\mu \\mathrm{C}$. It is suspended by a thread in a downward directed electric field of intensity $300 \\mathrm{~N} / \\mathrm{C}$. What is the tension in the thread if the charge on the ball is positive?\nA: $2.40 \\times 10^{-3} \\mathrm{~N}$\nB: $6.00 \\times 10^{-3} \\mathrm{~N}$\nC: $8.40 \\times 10^{-3} \\mathrm{~N}$\nD: $6.00 \\times 10^{-2} \\mathrm{~N}$\nE: $6.24 \\times 10^{-3} \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA tiny ball with a mass of $0.6 \\mathrm{~g}$ carries a charge of magnitude $8 \\mu \\mathrm{C}$. It is suspended by a thread in a downward directed electric field of intensity $300 \\mathrm{~N} / \\mathrm{C}$. What is the tension in the thread if the charge on the ball is positive?\n\nA: $2.40 \\times 10^{-3} \\mathrm{~N}$\nB: $6.00 \\times 10^{-3} \\mathrm{~N}$\nC: $8.40 \\times 10^{-3} \\mathrm{~N}$\nD: $6.00 \\times 10^{-2} \\mathrm{~N}$\nE: $6.24 \\times 10^{-3} \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1466", "problem": "如图, 在光滑水平桌面上有一长为 $L$ 的轻杆, 轻杆两固定一质量均为 $M$ 的小球 $\\mathrm{A}$ 和 B。开始时细杆静止; 有一质量为 $m$ 的 $\\mathrm{C}$ 以垂直于杆的速度 $v_{0}$ 运动, 与 $\\mathrm{A}$ 球碰撞。将小球和细杆视为一个\n\n[图1]若碰后系统动能恰好达到极小值, 求此时球 C 的速度和系统的动能。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 在光滑水平桌面上有一长为 $L$ 的轻杆, 轻杆两固定一质量均为 $M$ 的小球 $\\mathrm{A}$ 和 B。开始时细杆静止; 有一质量为 $m$ 的 $\\mathrm{C}$ 以垂直于杆的速度 $v_{0}$ 运动, 与 $\\mathrm{A}$ 球碰撞。将小球和细杆视为一个\n\n[图1]\n\n问题:\n若碰后系统动能恰好达到极小值, 求此时球 C 的速度和系统的动能。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[the speed of C in y axis, the speed of C in x axis, 系统的动能]\n它们的答案类型依次是[数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e3a9fdbbef225ad3aefbg-01.jpg?height=319&width=417&top_left_y=1785&top_left_x=1482" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "the speed of C in y axis", "the speed of C in x axis", "系统的动能" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_694", "problem": "A uniform wire of cross-sectional area $1 \\mathrm{~mm}^{2}$ has resistance $3 \\Omega$. A voltage difference of $6 \\mathrm{~V}$ is applied across its two ends. Given that the number of conduction electron per volume of the wire is $1.00 \\times 10^{28} \\mathrm{~m}^{-3}$, what is the average velocity of electron along the wire?\nA: $3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$\nB: $3.1 \\times 10^{-3} \\mathrm{~m} / \\mathrm{s}$\nC: $3.1 \\times 10^{-4} \\mathrm{~m} / \\mathrm{s}$\nD: $1.3 \\times 10^{-3} \\mathrm{~m} / \\mathrm{s}$\nE: $1.3 \\times 10^{-4} \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA uniform wire of cross-sectional area $1 \\mathrm{~mm}^{2}$ has resistance $3 \\Omega$. A voltage difference of $6 \\mathrm{~V}$ is applied across its two ends. Given that the number of conduction electron per volume of the wire is $1.00 \\times 10^{28} \\mathrm{~m}^{-3}$, what is the average velocity of electron along the wire?\n\nA: $3 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$\nB: $3.1 \\times 10^{-3} \\mathrm{~m} / \\mathrm{s}$\nC: $3.1 \\times 10^{-4} \\mathrm{~m} / \\mathrm{s}$\nD: $1.3 \\times 10^{-3} \\mathrm{~m} / \\mathrm{s}$\nE: $1.3 \\times 10^{-4} \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_402", "problem": "The black box has three terminal wires: \"blue\", \"black\" and \"white\", and contains n a star configuration: a battery, a capacitor, an inductor in series with a diode. You may consider the diode to be \"ideal\" - it conducts current perfectly one way and not at all the other way. You may neglect internal resistance of the battery and capacitor, but the inductor has considerable internal resistance. The multimeter's internal resistance when measuring voltages is $R_{m}=10 \\mathrm{M} \\Omega$ and it displays a new reading every $t=0.4 \\mathrm{~s}$.\n\nDetermine the internal resistance of he inductor.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nThe black box has three terminal wires: \"blue\", \"black\" and \"white\", and contains n a star configuration: a battery, a capacitor, an inductor in series with a diode. You may consider the diode to be \"ideal\" - it conducts current perfectly one way and not at all the other way. You may neglect internal resistance of the battery and capacitor, but the inductor has considerable internal resistance. The multimeter's internal resistance when measuring voltages is $R_{m}=10 \\mathrm{M} \\Omega$ and it displays a new reading every $t=0.4 \\mathrm{~s}$.\n\nDetermine the internal resistance of he inductor.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $ \\mathrm{ohm}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$ \\mathrm{ohm}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1689", "problem": "如气体压强-体积 $(P-V)$ 图所示, 摩尔数为 $v$ 的双原子理想气体构成的系统经历一正循环过程（正循环指沿图中箭头所示的循环), 其中自 A 到 B 为直线过程, 自 B 到 $A$ 为等温过程。双原子理想气体的定容摩尔热容量为 $\\frac{5}{2} R, R$ 为气体常量。\n\n[图1]求整个直线 $A B$ 过程中所吸收的净热量和一个正循环过程中气体对外所作的净功。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如气体压强-体积 $(P-V)$ 图所示, 摩尔数为 $v$ 的双原子理想气体构成的系统经历一正循环过程（正循环指沿图中箭头所示的循环), 其中自 A 到 B 为直线过程, 自 B 到 $A$ 为等温过程。双原子理想气体的定容摩尔热容量为 $\\frac{5}{2} R, R$ 为气体常量。\n\n[图1]\n\n问题:\n求整个直线 $A B$ 过程中所吸收的净热量和一个正循环过程中气体对外所作的净功。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_2a6edd00c283cc2a8114g-04.jpg?height=422&width=494&top_left_y=266&top_left_x=1249" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_254", "problem": "The characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nStart with a case, where a point small charged object with a charge $+Q$ is fastened to the table. The center of the dipole is fixed at the distance $L$ from the charged object (see Figure 1). The dipole consists of two identical small balls fastened to the tiny, rigid rod with a length $d, d \\ll L$, so that the moment of inertia can be ignored. Each of the balls has a mass $m$ and have charge $+q$ and $-q$. The dipole can rotate around its center in a plane parallel to the surface of the smooth table.\n\n[figure1]\n\nFigure 1: Schematic representation of the system used in section 1.1\n\nWhat time $t_{1}$ is required to reduce the distance between the dipole and the charged body to half of the original distance?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nThe characterization of point object motion, when both radial and tangential forces are applied, is usually rather complicated, and requires advanced mathematical tools. However, some systems such as a motion of a point charge near an electric dipole, which have a very specific electrostatic field, provide interesting results, even for the case when an angular momentum is not conserved.\n\nAssume that the relativistic and the electromagnetic radiation effects can be neglected, unless otherwise stated.\n\nStart with a case, where a point small charged object with a charge $+Q$ is fastened to the table. The center of the dipole is fixed at the distance $L$ from the charged object (see Figure 1). The dipole consists of two identical small balls fastened to the tiny, rigid rod with a length $d, d \\ll L$, so that the moment of inertia can be ignored. Each of the balls has a mass $m$ and have charge $+q$ and $-q$. The dipole can rotate around its center in a plane parallel to the surface of the smooth table.\n\n[figure1]\n\nFigure 1: Schematic representation of the system used in section 1.1\n\nWhat time $t_{1}$ is required to reduce the distance between the dipole and the charged body to half of the original distance?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ed4e92416bdbac30298dg-1.jpg?height=179&width=1171&top_left_y=1187&top_left_x=477" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_387", "problem": "Tools At least three pairs of large color less medical semi-transparent latex rubber gloves; a roll of transparent and strong office tape; a pair of sharp scissors; at least four sheets of A4 or larger size graphing paper three rulers; a flexible measuring tape with a length of at least one meter; extra-fine poin universal surface marker. Rubber gloves can be cut as needed into pieces. The pieces of gloves can be fixed to your working table either directly using the tape and/or with the help of a ruler (to achieve a firmer fixing).\n\nLatex is a highly stretchable elastic ma terial for which it can be assumed that its volume remains constant during stretching up to the breaking point.\n\nFor each of the tasks, sketch your experimental setup and explain the steps you made to obtain the best possible precision, and tab ulate the directly measured data\n\nDetermine the maximal strain $\\varepsilon_{m}$ of the latex film band (i.e. the strain by which the band breaks). Strain is defined as the rel ative change in length, $\\varepsilon=\\left(l-l_{0}\\right) / l_{0}$, where $l$ and $l_{0}$ are the stretched and unstretched lengths of the band.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nTools At least three pairs of large color less medical semi-transparent latex rubber gloves; a roll of transparent and strong office tape; a pair of sharp scissors; at least four sheets of A4 or larger size graphing paper three rulers; a flexible measuring tape with a length of at least one meter; extra-fine poin universal surface marker. Rubber gloves can be cut as needed into pieces. The pieces of gloves can be fixed to your working table either directly using the tape and/or with the help of a ruler (to achieve a firmer fixing).\n\nLatex is a highly stretchable elastic ma terial for which it can be assumed that its volume remains constant during stretching up to the breaking point.\n\nFor each of the tasks, sketch your experimental setup and explain the steps you made to obtain the best possible precision, and tab ulate the directly measured data\n\nDetermine the maximal strain $\\varepsilon_{m}$ of the latex film band (i.e. the strain by which the band breaks). Strain is defined as the rel ative change in length, $\\varepsilon=\\left(l-l_{0}\\right) / l_{0}$, where $l$ and $l_{0}$ are the stretched and unstretched lengths of the band.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1738", "problem": "图 a 和图 b 分别表示某理想气体体系经历的两个循环过程;前者由甲过程 (实线) 和绝热过程 (虚线) 组成, 后者由乙过程 (实线) 和等温过程 (虚线) 组成。下列说法正确的是\n[图1]\n\n图a\n\n[图2]\n\n图 b\nA: 甲、乙两过程均放热\nB: 甲、乙两过程均吸热\nC: 甲过程放热, 乙过程吸热\nD: 甲过程吸热, 乙过程放热\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n图 a 和图 b 分别表示某理想气体体系经历的两个循环过程;前者由甲过程 (实线) 和绝热过程 (虚线) 组成, 后者由乙过程 (实线) 和等温过程 (虚线) 组成。下列说法正确的是\n[图1]\n\n图a\n\n[图2]\n\n图 b\n\nA: 甲、乙两过程均放热\nB: 甲、乙两过程均吸热\nC: 甲过程放热, 乙过程吸热\nD: 甲过程吸热, 乙过程放热\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-02.jpg?height=251&width=277&top_left_y=1188&top_left_x=1363", "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-02.jpg?height=245&width=285&top_left_y=1188&top_left_x=1702" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_738", "problem": "A current of $12 \\mathrm{~mA}$ passes through a red LED diode with a forward voltage of $3 \\mathrm{~V}$. Which of the following is a good estimate for the number of photons per second this LED produces?\nA: $1 \\times 10^{12}$\nB: $3 \\times 10^{14}$\nC: $1 \\times 10^{17}$\nD: $3 \\times 10^{19}$\nE: $1 \\times 10^{22}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA current of $12 \\mathrm{~mA}$ passes through a red LED diode with a forward voltage of $3 \\mathrm{~V}$. Which of the following is a good estimate for the number of photons per second this LED produces?\n\nA: $1 \\times 10^{12}$\nB: $3 \\times 10^{14}$\nC: $1 \\times 10^{17}$\nD: $3 \\times 10^{19}$\nE: $1 \\times 10^{22}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_242", "problem": "Induction (or asynchronous) motors are the simplest and most reliable electric motors. They are powered by alternating current, and they do not contain commutators, slip rings or brushes. They consist of a stator and a rotor (see fig. 1). The stator is a fixed set of coils, which produces a rotating magnetic field in the plane perpendicular to the axis of the motor. The rotor is just a cage, i.e., a set of closed metallic loops attached to the axis of the motor. The rotating magnetic field produced by the stator induces electric current in the loops of the cage, which behave as magnetic dipoles, and interact with the external field of the stator. As a result, a torque is exerted on the rotor, and it starts rotating.\n\n[figure1]\n\nFigure 1: The structure of an induction motor\n\nIn a simplified model (see fig. 2) we assume that the magnetic induction vector $\\mathbf{B}$ produced by the stator is rotating in the $x-y$ plane at a constant angular velocity $\\Omega$, and it has a constant magnitude $\\mathbf{B}$. The axis of the rotor is in the $z$ direction. The rotor is assumed to be a flat coil of area $A$, winding number $N$, Ohmic resistance $R$ and self inductance $L$. The vector $\\mathbf{n}$ perpendicular to this coil is rotating also in the $x-y$ plane.\n\n[figure2]\n\nFigure 2: The simplified model, seen from the $z$ axis\n\nFirst we study the stationary operation of the motor, when its angular velocity $\\omega$ and torque $T$ are constant. As we shall see, when the motor is loaded, its angular velocity $\\omega$ gets smaller than $\\Omega$. It is convenient to characterize this shift by the slip\n\n$$\ns=\\frac{\\Omega-\\omega}{\\Omega}\n$$\n\nwhich is a dimensionless number between 0 and 1 .\n\nFind the maximal stationary torque $T_{\\max }$ and the corresponding slip $s_{0}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nInduction (or asynchronous) motors are the simplest and most reliable electric motors. They are powered by alternating current, and they do not contain commutators, slip rings or brushes. They consist of a stator and a rotor (see fig. 1). The stator is a fixed set of coils, which produces a rotating magnetic field in the plane perpendicular to the axis of the motor. The rotor is just a cage, i.e., a set of closed metallic loops attached to the axis of the motor. The rotating magnetic field produced by the stator induces electric current in the loops of the cage, which behave as magnetic dipoles, and interact with the external field of the stator. As a result, a torque is exerted on the rotor, and it starts rotating.\n\n[figure1]\n\nFigure 1: The structure of an induction motor\n\nIn a simplified model (see fig. 2) we assume that the magnetic induction vector $\\mathbf{B}$ produced by the stator is rotating in the $x-y$ plane at a constant angular velocity $\\Omega$, and it has a constant magnitude $\\mathbf{B}$. The axis of the rotor is in the $z$ direction. The rotor is assumed to be a flat coil of area $A$, winding number $N$, Ohmic resistance $R$ and self inductance $L$. The vector $\\mathbf{n}$ perpendicular to this coil is rotating also in the $x-y$ plane.\n\n[figure2]\n\nFigure 2: The simplified model, seen from the $z$ axis\n\nFirst we study the stationary operation of the motor, when its angular velocity $\\omega$ and torque $T$ are constant. As we shall see, when the motor is loaded, its angular velocity $\\omega$ gets smaller than $\\Omega$. It is convenient to characterize this shift by the slip\n\n$$\ns=\\frac{\\Omega-\\omega}{\\Omega}\n$$\n\nwhich is a dimensionless number between 0 and 1 .\n\nFind the maximal stationary torque $T_{\\max }$ and the corresponding slip $s_{0}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [$s_0$, $T\\_max$].\nTheir answer types are, in order, [expression, expression].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_84b5561c4961da250149g-1.jpg?height=417&width=849&top_left_y=892&top_left_x=627", "https://cdn.mathpix.com/cropped/2024_03_06_84b5561c4961da250149g-1.jpg?height=441&width=570&top_left_y=1796&top_left_x=772" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "$s_0$", "$T\\_max$" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1482", "problem": "如图所示, 有一辆左右对称的光滑小车, 质量为 $M$, 放在光滑立平面上, 不考虑轮子质量, 重力加速度为 $g$, 将一个质点 $m=\\sqrt{2} M$ 的小球如图放置, 初态质点和小车都静止, 然后自由释放,\n小球下降 $r$ 之后进入半径为 $r$ 的圆弧, 经过圆心角为 $\\theta=\\frac{3 \\pi}{4}$ 后腾空一段距离 $l$ 后恰好对小车沿切线进入右侧圆弧, 最终上升到右侧与初态相同高度点。\n\n[图1]求出质点刚开始腾空时, 小车的速度大小;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图所示, 有一辆左右对称的光滑小车, 质量为 $M$, 放在光滑立平面上, 不考虑轮子质量, 重力加速度为 $g$, 将一个质点 $m=\\sqrt{2} M$ 的小球如图放置, 初态质点和小车都静止, 然后自由释放,\n小球下降 $r$ 之后进入半径为 $r$ 的圆弧, 经过圆心角为 $\\theta=\\frac{3 \\pi}{4}$ 后腾空一段距离 $l$ 后恰好对小车沿切线进入右侧圆弧, 最终上升到右侧与初态相同高度点。\n\n[图1]\n\n问题:\n求出质点刚开始腾空时, 小车的速度大小;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_176a47d8c6a915efafceg-07.jpg?height=363&width=697&top_left_y=401&top_left_x=722" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_689", "problem": "A passenger of a car is holding a cylindrical cup of tea (shown below). The passenger keeps the cup held upright. Neglecting the vibration of the car, what is the maximum acceleration that the car can go without spilling any tea?\n\n[figure1]\nA: $\\frac{g}{3}$\nB: $\\frac{g}{1.5}$\nC: $1.5 \\mathrm{~g}$\nD: $g$\nE: $3 g$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA passenger of a car is holding a cylindrical cup of tea (shown below). The passenger keeps the cup held upright. Neglecting the vibration of the car, what is the maximum acceleration that the car can go without spilling any tea?\n\n[figure1]\n\nA: $\\frac{g}{3}$\nB: $\\frac{g}{1.5}$\nC: $1.5 \\mathrm{~g}$\nD: $g$\nE: $3 g$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-05.jpg?height=260&width=326&top_left_y=1932&top_left_x=1322" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_278", "problem": "A student measures the density of water by determining the mass and volume of several different quantities of water.\n\nThe student plots the measured values of mass and volume on a graph.\n\nHow should the student determine the density of water from the graph?\n\n| | Quantity on the y-axis | Quantity on the x-axis | Determine the density from
the: |\n| :--- | :--- | :--- | :--- |\n| A. | volume | mass | gradient |\n| B. | volume | mass | area |\n| C. | mass | volume | gradient |\n| D. | mass | volume | area |\nA: A\nB: B\nC: C\nD: D\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (more than one correct answer).\n\nproblem:\nA student measures the density of water by determining the mass and volume of several different quantities of water.\n\nThe student plots the measured values of mass and volume on a graph.\n\nHow should the student determine the density of water from the graph?\n\n| | Quantity on the y-axis | Quantity on the x-axis | Determine the density from
the: |\n| :--- | :--- | :--- | :--- |\n| A. | volume | mass | gradient |\n| B. | volume | mass | area |\n| C. | mass | volume | gradient |\n| D. | mass | volume | area |\n\nA: A\nB: B\nC: C\nD: D\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be two or more of the options: [A, B, C, D].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1653", "problem": "一电路包含内阻为 $R_{E}$ 、电动势为 $E$ 的直流电源和 $N$ 个阻值均为 $R$ 的相同电阻, 有 $N+1$ 个半径为 $r$ 的相同导体球通过细长导线与电路连接起来. 为消除导体球之间的互相影响, 每个导体球的外边都用内半径为 $r_{0}(>r)$ 的同心接地导体薄球壳包围起来, 球壳上有小缺口容许细长导线进入但与其绝缘, 如图所示. 把导体球按照从左向右的顺序依\n次编号为 1 到 $N+1$. 所有导体球起初不带电, 开关闭合并达到稳定状态后, 导体球上所带的总电量为 $Q$. 问导体球的半径是多少？已知静电力常量为 $k$.\n\n[图1]", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n\n问题:\n一电路包含内阻为 $R_{E}$ 、电动势为 $E$ 的直流电源和 $N$ 个阻值均为 $R$ 的相同电阻, 有 $N+1$ 个半径为 $r$ 的相同导体球通过细长导线与电路连接起来. 为消除导体球之间的互相影响, 每个导体球的外边都用内半径为 $r_{0}(>r)$ 的同心接地导体薄球壳包围起来, 球壳上有小缺口容许细长导线进入但与其绝缘, 如图所示. 把导体球按照从左向右的顺序依\n次编号为 1 到 $N+1$. 所有导体球起初不带电, 开关闭合并达到稳定状态后, 导体球上所带的总电量为 $Q$. 问导体球的半径是多少？已知静电力常量为 $k$.\n\n[图1]\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e680ccd11e7de3ee63f1g-04.jpg?height=462&width=988&top_left_y=403&top_left_x=543" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_691", "problem": "A video camera that takes 15 frames per second is filming the rotation of a bicycle wheel. If the bicycle wheel is rotating at 2 rotations per second and has 10 spokes, at what speed would it seem to be rotating in the movie?\n\n[figure1]\nA: 0.25 rotations per sec in the same direction as the actual rotation.\nB: 0.25 rotations per sec in the opposite direction as the actual rotation.\nC: 0.33 rotations per sec in the same direction as the actual rotation.\nD: 0.5 rotations per sec in the same direction as the actual rotation.\nE: 0.5 rotations per sec in the opposite direction as the actual rotation.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA video camera that takes 15 frames per second is filming the rotation of a bicycle wheel. If the bicycle wheel is rotating at 2 rotations per second and has 10 spokes, at what speed would it seem to be rotating in the movie?\n\n[figure1]\n\nA: 0.25 rotations per sec in the same direction as the actual rotation.\nB: 0.25 rotations per sec in the opposite direction as the actual rotation.\nC: 0.33 rotations per sec in the same direction as the actual rotation.\nD: 0.5 rotations per sec in the same direction as the actual rotation.\nE: 0.5 rotations per sec in the opposite direction as the actual rotation.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_ad9879f18be53dac77a6g-06.jpg?height=340&width=348&top_left_y=1001&top_left_x=1302" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1577", "problem": "如图, 一半径为 $R$ 的固定的光滑绝缘圆环, 位于坚直平面内; 环上有两个相同的带电小球 $\\mathrm{a}$ 和 $\\mathrm{b}$ (可视为质点), 只能在环上移动,静止时两小球之间的距离为 $R$ 。现用外力缓慢推左球 $\\mathrm{a}$ 使其到达圆环最低点 $\\mathrm{C}$, 然后撤除外力。下列说法正确的是\n\n[图1]\nA: 在左球 $\\mathrm{a}$ 到达 $\\mathrm{c}$ 点的过程中, 圆环对 $\\mathrm{b}$ 球的支持力变大\nB: 在左球 $\\mathrm{a}$ 到达 $\\mathrm{c}$ 点的过程中, 外力做正功, 电势能增加\nC: 在左球 $\\mathrm{a}$ 到达 $\\mathrm{c}$ 点的过程中, $\\mathrm{a} 、 \\mathrm{~b}$ 两球的重力势能之和不变\nD: 撤除外力后, $\\mathrm{a} 、 \\mathrm{~b}$ 两球在轨道上运动过程中系统的能量守恒\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图, 一半径为 $R$ 的固定的光滑绝缘圆环, 位于坚直平面内; 环上有两个相同的带电小球 $\\mathrm{a}$ 和 $\\mathrm{b}$ (可视为质点), 只能在环上移动,静止时两小球之间的距离为 $R$ 。现用外力缓慢推左球 $\\mathrm{a}$ 使其到达圆环最低点 $\\mathrm{C}$, 然后撤除外力。下列说法正确的是\n\n[图1]\n\nA: 在左球 $\\mathrm{a}$ 到达 $\\mathrm{c}$ 点的过程中, 圆环对 $\\mathrm{b}$ 球的支持力变大\nB: 在左球 $\\mathrm{a}$ 到达 $\\mathrm{c}$ 点的过程中, 外力做正功, 电势能增加\nC: 在左球 $\\mathrm{a}$ 到达 $\\mathrm{c}$ 点的过程中, $\\mathrm{a} 、 \\mathrm{~b}$ 两球的重力势能之和不变\nD: 撤除外力后, $\\mathrm{a} 、 \\mathrm{~b}$ 两球在轨道上运动过程中系统的能量守恒\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_7973e92ea1a5d86ba5a6g-01.jpg?height=392&width=391&top_left_y=2117&top_left_x=1415" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_625", "problem": "A newly discovered subatomic particle, the $S$ meson, has a mass $M$. When at rest, it lives for exactly $\\tau=3 \\times 10^{-8}$ seconds before decaying into two identical particles called $P$ mesons (peons?) that each have a mass of $\\alpha M$.\n\nIn a reference frame where the $\\mathrm{S}$ meson travels 9 meters between creation and decay, determine the kinetic energy of the $\\mathrm{S}$ meson. Write the answers in terms of $M$, the speed of light $c$, and any numerical constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA newly discovered subatomic particle, the $S$ meson, has a mass $M$. When at rest, it lives for exactly $\\tau=3 \\times 10^{-8}$ seconds before decaying into two identical particles called $P$ mesons (peons?) that each have a mass of $\\alpha M$.\n\nIn a reference frame where the $\\mathrm{S}$ meson travels 9 meters between creation and decay, determine the kinetic energy of the $\\mathrm{S}$ meson. Write the answers in terms of $M$, the speed of light $c$, and any numerical constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_356", "problem": "A submarine of unknown nationality is travelling near the bottom of the Baltic sea, at the depth of $h=300 \\mathrm{~m}$. Its interior is one big room of volume $V=10 \\mathrm{~m}^{3}$ filled with air $(M=29 \\mathrm{~g} / \\mathrm{mol})$ at pressure $p_{0}=100 \\mathrm{kPa}$ and temperature $t_{0}=20^{\\circ} \\mathrm{C}$. Suddenly it hits a rock and a large hole of area $A=20 \\mathrm{~cm}^{2}$ is formed at the bottom of the submarine. As a result, the submarine sinks to the bottom and most of it is filled fast with water, leaving a bubble of air at increased pressure (no air escapes the submarine). The density of water $\\rho=1000 \\mathrm{~kg} / \\mathrm{m}^{3}$ and free fall acceleration $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$. Molar heat capacitance of air by constant volume $c_{V}=\\frac{5}{2} R$, where $R=8.31 \\mathrm{~J} / \\mathrm{Kmol}$ is the gas constant.\n\nThe flow rate is so large that the submarine is filled with water so fast that the heat exchange between the gas and the water can be neglected (this applies also to the next question).\n\nWhat is the volume of the air bubble once water watch. flow has stopped?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA submarine of unknown nationality is travelling near the bottom of the Baltic sea, at the depth of $h=300 \\mathrm{~m}$. Its interior is one big room of volume $V=10 \\mathrm{~m}^{3}$ filled with air $(M=29 \\mathrm{~g} / \\mathrm{mol})$ at pressure $p_{0}=100 \\mathrm{kPa}$ and temperature $t_{0}=20^{\\circ} \\mathrm{C}$. Suddenly it hits a rock and a large hole of area $A=20 \\mathrm{~cm}^{2}$ is formed at the bottom of the submarine. As a result, the submarine sinks to the bottom and most of it is filled fast with water, leaving a bubble of air at increased pressure (no air escapes the submarine). The density of water $\\rho=1000 \\mathrm{~kg} / \\mathrm{m}^{3}$ and free fall acceleration $g=9.81 \\mathrm{~m} / \\mathrm{s}^{2}$. Molar heat capacitance of air by constant volume $c_{V}=\\frac{5}{2} R$, where $R=8.31 \\mathrm{~J} / \\mathrm{Kmol}$ is the gas constant.\n\nThe flow rate is so large that the submarine is filled with water so fast that the heat exchange between the gas and the water can be neglected (this applies also to the next question).\n\nWhat is the volume of the air bubble once water watch. flow has stopped?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $m^3$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$m^3$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_228", "problem": "Induction (or asynchronous) motors are the simplest and most reliable electric motors. They are powered by alternating current, and they do not contain commutators, slip rings or brushes. They consist of a stator and a rotor (see fig. 1). The stator is a fixed set of coils, which produces a rotating magnetic field in the plane perpendicular to the axis of the motor. The rotor is just a cage, i.e., a set of closed metallic loops attached to the axis of the motor. The rotating magnetic field produced by the stator induces electric current in the loops of the cage, which behave as magnetic dipoles, and interact with the external field of the stator. As a result, a torque is exerted on the rotor, and it starts rotating.\n\n[figure1]\n\nFigure 1: The structure of an induction motor\n\nIn a simplified model (see fig. 2) we assume that the magnetic induction vector $\\mathbf{B}$ produced by the stator is rotating in the $x-y$ plane at a constant angular velocity $\\Omega$, and it has a constant magnitude $\\mathbf{B}$. The axis of the rotor is in the $z$ direction. The rotor is assumed to be a flat coil of area $A$, winding number $N$, Ohmic resistance $R$ and self inductance $L$. The vector $\\mathbf{n}$ perpendicular to this coil is rotating also in the $x-y$ plane.\n\n[figure2]\n\nFigure 2: The simplified model, seen from the $z$ axis\n\nFirst we study the stationary operation of the motor, when its angular velocity $\\omega$ and torque $T$ are constant. As we shall see, when the motor is loaded, its angular velocity $\\omega$ gets smaller than $\\Omega$. It is convenient to characterize this shift by the slip\n\n$$\ns=\\frac{\\Omega-\\omega}{\\Omega}\n$$\n\nwhich is a dimensionless number between 0 and 1 .\n\n\nb. Determine the average torque $T$ as a function of an arbitrary slip value $s$ between 0 and 1.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nInduction (or asynchronous) motors are the simplest and most reliable electric motors. They are powered by alternating current, and they do not contain commutators, slip rings or brushes. They consist of a stator and a rotor (see fig. 1). The stator is a fixed set of coils, which produces a rotating magnetic field in the plane perpendicular to the axis of the motor. The rotor is just a cage, i.e., a set of closed metallic loops attached to the axis of the motor. The rotating magnetic field produced by the stator induces electric current in the loops of the cage, which behave as magnetic dipoles, and interact with the external field of the stator. As a result, a torque is exerted on the rotor, and it starts rotating.\n\n[figure1]\n\nFigure 1: The structure of an induction motor\n\nIn a simplified model (see fig. 2) we assume that the magnetic induction vector $\\mathbf{B}$ produced by the stator is rotating in the $x-y$ plane at a constant angular velocity $\\Omega$, and it has a constant magnitude $\\mathbf{B}$. The axis of the rotor is in the $z$ direction. The rotor is assumed to be a flat coil of area $A$, winding number $N$, Ohmic resistance $R$ and self inductance $L$. The vector $\\mathbf{n}$ perpendicular to this coil is rotating also in the $x-y$ plane.\n\n[figure2]\n\nFigure 2: The simplified model, seen from the $z$ axis\n\nFirst we study the stationary operation of the motor, when its angular velocity $\\omega$ and torque $T$ are constant. As we shall see, when the motor is loaded, its angular velocity $\\omega$ gets smaller than $\\Omega$. It is convenient to characterize this shift by the slip\n\n$$\ns=\\frac{\\Omega-\\omega}{\\Omega}\n$$\n\nwhich is a dimensionless number between 0 and 1 .\n\n\nb. Determine the average torque $T$ as a function of an arbitrary slip value $s$ between 0 and 1.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_84b5561c4961da250149g-1.jpg?height=417&width=849&top_left_y=892&top_left_x=627", "https://cdn.mathpix.com/cropped/2024_03_06_84b5561c4961da250149g-1.jpg?height=441&width=570&top_left_y=1796&top_left_x=772" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_926", "problem": "If the medium is isotropic, we have $\\vec{P}=\\chi \\epsilon_{0} \\vec{E}$ and $\\vec{D}=\\epsilon \\vec{E}$, with $\\chi$ and $\\epsilon=\\epsilon_{0}(1+\\chi)$ being the electric susceptibility and permittivity, respectively, of the medium. For a light wave of angular frequency $\\omega$ in such a medium, a given phase will propagate in the direction $\\vec{k}$ with a velocity (called phase velocity) $v_{p}=c / n$. Here $c$ is the speed of light in vacuum and $n$ is the refractive index of the medium. One can also use rays to represent a train of light waves. The propagation of a light ray is characterized by the direction and speed $v_{r}$ of the electromagnetic energy flow.\n\nConsider a plane wave of light with angular frequency $\\omega$ and wave vector $\\vec{k}$ in a homogeneous isotropic dielectric medium.\n\nExpress its phase velocity $v_{p}$ in terms of $\\epsilon$ and $\\mu_{0}$. $0.4 \\mathrm{pt}$", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIf the medium is isotropic, we have $\\vec{P}=\\chi \\epsilon_{0} \\vec{E}$ and $\\vec{D}=\\epsilon \\vec{E}$, with $\\chi$ and $\\epsilon=\\epsilon_{0}(1+\\chi)$ being the electric susceptibility and permittivity, respectively, of the medium. For a light wave of angular frequency $\\omega$ in such a medium, a given phase will propagate in the direction $\\vec{k}$ with a velocity (called phase velocity) $v_{p}=c / n$. Here $c$ is the speed of light in vacuum and $n$ is the refractive index of the medium. One can also use rays to represent a train of light waves. The propagation of a light ray is characterized by the direction and speed $v_{r}$ of the electromagnetic energy flow.\n\nConsider a plane wave of light with angular frequency $\\omega$ and wave vector $\\vec{k}$ in a homogeneous isotropic dielectric medium.\n\nExpress its phase velocity $v_{p}$ in terms of $\\epsilon$ and $\\mu_{0}$. $0.4 \\mathrm{pt}$\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1026", "problem": "# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nAfter the bubble is formed for a while, it will be in thermal equilibrium with the surrounding. This bubble in still air will naturally fall towards the ground.\n\nThe above calculations suggest that the terms involving the surface tension $\\gamma$ add very little to the accuracy of the result. In all of the questions below, you can neglect the surface tension terms.\n\nWhat must be the magnitude of this charge $q$ in terms of $t, \\rho_{a}, \\rho_{s}, \\varepsilon_{0}, R_{0}, P_{a}$ in order that the bubble will float motionlessly in still air? Calculate also the numerical value of $q$. The permittivity of free space $\\varepsilon_{0}=8.85 \\times 10^{-12} \\mathrm{farad} / \\mathrm{m}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\n# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nAfter the bubble is formed for a while, it will be in thermal equilibrium with the surrounding. This bubble in still air will naturally fall towards the ground.\n\nThe above calculations suggest that the terms involving the surface tension $\\gamma$ add very little to the accuracy of the result. In all of the questions below, you can neglect the surface tension terms.\n\nWhat must be the magnitude of this charge $q$ in terms of $t, \\rho_{a}, \\rho_{s}, \\varepsilon_{0}, R_{0}, P_{a}$ in order that the bubble will float motionlessly in still air? Calculate also the numerical value of $q$. The permittivity of free space $\\varepsilon_{0}=8.85 \\times 10^{-12} \\mathrm{farad} / \\mathrm{m}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of nC, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "nC" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_970", "problem": "A bulldozer runs on a continuous track, sometimes called a caterpillar track, as shown in the image of Fig. 6. The driving wheel at the front has a diameter of $1.0 \\mathrm{~m}$ and rotates once in $0.84 \\mathrm{~s}$. A person standing at the side of the bulldozer as it drives past sees a large piece of mud stuck to the top side of the moving track (at about $1 \\mathrm{~m}$ above the ground). At what speed relative to the person is the mud moving past them?\n\n[figure1]\n\nFigure 6: The moving caterpillar track on a bulldozer.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nA bulldozer runs on a continuous track, sometimes called a caterpillar track, as shown in the image of Fig. 6. The driving wheel at the front has a diameter of $1.0 \\mathrm{~m}$ and rotates once in $0.84 \\mathrm{~s}$. A person standing at the side of the bulldozer as it drives past sees a large piece of mud stuck to the top side of the moving track (at about $1 \\mathrm{~m}$ above the ground). At what speed relative to the person is the mud moving past them?\n\n[figure1]\n\nFigure 6: The moving caterpillar track on a bulldozer.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\mathrm{~m} / \\mathrm{s}$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_9de0d5715d0b2f377364g-08.jpg?height=251&width=340&top_left_y=297&top_left_x=1469", "https://cdn.mathpix.com/cropped/2024_03_06_38b9374d4ba914cc7d40g-3.jpg?height=223&width=397&top_left_y=2207&top_left_x=264" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m} / \\mathrm{s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_294", "problem": "Note: In this question all of the diagrams are not to scale. The planets and the sun are very small compared to the distances between them.\n\nIn 1619 Johannes Kepler published his $3^{\\text {rd }}$ Law of planetary motion. The $3^{\\text {rd }}$ Law stated that the square of the orbital period of a planet is proportional to the cube of the mean orbital radius around the Sun. This is written as:\n\n$$\nT^{2} \\propto R^{3} \\quad T=\\text { period of orbit } \\quad R=\\text { mean radius of orbit }\n$$\n\nThe mean orbital radius of the Earth is defined as 1 Astronomical Unit (1 AU) i.e. the distance between the Earth and the Sun is 1 AU.\n\nKepler's Laws allowed early astronomers to find the relative positions of the planets in the solar system but finding the actual (absolute) distances was more challenging.\n\nEdmund Halley (1656-1742) suggested observing the transit of Venus across the Sun from different locations on Earth would allow the absolute scale of the solar system to be determined. Two different observers on opposite sides of the planet would see Venus follow slightly different paths across the face of the sun. The apparent difference in position that comes from viewing an object from two different points is called parallax. Accurately measuring the different transit times would allow the parallax angle to be determined.\n\n[figure1]\n\nNOT to scale\n\nIn 1768, Captain James Cook sailed to the island of Tahiti in the southern hemisphere to observe the 1769 transit of Venus.\n\nFrom two different locations separated by a distance of $d=9560 \\mathrm{~km}$, the parallax angle was determined to be $0.0130^{\\circ}$.\n\n[figure2]\n\nNote: $d=9560 \\mathrm{~km}$ is the straight line distance between the two observation locations, not the distance covered by travelling along the curve of the Earth's surface.\n\nc) Use these values to calculate the distance between the Earth and Venus", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nNote: In this question all of the diagrams are not to scale. The planets and the sun are very small compared to the distances between them.\n\nIn 1619 Johannes Kepler published his $3^{\\text {rd }}$ Law of planetary motion. The $3^{\\text {rd }}$ Law stated that the square of the orbital period of a planet is proportional to the cube of the mean orbital radius around the Sun. This is written as:\n\n$$\nT^{2} \\propto R^{3} \\quad T=\\text { period of orbit } \\quad R=\\text { mean radius of orbit }\n$$\n\nThe mean orbital radius of the Earth is defined as 1 Astronomical Unit (1 AU) i.e. the distance between the Earth and the Sun is 1 AU.\n\nKepler's Laws allowed early astronomers to find the relative positions of the planets in the solar system but finding the actual (absolute) distances was more challenging.\n\nEdmund Halley (1656-1742) suggested observing the transit of Venus across the Sun from different locations on Earth would allow the absolute scale of the solar system to be determined. Two different observers on opposite sides of the planet would see Venus follow slightly different paths across the face of the sun. The apparent difference in position that comes from viewing an object from two different points is called parallax. Accurately measuring the different transit times would allow the parallax angle to be determined.\n\n[figure1]\n\nNOT to scale\n\nIn 1768, Captain James Cook sailed to the island of Tahiti in the southern hemisphere to observe the 1769 transit of Venus.\n\nFrom two different locations separated by a distance of $d=9560 \\mathrm{~km}$, the parallax angle was determined to be $0.0130^{\\circ}$.\n\n[figure2]\n\nNote: $d=9560 \\mathrm{~km}$ is the straight line distance between the two observation locations, not the distance covered by travelling along the curve of the Earth's surface.\n\nc) Use these values to calculate the distance between the Earth and Venus\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of km, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-13.jpg?height=440&width=1574&top_left_y=2030&top_left_x=238", "https://cdn.mathpix.com/cropped/2024_03_06_c45e60794812d80909bfg-14.jpg?height=440&width=945&top_left_y=1165&top_left_x=561" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "km" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_255", "problem": "Two uniform balls 1 and 2 of radii $R_{1}=2.00 \\mathrm{~cm}$ and $R_{2}=4.00 \\mathrm{~cm}$ respectively are made of the same material of the mass density $\\rho=1.50 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$. They are firmly glued together to form a rigid body as shown in Figure 1. In this problem you will have to investigate various kinds of motions of that rigid body called the two-ball body.\n\n[figure1]\n\nFigure 1: Two-ball body lying at rest on the flat horizontal surface.\n\nFind the distance $x_{0}$ from the center-of-mass of the two-ball body to the center of ball 1 .", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nTwo uniform balls 1 and 2 of radii $R_{1}=2.00 \\mathrm{~cm}$ and $R_{2}=4.00 \\mathrm{~cm}$ respectively are made of the same material of the mass density $\\rho=1.50 \\times 10^{3} \\mathrm{~kg} / \\mathrm{m}^{3}$. They are firmly glued together to form a rigid body as shown in Figure 1. In this problem you will have to investigate various kinds of motions of that rigid body called the two-ball body.\n\n[figure1]\n\nFigure 1: Two-ball body lying at rest on the flat horizontal surface.\n\nFind the distance $x_{0}$ from the center-of-mass of the two-ball body to the center of ball 1 .\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of cm, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_57b03088b694c15dae09g-1.jpg?height=501&width=900&top_left_y=671&top_left_x=604" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "cm" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_610", "problem": "For large $V_{0}$, the velocity of the positive charges is determined by a strong drag force, so that\n\n$$\nv=\\mu E\n$$\n\nwhere $E$ is the local electric field and $\\mu$ is the charge mobility.\n\nIn the steady state, the current is the same everywhere. Consider the region $(x, x+d x)$. The time it takes for the charge in the second region to leave is $\\frac{\\mathrm{d} x}{v(x)}$. The amount of charge that leaves is $\\rho A \\mathrm{~d} x$. The current is thus given by $\\rho A v$, so $\\rho v$ is constant. Alternatively, one can write this as\n\n$$\nv \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}+\\rho \\frac{\\mathrm{d} v}{\\mathrm{~d} x}=0\n$$\n\nThe position $x$ is effectively in between two uniform sheets of charge density. The sheet on the left has charge density $\\int_{0}^{x} \\rho \\mathrm{d} x+\\sigma_{0}$, where $\\sigma_{0}$ is the charge density on the left plate, and the sheet on the right has charge density $\\int_{x}^{d} \\rho \\mathrm{d} x+\\sigma_{d}$, where $\\sigma_{d}$ is the charge density on the left plate. Then, the electric field is given by\n\n$$\nE=\\sigma_{0} /\\left(2 \\epsilon_{0}\\right)+\\int_{0}^{x} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\int_{x}^{d} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\sigma_{d} /\\left(2 \\epsilon_{0}\\right)\n$$\n\nThen, by the Fundamental Theorem of Calculus\n\n$$\n\\frac{d E}{d x}=\\frac{\\rho}{\\epsilon_{0}}\n$$\nso\n\n$$\n\\frac{d^{2} V}{d x^{2}}=-\\frac{\\rho}{\\epsilon_{0}}\n$$\n\nWe have that\n\n$$\n\\rho \\frac{d v}{d x}+v \\frac{d \\rho}{d x}=0\n$$\n\nand now that $v=-\\mu \\frac{d V}{d x}$, so substituting in Poisson's equation gives us that\n\n$$\n\\left(\\frac{d^{2} V}{d x^{2}}\\right)^{2}+\\frac{d V}{d x}\\left(\\frac{d^{3} V}{d x^{3}}\\right)=0\n$$\n\nUsing $V(x)=-V_{0}(x / d)^{b}$ gives\n\n$$\nb(b-1) b(b-1)=-b b(b-1)(b-2) \\text {. }\n$$\n\nThe solution with $b=0$ cannot satisfy the boundary conditions, while $b=1$ has zero current. Assuming $b$ is neither of these values, we have $b-1=-(b-2)$, so $b=3 / 2$. Substituting gives\n\n$$\nv=-\\frac{3 V_{0} \\mu x^{1 / 2}}{2 d^{3 / 2}}\n$$\n\nand\n\n$$\n\\rho=-\\frac{3 V_{0} \\epsilon_{0}}{4 d^{3 / 2} x^{1 / 2}}\n$$\n\nso\n\n$$\nI=\\rho A v=\\frac{9 \\epsilon_{0} \\mu A V_{0}^{2}}{8 d^{3}}\n$$\n\nwith the current flowing from left to right.Two large parallel plates of area $A$ are placed at $x=0$ and $x=d \\ll \\sqrt{A}$ in a semiconductor medium. The plate at $x=0$ is grounded, and the plate at $x=d$ is at a fixed potential $-V_{0}$, where $V_{0}>0$. Particles of positive charge $q$ flow between the two plates. You may neglect any dielectric effects of the medium.\n\nFor small $V_{0}$, the positive charges move by diffusion. The current due to diffusion is given by Fick's Law,\n\n$$\nI=-A D \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}\n$$\n\nHere, $D$ is the diffusion constant, which you can assume to be described by the Einstein relation\n\n$$\nD=\\frac{\\mu k_{B} T}{q}\n$$\n\nwhere $T$ is the temperature of the system.\n\nAssume that in the steady state, conditions have been established so that a nonzero, steady current flows, and the electric potential again satisfies $V(x) \\propto x^{b^{\\prime}}$, where $b^{\\prime}$ is another exponent you must find. Derive an expression for the current in terms of $V_{0}$ and the other given parameters.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nFor large $V_{0}$, the velocity of the positive charges is determined by a strong drag force, so that\n\n$$\nv=\\mu E\n$$\n\nwhere $E$ is the local electric field and $\\mu$ is the charge mobility.\n\nIn the steady state, the current is the same everywhere. Consider the region $(x, x+d x)$. The time it takes for the charge in the second region to leave is $\\frac{\\mathrm{d} x}{v(x)}$. The amount of charge that leaves is $\\rho A \\mathrm{~d} x$. The current is thus given by $\\rho A v$, so $\\rho v$ is constant. Alternatively, one can write this as\n\n$$\nv \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}+\\rho \\frac{\\mathrm{d} v}{\\mathrm{~d} x}=0\n$$\n\nThe position $x$ is effectively in between two uniform sheets of charge density. The sheet on the left has charge density $\\int_{0}^{x} \\rho \\mathrm{d} x+\\sigma_{0}$, where $\\sigma_{0}$ is the charge density on the left plate, and the sheet on the right has charge density $\\int_{x}^{d} \\rho \\mathrm{d} x+\\sigma_{d}$, where $\\sigma_{d}$ is the charge density on the left plate. Then, the electric field is given by\n\n$$\nE=\\sigma_{0} /\\left(2 \\epsilon_{0}\\right)+\\int_{0}^{x} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\int_{x}^{d} \\rho /\\left(2 \\epsilon_{0}\\right) \\mathrm{d} x-\\sigma_{d} /\\left(2 \\epsilon_{0}\\right)\n$$\n\nThen, by the Fundamental Theorem of Calculus\n\n$$\n\\frac{d E}{d x}=\\frac{\\rho}{\\epsilon_{0}}\n$$\nso\n\n$$\n\\frac{d^{2} V}{d x^{2}}=-\\frac{\\rho}{\\epsilon_{0}}\n$$\n\nWe have that\n\n$$\n\\rho \\frac{d v}{d x}+v \\frac{d \\rho}{d x}=0\n$$\n\nand now that $v=-\\mu \\frac{d V}{d x}$, so substituting in Poisson's equation gives us that\n\n$$\n\\left(\\frac{d^{2} V}{d x^{2}}\\right)^{2}+\\frac{d V}{d x}\\left(\\frac{d^{3} V}{d x^{3}}\\right)=0\n$$\n\nUsing $V(x)=-V_{0}(x / d)^{b}$ gives\n\n$$\nb(b-1) b(b-1)=-b b(b-1)(b-2) \\text {. }\n$$\n\nThe solution with $b=0$ cannot satisfy the boundary conditions, while $b=1$ has zero current. Assuming $b$ is neither of these values, we have $b-1=-(b-2)$, so $b=3 / 2$. Substituting gives\n\n$$\nv=-\\frac{3 V_{0} \\mu x^{1 / 2}}{2 d^{3 / 2}}\n$$\n\nand\n\n$$\n\\rho=-\\frac{3 V_{0} \\epsilon_{0}}{4 d^{3 / 2} x^{1 / 2}}\n$$\n\nso\n\n$$\nI=\\rho A v=\\frac{9 \\epsilon_{0} \\mu A V_{0}^{2}}{8 d^{3}}\n$$\n\nwith the current flowing from left to right.\n\nproblem:\nTwo large parallel plates of area $A$ are placed at $x=0$ and $x=d \\ll \\sqrt{A}$ in a semiconductor medium. The plate at $x=0$ is grounded, and the plate at $x=d$ is at a fixed potential $-V_{0}$, where $V_{0}>0$. Particles of positive charge $q$ flow between the two plates. You may neglect any dielectric effects of the medium.\n\nFor small $V_{0}$, the positive charges move by diffusion. The current due to diffusion is given by Fick's Law,\n\n$$\nI=-A D \\frac{\\mathrm{d} \\rho}{\\mathrm{d} x}\n$$\n\nHere, $D$ is the diffusion constant, which you can assume to be described by the Einstein relation\n\n$$\nD=\\frac{\\mu k_{B} T}{q}\n$$\n\nwhere $T$ is the temperature of the system.\n\nAssume that in the steady state, conditions have been established so that a nonzero, steady current flows, and the electric potential again satisfies $V(x) \\propto x^{b^{\\prime}}$, where $b^{\\prime}$ is another exponent you must find. Derive an expression for the current in terms of $V_{0}$ and the other given parameters.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_927", "problem": "The fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nWrite down the equation of motion of an electron in perpendicular electric field $\\vec{E}=-E_{y} \\hat{y}$ and magnetic field $\\vec{B}=B \\hat{z}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nThe fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nWrite down the equation of motion of an electron in perpendicular electric field $\\vec{E}=-E_{y} \\hat{y}$ and magnetic field $\\vec{B}=B \\hat{z}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_821", "problem": "In this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\nFind the orbital velocity $v_{b i n}$ of each SBH. Find the total energy $E$ of the SBH binary. Express it in terms of $a, G$ and $M$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nIn this part, we will consider the system of two SBHs with equal masses $M \\gg m$ located in the center of the galaxy. Let's call this system a SBH binary. We will assume that there are no stars near the SBH binary, each SBH has a circular orbit of radius $a$ in the gravitational field of another SBH.\nFind the orbital velocity $v_{b i n}$ of each SBH. Find the total energy $E$ of the SBH binary. Express it in terms of $a, G$ and $M$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1533", "problem": "有一块横截面为矩形的长板, 长度在 $81 \\mathrm{~cm}$ 与 $82 \\mathrm{~cm}$ 之间, 宽度在 $5 \\mathrm{~cm}$ 与 $6 \\mathrm{~cm}$ 之间, 厚度在 $1 \\mathrm{~cm}$ 与 $2 \\mathrm{~cm}$ 之间。现用直尺 (最小刻度为 $\\mathrm{mm}$ ) 、卡尺 (游标为 50 分度) 和千分尺 (螺旋测微器)去测量此板的长度、宽度和厚度, 要求测出的最后一位有效数字是估读的。试设想一组可能的数据填在下面的空格处。\n\n板的长度 $\\mathrm{cm}$ ，\n\n板的宽度 $\\mathrm{cm}$, \n\n板的厚度 $\\mathrm{cm}$ 。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个元组。\n\n问题:\n有一块横截面为矩形的长板, 长度在 $81 \\mathrm{~cm}$ 与 $82 \\mathrm{~cm}$ 之间, 宽度在 $5 \\mathrm{~cm}$ 与 $6 \\mathrm{~cm}$ 之间, 厚度在 $1 \\mathrm{~cm}$ 与 $2 \\mathrm{~cm}$ 之间。现用直尺 (最小刻度为 $\\mathrm{mm}$ ) 、卡尺 (游标为 50 分度) 和千分尺 (螺旋测微器)去测量此板的长度、宽度和厚度, 要求测出的最后一位有效数字是估读的。试设想一组可能的数据填在下面的空格处。\n\n板的长度 $\\mathrm{cm}$ ，\n\n板的宽度 $\\mathrm{cm}$, \n\n板的厚度 $\\mathrm{cm}$ 。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个元组，例如ANSWER=(3, 5)", "figure_urls": null, "answer": null, "solution": null, "answer_type": "TUP", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_221", "problem": "A spherical cloud of dust has uniform mass density $\\rho$ and radius $R$. Satellite A of negligible mass is orbiting the cloud at its edge, in a circular orbit of radius $R$, and satellite $\\mathrm{B}$ is orbiting the cloud just inside the cloud, in a circular orbit of radius $r$, with $rT_{B}$ and $v_{A}>v_{B}$\nB: $T_{A}>T_{B}$ and $v_{A}v_{B}$\nD: $T_{A}v_{B} $ \n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA spherical cloud of dust has uniform mass density $\\rho$ and radius $R$. Satellite A of negligible mass is orbiting the cloud at its edge, in a circular orbit of radius $R$, and satellite $\\mathrm{B}$ is orbiting the cloud just inside the cloud, in a circular orbit of radius $r$, with $rT_{B}$ and $v_{A}>v_{B}$\nB: $T_{A}>T_{B}$ and $v_{A}v_{B}$\nD: $T_{A}v_{B} $ \n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_999", "problem": "Gaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\nAssume that two chimneys are built to serve exactly the same purpose. Their cross sections are identical, but are designed to work in different parts of the world: one in cold regions, designed to work at an average atmospheric temperature of $-30{ }^{\\circ} \\mathrm{C}$ and the other in warm regions, designed to work at an average atmospheric temperature of $30^{\\circ} \\mathrm{C}$. The temperature of the furnace is $400{ }^{\\circ} \\mathrm{C}$. It was calculated that the height of the chimney designed to work in cold regions is $100 \\mathrm{~m}$. How high is the other chimney?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nGaseous products of burning are released into the atmosphere of temperature $T_{\\text {Air }}$ through a high chimney of cross-section $A$ and height $h$ (see Fig. 1). The solid matter is burned in the furnace which is at temperature $T_{\\text {smoke. }}$. The volume of gases produced per unit time in the furnace is $B$.\n\nAssume that:\n\n- the velocity of the gases in the furnace is negligibly small\n- the density of the gases (smoke) does not differ from that of the air at the same temperature and pressure; while in furnace, the gases can be treated as ideal\n- the pressure of the air changes with height in accordance with the hydrostatic law; the change of the density of the air with height is negligible\n- the flow of gases fulfills the Bernoulli equation which states that the following quantity is conserved in all points of the flow:\n\n$\\frac{1}{2} \\rho v^{2}(z)+\\rho g z+p(z)=$ const,\n\nwhere $\\rho$ is the density of the gas, $v(z)$ is its velocity, $p(z)$ is pressure, and $z$ is the height\n\n- the change of the density of the gas is negligible throughout the chimney\n\n[figure1]\n\nAssume that two chimneys are built to serve exactly the same purpose. Their cross sections are identical, but are designed to work in different parts of the world: one in cold regions, designed to work at an average atmospheric temperature of $-30{ }^{\\circ} \\mathrm{C}$ and the other in warm regions, designed to work at an average atmospheric temperature of $30^{\\circ} \\mathrm{C}$. The temperature of the furnace is $400{ }^{\\circ} \\mathrm{C}$. It was calculated that the height of the chimney designed to work in cold regions is $100 \\mathrm{~m}$. How high is the other chimney?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of m, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_fd946cfac82ef740b1dag-1.jpg?height=977&width=1644&top_left_y=1453&top_left_x=206" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "m" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_720", "problem": "Three planets, each with same mass, orbit in different orbits around the same star. Which statement is true about the total energy of the three orbits? Assume that $m_{\\text {planets }} \\ll M_{\\text {star }}$.\n\n[figure1]\nA: $E_{1}0$. Write your answer in terms of $I, r, z$, and any necessary fundamental constants.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nAn \"Ampère\" dipole is a magnetic dipole produced by a current loop $I$ around a circle of radius $r$, where $r$ is small. Assume the that the $z$ axis is the axis of rotational symmetry for the circular loop, and the loop lies in the $x y$ plane at $z=0$.\n\n[figure1]\n\nWrite an exact expression for the magnetic field strength $B(z)$ along the $z$ axis as a function of $z$ for $z>0$. Write your answer in terms of $I, r, z$, and any necessary fundamental constants.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_d2de61e197238a85a597g-19.jpg?height=296&width=653&top_left_y=421&top_left_x=779" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_110", "problem": "Two masses are attached with pulleys by a massless rope on an inclined plane as shown. All surfaces are frictionless. If the masses are released from rest, then the inclined plane\n\n[figure1]\nA: accelerates to the left if $m_{1}2$ 的能级向 $k=2$ 的能级跃迁而产生的光谱. (已知氢原子的基态能量 $E_{0}=-13.60 \\mathrm{eV}$, 真空中光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 普朗克常量 $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 电子电荷量 $e=1.602 \\times 10^{-19} \\mathrm{C}$ )该星系发出的光咝线对应于实验室中测出的氢原子的哪两条咝线? 它们在实验室中的波长分别是多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n天文观测表明, 远处的星系均离我们而去. 著名的哈勃定律指出, 星系离开我们的速度大小 $v=H D$, 其中 $D$ 为星系与我们之间的距离, 该距离通常以百方秒差距 ( $\\mathrm{Mpc}$ ) 为单位: $H$ 为哈勃常数, 最新的测量结果为 $H=67.80 \\mathrm{~km} /(\\mathrm{s} \\cdot \\mathrm{Mpc})$. 当星系离开我们远去时, 它发出的光谱线的波长会变长 (称为红移). 红移量 $z$ 被定义为 $z=\\frac{\\lambda^{\\prime}-\\lambda}{\\lambda}$, 其中 $\\lambda^{\\prime}$ 是我们观测到的星系中某恒星发出的谱线的波长, 而 $\\lambda$ 是实验室中测得的同种原子发出的相应的谱线的波长, 该红移可用多普勒效应解释. 绝大部分星系的红移量 $z$ 远小于 1 , 即星系退行的速度远小于光速. 在一次天文观测中发现从天鹰座的一个星系中射来的氢原子光谱中有两条谱线, 它们的频率 $v^{\\prime}$ 分别为 $4.549 \\times 10^{14} \\mathrm{~Hz}$ 和 $6.141 \\times 10^{14} \\mathrm{~Hz}$. 由于这两条谱线处于可见光频率区间，可假设它们属于氢原子的巴尔末系, 即为由 $n>2$ 的能级向 $k=2$ 的能级跃迁而产生的光谱. (已知氢原子的基态能量 $E_{0}=-13.60 \\mathrm{eV}$, 真空中光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 普朗克常量 $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 电子电荷量 $e=1.602 \\times 10^{-19} \\mathrm{C}$ )\n\n问题:\n该星系发出的光咝线对应于实验室中测出的氢原子的哪两条咝线? 它们在实验室中的波长分别是多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[氢原子咝线, 波长1, 波长2]\n它们的答案类型依次是[数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "氢原子咝线", "波长1", "波长2" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1676", "problem": "制冷机是通过外界对机器做功, 把从低温吸取的热量连同外界对机器做功得到的能量一起送到高温处的机器。它能使低温处的温度降低, 高温处的温度升高。已知当制冷机工作在绝对温度为 $T_{1}$ 的高温处和绝对温度为 $T_{2}$ 的低温处之间时, 若制冷机从低温处吸取的热量为 $Q$,外界对制冷机做的功为 $W$, 则有\n\n$$\n\\frac{Q}{W} \\leq \\frac{T_{2}}{T_{1}-T_{2}}\n$$\n\n式中等号对应于理论上的理想情况。\n\n某制冷机在冬天作热永使用 (即取暖空调机), 在室外温度为 $-5.00^{\\circ} \\mathrm{C}$ 的情况下, 使某房间内的温度保持在 $20.00^{\\circ} \\mathrm{C}$ 。由于室内温度高于室外, 故将有热量从室内传递到室外。本题只考虑传导方式的传热, 它服从以下的规律: 设一块导热层, 其厚度为 $l$, 面积为 $S$, 两侧温度差的大小为 $\\Delta T$ ，则单位时间内通过导热层由高温处传导到低温处的热量为\n\n$$\nH=\\kappa \\frac{\\Delta T}{l} S\n$$\n\n其中 $\\kappa$ 为导热率, 取决于导热层材料的性质。若将上述玻璃板换为“双层玻璃板”, 两层玻璃的厚度均为 $2.00 \\mathrm{~mm}$, 玻璃板之间夹有厚度为 $l_{0}=0.50 \\mathrm{~mm}$ 的空气层, 假设空气的导热率为 $\\kappa=0.025 \\mathrm{~W} \\cdot \\mathrm{m}^{-1} \\cdot \\mathrm{k}^{-1}$, 电费仍为每度 0.50 元,若该热泉仍然工作 12 小时，问这时的电费比上一问单层玻璃情形节省多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n制冷机是通过外界对机器做功, 把从低温吸取的热量连同外界对机器做功得到的能量一起送到高温处的机器。它能使低温处的温度降低, 高温处的温度升高。已知当制冷机工作在绝对温度为 $T_{1}$ 的高温处和绝对温度为 $T_{2}$ 的低温处之间时, 若制冷机从低温处吸取的热量为 $Q$,外界对制冷机做的功为 $W$, 则有\n\n$$\n\\frac{Q}{W} \\leq \\frac{T_{2}}{T_{1}-T_{2}}\n$$\n\n式中等号对应于理论上的理想情况。\n\n某制冷机在冬天作热永使用 (即取暖空调机), 在室外温度为 $-5.00^{\\circ} \\mathrm{C}$ 的情况下, 使某房间内的温度保持在 $20.00^{\\circ} \\mathrm{C}$ 。由于室内温度高于室外, 故将有热量从室内传递到室外。本题只考虑传导方式的传热, 它服从以下的规律: 设一块导热层, 其厚度为 $l$, 面积为 $S$, 两侧温度差的大小为 $\\Delta T$ ，则单位时间内通过导热层由高温处传导到低温处的热量为\n\n$$\nH=\\kappa \\frac{\\Delta T}{l} S\n$$\n\n其中 $\\kappa$ 为导热率, 取决于导热层材料的性质。\n\n问题:\n若将上述玻璃板换为“双层玻璃板”, 两层玻璃的厚度均为 $2.00 \\mathrm{~mm}$, 玻璃板之间夹有厚度为 $l_{0}=0.50 \\mathrm{~mm}$ 的空气层, 假设空气的导热率为 $\\kappa=0.025 \\mathrm{~W} \\cdot \\mathrm{m}^{-1} \\cdot \\mathrm{k}^{-1}$, 电费仍为每度 0.50 元,若该热泉仍然工作 12 小时，问这时的电费比上一问单层玻璃情形节省多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以元为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "元" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_18", "problem": "Two forces have magnitudes of $11.0 \\mathrm{~N}$ and $5.0 \\mathrm{~N}$. The magnitude of their sum could NOT be equal to which of the following values?\nA: $16.0 \\mathrm{~N}$\nB: $9.0 \\mathrm{~N}$\nC: $7.0 \\mathrm{~N}$\nD: $5.0 \\mathrm{~N}$\nE: $6.0 \\mathrm{~N}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nTwo forces have magnitudes of $11.0 \\mathrm{~N}$ and $5.0 \\mathrm{~N}$. The magnitude of their sum could NOT be equal to which of the following values?\n\nA: $16.0 \\mathrm{~N}$\nB: $9.0 \\mathrm{~N}$\nC: $7.0 \\mathrm{~N}$\nD: $5.0 \\mathrm{~N}$\nE: $6.0 \\mathrm{~N}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_520", "problem": "A room air conditioner is modeled as a heat engine run in reverse: an amount of heat $Q_{L}$ is absorbed from the room at a temperature $T_{L}$ into cooling coils containing a working gas; this gas is compressed adiabatically to a temperature $T_{H}$; the gas is compressed isothermally in a coil outside the house, giving off an amount of heat $Q_{H}$; the gas expands adiabatically back to a temperature $T_{L}$; and the cycle repeats. An amount of energy $W$ is input into the system every cycle through an electric pump. This model describes the air conditioner with the best possible efficiency.\n\n[figure1]\n\nAssume that the outside air temperature is $T_{H}$ and the inside air temperature is $T_{L}$. The air-conditioner unit consumes electric power $P$. Assume that the air is sufficiently dry so that no condensation of water occurs in the cooling coils of the air conditioner. Water boils at $373 \\mathrm{~K}$ and freezes at $273 \\mathrm{~K}$ at normal atmospheric pressure.\n\nDerive an expression for the maximum rate at which heat is removed from the room in terms of the air temperatures $T_{H}, T_{L}$, and the power consumed by the air conditioner $P$. Your derivation must refer to the entropy changes that occur in a Carnot cycle in order to receive full marks for this part.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nA room air conditioner is modeled as a heat engine run in reverse: an amount of heat $Q_{L}$ is absorbed from the room at a temperature $T_{L}$ into cooling coils containing a working gas; this gas is compressed adiabatically to a temperature $T_{H}$; the gas is compressed isothermally in a coil outside the house, giving off an amount of heat $Q_{H}$; the gas expands adiabatically back to a temperature $T_{L}$; and the cycle repeats. An amount of energy $W$ is input into the system every cycle through an electric pump. This model describes the air conditioner with the best possible efficiency.\n\n[figure1]\n\nAssume that the outside air temperature is $T_{H}$ and the inside air temperature is $T_{L}$. The air-conditioner unit consumes electric power $P$. Assume that the air is sufficiently dry so that no condensation of water occurs in the cooling coils of the air conditioner. Water boils at $373 \\mathrm{~K}$ and freezes at $273 \\mathrm{~K}$ at normal atmospheric pressure.\n\nDerive an expression for the maximum rate at which heat is removed from the room in terms of the air temperatures $T_{H}, T_{L}$, and the power consumed by the air conditioner $P$. Your derivation must refer to the entropy changes that occur in a Carnot cycle in order to receive full marks for this part.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_0256c432f4019b26894dg-04.jpg?height=479&width=1006&top_left_y=1403&top_left_x=554" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1622", "problem": "如图 7a, 导电物质为电子的霍尔元件长方体样品置于磁场中, 其上下表面均与磁场方向垂直, 其中的 $1 、 2 、 3 、 4$ 是霍尔元件上的四个接线端。若开关 $\\mathrm{S}_{1}$ 处于断开状态、开关 $\\mathrm{S}_{2}$ 处于闭合状态,电压表示数为 0 ; 当开关 $\\mathrm{S}_{1} 、 \\mathrm{~S}_{2}$ 闭合后, 三个电表都有明显示数。已知由于温度非均匀性等因素引起的其它效应可忽略, 则接线端 2的电势—（填 “低于”、“等于” 或 “高于”) 接线端 4 的电势; 若将电源 $E_{1} 、 E_{2}$ 均反向接入电路, 电压表的示数 （填“正负号改变，大小不变”、“正负号和大小都不变”或 “正负号不变，大小改变”）。\n\n[图1]\n\n图 7a", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n\n问题:\n如图 7a, 导电物质为电子的霍尔元件长方体样品置于磁场中, 其上下表面均与磁场方向垂直, 其中的 $1 、 2 、 3 、 4$ 是霍尔元件上的四个接线端。若开关 $\\mathrm{S}_{1}$ 处于断开状态、开关 $\\mathrm{S}_{2}$ 处于闭合状态,电压表示数为 0 ; 当开关 $\\mathrm{S}_{1} 、 \\mathrm{~S}_{2}$ 闭合后, 三个电表都有明显示数。已知由于温度非均匀性等因素引起的其它效应可忽略, 则接线端 2的电势—（填 “低于”、“等于” 或 “高于”) 接线端 4 的电势; 若将电源 $E_{1} 、 E_{2}$ 均反向接入电路, 电压表的示数 （填“正负号改变，大小不变”、“正负号和大小都不变”或 “正负号不变，大小改变”）。\n\n[图1]\n\n图 7a\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[接线端 2的电势—（填 “低于”、“等于” 或 “高于”) 接线端 4 的电势, 接线端 4 的电势; 若将电源 $E_{1} 、 E_{2}$ 均反向接入电路, 电压表的示数 （填“正负号改变，大小不变”、“正负号和大小都不变”或 “正负号不变，大小改变”）。]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-03.jpg?height=309&width=606&top_left_y=1236&top_left_x=1339" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "接线端 2的电势—（填 “低于”、“等于” 或 “高于”) 接线端 4 的电势", "接线端 4 的电势; 若将电源 $E_{1} 、 E_{2}$ 均反向接入电路, 电压表的示数 （填“正负号改变，大小不变”、“正负号和大小都不变”或 “正负号不变，大小改变”）。" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_133", "problem": "A projectile is launched with speed $v_{0}$ off the edge of a cliff of height $h$, at an angle $\\theta$ from the horizontal. Air friction is negligible. To maximize the horizontal range of the projectile, $\\theta$ should satisfy\nA: $45^{\\circ}<\\theta<90^{\\circ}$\nB: $\\theta=45^{\\circ}$\nC: $0^{\\circ}<\\theta<45^{\\circ} $ \nD: $\\theta=0^{\\circ}$\nE: $\\theta<45^{\\circ}$ or $\\theta>45^{\\circ}$, depending on the values of $h$ and $v_{0}$.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA projectile is launched with speed $v_{0}$ off the edge of a cliff of height $h$, at an angle $\\theta$ from the horizontal. Air friction is negligible. To maximize the horizontal range of the projectile, $\\theta$ should satisfy\n\nA: $45^{\\circ}<\\theta<90^{\\circ}$\nB: $\\theta=45^{\\circ}$\nC: $0^{\\circ}<\\theta<45^{\\circ} $ \nD: $\\theta=0^{\\circ}$\nE: $\\theta<45^{\\circ}$ or $\\theta>45^{\\circ}$, depending on the values of $h$ and $v_{0}$.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1720", "problem": "有两个同样的梯子, 其顶部用活页连在一起, 在两梯中间某相对的位置用一轻绳系住,便形成了人字梯。如图 1a 所示，将两个同样的人字梯甲、乙放置于水平地面上，甲梯用的绳更长一些。当某人先、后站在甲、乙两梯顶端时，下述说法正确的是\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\n图 1a\nA: 甲梯所受地面的支持力一定较大\nB: 甲、乙两梯所受地面的支持力一定相等\nC: 绳子被张紧时，甲梯所受地面的摩擦力一定比乙梯的大\nD: 绳子被张紧时，甲梯所受地面的摩擦力一定比乙梯的小\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个单选题（只有一个正确答案）。\n\n问题:\n有两个同样的梯子, 其顶部用活页连在一起, 在两梯中间某相对的位置用一轻绳系住,便形成了人字梯。如图 1a 所示，将两个同样的人字梯甲、乙放置于水平地面上，甲梯用的绳更长一些。当某人先、后站在甲、乙两梯顶端时，下述说法正确的是\n\n[图1]\n\n甲\n\n[图2]\n\n乙\n\n图 1a\n\nA: 甲梯所受地面的支持力一定较大\nB: 甲、乙两梯所受地面的支持力一定相等\nC: 绳子被张紧时，甲梯所受地面的摩擦力一定比乙梯的大\nD: 绳子被张紧时，甲梯所受地面的摩擦力一定比乙梯的小\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为以下选项之一：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-01.jpg?height=306&width=280&top_left_y=1729&top_left_x=1413", "https://cdn.mathpix.com/cropped/2024_03_31_6557cde761f93ea06130g-01.jpg?height=340&width=229&top_left_y=1732&top_left_x=1710" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_102", "problem": "A flat uniform disk of radius $2 R$ has a hole of radius $R$ removed from the center. The resulting annulus is then cut in half along the diameter. The remaining shape has mass $M$. What is the moment of inertia of this shape, about the axis of rotational symmetry of the original disk?\nA: $\\frac{45}{32} M R^{2}$\nB: $\\frac{7}{6} M R^{2}$\nC: $\\frac{8}{5} M R^{2}$\nD: $\\frac{5}{2} M R^{2} $ \nE: $\\frac{15}{8} M R^{2}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA flat uniform disk of radius $2 R$ has a hole of radius $R$ removed from the center. The resulting annulus is then cut in half along the diameter. The remaining shape has mass $M$. What is the moment of inertia of this shape, about the axis of rotational symmetry of the original disk?\n\nA: $\\frac{45}{32} M R^{2}$\nB: $\\frac{7}{6} M R^{2}$\nC: $\\frac{8}{5} M R^{2}$\nD: $\\frac{5}{2} M R^{2} $ \nE: $\\frac{15}{8} M R^{2}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1692", "problem": "如图所示, 某绝热熔器被两块装有阀门 $K_{1}$ 和 $K_{2}$ 的固定绝热隔板分割成相等体积 $V_{0}$ 的三室 A、B、C, $V_{A}=V_{B}=V_{C}=V_{0}$. 容器左端用绝热活塞 $\\mathrm{H}$ 封闭, 左侧 $\\mathrm{A}$ 室装有 $v_{1}=1$ 摩尔单原子分子气体, 处在压强为 $P_{0}$ 、温度为 $T_{0}$ 的平衡态; 中段 $B$ 室为真空; 右侧 $C$ 室装有 $v_{2}=2$摩尔双原子分子气体, 测得其平衡态温度为 $T \\mathrm{~T}=0.50 \\mathrm{~T}_{0}$. 初始时刻 $\\mathrm{K}_{1}$ 和 $\\mathrm{K}_{2}$ 都处在关闭状态. 然后系统依次经历如下与外界无热量交换的热力学过程:\n\n提示: 上述所有过程中, 气体均可视为理想气体, 计算结果可含数值的指数式或分式; 根据热力学第二定律, 当一种理想气体构成的热力学系统从初态 $\\left(p_{\\mathrm{i}}, T_{\\mathrm{i}}, V_{\\mathrm{i}}\\right)$ 经过一个绝热可逆过程 (准静态绝热过程) 到达终态 $\\left(p_{\\mathrm{f}}, T_{\\mathrm{f}}, V_{\\mathrm{f}}\\right)$ 时, 其状态参数满足方程:\n\n$$\n(\\Delta S)_{i f}=v_{1} C_{V_{1}} \\ln \\left(\\frac{T_{f}}{T_{i}}\\right)+v_{1} R \\ln \\left(\\frac{T_{f}}{T_{i}}\\right)=0\n$$\n\n其中, $v_{1}$ 为该气体的摩尔数, $C_{V 1}$ 为它的定容摩尔热容量, $R$ 为普适气体常量. 当热力学系统由两种理想气体组成, 则方程（1）需修改为\n\n$$\n\\left(\\Delta S_{1}\\right)_{i f}+\\left(\\Delta S_{2}\\right)_{i f}=0\n$$\n\n[图1]保持 $K_{1}$ 开放, 打开 $K_{2}$, 让容器中的两种气体自由混合后共同达到平衡态. 求此时混合气体的温度和压强", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图所示, 某绝热熔器被两块装有阀门 $K_{1}$ 和 $K_{2}$ 的固定绝热隔板分割成相等体积 $V_{0}$ 的三室 A、B、C, $V_{A}=V_{B}=V_{C}=V_{0}$. 容器左端用绝热活塞 $\\mathrm{H}$ 封闭, 左侧 $\\mathrm{A}$ 室装有 $v_{1}=1$ 摩尔单原子分子气体, 处在压强为 $P_{0}$ 、温度为 $T_{0}$ 的平衡态; 中段 $B$ 室为真空; 右侧 $C$ 室装有 $v_{2}=2$摩尔双原子分子气体, 测得其平衡态温度为 $T \\mathrm{~T}=0.50 \\mathrm{~T}_{0}$. 初始时刻 $\\mathrm{K}_{1}$ 和 $\\mathrm{K}_{2}$ 都处在关闭状态. 然后系统依次经历如下与外界无热量交换的热力学过程:\n\n提示: 上述所有过程中, 气体均可视为理想气体, 计算结果可含数值的指数式或分式; 根据热力学第二定律, 当一种理想气体构成的热力学系统从初态 $\\left(p_{\\mathrm{i}}, T_{\\mathrm{i}}, V_{\\mathrm{i}}\\right)$ 经过一个绝热可逆过程 (准静态绝热过程) 到达终态 $\\left(p_{\\mathrm{f}}, T_{\\mathrm{f}}, V_{\\mathrm{f}}\\right)$ 时, 其状态参数满足方程:\n\n$$\n(\\Delta S)_{i f}=v_{1} C_{V_{1}} \\ln \\left(\\frac{T_{f}}{T_{i}}\\right)+v_{1} R \\ln \\left(\\frac{T_{f}}{T_{i}}\\right)=0\n$$\n\n其中, $v_{1}$ 为该气体的摩尔数, $C_{V 1}$ 为它的定容摩尔热容量, $R$ 为普适气体常量. 当热力学系统由两种理想气体组成, 则方程（1）需修改为\n\n$$\n\\left(\\Delta S_{1}\\right)_{i f}+\\left(\\Delta S_{2}\\right)_{i f}=0\n$$\n\n[图1]\n\n问题:\n保持 $K_{1}$ 开放, 打开 $K_{2}$, 让容器中的两种气体自由混合后共同达到平衡态. 求此时混合气体的温度和压强\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[温度, 压强]\n它们的答案类型依次是[数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_bc0dbcee918772e93d28g-02.jpg?height=511&width=1078&top_left_y=1980&top_left_x=383" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "温度", "压强" ], "type_sequence": [ "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1310", "problem": "两颗人造地球卫星 A 和 B 都在同一平面内的圆轨道上运行, 绕向相同, 卫星 A 的轨道半径为 $r$ 。某时刻, $\\mathrm{B}$ 恰好在 $\\mathrm{A}$ 的正上方 $h$ 高处, $h<2$ 的能级向 $k=2$ 的能级跃迁而产生的光谱. (已知氢原子的基态能量 $E_{0}=-13.60 \\mathrm{eV}$, 真空中光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 普朗克常量 $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 电子电荷量 $e=1.602 \\times 10^{-19} \\mathrm{C}$ )求该星系与我们的距离 $D$", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n天文观测表明, 远处的星系均离我们而去. 著名的哈勃定律指出, 星系离开我们的速度大小 $v=H D$, 其中 $D$ 为星系与我们之间的距离, 该距离通常以百方秒差距 ( $\\mathrm{Mpc}$ ) 为单位: $H$ 为哈勃常数, 最新的测量结果为 $H=67.80 \\mathrm{~km} /(\\mathrm{s} \\cdot \\mathrm{Mpc})$. 当星系离开我们远去时, 它发出的光谱线的波长会变长 (称为红移). 红移量 $z$ 被定义为 $z=\\frac{\\lambda^{\\prime}-\\lambda}{\\lambda}$, 其中 $\\lambda^{\\prime}$ 是我们观测到的星系中某恒星发出的谱线的波长, 而 $\\lambda$ 是实验室中测得的同种原子发出的相应的谱线的波长, 该红移可用多普勒效应解释. 绝大部分星系的红移量 $z$ 远小于 1 , 即星系退行的速度远小于光速. 在一次天文观测中发现从天鹰座的一个星系中射来的氢原子光谱中有两条谱线, 它们的频率 $v^{\\prime}$ 分别为 $4.549 \\times 10^{14} \\mathrm{~Hz}$ 和 $6.141 \\times 10^{14} \\mathrm{~Hz}$. 由于这两条谱线处于可见光频率区间，可假设它们属于氢原子的巴尔末系, 即为由 $n>2$ 的能级向 $k=2$ 的能级跃迁而产生的光谱. (已知氢原子的基态能量 $E_{0}=-13.60 \\mathrm{eV}$, 真空中光速 $c=2.998 \\times 10^{8} \\mathrm{~m} / \\mathrm{s}$, 普朗克常量 $h=6.626 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 电子电荷量 $e=1.602 \\times 10^{-19} \\mathrm{C}$ )\n\n问题:\n求该星系与我们的距离 $D$\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以Mpc为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "Mpc" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_183", "problem": "Several identical cars are standing at a red light on a one-lane road, one behind the other, with negligible (and equal) distance between adjacent cars. When the green light comes up, the first car takes off to the right with constant acceleration. The driver in the second car reacts and does the same $0.2 \\mathrm{~s}$ later. The third driver starts moving $0.2 \\mathrm{~s}$ after the second one and so on. All cars accelerate until they reach the speed limit of $45 \\mathrm{~km} / \\mathrm{hr}$, after which they move to the right at a constant speed. Consider the following patterns of cars.\n\n[figure1]\n\nJust before the first car starts accelerating to the right, the car pattern will qualitatively look like the pattern in I. After that, the pattern will qualitatively evolve according to\nA: First I, then II, and then III.\nB: First I, then II, and then IV. \nC: First I, and then IV, with neither II nor III as intermediate stage.\nD: First I, and then II.\nE: First I, and then III.\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nSeveral identical cars are standing at a red light on a one-lane road, one behind the other, with negligible (and equal) distance between adjacent cars. When the green light comes up, the first car takes off to the right with constant acceleration. The driver in the second car reacts and does the same $0.2 \\mathrm{~s}$ later. The third driver starts moving $0.2 \\mathrm{~s}$ after the second one and so on. All cars accelerate until they reach the speed limit of $45 \\mathrm{~km} / \\mathrm{hr}$, after which they move to the right at a constant speed. Consider the following patterns of cars.\n\n[figure1]\n\nJust before the first car starts accelerating to the right, the car pattern will qualitatively look like the pattern in I. After that, the pattern will qualitatively evolve according to\n\nA: First I, then II, and then III.\nB: First I, then II, and then IV. \nC: First I, and then IV, with neither II nor III as intermediate stage.\nD: First I, and then II.\nE: First I, and then III.\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_f726c6cf4a23f08e0214g-03.jpg?height=417&width=943&top_left_y=548&top_left_x=583" ], "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_3", "problem": "A person has a height $1.8 \\mathrm{~m}$. Beginning directly under a streetlight that is $6 \\mathrm{~m}$ above the ground, they walk away from their starting position at $7 \\mathrm{~m} / \\mathrm{s}$. Find the speed at which the top of the person's shadow moves.\nA: $6 \\mathrm{~m} / \\mathrm{s}$.\nB: $8 \\mathrm{~m} / \\mathrm{s}$\nC: $10 \\mathrm{~m} / \\mathrm{s}$\nD: $12 \\mathrm{~m} / \\mathrm{s}$\nE: $14 \\mathrm{~m} / \\mathrm{s}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA person has a height $1.8 \\mathrm{~m}$. Beginning directly under a streetlight that is $6 \\mathrm{~m}$ above the ground, they walk away from their starting position at $7 \\mathrm{~m} / \\mathrm{s}$. Find the speed at which the top of the person's shadow moves.\n\nA: $6 \\mathrm{~m} / \\mathrm{s}$.\nB: $8 \\mathrm{~m} / \\mathrm{s}$\nC: $10 \\mathrm{~m} / \\mathrm{s}$\nD: $12 \\mathrm{~m} / \\mathrm{s}$\nE: $14 \\mathrm{~m} / \\mathrm{s}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_642", "problem": "Relativistic particles obey the mass energy relation\n\n$$\nE^{2}=(p c)^{2}+\\left(m c^{2}\\right)^{2}\n$$\n\nwhere $E$ is the relativistic energy of the particle, $p$ is the relativistic momentum, $m$ is the mass, and $c$ is the speed of light.\n\nA proton with mass $m_{p}$ and energy $E_{p}$ collides head on with a photon which is massless and has energy $E_{b}$. The two combine and form a new particle with mass $m_{\\Delta}$ called $\\Delta$, or \"delta\". It is a one dimensional collision that conserves both relativistic energy and relativistic momentum.\n\nDetermine $E_{p}$ in terms of $m_{p}, m_{\\Delta}$, and $E_{b}$. You may assume that $E_{b}$ is small.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nRelativistic particles obey the mass energy relation\n\n$$\nE^{2}=(p c)^{2}+\\left(m c^{2}\\right)^{2}\n$$\n\nwhere $E$ is the relativistic energy of the particle, $p$ is the relativistic momentum, $m$ is the mass, and $c$ is the speed of light.\n\nA proton with mass $m_{p}$ and energy $E_{p}$ collides head on with a photon which is massless and has energy $E_{b}$. The two combine and form a new particle with mass $m_{\\Delta}$ called $\\Delta$, or \"delta\". It is a one dimensional collision that conserves both relativistic energy and relativistic momentum.\n\nDetermine $E_{p}$ in terms of $m_{p}, m_{\\Delta}$, and $E_{b}$. You may assume that $E_{b}$ is small.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1099", "problem": "# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nAfter the bubble is formed for a while, it will be in thermal equilibrium with the surrounding. This bubble in still air will naturally fall towards the ground.\n\nThe above calculations suggest that the terms involving the surface tension $\\gamma$ add very little to the accuracy of the result. In all of the questions below, you can neglect the surface tension terms.\n\nIf this spherical bubble is now electrified uniformly with a total charge $q$, find an equation describing the new radius $R_{1}$ in terms of $R_{0}, P_{a}, q$ and the permittivity of free space $\\varepsilon_{0}$.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\n# An Electrified Soap Bubble \n\nA spherical soap bubble with internal air density $\\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\\gamma$, density $\\rho_{s}$ and thickness $t$. The mass and the surface tension of the soap do not change with the temperature. Assume that $R_{0} \\gg t$.\n\nThe increase in energy, $d E$, that is needed to increase the surface area of a soap-air interface by $d A$, is given by $d E=\\gamma d A$ where $\\gamma$ is the surface tension of the film.\n\nAfter the bubble is formed for a while, it will be in thermal equilibrium with the surrounding. This bubble in still air will naturally fall towards the ground.\n\nThe above calculations suggest that the terms involving the surface tension $\\gamma$ add very little to the accuracy of the result. In all of the questions below, you can neglect the surface tension terms.\n\nIf this spherical bubble is now electrified uniformly with a total charge $q$, find an equation describing the new radius $R_{1}$ in terms of $R_{0}, P_{a}, q$ and the permittivity of free space $\\varepsilon_{0}$.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_1a896b307b2dae9945c3g-4.jpg?height=304&width=1090&top_left_y=558&top_left_x=469", "https://cdn.mathpix.com/cropped/2024_03_14_1a896b307b2dae9945c3g-4.jpg?height=532&width=510&top_left_y=1897&top_left_x=599" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1506", "problem": "1958 年穆斯堡尔发现的原子核无反冲共振吸收效应（即穆斯堡尔效应）可用于测量光子频率极微小的变化, 穆斯堡尔因此荣获 1961 年诺贝尔物理学奖。类似于原子的能级结构,原子核也具有分立的能级, 并能通过吸收或放出光子在能级间跃迁。原子核在吸收和放出光子时会有反冲, 部分能量转化为原子核的动能（即反冲能）。此外, 原子核的激发态相对于其基态的能量差并不是一个确定值, 而是在以 $E_{0}$ 为中心、宽度为 $2 \\Gamma$ 的范围内取值的。对于 ${ }^{57} \\mathrm{Fe}$ 从第一激发态到基态的跃迁, $E_{0}=2.31 \\times 10^{-15} \\mathrm{~J}, \\Gamma \\simeq 3.2 \\times 10^{-13} E_{0}$ 。已知质量 $m_{\\mathrm{Fe}} \\simeq 9.5 \\times 10^{-26} \\mathrm{~kg}$, 普朗克常量 $h=6.6 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 真空中的光速 $c=3.0 \\times 10^{8} \\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ 。忽略激发态的能级宽度, 求反冲能, 以及在考虑核反冲和不考虑核反冲的情形下, ${ }^{57} \\mathrm{Fe}$从第一激发态跃迁到基态发出的光子的频率之差;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n1958 年穆斯堡尔发现的原子核无反冲共振吸收效应（即穆斯堡尔效应）可用于测量光子频率极微小的变化, 穆斯堡尔因此荣获 1961 年诺贝尔物理学奖。类似于原子的能级结构,原子核也具有分立的能级, 并能通过吸收或放出光子在能级间跃迁。原子核在吸收和放出光子时会有反冲, 部分能量转化为原子核的动能（即反冲能）。此外, 原子核的激发态相对于其基态的能量差并不是一个确定值, 而是在以 $E_{0}$ 为中心、宽度为 $2 \\Gamma$ 的范围内取值的。对于 ${ }^{57} \\mathrm{Fe}$ 从第一激发态到基态的跃迁, $E_{0}=2.31 \\times 10^{-15} \\mathrm{~J}, \\Gamma \\simeq 3.2 \\times 10^{-13} E_{0}$ 。已知质量 $m_{\\mathrm{Fe}} \\simeq 9.5 \\times 10^{-26} \\mathrm{~kg}$, 普朗克常量 $h=6.6 \\times 10^{-34} \\mathrm{~J} \\cdot \\mathrm{s}$, 真空中的光速 $c=3.0 \\times 10^{8} \\mathrm{~m} \\cdot \\mathrm{s}^{-1}$ 。\n\n问题:\n忽略激发态的能级宽度, 求反冲能, 以及在考虑核反冲和不考虑核反冲的情形下, ${ }^{57} \\mathrm{Fe}$从第一激发态跃迁到基态发出的光子的频率之差;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~J}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": null, "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~J}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "text-only" }, { "id": "Physics_1642", "problem": "2016 年 9 月，G20 峰会在杭州隆重召开，其会议厅的装饰设计既展示出中国建筑的节能环保理念，又体现了浙江的竹文化特色。图 a 给出了其部分墙面采用的微孔竹板装饰的局部放大照片，该装饰同时又实现了对声波的共振吸收. 竹板上有一系列不同面积、周期性排列的长方形微孔，声波进入微孔后导致微孔中的空气柱做简谐振动. 单个微孔和竹板后的空气层, 可简化成一个亥姆霍兹共振器, 如图 b 所示. 假设微孔深度均为 1 、单个微孔后的空气腔体体积均为 $V_{0}$ 、微孔横截面积记为 $\\mathrm{S}$. 声波在空气层中传播可视为绝热过程, 声波传播速度 $v_{s}$ 与空气密度 $\\rho$ 及体积弹性模量 $\\kappa$ 的关系为\n\n$v_{s}=\\sqrt{\\frac{\\kappa}{\\rho}}$\n\n其中 $\\kappa$ 是气体压强的增加量 $\\Delta p$ 与其体积 $\\mathrm{V}$ 相对变化量之比\n\n$\\kappa=-\\frac{\\Delta p}{\\Delta V / V}=-V \\frac{\\Delta p}{\\Delta V}$\n\n已知标准状态 $\\left(273 \\mathrm{~K}, \\quad l \\mathrm{~atm}=1.01 \\times 10^{5} \\mathrm{~Pa}\\right)$ 下空气 (可视为理想气体) 的摩尔质量 $\\mathrm{M}_{\\mathrm{mol}}=29.0 \\mathrm{~g} / \\mathrm{mol}$, 热容比\n$\\gamma=\\frac{7}{5}$, 气体普适常量 $\\mathrm{R}=8.31 \\mathrm{~J} /(\\mathrm{K} \\cdot \\mathrm{mol})$.\n\n[图1]\n\n图 a. 微孔竹板墙照片 (局部)\n\n[图2]\n\n图 b. 微孔竹板装置简化图及对应的亥㛂霍兹共振器模型为了吸收频率分别为 $120 \\mathrm{~Hz}$ 和 $200 \\mathrm{~Hz}$ 的声波, 相应的两种微孔横截面积之比应为多少?", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n2016 年 9 月，G20 峰会在杭州隆重召开，其会议厅的装饰设计既展示出中国建筑的节能环保理念，又体现了浙江的竹文化特色。图 a 给出了其部分墙面采用的微孔竹板装饰的局部放大照片，该装饰同时又实现了对声波的共振吸收. 竹板上有一系列不同面积、周期性排列的长方形微孔，声波进入微孔后导致微孔中的空气柱做简谐振动. 单个微孔和竹板后的空气层, 可简化成一个亥姆霍兹共振器, 如图 b 所示. 假设微孔深度均为 1 、单个微孔后的空气腔体体积均为 $V_{0}$ 、微孔横截面积记为 $\\mathrm{S}$. 声波在空气层中传播可视为绝热过程, 声波传播速度 $v_{s}$ 与空气密度 $\\rho$ 及体积弹性模量 $\\kappa$ 的关系为\n\n$v_{s}=\\sqrt{\\frac{\\kappa}{\\rho}}$\n\n其中 $\\kappa$ 是气体压强的增加量 $\\Delta p$ 与其体积 $\\mathrm{V}$ 相对变化量之比\n\n$\\kappa=-\\frac{\\Delta p}{\\Delta V / V}=-V \\frac{\\Delta p}{\\Delta V}$\n\n已知标准状态 $\\left(273 \\mathrm{~K}, \\quad l \\mathrm{~atm}=1.01 \\times 10^{5} \\mathrm{~Pa}\\right)$ 下空气 (可视为理想气体) 的摩尔质量 $\\mathrm{M}_{\\mathrm{mol}}=29.0 \\mathrm{~g} / \\mathrm{mol}$, 热容比\n$\\gamma=\\frac{7}{5}$, 气体普适常量 $\\mathrm{R}=8.31 \\mathrm{~J} /(\\mathrm{K} \\cdot \\mathrm{mol})$.\n\n[图1]\n\n图 a. 微孔竹板墙照片 (局部)\n\n[图2]\n\n图 b. 微孔竹板装置简化图及对应的亥㛂霍兹共振器模型\n\n问题:\n为了吸收频率分别为 $120 \\mathrm{~Hz}$ 和 $200 \\mathrm{~Hz}$ 的声波, 相应的两种微孔横截面积之比应为多少?\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-04.jpg?height=340&width=408&top_left_y=635&top_left_x=341", "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-04.jpg?height=325&width=762&top_left_y=637&top_left_x=841" ], "answer": null, "solution": null, "answer_type": "NV", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_462", "problem": "Beloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nHollow pipe is used instead of solid conductors in order to allow for cooling of the magnet. If the resistivity of copper is $\\rho=1.7 \\times 10^{-8} \\Omega \\cdot \\mathrm{m}$, determine the electrical resistance of one spiral.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is a numerical value.\n\nproblem:\nBeloit College has a \"homemade\" $500 \\mathrm{kV}$ VanDeGraff proton accelerator, designed and constructed by the students and faculty.\n[figure1]\n\nAccelerator dome (assume it is a sphere); accelerating column; bending electromagnet\n\nThe accelerator dome, an aluminum sphere of radius $a=0.50$ meters, is charged by a rubber belt with width $w=10 \\mathrm{~cm}$ that moves with speed $v_{b}=20 \\mathrm{~m} / \\mathrm{s}$. The accelerating column consists of 20 metal rings separated by glass rings; the rings are connected in series with $500 \\mathrm{M} \\Omega$ resistors. The proton beam has a current of $25 \\mu \\mathrm{A}$ and is accelerated through $500 \\mathrm{kV}$ and then passes through a tuning electromagnet. The electromagnet consists of wound copper pipe as a conductor. The electromagnet effectively creates a uniform field $B$ inside a circular region of radius $b=10 \\mathrm{~cm}$ and zero outside that region.\n\n[figure2]\n\nOnly six of the 20 metals rings and resistors are shown in the figure. The fuzzy grey path is the path taken by the protons as they are accelerated from the dome, through the electromagnet, into the target.\n\nHollow pipe is used instead of solid conductors in order to allow for cooling of the magnet. If the resistivity of copper is $\\rho=1.7 \\times 10^{-8} \\Omega \\cdot \\mathrm{m}$, determine the electrical resistance of one spiral.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nRemember, your answer should be calculated in the unit of $\\Omega$, but when concluding your final answer, do not include the unit.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is the numerical value without any units.", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=468&width=1592&top_left_y=438&top_left_x=259", "https://cdn.mathpix.com/cropped/2024_03_06_8c5c0b6efdeaae46cc88g-19.jpg?height=493&width=1268&top_left_y=1339&top_left_x=426" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\Omega$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_151", "problem": "A ball of mass $m$ moving at speed $v$ collides with a massless spring of spring constant $k$ mounted on a stationary box of mass $M$ in free space. No mechanical energy is lost in the collision. If the system does not rotate, what is the maximum compression $x$ of the spring?\nA: $x=v \\sqrt{\\frac{m M}{(m+M) k}} $ \nB: $x=v \\sqrt{\\frac{m}{k}}$\nC: $x=v \\sqrt{\\frac{M}{k}}$\nD: $x=v \\sqrt{\\frac{m+M}{k}}$\nE: $x=v \\sqrt{\\frac{(m+M)^{3}}{m M k}}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA ball of mass $m$ moving at speed $v$ collides with a massless spring of spring constant $k$ mounted on a stationary box of mass $M$ in free space. No mechanical energy is lost in the collision. If the system does not rotate, what is the maximum compression $x$ of the spring?\n\nA: $x=v \\sqrt{\\frac{m M}{(m+M) k}} $ \nB: $x=v \\sqrt{\\frac{m}{k}}$\nC: $x=v \\sqrt{\\frac{M}{k}}$\nD: $x=v \\sqrt{\\frac{m+M}{k}}$\nE: $x=v \\sqrt{\\frac{(m+M)^{3}}{m M k}}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_441", "problem": "In the future, as the miniaturisation of microprocessors continues, an obvious limit to the technology is when the size of one active unit in a device approaches the size of an atom. We will explore, speculatively, an aspect of passing a current into and out of the structure of graphene.\n\nThe following Figures represent part of an infinite graphene layer for which we wish to find the measured resistance between a pair of atoms located at points $\\mathbf{A}$ and $\\mathbf{B}$. Each straight line (bond) between two nodes represents a resistance $R$.\n\nFigure: The single layer hexagonal structure of graphene.\n(a)\n[figure1]\n\n(b)\n[figure2]\n\n(c)\n[figure3]\n\nNow a current $I$ flows through the contact wires connected so that $I$ flows in at $\\mathbf{A}$ and out at B, as in Fig. 6(c). The principle of [linear] superposition (which will be more familiar to you in your studies of the interference of waves) can also be applied to the currents you have postulated. State the total current through the bond AB.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nIn the future, as the miniaturisation of microprocessors continues, an obvious limit to the technology is when the size of one active unit in a device approaches the size of an atom. We will explore, speculatively, an aspect of passing a current into and out of the structure of graphene.\n\nThe following Figures represent part of an infinite graphene layer for which we wish to find the measured resistance between a pair of atoms located at points $\\mathbf{A}$ and $\\mathbf{B}$. Each straight line (bond) between two nodes represents a resistance $R$.\n\nFigure: The single layer hexagonal structure of graphene.\n(a)\n[figure1]\n\n(b)\n[figure2]\n\n(c)\n[figure3]\n\nNow a current $I$ flows through the contact wires connected so that $I$ flows in at $\\mathbf{A}$ and out at B, as in Fig. 6(c). The principle of [linear] superposition (which will be more familiar to you in your studies of the interference of waves) can also be applied to the currents you have postulated. State the total current through the bond AB.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-7.jpg?height=360&width=366&top_left_y=188&top_left_x=434", "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-7.jpg?height=363&width=366&top_left_y=187&top_left_x=868", "https://cdn.mathpix.com/cropped/2024_03_06_22bb4cfada5ea54fde10g-7.jpg?height=365&width=349&top_left_y=183&top_left_x=1276" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1733", "problem": "如图, 一带正电荷 $Q$ 的绝缘小球（可视为点电荷）固定在光滑绝缘平板上，另一绝缘小球\n\n[图1]\n\n(可视为点电荷) 所带电荷用 $q$ (其值可任意选择) 表示, 可在平板上移动, 并连在轻弹簧的一端, 轻弹簧的另一端连在固定挡板上; 两小球的球心在弹簧的轴线上。不考虑可移动小球与固定小球相互接触的情形, 且弹簧的形变处于弹性限度内。关于可移动小球的平衡位置, 下列说法正确的是\nA: 若 $q>0$, 总有一个平衡的位置\nB: 若 $q>0$, 没有平衡位置\nC: 若 $q<0$, 可能有一个或两个平衡位置\nD: 若 $q<0$, 没有平衡位置\n", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这是一个多选题（有多个正确答案）。\n\n问题:\n如图, 一带正电荷 $Q$ 的绝缘小球（可视为点电荷）固定在光滑绝缘平板上，另一绝缘小球\n\n[图1]\n\n(可视为点电荷) 所带电荷用 $q$ (其值可任意选择) 表示, 可在平板上移动, 并连在轻弹簧的一端, 轻弹簧的另一端连在固定挡板上; 两小球的球心在弹簧的轴线上。不考虑可移动小球与固定小球相互接触的情形, 且弹簧的形变处于弹性限度内。关于可移动小球的平衡位置, 下列说法正确的是\n\nA: 若 $q>0$, 总有一个平衡的位置\nB: 若 $q>0$, 没有平衡位置\nC: 若 $q<0$, 可能有一个或两个平衡位置\nD: 若 $q<0$, 没有平衡位置\n\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER应为两个或更多的选项：[A, B, C, D]", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_17b1131fe8d911867aa0g-02.jpg?height=140&width=777&top_left_y=501&top_left_x=1071" ], "answer": null, "solution": null, "answer_type": "MC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_117", "problem": "An object of mass $m_{1}$ initially moving at speed $v_{0}$ collides with an originally stationary object of mass $m_{2}=\\alpha m_{1}$, where $\\alpha<1$. The collision could be completely elastic, completely inelastic, or partially inelastic. After the collision the two objects move at speeds $v_{1}$ and $v_{2}$. Assume that the collision is one dimensional, and that object one cannot pass through object two.\n\nAfter the collision, the speed ratio $r_{2}=v_{2} / v_{0}$ of object 2 is bounded by\nA: $(1-\\alpha) /(1+\\alpha) \\leq r_{2} \\leq 1$\nB: $(1-\\alpha) /(1+\\alpha) \\leq r_{2} \\leq 1 /(1+\\alpha)$\nC: $\\alpha /(1+\\alpha) \\leq r_{2} \\leq 1$\nD: $0 \\leq r_{2} \\leq 2 \\alpha /(1+\\alpha)$\nE: $1 /(1+\\alpha) \\leq r_{2} \\leq 2 /(1+\\alpha) $ \n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nAn object of mass $m_{1}$ initially moving at speed $v_{0}$ collides with an originally stationary object of mass $m_{2}=\\alpha m_{1}$, where $\\alpha<1$. The collision could be completely elastic, completely inelastic, or partially inelastic. After the collision the two objects move at speeds $v_{1}$ and $v_{2}$. Assume that the collision is one dimensional, and that object one cannot pass through object two.\n\nAfter the collision, the speed ratio $r_{2}=v_{2} / v_{0}$ of object 2 is bounded by\n\nA: $(1-\\alpha) /(1+\\alpha) \\leq r_{2} \\leq 1$\nB: $(1-\\alpha) /(1+\\alpha) \\leq r_{2} \\leq 1 /(1+\\alpha)$\nC: $\\alpha /(1+\\alpha) \\leq r_{2} \\leq 1$\nD: $0 \\leq r_{2} \\leq 2 \\alpha /(1+\\alpha)$\nE: $1 /(1+\\alpha) \\leq r_{2} \\leq 2 /(1+\\alpha) $ \n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1690", "problem": "Z-籍缩作为惯性约束核聚变的一种可能方式, 近年来受到特别重视, 其原理如图所示. 图中, 长 $20 \\mathrm{~mm}$ 、直径为 $5 \\mu \\mathrm{m}$ 的铇丝组成的两个共轴的圆柱面阵列, 殠间通以超强电流,钨丝阵列在安培力的作用下以极大的加速度向内运动, 即所谓自笣缩效应; 铇丝的巨大动量转移到处于阵列中心的直径为豪米量级的氛氛靶球上, 可以使靯球压缩后达到高温高密度状态, 实现核聚变. 设内圈有 $N$ 根钨丝 (可视为长直导线) 均匀地分布在半径为 $r$ 的圆周上, 通有总电流 $I_{\\text {内 }}=2 \\times 10^{7} \\mathrm{~A}$; 外圈有 $M$ 根钨丝, 均匀地分布在半径为 $R$ 的圆周上, 每根钨丝所通过的电流同内圈铇丝. 已知通有电流 $i$ 的长直导线在距其 $r$ 处产生的磁感应强度大小为 $k_{m} \\frac{i}{r}$, 式中比例常量 $k_{\\mathrm{m}}=2 \\times 10^{-7} \\mathrm{~T} \\cdot \\mathrm{m} / \\mathrm{A}=2 \\times 10^{-7} \\mathrm{~N} / \\mathrm{A}^{2}$.\n\n[图1]若不考虑外圈铇丝, 计算内圈某一根通电钨丝中间长为 $\\Delta L$ 的一小段铇丝所受到的安培力", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\nZ-籍缩作为惯性约束核聚变的一种可能方式, 近年来受到特别重视, 其原理如图所示. 图中, 长 $20 \\mathrm{~mm}$ 、直径为 $5 \\mu \\mathrm{m}$ 的铇丝组成的两个共轴的圆柱面阵列, 殠间通以超强电流,钨丝阵列在安培力的作用下以极大的加速度向内运动, 即所谓自笣缩效应; 铇丝的巨大动量转移到处于阵列中心的直径为豪米量级的氛氛靶球上, 可以使靯球压缩后达到高温高密度状态, 实现核聚变. 设内圈有 $N$ 根钨丝 (可视为长直导线) 均匀地分布在半径为 $r$ 的圆周上, 通有总电流 $I_{\\text {内 }}=2 \\times 10^{7} \\mathrm{~A}$; 外圈有 $M$ 根钨丝, 均匀地分布在半径为 $R$ 的圆周上, 每根钨丝所通过的电流同内圈铇丝. 已知通有电流 $i$ 的长直导线在距其 $r$ 处产生的磁感应强度大小为 $k_{m} \\frac{i}{r}$, 式中比例常量 $k_{\\mathrm{m}}=2 \\times 10^{-7} \\mathrm{~T} \\cdot \\mathrm{m} / \\mathrm{A}=2 \\times 10^{-7} \\mathrm{~N} / \\mathrm{A}^{2}$.\n\n[图1]\n\n问题:\n若不考虑外圈铇丝, 计算内圈某一根通电钨丝中间长为 $\\Delta L$ 的一小段铇丝所受到的安培力\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_ea729d4659bcaa2c4b91g-04.jpg?height=677&width=1188&top_left_y=632&top_left_x=454" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_871", "problem": "An electron with charge $-e(e>0)$ experiences the Lorentz force due to the perpendicular magnetic field and the electric force\n\n$$\nm^{*} \\frac{d \\vec{v}}{d t}=-e(\\vec{v} \\times \\vec{B}+\\vec{E})\n$$\n\nwhere $\\vec{v}$ is the velocity of the electron.The fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nDetermine the velocity $v_{\\mathrm{s}}$ of the electrons in the stationary case.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\nHere is some context information for this question, which might assist you in solving it:\nAn electron with charge $-e(e>0)$ experiences the Lorentz force due to the perpendicular magnetic field and the electric force\n\n$$\nm^{*} \\frac{d \\vec{v}}{d t}=-e(\\vec{v} \\times \\vec{B}+\\vec{E})\n$$\n\nwhere $\\vec{v}$ is the velocity of the electron.\n\nproblem:\nThe fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and H. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.\n\nIn a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.\n\nDetermine the velocity $v_{\\mathrm{s}}$ of the electrons in the stationary case.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_834", "problem": "When modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nIf there is a charge $\\Delta q$ on a length $\\Delta x$ of the inner core of the co-axial cable, and the outer shield is grounded, find the electric field in the region between the inner core and the shield.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\n\nproblem:\nWhen modelling DC or low frequency signals, one often assumes that a voltage pulse travels instantaneously throughout the circuit. This assumption is valid when the wavelength of such signals is much longer than the size of the circuit, however when working with radio frequency signals, the dynamics are more complex, and we need to account for the intrinsic capacitance and inductance of our cables in our model. We model a co-axial transmission line which acts as a waveguide as described below, ignoring the small resistance of the copper and the small conductance through the dielectric. Throughout the problem, we consider the large-wavelength limit of electromagnetic waves in the co-axial cable such that electric and magnetic fields are perpendicular to the axis of the cable everywhere (the so-called transverse electromagnetic mode).\n[figure1]\n\nDiagram of a coaxial cable showing C - the centre core, I - the dielectric insulator, S - the metallic shield and J - the plastic jacket.\n\nConsider a co-axial cable consisting of a copper inner core of negligible resistance, negligible magnetic permeability and radius $a$, covered by an outer co-axial copper shield with inner radius $b$. A dielectric of dimensionless relative permittivity $\\varepsilon_{\\mathrm{r}}$ and dimensionless relative permeability $\\mu_{\\mathrm{r}}$ separates the layers. When electromagnetic signals propagate through the co-axial cable, they are confined between the inner core and outer shielding.\n\nIf there is a charge $\\Delta q$ on a length $\\Delta x$ of the inner core of the co-axial cable, and the outer shield is grounded, find the electric field in the region between the inner core and the shield.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_c16db445016d4c0e9a0ag-1.jpg?height=358&width=844&top_left_y=1688&top_left_x=617" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_177", "problem": "A ball rolls from the back of a large truck traveling $10.0 \\mathrm{~m} / \\mathrm{s}$ to the right. The ball is traveling horizontally at $8.0 \\mathrm{~m} / \\mathrm{s}$ to the left relative to an observer in the truck. The ball lands on the roadway $1.25 \\mathrm{~m}$ below its starting level. How far behind the truck does it land?\nA: $0.50 \\mathrm{~m}$\nB: $1.0 \\mathrm{~m}$\nC: $4.0 \\mathrm{~m} $ \nD: $5.0 \\mathrm{~m}$\nE: $9.0 \\mathrm{~m}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nA ball rolls from the back of a large truck traveling $10.0 \\mathrm{~m} / \\mathrm{s}$ to the right. The ball is traveling horizontally at $8.0 \\mathrm{~m} / \\mathrm{s}$ to the left relative to an observer in the truck. The ball lands on the roadway $1.25 \\mathrm{~m}$ below its starting level. How far behind the truck does it land?\n\nA: $0.50 \\mathrm{~m}$\nB: $1.0 \\mathrm{~m}$\nC: $4.0 \\mathrm{~m} $ \nD: $5.0 \\mathrm{~m}$\nE: $9.0 \\mathrm{~m}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_468", "problem": "In this problem, we will investigate a simple thermodynamic model for the conversion of solar energy into wind. Consider a planet of radius $R$, and assume that it rotates so that the same side always faces the Sun. The bright side facing the Sun has a constant uniform temperature $T_{1}$, while the dark side has a constant uniform temperature $T_{2}$. The orbit radius of the planet is $R_{0}$, the Sun has temperature $T_{s}$, and the radius of the Sun is $R_{s}$. Assume that outer space has zero temperature, and treat all objects as ideal blackbodies.\n\nIn order to keep both $T_{1}$ and $T_{2}$ constant, heat must be continually transferred from the bright side to the dark side. By viewing the two hemispheres as the two reservoirs of a reversible heat engine, work can be performed from this temperature difference, which appears in the form of wind power. For simplicity, we assume all of this power is immediately captured and stored by windmills.\n\nThe equilibrium temperature ratio $T_{2} / T_{1}$ depends on the heat transfer rate between the hemispheres. Find the minimum and maximum possible values of $T_{2} / T_{1}$. In each case, what is the wind power $P_{w}$ produced?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis question involves multiple quantities to be determined.\n\nproblem:\nIn this problem, we will investigate a simple thermodynamic model for the conversion of solar energy into wind. Consider a planet of radius $R$, and assume that it rotates so that the same side always faces the Sun. The bright side facing the Sun has a constant uniform temperature $T_{1}$, while the dark side has a constant uniform temperature $T_{2}$. The orbit radius of the planet is $R_{0}$, the Sun has temperature $T_{s}$, and the radius of the Sun is $R_{s}$. Assume that outer space has zero temperature, and treat all objects as ideal blackbodies.\n\nIn order to keep both $T_{1}$ and $T_{2}$ constant, heat must be continually transferred from the bright side to the dark side. By viewing the two hemispheres as the two reservoirs of a reversible heat engine, work can be performed from this temperature difference, which appears in the form of wind power. For simplicity, we assume all of this power is immediately captured and stored by windmills.\n\nThe equilibrium temperature ratio $T_{2} / T_{1}$ depends on the heat transfer rate between the hemispheres. Find the minimum and maximum possible values of $T_{2} / T_{1}$. In each case, what is the wind power $P_{w}$ produced?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nYour final quantities should be output in the following order: [the minimum possible values of $T_{2} / T_{1}$, the maximum possible values of $T_{2} / T_{1}$, the wind power $P_{w}$].\nTheir answer types are, in order, [numerical value, numerical value, numerical value].\nPlease end your response with: \"The final answers are \\boxed{ANSWER}\", where ANSWER should be the sequence of your final answers, separated by commas, for example: 5, 7, 2.5", "figure_urls": null, "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null, null ], "answer_sequence": [ "the minimum possible values of $T_{2} / T_{1}$", "the maximum possible values of $T_{2} / T_{1}$", "the wind power $P_{w}$" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_945", "problem": "Energies in particle accelerators are measured in $\\mathrm{eV}$. What is the kinetic energy to an order of magnitude, in $\\mathrm{eV}$, of a snail of mass $1 \\mathrm{~g}$ which crawls along at a rate of $1 \\mathrm{~cm}$ in $10 \\mathrm{~s}$ ?\nA: $1 \\mathrm{eV}$\nB: $1 \\mathrm{keV}$\nC: $1 \\mathrm{MeV}$\nD: $1 \\mathrm{GeV}$\nE: $1 \\mathrm{TeV}$\n", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThis is a multiple choice question (only one correct answer).\n\nproblem:\nEnergies in particle accelerators are measured in $\\mathrm{eV}$. What is the kinetic energy to an order of magnitude, in $\\mathrm{eV}$, of a snail of mass $1 \\mathrm{~g}$ which crawls along at a rate of $1 \\mathrm{~cm}$ in $10 \\mathrm{~s}$ ?\n\nA: $1 \\mathrm{eV}$\nB: $1 \\mathrm{keV}$\nC: $1 \\mathrm{MeV}$\nD: $1 \\mathrm{GeV}$\nE: $1 \\mathrm{TeV}$\n\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER should be one of the options: [A, B, C, D, E].", "figure_urls": null, "answer": null, "solution": null, "answer_type": "SC", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_598", "problem": "Two masses $m$ separated by a distance $l$ are given initial velocities $v_{0}$ as shown in the diagram. The masses interact only through universal gravitation.\n\n[figure1]\n\nUnder what conditions will the masses eventually collide?", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an equation.\n\nproblem:\nTwo masses $m$ separated by a distance $l$ are given initial velocities $v_{0}$ as shown in the diagram. The masses interact only through universal gravitation.\n\n[figure1]\n\nUnder what conditions will the masses eventually collide?\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an equation, e.g. ANSWER=\\frac{x^2}{4}+\\frac{y^2}{2}=1", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_14_05f4d0107dd82de1849ag-09.jpg?height=613&width=791&top_left_y=436&top_left_x=662" ], "answer": null, "solution": null, "answer_type": "EQ", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "multi-modal" }, { "id": "Physics_1612", "problem": "真空中平行板电容器两极板的面积均为 $S$, 相距 $d$, 上、下极\n\n板所带电量分别为 $Q$ 和 $-Q(Q>0)$ 。现将一厚度为 $t$ 、面积为 $S / 2$\n\n（宽度和原来的极板相同, 长度是原来极板的一半）的金属片在上极板的正下方平行插入电容器, 将电容器分成如图所示的 $1 、 2 、 3$ 三部分。不考虑边缘效应。静电力常量为 $k$ 。试求\n\n\n[图1]金属片上表面所带电量;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n真空中平行板电容器两极板的面积均为 $S$, 相距 $d$, 上、下极\n\n板所带电量分别为 $Q$ 和 $-Q(Q>0)$ 。现将一厚度为 $t$ 、面积为 $S / 2$\n\n（宽度和原来的极板相同, 长度是原来极板的一半）的金属片在上极板的正下方平行插入电容器, 将电容器分成如图所示的 $1 、 2 、 3$ 三部分。不考虑边缘效应。静电力常量为 $k$ 。试求\n\n\n[图1]\n\n问题:\n金属片上表面所带电量;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[极板1上的电荷量, 极板3上的电荷量]\n它们的答案类型依次是[表达式, 表达式]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://i.postimg.cc/c4YQHfdj/2016-CPho-Q13.png" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ null, null ], "answer_sequence": [ "极板1上的电荷量", "极板3上的电荷量" ], "type_sequence": [ "EX", "EX" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1451", "problem": "如图, 左边试管由一段 $24 \\mathrm{~cm}$ 长的水银柱封住一段高为 $60 \\mathrm{~cm}$ 、温度为 $300 \\mathrm{~K}$ 的理想气体柱，上水银面与管侧面小孔相距 $16 \\mathrm{~cm}$ ，小孔右边用一软管连接一空的试管。一控温系统可持续升高或降低被封住的气体柱的温度,当气体温度升高到一定值时水银会从左边试管通过小孔溢出到右边试管中。左边试管坚直放置，右边试管可上下移动，上移时可使右边试管中的水银回流到左边试管内，从而控制左边试管中水银柱的高度。大气压强为 $76 \\mathrm{cmHg}$ 。\n\n[图1]已知被封住的气体处在温度为 $384 \\mathrm{~K}$ 的平衡状态, 求左边试管中水银柱可能的高度。", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 左边试管由一段 $24 \\mathrm{~cm}$ 长的水银柱封住一段高为 $60 \\mathrm{~cm}$ 、温度为 $300 \\mathrm{~K}$ 的理想气体柱，上水银面与管侧面小孔相距 $16 \\mathrm{~cm}$ ，小孔右边用一软管连接一空的试管。一控温系统可持续升高或降低被封住的气体柱的温度,当气体温度升高到一定值时水银会从左边试管通过小孔溢出到右边试管中。左边试管坚直放置，右边试管可上下移动，上移时可使右边试管中的水银回流到左边试管内，从而控制左边试管中水银柱的高度。大气压强为 $76 \\mathrm{cmHg}$ 。\n\n[图1]\n\n问题:\n已知被封住的气体处在温度为 $384 \\mathrm{~K}$ 的平衡状态, 求左边试管中水银柱可能的高度。\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~cm}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-05.jpg?height=545&width=488&top_left_y=1201&top_left_x=1521" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~cm}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1384", "problem": "某电磁轨道炮的简化模型如图 a 所示, 两圆柱形固定导轨相互平行, 其对称轴所在平面与水平面的夹角为 $\\theta$, 两导轨的长均为 $\\mathrm{L}$ 、半径均为 $\\mathrm{b}$ 、每单位长度的电阻均为 $\\lambda$, 两导轨之间的最近距离为 $\\mathrm{d}(\\mathrm{d}$很小). 一弹丸质量为 $\\mathrm{m}$ ( $\\mathrm{m}$ 较小) 的金属弹丸 (可视为薄片) 置于两导轨之间, 弹丸直径为 $\\mathrm{d}$ 、电阻为 $\\mathrm{R}$, 与导轨保持良好接触. 两导轨下端横截面共面, 下端 (通过两根与相应导轨同轴的、较长的硬导线) 与一电流为 I 的理想恒流源 (恒流源内部的能量损耗可不计) 相连, 不考虑空气阻力和摩擦阻力, 重力加速度大小图 a. 某电磁轨道炮的简化模型为 $g$, 真空磁导率为 $\\mu_{0}$. 考虑一弹丸自导轨下端从静止开始被磁场加速直至射出的过程.\n\n[图1]\n\n图 a. 某电磁轨道炮的简化模型求在弹丸的整个加速过程中理想恒流源所做的功:", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个表达式。\n这里是一些可能会帮助你解决问题的先验信息提示：\n某电磁轨道炮的简化模型如图 a 所示, 两圆柱形固定导轨相互平行, 其对称轴所在平面与水平面的夹角为 $\\theta$, 两导轨的长均为 $\\mathrm{L}$ 、半径均为 $\\mathrm{b}$ 、每单位长度的电阻均为 $\\lambda$, 两导轨之间的最近距离为 $\\mathrm{d}(\\mathrm{d}$很小). 一弹丸质量为 $\\mathrm{m}$ ( $\\mathrm{m}$ 较小) 的金属弹丸 (可视为薄片) 置于两导轨之间, 弹丸直径为 $\\mathrm{d}$ 、电阻为 $\\mathrm{R}$, 与导轨保持良好接触. 两导轨下端横截面共面, 下端 (通过两根与相应导轨同轴的、较长的硬导线) 与一电流为 I 的理想恒流源 (恒流源内部的能量损耗可不计) 相连, 不考虑空气阻力和摩擦阻力, 重力加速度大小图 a. 某电磁轨道炮的简化模型为 $g$, 真空磁导率为 $\\mu_{0}$. 考虑一弹丸自导轨下端从静止开始被磁场加速直至射出的过程.\n\n[图1]\n\n图 a. 某电磁轨道炮的简化模型\n\n问题:\n求在弹丸的整个加速过程中理想恒流源所做的功:\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是一个不含等号的表达式，例如ANSWER=\\frac{1}{2} g t^2", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_0207e542b9f7c87dc04dg-03.jpg?height=539&width=808&top_left_y=250&top_left_x=521" ], "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_567", "problem": "The intensity of solar radiation at the surface of the sun is $\\sigma T_{s}^{4}$, so the intensity at the planet's orbit radius is\n\n$$\nI=\\sigma T_{s}^{4} \\frac{R_{s}^{2}}{R_{0}^{2}}\n$$\n\nThe area subtended by the planet is $\\pi R^{2}$, so\n\n$$\nP=\\pi \\sigma T_{s}^{4} \\frac{R^{2} R_{s}^{2}}{R_{0}^{2}}\n$$\n\nLet heat be transferred from the bright side at a rate $Q_{1}$ and transferred to the dark side at a rate $Q_{2}$. Then by conservation of energy,\n\n$$\nQ_{1}=P_{w}+Q_{2} .\n$$\n\nSince the two hemispheres have constant temperatures, energy balance for each gives\n\n$$\nP=Q_{1}+\\left(2 \\pi R^{2} \\sigma\\right) T_{1}^{4}, \\quad Q_{2}=A T_{2}^{4}\n$$\n\nFinally, since the engine is reversible,\n\n$$\n\\frac{Q_{1}}{T_{1}}=\\frac{Q_{2}}{T_{2}}\n$$\n\nBy combining the first three equations, and defining $x=T_{2} / T_{1}$, we have\n\n$$\nP_{w}=Q_{1}-Q_{2}=P-\\left(2 \\pi R^{2} \\sigma\\right)\\left(T_{1}^{4}+T_{2}^{4}\\right)=P-\\left(2 \\pi R^{2} \\sigma\\right) T_{1}^{4}\\left(1+x^{4}\\right) .\n$$\n\nThis is not yet in terms of $P$ and $x$, so now we use the reversibility condition,\n\n$$\n\\frac{P-A T_{1}^{4}}{T_{1}}=\\frac{A T_{2}^{4}}{T_{2}}\n$$\n\nwhich simplifies to\n\n$$\nP=\\left(2 \\pi R^{2} \\sigma\\right)\\left(T_{1}^{4}+T_{1} T_{2}^{3}\\right)=\\left(2 \\pi R^{2} \\sigma\\right) T_{1}^{4}\\left(1+x^{3}\\right)\n$$\n\nPlugging this in above, we find\n\n$$\nP_{w}=P-\\frac{P}{1+x^{3}}\\left(1+x^{4}\\right)=\\frac{x^{3}(1-x)}{1+x^{3}} P\n$$In this problem, we will investigate a simple thermodynamic model for the conversion of solar energy into wind. Consider a planet of radius $R$, and assume that it rotates so that the same side always faces the Sun. The bright side facing the Sun has a constant uniform temperature $T_{1}$, while the dark side has a constant uniform temperature $T_{2}$. The orbit radius of the planet is $R_{0}$, the Sun has temperature $T_{s}$, and the radius of the Sun is $R_{s}$. Assume that outer space has zero temperature, and treat all objects as ideal blackbodies.\n\nIn order to keep both $T_{1}$ and $T_{2}$ constant, heat must be continually transferred from the bright side to the dark side. By viewing the two hemispheres as the two reservoirs of a reversible heat engine, work can be performed from this temperature difference, which appears in the form of wind power. For simplicity, we assume all of this power is immediately captured and stored by windmills.\n\nEstimate the maximum possible value of $P_{w}$ as a fraction of $P$, to one significant figure. Briefly explain how you obtained this estimate.", "prompt": "You are participating in an international Physics competition and need to solve the following question.\nThe answer to this question is an expression.\nHere is some context information for this question, which might assist you in solving it:\nThe intensity of solar radiation at the surface of the sun is $\\sigma T_{s}^{4}$, so the intensity at the planet's orbit radius is\n\n$$\nI=\\sigma T_{s}^{4} \\frac{R_{s}^{2}}{R_{0}^{2}}\n$$\n\nThe area subtended by the planet is $\\pi R^{2}$, so\n\n$$\nP=\\pi \\sigma T_{s}^{4} \\frac{R^{2} R_{s}^{2}}{R_{0}^{2}}\n$$\n\nLet heat be transferred from the bright side at a rate $Q_{1}$ and transferred to the dark side at a rate $Q_{2}$. Then by conservation of energy,\n\n$$\nQ_{1}=P_{w}+Q_{2} .\n$$\n\nSince the two hemispheres have constant temperatures, energy balance for each gives\n\n$$\nP=Q_{1}+\\left(2 \\pi R^{2} \\sigma\\right) T_{1}^{4}, \\quad Q_{2}=A T_{2}^{4}\n$$\n\nFinally, since the engine is reversible,\n\n$$\n\\frac{Q_{1}}{T_{1}}=\\frac{Q_{2}}{T_{2}}\n$$\n\nBy combining the first three equations, and defining $x=T_{2} / T_{1}$, we have\n\n$$\nP_{w}=Q_{1}-Q_{2}=P-\\left(2 \\pi R^{2} \\sigma\\right)\\left(T_{1}^{4}+T_{2}^{4}\\right)=P-\\left(2 \\pi R^{2} \\sigma\\right) T_{1}^{4}\\left(1+x^{4}\\right) .\n$$\n\nThis is not yet in terms of $P$ and $x$, so now we use the reversibility condition,\n\n$$\n\\frac{P-A T_{1}^{4}}{T_{1}}=\\frac{A T_{2}^{4}}{T_{2}}\n$$\n\nwhich simplifies to\n\n$$\nP=\\left(2 \\pi R^{2} \\sigma\\right)\\left(T_{1}^{4}+T_{1} T_{2}^{3}\\right)=\\left(2 \\pi R^{2} \\sigma\\right) T_{1}^{4}\\left(1+x^{3}\\right)\n$$\n\nPlugging this in above, we find\n\n$$\nP_{w}=P-\\frac{P}{1+x^{3}}\\left(1+x^{4}\\right)=\\frac{x^{3}(1-x)}{1+x^{3}} P\n$$\n\nproblem:\nIn this problem, we will investigate a simple thermodynamic model for the conversion of solar energy into wind. Consider a planet of radius $R$, and assume that it rotates so that the same side always faces the Sun. The bright side facing the Sun has a constant uniform temperature $T_{1}$, while the dark side has a constant uniform temperature $T_{2}$. The orbit radius of the planet is $R_{0}$, the Sun has temperature $T_{s}$, and the radius of the Sun is $R_{s}$. Assume that outer space has zero temperature, and treat all objects as ideal blackbodies.\n\nIn order to keep both $T_{1}$ and $T_{2}$ constant, heat must be continually transferred from the bright side to the dark side. By viewing the two hemispheres as the two reservoirs of a reversible heat engine, work can be performed from this temperature difference, which appears in the form of wind power. For simplicity, we assume all of this power is immediately captured and stored by windmills.\n\nEstimate the maximum possible value of $P_{w}$ as a fraction of $P$, to one significant figure. Briefly explain how you obtained this estimate.\n\nAll mathematical formulas and symbols you output should be represented with LaTeX!\nYou can solve it step by step.\nPlease end your response with: \"The final answer is $\\boxed{ANSWER}$\", where ANSWER is an expression without equals signs, e.g. ANSWER=\\frac{1}{2} g t^2", "figure_urls": null, "answer": null, "solution": null, "answer_type": "EX", "unit": null, "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "EN", "modality": "text-only" }, { "id": "Physics_1471", "problem": "用波长为 $633 \\mathrm{~nm}$ 的激光水平照射坚直圆珠笔中的小弹簧, 在距离弹簧 $4.2 \\mathrm{~m}$ 的光屏 (与激光水平照射方向垂直) 上形成衍射图像, 如图 a 所示。其右图与 1952 年拍摄的首张 DNA 分子双螺旋结构 X 射线衍射图像（图 b）十分相似。\n\n[图1]\n\n图 b\n\n说明: 由光学原理可知, 弹簧上两段互成角度的细铁丝的衍射、干涉图像与两条成同样角度、相同宽度的狭缝的衍射、干涉图像一致。利用图 a 右图中给出的尺寸信息, 通过测量估算弹簧钢丝的直径 $d_{1}$ 、弹簧圈的半径 $R$和弹簧的螺距 $p$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题包含多个待求解的量。\n这里是一些可能会帮助你解决问题的先验信息提示：\n用波长为 $633 \\mathrm{~nm}$ 的激光水平照射坚直圆珠笔中的小弹簧, 在距离弹簧 $4.2 \\mathrm{~m}$ 的光屏 (与激光水平照射方向垂直) 上形成衍射图像, 如图 a 所示。其右图与 1952 年拍摄的首张 DNA 分子双螺旋结构 X 射线衍射图像（图 b）十分相似。\n\n[图1]\n\n图 b\n\n说明: 由光学原理可知, 弹簧上两段互成角度的细铁丝的衍射、干涉图像与两条成同样角度、相同宽度的狭缝的衍射、干涉图像一致。\n\n问题:\n利用图 a 右图中给出的尺寸信息, 通过测量估算弹簧钢丝的直径 $d_{1}$ 、弹簧圈的半径 $R$和弹簧的螺距 $p$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n你的最终解答的量应该按以下顺序输出：[估算弹簧钢丝的直径 $d_{1}$ , 估算弹簧圈的半径 $R$, 估算弹簧的螺距 $p$]\n它们的单位依次是[$\\mathrm{~mm}$, $\\mathrm{~mm}$, $\\mathrm{~mm}$]，但在你给出最终答案时不应包含单位。\n它们的答案类型依次是[数值, 数值, 数值]\n你需要在输出的最后用以下格式总结答案：“最终答案是\\boxed{ANSWER}”，其中ANSWER应为你的最终答案序列，用英文逗号分隔，例如：5, 7, 2.5", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-04.jpg?height=960&width=1445&top_left_y=1810&top_left_x=297", "https://cdn.mathpix.com/cropped/2024_03_31_e6ca7470b8c778c14f49g-34.jpg?height=302&width=242&top_left_y=1174&top_left_x=1478" ], "answer": null, "solution": null, "answer_type": "MPV", "unit": [ "$\\mathrm{~mm}$", "$\\mathrm{~mm}$", "$\\mathrm{~mm}$" ], "answer_sequence": [ "估算弹簧钢丝的直径 $d_{1}$ ", "估算弹簧圈的半径 $R$", "估算弹簧的螺距 $p$" ], "type_sequence": [ "NV", "NV", "NV" ], "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" }, { "id": "Physics_1442", "problem": "如图, 两根内径相同的绝缘细管 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 连接成倒 $\\mathrm{V}$ 字形, 坚直放置,连接点 $\\mathrm{B}$ 处可视为一段很短的圆弧; 两管长度均为 $l=2.0 \\mathrm{~m}$, 倾角 $\\alpha=37^{\\circ}$,处于方向坚直向下的匀强电场中, 场强大小 $E=10000 \\mathrm{~V} / \\mathrm{m}$ 。一质量 $m=1.0 \\times 10^{-4} \\mathrm{~kg}$ 、带电量 $-q=-2.0 \\times 10^{-7} \\mathrm{C}$ 的小球 (小球直径比细管内径稍小,可视为质点), 从 $\\mathrm{A}$ 点由静止开始在管内运动, 小球与 $\\mathrm{AB}$ 管壁间的动摩擦因数为 $\\mu_{1}=0.50$,小球与 $\\mathrm{BC}$ 管壁间的动摩擦因数为 $\\mu_{2}=0.25$ 。小球在运动过程中带电量保持不变。已知重力加速度大小 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}, \\sin 37^{\\circ}=\\frac{3}{5}$ 。求\n\n[图1]小球第一次运动到 $\\mathrm{B}$ 点时的速率 $v_{B}$;", "prompt": "你正在参加一个国际物理竞赛，并需要解决以下问题。\n这个问题的答案是一个数值。\n这里是一些可能会帮助你解决问题的先验信息提示：\n如图, 两根内径相同的绝缘细管 $\\mathrm{AB}$ 和 $\\mathrm{BC}$, 连接成倒 $\\mathrm{V}$ 字形, 坚直放置,连接点 $\\mathrm{B}$ 处可视为一段很短的圆弧; 两管长度均为 $l=2.0 \\mathrm{~m}$, 倾角 $\\alpha=37^{\\circ}$,处于方向坚直向下的匀强电场中, 场强大小 $E=10000 \\mathrm{~V} / \\mathrm{m}$ 。一质量 $m=1.0 \\times 10^{-4} \\mathrm{~kg}$ 、带电量 $-q=-2.0 \\times 10^{-7} \\mathrm{C}$ 的小球 (小球直径比细管内径稍小,可视为质点), 从 $\\mathrm{A}$ 点由静止开始在管内运动, 小球与 $\\mathrm{AB}$ 管壁间的动摩擦因数为 $\\mu_{1}=0.50$,小球与 $\\mathrm{BC}$ 管壁间的动摩擦因数为 $\\mu_{2}=0.25$ 。小球在运动过程中带电量保持不变。已知重力加速度大小 $g=10 \\mathrm{~m} / \\mathrm{s}^{2}, \\sin 37^{\\circ}=\\frac{3}{5}$ 。求\n\n[图1]\n\n问题:\n小球第一次运动到 $\\mathrm{B}$ 点时的速率 $v_{B}$;\n\n你输出的所有数学公式和符号应该使用LaTeX表示！\n你可以一步一步来解决这个问题，并输出详细的解答过程。\n请记住，你的答案应以$\\mathrm{~m} / \\mathrm{s}$为单位计算，但在给出最终答案时，请不要包含单位。\n你需要在输出的最后用以下格式总结答案：“最终答案是$\\boxed{ANSWER}$”，其中ANSWER是不包含任何单位的数值。", "figure_urls": [ "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-04.jpg?height=197&width=414&top_left_y=1735&top_left_x=1569", "https://cdn.mathpix.com/cropped/2024_03_31_65f5c8a7e52f16edf8b7g-10.jpg?height=232&width=676&top_left_y=792&top_left_x=1064" ], "answer": null, "solution": null, "answer_type": "NV", "unit": [ "$\\mathrm{~m} / \\mathrm{s}$" ], "answer_sequence": null, "type_sequence": null, "test_cases": null, "subject": "Physics", "language": "ZH", "modality": "multi-modal" } ]