question_id string | subfield string | context string | question string | images images list | final_answer list | is_multiple_answer bool | unit string | answer_type string | error string | source string |
|---|---|---|---|---|---|---|---|---|---|---|
1466 | Electromagnetism | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | ii. At roughly what voltage $V_{0}$ does the system transition from this regime to the high voltage regime of the previous part? | [
"$V_{0}=\\frac{16 k_{B} T}{3 q}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1475 | Mechanics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | a. Suppose you drop a block of mass $m$ vertically onto a fixed ramp with angle $\theta$ with coefficient of static and kinetic friction $\mu$. The block is dropped in such a way that it does not rotate after colliding with the ramp. Throughout this problem, assume the time of the collision is negligible.
i. Suppose ... | [
"$u=v(\\sin \\theta-\\mu \\cos \\theta)$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1476 | Mechanics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | ii. What is the minimum $\mu$ such that the speed of the block right after the collision is 0 ? | [
"$\\mu=\\tan \\theta$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1477 | Mechanics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | b. Now 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 $v$.
i. Suppose the sphere immediately begins to roll without slipping. What is the new speed of the sphere in this case? | [
"$u=\\frac{v \\sin \\theta}{1+\\beta}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1478 | Mechanics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | ii. What is the minimum coefficient of friction such that the sphere rolls without slipping immediately after the collision? | [
"$\\mu=\\frac{\\beta \\tan \\theta}{1+\\beta}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1482 | Electromagnetism | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | a. What 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. | [
"$\\Phi_{c}(a, \\rho) \\approx \\frac{C \\rho a^{2}}{\\epsilon_{0}}$, $C=0.0947$"
] | false | null | Equation,Numerical | ,1e-3 | OE_TO_physics_en_COMP | |
1486 | Optics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | a. Find the standard deviation $\Delta r$ of the distribution for $r$, the distance from the center of the telescope mirror to the point of reflection of the photon. | [
"$\\Delta r=\\frac{R}{\\sqrt{18}}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1488 | Modern Physics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | i. Given that the total power radiated from the sun is given by $L_{\odot}$, find an expression for the radiation pressure a distance $R$ from the sun. | [
"$P=\\frac{L \\odot}{4 \\pi R^{2} c}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1489 | Modern Physics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | ii. Assuming that the particle has a density $\rho$, derive an expression for the ratio $\frac{F_{\text {radiation }}}{F_{\text {gravity }}}$ in terms of $L_{\odot}$, mass of sun $M_{\odot}, \rho$, particle radius $r$, and quality factor $Q$. | [
"$\\frac{F_{\\text {radiation }}}{F_{\\text {gravity }}}=\\frac{3 L_{\\odot}}{16 \\pi G c M_{\\odot} \\rho} \\frac{Q}{r}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1491 | Modern Physics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | i. Assume 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. | [
"$v=\\sqrt{\\frac{G M_{\\odot}}{R}}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1492 | Modern Physics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | ii. Because the particle is moving, the radiation force is not directed directly away from the sun. Find the torque $\tau$ on the particle because of radiation pressure. You may assume that $v \ll c$. | [
"$\\tau=-\\frac{v}{c} \\frac{L \\odot}{4 R c} Q r^{2}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1493 | Modern Physics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | iii. Since $\tau=d L / d t$, the angular momentum $L$ of the particle changes with time. As such, develop a differential equation to find $d R / d t$, the rate of change of the radial location of the particle. You may assume the orbit is always quasi circular. | [
"$-\\frac{1}{c^{2}} \\frac{L_{\\odot}}{R} Q=\\frac{8}{3} \\pi \\rho r \\frac{d R}{d t}$"
] | false | null | Equation | null | OE_TO_physics_en_COMP | |
1494 | Modern Physics | $$
\begin{array}{ll}
g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\
k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\
c=3.... | iv. Develop 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. | [
"$2 \\times 10^{14}$"
] | false | s | Numerical | 5e13 | OE_TO_physics_en_COMP | |
1502 | Thermodynamics | A solid, uniform cylinder of height $h=10 \mathrm{~cm}$ and base area $s=100 \mathrm{~cm}^{2}$ floats in a cylindrical beaker of height $H=20 \mathrm{~cm}$ and inner bottom area $S=102 \mathrm{~cm}^{2}$ filled with a liquid. The ratio between the density of the cylinder and that of the liquid is $\gamma=0.70$. The bott... | Find the period of the motion $T$. Neglect the viscosity of the liquid. | [
"0.53"
] | false | s | Numerical | 5e-2 | OE_TO_physics_en_COMP | |
1565 | Electromagnetism | A system consisted of two conductor bodies is immersed in a uniform dielectric and weakly conducting liquid. When a constant voltage difference is applied between both conductors, the system has both electric and magnetic fields. In this problem we will investigate this system. | 1. First consider an infinitely long line with charge per unit length $\lambda$ in vacuum. Calculate the electric field $\mathbf{E}(\mathbf{r})$ due to the line. | [
"$\\mathbf{E}=\\hat{r} \\frac{\\lambda}{2 \\pi \\epsilon_{0} r}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1566 | Electromagnetism | A system consisted of two conductor bodies is immersed in a uniform dielectric and weakly conducting liquid. When a constant voltage difference is applied between both conductors, the system has both electric and magnetic fields. In this problem we will investigate this system.
Context question:
1. First consider an i... | 2. The potential due to the line charge could be written as
$$
V(r)=f(r)+K,
$$
where $K$ is a constant. Determine $f(r)$. | [
"$f(r)=-\\frac{\\lambda}{2 \\pi \\epsilon_{0}} \\ln r$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1572 | Modern Physics | Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a... | 1. When the velocity of the particle is $v$, calculate the acceleration of the particle, $a$ (with respect to the rest frame). | [
"$a=\\frac{F}{\\gamma^{3} m}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1573 | Modern Physics | Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a... | 2. Calculate the velocity of the particle $\beta(t)=\frac{v(t)}{c}$ at time $t$ (in rest frame), in terms of $F, m, t$ and $c$. | [
"$\\beta=\\frac{\\frac{F t}{m c}}{\\sqrt{1+\\left(\\frac{F t}{m c}\\right)^{2}}}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1574 | Modern Physics | Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a... | 3. Calculate the position of the particle $x(t)$ at time $t$, in term of $F, m, t$ and $c$. | [
"$x=\\frac{m c^{2}}{F}\\left(\\sqrt{1+\\left(\\frac{F t}{m c}\\right)^{2}}-1\\right)$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1576 | Modern Physics | Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a... | 5. Calculate the velocity of the particle $\beta(\tau)$, when the time as experienced by the particle is $\tau$. Express the answer in $g, \tau$, and $c$. | [
"$\\beta=\\tanh \\frac{g \\tau}{c}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1577 | Modern Physics | Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a... | 6. ( $\mathbf{0 . 4} \mathbf{~ p t s )}$ Also calculate the time $t$ in the rest frame in terms of $g, \tau$, and $c$. | [
"$t=\\frac{c}{g} \\sinh \\frac{g \\tau}{c}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1592 | Modern Physics | All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formali... | 2. A uniform magnetic field $\boldsymbol{B}$ exists and it makes an angle $\phi$ with a particle's magnetic moment $\boldsymbol{\mu}$. Due to the torque by the magnetic field, the magnetic moment $\boldsymbol{\mu}$ rotates around the field $\boldsymbol{B}$, which is also known as Larmor precession. Determine the Larmo... | [
"$\\omega_{0}=\\gamma B_{0}$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1594 | Modern Physics | All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formali... | 2. For $\boldsymbol{B}=B_{0} \boldsymbol{k}$, what is the new precession frequency $\Delta$ in terms of $\omega_{0}$ and $\omega$ ? | [
"$\\Delta =\\gamma B_{0}-\\omega$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1596 | Modern Physics | All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formali... | 4. Instead of applying the field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, now we apply $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}-\sin \omega t \boldsymbol{j})$, which rotates in the opposite direction and hence $\boldsymbol{B}=B_{0} \boldsymbol{k}+b(\cos \omega t \bolds... | [
"$\\mathbf{B}_{\\mathrm{eff}}=\\left(B_{0}-\\frac{\\omega}{\\gamma}\\right) \\mathbf{k}^{\\prime}+b\\left(\\cos 2 \\omega t \\mathbf{i}^{\\prime}-\\sin 2 \\omega t \\mathbf{j}^{\\prime}\\right)$ , $\\overline{\\mathbf{B}_{\\mathrm{eff}}}=\\left(B_{0}-\\frac{\\omega}{\\gamma}\\right) \\mathbf{k}^{\\prime}$"
] | true | null | Expression | null | OE_TO_physics_en_COMP | |
1597 | Modern Physics | All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formali... | 1. In the rotating frame $S^{\prime}$, show that the effective field can be approximated by
$$
\boldsymbol{B}_{\text {eff }} \approx b \boldsymbol{i}^{\prime},
$$
which is commonly known as rotating wave approximation. What is the precession frequency $\Omega$ in frame $S^{\prime}$ ? | [
"$\\Omega=\\gamma b$"
] | false | null | Expression | null | OE_TO_physics_en_COMP | |
1599 | Modern Physics | All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formali... | 3. Under the application of magnetic field described above, determine the fractional population of each spin up $P_{\uparrow}=N_{\uparrow} / N$ and spin down $P_{\downarrow}=N_{\downarrow} / N$ as a function of time. Plot $P_{\uparrow}(t)$ and $P_{\downarrow}(t)$ on the same graph vs. time $t$. The alternating spin up... | [
"$P_{\\downarrow}=\\sin ^{2} \\frac{\\Omega t}{2}$ , $P_{\\uparrow}=\\cos ^{2} \\frac{\\Omega t}{2}$"
] | true | null | Expression | null | OE_TO_physics_en_COMP |
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