id int64 | question string | final_answer list | embedding list |
|---|---|---|---|
1,509 | (d) (1 pt) If the dipole starts with $\vec{v}_{0}=0$ and $\omega_{0}=\omega_{\text {min }}$ found in part (c), the trajectory of its center of mass has an asymptote. Find the distance $D$ from the origin to the asymptote. | [
"$D=d$"
] | [
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1,510 | a. Determine the mass fraction of the droplet that freezes before reaching the ground. | [
"$\\frac{4}{5}$"
] | [
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0,
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1,511 | b. Find, 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 $2 \mathrm{~km}$.
Neglect evaporation, condensation and size changes of the droplet. Assume that water and ice have very high thermal conduc... | [
"$8$"
] | [
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1,513 | (ii) Determine the magnitude of the velocity $v_{\mathrm{s}}$ of the electrons in the stationary case. | [
"$E / B$."
] | [
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1,515 | (b) The 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, respective... | [
"$R_{H}=\\frac{1}{e} \\frac{\\phi}{N}$"
] | [
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1,516 | (c) We know that electrons move in circular orbits in the magnetic field. In the quantum mechanical picture, the impinging magnetic field $B$ could be viewed as creating tiny whirlpools, so-called vortices, in the sea of electrons-one whirlpool for each flux quantum $h / e$ of the magnetic field, where $h$ is the Plan... | [
"$\\frac{1}{3}$"
] | [
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1,517 | (d) It 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 Coulomb energy at the corresponding filling factor. Determine the scaling exponent $\al... | [
"$\\frac{1}{2}$"
] | [
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1,518 | (e) As 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 electro... | [
"$\\Delta B=\\frac{h }{e W L}$"
] | [
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1,519 | (i) Calculate the thermal energy $E_{\text {th }}$ at temperature $T=1.0 \mathrm{~K}$. | [
"$1.38 \\times 10^{-23}$"
] | [
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1,520 | (ii) 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 wh... | [
"$1.3 \\times 10^{-20}$"
] | [
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1,521 | (i) Determine the charge current $I_{\mathrm{B}}$, which measures total charge per unit of time, in terms of $\lambda$ and $\tau$. | [
"$I_{B}=\\frac{\\nu e \\lambda}{\\tau}$"
] | [
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1,522 | (ii) Current noise is defined as the charge fluctuations per unit of time. One can analyze the noise by measuring the mean square deviation of the number of current-carrying charges. Determine the current noise $S_{I}$ due to the discreteness of the current-carrying charges in terms of $\lambda$ and $\tau$. | [
"$S_{I}=\\frac{(\\nu e)^{2} \\lambda}{\\tau}$"
] | [
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1,523 | (iii) Calculate the noise-to-current ratio $S_{I} / I_{\mathrm{B}}$, which was verified by R. de-Picciotto et al. and L. Saminadayar et al. in 1997. (One year later, Tsui and Stormer shared the Nobel Prize in Physics with R. B. Laughlin, who proposed an elegant ansatz for the ground state wave function at $v=1 / 3$.) | [
"$\\nu e$"
] | [
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1,525 | (ii) Please determine the electric current of the electron motion and calculate the magnetic moment $\vec{\mu}=I \vec{A}$, where $\stackrel{\mathrm{I}}{A}$ is the area of the electron circular orbit and the direction of $\vec{A}$ is determined by the right-hand rule of theelectric current. | [
"$I=\\frac{e^{2} B}{2 \\pi m}$ , $\\vec{\\mu}=-\\frac{m v_{\\perp}^{2}}{2 B^{2}} \\vec{B}$"
] | [
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1,526 | (iii) If the initial electron velocity $\vec{v}$ is not perpendicular to the uniform magnetic field, i.e, the angle $\theta$ between $\vec{B}$ and $\vec{v}$ is $0^{0}<\theta<90^{\circ}$, please give the screw pitch (the distance along the z-axis between successive orbits) of the electron trajectory. | [
"$2 \\pi \\frac{m v}{e B} \\cos \\theta$"
] | [
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1,528 | (ii) If both the ions and electrons have a Maxwellian distribution, the ion distribution is
$f_{i}\left(x, v_{\perp}, v_{\|}\right)=n_{i}(x)\left(\frac{m_{i}}{2 \pi k T}\right)^{3 / 2} e^{-m_{i}\left(v_{\perp}^{2}+v_{\|}^{2}\right) / 2 k T}$,
please calculate the constant $\beta$ in the magnetization $M=\beta n(x) \... | [
"$\\beta=-2$"
] | [
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1,529 | (c) Now let's go back to the Earth's dipole magnetic field. Please apply the result from Question(b) to calculate the ratio of the diamagnetic field and the Earth's dipole magnetic field in Equation (1) at the position $\left(x=10 \mathrm{R}_{\mathrm{E}}, y=0, z=1 \mathrm{R}_{\mathrm{E}}\right)$. The plasma pressure is... | [
"1.0"
] | [
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1,530 | (i) Please give the gyro-averaged magnetic-field force along the magnetic field lines on an electron and show that the magnetic moment is a motion constant, i.e., $\frac{d \mu}{d t}=0$, based on the law of the total kinetic energy conservation. | [
"$-\\mu \\frac{d B}{d s}$"
] | [
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1,533 | (i) Please give the atmospheric density as a function of the altitude and the ratio of the oxygen density at the altitudes $\mathrm{H}=160 \mathrm{~km}$ and $\mathrm{H}=220 \mathrm{~km}$. For simplicity, we assume that the atmospheric temperature is independent of the altitude and the air is an ideal gas.
( $\rho_{0}... | [
"$\\rho_{0} e^{-\\frac{\\rho_{0} g R_{E}}{P_{0}}\\left(1-\\frac{R_{E}}{r}\\right)} $",
"$ \\rho_{0} e^{-\\frac{\\rho_{0} g H}{P_{0}}}$ , $2.44 \\times 10^{3}$"
] | [
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1,535 | (g) As mentioned above, a powerful solar wind can push the dayside magnetopause tovery close to the Earth, which could cause a high-orbit satelliteto be fully exposed to the solar wind. The energetic particles in the solar wind could damage high-tech electronic components in a satellite.For simplicity, the Earth's dip... | [
"330"
] | [
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1,537 | (i) the width $\varepsilon$ of the spectral line. | [
"0.21"
] | [
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1,538 | (ii) the resolution $\lambda / \Delta \lambda$ of the etalon. | [
"$1.01 \\times 10^{6}$"
] | [
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1,539 | (c) As shown in Fig. 1, the initial air pressure is zero. By slowly tuning the pin valve, air is gradually injected into the F-P etalon and finally the air pressure reaches the standard atmospheric pressure. On the same time, ten new fringes are observed to produce from the center of the ring patterns on the focal pla... | [
"$1.00029$"
] | [
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1,540 | (d) Energy levels splitting of Sodium atoms occurs when they are placed in a magnetic field. This is called as the Zeeman effect. The energy shift given by $\Delta E=m_{j} g_{k} \mu_{B} B$, where the quantum number $\mathrm{m}_{\mathrm{j}}$ can be $\mathrm{J}, \mathrm{J}-1, \ldots,-\mathrm{J}+1,-\mathrm{J}, \mathrm{J}... | [
"$v_{a}=v_{0}-\\frac{1}{2 h}\\left(\\Delta E_{1}+\\Delta E_{2}\\right)$,$v_{b}=v_{0}-\\frac{1}{2 h}\\left(\\Delta E_{2}-\\Delta E_{1}\\right)$,$v_{c}=v_{0}+\\frac{1}{2 h}\\left(\\Delta E_{2}-\\Delta E_{1}\\right)$,$v_{d}=v_{0}+\\frac{1}{2 h}\\left(\\Delta E_{1}+\\Delta E_{2}\\right)$"
] | [
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1,541 | (e) As shown in Fig. 4, when the magnetic field is turned on, each fringe of the D1 line will split into four sub-fringes $(1,2,3$, and 4$)$. The diameter of the four sub-fringes near the center is measured as $D_{1}, D_{2}, D_{3}$, and $D_{4}$. Please give the expression of the splitting energy gap $\Delta \mathrm{E}... | [
"$\\Delta E_{1}=\\frac{h c}{\\lambda} \\cdot \\frac{D_{2}^{2}-D_{1}^{2}}{8 f^{2}}$,$\\Delta E_{2}=\\frac{h c}{\\lambda} \\cdot \\frac{D_{4}^{2}-D_{2}^{2}}{8 f^{2}}$"
] | [
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1,542 | (f) For the magnetic field $B=0.1 \mathrm{~T}$, the diameter of four sub-fringes is measured as: $D_{1}=3.88 \mathrm{~mm}, D_{2}=4.05 \mathrm{~mm}, D_{3}=4.35 \mathrm{~mm}$, and $D_{4}=4.51 \mathrm{~mm}$. Please calculate the Landé factor $\mathrm{g}_{\mathrm{k} 1}$ of ${ }^{2} \mathrm{P}_{1 / 2}$ and $\mathrm{g}_{\ma... | [
"0.68, 1.99"
] | [
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1,543 | (g) The magnetic field on the sun can be determined by measuring the Zeeman effect of the Sodium D1 line on some special regions of the sun. One observes that, in the four split lines, the wavelength difference between the shortest and longest wavelength is $0.012 \mathrm{~nm}$ by a solar spectrograph. What is the magn... | [
"0.2772"
] | [
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1,544 | (h) A Light- Emitting Diode (LED) source with a central wavelength $\lambda=650 \mathrm{~nm}$ and spectral width $\Delta \lambda=20 \mathrm{~nm}$ is normally incident $(\theta=0)$ into the F-P etalon shown in Fig. 1. For the vacuum case, find (i) the number of lines in transmitted spectrum and (ii) the frequency width... | [
"946 , $5.0 \\times 10^{8}$"
] | [
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1,545 | (a) 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.
(The angular velocity of the Earth about its spin axis is $\Omega$, the radius of... | [
"$u_{Y}=\\Omega a\\left(\\frac{1}{\\cos \\varphi_{d}}-\\cos \\varphi_{d}\\right)$"
] | [
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-0.0142822265625,
-0.0191650390625,
0.01373291015625,
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0.01495361328125,
0.0106201171875,
0.0224609375,
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0.02294921875,
-0.005889892578125,
-0... |
1,549 | (g) Let the temperature next to the surface and at the top of the atmosphere be $T_{H}$ and $T_{C}$ respectively. Given that the pressure difference between points $\mathrm{A}$ and $\mathrm{E}$ is $20 \mathrm{hPa}$, calculate $T_{C}$ for $T_{H}=300 \mathrm{~K}$.
Note that the ratio of molar gas constant $(R)$ to mola... | [
"195"
] | [
0.01055908203125,
-0.0079345703125,
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0.013427734375,
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0.0118408203125,
0.00421142578125,
-0... |
1,550 | (h) Calculate the pressure $p_{B}$. | [
"220"
] | [
0.01055908203125,
0.0128173828125,
-0.009765625,
0.0126953125,
0.003997802734375,
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0.00958251953125,
-0.005615234375,
-... |
1,551 | (i) For an air mass moving once around the winter Hadley circulation, using the molar gas constant, $R$, and the quantities defined above, obtain expressions for
(A) the net work done per unit mole $W_{\text {net }}$ ignoring surface friction; | [
"$R\\left(T_{H}-T_{C}\\right) \\ln \\left(\\frac{p_{E}}{p_{A}}\\right)$",
"$R\\left(T_{H}-T_{C}\\right) \\ln \\left(\\frac{p_{D}}{p_{B}}\\right)$"
] | [
0.024658203125,
0.004638671875,
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0.0257568359375,
-0.01141357421875,
-0.00994873046875,
0.04443359375,
-0.002716064453125,
-0... |
1,552 | (B) the heat loss per unit mole $Q_{\text {loss }}$ at the top of the atmosphere. | [
"$R T_{C} \\ln \\left(\\frac{p_{D}}{p_{C}}\\right)$"
] | [
0.0201416015625,
-0.0076904296875,
-0.01458740234375,
0.0167236328125,
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0.0224609375,
-0.023681640625,
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-0.00750732421875,
-0.020751953125,
... |
1,553 | (j) What is the value of the ideal thermodynamic efficiency $\varepsilon_{i}$ for the winter Hadley circulation? | [
"0.35"
] | [
0.0089111328125,
-0.01055908203125,
-0.01373291015625,
-0.000637054443359375,
0.0015106201171875,
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-0.02294921875,
0.006927490234375,
0.009033203125,
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-0.033203125,
0.0084228515625,
-0.0086669921875,
-0.0185546875,
0.0283203125,
0.0003070831298828125... |
1,556 | (a) Taking the center of the circular cross section of the filament as the origin $O$, find the electric potential at any point $(x, z)$ very near the filament in terms of $V_{a}, a$ and $b$ where $V_{a}$ is the electric potential of the surface of the filament, $a$ is the radius of the filament and $b$ is the distanc... | [
"$V(r)=V_{a} \\frac{\\ln (b / \\sqrt{x^{2}+z^{2}})}{\\ln (b / a)}$"
] | [
0.0279541015625,
-0.00086212158203125,
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-0.0205078125,
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0.007415771484375,
0.01953125,
0.0174560546875,
... |
1,557 | (b) An 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, $e, v_{\mathrm{z}}, V_{a}, k_{\mathrm{z}}, a$ and $b$, w... | [
"$k_{x} =\\frac{e_{0} V_{a} \\pi}{\\hbar v_{z} \\ln \\frac{b}{a}}$"
] | [
0.0106201171875,
-0.010009765625,
0.01104736328125,
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-0.0152587890625,
-0.0042724609375,
-0.004730224609375,
0.026123046875,
... |
1,558 | (i) assuming relativistic effects can be ignored | [
"$\\frac{h}{\\sqrt{2 m e_{0} V_{0}}}$"
] | [
0.0269775390625,
-0.0184326171875,
-0.03564453125,
0.01025390625,
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-0.013427734375,
0.0164794921875,
0.01312255859375,
-0.00183868408203... |
1,559 | (ii) taking relativistic effects into consideration. | [
"$\\frac{h}{\\sqrt{2 m e_{0} V_{0}\\left(1+\\frac{e_{0} V_{0}}{2 m_{0} c^{2}}\\right)}}$"
] | [
0.0181884765625,
-0.0157470703125,
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0.024658203125,
-0.0086669921875,
0.022216796875,
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-0.02392578125,
-0.010009765625,
0.03662109375,
0.0130615234375,
-0.01940... |
1,560 | (i) calculate the value of $k_{x}$, | [
"$3.46 \\times 10^{7} $"
] | [
-0.0014190673828125,
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0.013916015625,
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0.011474609375,
-0.02294921875,
-0.0302734375,
0.0068359375,
-0.0108642578125,
-0.01879882812... |
1,561 | (ii) determine the fringe separation of the interference pattern on the screen, | [
"$907$"
] | [
0.010009765625,
0.00970458984375,
0.000457763671875,
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-0.0286865234375,
0.01373291015625,
-0.0005912780761... |
1,563 | (iv) In part (c), determine the percentage error in the wavelength of the electron using non-relativistic approximation. | [
"2.4"
] | [
0.0302734375,
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0.003448486328125,
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-0.01708984375,
0.0252685546875,
-0.00... |
1,564 | (v) the distance $d$ between the apparent double slits. | [
"$1.03 \\times 10^{-4}$"
] | [
0.0361328125,
0.01397705078125,
-0.0025177001953125,
-0.004364013671875,
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0.0033721923828125,
-0.0137939453125,
-0.0072631835937... |
1,567 | 4. Now consider two identical conducting cylinders, both with radius $R=3 a$ in vacuum. The length of each cylinders are the same and much larger than its radius $(l \gg R)$. The axis of both cylinders are on the $x z$-plane and parallel to the $z$-axis, one at $x=-5 a, y=0$ and the other at $x=5 a, y=0$. An electrica... | [
"$\\frac{V_{0}}{4 \\ln 3} \\ln \\frac{(4 a-x)^{2}+y^{2}}{(4 a+x)^{2}+y^{2}}$, $-V_{0} / 2$, $V_{0} / 2$"
] | [
0.027099609375,
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-0.001434326171875,
0.006988525390625,
0.007263183593... |
1,568 | 5. Calculate the capacitance $C$ of the system. | [
"$\\frac{l \\pi \\epsilon_{0}}{\\ln 3}$"
] | [
0.03759765625,
-0.0003070831298828125,
-0.0216064453125,
0.0019683837890625,
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0.0093994140625,
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-0.03125,
0.0128173828125,
0.005523681640625,
0.00... |
1,569 | 6. Now both cylinders are totally immersed in a weakly conducting liquid with conductivity $\sigma$. Calculate the total current that flows between both cylinders. Assume the permittivity of the liquid is equal to that of vacuum, $\epsilon=\epsilon_{0}$. | [
"$I=\\frac{V_{0} \\pi \\sigma l}{\\ln 3}$"
] | [
0.0179443359375,
0.00628662109375,
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0.01397705078125,
0.0224609375,
-0.... |
1,570 | 7. Calculate the resistance $R$ of the system. Calculate $R C$ of the system. | [
"$R C=\\frac{\\epsilon_{0}}{\\sigma}$"
] | [
0.045654296875,
0.01324462890625,
0.00347900390625,
0.0166015625,
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0.0189208984375,
-0.0021514892578125,
-0.013... |
1,571 | 8. Calculate the magnetic field due to the current in question 6. Assume that the permeability of the liquid is equal to that of vacuum $\mu=\mu_{0}$.
Notes $\int \frac{\alpha d x}{\alpha^{2}+x^{2}}=\arctan \frac{x}{a}+$ const | [
"$\\mathbf{B}=\\hat{z} \\frac{\\mu_{0} V_{0} \\sigma}{2 \\ln 3}\\left(\\arctan \\frac{y}{4 a+x}+\\arctan \\frac{y}{4 a-x}\\right)$"
] | [
0.0289306640625,
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0.0030364990234375,
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0.0048828125,
0.0181884765625,
-0.020... |
1,580 | 1. Using a Minkowski diagram, calculate the length of a stick with proper length $L$ in the rest frame, as measured in the moving frame. | [
"$\\sqrt{1-\\beta^{2}} L$"
] | [
0.0260009765625,
-0.025634765625,
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0.009765625,
-0.0142822265625,
-0.00836181640625... |
1,582 | 1. After a while, an observer in the rest frame make an observation. The first particle's clock shows time at $\tau_{A}$. What is the reading of the second clock $\tau_{B}$, according to the observer in the rest frame. | [
"$\\tau_{B}=\\tau_{A}$"
] | [
0.0135498046875,
-0.00970458984375,
-0.01080322265625,
0.000278472900390625,
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0.025146484375,
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... |
1,583 | 2. Now consider the observation from the first particle's frame. At a certain moment, an observer that move together with the first particle observed that the reading of his own clock is $\tau_{1}$. At the same time, he observed the second particle's clock, and the reading is $\tau_{2}$. Show that
$$
\sinh \frac{g}{c... | [
"$C_{1}=\\frac{g L}{c^{2}}$"
] | [
0.03564453125,
0.00445556640625,
0.000141143798828125,
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0.01080322265625,
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0.0167236328125,
-0.03369140625,
-0.0186767578125,
0.0196533203125,
-0.0185546875,
-0... |
1,584 | 3. The first particle will see the second particle move away from him. Show that the rate of change of the distance between the two particles according to the first particle is
$$
\frac{d L^{\prime}}{d \tau_{1}}=C_{2} \frac{\sinh \frac{g \tau_{2}}{c}}{\cosh \frac{g}{c}\left(\tau_{2}-\tau_{1}\right)}
$$
where $C_{2}$... | [
"$C_{2}=\\frac{g L}{c}$"
] | [
0.0302734375,
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0.013427734375,
-0.003570556640625,
0.001274... |
1,585 | 1. The first particle has a proper acceleration $g_{1}$ in the positive $x$ direction. When it is being accelerated, there exists a fixed point in the rest frame at $x=x_{\mathrm{p}}$ that has a constant distance from the first particle, according to the first particle thoughout the motion. Determine $x_{p}$. | [
"$x_{p}=-\\frac{c^{2}}{g_{1}}$"
] | [
0.00921630859375,
-0.0030517578125,
-0.005706787109375,
0.004150390625,
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0.0233154296875,
-0.01385498046875,
0.006896... |
1,586 | 2. Given the proper acceleration of the first particle is $g_{1}$, determine the proper acceleration of the second particle $g_{2}$, so that the distance between the two particles are constant according to the first particle. | [
"$g_{2}=\\frac{g_{1}}{1+\\frac{g_{1} L}{c^{2}}}$"
] | [
0.0390625,
0.00897216796875,
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-0.01177978515625,
0.00133514404296875,
0.031494140625,
-0.0172119140625,
0.0... |
1,587 | 3. What is the ratio of time rate of the second particle to the first particle $\frac{d \tau_{2}}{d \tau_{1}}$, according to the first particle. | [
"$1+\\frac{g_{1} L}{c^{2}}$"
] | [
0.01397705078125,
-0.00909423828125,
-0.03125,
-0.005157470703125,
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0.01446533203125,
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0.0211181640625,
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0.0189208984375,
0.000457763671875,
-0.0090... |
1,588 | 1. If the gravitational acceleration on the Earth's surface is $9.78 \mathrm{~m} \cdot \mathrm{s}^{-2}$, and the Earth's radius is $6380 \mathrm{~km}$, what is the radius of the GPS satellite orbit? What is the velocity of the satellite? Calculate the numerical values of the radius and the velocity. | [
"$r =2.66 \\times 10^{7}$ and $v=3.87 \\times 10^{3}$"
] | [
0.00836181640625,
-0.0164794921875,
0.015625,
0.002960205078125,
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0.0017242431640625,
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0.0228271484375,
-0.0281982421875,
-0.02880859375,
0.0264892578125,
-0.00799560546875,
-... |
1,590 | 3. After one day, estimate the error in position due to this effect? | [
"11.5"
] | [
0.007293701171875,
-0.037353515625,
0.00872802734375,
0.017822265625,
-0.0093994140625,
0.02685546875,
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0.021484375,
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-0.... |
802 | In an old coal factory, a conveyor belt will move at a constant velocity of $20.3 \mathrm{~m} / \mathrm{s}$ and can deliver a maximum power of $15 \mathrm{MW}$. Each wheel in the conveyor belt has a diameter of $2 \mathrm{~m}$. However a changing demand has pushed the coal factory to fill their coal hoppers with a diff... | [
"$2022.2$"
] | [
0.005279541015625,
-0.00933837890625,
-0.003997802734375,
0.01904296875,
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0.01513671875,
-0.003326416015625,
... |
803 | Neutrinos are extremely light particles and rarely interact with matter. The Sun emits neutrinos, each with an energy of $8 \times 10^{-14} \mathrm{~J}$ and reaches a flux density of $10^{11}$ neutrinos $/\left(\mathrm{s} \mathrm{cm}^{2}\right)$ at Earth's surface.
In the movie 2012, neutrinos have mutated and now are... | [
"$1 \\times 10^{14}$"
] | [
0.0196533203125,
0.0284423828125,
-0.038330078125,
0.0283203125,
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0.0184326171875,
-0.03076171875,
-0.... |
806 | Eddie is experimenting with his sister's violin. Allow the "A" string of his sister's violin have an ultimate tensile strength $\sigma_{1}$. He tunes a string up to its highest possible frequency $f_{1}$ before it breaks. He then builds an exact copy of the violin, where all lengths have been increased by a factor of $... | [
"$\\frac{\\sqrt{2}}{2}$"
] | [
0.013427734375,
-0.01171875,
0.021728515625,
0.0042724609375,
0.00396728515625,
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0.004608154296875,
0.0142822265625,
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0.00543212890625,
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-0.0004634857177734375,
0.00701904296875,
-0.0179443359375,... |
807 | A one horsepower propeller powered by a battery and is used to propel a small boat initially at rest. You have two options:
1. Put the propeller on top of the boat and push on the air with an initial force $F_{1}$
2. Put the propeller underwater and push on the water with an initial force $F_{2}$.
The density of wate... | [
"9.26"
] | [
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808 | A professional pastry chef is making a sweet which consists of 3 sheets of chocolate. The chef leaves a gap with width $d_{1}=0.1 \mathrm{~m}$ between the top and middle layers and fills it with a chocolate syrup with uniform viscosity $\eta_{1}=10 \mathrm{~Pa} \cdot \mathrm{s}$ and a gap with width $d_{2}=0.2 \mathrm{... | [
"$2.667$"
] | [
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810 | A magnetic field is located within a region enclosed by an elliptical island with semi-minor axis of $a=100 \mathrm{~m}$ and semi-major axis of $b=200 \mathrm{~m}$. A car carrying charge $+Q=1.5 \mathrm{C}$ drives on the boundary of the island at a constant speed of $v=5 \mathrm{~m} / \mathrm{s}$ and has mass $m=2000 \... | [
"70.7"
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811 | Inside a laboratory at room temperature, a steel tuning fork in the shape of a $\mathrm{U}$ is struck and begins to vibrate at $f=426 \mathrm{~Hz}$. The tuning fork is then brought outside where it is $10^{\circ} \mathrm{C}$ hotter and the experiment is performed again. What is the change in frequency, $\Delta f$ of th... | [
"0.0320"
] | [
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812 | A large metal conducting sphere with radius $10 \mathrm{~m}$ at an initial potential of 0 and an infinite supply of smaller conducting spheres of radius $1 \mathrm{~m}$ and potential $10 \mathrm{~V}$ are placed into contact in such a way: the large metal conducting sphere is contacted with each smaller sphere one at a ... | [
"25"
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813 | During high speed motion in a strong electric field, a charged particle can ionize air molecules it collides with.
A charged particle of mass $m=0.1 \mathrm{~kg}$ and charge $q=0.5 \mu \mathrm{C}$ is located in the center of a cubical box. Each vertex of the box is fixed in space and has a charge of $Q=-4 \mu \mathrm{... | [
"0.354"
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814 | Max finds himself trapped in the center of a mirror walled equilateral triangular room. What minimum beam angle must his flashlight have so that any point of illumination in the room can be traced back to his flashlight with at most 1 bounce? (Answer in degrees.) Since the room is large, assume the person is a point do... | [
"$120$"
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816 | Kushal finds himself trapped in a large room with mirrors as walls. Being scared of the dark, he has a powerful flashlight to light the room. All references to "percent" refer to area. Since the room is large, assume the person is a point does not block light. Visualize the questions in a 2D setup. The floor/ceiling is... | [
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817 | Two identical neutron stars with mass $m=4 \times 10^{30} \mathrm{~kg}$ and radius $15 \mathrm{~km}$ are orbiting each other a distance $d=700 \mathrm{~km}$ away from each other ( $d$ refers to the initial distance between the cores of the neutron stars). Assume that they orbit as predicted by classical mechanics, exce... | [
"590"
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819 | In the cosmic galaxy, the Sun is a mainsequence star, generating its energy mainly by nuclear fusion of hydrogen nuclei into helium. In its core, the Sun fuses hydrogen to produce deuterium $(2 \mathrm{H})$ and tritium $(3 \mathrm{H})$, then makes about 600 million metric tons of helium (4He) per second. Of course, the... | [
"17.51"
] | [
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821 | A particle of rest mass $m$ moving at a speed $v=0.7 c$ decomposes into two photons which fly off at a separated angle $\theta$. What is the minimum value of the angle of separation assuming that the two photons have equal wavelength. (Answer in degrees) | [
"$91.1$"
] | [
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823 | At an amusement park, there is a ride with three "teacups" that are circular with identical dimensions. Three friends, Ethan, Rishab, and Kushal, all pick a teacup and sit at the edge. Each teacup rotates about its own axis clockwise at an angular speed $\omega=1 \mathrm{rad} / \mathrm{s}$ and can also move linearly at... | [
"2"
] | [
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826 | Life on Earth would not exist as we know it without the atmosphere. There are many reasons for this, but one of which is temperature. Let's explore how the atmosphere affects the temperature on Earth. Assume that all thermal energy striking the earth uniformly and ideally distributes itself across the Earth's surface.
... | [
"289.601"
] | [
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828 | Mountains have two sides: windward and leeward. The windward side faces the wind and typically receives warm, moist air, often from an ocean. As wind hits a mountain, it is forced upward and begins to move towards the leeward side. During social distancing, Rishab decides to cross a mountain from the windward side to t... | [
"7.41"
] | [
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835 | A planet has a radius of $10 \mathrm{~km}$ and a uniform density of $5 \mathrm{~g} / \mathrm{cm}^{3}$. A powerful bomb detonates at the center of the planet, releasing $8.93 \times 10^{17} \mathrm{~J}$ of energy, causing the planet to separate into three large sections each with equal masses. You may model each section... | [
"136000"
] | [
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836 | A point charge $+q$ is placed a distance $a$ away from an infinitely large conducting plate. The force of the electrostatic interaction is $F_{0}$. Then, an identical conducting plate is placed a distance $3 a$ from the charge, parallel to the first one such that the charge is "sandwiched in." The new electrostatic for... | [
"0.916"
] | [
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... |
837 | Jerry spots a truckload of his favourite golden yellow Swiss cheese being transported on a cart moving at a constant velocity $v_{0}=5 \mathrm{~m} / \mathrm{s} \hat{i}$ along the x-axis, which is initially placed at $(0,0)$. Jerry, driven by desire immediately starts pursuing the cheese-truck in such a way that his vel... | [
"10.9375"
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0.... |
840 | Consider an LC circuit with one inductor and one capacitor. The amplitude of the charge on the plates of the capacitor is $Q=10 \mathrm{C}$ and the two plates are initially at a distance $d=1 \mathrm{~cm}$ away from each other. The plates are then slowly pushed together to a distance $0.5 \mathrm{~cm}$ from each other.... | [
"$11.892$"
] | [
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842 | A child attaches a small rock of mass $M=0.800 \mathrm{~kg}$ to one end of a uniform elastic string of mass $m=0.100 \mathrm{~kg}$ and natural length $L=0.650 \mathrm{~m}$. He grabs the other end and swings the rock in uniform circular motion around his hand, with angular velocity $\omega=6.30 \mathrm{rad} / \mathrm{s}... | [
"1.903"
] | [
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844 | What is the smallest number of $1 \Omega$ resistors needed such that when arranged in a certain arrangement involving only series and parallel connections, that the equivalent resistance is $\frac{7}{6} \Omega$ ? | [
"5"
] | [
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847 | A coaxial cable is cylindrically symmetric and consists of a solid inner cylinder of radius $a=2 \mathrm{~cm}$ and an outer cylindrical shell of inner radius $b=5 \mathrm{~cm}$ and outer radius $c=7 \mathrm{~cm}$. A uniformly distributed current of total magnitude $I=5 \mathrm{~A}$ is flowing in the inner cylinder and ... | [
"$1.6 \\times 10^{-8}$"
] | [
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0.008056640625... |
849 | A train of length $100 \mathrm{~m}$ and mass $10^{5} \mathrm{~kg}$ is travelling at $20 \mathrm{~m} / \mathrm{s}$ along a straight track. The driver engages the brakes and the train starts deccelerating at a constant rate, coming to a stop after travelling a distance $d=2000 \mathrm{~m}$. As the train decelerates, ener... | [
"$63.98$"
] | [
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851 | Consider a gas of mysterious particles called nieons that all travel at the same speed, $v$. They are enclosed in a cubical box, and there are $\rho$ nieons per unit volume. A very small hole of area $A$ is punched in the side of the box. The number of nieons that escape the box per unit time is given by
$$
\alpha v^{\... | [
"3.25"
] | [
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0.00010013580322265625... |
853 | Now, Poncho has encountered a different Pico-Pico game that uses the same shaped frictionless track, but lays it horizontally on a table with friction and coefficient of friction $\mu=0.8$. In addition, the ball, which can once again be considered a point mass, is placed on the other side of the track as the ball in pa... | [
"$13.1$"
] | [
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856 | Let $A B C$ be a solid right triangle $(A B=5 s, A C=12 s$, and $B C=13 s$ ) with uniform charge density $\sigma$. Let $D$ be the midpoint of $B C$. We denote the electric potential of a point $P$ by $\phi(P)$. The electric potential at infinity is 0 . If $\phi(B)+\phi(C)+\phi(D)=\frac{k \sigma s}{\epsilon_{0}}$ where ... | [
"2.055"
] | [
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... |
860 | A straight ladder $A B$ of mass $m=1 \mathrm{~kg}$ is positioned almost vertically such that point $B$ is in contact with the ground with a coefficient of friction $\mu=0.15$. It is given an infinitesimal kick at the point $A$ so that the ladder begins rotating about point $B$. Find the value $\phi_{m}$ of angle $\phi$... | [
"$11.5$"
] | [
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0.031982421875,... |
861 | Two Ladders Two straight ladders $A B$ and $C D$, each with length $1 \mathrm{~m}$, are symmetrically placed on smooth ground, leaning on each other, such that they are touching with their ends $B$ and $C$, ends $A$ and $D$ are touching the floor. The friction at any two surfaces is negligible. Initially both ladders a... | [
"$\\frac{2\\sqrt{5}}{3}$"
] | [
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863 | An evil gamma photon of energy $E_{\gamma 1}=200 \mathrm{keV}$ is heading towards a spaceship. The commander's only choice is shooting another photon in the direction of the gamma photon such that they 'collide' head on and produce an electron-positron pair (both have mass $m_{e}$ ). Find the lower bound on the energy ... | [
"1306"
] | [
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865 | Adithya is launching a package from New York City $\left(40^{\circ} 43^{\prime} \mathrm{N}\right.$ and $\left.73^{\circ} 56^{\prime} \mathrm{W}\right)$ to Guam $\left(13^{\circ} 27^{\prime} \mathrm{N}\right.$ and $\left.144^{\circ} 48^{\prime} \mathrm{E}\right)$. Find the minimal launch velocity $v_{0}$ from New York C... | [
"$7564$"
] | [
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0.0238037109375,
-0.00592041015625,... |
866 | Consider a container filled with argon, with molar mass $39.9 \mathrm{~g} \mathrm{~mol}^{-1}$ whose pressure is much smaller than that of atmospheric pressure. Suppose there is a plate of area $A=10 \mathrm{~mm}^{2}$ moving with a speed $v$ perpendicular to its plane. If the gas has density $\rho=4.8 \times 10^{-7} \ma... | [
"$2.41\\times 10^{-4}$"
] | [
0.03662109375,
0.00140380859375,
-0.01483154296875,
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-0.0028076... |
868 | Consider a $1 \mathrm{~cm}$ long slit with negligible height. First, we divide the slit into thirds and cover the middle third. Then, we perform the same steps on the two shorter slits. Again, we perform the same steps on the four even shorter slits and continue for a very long time.
Then, we shine a monochromatic, coh... | [
"$0.647$"
] | [
0.014404296875,
-0.006439208984375,
-0.005767822265625,
-0.0294189453125,
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-0.025146484375,
-0.0211181640625,
... |
869 | A certain planet with radius $R=$ $3 \times 10^{4} \mathrm{~km}$ is made of a liquid with constant density $\rho=1.5 \mathrm{~g} / \mathrm{cm}^{3}$ with the exception of a homogeneous solid core of radius $r=10 \mathrm{~km}$ and mass $m=2.4 \times 10^{16} \mathrm{~kg}$. Normally, the core is situated at the geometric c... | [
"$1.0058 \\times 10^{13}$"
] | [
-0.006591796875,
-0.016845703125,
0.0032806396484375,
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0.005859375,
-0.0172119140625,
-0.02880859375,
0.0186767578125,
0.000858306884765625,
-0.01... |
870 | A scientist is doing an experiment with a setup consisting of 2 ideal solenoids that share the same axis. The lengths of the solenoids are both $\ell$, the radii of the solenoids are $r$ and $2 r$, and the smaller solenoid is completely inside the larger one. Suppose that the solenoids share the same (constant) current... | [
"90"
] | [
-0.0028839111328125,
-0.001800537109375,
-0.00005221366882324219,
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-0.03759765625,
0.0111083984375,
0.00579833984375,
-0.017333984375,
0.0022125244140625,
0.0... |
871 | Adithya is in a rocket with proper acceleration $a_{0}=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}^{2}$ to the right, and Eddie is in a rocket with proper acceleration $\frac{a_{0}}{2}$ to the left. Let the frame of Adithya's rocket be $S_{1}$, and the frame of Eddie's rocket be $S_{2}$. Initially, both rockets are at ... | [
"$2.564 \\times 10^{8}$"
] | [
0.03857421875,
-0.01348876953125,
-0.0076904296875,
-0.00872802734375,
-0.0005950927734375,
0.0133056640625,
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0.00537109375,
-0.0147705078125,
0.00006723403930664062,
0.023681640625,
-0.0341796875,
-0.00854... |
872 | Suppose a ping pong ball of radius $R$, thickness $t$, made out of a material with density $\rho_{b}$, and Young's modulus $Y$, is hit so that it resonates in mid-air with small amplitude oscillations. Assume $t \ll R$. The density of air around (and inside) the ball is $\rho_{a}$, and the air pressure is $p$, where $\... | [
"19.75"
] | [
0.00762939453125,
-0.00616455078125,
0.00860595703125,
0.02001953125,
0.0045166015625,
0.01422119140625,
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-0.010986328125,
0.0087890625,
0.00347900390625,
-0.002777099609375,
-0.0... |
873 | A player throws two tennis balls on a level ground at $v=20 \mathrm{~m} / \mathrm{s}$ in the same direction, once at an angle of $\alpha=35^{\circ}$ and once at an angle $\beta=55^{\circ}$ to the horizontal. The distance between the landing spots of the two balls is $d$. Find $d$ in meters.
Assume the height of the pla... | [
"0"
] | [
0.0281982421875,
-0.00732421875,
-0.021240234375,
0.00048828125,
0.0076904296875,
0.01495361328125,
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0.01104736328125,
0.004180908203125,
0.031982421875,
0.019287109375,
-0.0137939453125,
-0.007... |
876 | For this problem, assume the Earth moves in a perfect circle around the sun in the $x y$ plane, with a radius of $r=1.496 \times 10^{11} \mathrm{~m}$, and the Earth has a mass $m=5.972 \times 10^{24} \mathrm{~kg}$. An alien stands far away from our solar system on the $x$ axis such that it appears the Earth is moving a... | [
"-2"
] | [
0.00531005859375,
-0.0291748046875,
0.002655029296875,
0.00787353515625,
-0.0142822265625,
0.00872802734375,
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-0.00640869140625,
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-0.056640625,
-0.01092529296875,
-0.03759765625,
-0.041748046875,
0.0194091796875,
-0.000835418701171... |
877 | Battle ropes can be used as a full body workout (see photo). It consists of a long piece of thick rope (ranging from $35 \mathrm{~mm}$ to $50 \mathrm{~mm}$ in diameter), wrapped around a stationary pole. The athlete grabs on to both ends, leans back, and moves their arms up and down in order to create waves, as shown i... | [
"1.43"
] | [
-0.00518798828125,
-0.0029144287109375,
-0.016357421875,
0.00860595703125,
-0.00250244140625,
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-0.0272216796875,
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-0.0079345703125,
-0.0164794921875,
-0.005126953125,
0.0206298828125,
0.010314941406... |
878 | Given vertically polarized light, you're given the task of changing it to horizontally polarized light by passing it through a series of $N=5$ linear polarizers. What is the maximum possible efficiency of this process? (Here, efficiency is defined as the ratio between output light intensity and input light intensity.) | [
"$\\cos ^{10}(\\frac{\\pi}{10})$"
] | [
0.004302978515625,
0.00897216796875,
-0.005889892578125,
0,
-0.01092529296875,
0.013671875,
-0.01116943359375,
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-0.0078125,
-0.00064849853515625,
-0.058349609375,
0.00946044921875,
0.00102996826171875,
-0.037109375,
0.0030517578125,
0.00015354156494140625,
0.01116943... |
881 | In this problem, we explore how fast an iceberg can melt, through the dominant mode of forced convection. For simplicity, consider a very thin iceberg in the form of a square with side lengths $L=100 \mathrm{~m}$ and a height of $1 \mathrm{~m}$, moving in the arctic ocean at a speed of $0.2 \mathrm{~m} / \mathrm{s}$ wi... | [
"60"
] | [
0.0262451171875,
-0.01416015625,
-0.0240478515625,
0.02734375,
0.01409912109375,
0.041015625,
0.0126953125,
0.00286865234375,
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-0.03564453125,
0.0283203125,
-0.00732421875,
-0.02197265625,
0.0030975341796875,
-0.007415771484375,
-0.01483154296875,... |
883 | In a galaxy far, far away, there is a planet of mass $M=6 \cdot 10^{27} \mathrm{~kg}$ which is a sphere of radius $R$ and charge $Q=10^{3} \mathrm{C}$ uniformly distributed. Aliens on this planet have devised a device for transportation, which is an insulating rectangular plate with mass $m=1 \mathrm{~kg}$ and charge $... | [
"$0.522$"
] | [
0.01226806640625,
-0.0269775390625,
-0.006866455078125,
0.013427734375,
-0.015625,
0.010498046875,
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-0.040283203125,
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0.0269775390625,
-0.05322265625,
-0.041259765625,
0.0140380859375,
-0.0213623046875,
-0.0231933... |
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