Datasets:
ttlynne
/
physics_en_no_proof_embedding

Dataset card Data Studio Files
xet
 Community
1
id
int64
question
string
final_answer
list
embedding
list
1,191
A.1 Find an expression for $b$ as a function of the quantities (1), the angle $\phi$ and the $0.8 p t$ tilting angle $\Theta$ of the base.
[ "$b=\\frac{r_{1} \\sin \\Theta}{\\sin \\phi}$" ]
[ -0.0135498046875, -0.004119873046875, 0.00018787384033203125, -0.020751953125, -0.0179443359375, 0.0184326171875, -0.005218505859375, -0.007537841796875, -0.0184326171875, 0.0157470703125, -0.031005859375, 0.021240234375, -0.0155029296875, -0.0166015625, 0.03564453125, -0.012939453125,...
1,193
A.3 Find an expression for the distance $d$ as a function of $b$ and the quantities (1). You may also include $r_{2}$ and $h_{2}$ as variables in your expression, as they will be calculated in subtask A.5.
[ "$d=\\frac{b M}{\\pi h_{2} r_{2}^{2}(\\rho_{2}-\\rho_{1})}$" ]
[ 0.0291748046875, 0.004302978515625, -0.023681640625, -0.03173828125, -0.01416015625, 0.007080078125, 0.0198974609375, -0.00860595703125, -0.0283203125, 0.01318359375, -0.036376953125, 0.016845703125, -0.00421142578125, -0.0166015625, 0.00592041015625, -0.004638671875, -0.004547119140...
1,194
A.4 Find an expression for the moment of inertia $I_{S}$ in terms of $b$ and the known quantities (1). You may also include $r_{2}$ and $h_{2}$ as variables in your expression, as they will be calculated in subtask A.5.
[ "$I_{S}=\\frac{1}{2} \\pi h_{1} \\rho_{1} r_{1}^{4}+\\frac{1}{2} \\pi h_{2}(\\rho_{2}-\\rho_{1}) r_{2}^{4}+d^{2} \\pi r_{2}^{2} h_{2}(\\rho_{2}-\\rho_{1})$" ]
[ 0.0224609375, -0.017578125, -0.0181884765625, 0.001251220703125, -0.01495361328125, 0.03662109375, 0.0155029296875, -0.0245361328125, -0.022216796875, 0.002716064453125, -0.0267333984375, 0.013427734375, -0.005279541015625, -0.0228271484375, 0.0185546875, 0.00823974609375, 0.02172851...
1,195
A.5 Using all the above results, write down an expression for $h_{2}$ and $r_{2}$ in terms of $b, T$ and the known quantities (1). You may express $h_{2}$ as a function of $r_{2}$.
[ "$r_{2}=\\sqrt{\\frac{2}{M-\\pi r_{1}^{2} h_{1} \\rho_{1}}(M \\frac{b g T^{2}}{4 \\pi^{2}}-\\frac{1}{2} \\pi h_{1} \\rho_{1} r_{1}^{4}-b^{2} \\frac{M^{2}}{M-\\pi r_{1}^{2} h_{1} \\rho_{1}})}$ , $h_{2}=\\frac{M-\\pi r_{1}^{2} \\rho_{1} h_{1}}{\\pi r_{2}^{2}(\\rho_{2}-\\rho_{1})}$" ]
[ 0.0235595703125, 0.000736236572265625, -0.01123046875, 0.0146484375, -0.00104522705078125, 0.015380859375, 0.01483154296875, -0.01080322265625, -0.02587890625, 0.031494140625, -0.0255126953125, 0.017822265625, -0.01513671875, -0.00726318359375, 0.00189208984375, -0.009521484375, -0.0...
1,196
B.1 At what angular frequency $\omega_{s s}$ does the space station rotate so that the astronauts experience the same gravity $g_{E}$ as on the Earth's surface?
[ "$\\omega_{s s}=\\sqrt{\\frac{g_{E}}{R}}$" ]
[ 0.01483154296875, -0.02001953125, 0.00787353515625, 0.01806640625, -0.021728515625, 0.0224609375, -0.0186767578125, -0.01177978515625, -0.00653076171875, -0.0030670166015625, -0.033203125, -0.028564453125, -0.021728515625, 0.00153350830078125, 0.023681640625, -0.0284423828125, -0.027...
1,197
B.2 Assuming that on Earth gravity is constant with acceleration $g_{E}$, what would be the angular oscillation frequency $\omega_{E}$ that a person on Earth would measure?
[ "$\\omega_{E}=\\sqrt{k / m}$" ]
[ 0.0234375, -0.0181884765625, 0.004638671875, 0.02294921875, -0.0037841796875, 0.013427734375, -0.024658203125, -0.00921630859375, -0.00098419189453125, -0.00093841552734375, -0.01409912109375, -0.013427734375, -0.0128173828125, -0.007476806640625, 0.0186767578125, -0.01397705078125, ...
1,198
B.3 What angular oscillation frequency $\omega$ does Alice measure on the space station?
[ "$\\sqrt{k / m-\\omega_{s s}^{2}}$", "$\\sqrt{k / m-\\frac{g_E}{R}}$" ]
[ 0.0054931640625, -0.0277099609375, -0.0034332275390625, 0.002685546875, -0.0220947265625, 0.020263671875, -0.0260009765625, 0.00921630859375, -0.00628662109375, -0.00958251953125, -0.047607421875, 0.01104736328125, -0.0257568359375, 0.003997802734375, 0.00738525390625, -0.0052795410156...
1,199
B.4 Derive an expression of the gravity $g_{E}(h)$ for small heights $h$ above the surface of the Earth and compute the oscillation frequency $\tilde{\omega}_{F}$ of the oscillating mass (linear approximation is enough). Denote the radius of the Earth by $R_{E}$. Neglect the rotation of Earth.
[ "$g_{E}(h)=\\frac{-G M}{(R_{E}+h)^{2}}$ , $\\tilde{\\omega}_{E}=\\sqrt{k / m-2 g_{E} / R_{E}}$" ]
[ 0.016357421875, -0.0145263671875, 0.0067138671875, 0.0159912109375, -0.017822265625, 0.01055908203125, -0.005828857421875, -0.0130615234375, -0.01007080078125, -0.00701904296875, -0.03173828125, -0.0166015625, -0.00531005859375, -0.037841796875, 0.00830078125, -0.005706787109375, -0....
1,200
B.5 For what radius $R$ of the space station does the oscillation frequency $\omega$ match the oscillation frequency $\tilde{\omega}_{E}$ on the Earth? Express your answer in terms of $R_{E}$.
[ "$R=R_{E} / 2$" ]
[ 0.0220947265625, -0.027099609375, 0.00096893310546875, 0.018798828125, -0.0087890625, 0.0059814453125, -0.02197265625, 0.0145263671875, -0.01300048828125, -0.003814697265625, -0.018798828125, -0.0130615234375, -0.002227783203125, -0.0179443359375, 0.005767822265625, -0.022216796875, ...
1,201
B.6 Calculate the horizontal velocity $v_{x}$ and the horizontal displacement $d_{x}$ (relative to the base of the tower, in the direction perpendicular to the tower) of the mass at the moment it hits the floor. You may assume that the height $H$ of the tower is small, so that the acceleration as measured by the astron...
[ "$2 H \\omega_{s s}$,$\\frac{1}{3} \\sqrt{\\frac{8 H^{3}}{R}}$", "$-2 H \\omega_{s s}$,$\\frac{1}{3} \\sqrt{\\frac{8 H^{3}}{R}}$" ]
[ 0.031005859375, -0.03515625, 0.003814697265625, 0.0181884765625, -0.021240234375, 0.0189208984375, 0.00180816650390625, 0.0220947265625, -0.01318359375, 0.002105712890625, -0.03955078125, -0.00604248046875, -0.0167236328125, -0.01129150390625, 0.0458984375, -0.012939453125, -0.018554...
1,202
B.7 Find a lower bound for the height of the tower for which it can happen that $d_{x}=0$.
[ "0.871" ]
[ 0.034912109375, -0.01214599609375, -0.00982666015625, -0.0185546875, -0.00848388671875, 0.004608154296875, 0.0224609375, 0.0027923583984375, -0.00823974609375, -0.010009765625, -0.044921875, -0.0196533203125, -0.0037841796875, -0.030517578125, 0.0133056640625, -0.0263671875, -0.03735...
1,203
B.8 Alice pulls the mass a distance $d$ downwards from the equilibrium point $x=0$, $y=0$, and then lets it go (see figure 5). - Give an algebraic expression of $x(t)$ and $y(t)$. You may assume that $\omega_{s s} d$ is small, and neglect the Coriolis force for motion along the $y$-axis.
[ "$y(t)=-d \\cos \\omega t$ , $x(t)=\\frac{2 \\omega_{s s} d}{\\omega} \\sin \\omega t-2 \\omega_{s s} d t$" ]
[ 0.01409912109375, -0.0220947265625, 0.004425048828125, -0.0027618408203125, -0.025634765625, 0.031494140625, -0.012451171875, 0.002899169921875, -0.01318359375, -0.009033203125, -0.04052734375, -0.01092529296875, -0.040771484375, 0.0098876953125, 0.01397705078125, 0.01385498046875, -...
1,204
A.1 Using the graph, determine the resistance $R_{\mathrm{on}}$ of the element $X$ on the upper branch of the $I-V$ characteristics, and $R_{\text {off }}$ on the lower branch, respectively. The middle branch is described by the equation $$ I=I_{0}-\frac{U}{R_{\text {int }}} \tag{1} $$ Find the values of the paramete...
[ "$R_{int}=2.00 $ , $I_{0}=6.00$" ]
[ 0.033447265625, -0.01495361328125, -0.006988525390625, -0.00775146484375, 0.02490234375, 0.00592041015625, 0.0225830078125, -0.0025482177734375, -0.02294921875, 0.004058837890625, -0.060302734375, -0.002044677734375, -0.027099609375, 0.01220703125, 0.0067138671875, -0.0181884765625, ...
1,206
A.3 Let $R=3.00 \Omega, L=1.00 \mu \mathrm{H}$ and $\mathcal{E}=15.0 \mathrm{~V}$ in the circuit shown in Fig.2 Determine the values of the current $I_{\text {stationary }}$ and the voltage $V_{\text {stationary }}$ on the non-linear element $X$ in the stationary state.
[ "$I_{\\text {stationary }}=3.00$ , $U_{\\text {stationary }}=6.00$" ]
[ 0.057861328125, -0.00634765625, -0.01202392578125, -0.0137939453125, 0.00811767578125, 0.0198974609375, 0.0037078857421875, 0.007171630859375, -0.02783203125, -0.0003528594970703125, -0.0771484375, 0.01513671875, 0.006134033203125, 0.0027618408203125, 0.0201416015625, -0.00059127807617...
1,209
B.3 Estimate the average power $P$ dissipated by the non-linear element over the course of one oscillation. An order of magnitude is sufficient.
[ "19.3" ]
[ 0.03466796875, 0.00103759765625, -0.00946044921875, 0.01361083984375, 0.01416015625, 0.041259765625, -0.006591796875, -0.0152587890625, -0.00811767578125, 0.0224609375, -0.046875, 0.00823974609375, -0.00408935546875, -0.01263427734375, 0.0341796875, 0.0137939453125, -0.0166015625, ...
1,210
B.4 What is the optimal value of $s$ assuming that it cannot exceed $1 \mathrm{~km}$ ?
[ "459" ]
[ 0.011962890625, -0.015625, 0.0164794921875, -0.003387451171875, -0.0135498046875, 0.0108642578125, 0.005523681640625, 0.0014190673828125, -0.0078125, 0.001953125, -0.0267333984375, 0.03173828125, -0.03173828125, -0.01263427734375, 0.01373291015625, -0.012451171875, -0.036376953125, ...
1,211
C.2 Find the expression and the numerical value of the critical time $\tau_{\text {crit }}$ for which the scenario switches.
[ "$\\tau_{\\text {crit }}=9.36 \\cdot 10^{-7}$" ]
[ 0.040771484375, -0.000598907470703125, 0.018798828125, -0.00567626953125, 0.0194091796875, 0.0301513671875, 0.00830078125, -0.023681640625, -0.0126953125, -0.01251220703125, -0.0380859375, -0.002655029296875, -0.0152587890625, -0.0123291015625, 0.03125, -0.013427734375, -0.0103149414...
1,218
A.6 Determine the time T that the protons take to pass through this field <image_1> Figure 1: Sketch of an accelerator module.
[ "218" ]
[ 0.036865234375, -0.0103759765625, -0.004638671875, 0, 0.00909423828125, 0.0296630859375, 0.030517578125, -0.018310546875, -0.003631591796875, 0.0146484375, -0.0255126953125, 0.0098876953125, -0.01141357421875, -0.016357421875, -0.01312255859375, -0.0220947265625, -0.0302734375, 0.0...
1,219
B.1 Express the particle mass $m$ in terms of of the momentum $p$, the flight length $l$ and the flight time $t$, assuming that particles have elementary charge $e$ and travel with velocity close to $c$ on straight tracks in the ToF detector and that they travel perpendicular to the two detection planes (see figure 2).
[ "$m=\\frac{p}{l \\cdot c} \\cdot \\sqrt{t^{2} \\cdot c^{2}-l^{2}}$" ]
[ 0.030517578125, -0.0054931640625, 0.00677490234375, -0.000514984130859375, 0.000934600830078125, 0.032958984375, 0.0286865234375, -0.0157470703125, 0.004425048828125, -0.01116943359375, -0.0296630859375, 0.00075531005859375, -0.04296875, -0.0301513671875, 0.007476806640625, -0.01239013...
1,220
B.2 Calculate the minimal length $l$ of a ToF detector that allows to safely distinguish a charged kaon from a charged pion, given both their momenta are measured to be $1.00 \mathrm{GeV} / \mathrm{c}$. For a good separation it is required that the difference in the time-of-flight is larger than three times the time re...
[ "1.28" ]
[ 0.004669189453125, -0.00537109375, -0.007232666015625, 0.005096435546875, 0.0014495849609375, 0.021240234375, 0.01611328125, -0.0198974609375, 0.00074005126953125, -0.01470947265625, -0.04931640625, 0.01275634765625, -0.0140380859375, -0.02587890625, -0.010986328125, -0.0123291015625, ...
1,226
1a Write down the equation for the present total angular momentum of the Earth-Moon system. Set this equation in terms of $I_{E}$, the moment of inertia of the Earth; $a_{E 1}$, the present angular frequency of the Earth's rotation; $I_{M 1}$, the present moment of inertia of the Moon with respect to the Earth's axis; ...
[ "$L_{1}=I_{E} \\omega_{E 1}+I_{M 1} \\omega_{M 1}$" ]
[ -0.0018310546875, -0.01171875, -0.0006256103515625, -0.005096435546875, -0.0022735595703125, 0.0267333984375, -0.00799560546875, 0.01190185546875, -0.0302734375, -0.007293701171875, -0.006378173828125, 0.004241943359375, -0.0201416015625, -0.030517578125, -0.007293701171875, 0.00811767...
1,227
1b Write down the equation for the final total angular momentum $L_{2}$ of the Earth-Moon system. Make the same assumptions as in Question 1a. Set this equation in terms of $I_{E}$, the moment of inertia of the Earth; $a_{2}$, the final angular frequency of the Earth's rotation and Moon's translation; and $I_{M 2}$, th...
[ "$L_{2}=I_{E} \\omega_{2}+I_{M 2} \\omega_{2}$" ]
[ 0.01495361328125, -0.02001953125, -0.006256103515625, -0.01214599609375, 0.01324462890625, 0.01953125, -0.0028533935546875, 0.006195068359375, -0.0439453125, -0.00897216796875, -0.01141357421875, -0.00028228759765625, -0.03076171875, -0.0238037109375, -0.004913330078125, 0.005340576171...
1,228
1c Neglecting the contribution of the Earth's rotation to the final total angular momentum, write down the equation that expresses the angular momentum conservation for this problem.
[ "$I_{E} \\omega_{E 1}+I_{M 1} \\omega_{M 1}=I_{M 2} \\omega_{2}=L_{1}$" ]
[ 0.0103759765625, -0.00933837890625, -0.007537841796875, 0.0106201171875, 0.0034332275390625, 0.00799560546875, -0.0086669921875, 0.01080322265625, -0.04541015625, -0.0017547607421875, -0.03271484375, 0.006927490234375, -0.033935546875, -0.01806640625, 0.0244140625, -0.007171630859375, ...
1,229
2a Write down the gravitational equation for the circular orbit of the Moon around the Earth, at the final state, in terms of $M_{E}, a_{2}, G$ and the final separation $D_{2}$ between the Earth and the Moon. $M_{E}$ is the mass of the Earth and $G$ is the gravitational constant.
[ "$\\omega_{2}^{2} D_{2}^{3}=G M_{E}$" ]
[ 0.007568359375, -0.026611328125, 0.00421142578125, -0.008544921875, 0.000457763671875, -0.003570556640625, 0.00811767578125, -0.007476806640625, -0.055908203125, -0.0216064453125, -0.0260009765625, -0.00335693359375, -0.035400390625, -0.0361328125, 0.0159912109375, 0.00543212890625, ...
1,230
2b Write down the equation for the final separation $D_{2}$ between the Earth and the Moon in terms of the known parameters, $L_{1}$, the total angular momentum of the system, $M_{E}$ and $M_{M}$, the masses of the Earth and Moon, respectively, and $G$.
[ "$D_{2}=\\frac{L_{1}^{2}}{G M_{E} M_{M}^{2}}$" ]
[ 0.000591278076171875, -0.00830078125, -0.0072021484375, -0.0216064453125, 0.009033203125, 0.003997802734375, -0.0023651123046875, 0.01171875, -0.03955078125, -0.0235595703125, -0.03076171875, 0.00139617919921875, -0.034423828125, -0.01416015625, 0.006103515625, -0.0032501220703125, -...
1,231
2c Write down the equation for the final angular frequency $a_{2}$ of the Earth-Moon system in terms of the known parameters $L_{1}, M_{E}, M_{M}$ and $G$.
[ "$\\omega_{2}=\\frac{G^{2} M_{E}^{2} M_{M}^{3}}{L_{1}^{3}}$" ]
[ 0.009521484375, -0.021240234375, -0.00384521484375, -0.004058837890625, 0.00860595703125, 0.00927734375, -0.006866455078125, 0.005615234375, -0.0269775390625, -0.007110595703125, -0.0145263671875, 0.00124359130859375, -0.029296875, -0.01300048828125, -0.005462646484375, -0.001892089843...
1,232
2d Write down the equation for the moment of inertia of the Earth $I_{E}$ assuming it is a sphere with inner density $\rho_{i}$ from the center to a radius $r_{i}$, and with outer density $\rho_{o}$ from the radius $r_{i}$ to the surface at a radius $r_{o}$ (see Figure 3).
[ "$I_{E}=\\frac{2}{5} \\frac{4 \\pi}{3}\\left[r_{o}^{5} \\rho_{o}+r_{i}^{5}\\left(\\rho_{i}-\\rho_{o}\\right)\\right]$" ]
[ 0.023193359375, -0.00970458984375, -0.017822265625, -0.004425048828125, 0.006011962890625, 0.0027008056640625, 0.00469970703125, -0.0155029296875, -0.03857421875, 0.00982666015625, -0.0021209716796875, -0.005126953125, -0.026123046875, -0.030029296875, -0.00201416015625, 0.013671875, ...
1,233
2e Evaluate the moment of inertia of the Earth $I_{E}$, using $\rho_{i}=1.3 \times 0^{4} \mathrm{~kg} \mathrm{~m}^{-3}, \quad 0.2$ $r_{i}=3.5 \times 0^{6} \mathrm{~m}, \rho_{o}=4.0 \times 0^{3} \mathrm{~kg} \mathrm{~m}^{-3}$, and $r_{o}=6.4 \times 0^{6} \mathrm{~m}$.
[ "$8.0 \\times 10^{37} $" ]
[ 0.022705078125, 0.005340576171875, -0.001434326171875, 0.01171875, 0.01300048828125, 0.01019287109375, 0.01300048828125, -0.00103759765625, -0.033935546875, 0.0022430419921875, -0.01373291015625, 0.00110626220703125, -0.0233154296875, -0.03125, -0.007232666015625, 0.01953125, -0.0152...
1,234
2f Evaluate the numerical value of the total angular momentum of the system, $L_{1}$.
[ "$3.4 \\times 10^{34}$" ]
[ 0.01116943359375, 0.0152587890625, 0.011474609375, -0.006744384765625, -0.002105712890625, 0.031494140625, -0.0025787353515625, -0.00927734375, -0.019775390625, 0.000720977783203125, -0.035400390625, 0.0086669921875, -0.032958984375, -0.01129150390625, 0.0184326171875, -0.0036468505859...
1,235
2g Find the final separation $D_{2}$ in meters and in units of the present separation $D_{1}$.
[ "$5.4 \\times 10^{8}$" ]
[ -0.0048828125, 0.01068115234375, -0.007171630859375, -0.0216064453125, 0.00799560546875, -0.00860595703125, -0.005859375, -0.025634765625, -0.019287109375, -0.02392578125, -0.04345703125, 0.01220703125, -0.003265380859375, -0.01214599609375, 0.036376953125, -0.01470947265625, -0.0334...
1,237
2i Find the ratio of the final angular momentum of the Earth to that of the <br> Moon.
[ "$\\frac{1}{260}$" ]
[ -0.0087890625, -0.0294189453125, -0.01092529296875, -0.01019287109375, 0.005096435546875, 0.020751953125, -0.0301513671875, 0.01239013671875, -0.044921875, -0.01495361328125, -0.0277099609375, -0.01251220703125, -0.029052734375, -0.022705078125, 0.01025390625, 0.005859375, -0.0170898...
1,238
3a Find $F_{c}$, the magnitude of the force produced on the Moon by the closest point mass.
[ "$F_{c}=\\frac{G m M_{M}}{D_{1}^{2}+r_{o}^{2}-2 D_{1} r_{o} \\cos (\\theta)}$" ]
[ -0.004058837890625, -0.0289306640625, -0.0074462890625, 0.0006256103515625, -0.00823974609375, 0.0177001953125, 0.00494384765625, -0.00762939453125, -0.0296630859375, -0.01422119140625, -0.0225830078125, -0.0062255859375, -0.032470703125, -0.033935546875, -0.00933837890625, -0.00631713...
1,239
3b Find $F_{f}$, the magnitude of the force produced on the Moon by the farthest point mass.
[ "$F_{f}=\\frac{G m M_{M}}{D_{1}^{2}+r_{o}^{2}+2 D_{1} r_{o} \\cos (\\theta)}$" ]
[ -0.01019287109375, -0.0283203125, -0.01470947265625, 0.0006103515625, -0.01611328125, 0.0028228759765625, -0.0024871826171875, -0.016357421875, -0.0206298828125, -0.01806640625, -0.0269775390625, -0.006195068359375, -0.01458740234375, -0.030517578125, -0.025390625, -0.01806640625, -0...
1,240
3c Find the magnitude of $\boldsymbol{\tau}_{c}$, the torque produced by the closest point mass.
[ "$\\tau_{c}=\\frac{G m M_{M} \\sin (\\theta) r_{0} D_{1}}{\\left[D_{1}^{2}+r_{o}^{2}-2 D_{1} r_{o} \\cos (\\theta)\\right]^{3 / 2}}$" ]
[ 0.000133514404296875, -0.0091552734375, -0.004913330078125, 0.00537109375, 0.0133056640625, 0.021728515625, -0.0032958984375, -0.01025390625, -0.013671875, -0.0101318359375, -0.0286865234375, -0.00872802734375, -0.021484375, -0.0301513671875, 0.0198974609375, -0.0172119140625, -0.012...
1,241
3d Find the magnitude of $\tau_{f}$, the torque produced by the farthest point mass.
[ "$\\tau_{f}=\\frac{G m M_{M} \\sin (\\theta) r_{0} D_{1}}{\\left[D_{1}^{2}+r_{o}^{2}+2 D_{1} r_{o} \\cos (\\theta)\\right]^{3 / 2}}$" ]
[ 0.00115203857421875, -0.016845703125, -0.0093994140625, 0.00775146484375, 0.004974365234375, 0.00921630859375, -0.0118408203125, -0.0174560546875, -0.0186767578125, -0.006103515625, -0.049072265625, -0.006744384765625, -0.00982666015625, -0.026123046875, 0.00830078125, -0.03076171875, ...
1,242
3e Find the magnitude of the total torque $\tau$ produced by the two masses. Since $r_{o} \ll D_{1}$ you should approximate your expression to lowest significant order in $r_{o} / D_{1}$. You may use that $(1+x)^{a} \approx 1+a x$, if $x \ll 1$.
[ "$\\frac{6 G m M_{M} r_{o}^{2} \\sin (\\theta) \\cos (\\theta)}{D_{1}^{3}}$" ]
[ -0.001922607421875, -0.005035400390625, 0.005706787109375, -0.0084228515625, 0.009521484375, 0.026123046875, 0.00011444091796875, -0.0234375, -0.0380859375, 0.01446533203125, -0.04052734375, -0.0032806396484375, -0.0152587890625, 0.0011138916015625, 0.0284423828125, -0.006591796875, ...
1,243
3f Calculate the numerical value of the total torque $\tau$, taking into account 0.5 that $\theta=3^{\circ}$ and that $m=3.6 \times 10^{16} \mathrm{~kg}$ (note that this mass is of the order of $10^{-8}$ times the mass of the Earth).
[ "$4.1 \\times 10^{16} $" ]
[ 0.019775390625, 0.0027313232421875, -0.0216064453125, 0.01495361328125, -0.01409912109375, 0.031005859375, -0.004364013671875, 0.01324462890625, -0.03955078125, -0.001312255859375, -0.049072265625, 0.013427734375, -0.0208740234375, -0.028076171875, 0.01953125, -0.0185546875, -0.01074...
1,244
3g Find the increase in the distance Earth-Moon at present, per year.
[ "0.034" ]
[ -0.00396728515625, -0.029052734375, -0.023193359375, -0.01519775390625, -0.00830078125, 0.018310546875, 0.0014495849609375, 0.0012359619140625, -0.027587890625, -0.028076171875, -0.03076171875, 0.024658203125, -0.01470947265625, -0.0289306640625, -0.004791259765625, -0.009521484375, ...
1,245
3h Find the decrease of $\omega_{E1}$ per year and how much is the length of the day at present increasing each year.
[ "$\\Delta \\omega_{E1}=-\\frac{\\tau \\Delta t}{I_E}$ , $\\Delta P_E=1.9\\times 10^{-5}s$" ]
[ -0.00555419921875, -0.036865234375, -0.026123046875, 0.01251220703125, 0.002197265625, 0.024658203125, -0.0040283203125, -0.0205078125, -0.0011138916015625, -0.01495361328125, -0.0291748046875, 0.0191650390625, 0.0069580078125, -0.0302734375, 0.01177978515625, 0.00970458984375, -0.02...
1,246
4a Write down an equation for the total (rotational plus gravitational) energy of the Earth-Moon system at present, $E$. Put this equation in terms of $I_{E}$, $u_{E 1}, M_{M}, M_{E}, D_{1}$ and $G$ only.
[ "$E=\\frac{1}{2} I_{E} \\omega_{E 1}^{2}-\\frac{1}{2} \\frac{G M_{E} M_{M}}{D_{1}}$" ]
[ -0.007568359375, -0.0262451171875, 0.005615234375, 0.005615234375, -0.000568389892578125, 0.0118408203125, -0.01031494140625, 0.01123046875, -0.043212890625, 0.00531005859375, -0.0211181640625, -0.002227783203125, -0.037353515625, -0.04150390625, 0.00799560546875, -0.0016632080078125, ...
1,247
4b Write down an equation for the change in $E, \Delta E$, as a function of the changes in $D_{1}$ and in $\boldsymbol{a}_{E 1}$. Evaluate the numerical value of $\Delta E$ for a year, using the values of changes in $D_{1}$ and in $a_{E 1}$ found in questions $3 \mathrm{~g}$ and $3 h$.
[ "$\\Delta E=I_{E} \\omega_{E 1} \\Delta \\omega_{E 1}+\\frac{1}{2} \\frac{G M_{E} M_{M}}{D_{1}^{2}} \\Delta D_{1}$ , $-9.0 \\times 10^{19}$" ]
[ -0.000362396240234375, -0.023193359375, -0.004608154296875, -0.022216796875, 0.016357421875, 0.0045166015625, -0.0004138946533203125, -0.001922607421875, -0.00897216796875, 0.009765625, -0.045654296875, 0.02587890625, -0.03564453125, -0.0286865234375, 0.007476806640625, 0.0087890625, ...
1,248
4c What is the mass of this surface layer of water?
[ "$2.6 \\times 10^{17} $" ]
[ 0.0194091796875, -0.0289306640625, -0.0174560546875, 0.0177001953125, -0.01611328125, 0.02001953125, -0.0096435546875, -0.0179443359375, -0.013916015625, -0.0234375, -0.01397705078125, -0.00909423828125, -0.0234375, -0.032470703125, 0.0020751953125, -0.01153564453125, -0.014892578125...
1,250
1a Write down the resonance condition for the absorption of the photon.
[ "$\\omega_{L}\\left(1+\\frac{v}{C}\\right)$" ]
[ 0.01287841796875, -0.026611328125, 0.0240478515625, -0.00182342529296875, 0.00021648406982421875, 0.00095367431640625, -0.004241943359375, -0.0213623046875, 0.001678466796875, -0.00640869140625, -0.0322265625, 0.00274658203125, -0.004486083984375, -0.01043701171875, -0.017333984375, -0...
1,251
1b Write down the momentum $p_{a t}$ of the atom after absorption, as seen in the laboratory.
[ "$m v-\\frac{\\hbar \\omega_{L}}{c}$" ]
[ 0.0247802734375, -0.006744384765625, 0.04296875, 0.00958251953125, 0.0076904296875, 0.007476806640625, -0.00445556640625, 0.00830078125, -0.0037384033203125, 0.00020694732666015625, -0.037353515625, 0.0091552734375, -0.0284423828125, -0.0203857421875, 0.0113525390625, -0.00055313110351...
1,252
1c Write down the total energy $\varepsilon_{a t}$ of the atom after absorption, as seen in the 0.2 laboratory.
[ "$\\frac{m v^{2}}{2}+\\hbar \\omega_{L}$" ]
[ 0.019775390625, -0.0098876953125, 0.0380859375, 0.00726318359375, 0.019775390625, -0.013427734375, 0.0030364990234375, -0.01300048828125, -0.0091552734375, 0.0218505859375, -0.0380859375, 0.00099945068359375, -0.025146484375, -0.02490234375, 0.0234375, -0.017333984375, -0.01843261718...
1,253
2a Write down the energy of the emitted photon, $\varepsilon_{p h}$, after the emission process in the $-x$ direction, as seen in the laboratory.
[ "$\\hbar \\omega_{L}$" ]
[ -0.0050048828125, -0.0224609375, 0.036376953125, 0.0152587890625, 0.02197265625, 0.0003719329833984375, -0.0081787109375, 0.00689697265625, -0.0164794921875, 0.0179443359375, -0.03076171875, 0.0027313232421875, -0.0247802734375, -0.0179443359375, 0.000701904296875, -0.0005340576171875,...
1,254
2b Write down the momentum of the emitted photon $p_{p h}$, after the emission process in the $-x$ direction, as seen in the laboratory.
[ "$-\\hbar \\omega_{L} / c$" ]
[ -0.004547119140625, -0.0126953125, 0.029052734375, 0.01220703125, 0.0166015625, 0.00927734375, -0.0123291015625, 0.01116943359375, -0.01446533203125, 0.00811767578125, -0.0283203125, 0.0004711151123046875, -0.0284423828125, -0.01470947265625, -0.01409912109375, 0.008056640625, -0.009...
1,255
2c Write down the momentum of the atom $p_{a t}$, after the emission process in the $-x$ direction, as seen in the laboratory.
[ "$m v$" ]
[ 0.009521484375, 0.00186920166015625, 0.037109375, 0.013916015625, 0.0111083984375, 0.00848388671875, -0.017822265625, 0.009521484375, -0.0113525390625, 0.006072998046875, -0.0361328125, -0.0030364990234375, -0.01953125, -0.0250244140625, 0.0020904541015625, 0.006439208984375, 0.00396...
1,256
2d Write down the total energy of the atom $\varepsilon_{a t}$, after the emission process in the $-x$ direction, as seen in the laboratory.
[ "$\\frac{m v^{2}}{2}$" ]
[ 0.0035552978515625, -0.00897216796875, 0.036865234375, 0.01287841796875, 0.0172119140625, -0.00152587890625, -0.017578125, -0.004241943359375, -0.0162353515625, 0.0230712890625, -0.03857421875, -0.0048828125, -0.0247802734375, -0.0244140625, 0.0252685546875, -0.0048828125, -0.0267333...
1,257
3a Write down the energy of the emitted photon, $\varepsilon_{p h}$, after the emission process in the $+x$ direction, as seen in the laboratory.
[ "$\\hbar \\omega_{L}\\left(1+2 \\frac{v}{c}\\right)$" ]
[ -0.004791259765625, -0.024658203125, 0.030517578125, 0.01214599609375, 0.0201416015625, -0.00186920166015625, -0.0062255859375, 0.0036773681640625, -0.004638671875, 0.01708984375, -0.0322265625, 0.012451171875, -0.025634765625, -0.0341796875, 0.006439208984375, 0.000705718994140625, ...
1,258
3b Write down the momentum of the emitted photon $p_{p h}$, after the emission process in the $+x$ direction, as seen in the laboratory.
[ "$\\frac{\\hbar \\omega_{L}}{c}\\left(1+2 \\frac{v}{c}\\right)$" ]
[ -0.003997802734375, -0.0166015625, 0.0260009765625, 0.0086669921875, 0.01507568359375, 0.00830078125, -0.00787353515625, 0.01141357421875, -0.00421142578125, 0.006591796875, -0.0283203125, 0.0081787109375, -0.02783203125, -0.0296630859375, -0.00787353515625, 0.01220703125, -0.0085449...
1,259
3c Write down the momentum of the atom $p_{a t}$, after the emission process in the $+x$ direction, as seen in the laboratory.
[ "$m v-2 \\frac{\\hbar \\omega_{L}}{c}$" ]
[ 0.0125732421875, -0.002288818359375, 0.03271484375, 0.01092529296875, 0.00958251953125, 0.00872802734375, -0.0157470703125, 0.00799560546875, -0.0020599365234375, 0.005126953125, -0.039306640625, 0.006103515625, -0.019775390625, -0.038818359375, 0.006744384765625, 0.00958251953125, 0...
1,260
3d Write down the total energy of the atom $\varepsilon_{a t}$, after the emission process in the $+x$ direction, as seen in the laboratory.
[ "$\\frac{m v^{2}}{2}\\left(1-2 \\frac{\\hbar q}{m v}\\right)$" ]
[ 0.005218505859375, -0.01080322265625, 0.034912109375, 0.012939453125, 0.0152587890625, -0.002288818359375, -0.0126953125, -0.0019073486328125, -0.00830078125, 0.02099609375, -0.040283203125, 0.00189971923828125, -0.02294921875, -0.03564453125, 0.02685546875, -0.003875732421875, -0.02...
1,261
4a Write down the average energy of an emitted photon, $\varepsilon_{p h}$, after the <br> emission process.
[ "$\\hbar \\omega_{L}\\left(1+\\frac{v}{c}\\right)$" ]
[ 0.00099945068359375, -0.0191650390625, 0.0262451171875, 0.00160980224609375, 0.0234375, 0.0079345703125, -0.0029144287109375, -0.00113677978515625, -0.01446533203125, 0.007049560546875, -0.0301513671875, 0.01068115234375, -0.032470703125, -0.0289306640625, -0.0028076171875, 0.006011962...
1,262
4b Write down the average momentum of an emitted photon $p_{p h}$, after the <br> emission process.
[ "$m v\\left(\\frac{\\hbar q}{m v} \\frac{v}{c}\\right)$", "$0$" ]
[ 0.0030059814453125, -0.00653076171875, 0.021240234375, 0.0023040771484375, 0.015380859375, 0.025146484375, -0.0062255859375, 0.0106201171875, -0.01611328125, -0.00136566162109375, -0.0289306640625, 0.0034332275390625, -0.03857421875, -0.0247802734375, -0.01806640625, 0.0155029296875, ...
1,263
4c Write down the average total energy of the atom $\varepsilon_{a t}$, after the emission process.
[ "$\\frac{m v^{2}}{2}\\left(1-\\frac{\\hbar q}{m v}\\right)$" ]
[ 0.00113677978515625, -0.00408935546875, 0.0341796875, 0.007080078125, 0.018310546875, 0.006317138671875, -0.0123291015625, -0.0064697265625, -0.0177001953125, 0.0125732421875, -0.0341796875, -0.003387451171875, -0.0244140625, -0.040283203125, 0.0174560546875, -0.000598907470703125, -...
1,264
4d Write down the average momentum of the atom $p_{a t}$, after the emission process.
[ "$p-\\frac{\\hbar \\omega_{L}}{c}$" ]
[ 0.014404296875, 0.00927734375, 0.0322265625, 0.00384521484375, 0.01068115234375, 0.0257568359375, -0.0184326171875, 0.00677490234375, -0.01458740234375, 0.0000667572021484375, -0.03857421875, -0.000255584716796875, -0.0284423828125, -0.03466796875, 0.0010528564453125, 0.01043701171875,...
1,265
5a Write down the average energy change $\Delta \varepsilon$ of the atom after a complete <br> one-photon absorption-emission process.
[ "$-\\frac{1}{2} \\hbar \\omega_{L} \\frac{v}{C}$" ]
[ 0.0257568359375, -0.008056640625, 0.0247802734375, -0.01153564453125, 0.01385498046875, -0.0084228515625, -0.0027313232421875, -0.00860595703125, -0.0015106201171875, 0.0157470703125, -0.0361328125, 0.0091552734375, -0.0005645751953125, -0.038818359375, -0.00604248046875, 0.00154113769...
1,266
5b Write down the average momentum change $\Delta p$ of the atom after a complete one-photon absorption-emission process.
[ "$-\\frac{\\hbar \\omega_{L}}{c}$" ]
[ 0.0216064453125, -0.00537109375, 0.0233154296875, -0.006378173828125, 0.01104736328125, 0.0096435546875, -0.007080078125, 0.0059814453125, -0.0054931640625, 0.008544921875, -0.044189453125, 0.0036163330078125, -0.01519775390625, -0.03271484375, -0.0194091796875, 0.01123046875, 0.0126...
1,267
6a Write down the average energy change $\Delta \varepsilon$ of the atom after a complete one-photon absorption-emission process.
[ "$+\\frac{1}{2} \\hbar \\omega_{L}^{\\prime} \\frac{v}{c}$" ]
[ 0.0228271484375, -0.008056640625, 0.02197265625, -0.0002613067626953125, 0.0135498046875, -0.00592041015625, -0.00933837890625, -0.0145263671875, 0.004974365234375, 0.01458740234375, -0.04345703125, 0.00714111328125, -0.0084228515625, -0.037353515625, -0.002685546875, 0.0045166015625, ...
1,268
6b Write down the average momentum change $\Delta p$ of the atom after a complete one-photon absorption-emission process.
[ "$+\\frac{\\hbar \\omega_{L}^{\\prime}}{c}$" ]
[ 0.0198974609375, -0.00421142578125, 0.0184326171875, 0.0037384033203125, 0.0120849609375, 0.00982666015625, -0.01611328125, 0.0032501220703125, -0.00078582763671875, 0.006195068359375, -0.048583984375, 0.004974365234375, -0.018798828125, -0.034423828125, -0.0181884765625, 0.01214599609...
1,269
7a With the information found so far, find the force that the lasers exert on the atomic beam. You should assume that $m v \gg \hbar q$.
[ "$F=\\left(\\frac{\\Omega^2_R}{(\\omega_0 - \\omega_L + \\omega_L \\frac{v}{c})^2+\\frac{\\Gamma^2}{4}+2\\Omega^2_R} - \\frac{\\Omega^2_R}{(\\omega_0 - \\omega_L - \\omega_L \\frac{v}{c})^2+\\frac{\\Gamma^2}{4}+2\\Omega^2_R} \\right) N\\Gamma\\hbar q$" ]
[ 0.020751953125, -0.0118408203125, 0.003570556640625, -0.0011444091796875, 0.006988525390625, 0.0152587890625, 0.002349853515625, -0.02685546875, 0.0020599365234375, 0.0029449462890625, -0.049072265625, 0.0247802734375, 0.0020599365234375, -0.006103515625, 0.0120849609375, 0.00698852539...
1,270
8a Find an expression for the force found in Question (7a), in this limit.
[ "$-\\frac{4 N \\hbar q^{2} \\Omega_{R}^{2} \\Gamma}{\\left(\\left(\\omega_{0}-\\omega_{L}\\right)^{2}+\\frac{\\Gamma^{2}}{4}+2 \\Omega_{R}^{2}\\right)^{2}}\\left(\\omega_{0}-\\omega_{L}\\right) v$" ]
[ 0.025390625, -0.0189208984375, 0.0001049041748046875, -0.016357421875, -0.004425048828125, 0.0009765625, -0.00872802734375, -0.0179443359375, -0.0091552734375, -0.00555419921875, -0.056884765625, 0.01171875, -0.004180908203125, -0.01513671875, 0.0284423828125, 0.004425048828125, -0.0...
1,271
8b Write down the condition to obtain a positive force (speeding up the atoms).
[ "$\\omega_{0}<\\omega_{L}$" ]
[ 0.010498046875, -0.00836181640625, -0.00323486328125, 0.0089111328125, -0.0230712890625, 0.000835418701171875, -0.0028228759765625, -0.030517578125, 0.015625, 0.0042724609375, -0.0274658203125, 0.0172119140625, -0.012451171875, -0.035400390625, 0.0115966796875, 0.00592041015625, 0.01...
1,272
8c Write down the condition to obtain a zero force.
[ "$\\omega_{0}=\\omega_{L}$" ]
[ 0.0250244140625, -0.01116943359375, -0.0264892578125, 0.01025390625, 0.00628662109375, -0.0025482177734375, 0.004852294921875, -0.01165771484375, -0.0224609375, -0.0283203125, -0.041259765625, 0.0023651123046875, -0.00152587890625, -0.0026702880859375, 0.014404296875, 0.000595092773437...
1,273
8d Write down the condition to obtain a negative force (slowing down the atoms).
[ "$\\omega_{0}>\\omega_{L}$" ]
[ 0.01055908203125, 0.00069427490234375, 0.00262451171875, -0.00148773193359375, -0.01361083984375, -0.00787353515625, -0.0037841796875, -0.0208740234375, -0.010009765625, -0.014404296875, -0.029052734375, -0.00086212158203125, -0.009033203125, -0.0235595703125, 0.006500244140625, -0.000...
1,274
8e Consider now that the atoms are moving with a velocity $-v$ (in the $-x$ direction). Write down the condition to obtain a slowing down force on the atoms.
[ "$\\omega_{0}>\\omega_{L}$" ]
[ 0.0247802734375, 0.00421142578125, 0.0033721923828125, 0.018798828125, -0.0133056640625, -0.00335693359375, 0.004058837890625, -0.01025390625, -0.00286865234375, 0.00897216796875, -0.041015625, -0.00049591064453125, -0.01092529296875, -0.0189208984375, 0.014892578125, 0.0179443359375, ...
1,275
9a In the limit of low velocities, find the velocity of the atoms after the laser beams have been on for a time $\tau$.
[ "$v=v_{0} e^{-\\beta t / m}$" ]
[ 0.03125, 0.0087890625, 0.00909423828125, 0.0028076171875, 0.00653076171875, 0.006744384765625, 0.01275634765625, -0.01513671875, 0.01007080078125, 0.0023345947265625, -0.056640625, -0.00579833984375, -0.0029754638671875, -0.0167236328125, 0.02001953125, 0.01116943359375, -0.013244628...
1,276
9b Assume now that the gas of atoms is in thermal equilibrium at a temperature $T_{0}$. Find the temperature $T$ after the laser beams have been on for a time $\tau$.
[ "$T=T e^{-2 \\beta t / m}$" ]
[ 0.0135498046875, 0.00156402587890625, -0.0025787353515625, 0.00118255615234375, 0.0009765625, 0.0113525390625, 0.005645751953125, -0.0057373046875, -0.000125885009765625, 0.0196533203125, -0.04296875, -0.0145263671875, -0.01007080078125, -0.0068359375, 0.00640869140625, 0.0062561035156...
1,277
1a What has to be the temperature of the gas, $T_{c}$, so that the distance of closest approach of the protons, $d_{c}$, equals $10^{-15} \mathrm{~m}$ ? Give this and all numerical values in this problem up to two significant figures.
[ "$5.5 \\times 10^{9}$" ]
[ 0.00933837890625, 0.002593994140625, 0.006256103515625, 0.0103759765625, -0.0130615234375, 0.0205078125, 0.00823974609375, 0.0091552734375, -0.006988525390625, 0.0023193359375, -0.041015625, 0.01483154296875, -0.02294921875, -0.00823974609375, 0.010986328125, -0.0142822265625, -0.015...
1,278
2a Find an equation for the temperature at the center of the star, $T_{c}$, in terms of the radius and mass of the star and of physical constants only.
[ "$T_{c}=\\frac{G M m_{p}}{2 k R}$" ]
[ 0.01495361328125, -0.018310546875, 0.028076171875, 0.006317138671875, 0.019287109375, -0.0050048828125, -0.01177978515625, -0.0284423828125, -0.02099609375, 0.01153564453125, -0.025634765625, -0.01397705078125, -0.01007080078125, -0.01092529296875, 0.00811767578125, -0.020263671875, ...
1,279
2b Using the equation found in (2a) write down the ratio $M / R$ expected for a star in terms of physical constants and $T_{c}$ only.
[ "$\\frac{M}{R}=\\frac{2 k T_{c}}{G m_{p}}$" ]
[ 0.0152587890625, -0.0224609375, 0.01165771484375, -0.01708984375, 0.02734375, 0.00433349609375, -0.021728515625, -0.0206298828125, -0.03369140625, 0.005706787109375, -0.027099609375, 0.0118408203125, -0.00083160400390625, -0.0146484375, 0.006622314453125, -0.0208740234375, -0.0012893...
1,280
2c Use the value of $T_{c}$ derived in section (1a) and find the numerical value of the ratio $M / R$ expected for a star.
[ "$1.4 \\times 10^{24} $" ]
[ 0.0205078125, -0.010986328125, 0.0174560546875, -0.0230712890625, 0.025634765625, 0.0108642578125, -0.0194091796875, -0.017578125, -0.0380859375, 0.00164031982421875, -0.03759765625, 0.025634765625, 0.00250244140625, -0.033203125, 0.00151824951171875, -0.021240234375, -0.017333984375...
1,281
2d Now, calculate the ratio $M($ Sun $) / R($ Sun $)$, and verify that this value is much smaller than the one found in $(2 \mathrm{c})$.
[ "$2.9 \\times 10^{21}$" ]
[ 0.004150390625, 0.000629425048828125, 0.0001850128173828125, 0.00982666015625, 0.0146484375, 0.0164794921875, -0.0167236328125, -0.0201416015625, -0.03369140625, 0.0087890625, -0.037841796875, 0.026123046875, -0.01153564453125, -0.017333984375, 0.0220947265625, -0.0081787109375, -0.0...
1,282
3a Assuming that $d_{c}=\frac{\lambda_{p}}{2^{1 / 2}}$ is the condition that allows fusion, for a proton <br> with velocity $v_{r m s}$, find an equation for $T_{c}$ in terms of physical constants <br> only.
[ "$T_{c}=\\frac{q^{4} m_{p}}{24 \\pi^{2} \\varepsilon_{0}^{2} k h^{2}}$" ]
[ 0.0235595703125, -0.0191650390625, -0.0001277923583984375, 0.01507568359375, 0.0228271484375, 0.005401611328125, -0.0014190673828125, 0.00086212158203125, -0.0166015625, 0.0135498046875, -0.0257568359375, 0.004669189453125, -0.0052490234375, -0.013671875, -0.002227783203125, 0.00207519...
1,283
3b Evaluate numerically the value of $T_{c}$ obtained in (3a).
[ "$9.7 \\times 10^{6}$" ]
[ 0.032470703125, 0.00677490234375, 0.0208740234375, 0.0096435546875, 0.034423828125, 0.017822265625, 0.00848388671875, -0.00091552734375, -0.001251220703125, 0.00445556640625, -0.0228271484375, 0.01116943359375, 0.005126953125, -0.023193359375, 0.00836181640625, -0.00830078125, -0.037...
1,284
3c Use the value of $T_{c}$ derived in (3b) to find the numerical value of the 0.5 ratio $M / R$ expected for a star, using the formula derived in (2b). Verify that this value is quite similar to the ratio $M($ Sun $) / R($ Sun $)$ observed.
[ "$2.4 \\times 10^{21}$" ]
[ 0.018310546875, -0.01226806640625, 0.0191650390625, -0.01129150390625, 0.025634765625, 0.01092529296875, -0.00860595703125, -0.0101318359375, -0.034423828125, 0.009765625, -0.03076171875, 0.0247802734375, 0.0018157958984375, -0.03564453125, -0.0031280517578125, -0.0260009765625, -0.0...
1,285
4a Use the previous results to demonstrate that for any star fusing hydrogen, the ratio of mass $M$ to radius $R$ is the same and depends only on physical constants. Find the equation for the ratio $M / R$ for stars fusing hydrogen.
[ "$\\frac{M}{R}=\\frac{q^{4}}{12 \\pi^{2} \\varepsilon_{0}^{2} G h^{2}}$" ]
[ 0, -0.0157470703125, 0.00439453125, -0.008056640625, 0.007415771484375, 0.0230712890625, -0.01275634765625, -0.016357421875, -0.026611328125, 0.007110595703125, -0.038818359375, 0.00531005859375, -0.0194091796875, -0.0279541015625, -0.00106048583984375, -0.034423828125, -0.0152587890...
1,286
5a Find an equation for $n_{e}$, the average electron number density inside the <br> star.
[ "$n_{e}=\\frac{M}{(4 / 3) \\pi R^{3} m_{p}}$" ]
[ 0.01708984375, 0.01434326171875, 0.021728515625, -0.0108642578125, -0.00165557861328125, 0.0206298828125, -0.015869140625, -0.017333984375, -0.0247802734375, 0.000194549560546875, -0.033935546875, -0.0078125, -0.01226806640625, -0.0208740234375, -0.015869140625, -0.006988525390625, -...
1,287
5b Find an equation for $d_{e}$, the typical separation between electrons inside <br> the star.
[ "$d_{e}=\\left(\\frac{M}{(4 / 3) \\pi R^{3} m_{p}}\\right)^{-1 / 3}$" ]
[ 0.022705078125, 0.00787353515625, 0.0177001953125, -0.021484375, -0.006500244140625, -0.00799560546875, -0.00775146484375, -0.0024261474609375, -0.0169677734375, -0.0096435546875, -0.036865234375, 0.0028533935546875, -0.00811767578125, -0.017578125, 0.01019287109375, 0.00860595703125, ...
1,289
5d Find the numerical value of the radius of the smallest normal star possible, in meters.
[ "$6.9 \\times 10^{7}$" ]
[ -0.006439208984375, 0.0089111328125, 0.00421142578125, -0.028564453125, -0.02294921875, 0.03125, -0.0172119140625, -0.007080078125, -0.030517578125, -0.005645751953125, -0.033447265625, 0.010986328125, -0.003936767578125, -0.033447265625, -0.0169677734375, -0.0186767578125, -0.022460...
1,290
5e Find the numerical value of the mass of the smallest normal star possible, in $\mathrm{kg}$.
[ "$1.7 \\times 10^{29}$" ]
[ 0.0033111572265625, 0.005706787109375, 0.0218505859375, -0.00830078125, -0.016357421875, 0.0458984375, 0.003265380859375, -0.0034332275390625, -0.0196533203125, -0.00531005859375, -0.0361328125, -0.01025390625, -0.01031494140625, -0.0257568359375, -0.019287109375, -0.01287841796875, ...
1,291
6a Set the equivalent condition for helium nuclei and find $v_{r m s}(H e)$, the rms velocity of the helium nuclei and $T(\mathrm{He})$, the temperature needed for helium fusion.
[ "$2.0 \\times 10^{6}$ , $6.5 \\times 10^{8}$" ]
[ 0.026611328125, -0.0048828125, -0.0233154296875, 0.051025390625, -0.009033203125, 0.02197265625, 0.005950927734375, 0.0223388671875, -0.004119873046875, 0.0029296875, -0.0162353515625, 0.00060272216796875, -0.0118408203125, -0.01422119140625, -0.0020294189453125, -0.0108642578125, -0...
1,292
A.1 Consider a segment of length $\Delta \ell$ of the unstretched ZLS spring which is then stretched by a force $F$, under weightless conditions. What is the length $\Delta y$ of this segment as a function of $F, \Delta \ell$ and the parameters of the spring?
[ "$\\Delta y=\\max \\left\\{\\frac{F}{k L_{0}} \\Delta l, \\Delta l\\right\\}$ , $k^{*}=k \\frac{L_{0}}{\\Delta l}$" ]
[ 0.03662109375, -0.0096435546875, -0.00872802734375, -0.01397705078125, 0.00762939453125, -0.00018310546875, -0.004730224609375, 0.0079345703125, 0.00213623046875, -0.017822265625, -0.051513671875, -0.0106201171875, 0.003692626953125, -0.01336669921875, 0.00555419921875, 0.0134887695312...
1,293
A.2 For a segment of length $\Delta \ell$, calculate the work $\Delta W$ required to stretch it from its original length $\Delta \ell$ to a length $\Delta y$.
[ "$\\Delta W=\\frac{k L_{0}}{2 \\Delta l}\\left(\\Delta y^{2}-\\Delta l^{2}\\right)$" ]
[ 0.02880859375, -0.01361083984375, -0.01507568359375, -0.0091552734375, -0.023193359375, -0.0216064453125, -0.020263671875, 0.0028533935546875, 0.0125732421875, -0.0247802734375, -0.0302734375, 0.00160980224609375, 0.005889892578125, -0.0130615234375, 0.00628662109375, -0.00405883789062...
1,294
A.3 Suppose that we hang the spring by its top end, so that it stretches under its own weight. What is the total length $H$ of the suspended spring in equilibrium? Express your answers in terms of $L_{0}$ and $\alpha$.
[ "$H=\\frac{L_{0}}{2}(\\alpha+\\frac{1}{\\alpha})$" ]
[ 0.017333984375, -0.02783203125, 0.0159912109375, -0.0074462890625, -0.01611328125, 0.004852294921875, -0.01129150390625, -0.0115966796875, -0.01275634765625, -0.02734375, -0.03125, -0.013916015625, -0.0179443359375, -0.0125732421875, 0.01409912109375, 0.00811767578125, -0.02294921875...
1,295
B.1 Calculate the time $t_{c}$ it takes from the moment the spring is released, until it fully collapses back to its minimal length $L_{0}$. Express your answer in terms of $L_{0}, g$ and $\alpha$. Compute the numerical value of $t_{c}$ for a spring with $k=1.02 \mathrm{~N} / \mathrm{m}, L_{0}=0.055 \mathrm{~m}$ and $...
[ "$t_{c}=\\sqrt{\\frac{L_{0}}{3 g \\alpha}(1-\\alpha)^{3}}$ , $t_{c}=0.245$" ]
[ 0.01806640625, -0.0146484375, -0.00335693359375, 0.0185546875, 0.01165771484375, 0.0181884765625, -0.01116943359375, -0.002655029296875, -0.007293701171875, -0.0179443359375, -0.03271484375, 0.0010986328125, -0.03076171875, -0.02001953125, 0.0185546875, -0.01129150390625, -0.00463867...
1,296
B.2 In this task $\ell$ is used to denote the coordinate of the boundary between parts I (in figure 2, the moving part) and II (the stationary part). At a certain moment, while a stationary part still exists its mass is $m(\ell)=\frac{\ell}{L_{0}} M$, and the moving part moves with uniform instantaneous velocity $v_{I}...
[ "$A=\\frac{2 g}{3 \\alpha}$ , $B=(\\frac{1}{3 \\alpha}-1) g L_{0}$" ]
[ 0.05078125, -0.0299072265625, -0.0191650390625, -0.024658203125, -0.00677490234375, 0.0234375, 0.00506591796875, 0.007293701171875, 0.0067138671875, -0.037109375, -0.03466796875, 0.0033111572265625, -0.01031494140625, -0.021484375, 0.03271484375, 0.017333984375, 0.009033203125, 0.0...
1,297
B.3 Based on B.2, find the minimum speed $v_{\min }$ of the moving part of the spring in the course of its motion, after its release and before it hits the ground. Express your answer in terms of $L_{0}, \alpha, A$ and $B$.
[ "$v_{min}=(1-\\alpha) \\sqrt{A \\alpha L_{0}+B}$" ]
[ 0.032958984375, -0.03125, -0.01214599609375, -0.0194091796875, -0.005584716796875, 0.0208740234375, 0.003875732421875, 0.008056640625, -0.00408935546875, -0.014404296875, -0.031005859375, -0.005340576171875, -0.036376953125, -0.00159454345703125, 0.041748046875, -0.00001549720764160156...
1,298
C.1 Calculate the amount of mechanical energy $Q$ that was lost by generating heat, from the moment the spring is released until just before the spring hits the ground. Express your answer in terms of $L_{0}, M, g$ and $\alpha$.
[ "$W=M g L_{0} \\frac{(1-\\alpha)^{2}(2 \\alpha+1)}{6 \\alpha}$" ]
[ 0.0235595703125, -0.028564453125, -0.00946044921875, 0.0166015625, 0.007354736328125, -0.0004062652587890625, 0.0128173828125, 0.00555419921875, 0.0020294189453125, -0.015869140625, -0.03857421875, -0.001220703125, -0.0230712890625, -0.0244140625, 0.0220947265625, -0.0031585693359375, ...
1,299
A.1 Use the above data to estimate the frequency $f_{\text {est }}$ of a single resonator. (Your result may differ from the actual value, $f=2.45 \mathrm{GHz}$. Use the actual value in the remainder of the question.)
[ "$2.0 \\times 10^{9}$" ]
[ 0.017578125, 0.00146484375, 0.0213623046875, 0.01904296875, 0.0166015625, 0.0286865234375, 0.032470703125, -0.0235595703125, -0.01153564453125, 0.00830078125, -0.0301513671875, 0.0155029296875, 0.00469970703125, -0.036865234375, 0.0012664794921875, -0.021240234375, -0.041748046875, ...
1,300
A.2 In each of the following two cases, find $\vec{u}_{D}$ if: 1. at $t=0$ the electron velocity is $\vec{u}(0)=\left(3 E_{0} / B_{0}\right) \hat{x}$ 2. at $t=0$ the electron velocity is $\vec{u}(0)=-\left(3 E_{0} / B_{0}\right) \hat{x}$.
[ "$u^{\\prime}=4 E_{0} / B_{0}$,$u^{\\prime}=2 E_{0} / B_{0}$" ]
[ 0.0244140625, -0.0185546875, -0.002685546875, 0.0023651123046875, 0.000823974609375, 0.0264892578125, 0.015869140625, -0.006195068359375, -0.0224609375, -0.005645751953125, -0.055419921875, -0.012939453125, -0.0302734375, -0.0096435546875, 0.01226806640625, 0.00762939453125, -0.01647...
1,301
A.3 Numerically estimate the maximal radius $r$ of the electron motion trajectory in the reference frame in which this motion is approximately circular, considering this reference frame as approximately inertial.
[ "0.3" ]
[ 0.037841796875, -0.021728515625, 0.0054931640625, 0.0213623046875, -0.0289306640625, 0.0213623046875, -0.0045166015625, -0.004119873046875, -0.04345703125, 0.000614166259765625, -0.042236328125, 0.036376953125, -0.021728515625, -0.046875, 0.03076171875, -0.00421142578125, -0.01599121...
1,302
A.7 Find an approximate expression for the static voltage $V_{0}$ required for operating the magnetron in the manner described. (The expression you will find gives an approximation for the minimal value required for the magnetron operation; the optimal voltage is somewhat higher.)
[ "$V_{0}=\\pi f B_{0}\\left(b^{2}-a^{2}\\right) / 4$" ]
[ 0.0096435546875, 0.0147705078125, 0.00009012222290039062, -0.0087890625, -0.0147705078125, 0.01458740234375, 0.02685546875, -0.00130462646484375, -0.013427734375, 0.005889892578125, -0.043212890625, 0.01611328125, 0.0003719329833984375, -0.004913330078125, -0.002899169921875, -0.016479...
1,303
B.1 Write expressions for both the magnitude of the torque $\tau(t)$ applied by the electric field on the dipole and the power $H_{i}(t)$ delivered by the field to the dipole, in terms of $p_{0}, E(t), \theta(t)$ and their derivatives.
[ "$\\tau(t)=-p_{0} \\sin [\\theta(t)] E(t)$ , $H_{i}(t)=-p_{0} E(t) \\sin \\theta(t) \\dot{\\theta}(t)$" ]
[ 0.01513671875, -0.01129150390625, -0.0272216796875, 0.0087890625, 0.0205078125, 0.0162353515625, -0.01422119140625, -0.0235595703125, -0.004974365234375, -0.0181884765625, -0.04833984375, 0.0152587890625, -0.00994873046875, -0.01220703125, 0.01953125, -0.0062255859375, -0.00473022460...
1,304
B.2 Find an expression for the time-averaged power $\langle H(t)\rangle$ per unit volume absorbed by the water. The time-average for a time dependent periodic variable $f(t)$ over its period $T$ is defined as: $$ \langle f(t)\rangle=\frac{1}{T} \int_{t_{0}}^{t_{0}+T} f(t) \mathrm{d} t \tag{1} $$
[ "$\\langle H(t)\\rangle=0.5 E_{0}^{2} \\beta \\varepsilon_{0} \\omega_{f} \\sin \\delta$" ]
[ 0.031494140625, -0.001251220703125, -0.006622314453125, 0.007110595703125, 0.01220703125, 0.0198974609375, -0.0123291015625, -0.046142578125, 0.00897216796875, 0.0024566650390625, -0.025146484375, 0.0185546875, -0.01300048828125, -0.0283203125, 0.01806640625, -0.0240478515625, -0.032...
1,305
B.3 Let us denote the time-averaged radiation energy flux density by $I(z)$ (average radiation power flow per unit area). Here $z$ is the depth of penetration into the water, and the radiation propagates in the $z$ direction. Find an expression for the dependence of the flux density $I(z)$ on $z$. The flux density at t...
[ "$I(z)=I(0) e^{[-z \\beta \\omega \\sin \\delta /(c \\sqrt{\\varepsilon_{r}})]}$" ]
[ 0.0028076171875, -0.005035400390625, -0.0093994140625, 0.021484375, -0.0157470703125, 0.00592041015625, 0.01104736328125, -0.022216796875, 0.003570556640625, 0.00714111328125, -0.01513671875, 0.00946044921875, -0.00726318359375, -0.0216064453125, 0.00091552734375, 0.00390625, -0.0312...
1,306
B.4 Employ the approximation $\tan \delta \approx \sin \delta$ and find an expression for the coefficient $\beta$ defined in Task B. 2 in terms of the other parameters.
[ "$\\beta=\\varepsilon_{r}$" ]
[ 0.01397705078125, -0.001312255859375, -0.0157470703125, 0.00188446044921875, -0.004852294921875, 0.0078125, 0.002685546875, -0.0023193359375, -0.0013580322265625, -0.01116943359375, -0.048095703125, 0.046875, -0.0034942626953125, -0.01953125, 0.00177764892578125, -0.007415771484375, ...
1,307
B.5 Use Figure 5 to address the following question: For water at $20^{\circ} \mathrm{C}$, find the penetration depth $z_{1 / 2}$ at which the power per unit volume is reduced to half of its value at $z=0$.
[ "$12$" ]
[ 0.0107421875, -0.00616455078125, -0.0184326171875, 0.01446533203125, 0.0089111328125, 0.02001953125, 0.0111083984375, -0.03271484375, -0.0103759765625, -0.01251220703125, -0.036865234375, 0.003997802734375, 0.0032806396484375, -0.004486083984375, -0.0101318359375, -0.000135421752929687...
1,308
A.1 When a standing sound wave forms, the gas elements oscillate in the $x$ direction with angular frequency $\omega$. The amplitude of the oscillations depends on each element's equilibrium position $x$ along the tube. The longitudinal displacement of each gas element from its equilibrium position $x$ is given by $$ ...
[ "$\\lambda_{\\max }=2 L$" ]
[ 0.0128173828125, 0.0172119140625, 0.01416015625, 0.000782012939453125, -0.0177001953125, -0.00531005859375, -0.00933837890625, -0.0228271484375, 0.035400390625, 0.0002002716064453125, -0.043701171875, 0.0025482177734375, -0.0150146484375, -0.0238037109375, 0.0042724609375, 0.0016174316...