shayekh/physics_gre huggingface dataset:
Two wedges, each of mass $m$, are placed next to each other on a flat floor. A cube of mass $M$ is balanced on the wedges as shown above. Assume no friction between the cube and the wedges, but a coefficient of static friction $\mu<1$ between the wedges and the floor. What is the largest $M$ that can be balanced as shown without motion of the wedges?

A. $\frac{m}{\sqrt{2}}$

B. $\frac{\mu m}{\sqrt{2}}$

C. $\frac{\mu m}{1-\mu}$

D. $\frac{2 \mu m}{1-\mu}$

E. All $M$ will balance.


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A cylindrical tube of mass $M$ can slide on a horizontal wire. Two identical pendulums, each of mass $m$ and length $\ell$, hang from the ends of the tube, as shown above. For small oscillations of the pendulums in the plane of the paper, the eigenfrequencies of the normal modes of oscillation of this system

are $0, \sqrt{\frac{g(M+2 m)}{\ell M}}$, and

A. $\sqrt{\frac{g}{\ell}}$

B. $\sqrt{\frac{g}{\ell} \frac{M+m}{M}}$

C. $\sqrt{\frac{g}{\ell} \frac{m}{M}}$

D. $\sqrt{\frac{g}{\ell} \frac{m}{M+m}}$

E. $\sqrt{\frac{g}{\ell} \frac{m}{M+2 m}}$


Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
A solid cone hangs from a frictionless pivot at the origin $O$, as shown above. If $\hat{\mathbf{i}}, \hat{\mathbf{j}}$, and $\hat{\mathbf{k}}$ are unit vectors, and $a, b$, and $c$ are positive constants, which of the following forces $F$ applied to the rim of the cone at a point $P$ results in a torque $\tau$ on the cone with a negative component $\tau_{z}$ ?

A. $\mathbf{F}=a \hat{\mathbf{k}}, P$ is $(0, b,-c)$

B. $\mathbf{F}=-a \hat{\mathbf{k}}, P$ is $(0,-b,-c)$

C. $\mathbf{F}=a \hat{\mathbf{j}}, P$ is $(-b, 0,-c)$

D. $\mathbf{F}=a \hat{\mathbf{j}}, P$ is $(b, 0,-c)$

E. $\mathbf{F}=-a \hat{\mathbf{k}}, P$ is $(-b, 0,-c)$


Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
A coaxial cable having radii $a, b$, and $c$ carries equal and opposite currents of magnitude $i$ on the inner and outer conductors. What is the magnitude of the magnetic induction at point $P$ outside of the cable at a distance $r$ from the axis?
A. Zero
B. $\frac{\mu_{0} i r}{2 \pi a^{2}}$
C. $\frac{\mu_{0} i}{2 \pi r}$
D. $\frac{\mu_{0} i}{2 \pi r} \frac{c^{2}-r^{2}}{c^{2}-b^{2}}$
E. $\frac{\mu_{0} i}{2 \pi r} \frac{r^{2}-b^{2}}{c^{2}-b^{2}}$


Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
Two positive charges of $q$ and $2 q$ coulombs are located on the $x$-axis at $x=0.5 a$ and $1.5 a$, respectively, as shown above. There is an infinite, grounded conducting plane at $x=0$. What is the magnitude of the net force on the charge $q$ ?

A. $\frac{1}{4 \pi \epsilon_{0}} \frac{q^{2}}{a^{2}}$

B. $\frac{1}{4 \pi \epsilon_{0}} \frac{3 q^{2}}{2 a^{2}}$

C. $\frac{1}{4 \pi \epsilon_{0}} \frac{2 q^{2}}{a^{2}}$

D. $\frac{1}{4 \pi \epsilon_{0}} \frac{3 q^{2}}{a^{2}}$

E. $\frac{1}{4 \pi \epsilon_{0}} \frac{7 q^{2}}{2 a^{2}}$


Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
The capacitor in the circuit shown above is initially charged. After closing the switch, how much time elapses until one-half of the capacitor's initial stored energy is dissipated?

A. $R C$

B. $\frac{R C}{2}$

C. $\frac{R C}{4}$

D. $2 R C \ln (2)$

E. $\frac{R C \ln (2)}{2}$


Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
Two large conducting plates form a wedge of angle $\alpha$ as shown in the diagram above. The plates are insulated from each other; one has a potential $V_{0}$ and the other is grounded. Assuming that the plates are large enough so that the potential difference between them is independent of the cylindrical coordinates $z$ and $\rho$, the potential anywhere between the plates as a function of the angle $\varphi$ is

A. $\frac{V_{0}}{\alpha}$

B. $\frac{V_{0} \varphi}{\alpha}$

C. $\frac{V_{0} \alpha}{\varphi}$

D. $\frac{V_{0} \varphi^{2}}{\alpha}$

E. $\frac{V_{0} \alpha}{\varphi^{2}}$

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
Listed below are Maxwell's equations of electromagnetism. If magnetic monopoles exist, which of these equations would be INCORRECT?
I. $\operatorname{Curl} \mathbf{H}=\mathbf{J}+\frac{\partial \mathbf{D}}{\partial t}$

II. $\operatorname{Curl} \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

III. $\operatorname{div} \mathbf{D}=\rho$

IV. $\operatorname{div} B=0$

A. IV only

B. I and II

C. I and III

D. II and IV

E. III and IV

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
The total energy of a blackbody radiation source is collected for one minute and used to heat water. The temperature of the water increases from $20.0^{\circ} \mathrm{C}$ to $20.5^{\circ} \mathrm{C}$. If the absolute temperature of the blackbody

ie doubled and the experiment repeated, which of the following statements would be most nearly corre t?

A. I the temperature of the water would increase from $20^{\circ} \mathrm{C}$ to a final temperature of $21^{\circ} \mathrm{C}$.

B. The temperature of the water would increase from $20^{\circ} \mathrm{C}$ to a final temperature of $24^{\circ} \mathrm{C}$.

C. The temperature of the water would increase from $20^{\circ} \mathrm{C}$ to a final temperature of $28^{\circ} \mathrm{C}$.

D. The temperature of the water would increase from $20^{\circ} \mathrm{C}$ to a final temperature of $36^{\circ} \mathrm{C}$.

E. The water would boil within the one-minute time period.

\section*{Orovono}

Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
A classical model of a diatomic molecule is a springy dumbbell, as shown above, where the dumbbell is free to rotate about axes perpendicular to the spring. In the limit of high temperature, what is the specific heat per mole at constant volume?

A. $\frac{3}{2} R$

B. $\frac{5}{2} R$

C. $\frac{7}{2} R$

D. $\frac{9}{2} R$

E. $\frac{11}{2} R$

Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
An engine absorbs heat at a temperature of $727^{\circ} \mathrm{C}$ and exhausts heat at a temperature of $527^{\circ} \mathrm{C}$. If the engine operates at maximum possible efficiency, for 2000 joules of heat input the amount of work the engine performs is most nearly

A. $400 \mathrm{~J}$

B. $1450 \mathrm{~J}$

C. $1600 \mathrm{~J}$

D. $2000 \mathrm{~J}$

E. $2760 \mathrm{~J}$

Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
The outputs of two electrical oscillators are compared on an oscilloscope screen. The oscilloscope spot is initially at the center of the screen. Oscillator $Y$ is connected to the vertical terminals of the oscilloscope and oscillator $X$ to the horizontal terminals. Which of the following patterns could appear on the oscilloscope screen, if the frequency of oscillator $Y$ is twice that of oscillator $X$ ?

A.


Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
In transmitting high frequency signals on a coaxial cable, it is important that the cable be terminated at an end with its characteristic impedance in order to avoid

A. leakage of the signal out of the cable

B. overheating of the cable

C. reflection of signals from the terminated end of the cable

D. attenuation of the signal propagating in the cable

E. production of image currents in the outer conductor

Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
Which of the following is most nearly the mass of the Earth? (The radius of the Earth is about $6.4 \times 10^{6}$ meters.)

A. $6 \times 10^{24} \mathrm{~kg}$

B. $6 \times 10^{27} \mathrm{~kg}$

C. $6 \times 10^{30} \mathrm{~kg}$

D. $6 \times 10^{33} \mathrm{~kg}$

E. $6 \times 10^{36} \mathrm{~kg}$


Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
In a double-slit interference experiment, $d$ is the distance between the centers of the slits and $w$ is the width of each slit, as shown in the figure above. For incident plane waves, an interference maximum on a distant screen will be "missing" when

A. $d=\sqrt{2} w$

B. $d=\sqrt{3} w$

C. $2 d=w$

D. $2 d=3 w$

E. $3 d=2 w$
Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A soap film with index of refraction greater than air is formed on a circular wire frame that is held in a vertical plane. The film is viewed by reflected light from a white-light source. Bands of color are observed at the lower parts of the soap film, but the area near the top appears black. A correct explanation for this phenomenon would involve which of the following?

I. The top of the soap film absorbs all of the light incident on it; none is transmitted.

II. The thickness of the top part of the soap film has become much less than a wavelength of visible light.

III. There is a phase change of $180^{\circ}$ for all wavelengths of light reflected from the front surface of the soap film.

IV. There is no phase change for any wavelength of light reflected from the back surface of the soap film.

A. I only

B. II and III only

C. III and IV only

D. I, II, and III

E. II, III, and IV


Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
A simple telescope consists of two convex lenses, the objective and the eyepiece, which have a common focal point $P$, as shown in the figure above. If the focal length of the objective is 1.0 meter and the angular magnification of the telescope is 10 , what is the optical path length between objective and eyepiece?

A. $0.1 \mathrm{~m}$

B. $0.9 \mathrm{~m}$

C. $1.0 \mathrm{~m}$

D. $1.1 \mathrm{~m}$

E. $10 \mathrm{~m}$

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
The Fermi temperature of $\mathrm{Cu}$ is about $80,000 \mathrm{~K}$. Which of the following is most nearly equal to the average speed of a conduction electron in $\mathrm{Cu}$ ?

A. $2 \times 10^{-2} \mathrm{~m} / \mathrm{s}$

B. $2 \mathrm{~m} / \mathrm{s}$

C. $2 \times 10^{2} \mathrm{~m} / \mathrm{s}$

D. $2 \times 10^{4} \mathrm{~m} / \mathrm{s}$

E. $2 \times 10^{6} \mathrm{~m} / \mathrm{s}$

Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
Solid argon is held together by which of the following bonding mechanisms?

A. Ionic bond only

B. Covalent bond only

C. Partly covalent and partly ionic bond

D. Metallic bond

E. van der Waals bond

Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
In experiments located deep underground, the two types of cosmic rays that most commonly reach the experimental apparatus are

A. alpha particles and neutrons

B. protons and electrons

C. iron nuclei and carbon nuclei

D. muons and neutrinos

E. positrons and electrons


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A radioactive nucleus decays, with the activity shown in the graph above. The half-life of the nucleus is

A. 2 min

B. $7 \mathrm{~min}$

C. $11 \mathrm{~min}$

D. $18 \mathrm{~min}$

E. $23 \mathrm{~min}$
Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
If a freely moving electron is localized in space to within $\Delta x_{0}$ of $x_{0}$, its wave function can be described by a wave packet $\psi(x, t)=\int_{-\infty}^{\infty} e^{i(k x-\omega t)} f(k) d k$, where $f(k)$ is peaked around a central value $k_{0}$. Which of the following is most nearly the width of the peak in $k$ ?

A. $\Delta k=\frac{1}{x_{0}}$

B. $\Delta k=\frac{1}{\Delta x_{0}}$

C. $\Delta k=\frac{\Delta x_{0}}{x_{0}{ }^{2}}$

D. $\Delta k=\left(\frac{\Delta x_{0}}{x_{0}}\right) k_{0}$

E. $\Delta k=\sqrt{k_{0}^{2}+\left(\frac{1}{x_{0}}\right)^{2}}$

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A system is known to be in the normalized state described by the wave function

$$
\psi(\theta, \varphi)=\frac{1}{\sqrt{30}}\left(5 Y_{4}^{3}+Y_{6}^{3}-2 Y_{6}^{0}\right)
$$

where the $Y_{l}{ }^{m}(\theta, \varphi)$ are the spherical harmonics.

The probability of finding the system in a state with azimuthal orbital quantum number $m=3$ is

A. 0

B. $\frac{1}{15}$

C. $\frac{1}{6}$

D. $\frac{1}{3}$

E. $\frac{13}{15}$


Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
An attractive, one-dimensional square well has depth $V_{0}$ as shown above. Which of the following best shows a possible wave function for a bound state?

A.


Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
Given that the binding energy of the hydrogen atom ground state is $E_{0}=13.6 \mathrm{eV}$, the binding energy of the $n=2$ state of positronium (positron-electron system) is

A. $8 E_{0}$

B. $4 E_{0}$

C. $E_{0}$

D. $\frac{E_{0}}{4}$

E. $\frac{E_{0}}{8}$

Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
In a ${ }^{3} S$ state of the helium atom, the possible values of the total electronic angular momentum quantum number are

A. 0 only

B. 1 only

C. 0 and 1 only

D. $0, \frac{1}{2}$, and 1

E. 0,1 , and 2


Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
In the circuit shown above, the resistances are given in ohms and the battery is assumed ideal with emf equal to 3.0 volts.
The resistor that dissipates the most power is
A. $R_{1}$
B. $R_{2}$
C. $R_{3}$
D. $R_{4}$
E. $R_{5}$

Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
In the circuit shown above, the resistances are given in ohms and the battery is assumed ideal with emf equal to 3.0 volts.
The voltage across resistor $R_{4}$ is

A. $0.4 \mathrm{~V}$

B. $0.6 \mathrm{~V}$

C. $1.2 \mathrm{~V}$

D. $1.5 \mathrm{~V}$

E. $3.0 \mathrm{~V}$

Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
A conducting cavity is driven as an electromagnetic resonator. If perfect conductivity is assumed, the transverse and normal field components must obey which of the following conditions at the inner cavity walls?

A. $E_{n}=0, B_{n}=0$

B. $E_{n}=0, B_{t}=0$

C. $E_{t}=0, B_{t}=0$

D. $E_{t}=0, B_{n}=0$

E. None of the above

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
Light of wavelength 5200 angstroms is incident normally on a transmission diffraction grating with 2000 lines per centimeter. The first-order diffraction maximum is at an angle, with respect to the incident beam, that is most nearly

A. $3^{\circ}$

B. $6^{\circ}$

C. $9^{\circ}$

D. $12^{\circ}$

E. $15^{\circ}$

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A plane-polarized electromagnetic wave is incident normally on a flat, perfectly conducting surface. Upon reflection at the surface, which of the following is true?

A. Both the electric vector and magnetic vector are reversed.

B. Neither the electric vector nor the magnetic vector is reversed.

C. The electric vector is reversed; the magnetic vector is not.

D. The magnetic vector is reversed; the electric vector is not.

E. The directions of the electric and magnetic vectors are interchanged.


Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
A $\pi^{0}$ meson (rest-mass energy $135 \mathrm{MeV}$ ) is moving with velocity $0.8 c \hat{\mathbf{k}}$ in the laboratory rest frame when it decays into two photons, $\gamma_{1}$ and $\gamma_{2}$. In the $\pi^{0}$ rest frame, $\gamma_{1}$ is emitted forward and $\gamma_{2}$ is emitted backward relative to the $\pi^{0}$ direction of flight. The velocity of $\gamma_{2}$ in the laboratory rest frame is

A. $-1.0 c \hat{\mathbf{k}}$

B. $-0.2 c \hat{\mathbf{k}}$

C. $+0.8 c \hat{\mathbf{k}}$

D. $+1.0 c \hat{\mathbf{k}}$

E. $+1.8 c \hat{\mathbf{k}}$

Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
Tau leptons are observed to have an average half-life of $\Delta t_{1}$ in the frame $S_{1}$ in which the leptons are at rest. In an inertial frame $S_{2}$, which is moving at a speed $v_{12}$ relative to $S_{1}$, the leptons are observed to have an average half-life of $\Delta t_{2}$. In another inertial reference frame $S_{3}$, which is moving at a speed $v_{13}$ relative to $S_{1}$ and $v_{23}$ relative to $S_{2}$, the leptons have an observed half-life of $\Delta t_{3}$. Which of the following is a correct relationship among two of the half-lives, $\Delta t_{1}, \Delta t_{2}$, and $\Delta t_{3}$ ?

A. $\Delta t_{2}=\Delta t_{1} \sqrt{1-\left(v_{12}\right)^{2} / c^{2}}$

B. $\Delta t_{1}=\Delta t_{3} \sqrt{1-\left(v_{13}\right)^{2} / c^{2}}$

C. $\Delta t_{2}=\Delta t_{3} \sqrt{1-\left(v_{23}\right)^{2} / c^{2}}$

D. $\Delta t_{3}=\Delta t_{2} \sqrt{1-\left(v_{23}\right)^{2} / c^{2}}$

E. $\Delta t_{1}=\Delta t_{2} \sqrt{1-\left(v_{23}\right)^{2} / c^{2}}$


Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
If $n$ is an integer ranging from 1 to infinity, $\omega$ is an angular frequency, and $t$ is time, then the Fourier series for a square wave, as shown above, is given by which of the following?

A. $V(t)=\frac{4}{\pi} \sum_{1}^{\infty} \frac{1}{n} \sin (n \omega t)$

B. $V(t)=\frac{4}{\pi} \sum_{0}^{\infty} \frac{1}{(2 n+1)} \sin ((2 n+1) \omega t)$

C. $V(t)=\frac{4}{\pi} \sum_{1}^{\infty} \frac{1}{n} \cos (n \omega t)$

D. $V(t)=\frac{4}{\pi} \sum_{0}^{\infty} \frac{1}{(2 n+1)} \cos ((2 n+1) \omega t)$

E. $V(t)=-\frac{4}{\pi}+\frac{4}{\pi} \sum_{i}^{\infty} \frac{1}{n^{2}} \cos (n \omega t)$

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A rigid cylinder rolls at constant speed without slipping on top of a horizontal plane surface. The acceleration of a point on the circumference of the cylinder at the moment when the point touches the plane is

A. directed forward

B. directed backward

C. directed up

D. directed down

E. zero


Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
A cylinder with moment of inertia $4 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about a fixed axis initially rotates at 80 radians per second about this axis. A constant torque is applied to slow it down to 40 radians per second.
The kinetic energy lost by the cylinder is

A. $80 \mathrm{~J}$

B. $800 \mathrm{~J}$

C. $4000 \mathrm{~J}$

D. $9600 \mathrm{~J}$

E. $19,200 \mathrm{~J}$

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A cylinder with moment of inertia $4 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about a fixed axis initially rotates at 80 radians per second about this axis. A constant torque is applied to slow it down to 40 radians per second.
If the cylinder takes 10 seconds to reach 40 radians per second, the magnitude of the applied torque is

A. $80 \mathrm{~N} \cdot \mathrm{m}$

B. $40 \mathrm{~N} \cdot \mathrm{m}$

C. $32 \mathrm{~N} \cdot \mathrm{m}$

D. $16 \mathrm{~N} \cdot \mathrm{m}$

E. $8 \mathrm{~N} \cdot \mathrm{m}$

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
If $\frac{\partial L}{\partial q_{n}}=0$, where $L$ is the Lagrangian for a conservative system without constraints and $q_{n}$ is a generalized coordinate, then the generalized momentum $p_{n}$ is

A. an ignorable coordinate

B. constant

C. undefined

D. equal to $\frac{d}{d t}\left(\frac{\partial L}{\partial q_{n}}\right)$

E. equal to the Hamiltonian for the system

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A particle of mass $m$ on the Earth's surface is confined to move on the parabolic curve $y=a x^{2}$, where $y$ is up. Which of the following is a Lagrangian for the particle?

A. $L=\frac{1}{2} m \dot{y}^{2}\left(1+\frac{1}{4 a y}\right)-m g y$

B. $L=\frac{1}{2} m \dot{y}^{2}\left(1-\frac{1}{4 a y}\right)-m g y$

C. $L=\frac{1}{2} m \dot{x}^{2}\left(1+\frac{1}{4 a x}\right)-m g x$

D. $L=\frac{1}{2} m \dot{x}^{2}\left(1+4 a^{2} x^{2}\right)+m g x$

E. $L=\frac{1}{2} m \dot{x}^{2}+\frac{1}{2} m \dot{y}^{2}+m g y$

Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
A ball is dropped from a height $h$. As it bounces off the floor, its speed is 80 percent of what it was just before it hit the floor. The ball will then rise to a height of most nearly

A. $0.94 h$

B. $0.80 h$

C. $0.75 h$

D. $0.64 h$

E. $0.50 h$


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
Isotherms and coexistence curves are shown in the $p V$ diagram above for a liquid-gas system. The dashed lines are the boundaries of the labeled regions.
Which numbered curve is the critical isotherm?

A. 1

B. 2

C. 3

D. 4

E. 5

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
Isotherms and coexistence curves are shown in the $p V$ diagram above for a liquid-gas system. The dashed lines are the boundaries of the labeled regions.
In which region are the liquid and the vapor in equilibrium with each other?

A. $A$

B. $B$

C. $C$

D. $D$

E. $E$

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
The magnitude of the force $F$ on an object can be determined by measuring both the mass $m$ of an object and the magnitude of its acceleration $a$, where $F=m a$. Assume that these measurements are uncorrelated and normally distributed. If the standard deviations of the measurements of the mass and the acceleration are $\sigma_{m}$ and $\sigma_{a}$, respectively, then $\sigma_{F} / F$ is

A. $\left(\frac{\sigma_{m}}{m}\right)^{2}+\left(\frac{\sigma_{a}}{a}\right)^{2}$

B. $\left(\frac{\sigma_{m}}{m}+\frac{\sigma_{a}}{a}\right)^{\frac{1}{2}}$

C. $\left[\left(\frac{\sigma_{m}}{m}\right)^{2}+\left(\frac{\sigma_{a}}{a}\right)^{2}\right]^{\frac{1}{2}}$

D. $\frac{\sigma_{m} \sigma_{a}}{m a}$

E. $\frac{\sigma_{m}}{m}+\frac{\sigma_{a}}{a}$

Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
Two horizontal scintillation counters are located near the Earth's surface. One is 3.0 meters directly above the other. Of the following, which is the largest scintillator resolving time that can be used to distinguish downward-going relativistic muons from upwardgoing relativistic muons using the relative time of the scintillator signals?

A. 1 picosecond

B. 1 nanosecond

C. 1 microsecond

D. 1 millisecond

E. 1 second

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
The state of a quantum mechanical system is described by a wave function $\psi$. Consider two physical observables that have discrete eigenvalues: observable $A$ with eigenvalues $\{\alpha\}$, and observable $B$ with eigenvalues $\{\beta\}$. Under what circumstances can all wave functions be expanded in a set of basis states, each of which is a simultaneous eigenfunction of both $A$ and $B$ ?

A. Only if the values $\{\alpha\}$ and $\{\beta\}$ are nondegenerate

B. Only if $A$ and $B$ commute

C. Only if $A$ commutes with the Hamiltonian of the system

D. Only if $B$ commutes with the Hamiltonian of the system

E. Under all circumstances


Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A particle of mass $m$ is confined to an infinitely deep square-well potential:

$$
egin{aligned}
& V(x)=\infty, x \leq 0, x \geq a \
& V(x)=0,0<x<a .
\end{aligned}
$$

The normalized eigenfunctions, labeled by the quantum number $n$, are $\psi_{n}=\sqrt{rac{2}{a}} \sin rac{n \pi x}{a}$.
For any state $n$, the expectation value of the momentum of the particle is

A. 0

B. $\frac{\hbar n \pi}{a}$

C. $\frac{2 h n \pi}{a}$

D. $\frac{\hbar n \pi}{a}(\cos n \pi-1)$

E. $\frac{-i \hbar n \pi}{a}(\cos n \pi-1)$

Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
A particle of mass $m$ is confined to an infinitely deep square-well potential:

$$
egin{aligned}
& V(x)=\infty, x \leq 0, x \geq a \
& V(x)=0,0<x<a .
\end{aligned}
$$

The normalized eigenfunctions, labeled by the quantum number $n$, are $\psi_{n}=\sqrt{rac{2}{a}} \sin rac{n \pi x}{a}$.
The eigenfunctions satisfy the condition $\int_{0}^{a} \psi_{n}{ }^{*}(x) \psi_{\ell}(x) d x=\delta_{n \ell}, \delta_{n \ell}=1$ if $n=\ell$, otherwise $\delta_{n \ell}=0$. This is a statement that the eigenfunctions are

A. solutions to the Schrödinger equation

B. orthonormal

C. bounded

D. linearly dependent

E. symmetric

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A particle of mass $m$ is confined to an infinitely deep square-well potential:

$$
egin{aligned}
& V(x)=\infty, x \leq 0, x \geq a \
& V(x)=0,0<x<a .
\end{aligned}
$$

The normalized eigenfunctions, labeled by the quantum number $n$, are $\psi_{n}=\sqrt{rac{2}{a}} \sin rac{n \pi x}{a}$.
A measurement of energy $E$ will always satisfy which of the following relationships?

A. $E \leq \frac{\pi^{2} \hbar^{2}}{8 m a^{2}}$

B. $E \geq \frac{\pi^{2} \hbar^{2}}{2 m a^{2}}$

C. $E=\frac{\pi^{2} \hbar^{2}}{8 m a^{2}}$

D. $E=\frac{n^{2} \pi^{2} \hbar^{2}}{8 m a^{2}}$

E. $E=\frac{\pi^{2} \hbar^{2}}{2 m a^{2}}$


Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A rectangular loop of wire with dimensions shown above is coplanar with a long wire carrying current $I$. The distance between the wire and the left side of the loop is $r$. The loop is pulled to the right as indicated.
What are the directions of the induced current in the loop and the magnetic forces on the left and the right sides of the loop as the loop is pulled?

\begin{tabular}{lll} 
Induced Current & \begin{tabular}{l} 
Force on \\
Left Side
\end{tabular} & \begin{tabular}{l} 
Force on \\
Right Side
\end{tabular} \\
A. Counterclockwise & To the left & To the right \\
B. Counterclockwise & To the left & To the left \\
C. Counterclockwise & To the right & To the left \\
D. Clockwise & To the right & To the left \\
E. Clockwise & To the left & To the right
\end{tabular}

Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1