Input:
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. $\frac{13}{15}$


Input:
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.


Input:
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. $\frac{E_{0}}{8}$


Input:
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


Input:
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. $R_{5}$


Input:
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. $3.0 \mathrm{~V}$


Input:
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


Input:
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. $15^{\circ}$


Input:
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.


Input:
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. $+1.8 c \hat{\mathbf{k}}$


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


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


Input:
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


Input:
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. $19,200 \mathrm{~J}$


Input:
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. $8 \mathrm{~N} \cdot \mathrm{m}$


Input:
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


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


Input:
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. $0.50 h$


Input:
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


Input:
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. $E$


Input:
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. $\frac{\sigma_{m}}{m}+\frac{\sigma_{a}}{a}$


Input:
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


Input:
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


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


Input:
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


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


Input:
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}


Input:
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 is the magnitude of the net force on the loop when the induced current is $i$ ?

A. $\frac{\mu_{0} i I}{2 \pi} \ln \left(\frac{r+a}{r}\right)$

B. $\frac{\mu_{0} i I}{2 \pi} \ln \left(\frac{r}{r+a}\right)$

C. $\frac{\mu_{0} i I}{.2 \pi} \frac{b}{a}$

D. $\frac{\mu_{0} i I}{2 \pi} \frac{a b}{r(r+a)}$

E. $\frac{\mu_{0} i I}{2 \pi} \frac{r(r+a)}{a b}$


Input:
If $v$ is frequency and $h$ is Planck's constant, the ground state energy of a one-dimensional quantum mechanical harmonic oscillator is

A. 0

B. $\frac{1}{3} h v$

C. $\frac{1}{2} h v$

D. $h v$

E. $\frac{3}{2} h v$


Input:
A uniform and constant magnetic field $B$ is directed perpendicularly into the plane of the page everywhere within a rectangular region as shown above. A wire circuit in the shape of a semicircle is uniformly rotated counterclockwise in the plane of the page about an axis $A$. The axis $A$ is perpendicular to the page at the edge of the field and directed through the center of the straight-line portion of the circuit. Which of the following graphs best approximates the emf $\varepsilon$ induced in the circuit as a function of time $t$ ?

A.


Input:
The ground state configuration of a neutral sodium atom $(Z=11)$ is

A. $1 s^{2} 2 s^{2} 2 p^{5} 3 s^{2}$

B. $1 s^{2} 2 s^{3} 2 p^{6}$

C. $1 s^{2} 2 s^{2} 2 p^{6} 3 s$

D. $1 s^{2} 2 s^{2} 2 p^{6} 3 p$

E. $l s^{2} 2 s^{2} 2 p^{5}$


Input:
The ground state of the helium atom is a spin

A. singlet

B. doublet

C. triplet

D. quartet

E. quintuplet


Input:
An electron in a metal has an effective mass $m^{*}=0.1 m_{e}$. If this metal is placed in a magnetic field of magnitude 1 tesla, the cyclotron resonance frequency, $\omega_{c}$, is most nearly

A. $930 \mathrm{rad} / \mathrm{s}$

B. $8.5 \times 10^{6} \mathrm{rad} / \mathrm{s}$

C. $2.8 \times 10^{11} \mathrm{rad} / \mathrm{s}$

D. $1.8 \times 10^{12} \mathrm{rad} / \mathrm{s}$

E. $7.7 \times 10^{20} \mathrm{rad} / \mathrm{s}$


Input:
A long, straight, and massless rod pivots about one end in a vertical plane. In configuration I, shown above, two small identical masses are attached to the free end; in configuration II, one mass is moved to the center of the rod. What is the ratio of the frequency of small oscillations of configuration II to that of configuration I ?

A. $(6 / 5)^{\frac{1}{2}}$

B. $(3 / 2)^{\frac{1}{2}}$

C. $6 / 5$

D. $3 / 2$

E. $5 / 3$


Input:
A mole of ideal gas initially at temperature $T_{0}$ and volume $V_{0}$ undergoes a reversible isothermal expansion to volume $V_{1}$. If the ratio of specific heats is $c_{p} / c_{v}=\gamma$ and if $R$ is the gas constant, the work done by the gas is

A. zero

B. $R T_{0}\left(V_{1} / V_{0}\right)^{\gamma}$

C. $R T_{0}\left(V_{1} / V_{0}-1\right)$

D. $c_{v} T_{0}\left[1-\left(V_{0} / V_{1}\right)^{\gamma^{-1}}\right]$

E. $R T_{0} \ln \left(V_{1} / V_{0}\right)$


Input:
Which of the following is true if the arrangement of an isolated thermodynamic system is of maximal probability?

A. Spontaneous change to a lower probability occurs.

B. The entropy is a minimum.

C. Boltzmann's constant approaches zero.

D. No spontaneous change occurs.

E. The entropy is zero.


Input:
If an electric field is given in a certain region by $E_{x}=0, E_{y}=0, E_{z}=k z$, where $k$ is a nonzero constant, which of the following is true?

A. There is a time-varying magnetic field.

B. There is charge density in the region.

C. The electric field cannot be constant in time.

D. The electric field is impossible under any circumstances.

E. None of the above


Input:
Two point charges with the same charge $+Q$ are fixed along the $x$-axis and are a distance $2 R$ apart as shown. A small particle with mass $m$ and charge $-q$ is placed at the midpoint between them. What is the angular frequency $\omega$ of small oscillations of this particle along the $y$-direction?

A. $\frac{Q q}{2 \pi \epsilon_{0} m R^{2}}$

B. $\frac{Q q}{4 \pi \varepsilon_{0} m R^{2}}$

C. $\frac{Q q}{2 \pi \epsilon_{0} m R^{3}}$

D. $\left(\frac{Q q}{4 \pi \epsilon_{0} m R^{2}}\right)^{\frac{1}{2}}$

E. $\left(\frac{Q q}{2 \pi \epsilon_{0} m R^{3}}\right)^{\frac{1}{2}}$


Input:
A thin uniform steel chain is 10 meters long with a mass density of 2 kilograms per meter. One end of the chain is attached to a horizontal axle having a radius that is small compared to the length of the chain. If the chain initially hangs vertically, the work required to slowly wind it up on to the axle is closest to

A. $100 \mathrm{~J}$

B. $200 \mathrm{~J}$

C. $1,000 \mathrm{~J}$

D. $2,000 \mathrm{~J}$

E. $10,000 \mathrm{~J}$


Input:
A steady beam of light is normally incident on a piece of polaroid. As the polaroid is rotated around the beam axis, the transmitted intensity varies as $A+B \cos 2 \theta$, where $\theta$ is the angle of rotation, and $A$ and $B$ are constants with $A>B>0$. Which of the following may be correctly concluded about the incident light?

A. The light is completely unpolarized.

B. The light is completely plane polarized.

C. The light is partly plane polarized and partly unpolarized.

D. The light is partly circularly polarized and partly unpolarized.

E. The light is completely circularly polarized.


Input:
The angular separation of the two components of a double star is 8 microradians, and the light from the double star has a wavelength of 5500 angstroms. The smallest diameter of a telescope mirror that will resolve the double star is most nearly

A. $1 \mathrm{~mm}$

B. $1 \mathrm{~cm}$

C. $10 \mathrm{~cm}$

D. $1 \mathrm{~m}$

E. $100 \mathrm{~m}$


Input:
A fast charged particle passes perpendicularly through a thin glass sheet of index of refraction 1.5. The particle emits light in the glass. The minimum speed of the particle is

A. $\frac{1}{3} c$

B. $\frac{4}{9} c$

C. $\frac{5}{9} c$

D. $\frac{2}{3} c$

E. $c$


Input:
A monoenergetic beam consists of unstable particles with total energies 100 times their rest energy. If the particles have rest mass $m$, their momentum is most nearly

A. $m c$

B. $10 \mathrm{mc}$

C. $70 \mathrm{mc}$

D. $100 m c$

E. $10^{4} \mathrm{mc}$


Input:
A system in thermal equilibrium at temperature $T$ consists of a large number $N_{0}$ of subsystems, each of which can exist only in two states of energy $E_{1}$ and $E_{2}$, where $E_{2}-E_{1}=\epsilon>0$. In the expressions that follow, $k$ is the Boltzmann constant.
For a system at temperature $T$, the average number of subsystems in the state of energy $E_{1}$ is given by

A. $\frac{N_{0}}{2}$

B. $\frac{N_{0}}{1+e^{-\epsilon / k T}}$

C. $N_{0} e^{-\epsilon / k T}$

D. $\frac{N_{0}}{1+e^{\epsilon / k T}}$

E. $\frac{N_{0} e^{\epsilon / k T}}{2}$


Input:
A system in thermal equilibrium at temperature $T$ consists of a large number $N_{0}$ of subsystems, each of which can exist only in two states of energy $E_{1}$ and $E_{2}$, where $E_{2}-E_{1}=\epsilon>0$. In the expressions that follow, $k$ is the Boltzmann constant.
The internal energy of this system at any temperature $T$ is given by $E_{1} N_{0}+\frac{N_{0} \epsilon}{1+e^{\epsilon / k T}}$. The heat capacity of the system is given by which of the following expressions?

A. $N_{0} k\left(\frac{\epsilon}{k T}\right)^{2} \frac{e^{\epsilon / k T}}{\left(1+e^{\epsilon / k T}\right)^{2}}$

B. $N_{0} k\left(\frac{\epsilon}{k T}\right)^{2} \frac{1}{\left(1+e^{\epsilon / k T}\right)^{2}}$

C. $N_{0} k\left(\frac{\epsilon}{k T}\right)^{2} e^{-\epsilon / k T}$

D. $\frac{N_{0} k}{2}\left(\frac{\epsilon}{k T}\right)^{2}$

E. $\frac{3}{2} N_{0} k$


Input:
A system in thermal equilibrium at temperature $T$ consists of a large number $N_{0}$ of subsystems, each of which can exist only in two states of energy $E_{1}$ and $E_{2}$, where $E_{2}-E_{1}=\epsilon>0$. In the expressions that follow, $k$ is the Boltzmann constant.
Which of the following is true of the entropy of the system?

A. It increases without limit with $T$ from zero at $T=0$.

B. It decreases with increasing $T$.

C. It increases from zero at $T=0$ to $N_{0} k \ln 2$ at arbitrarily high temperatures.

D. It is given by $N_{0} k\left[\frac{5}{2} \ln T-\ln p+\right.$ constant $]$.

E. It cannot be calculated from the information given.


Input:
Two circular hoops, $X$ and $Y$, are hanging on nails in a wall. The mass of $X$ is four times that of $Y$, and the diameter of $X$ is also four times that of $Y$. If the period of small oscillations of $X$ is $T$, the period of small oscillations of $Y$ is
A. $T$
B. $T / 2$
C. $T / 4$
D. $T / 8$
E. $T / 16$

$$
{ }_{92}^{235} \mathrm{U} \rightarrow{ }_{90}^{231} \mathrm{Th}+{ }_{2}^{4} \mathrm{He}
$$


Input:
A uranium nucleus decays at rest into a thorium nucleus and a helium nucleus, as shown above. Which of the following is true?

A. Each decay product has the same kinetic energy.

B. Each decay product has the same speed.

C. The decay products tend to go in the same direction.

D. The thorium nucleus has more momentum than the helium nucleus.

E. The helium nucleus has more kinetic energy than the thorium nucleus.


Input:
The configuration of three electrons $1 \mathrm{~s} 2 \mathrm{p} 3 \mathrm{p}$ has which of the following as the value of its maximum possible total angular momentum quantum number?

A. $\frac{7}{2}$

B. 3

C. $\frac{5}{2}$

D. 2

E. $\frac{3}{2}$


Input:
Consider a heavy nucleus with spin $\frac{1}{2}$. The magnitude of the ratio of the intrinsic magnetic moment of this nucleus to that of an electron is

A. zero, because the nucleus has no intrinsic magnetic moment

B. greater than 1, because the nucleus contains many protons

C. greater than 1, because the nucleus is so much larger in diameter than the electron

D. less than 1, because of the strong interactions among the nucleons in a nucleus

E. less than 1, because the nucleus has a mass much larger than that of the electron


Input:
One ice skater of mass $m$ moves with speed $2 v$ to the right, while another of the same mass $m$ moves with speed $v$ toward the left, as shown in Figure I. Their paths are separated by a distance $b$. At $t=0$, when they are both at $x=0$, they grasp a pole of length $b$ and negligible mass. For $t>0$, consider the system as a rigid body of two masses $m$ separated by distance $b$, as shown in Figure II. Which of the following is the correct formula for the motion after $t=0$ of the skater initially at $y=b / 2$ ?

A. $x=2 v t, y=b / 2$

B. $x=v t+0.5 b \sin (3 v t / b), \quad y=0.5 b \cos (3 v t / b)$

C. $x=0.5 v t+0.5 b \sin (3 v t / b), \quad y=0.5 b \cos (3 v t / b)$

D. $x=v t+0.5 b \sin (6 v t / b), \quad y=0.5 b \cos (6 v t / b)$

E. $x=0.5 v t+0.5 b \sin (6 v t / b), \quad y=0.5 b \cos (6 v t / b)$


Input:
The dispersion curve shown above relates the angular frequency $\omega$ to the wave number $k$. For waves with wave numbers lying in the range $k_{1}<k<k_{2}$, which of the following is true of the phase velocity and the group velocity?

A. They are in opposite directions.

B. They are in the same direction and the phase velocity is larger.

C. They are in the same direction and the group velocity is larger.

D. The phase velocity is infinite and the group velocity is finite.

E. They are the same in direction and magnitude.


Input:
A beam of electrons is accelerated through a potential difference of 25 kilovolts in an x-ray tube. The continuous $x$-ray spectrum emitted by the target of the tube will have a short wavelength limit of most nearly

A. $0.1 \AA$

B. $0.5 \AA$

C. $2 \AA$

D. $25 \AA$

E. $50 \AA$


Input:
In the $R L C$ circuit shown above, the applied voltage is

$$
\varepsilon(t)=\varepsilon_{\mathrm{m}} \cos \omega t .
$$

For a constant $\varepsilon_{\mathrm{m}}$, at what angular frequency $\omega$ does the current have its maximum steady-state amplitude after the transients have died out?

A. $\frac{1}{R C}$

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

C. $\frac{1}{\sqrt{L C}}$

D. $\sqrt{\frac{1}{L C}-\left(\frac{R}{2 L}\right)^{2}}$

E. $\sqrt{\left(\frac{1}{R C}\right)^{2}-\left(\frac{L}{R}\right)^{2}}$


Input:
A thin plate of mass $M$, length $L$, and width $2 d$ is mounted vertically on a frictionless axle along the $z$-axis, as shown above. Initially the object is at rest. It is then tapped with a hammer to provide a torque $\tau$, which produces an angular impulse $\mathbf{H}$ about the $z$-axis of magnitude $H=\int \tau d t$. What is the angular speed $\omega$ of the plate about the $z$-axis after the tap?

A. $\frac{H}{2 M d^{2}}$

B. $\frac{H}{M d^{2}}$

C. $\frac{2 H}{M d^{2}}$

D. $\frac{3 H}{M d^{2}}$

E. $\frac{4 H}{M d^{2}}$


Input:
Two pith balls of equal mass $M$ and equal charge $q$ are suspended from the same point on long massless threads of length $L$ as shown in the figure above. If $k$ is the Coulomb's law constant, then for small values of $\theta$, the distance $d$ between the charged pith balls at equilibrium is

A. $\left(\frac{2 k q^{2} L}{M g}\right)^{\frac{1}{3}}$

B. $\left(\frac{k q^{2} L}{M g}\right)^{\frac{1}{3}}$

C. $\left(\frac{2 k q^{2} L}{M g}\right)^{\frac{1}{2}}$

D. $\left(\frac{k q^{2} L}{M g}\right)^{\frac{1}{2}}$

E. $\frac{L}{4}$


Input:
An electron oscillates back and forth along the + and $-x$-axes, consequently emitting electromagnetic radiation. Which of the following statements concerning the radiation is NOT true?

A. The total rate of radiation of energy into all directions is proportional to the square of the electron's acceleration.

B. The total rate of radiation of energy into all directions is proportional to the square of the electron's charge.

C. Far from the electron, the rate at which radiated energy crosses a perpendicular unit area decreases as the inverse square of the distance from the electron.

D. Far from the electron, the rate at which radiated energy crosses a perpendicular unit area is a maximum when the unit area is located on the + or $-x$-axes.

E. Far from the electron, the radiated energy is carried equally by the transverse electric and the transverse magnetic fields.


Input:
A free electron (rest mass $m_{e}=0.5 \mathrm{MeV} / \mathrm{c}^{2}$ ) has a total energy of $1.5 \mathrm{MeV}$. Its momentum $p$ in units of $\mathrm{MeV} / \mathrm{c}$ is about

A. 0.86

B. 1.0

C. 1.4

D. 1.5

E. 2.0


Input:
The circuit shown above is used to measure the size of the capacitance $C$. The $y$-coordinate of the spot on the oscilloscope screen is proportional to the potential difference across $R$, and the $x$-coordinate of the spot is swept at a constant speed $s$. The switch is closed and then opened. One can then calculate $C$ from the shape and the size of the curve on the screen plus a knowledge of which of the following?

A. $V_{0}$ and $R$

B. $s$ and $R$

C. $s$ and $V_{0}$

D. $R$ and $R^{\prime}$

E. The sensitivity of the oscilloscope


Input:
A particle of mass $M$ moves in a circular orbit of radius $r$ around a fixed point under the influence of an attractive force $F=\frac{K}{r^{3}}$, where $K$ is a constant. If the potential energy of the particle is zero at an infinite distance from the force center, the total energy of the particle in the circular orbit is

A. $-\frac{K}{r^{2}}$

B. $-\frac{K}{2 r^{2}}$

C. 0

D. $\frac{K}{2 r^{2}}$

E. $\frac{K}{r^{2}}$


Input:
A parallel-plate capacitor is connected to a battery. $V_{0}$ is the potential difference between the plates, $Q_{0}$ the charge on the positive plate, $E_{0}$ the magnitude of the electric field, and $D_{0}$ the magnitude of the displacement vector. The original vacuum between the plates is filled with a dielectric and then the battery is disconnected. If the corresponding electrical parameters for the final state of the capacitor are denoted by a subscript $f$, which of the following is true?

A. $V_{f}>V_{0}$

B. $V_{f}<V_{0}$

C. $Q_{f}=Q_{0}$

D. $E_{f}>E_{0}$

E. $D_{f}>D_{0}$


Input:
The energy levels for the one-dimensional harmonic oscillator are $h v\left(n+\frac{1}{2}\right), n=0,1,2 \ldots$ How will the energy levels for the potential shown in the graph above differ from those for the harmonic oscillator?

A. The term $\frac{1}{2}$ will be changed to $\frac{3}{2}$.

B. The energy of each level will be doubled.

C. The energy of each level will be halved.

D. Only those for even values of $n$ will be present.

E. Only those for odd values of $n$ will be present.


Input:
The spacing of the rotational energy levels for the hydrogen molecule $\mathbf{H}_{2}$ is most nearly

A. $10^{-9} \mathrm{eV}$

B. $10^{-3} \mathrm{eV}$

C. $10 \mathrm{eV}$

D. $10 \mathrm{MeV}$

E. $100 \mathrm{MeV}$


Input:
The particle decay $\Lambda \rightarrow p+\pi^{-}$must be a weak interaction because

A. the $\pi^{-}$is a lepton

B. the $\Lambda$ has spin zero

C. no neutrino is produced in the decay

D. it does not conserve angular momentum

E. it does not conserve strangeness


Input:
A flat coil of wire is rotated at a frequency of 10 hertz in the magnetic field produced by three pairs of magnets as shown above. The axis of rotation of the coil lies in the plane of the coil and is perpendicular to the field lines. What is the frequency of the alternating voltage in the coil?

A. $\frac{10}{6} \mathrm{~Hz}$

B. $\frac{10}{3} \mathrm{~Hz}$

C. $10 \mathrm{~Hz}$

D. $30 \mathrm{~Hz}$

E. $60 \mathrm{~Hz}$


Input:
The figure above shows a small mass connected to a string, which is attached to a vertical post. If the mass is released when the string is horizontal as shown, the magnitude of the total acceleration of the mass as a function of the angle $\theta$ is

A. $g \sin \theta$

B. $2 g \cos \theta$

C. $2 g \sin \theta$

D. $g \sqrt{3 \cos ^{2} \theta+1}$

E. $g \sqrt{3 \sin ^{2} \theta+1}$


Input:
Which of the following is a Lorentz transformation? (Assume a system of units such that the velocity of light is 1 .)

A. $x^{\prime}=4 x$

$y^{\prime}=y$

$z^{\prime}=z$

$t^{\prime}=.25 t$

B. $x^{\prime}=x-.75 t$

$y^{\prime}=y$

$z^{\prime}=z$

$t^{\prime}=t$

C. $x^{\prime}=1.25 x-.75 t$

$y^{\prime}=y$

$z^{\prime}=z$

$t^{\prime}=1.25 t-.75 x$

D. $x^{\prime}=1.25 x-.75 t$

$y^{\prime}=y$

$z^{\prime}=z$

$t^{\prime}=.75 t-1.25 x$

E. None of the above


Input:
A beam of $10^{12}$ protons per second is incident on a target containing $10^{20}$ nuclei per square centimeter. At an angle of 10 degrees, there are $10^{2}$ protons per second elastically scattered into a detector that subtends a solid angle of $10^{-4}$ steradians. What is the differential elastic scattering cross section, in units of square centimeters per steradian?

A. $10^{-24}$

B. $10^{-25}$

C. $10^{-26}$

D. $10^{-27}$

E. $10^{-28}$


Input:
A gas-filled cell of length 5 centimeters is inserted in one arm of a Michelson interferometer, as shown in the figure above. The interferometer is illuminated by light of wavelength 500 nanometers. As the gas is evacuated from the cell, $\mathbf{4 0}$ fringes cross a point in the field of view. The refractive index of this gas is most nearly

A. 1.02

B. 1.002

C. 1.0002

D. 1.00002

E. 0.98


Input:
Lattice forces affect the motion of electrons in a metallic crystal, so that the relationship between the energy $E$ and wave number $k$ is not the classical equation $E=\hbar^{2} k^{2} / 2 m$, where $m$ is the electron mass. Instead, it is possible to use an effective mass $m^{*}$ given by which of the following?

A. $m^{*}=\frac{1}{2} \hbar^{2} k\left(\frac{d k}{d E}\right)$

B. $m^{*}=\frac{\hbar^{2} k}{\left(\frac{d k}{d E}\right)}$

C. $m^{*}=\hbar^{2} k\left(\frac{d^{2} k}{d E^{2}}\right)^{\frac{1}{3}}$

D. $m^{*}=\frac{\hbar^{2}}{\left(\frac{d^{2} E}{d k^{2}}\right)}$

E. $m^{*}=\frac{1}{2} \hbar^{2} m^{2}\left(\frac{d^{2} E}{d k^{2}}\right)$


Input:
The matrix $A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$

has three eigenvalues $\lambda_{i}$ defined by $A v_{i}=\lambda_{i} v_{i}$. Which of the following statements is NOT true?

A. $\lambda_{1}+\lambda_{2}+\lambda_{3}=0$

B. $\lambda_{1}, \lambda_{2}$, and $\lambda_{3}$ are all real numbers.

C. $\lambda_{2} \lambda_{3}=+1$ for some pair of roots.

D. $\lambda_{1} \lambda_{2}+\lambda_{2} \lambda_{3}+\lambda_{3} \lambda_{1}=0$

E. $\lambda_{i}{ }^{3}=+1, i=1,2,3$


Input:
In perturbation theory, what is the first order correction to the energy of a hydrogen atom (Bohr radius $a_{0}$ ) in its ground state due to the presence of a static electric field $E$ ?

A. Zero

B. $e E a_{0}$

C. $3 e E a_{0}$

D. $\frac{8 e^{2} E a_{0}^{3}}{3}$

E. $\frac{8 e^{2} E^{2} a_{0}^{3}}{3}$


Input:
A uniform rod of length 10 meters and mass 20 kilograms is balanced on a fulcrum with a 40 -kilogram mass on one end of the rod and a 20-kilogram mass on the other end, as shown above. How far is the fulcrum located from the center of the rod?

A. $0 \mathrm{~m}$

B. $1 \mathrm{~m}$

C. $1.25 \mathrm{~m}$

D. $1.5 \mathrm{~m}$

E. $2 \mathrm{~m}$


Input:
A ball is thrown out of the passenger window of a car moving to the right (ignore air resistance). If the ball is thrown out perpendicular to the velocity of the car, which of the following best depicts the path the ball takes, as viewed from above?

A.


Input:
An object is thrown horizontally from the open window of a building. If the initial speed of the object is $20 \mathrm{~m} / \mathrm{s}$ and it hits the ground $2.0 \mathrm{~s}$ later, from what height was it thrown? (Neglect air resistance and assume the ground is level.)

A. $4.9 \mathrm{~m}$

B. $9.8 \mathrm{~m}$

C. $10.0 \mathrm{~m}$

D. $19.6 \mathrm{~m}$

E. $39.2 \mathrm{~m}$


Input:
A resistor in a circuit dissipates energy at a rate of $1 \mathrm{~W}$. If the voltage across the resistor is doubled, what will be the new rate of energy dissipation?

A. $0.25 \mathrm{~W}$

B. $0.5 \mathrm{~W}$

C. $1 \mathrm{~W}$

D. $2 \mathrm{~W}$

E. $4 \mathrm{~W}$


Input:
An infinitely long, straight wire carrying current $I_{1}$ passes through the center of a circular loop of wire carrying current $I_{2}$, as shown above. The long wire is perpendicular to the plane of the loop. Which of the following describes the magnetic force on the loop?

A. Outward, along a radius of the loop.

B. Inward, along a radius of the loop.

C. Upward, along the axis of the loop.

D. Downward, along the axis of the loop.

E. There is no magnetic force on the loop.


Input:
De Broglie hypothesized that the linear momentum and wavelength of a free massive particle are related by which of the following constants?

A. Planck's constant

B. Boltzmann's constant

C. The Rydberg constant

D. The speed of light

E. Avogadro's number


Input:
An atom has filled $n=1$ and $n=2$ levels. How many electrons does the atom have?

A. 2

B. 4

C. 6

D. 8

E. 10


Input:
The root-mean-square speed of molecules of mass $m$ in an ideal gas at temperature $T$ is

A. 0

B. $\sqrt{\frac{2 k T}{m}}$

C. $\sqrt{\frac{3 k T}{m}}$

D. $\sqrt{\frac{8 k T}{\pi m}}$

E. $\frac{k T}{m}$


Input:
The energy from electromagnetic waves in equilibrium in a cavity is used to melt ice. If the Kelvin temperature of the cavity is increased by a factor of two, the mass of ice that can be melted in a fixed amount of time is increased by a factor of

A. 2

B. 4

C. 8

D. 16

E. 32


Input:
The figure above represents the orbit of a planet around a star, $S$, and the marks divide the orbit into 14 equal time intervals, $t=T / 14$, where $T$ is the orbital period. If the only force acting on the planet is Newtonian gravitation, then true statements about the situation include which of the following?

I. Area $A=\operatorname{area} B$

II. The star $S$ is at one focus of an elliptically shaped orbit.

III. $T^{2}=C a^{3}$, where $a$ is the semimajor axis of the ellipse and $C$ is a constant.

A. I only

B. II only

C. I and II only

D. II and III only

E. I, II, and III


Input:
A massless spring with force constant $k$ launches a ball of mass $m$. In order for the ball to reach a speed $v$, by what displacement $s$ should the spring be compressed?

A. $s=v \sqrt{\frac{k}{m}}$

B. $s=v \sqrt{\frac{m}{k}}$

C. $s=v \sqrt{\frac{2 k}{m}}$

D. $s=v \frac{m}{k}$

E. $s=v^{2} \frac{m}{2 k}$


Input:
A quantum mechanical harmonic oscillator has an angular frequency $\omega$. The Schrödinger equation predicts that the ground state energy of the oscillator will be

A. $-\frac{1}{2} \hbar \omega$

B. 0

C. $\frac{1}{2} \hbar \omega$

D. $\hbar \omega$

E. $\frac{3}{2} \hbar \omega$


Input:
In the Bohr model of the hydrogen atom, the linear momentum of the electron at radius $r_{n}$ is given by which of the following? ( $n$ is the principal quantum number.)

A. $n \hbar$

B. $n r_{n} \hbar$

C. $\frac{n \hbar}{r_{n}}$

D. $n^{2} r_{n} \hbar$

E. $\frac{n^{2} \hbar}{r_{n}}$


Input:
The figure above represents a log-log plot of variable $y$ versus variable $x$. The origin represents the point $x=1$ and $y=1$. Which of the following gives the approximate functional relationship between $y$ and $x$ ?

A. $y=6 \sqrt{x}$

B. $y=\frac{1}{2} x+6$

C. $y=6 x+0.5$

D. $y=\frac{1}{6} x^{2}$

E. $y=6 x^{2}$


Input:
Two experimental techniques determine the mass of an object to be $11 \pm 1 \mathrm{~kg}$ and $10 \pm 2 \mathrm{~kg}$. These two measurements can be combined to give a weighted average. The uncertainty of the weighted average is equal to which of the following?

A. $\frac{1}{2} \mathrm{~kg}$

B. $\frac{2}{\sqrt{5}} \mathrm{~kg}$

C. $\frac{2}{\sqrt{3}} \mathrm{~kg}$

D. $2 \mathrm{~kg}$

E. $\sqrt{5} \mathrm{~kg}$


Input:
If the five lenses shown below are made of the same material, which lens has the shortest positive focal length?

A.


Input:
Unpolarized light is incident on a pair of ideal linear polarizers whose transmission axes make an angle of $45^{\circ}$ with each other. The transmitted light intensity through both polarizers is what percentage of the incident intensity?

A. $100 \%$

B. $75 \%$

C. $50 \%$

D. $25 \%$

E. $0 \%$


Input:
A very long, thin, straight wire carries a uniform charge density of $\lambda$ per unit length. Which of the following gives the magnitude of the electric field at a radial distance $r$ from the wire?

A. $\frac{1}{2 \pi \varepsilon_{0}} \frac{\lambda}{r}$

B. $\frac{1}{2 \pi \varepsilon_{0}} \frac{r}{\lambda}$

C. $\frac{1}{2 \pi \varepsilon_{0}} \frac{\lambda}{r^{2}}$

D. $\frac{1}{4 \pi \varepsilon_{0}} \frac{\lambda^{2}}{r^{2}}$

E. $\frac{1}{4 \pi \varepsilon_{0}} \lambda \ln r$


Input:
The bar magnet shown in the figure above is moved completely through the loop. Which of the following is a true statement about the direction of the current flow between the two points $a$ and $b$ in the circuit?

A. No current flows between $a$ and $b$ as the magnet passes through the loop.

B. Current flows from $a$ to $b$ as the magnet passes through the loop.

C. Current flows from $b$ to $a$ as the magnet passes through the loop.

D. Current flows from $a$ to $b$ as the magnet enters the loop and from $b$ to $a$ as the magnet leaves the loop.

E. Current flows from $b$ to $a$ as the magnet enters the loop and from $a$ to $b$ as the magnet leaves the loop.


Input:
The surface of the Sun has a temperature close to $6,000 \mathrm{~K}$ and it emits a blackbody (Planck) spectrum that reaches a maximum near $500 \mathrm{~nm}$. For a body with a surface temperature close to $300 \mathrm{~K}$, at what wavelength would the thermal spectrum reach a maximum?

A. $10 \mu \mathrm{m}$

B. $100 \mu \mathrm{m}$

C. $10 \mathrm{~mm}$

D. $100 \mathrm{~mm}$

E. $10 \mathrm{~m}$


Input:
At the present time, the temperature of the universe (i.e., the microwave radiation background) is about $3 \mathrm{~K}$. When the temperature was $12 \mathrm{~K}$, typical objects in the universe, such as galaxies, were

A. one-quarter as distant as they are today

B. one-half as distant as they are today

C. separated by about the same distances as they are today

D. two times as distant as they are today

E. four times as distant as they are today


Input:
For an adiabatic process involving an ideal gas having volume $V$ and temperature $T$, which of the following is constant? $\left(\gamma=C_{P} / C_{V}\right)$

A. $T V$

B. $T V^{\gamma}$

C. $T V^{\gamma-1}$

D. $T^{\gamma} V$

E. $T^{\gamma} V^{-1}$


Input:
An electron has total energy equal to four times its rest energy. The momentum of the electron is

A. $m_{e} c$

B. $\sqrt{2} m_{e} c$

C. $\sqrt{15} m_{e} c$

D. $4 m_{e} c$

E. $2 \sqrt{15} m_{e} c$


Input:
Two spaceships approach Earth with equal speeds, as measured by an observer on Earth, but from opposite directions. A meterstick on one spaceship is measured to be $60 \mathrm{~cm}$ long by an occupant of the other spaceship. What is the speed of each spaceship, as measured by the observer on Earth?

A. $0.4 c$

B. $0.5 c$

C. $0.6 c$

D. $0.7 c$

E. $0.8 c$


Input:
A meter stick with a speed of $0.8 c$ moves past an observer. In the observer's reference frame, how long does it take the stick to pass the observer?

A. $1.6 \mathrm{~ns}$

B. $2.5 \mathrm{~ns}$

C. $4.2 \mathrm{~ns}$

D. $6.9 \mathrm{~ns}$

E. $8.3 \mathrm{~ns}$


Input:
Consider a set of wave functions $\psi_{i}(x)$. Which of the following conditions guarantees that the functions are normalized and mutually orthogonal? (The indices $i$ and $j$ take on the values in the set $\{1,2, \ldots, n\}$. )

A. $\psi_{i}^{*}(x) \psi_{j}(x)=0$

B. $\psi_{i}^{*}(x) \psi_{j}(x)=1$

C. $\int_{-\infty}^{\infty} \psi_{i}^{*}(x) \psi_{j}(x) d x=0$

D. $\int_{-\infty}^{\infty} \psi_{i}^{*}(x) \psi_{j}(x) d x=1$

E. $\int_{-\infty}^{\infty} \psi_{i}^{*}(x) \psi_{j}(x) d x=\delta_{i j}$


Input:
The normalized ground state wave function of hydrogen is $\psi_{100}=\frac{2}{(4 \pi)^{1 / 2} a_{0}^{3 / 2}} e^{-r / a_{0}}$, where $a_{0}$ is the Bohr radius. What is the most likely distance that the electron is from the nucleus?

A. 0

B. $\frac{a_{0}}{2}$

C. $\frac{a_{0}}{\sqrt{2}}$

D. $a_{0}$

E. $2 a_{0}$


Input:
The lifetime for the $2 p \rightarrow 1 s$ transition in hydrogen is $1.6 \times 10^{-9} \mathrm{~s}$. The natural line width for the radiation emitted during the transition is approximately

A. $100 \mathrm{~Hz}$

B. $100 \mathrm{kHz}$

C. $100 \mathrm{MHz}$

D. $100 \mathrm{GHz}$

E. $100 \mathrm{THz}$


Input:
A spring of force constant $k$ is stretched a certain distance. It takes twice as much work to stretch a second spring by half this distance. The force constant of the second spring is

A. $k$

B. $2 k$

C. $4 k$

D. $8 k$

E. $16 k$


Input:
On a frictionless surface, a block of mass $M$ moving at speed $v$ collides elastically with another block of the same mass that is initially at rest. After the collision, the first block moves at an angle $\theta$ to its initial direction and has a speed $v / 2$. The second block's speed after the collision is

A. $\frac{\sqrt{3}}{4} v$

B. $\frac{v}{2}$

C. $\frac{\sqrt{3}}{2} v$

D. $\frac{\sqrt{5}}{2} v$

E. $v+\frac{v}{2} \cos \theta$


Input:
Which of the following gives Hamilton's canonical equation(s) of motion? ( $H$ is the Hamiltonian, $q_{i}$ are the generalized coordinates, and $p_{i}$ are the generalized momenta.)

A. $q_{i}=\frac{\partial H}{\partial p_{i}}, \quad p_{i}=-\frac{\partial H}{\partial q_{i}}$

B. $q_{i}=\frac{\partial H}{\partial \dot{q}_{i}}, \quad p_{i}=\frac{\partial H}{\partial \dot{p}_{i}}$

C. $\dot{q}_{i}=\frac{\partial H}{\partial q_{i}}, \quad \dot{p}_{i}=-\frac{\partial H}{\partial p_{i}}$

D. $\dot{q}_{i}=\frac{\partial H}{\partial p_{i}}, \quad \dot{p}_{i}=-\frac{\partial H}{\partial q_{i}}$

E. $\frac{d}{d t}\left(\frac{\partial H}{\partial p_{i}}\right)-\frac{\partial H}{\partial q_{i}}=0$

\begin{tabular}{|l|}
\hline Oil \\
\hline Water \\
\hline
\end{tabular}


Input:
A layer of oil with density $800 \mathrm{~kg} / \mathrm{m}^{3}$ floats on top of a volume of water with density $1,000 \mathrm{~kg} / \mathrm{m}^{3}$. A block floats at the oil-water interface with $1 / 4$ of its volume in oil and 3/4 of its volume in water, as shown in the figure above. What is the density of the block?

A. $200 \mathrm{~kg} / \mathrm{m}^{3}$

B. $850 \mathrm{~kg} / \mathrm{m}^{3}$

C. $950 \mathrm{~kg} / \mathrm{m}^{3}$

D. $1,050 \mathrm{~kg} / \mathrm{m}^{3}$

E. $1,800 \mathrm{~kg} / \mathrm{m}^{3}$


Input:
An incompressible fluid of density $\rho$ flows through a horizontal pipe of radius $r$ and then passes through a constriction of radius $r / 2$. If the fluid has pressure $P_{0}$ and velocity $v_{0}$ before the constriction, the pressure in the constriction is

A. $P_{0}-\frac{15}{2} \rho v_{0}^{2}$

B. $P_{0}-\frac{3}{2} \rho v_{0}^{2}$

C. $\frac{P_{0}}{4}$

D. $P_{0}+\frac{3}{2} \rho v_{0}^{2}$

E. $P_{0}+\frac{15}{2} \rho v_{0}^{2}$


Input:
A thermodynamic system, initially at absolute temperature $T_{1}$, contains a mass $m$ of water with specific heat capacity $c$. Heat is added until the temperature rises to $T_{2}$. The change in entropy of the water is

A. 0

B. $T_{2}-T_{1}$

C. $m c T_{2}$

D. $m c\left(T_{2}-T_{1}\right)$

E. $m c \ln \left(\frac{T_{2}}{T_{1}}\right)$


Input:
Heat $Q$ is added to a monatomic ideal gas under conditions of constant volume, resulting in a temperature change $\Delta T$. How much heat will be required to produce the same temperature change, if it is added under conditions of constant pressure?

A. $\frac{3}{5} Q$

B. $Q$

C. $\frac{5}{3} Q$

D. $2 Q$

E. $\frac{10}{3} Q$


Input:
A heat pump is to extract heat from an outdoor environment at $7^{\circ} \mathrm{C}$ and heat the environment indoors to $27^{\circ} \mathrm{C}$. For each $15,000 \mathrm{~J}$ of heat delivered indoors, the smallest amount of work that must be supplied to the heat pump is approximately

A. $500 \mathrm{~J}$

B. $1,000 \mathrm{~J}$

C. $1,100 \mathrm{~J}$

D. $2,000 \mathrm{~J}$

E. $2,200 \mathrm{~J}$


Input:
The capacitor in the circuit above is charged. If switch $S$ is closed at time $t=0$, which of the following represents the magnetic energy, $U$, in the inductor as a function of time? (Assume that the capacitor and inductor are ideal.)

A.


Input:
A pair of electric charges of equal magnitude $q$ and opposite sign are separated by a distance $\ell$, as shown in the figure above. Which of the following gives the approximate magnitude and direction of the electric field set up by the two charges at a point $P$ on the $y$-axis, which is located a distance $r \gg \ell$ from the $x$-axis?
Magnitude
Direction
A. $\frac{1}{4 \pi \epsilon_{0}} \frac{2 q}{r^{2}}$
$+y$
B. $\frac{1}{4 \pi \epsilon_{0}} \frac{2 q}{r^{2}}$
$+x$
C. $\frac{1}{4 \pi \epsilon_{0}} \frac{2 q}{r^{2}}$
$-x$
D. $\frac{1}{4 \pi \epsilon_{0}} \frac{q \ell}{r^{3}}$
$+x$
E. $\frac{1}{4 \pi \epsilon_{0}} \frac{q \ell}{r^{3}}$


Input:
Consider two very long, straight, insulated wires oriented at right angles. The wires carry currents of equal magnitude $I$ in the directions shown in the figure above. What is the net magnetic field at point $P$ ?

A. $\frac{\mu_{0} I}{2 \pi a}(\hat{\mathbf{x}}+\hat{\mathbf{y}})$

B. $-\frac{\mu_{0} I}{2 \pi a}(\hat{\mathbf{x}}+\hat{\mathbf{y}})$

C. $\frac{\mu_{0} I}{\pi a} \hat{\mathbf{z}}$

D. $-\frac{\mu_{0} I}{\pi a} \hat{\mathbf{z}}$

E. 0


Input:
A beam of muons travels through the laboratory with speed $v=\frac{4}{5} c$. The lifetime of a muon in its rest frame is $\tau=2.2 \times 10^{-6} \mathrm{~s}$. The mean distance traveled by the muons in the laboratory frame is

A. $530 \mathrm{~m}$

B. $660 \mathrm{~m}$

C. $880 \mathrm{~m}$

D. $1,100 \mathrm{~m}$

E. $1,500 \mathrm{~m}$


Input:
A particle of mass $M$ decays from rest into two particles. One particle has mass $m$ and the other particle is massless. The momentum of the massless particle is

A. $\frac{\left(M^{2}-m^{2}\right) c}{4 M}$

B. $\frac{\left(M^{2}-m^{2}\right) c}{2 M}$

C. $\frac{\left(M^{2}-m^{2}\right) c}{M}$

D. $\frac{2\left(M^{2}-m^{2}\right) c}{M}$

E. $\frac{4\left(M^{2}-m^{2}\right) c}{M}$


Input:
In an experimental observation of the photoelectric effect, the stopping potential was plotted versus the light frequency, as shown in the figure above. The best straight line was fitted to the experimental points. Which of the following gives the slope of the line? (The work function of the metal is $\phi$.)

A. $\frac{h}{\phi}$

B. $\frac{h}{e}$

C. $\frac{e}{h}$

D. $\frac{e}{\phi}$

E. $\frac{\phi}{e}$


Input:
Two sinusoidal waveforms of the same frequency are displayed on an oscilloscope screen, as indicated above. The horizontal sweep of the oscilloscope is set to $100 \mathrm{~ns} / \mathrm{cm}$ and the vertical gains of channels 1 and 2 are each set to $2 \mathrm{~V} / \mathrm{cm}$. The zero-voltage level of each channel is given at the right in the figure. The phase difference between the two waveforms is most nearly

A. $30^{\circ}$

B. $45^{\circ}$

C. $60^{\circ}$

D. $90^{\circ}$

E. $120^{\circ}$


Input:
In the diamond structure of elemental carbon, the nearest neighbors of each $\mathrm{C}$ atom lie at the corners of a

A. square

B. hexagon

C. cube

D. tetrahedron

E. octahedron


Input:
According to the BCS theory, the attraction between Cooper pairs in a superconductor is due to

A. the weak nuclear force

B. the strong nuclear force

C. vacuum polarization

D. interactions with the ionic lattice

E. the Casimir effect


Input:
During a hurricane, a $1,200 \mathrm{~Hz}$ warning siren on the town hall sounds. The wind is blowing at $55 \mathrm{~m} / \mathrm{s}$ in a direction from the siren toward a person $1 \mathrm{~km}$ away. With what frequency does the sound wave reach the person? (The speed of sound in air is $330 \mathrm{~m} / \mathrm{s}$.)

A. $1,000 \mathrm{~Hz}$

B. $1,030 \mathrm{~Hz}$

C. $1,200 \mathrm{~Hz}$

D. $1,400 \mathrm{~Hz}$

E. $1,440 \mathrm{~Hz}$


Input:
Sound waves moving at $350 \mathrm{~m} / \mathrm{s}$ diffract out of a speaker enclosure with an opening that is a long rectangular slit $0.14 \mathrm{~m}$ across. At about what frequency will the sound first disappear at an angle of $45^{\circ}$ from the normal to the speaker face?

A. $500 \mathrm{~Hz}$

B. $1,750 \mathrm{~Hz}$

C. $2,750 \mathrm{~Hz}$

D. $3,500 \mathrm{~Hz}$

E. $5,000 \mathrm{~Hz}$


Input:
An organ pipe, closed at one end and open at the other, is designed to have a fundamental frequency of $\mathrm{C}(131 \mathrm{~Hz})$. What is the frequency of the next higher harmonic for this pipe?

A. $44 \mathrm{~Hz}$

B. $196 \mathrm{~Hz}$

C. $262 \mathrm{~Hz}$

D. $393 \mathrm{~Hz}$

E. $524 \mathrm{~Hz}$

Inputs


Input:
For the logic circuit shown above, which of the following Boolean statements gives the output $E$ in terms of inputs $A, B, C$, and $D$ ?

A. $E=\overline{A+B}+\overline{C \cdot D}$

B. $E=\overline{A+B} \cdot \overline{C \cdot D}$

C. $E=\overline{\bar{A}+\bar{B}} \cdot \overline{C \cdot D}$

D. $E=\overline{A \cdot B} \cdot \overline{C \cdot D}$

E. $E=\overline{\bar{A} \cdot \bar{B}} \cdot \overline{C \cdot D}$


Input:
Which of the following lasers utilizes transitions that involve the energy levels of free atoms?

A. Diode laser

B. Dye laser

C. Free-electron laser

D. Gas laser

E. Solid-state laser


Input:
Which of the following expressions is proportional to the total energy for the levels of a oneelectron Bohr atom? ( $m$ is the reduced mass, $Z$ is the number of protons in the nucleus, $-e$ is the charge on the electron, and $n$ is the principal quantum number.)

A. $\frac{m Z e^{2}}{n}$

B. $\frac{m Z e^{2}}{n^{2}}$

C. $\frac{m Z^{2} e^{4}}{n^{2}}$

D. $\frac{m^{2} Z^{2} e^{2}}{n^{2}}$

E. $\frac{m^{2} Z^{2} e^{4}}{n^{2}}$


Input:
True statements about the absorption and emission of energy by an atom include which of the following?

I. An atom can only absorb photons of light that have certain specific energies.

II. An atom can emit photons of light of any energy.

III. At low temperature, the lines in the absorption spectrum of an atom coincide with the lines in its emission spectrum that represent transitions to the ground state.

A. I only

B. III only

C. I and II only

D. I and III only

E. I, II, and III


Input:
$X$ rays of wavelength $\lambda=0.250 \mathrm{~nm}$ are incident on the face of a crystal at angle $\theta$, measured from the crystal surface. The smallest angle that yields an intense reflected beam is $\theta=14.5^{\circ}$. Which of the following gives the value of the interplanar spacing $d ?\left(\sin 14.5^{\circ} \approx 1 / 4\right)$

A. $0.125 \mathrm{~nm}$

B. $0.250 \mathrm{~nm}$

C. $0.500 \mathrm{~nm}$

D. $0.625 \mathrm{~nm}$

E. $0.750 \mathrm{~nm}$


Input:
Astronomers observe two separate solar systems, each consisting of a planet orbiting a sun. The two orbits are circular and have the same radius $R$. It is determined that the planets have angular momenta of the same magnitude $L$ about their suns, and that the orbital periods are in the ratio of three to one; i.e., $T_{1}=3 T_{2}$. The ratio $m_{1} / m_{2}$ of the masses of the two planets is

A. 1

B. $\sqrt{3}$

C. 2

D. 3

E. 9


Input:
If the Sun were suddenly replaced by a black hole of the same mass, it would have a Schwarzschild radius of $3,000 \mathrm{~m}$. What effect, if any, would this change have on the orbits of the planets?

A. The planets would move directly toward the Sun.

B. The planets would move in spiral orbits.

C. The planets would oscillate about their former elliptical orbits.

D. The orbits would precess much more rapidly.

E. The orbits would remain unchanged.


Input:
A distant galaxy is observed to have its hydrogen- $\beta$ line shifted to a wavelength of $580 \mathrm{~nm}$, away from the laboratory value of $434 \mathrm{~nm}$. Which of the following gives the approximate velocity of recession of the distant galaxy? (Note: $\frac{580}{434} \approx \frac{4}{3}$ )

A. $0.28 c$

B. $0.53 c$

C. $0.56 c$

D. $0.75 c$

E. $0.86 c$


Input:
A small plane can fly at a speed of $200 \mathrm{~km} / \mathrm{h}$ in still air. A $30 \mathrm{~km} / \mathrm{h}$ wind is blowing from west to east. How much time is required for the plane to fly $500 \mathrm{~km}$ due north?

A. $\frac{50}{23} \mathrm{~h}$

B. $\frac{50}{\sqrt{409}} \mathrm{~h}$

C. $\frac{50}{20} \mathrm{~h}$

D. $\frac{50}{\sqrt{391}} \mathrm{~h}$

E. $\frac{50}{17} \mathrm{~h}$


Input:
Each of the figures above shows blocks of mass $2 m$ and $m$ acted on by an external horizontal force $\mathbf{F}$. For each figure, which of the following statements about the magnitude of the force that one block exerts on the other $\left(F_{12}\right)$ is correct? (Assume that the surface on which the blocks move is frictionless.)

\section*{Figure 1}

A. $F_{12}=\frac{F}{3}$

B. $F_{12}=\frac{F}{3}$

C. $F_{12}=\frac{2 F}{3}$

D. $F_{12}=\frac{2 F}{3}$

E. $F_{12}=F$

\section*{Figure 2}

$$
F_{12}=\frac{F}{3}
$$

$F_{12}=\frac{2 F}{3}$

$F_{12}=\frac{F}{3}$

$F_{12}=\frac{2 F}{3}$

$F_{12}=F$


Input:
In the figure above, block $A$ has mass

$m_{A}=25 \mathrm{~kg}$ and block $B$ has mass $m_{B}=10 \mathrm{~kg}$.

Both blocks move with constant acceleration $a=2 \mathrm{~m} / \mathrm{s}^{2}$ to the right, and the coefficient of static friction between the two blocks is $\mu_{s}=0.8$. The static frictional force acting between the blocks is

A. $20 \mathrm{~N}$

B. $50 \mathrm{~N}$

C. $78 \mathrm{~N}$

D. $196 \mathrm{~N}$

E. $274 \mathrm{~N}$


Input:
A simple pendulum of length $l$ is suspended from the ceiling of an elevator that is accelerating upward with constant acceleration $a$. For small oscillations, the period, $T$, of the pendulum is

A. $T=2 \pi \sqrt{\frac{l}{g}}$

B. $T=2 \pi \sqrt{\frac{l}{g-a}}$

C. $T=2 \pi \sqrt{\frac{l}{g+a}}$

D. $T=2 \pi \sqrt{\frac{l}{g} \frac{a}{(g+a)}}$

E. $T=2 \pi \sqrt{\frac{l}{g} \frac{(g+a)}{a}}$


Input:
Three long, straight wires in the $x z$-plane, each carrying current $I$, cross at the origin of coordinates, as shown in the figure above. Let $\hat{\mathbf{x}}, \hat{\mathbf{y}}$, and $\hat{\mathbf{z}}$ denote the unit vectors in the $x$-, $y$-, and $z$-directions, respectively. The magnetic field $\mathbf{B}$ as a function of $x$, with $y=0$ and $z=0$, is

A. $\mathbf{B}=\frac{3 \mu_{0} I}{2 \pi x} \hat{\mathbf{x}}$

B. $\mathbf{B}=\frac{3 \mu_{0} I}{2 \pi x} \hat{\mathbf{y}}$

C. $\mathbf{B}=\frac{\mu_{0} I}{2 \pi x}(1+2 \sqrt{2}) \hat{\mathbf{y}}$

D. $\mathbf{B}=\frac{\mu_{0} I}{2 \pi x} \hat{\mathbf{x}}$

E. $\mathbf{B}=\frac{\mu_{0} I}{2 \pi x} \hat{\mathbf{y}}$


Input:
A particle with mass $m$ and charge $q$, moving with a velocity $\mathbf{v}$, enters a region of uniform magnetic field $\mathbf{B}$, as shown in the figure above. The particle strikes the wall at a distance $d$ from the entrance slit. If the particle's velocity stays the same but its charge-to-mass ratio is doubled, at what distance from the entrance slit will the particle strike the wall?

A. $2 d$

B. $\sqrt{2} d$

C. $d$

D. $\frac{1}{\sqrt{2}} d$

E. $\frac{1}{2} d$


Input:
Consider the closed cylindrical Gaussian surface above. Suppose that the net charge enclosed within this surface is $+1 \times 10^{-9} \mathrm{C}$ and the electric flux out through the portion of the surface marked $A$ is $-100 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$. The flux through the rest of the surface is most nearly given by which of the following?

A. $-100 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

B. $\quad 0 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

C. $10 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

D. $100 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

E. $200 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}$

$$
{ }^{13} \mathrm{~N} \rightarrow{ }^{13} \mathrm{C}+e^{+}+v_{e}
$$


Input:
The nuclear decay above is an example of a process induced by the

A. Mössbauer effect

B. Casimir effect

C. photoelectric effect

D. weak interaction

E. strong interaction


Input:
Consider a single electron atom with orbital angular momentum $L=\sqrt{2} \hbar$. Which of the following gives the possible values of a measurement of $L_{z}$, the $z$-component of $L$ ?

A. 0

B. $0, \hbar$

C. $0, \hbar, 2 \hbar$

D. $-\hbar, 0, \hbar$

E. $-2 \hbar,-\hbar, 0, \hbar, 2 \hbar$


Input:
Characteristics of the quantum harmonic oscillator include which of the following?

I. A spectrum of evenly spaced energy states

II. A potential energy function that is linear in the position coordinate

III. A ground state that is characterized by zero kinetic energy

IV. A nonzero probability of finding the oscillator outside the classical turning points

A. I only

B. IV only

C. I and IV only

D. II and III only

E. I, II, III, and IV


Input:
A muon can be considered to be a heavy electron with a mass $m_{\mu}=207 m_{e}$. Imagine replacing the electron in a hydrogen atom with a muon. What are the energy levels $E_{n}$ for this new form of hydrogen in terms of the binding energy of ordinary hydrogen $E_{0}$, the mass of the proton $m_{p}$, and the principal quantum number $n$ ?

A. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{m_{\mu}}{m_{e}}\right)$

B. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{m_{e}}{m_{\mu}}\right)$

C. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{\left(m_{p}+m_{e}\right)}{\left(m_{p}+m_{\mu}\right)}\right)$

D. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{m_{\mu}\left(m_{p}+m_{e}\right)}{m_{e}\left(m_{p}+m_{\mu}\right)}\right)$

E. $E_{n}=\frac{-E_{0}}{n^{2}}\left(\frac{m_{e}\left(m_{p}+m_{\mu}\right)}{m_{\mu}\left(m_{p}+m_{e}\right)}\right)$


Input:
A large, parallel-plate capacitor consists of two square plates that measure $0.5 \mathrm{~m}$ on each side. A charging current of $9 \mathrm{~A}$ is applied to the capacitor. Which of the following gives the approximate rate of change of the electric field between the plates?

A. $2 \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$

B. $40 \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$

C. $1 \times 10^{12} \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$

D. $4 \times 10^{12} \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$

E. $2 \times 10^{13} \frac{\mathrm{V}}{\mathrm{m} \cdot \mathrm{s}}$


Input:
The circuit shown in the figure above consists of eight resistors, each with resistance $R$, and a battery with terminal voltage $V$ and negligible internal resistance. What is the current flowing through the battery?

A. $\frac{1}{3} \frac{V}{R}$

B. $\frac{1}{2} \frac{V}{R}$

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

D. $\frac{3}{2} \frac{V}{R}$

E. $3 \frac{V}{R}$


Input:
In the AC circuit above, $V_{i}$ is the amplitude of the input voltage and $V_{o}$ is the amplitude of the output voltage. If the angular frequency $\omega$ of the input voltage is varied, which of the following gives the ratio $V_{o} / V_{i}=G$ as a function of $\omega$ ?

A.


Input:
A wire loop that encloses an area of $10 \mathrm{~cm}^{2}$ has a resistance of $5 \Omega$. The loop is placed in a magnetic field of $0.5 \mathrm{~T}$ with its plane perpendicular to the field. The loop is suddenly removed from the field. How much charge flows past a given point in the wire?

A. $10^{-4} \mathrm{C}$

B. $10^{-3} \mathrm{C}$

C. $10^{-2} \mathrm{C}$

D. $10^{-1} \mathrm{C}$

E. $1 \mathrm{C}$


Input:
Two nonrelativistic electrons move in circles under the influence of a uniform magnetic field $\mathbf{B}$, as shown in the figure above. The ratio $r_{1} / r_{2}$ of the orbital radii is equal to $1 / 3$. Which of the following is equal to the ratio $v_{1} / v_{2}$ of the speeds?

A. $1 / 9$

B. $1 / 3$

C. 1

D. 3

E. 9


Input:
Which of the following statements about bosons and/or fermions is true?

A. Bosons have symmetric wave functions and obey the Pauli exclusion principle.

B. Bosons have antisymmetric wave functions and do not obey the Pauli exclusion principle.

C. Fermions have symmetric wave functions and obey the Pauli exclusion principle.

D. Fermions have antisymmetric wave functions and obey the Pauli exclusion principle.

E. Bosons and fermions obey the Pauli exclusion principle.


Input:
The discovery of the $J / \psi$ particle was especially significant because it provided evidence for which of the following?

A. Parity violation in weak interactions

B. Massive neutrinos

C. Higgs bosons

D. Charmed quarks

E. Strange quarks


Input:
The figure above shows an object $O$ placed at a distance $R$ to the left of a convex spherical mirror that has a radius of curvature $R$. Point $C$ is the center of curvature of the mirror. The image formed by the mirror is at

A. infinity

B. a distance $R$ to the left of the mirror and inverted

C. a distance $R$ to the right of the mirror and upright

D. a distance $\frac{R}{3}$ to the left of the mirror and inverted

E. a distance $\frac{R}{3}$ to the right of the mirror and upright


Input:
A uniform thin film of soapy water with index of refraction $n=1.33$ is viewed in air via reflected light. The film appears dark for long wavelengths and first appears bright for $\lambda=540 \mathrm{~nm}$. What is the next shorter wavelength at which the film will appear bright on reflection?

A. $135 \mathrm{~nm}$

B. $180 \mathrm{~nm}$

C. $270 \mathrm{~nm}$

D. $320 \mathrm{~nm}$

E. $405 \mathrm{~nm}$


Input:
A model of an optical fiber is shown in the figure above. The optical fiber has an index of refraction, $n$, and is surrounded by free space. What angles of incidence, $\theta$, will result in the light staying in the optical fiber?

A. $\theta>\sin ^{-1}\left(\sqrt{n^{2}-1}\right)$

B. $\theta<\sin ^{-1}\left(\sqrt{n^{2}-1}\right)$

C. $\theta>\sin ^{-1}\left(\sqrt{n^{2}+1}\right)$

D. $\theta<\sin ^{-1}\left(\sqrt{n^{2}+1}\right)$

E. $\sin ^{-1}\left(\sqrt{n^{2}-1}\right)<\theta<\sin ^{-1}\left(\sqrt{n^{2}+1}\right)$


Input:
A gas at temperature $T$ is composed of molecules of mass $m$. Which of the following describes how the average time between intermolecular collisions varies with $m$ ?

A. It is proportional to $\frac{1}{m}$.

B. It is proportional to $\sqrt[4]{m}$.

C. It is proportional to $\sqrt{m}$.

D. It is proportional to $m$.

E. It is proportional to $m^{2}$.


Input:
A particle can occupy two possible states with energies $E_{1}$ and $E_{2}$, where $E_{2}>E_{1}$. At temperature $T$, the probability of finding the particle in state 2 is given by which of the following?

A. $\frac{\mathrm{e}^{-E_{1} / k T}}{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}$

B. $\frac{\mathrm{e}^{-E_{2} / k T}}{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}$

C. $\frac{\mathrm{e}^{-\left(E_{1}+E_{2}\right) / k T}}{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}$

D. $\frac{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}{\mathrm{e}^{-E_{2} / k T}}$

E. $\frac{\mathrm{e}^{-E_{1} / k T}+\mathrm{e}^{-E_{2} / k T}}{\mathrm{e}^{-E_{1} / k T}}$

$$
\left(p+\frac{a}{V^{2}}\right)(V-b)=R T
$$


Input:
Consider 1 mole of a real gas that obeys the van der Waals equation of state shown above. If the gas undergoes an isothermal expansion at temperature $T_{0}$ from volume $V_{1}$ to volume $V_{2}$, which of the following gives the work done by the gas?

A. 0

B. $R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)$

C. $R T_{0} \ln \left(\frac{V_{2}-b}{V_{1}-b}\right)$

D. $R T_{0} \ln \left(\frac{V_{2}-b}{V_{1}-b}\right)+a\left(\frac{1}{V_{2}}-\frac{1}{V_{1}}\right)$

E. $R T_{0}\left(\frac{1}{\left(V_{2}-b\right)^{2}}-\frac{1}{\left(V_{1}-b\right)^{2}}\right)+a\left(\frac{1}{V_{2}^{3}}-\frac{1}{V_{1}^{3}}\right)$


Input:
A $1 \mathrm{~kg}$ block attached to a spring vibrates with a frequency of $1 \mathrm{~Hz}$ on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an $8 \mathrm{~kg}$ block placed on the same table. Which of the following gives the frequency of vibration of the $8 \mathrm{~kg}$ block?

A. $\frac{1}{4} \mathrm{~Hz}$

B. $\frac{1}{2 \sqrt{2}} \mathrm{~Hz}$

C. $\frac{1}{2} \mathrm{~Hz}$

D. $1 \mathrm{~Hz}$

E. $2 \mathrm{~Hz}$


Input:
A uniform disk with a mass of $m$ and a radius of $r$ rolls without slipping along a horizontal surface and ramp, as shown above. The disk has an initial velocity of $v$. What is the maximum height $h$ to which the center of mass of the disk rises?

A. $h=\frac{v^{2}}{2 g}$

B. $h=\frac{3 v^{2}}{4 g}$

C. $h=\frac{v^{2}}{g}$

D. $h=\frac{3 v^{2}}{2 g}$

E. $h=\frac{2 v^{2}}{g}$


Input:
A mass, $m$, is attached to a massless spring fixed at one end. The mass is confined to move in a horizontal plane, and its position is given by the polar coordinates $r$ and $\theta$. Both $r$ and $\theta$ can vary. If the relaxed length of the spring is $s$ and the force constant is $k$, what is the Lagrangian, $L$, for the system?

A. $L=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}-\frac{1}{2} k(r \cos \theta-s)^{2}$

B. $L=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}-\frac{1}{2} k(r \sin \theta-s)^{2}$

C. $L=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}+\frac{1}{2} k(r-s)^{2}$

D. $L=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}-\frac{1}{2} k(r-s)^{2}$

E. $L=-\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}+\frac{1}{2} k(r-s)^{2}$


Input:
A mass $m$ attached to the end of a massless rod of length $L$ is free to swing below the plane of support, as shown in the figure above. The Hamiltonian for this system is given by

$$
H=\frac{p_{\theta}^{2}}{2 m L^{2}}+\frac{p_{\phi}^{2}}{2 m L^{2} \sin ^{2} \theta}-m g L \cos \theta
$$

where $\theta$ and $\phi$ are defined as shown in the figure. On the basis of Hamilton's equations of motion, the generalized coordinate or momentum that is a constant in time is

A. $\theta$

B. $\phi$

C. $\dot{\theta}$

D. $p_{\theta}$

E. $p_{\phi}$


Input:
A rod of length $L$ and mass $M$ is placed along the $x$-axis with one end at the origin, as shown in the figure above. The rod has linear mass density $\lambda=\frac{2 M}{L^{2}} x$, where $x$ is the distance from the origin. Which of the following gives the $x$-coordinate of the rod's center of mass?

A. $\frac{1}{12} L$

B. $\frac{1}{4} L$

C. $\frac{1}{3} L$

D. $\frac{1}{2} L$

E. $\frac{2}{3} L$


Input:
A particle is in an infinite square well potential with walls at $x=0$ and $x=L$. If the particle is in the state $\psi(x)=A \sin \left(\frac{3 \pi x}{L}\right)$, where $A$ is a constant, what is the probability that the particle is between $x=\frac{1}{3} L$ and $x=\frac{2}{3} L$ ?

A. 0

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

C. $\frac{1}{\sqrt{3}}$

D. $\frac{2}{3}$

E. 1


Input:
Which of the following are the eigenvalues of the Hermitian matrix $\left(\begin{array}{cc}2 & i \\ -i & 2\end{array}\right)$ ?

A. 1,0

B. 1,3

C. 2,2

D. $i,-i$

E. $1+i, 1-i$

$$
\sigma_{x}=\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right), \quad \sigma_{y}=\left(\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right), \quad \sigma_{z}=\left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right), \quad I=\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)
$$


Input:
Consider the Pauli spin matrices $\sigma_{x}, \sigma_{y}$, and $\sigma_{z}$ and the identity matrix $I$ given above. The commutator $\left[\sigma_{x}, \sigma_{y}\right] \equiv \sigma_{x} \sigma_{y}-\sigma_{y} \sigma_{x}$ is equal to which of the following?

A. $I$

B. $2 i \sigma_{x}$

C. $2 i \sigma_{y}$

D. $2 i \sigma_{z}$

E. 0


Input:
A spin- $\frac{1}{2}$ particle is in a state described by the spinor

$$
\chi=A\left(\begin{array}{c}
1+i \\
2
\end{array}\right)
$$

where $A$ is a normalization constant. The probability of finding the particle with spin projection $S_{z}=-\frac{1}{2} \hbar$ is

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

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

C. $\frac{1}{2}$

D. $\frac{2}{3}$

E. 1


Input:
An electron with total energy $E$ in the region $x<0$ is moving in the $+x$-direction. It encounters a step potential at $x=0$. The wave function for $x \leq 0$ is given by

$$
\psi=A \mathrm{e}^{i k_{1} x}+B \mathrm{e}^{-i k_{1} x}, \text { where } k_{1}=\sqrt{\frac{2 m E}{\hbar^{2}}}
$$

and the wave function for $x>0$ is given by

$$
\psi=C \mathrm{e}^{i k_{2} x}, \text { where } k_{2}=\sqrt{\frac{2 m\left(E-V_{0}\right)}{\hbar^{2}}}
$$

Which of the following gives the reflection coefficient for the system?

A. $R=0$

B. $R=1$

C. $R=\frac{k_{2}}{k_{1}}$

D. $R=\left(\frac{k_{1}-k_{2}}{k_{1}+k_{2}}\right)^{2}$

E. $R=\frac{4 k_{1} k_{2}}{\left(k_{1}+k_{2}\right)^{2}}$


Input:
Two thin, concentric, spherical conducting shells are arranged as shown in the figure above. The inner shell has radius $a$, charge $+Q$, and is at zero electric potential. The outer shell has radius $b$ and charge $-Q$. If $r$ is the radial distance from the center of the spheres, what is the electric potential in region I $(a<r<b)$ and in region II $(r>b)$ ?

\section*{Region I}

\section*{Region II}

A. $\frac{Q}{4 \pi \varepsilon_{0}} \frac{1}{r}$

B. $\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{1}{a}\right)$ 0

C. $\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{1}{b}\right)$ $-\frac{Q}{4 \pi \varepsilon_{0}} \frac{1}{r}$

D. $\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{1}{a}\right)$

$\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{b}-\frac{1}{a}\right)$

E. $\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{1}{b}\right)$

$\frac{Q}{4 \pi \varepsilon_{0}}\left(\frac{1}{a}-\frac{1}{b}\right)$


Input:
In static electromagnetism, let $\mathbf{E}, \mathbf{B}, \mathbf{J}$, and $\rho$ be the electric field, magnetic field, current density, and charge density, respectively. Which of the following conditions allows the electric field to be written in the form $\mathbf{E}=-\nabla \phi$, where $\phi$ is the electrostatic potential?

A. $\nabla \cdot \mathbf{J}=0$

B. $\nabla \cdot \mathbf{E}=\rho / \epsilon_{0}$

C. $\nabla \times \mathbf{E}=\mathbf{0}$

D. $\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}$

E. $\nabla \cdot \mathbf{B}=0$


Input:
A long, straight, hollow cylindrical wire with an inner radius $R$ and an outer radius $2 R$ carries a uniform current density. Which of the following graphs best represents the magnitude of the magnetic field as a function of the distance from the center of the wire?

A.


Input:
A parallel-plate capacitor has plate separation $d$. The space between the plates is empty. A battery supplying voltage $V_{0}$ is connected across the capacitor, resulting in electromagnetic energy $U_{0}$ stored in the capacitor. A dielectric, of dielectric constant $\kappa$, is inserted so that it just fills the space between the plates. If the battery is still connected, what are the electric field $E$ and the energy $U$ stored in the dielectric, in terms of $V_{0}$ and $U_{0}$ ?
$\underline{E}$
$\underline{U}$
A. $\frac{V_{0}}{d}$
$U_{0}$
B. $\frac{V_{0}}{d}$
$\kappa U_{0}$
C. $\frac{V_{0}}{d}$
$\kappa^{2} U_{0}$
D. $\frac{V_{0}}{\kappa d}$
$U_{0}$
E. $\frac{V_{0}}{\kappa d}$
$\kappa U_{0}$


Input:
An observer $O$ at rest midway between two sources of light at $x=0$ and $x=10 \mathrm{~m}$ observes the two sources to flash simultaneously. According to a second observer $O^{\prime}$, moving at a constant speed parallel to the $x$-axis, one source of light flashes $13 \mathrm{~ns}$ before the other. Which of the following gives the speed of $O^{\prime}$ relative to $O$ ?

A. $0.13 c$

B. $0.15 c$

C. $0.36 c$

D. $0.53 c$

E. $0.62 c$


Input:
Let $\hat{\mathbf{J}}$ be a quantum mechanical angular momentum operator. The commutator $\left[\hat{J}_{x} \hat{J}_{y}, \hat{J}_{x}\right]$ is equivalent to which of the following?

A. 0

B. $i \hbar \hat{J}_{z}$

C. $i \hbar \hat{J}_{z} \hat{J}_{x}$

D. $-i \hbar \hat{J}_{x} \hat{J}_{z}$

E. $i \hbar \hat{J}_{x} \hat{J}_{y}$


Input:
Which of the following ions CANNOT be used as a dopant in germanium to make an $n$-type semiconductor?

A. As

B. $\mathrm{P}$

C. $\mathrm{Sb}$

D. $\mathrm{B}$

E. $\mathrm{N}$


Input:
In the Compton effect, a photon with energy $E$ scatters through a $90^{\circ}$ angle from a stationary electron of mass $m$. The energy of the scattered photon is

A. $E$

B. $\frac{E}{2}$

C. $\frac{E^{2}}{m c^{2}}$

D. $\frac{E^{2}}{E+m c^{2}}$

E. $\frac{E \cdot m c^{2}}{E+m c^{2}}$


Input:
Which of the following is the principal decay mode of the positive muon $\mu^{+}$?

A. $\mu^{+} \rightarrow e^{+}+v_{e}$

B. $\mu^{+} \rightarrow p+v_{\mu}$

C. $\mu^{+} \rightarrow n+e^{+}+v_{e}$

D. $\mu^{+} \rightarrow e^{+}+v_{e}+\bar{v}_{\mu}$

E. $\mu^{+} \rightarrow \pi^{+}+\bar{v}_{e}+v_{\mu}$


Input:
A small particle of mass $m$ is at rest on a horizontal circular platform that is free to rotate about a vertical axis through its center. The particle is located at a radius $r$ from the axis, as shown in the figure above. The platform begins to rotate with constant angular acceleration $\alpha$. Because of friction between the particle and the platform, the particle remains at rest with respect to the platform. When the platform has reached angular speed $\omega$, the angle $\theta$ between the static frictional force $\mathbf{f}_{\mathrm{s}}$ and the inward radial direction is given by which of the following?

A. $\theta=\frac{\omega^{2} r}{g}$

B. $\theta=\frac{\omega^{2}}{\alpha}$

C. $\theta=\frac{\alpha}{\omega^{2}}$

D. $\theta=\tan ^{-1}\left(\frac{\omega^{2}}{\alpha}\right)$

E. $\theta=\tan ^{-1}\left(\frac{\alpha}{\omega^{2}}\right)$


Input:
The partition function $Z$ in statistical mechanics can be written as

$$
Z=\sum_{r} \mathrm{e}^{-E_{r} / k T}
$$

where the index $r$ ranges over all possible microstates of a system and $E_{r}$ is the energy of microstate $r$. For a single quantum mechanical harmonic oscillator with energies

$$
E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega, \text { where } n=0,1,2, \ldots,
$$

the partition function $Z$ is given by which of the following?

A. $Z=\mathrm{e}^{-\frac{1}{2} \hbar \omega / k T}$

B. $Z=\mathrm{e}^{\frac{1}{2} \hbar \omega / k T}$

C. $Z=\mathrm{e}^{\frac{1}{2} \hbar \omega / k T}-1$

D. $Z=\mathrm{e}^{\frac{1}{2} \hbar \omega / k T}+1$

E. $Z=\frac{\mathrm{e}^{\frac{1}{2} \hbar \omega / k T}}{\mathrm{e}^{\hbar \omega / k T}-1}$