Input:
A rock is thrown vertically upward with initial speed $v_{0}$. Assume a friction force proportional to $-v$, where $v$ is the velocity of the rock, and neglect the buoyant force exerted by air. Which of the following is correct?

A. The acceleration of the rock is always equal to $\mathbf{g}$.

B. The acceleration of the rock is equal to $\mathrm{g}$ only at the top of the flight.

C. The acceleration of the rock is always less than $g$.

D. The speed of the rock upon return to its starting point is $v_{0}$.

E. The rock can attain a terminal speed greater than $v_{0}$ before it returns to its starting point.


Input:
A satellite orbits the Earth in a circular orbit. An astronaut on board perturbs the orbit slightly by briefly firing a control jet aimed toward the Earth's center. Afterward, which of the following is true of the satellite's path?

A. It is an ellipse.

B. It is a hyperbola.

C. It is a circle with larger radius.

D. It is a spiral with increasing radius.

E. It exhibits many radial oscillations per revolution.


Input:
For blue light, a transparent material has a relative permittivity (dielectric constant) of 2.1 and a relative permeability of 1.0 . If the speed of light in a vacuum is $c$, the phase velocity of blue light in an unbounded medium of this material is

A. $\sqrt{3.1} c$

B. $\sqrt{2.1} c$

C. $\frac{c}{\sqrt{1.1}}$

D. $\frac{c}{\sqrt{2.1}}$

E. $\frac{c}{\sqrt{3.1}}$


Input:
The equation $y=A \sin 2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right)$, where $A, T$, and $\lambda$ are positive constants, represents a wave whose

A. amplitude is $2 \mathrm{~A}$

B. velocity is in the negative $x$-direction

C. period is $\frac{T}{\lambda}$

D. speed is $\frac{x}{t}$

E. speed is $\frac{\lambda}{T}$


Input:
Two small spheres of putty, $A$ and $B$, of mass $M$ and $3 M$, respectively, hang from the ceiling on strings of equal length $\ell$. Sphere $A$ is drawn aside so that it is raised to a height $h_{0}$ as shown above and then released. Sphere $A$ collides with sphere $B$; they stick together and swing to a maximum height $h$ equal to
A. $\frac{1}{16} h_{0}$
B. $\frac{1}{8} h_{0}$
C. $\frac{1}{4} h_{0}$
D. $\frac{1}{3} h_{0}$
E. $\frac{1}{2} h_{0}$


Input:
A particle is initially at rest at the top of a curved frictionless track. The $x$ - and $y$-coordinates of the track are related in dimensionless units by $y=\frac{x^{2}}{4}$, where the positive $y$-axis is in the vertical downward direction. As the particle slides down the track, what is its tangential acceleration?

A. 0

B. $g$

C. $\frac{g x}{2}$

D. $\frac{g x}{\sqrt{x^{2}+4}}$

E. $\frac{g x^{2}}{\sqrt{x^{2}+16}}$


Input:
A 2-kilogram box hangs by a massless rope from a ceiling. A force slowly pulls the box horizontally to the side until the horizontal force is 10 newtons. The box is then in equilibrium as shown above. The angle that the rope makes with the vertical is closest to

A. $\arctan 0.5$

B. $\arcsin 0.5$

C. $\arctan 2.0$

D. $\arcsin 2.0$

E. $45^{\circ}$


Input:
A 5-kilogram stone is dropped on a nail and drives the nail 0.025 meter into a piece of wood. If the stone is moving at 10 meters per second when it hits the nail, the average force exerted on the nail by the stone while the nail is going into the wood is most nearly

A. 10 N

B. 100 N

C. 1000 N

D. 10,000 N

E. 100,000 N


Input:
A wire of diameter 0.02 meter contains $1 \times 10^{28}$ free electrons per cubic meter. For an electric current of 100 amperes, the drift velocity for free electrons in the wire is most nearly

A. $0.6 \times 10^{-29} \mathrm{~m} / \mathrm{s}$

B. $1 \times 10^{-19} \mathrm{~m} / \mathrm{s}$

C. $5 \times 10^{-10} \mathrm{~m} / \mathrm{s}$

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

E. $8 \times 10^{3} \mathrm{~m} / \mathrm{s}$

Electric Field

Magnitude


Input:
An isolated sphere of radius $R$ contains a uniform volume distribution of positive charge. Which of the curves on the graph above correctly illustrates the dependence of the magnitude of the electric field of the sphere as a function of the distance $r$ from its center?
A. $A$
B. $B$
C. $C$
D. $D$
E. $E$


Input:
Which of the following equations is a consequence of the equation $\nabla \times \mathbf{H}=\dot{\mathbf{D}}+\mathbf{J}$ ?

A. $\nabla \cdot(\dot{\mathbf{D}}+\mathbf{J})=0$

B. $\nabla \times(\dot{\mathbf{D}}+\mathbf{J})=0$

C. $\nabla(\dot{\mathbf{D}} \cdot \mathbf{J})=0$

D. $\dot{\mathbf{D}}+\mathrm{J}=0$

E. $\dot{\mathbf{D}} \cdot \mathbf{J}=\mathbf{0}$


Input:
A source of 1-kilohertz sound is moving straight toward you at a speed 0.9 times the speed of sound. The frequency you receive is

A. $0.1 \mathrm{kHz}$

B. $0.5 \mathrm{kHz}$

C. $1.1 \mathrm{kHz}$

D. $1.9 \mathrm{kHz}$

E. $10 \mathrm{kHz}$


Input:
Two coherent sources of visible monochromatic light form an interference pattern on a screen. If the relative phase of the sources is varied from 0 to $2 \pi$ at a frequency of 500 hertz, which of the following best describes the effect, if any, on the interference pattern?

A. It is unaffected because the frequency of the phase change is very small compared to the frequency of visible light.

B. It is unaffected because the frequency of the phase change is an integral multiple of $\pi$.

C. It is destroyed except when the phase difference is 0 or $\pi$.

D. It is destroyed for all phase differences because the monochromaticity of the sources is destroyed.

E. It is not destroyed but simply shifts positions at a rate too rapid to be detected by the eye.


Input:
For an ideal gas, the specific heat at constant pressure $C_{p}$ is greater than the specific heat at constant volume $C_{\nu}$ because the

A. gas does work on its environment when its pressure remains constant while its temperature is increased

B. heat input per degree increase in temperature is the same in processes for which either the pressure or the volume is kept constant

C. pressure of the gas remains constant when its temperature remains constant

D. increase in the gas's internal energy is greater when the pressure remains constant than when the volume remains constant

E. heat needed is greater when the volume remains constant than when the pressure remains constant


Input:
A sample of $N$ atoms of helium gas is confined in a 1.0 cubic meter volume. The probability that none of the helium atoms is in a $1.0 \times 10^{-6}$ cubic meter volume of the container is
A. 0
B. $\left(10^{-6}\right)^{N}$
C. $\left(1-10^{-6}\right)^{N}$
D. $1-\left(10^{-6}\right)^{N}$
E. 1


Input:
Except for mass, the properties of the muon most closely resemble the properties of the
A. electron
B. graviton
C. photon
D. pion
E. proton


Input:
Suppose that ${ }_{Z}^{A} X$ decays by natural radioactivity in two stages to ${ }_{Z-1}^{A-4} Y$. The two stages would most likely be which of the following?
$\underline{\text { First Stage }}$
$\underline{\text { Second Stage }}$
A. $\beta^{-}$emission with an antineutrino
$\alpha$ emission
B. $\boldsymbol{\beta}^{-}$emission
$\alpha$ emission with a neutrino
C. $\boldsymbol{\beta}^{-}$emission
$\gamma$ emission
D. Emission of a deuteron Emission of two neutrons
E. $\alpha$ emission $\gamma$ emission


Input:
The wave function $\psi(x)=A \exp \left\{-\frac{b^{2} x^{2}}{2}\right\}$, where $A$ and $b$ are real constants, is a normalized eigenfunction of the Schrödinger equation for a particle of mass $M$ and energy $E$ in a one dimensional potential $V(x)$ such that $V(x)=0$ at $x=0$. Which of the following is correct?

A. $V=\frac{\hbar^{2} b^{4}}{2 M}$

B. $V=\frac{\hbar^{2} b^{4} x^{2}}{2 M}$

C. $V=\frac{\hbar^{2} b^{6} x^{4}}{2 M}$

D. $E=\hbar^{2} b^{2}\left(1-b^{2} x^{2}\right)$

E. $E=\frac{\hbar^{2} b^{4}}{2 M}$


Input:
The energy levels of the hydrogen atom are given in terms of the principal quantum number $n$ and a positive constant $A$ by the expression

A. $A\left(n+\frac{1}{2}\right)$

B. $A\left(1-n^{2}\right)$

C. $A\left(-\frac{1}{4}+\frac{1}{n^{2}}\right)$

D. $A n^{2}$

E. $-\frac{A}{n^{2}}$


Input:
A positive kaon $\left(K^{+}\right)$has a rest mass of $494 \mathrm{MeV} / c^{2}$, whereas a proton has a rest mass of $938 \mathrm{MeV} / c^{2}$. If a kaon has a total energy that is equal to the proton rest energy, the speed of the kaon is most nearly

A. $0.25 c$

B. $0.40 c$

C. $0.55 c$

D. $0.70 c$

E. $0.85 c$


Input:
Two observers $O$ and $O^{\prime}$ observe two events, $A$ and $B$. The observers have a constant relative speed of $0.8 c$. In units such that the speed of light is 1 , observer $O$ obtained the following coordinates:

Event $A: x=3, y=3, z=3, t=3$

Event $B: x=5, y=3, z=1, t=5$

What is the length of the space-time interval between these two events, as measured by $O^{\prime}$ ?
A. 1
B. $\sqrt{2}$
C. 2
D. 3
E. $2 \sqrt{3}$


Input:
Which of the following statements most accurately describes how an electromagnetic field behaves under a Lorentz transformation?

A. The electric field transforms completely into a magnetic field.

B. If initially there is only an electric field, after the transformation there may be both an electric and a magnetic field.

C. The electric field is unaltered.

D. The magnetic field is unaltered.

E. It cannot be determined unless a gauge transformation is also specified.


Input:
Which of the following statements concerning the electrical conductivities at room temperature of a pure copper sample and a pure silicon sample is NOT true?

A. The conductivity of the copper sample is many orders of magnitude greater than that of the silicon sample.

B. If the temperature of the copper sample is increased, its conductivity will decrease.

C. If the temperature of the silicon sample is increased, its conductivity will increase.

D. The addition of an impurity in the copper sample always decreases its conductivity.

E. The addition of an impurity in the silicon sample always decreases its conductivity.


Input:
The battery in the diagram above is to be charged by the generator $G$. The generator has a terminal voltage of 120 volts when the charging current is 10 amperes. The battery has an emf of 100 volts and an internal resistance of $1 \mathrm{ohm}$. In order to charge the battery at 10 amperes charging current, the resistance $R$ should be set at
A. $0.1 \Omega$
B. $0.5 \Omega$
D. $5.0 \Omega$
E. $10.0 \Omega$
C. $1.0 \Omega$


Input:
A charged particle is released from rest in a region where there is a constant electric field and a constant magnetic field. If the two fields are parallel to each other, the path of the particle is a

A. circle

B. parabola

C. helix

D. cycloid

E. straight line


Input:
A nickel target $(Z=28)$ is bombarded with fast electrons. The minimum electron kinetic energy needed to produce $\mathrm{x}$-rays in the $K$ series is most nearly

A. 10 eV

B. 100 eV

C. 1000 eV

D. 10,000 eV

E. 100,000 eV


Input:
The hypothesis that an electron possesses spin is qualitatively significant for the explanation of all of the following topics EXCEPT the

A. structure of the periodic table

B. specific heat of metals

C. anomalous Zeeman effect

D. deflection of a moving electron by a uniform magnetic field

E. fine structure of atomic spectra


Input:
Eigenfunctions for a rigid dumbbell rotating about its center have a $\phi$ dependence of the form $\psi(\phi)=A e^{i m \phi}$, where $m$ is a quantum number and $A$ is a constant. Which of the following values of $A$ will properly normalize the eigenfunction?

A. $\sqrt{2 \pi}$

B. $2 \pi$

C. $(2 \pi)^{2}$

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

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


Input:
A negative test charge is moving near a long straight wire in which there is a current. A force will act on the test charge in a direction parallel to the direction of the current if the motion of the charge is in a direction

A. toward the wire

B. away from the wire

C. the same as that of the current

D. opposite to that of the current

E. perpendicular to both the direction of the current and the direction toward the wire


Input:
The configuration of the potassium atom in its ground state is $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{1}$. Which of the following statements about potassium is true?

A. Its $n=3$ shell is completely filled.

B. Its $4 s$ subshell is completely filled.

C. Its least tightly bound electron has $\ell=4$.

D. Its atomic number is 17.

E. Its electron charge distribution is spherically symmetrical.


Input:
When the photoelectric equation is satisfied and applicable to this situation, $V$ is the

A. negative value at which the current stops

B. negative value at which the current starts

C. positive value at which the current stops

D. positive value at which the current starts

E. voltage induced when the light is on


Input:
The photoelectric equation is derived under the assumption that

A. electrons are restricted to orbits of angular momentum $n \hbar$, where $n$ is an integer

B. electrons are associated with waves of wavelength $\lambda=h / p$, where $p$ is momentum

C. light is emitted only when electrons jump between orbits

D. light is absorbed in quanta of energy $E=h \nu$

E. light behaves like a wave


Input:
The quantity $W$ in the photoelectric equation is the

A. energy difference between the two lowest electron orbits in the atoms of the photocathode

B. total light energy absorbed by the photocathode during the measurement

C. minimum energy a photon must have in order to be absorbed by the photocathode

D. minimum energy required to free an electron from its binding to the cathode material

E. average energy of all electrons in the photocathode


Input:
The potential energy of a body constrained to move on a straight line is $k x^{4}$ where $k$ is a constant. The position of the body is $x$, its speed $v$, its linear momentum $p$, and its mass $m$.

The force on the body is
A. $\frac{1}{2} m v^{2}$
B. $-4 k x^{3}$
C. $k x^{4}$
D. $-\frac{k x^{5}}{5}$
E. $m g$


Input:
The potential energy of a body constrained to move on a straight line is $k x^{4}$ where $k$ is a constant. The position of the body is $x$, its speed $v$, its linear momentum $p$, and its mass $m$.

The Hamiltonian function for this system is
A. $\frac{p^{2}}{2 m}+k x^{4}$
B. $\frac{p^{2}}{2 m}-k x^{4}$
C. $k x^{4}$
D. $\frac{1}{2} m v^{2}-k x^{4}$
E. $\frac{1}{2} m v^{2}$


Input:
The potential energy of a body constrained to move on a straight line is $k x^{4}$ where $k$ is a constant. The position of the body is $x$, its speed $v$, its linear momentum $p$, and its mass $m$.

The body moves from $x_{1}$ at time $t_{1}$ to $x_{2}$ at time $t_{2}$. Which of the following quantities is an extremum for the $x-t$ curve corresponding to this motion, if end points are fixed?

A. $\int_{t_{1}}^{t_{2}}\left(\frac{1}{2} m v^{2}-k x^{4}\right) d t$

B. $\int_{t_{1}}^{t_{2}}\left(\frac{1}{2} m v^{2}\right) d t$

C. $\int_{t_{1}}^{t_{2}}(m x v) d t$

D. $\int_{x_{1}}^{x_{2}}\left(\frac{1}{2} m v^{2}+k x^{4}\right) d x$

E. $\int_{x_{1}}^{x_{2}}(m v) d x$


Input:
The figure above represents a point mass $m$ attached to the ceiling by a cord of fixed length $\ell$. If the point mass moves in a horizontal circle of radius $r$ with uniform angular velocity $\omega$, the tension in the cord is

A. $m g\left(\frac{r}{\ell}\right)$

B. $m g \cos \left(\frac{\theta}{2}\right)$

C. $\frac{m \omega r}{\sin \left(\frac{\theta}{2}\right)}$

D. $m\left(\omega^{2} r^{2}+g^{2}\right)^{\frac{1}{2}}$

E. $m\left(\omega^{4} r^{2}+g^{2}\right)^{\frac{1}{2}}$


Input:
If logical 0 is 0 volts and logical 1 is +1 volt, the circuit shown above is a logic circuit commonly known as

A. an OR gate

B. an AND gate

C. a 2-bit adder

D. a flip-flop

E. a fanout


Input:
The gain of an amplifier is plotted versus angular frequency $\omega$ in the diagram above. If $K$ and $a$ are positive constants, the frequency dependence of the gain near $\omega=3 \times 10^{5}$ second $^{-1}$ is most accurately expressed by
A. $K e^{-a \omega}$
B. $K \omega^{2}$
C. $K \omega$
D. $K \omega^{-1}$
E. $K \omega^{-2}$


Input:
An experimenter measures 9934 counts during one hour from a radioactive sample. From this number the counting rate of the sample can be estimated with a standard deviation of most nearly
A. 100
B. 200
C. 300
D. 400
E. 500


Input:
Which of the following nuclei has the largest binding energy per nucleon? (Consider the most abundant isotope of each element.)

A. Helium

B. Carbon

C. Iron

D. Uranium

E. Plutonium


Input:
A proton beam is incident on a scatterer 0.1 centimeter thick. The scatterer contains $10^{20}$ target nuclei per cubic centimeter. In passing through the scatterer, one proton per incident million is scattered. The scattering cross section is

A. $10^{-29} \mathrm{~cm}^{2}$

B. $10^{-27} \mathrm{~cm}^{2}$

C. $10^{-25} \mathrm{~cm}^{2}$

D. $10^{-23} \mathrm{~cm}^{2}$

E. $10^{-21} \mathrm{~cm}^{2}$


Input:
Three masses are connected by two springs as shown above. A longitudinal normal mode with frequency $\frac{1}{2 \pi} \sqrt{\frac{k}{m}}$ is exhibited by

A. $A, B, C$ all moving in the same direction with equal amplitude

B. $A$ and $C$ moving in opposite directions with equal amplitude, and $B$ at rest

C. $A$ and $C$ moving in the same direction with equal amplitude, and $B$ moving in the opposite direction with the same amplitude

D. $A$ and $C$ moving in the same direction with equal amplitude, and $B$ moving in the opposite direction with twice the amplitude

E. none of the above


Input:
A uniform stick of length $L$ and mass $M$ lies on a frictionless horizontal surface. A point particle of mass $\boldsymbol{m}$ approaches the stick with speed $v$ on a straight line perpendicular to the stick that intersects the stick at one end, as shown above. After the collision, which is elastic, the particle is at rest. The speed $V$ of the center of mass of the stick after the collision is
A. $\frac{m}{M} v$
B. $\frac{m}{M+m} v$
C. $\sqrt{\frac{m}{M}} v$
D.<smiles>[Y14]1=CCC2CCCC1C2</smiles>
E. $\frac{3 m}{\bar{M}} v$


Input:
Photons of wavelength $\lambda$ scatter elastically on free protons initially at rest. The wavelength of the photons scattered at $90^{\circ}$ is increased by

A. $\lambda / 137$

B. $\lambda / 1836$

C. $h / m_{e} c$, where $h$ is Planck's constant, $m_{e}$ the rest mass of an electron, and $c$ the speed of light

D. $h / m_{p} c$, where $h$ is Planck's constant, $m_{p}$ the rest mass of a proton, and $c$ the speed of light

E. zero


Input:
A blackbody at temperature $T_{1}$ radiates energy at a power level of 10 milliwatts $(\mathrm{mW})$. The same blackbody, when at a temperature $2 T_{1}$, radiates energy at a power level of
A. $10 \mathrm{~mW}$
D. $80 \mathrm{~mW}$
B. $20 \mathrm{~mW}$
C. $40 \mathrm{~mW}$
E. $160 \mathrm{~mW}$


Input:
The Franck-Hertz experiment and related scattering experiments show that

A. electrons are always scattered elastically from atoms

B. electrons are never scattered elastically from atoms

C. electrons of a certain energy range can be scattered inelastically, and the energy lost by electrons is discrete

D. electrons always lose the same energy when they are scattered inelastically

E. there is no energy range in which the energy lost by electrons varies continuously


Input:
A transition in which one photon is radiated by the electron in a hydrogen atom when the electron's wave function changes from $\psi_{1}$ to $\psi_{2}$ is forbidden if $\psi_{1}$ and $\psi_{2}$

A. have opposite parity

B. are orthogonal to each other

C. are zero at the center of the atomic nucleus

D. are both spherically symmetrical

E. are associated with different angular momenta


Input:
The Hamiltonian operator in the Schrödinger equation can be formed from the classical Hamiltonian by substituting

A. wavelength and frequency for momentum and energy

B. a differential operator for momentum

C. transition probability for potential energy

D. sums over discrete eigenvalues for integrals over continuous variables

E. Gaussian distributions of observables for exact values


Input:
The Hall effect is used in solid-state physics to measure

A. ratio of charge to mass

B. magnetic susceptibility

C. the sign of the charge carriers

D. the width of the gap between the conduction and valence bands

E. Fermi energy


Input:
One feature common to both the Debye theory and the Einstein theory of the specific heat of a crystal composed of $N$ identical atoms is that the

A. average energy of each atom is $3 k T$

B. vibrational energy of the crystal is equivalent to the energy of $3 \mathrm{~N}$ independent harmonic oscillators

C. crystal is assumed to be continuous for all elastic waves

D. speed of the longitudinal elastic waves is less than the speed of the transverse elastic waves

E. upper cutoff frequency of the elastic waves is the same


Input:
A cube has a constant electric potential $V$ on its surface. If there are no charges inside the cube, the potential at the center of the cube is
A. zero
D. $V / 2$
B. $V / 8$
C. $V / 6$
E. $V$


Input:
A charged particle oscillates harmonically along the $x$-axis as shown above. The radiation from the particle is detected at a distant point $P$, which lies in the $x y$-plane. The electric field at $P$ is in the

A. $\pm z$ direction and has a maximum amplitude at $\theta=90^{\circ}$

B. $\pm z$ direction and has a minimum amplitude at $\theta=90^{\circ}$

C. $x y$-plane and has a maximum amplitude at $\theta=90^{\circ}$

D. $x y$-plane and has a minimum amplitude at $\theta=90^{\circ}$

E. $x y$-plane and has a maximum amplitude at $\theta=45^{\circ}$

Dielectric $K$


Input:
A dielectric of dielectric constant $K$ is placed in contact with a conductor having surface charge density $\sigma$, as shown above. What is the polarization (bound) charge density $\sigma_{p}$ on the surface of the dielectric at the interface between the two materials?
A. $\sigma \frac{K}{1-K}$
B. $\sigma \frac{K}{1+K}$
C. $\sigma K$
D. $\sigma \frac{1+K}{K}$
E. $\sigma \frac{1-K}{K}$


Input:
The mean kinetic energy of electrons in metals at room temperature is usually many times the thermal energy $k T$. Which of the following can best be used to explain this fact?

A. The energy-time uncertainty relation

B. The Pauli exclusion principle

C. The degeneracy of the energy levels

D. The Born approximation

E. The wave-particle duality


Input:
If $\psi$ is a normalized solution of the Schrödinger equation and $Q$ is the operator corresponding to a physical observable $x$, the quantity $\psi * Q \psi$ may be integrated in order to obtain the

A. normalization constant for $\psi$

B. spatial overlap of $Q$ with $\psi$

C. mean value of $x$

D. uncertainty in $x$

E. time derivative of $x$


Input:
Which of the following is an eigenfunction of the linear momentum operator $-i \hbar \frac{\partial}{\partial x}$ with a positive eigenvalue $\hbar k$; i.e., an eigenfunction that describes a particle that is moving in free space in the direction of positive $x$ with a precise value of linear momentum?
A. $\cos k x$
B. $\sin k x$
C. $\mathrm{e}^{-i k x}$
D. $\mathrm{e}^{i k x}$
E. $\mathrm{e}^{-k x}$


Input:
In an ordinary hologram, coherent monochromatic light produces a 3 -dimensional picture because wave information is recorded for which of the following?

1. Amplitude

II. Phase

III. Wave-front angular frequency

A. I only

B. I and II only

C. 1 and III only

D. II and III only

E. I, II, and III


Input:
The dispersion law for a certain type of wave motion is $\omega=\left(c^{2} k^{2}+m^{2}\right)^{\frac{1}{2}}$, where $\omega$ is the angular frequency, $k$ is the magnitude of the propagation vector, and $c$ and $m$ are constants. The group velocity of these waves approaches

A. infinity as $k \rightarrow 0$ and zero as $k \rightarrow \infty$

B. infinity as $k \rightarrow 0$ and $c$ as $k \rightarrow \infty$

C. $c$ as $k \rightarrow 0$ and zero as $k \rightarrow \infty$

D. zero as $k \rightarrow 0$ and infinity as $k \rightarrow \infty$

E. zero as $k \rightarrow 0$ and $c$ as $k \rightarrow \infty$


Input:
A particle of mass $m$ that moves along the $x$-axis has potential energy $V(x)=a+b x^{2}$, where $a$ and $b$ are positive constants. Its initial velocity is $v_{0}$ at $x=0$. It will execute simple harmonic motion with a frequency determined by the value of

A. $b$ alone

B. $b$ and $a$ alone

C. $b$ and $m$ alone

D. $b, a$, and $m$ alone

E. $b, a, m$, and $v_{0}$


Input:
The equation of motion of a rocket in free space can be written

$$
m rac{d v}{d t}+u rac{d m}{d t}=0
$$

where $m$ is the rocket's mass, $v$ is its velocity, $t$ is time, and $u$ is a constant.

The constant $u$ represents the speed of the

A. rocket at $t=0$

B. rocket after its fuel is spent

C. rocket in its instantaneous rest frame

D. rocket's exhaust in a stationary frame

E. rocket's exhaust relative to the rocket


Input:
The equation of motion of a rocket in free space can be written

$$
m rac{d v}{d t}+u rac{d m}{d t}=0
$$

where $m$ is the rocket's mass, $v$ is its velocity, $t$ is time, and $u$ is a constant.

The equation can be solved to give $v$ as a function of $m$. If the rocket has $m=m_{0}$ and $v=0$ when it starts, what is the solution?

A. $u m_{0} / m$

B. $u \exp \left(m_{0} / m\right)$

C. $u \sin \left(m_{0} / m\right)$

D. $u \tan \left(m_{0} / m\right)$

E. None of the above.


Input:
A point charge $-q$ coulombs is placed at a distance $d$ from a large grounded conducting plane. The surface charge density on the plane a distance $D$ from the point charge is

A. $\frac{q}{4 \pi D}$

B. $\frac{q D^{2}}{2 \pi}$

C. $\frac{q d}{2 \pi D^{2}}$

D. $\frac{g d}{2 \pi D^{3}}$

E. $\frac{q d}{4 \pi \epsilon_{0} D^{2}}$


Input:
An alternating current electrical generator has a fixed internal impedance $\boldsymbol{R}_{\boldsymbol{g}}+j X_{g}$ and is used to supply power to a passive load that has an impedance $\boldsymbol{R}_{g}+j X_{\ell}$, where $j=\sqrt{-1}$, $R_{g} \neq 0$, and $X_{g} \neq 0$. For maximum power transfer between the generator and the load, $X_{\ell}$ should be equal to
A. 0
B. $X_{g}$
C. $-X_{g}$
D. $\boldsymbol{R}_{\boldsymbol{g}}$
E. $-\boldsymbol{R}_{\boldsymbol{g}}$


Input:
A current $i$ in a circular loop of radius $b$ produces a magnetic field. At a fixed point far from the loop, the strength of the magnetic field is proportional to which of the following combinations of $i$ and $b$ ?
A. $i b$
B. $i b^{2}$
C. $i^{2} b$
D. $*-\frac{i}{b}$
E. $\frac{i}{b^{2}}$


Input:
For a system in which the number of particles is fixed, the reciprocal of the Kelvin temperature $T$ is given by which of the following derivatives? (Let $P=$ pressure, $V=$ volume, $S=$ entropy, and $U=$ internal energy.)

A. $\left(\frac{\partial P}{\partial V}\right)_{S}$

B. $\left(\frac{\partial P}{\partial S}\right)_{V}$

C. $\left(\frac{\partial S}{\partial P}\right)_{U}$

D. $\left(\frac{\partial V}{\partial P}\right)_{U}$

E. $\left(\frac{\partial S}{\partial U}\right)_{V}$


Input:
A large isolated system of $N$ weakly interacting particles is in thermal equilibrium. Each particle has only 3 possible nondegenerate states of energies $0, \epsilon$, and $3 \epsilon$. When the system is at an absolute temperature $T>\epsilon / k$, where $k$ is Boltzmann's constant, the average energy of each particle is
A. 0
B. $\epsilon$
C. $\frac{4}{3} \epsilon$
D. $2 \epsilon$
E. $3 \epsilon$


Input:
If a newly discovered particle $X$ moves with a speed equal to the speed of light in vacuum, then which of the following must be true?

A. The rest mass of $X$ is zero.

B. The spin of $X$ equals the spin of a photon.

C. The charge of $X$ is carried on its surface.

D. $X$ does not spin.

E. $X$ cannot be detected.


Input:
A car of rest length 5 meters passes through a garage of rest length 4 meters. Due to the relativistic Lorentz contraction, the car is only 3 meters long in the garage's rest frame. There are doors on both ends of the garage, which open automatically when the front of the car reaches them and close automatically when the rear passes them. The opening or closing of each door requires a negligible amount of time.

The velocity of the car in the garage's rest frame is

A. $0.4 c$

B. $0.6 c$

C. $0.8 c$

D. greater than $c$

E. not determinable from the data given


Input:
A car of rest length 5 meters passes through a garage of rest length 4 meters. Due to the relativistic Lorentz contraction, the car is only 3 meters long in the garage's rest frame. There are doors on both ends of the garage, which open automatically when the front of the car reaches them and close automatically when the rear passes them. The opening or closing of each door requires a negligible amount of time.

The length of the garage in the car's rest frame is

A. $2.4 \mathrm{~m}$

B. $4.0 \mathrm{~m}$

C. $5.0 \mathrm{~m}$

D. $8.3 \mathrm{~m}$

E. not determinable from the data given


Input:
A car of rest length 5 meters passes through a garage of rest length 4 meters. Due to the relativistic Lorentz contraction, the car is only 3 meters long in the garage's rest frame. There are doors on both ends of the garage, which open automatically when the front of the car reaches them and close automatically when the rear passes them. The opening or closing of each door requires a negligible amount of time.

Which of the following statements is the best response to the question:

"Was the car ever inside a closed garage?"

A. No, because the car is longer than the garage in all reference frames.

B. No, because the Lorentz contraction is not a "real" effect.

C. Yes, because the car is shorter than the garage in all reference frames.

D. Yes, because the answer to the question in the garage's rest frame must apply in all reference frames.

E. There is no unique answer to the question, as the order of door openings and closings depends on the reference frame.


Input:
The measured index of refraction of $x$-rays in rock salt is less than one. This is consistent with the theory of relativity because

A. relativity deals with light waves traveling in a vacuum only

B. $\mathrm{x}$-rays cannot transmit signals

C. $\mathrm{x}$-ray photons have imaginary mass

D. the theory of relativity predates the development of solid-state physics

E. the phase velocity and group velocity are different


Input:
It is necessary to coat a glass lens with a nonreflecting layer. If the wavelength of the light in the coating is $\lambda$, the best choice is a layer of material having an index of refraction between those of glass and air and a thickness of
A. $\frac{\lambda}{4}$
B. $\frac{\lambda}{2}$
C. $\frac{\lambda}{\sqrt{2}}$
D. $\lambda$
E. $1.5 \lambda$


Input:
Unpolarized light is incident on two ideal polarizers in series. The polarizers are oriented so that no light emerges through the second polarizer. A third polarizer is now inserted between the first two and its orientation direction is continuously rotated through $180^{\circ}$. The maximum fraction of the incident power transmitted through all three polarizers is
A. zero
B. $\frac{1}{8}$
C. $\frac{1}{2}$
D. $\frac{1}{\sqrt{2}}$
E. 1


Input:
The period of a hypothetical Earth satellite orbiting at sea level would be 80 minutes. In terms of the Earth's radius $\boldsymbol{R}_{e}$, the radius of a synchronous satellite orbit (period 24 hours) is most nearly

A. $3 R$

B.

C. $18 R$

D. $320 R_{e}$

E. $5800 R_{e}$


Input:
A hoop of mass $M$ and radius $R$ is at rest at the top of an inclined plane as shown above. The hoop rolls down the plane without slipping. When the hoop reaches the bottom, its angular momentum around its center of mass is

A. $M R \sqrt{g h}$

B. $\frac{1}{2} M R \sqrt{g h}$

C. $M \sqrt{2 g h}$

D. $M g h$

E. $\frac{1}{2} M g h$


Input:
A particle is constrained to move along the $x$-axis under the influence of the net force $\mathbf{F}=-k \mathbf{x}$ with amplitude $A$ and frequency $f$, where $k$ is a positive constant. When $x=A / 2$, the particle's speed is
A. $2 \pi f A$
B. $\sqrt{3} \pi f A$
C. $\sqrt{2} \pi f A$
D. $\pi f A$
E. $\frac{1}{3} \pi f A$


Input:
A system consists of two charged particles of equal mass. Initially the particles are far apart, have zero potential energy, and one particle has nonzero speed. If radiation is neglected, which of the following is true of the total energy of the system?

A. It is zero and remains zero.

B. It is negative and constant.

C. It is positive and constant.

D. It is constant, but the sign cannot be determined unless the initial velocities of both particles are known.

E. It cannot be a constant of the motion because the particles exert force on each other.


Input:
One of Maxwell's equations is $\nabla \cdot \mathbf{B}=\mathbf{0}$. Which of the following sketches shows magnetic field lines that clearly violate this equation within the region bounded by the dashed lines?

A.


Input:
Which of the following electric fields could exist in a finite region of space that contains no charges? (In these expressions, $A$ is a constant, and $\mathbf{i}, \mathbf{j}$, and $\mathbf{k}$ are unit vectors pointing in the $x, y$, and $z$ directions, respectively.)

A. $A(2 x y i-x z k)$

B. $A(-x y \mathbf{j}+x z \mathbf{k})$

C. $A(x z \mathbf{i}+x z \mathbf{j})$

D. $\operatorname{Axyz}(\mathbf{i}+\mathbf{j})$

E. $A x y z 1$


Input:
A small circular wire loop of radius $a$ is located at the center of much larger circular wire loop of radius $b$ as shown above. The larger loop carries an alternating current $I=I_{0} \cos \omega t$, where $I_{0}$ and $\omega$ are constants. The magnetic field generated by the current in the large loop induces in the small loop an emf that is approximately equal to which of the following? (Either use mks units and let $\mu_{0}$ be the permeability of free space, or use Gaussian units and let $\mu_{0}$ be $4 \pi / c^{2}$.)

A. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a^{2}}{b} \omega \cos \omega t$

B. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a^{2}}{b} \omega \sin \omega t$

C. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a}{b^{2}} \omega \sin \omega t$

D. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a}{b^{2}} \cos \omega t$

E. $\left(\frac{\pi \mu_{0} I_{0}}{2}\right) \frac{a}{b} \sin \omega t$


Input:
The emission spectrum of an atomic gas in a magnetic field differs from that of the gas in the absence of a magnetic field. Which of the following is true of the phenomenon?

A. It is called the Stern-Gerlach effect.

B. It is called the Stark effect.

C. It is due primarily to the nuclear magnetic moment of the atoms.

D. The number of emission lines observed for the gas in a magnetic field is always twice the number observed in the absence of a magnetic field.

E. The number of emission lines observed for the gas in a magnetic field is either greater than or equal to the number observed in the absence of a magnetic field.


Input:
A spectral line is produced by a gas that is sufficiently dense that the mean time between atomic collisions is much shorter than the mean lives of the atomic states responsible for the line. Compared with the same line produced by a low-density gas, the line produced by the higher-density gas will appear

A. the same

B. more highly polarized

C. broader

D. shifted toward the blue end of the spectrum

E. split into a doublet


Input:
Sodium has eleven electrons and the sequence in which energy levels fill in atoms is $1 s, 2 s, 2 p$, $3 s, 3 p, 4 s, 3 d$, etc. What is the ground state of sodium in the usual notation ${ }^{2 S+1} L_{J}$ ?
A. ${ }^{1} S_{0}$
B. ${ }^{2} S_{\frac{1}{2}}$
C. ${ }^{1} P_{0}$
D. ${ }^{2} P_{\frac{1}{2}}$
E. ${ }^{3} P_{\frac{1}{2}}$


Input:
The figure above shows the photon interaction cross sections for lead in the energy range where the Compton, photoelectric, and pair production processes all play a role. What is the correct identification of these cross sections?

A. 1 = photoelectric, 2 = Compton, 3 = pair production

B. $1=$ photoelectric, $2=$ pair production, $3=$ Compton

C. 1 = Compton, $2=$ pair production, $3=$ photoelectric

D. 1 = Compton, $2=$ photoelectric, $3=$ pair production

E. 1 = pair production, $2=$ photoelectric, $3=$ Compton


Input:
The exponent in Coulomb's inverse square law has been found to differ from two by less than one part in a billion by measuring which of the following?

A. The charge on an oil drop in the Millikan experiment

B. The deflection of an electron beam in an electric field

C. The neutrality of charge of an atom

D. The electric force between two charged objects

E. The electric field inside a charged conducting shell


Input:
In a gas of $N$ diatomic molecules, two possible models for a classical description of a diatomic molecule are: [FIGURE]. Which of the following statements about this gas is true?

A. Model I has a specific heat $c_{v}=rac{3}{2} N k$.

B. Model II has a smaller specific heat than Model I.

C. Model I is always correct.

D. Model II is always correct.

E. The choice between Models I and II dependes on the temperature.


Input:
Consider a system of $N$ noninteracting particles confined in a volume $V$ at a temperature such that the particles obey classical Boltzmann statistics. If the temperature is lowered to the point at which quantum effects become important, the pressure of the gas may differ depending on whether the particles are fermions or bosons. Let $P_{F}$ be the pressure exerted by the particles if they are fermions, $P_{B}$ be the pressure if they are bosons, and $P_{C}$ be the pressure the particles would exert if quantum effects are ignored. Which of the following is true?

A. $P_{F}=P_{B}=P_{C}$

B. $P_{F}>P_{C}>P_{B}$

C. $P_{F}>P_{B}>P_{C}$

D. $P_{F}<P_{B}<P_{C}$

E. $P_{F}<P_{C}<P_{B}$


Input:
A system containing two identical particles is described by a wave function of the form

$\psi=\frac{1}{\sqrt{2}}\left[\psi_{\alpha}\left(x_{1}\right) \psi_{\beta}\left(x_{2}\right)+\psi_{\beta}\left(x_{1}\right) \psi_{\alpha}\left(x_{2}\right)\right]$

where $x_{1}$ and $x_{2}$ represent the spatial coordinates of the particles and $\alpha$ and $\beta$ represent all the quantum numbers, including spin, of the states that they occupy. The particles might be

A. electrons

B. positrons

C. protons

D. neutrons

E. deuterons


Input:
The figure above shows one of the possible energy eigenfunctions $\psi(x)$ for a particle bouncing freely back and forth along the $x$-axis between impenetrable walls located at $x=-a$ and $x=+a$. The potential energy equals zero for $|x|<a$. If the energy of the particle is 2 electron volts when it is in the quantum state associated with this eigenfunction, what is its energy when it is in the quantum state of lowest possible energy?
A. $0 \mathrm{eV}$
B. $\frac{1}{\sqrt{2}} \mathrm{eV}$
C. $\frac{1}{2} \mathrm{eV}$
D. $1 \mathrm{eV}$
E. $2 \mathrm{eV}$


Input:
When a narrow beam of monoenergetic electrons impinges on the surface of a single metal crystal at an angle of 30 degrees with the plane of the surface, first-order reflection is observed. If the spacing of the reflecting crystal planes is known from $x$-ray measurements to be 3 angstroms, the speed of the electrons is most nearly

A. $1.4 \times 10^{-4} \mathrm{~m} / \mathrm{s}$

B. $2.4 \mathrm{~m} / \mathrm{s}$

C. $5.0 \times 10^{3} \mathrm{~m} / \mathrm{s}$

D. $2.4 \times 10^{6} \mathrm{~m} / \mathrm{s}$

E. $4.5 \times 10^{9} \mathrm{~m} / \mathrm{s}$


Input:
Which of the following is NOT compatible with the selection rule that controls electric dipole emission of photons by excited states of atoms?

A. $\Delta n$ may have any negative integral value.

B. $\Delta l= \pm 1$

C. $\Delta m_{\ell}=0, \pm 1$

D. $\Delta s= \pm 1$

E. $\Delta j= \pm 1$


Input:
An electric sander has a continuous belt that rubs against a wood surface as shown schematically above. The sander is 100 percent efficient and draws a current of 9 amperes from a 120 -volt line. The belt speed is 10 meters per second. If the sander is pushing against the wood with a normal force of 100 newtons, the coefficient of friction is most nearly
A. 0.02
B. 0.2
C. 0.4
D. 1.1
E. 10


Input:
In the circuit shown above, $R_{2}=3 R_{1}$ and the battery of emf $\varepsilon$ has negligible internal resistance. The resistance of the diode when it allows current to pass through it is also negligible. At time $t=0$, the switch $S$ is closed and the currents and voltages are allowed to reach their asymptotic values. Then at time $t_{1}$, the switch is opened. Which of the following curves most nearly represents the potential at point $A$ as a function of time $t$ ?

A.


Input:
In the cycle shown above, $K L$ and $N M$ represent isotherms, while $K N$ and $L M$ represent reversible adiabats. A system is carried through the Carnot cycle $K L M N$, taking in heat $Q_{2}$ from the hot reservoir $T_{2}$ and releasing heat $Q_{1}$ to the cold reservoir $T_{1}$. All of the following statements are true EXCEPT:

A. $Q_{1} / T_{1}=Q_{2} / T_{2}$

B. The entropy of the hot reservoir decreases.

C. The entropy of the system increases.

D. The work $W$ done is equal to the net heat absorbed, $Q_{2}-Q_{1}$.

E. The efficiency of the cycle is independent of the working substance.


Input:
A particle of mass $M$ is in an infinitely deep square well potential $V$ where

$$
\begin{array}{ll}
V=0 & \text { for }-a \leqq x \leqq a, \text { and } \\
V=\infty & \text { for } x<-a, a<x .
\end{array}
$$

A very small perturbing potential $V^{\prime}$ is superimposed on $V$ such that

$$
\begin{array}{ll}
V^{\prime}=\epsilon\left(\frac{a}{2}-|x|\right) & \text { for } \frac{-a}{2} \leqq x \leqq \frac{a}{2}, \text { and } \\
V^{\prime}=0 & \text { for } x<\frac{-a}{2}, \frac{a}{2}<x .
\end{array}
$$

If $\psi_{0}, \psi_{1}, \psi_{2}, \psi_{3}, \ldots$ are the energy eigenfunctions for a particle in the infinitely deep square well potential, with $\psi_{0}$ being the ground state, which of the following statements is correct about the eigenfunction $\psi_{0}^{\prime}$ of a particle in the perturbed potential $V+V^{\prime}$ ?

A. $\psi_{0}^{\prime}=a_{00} \psi_{0}, a_{00} \neq 0$

B. $\psi_{0}^{\prime}=\sum_{n=0}^{\infty} a_{0 n} \psi_{n}$ with $a_{0 n}=0$ for all odd values of $n$

C. $\psi_{0}^{\prime}=\sum_{n=0}^{\infty} a_{0 n} \psi_{n}$ with $a_{0 n}=0$ for all even values of $n$

D. $\psi_{0}^{\prime}=\sum_{n=0}^{\infty} a_{0 n} \psi_{n}$ with $a_{0 n} \neq 0$ for all values of $n$

E. None of the above


Input:
Two uniform cylindrical disks of identical mass $M$, radius $R$, and moment of inertia $\frac{1}{2} M R^{2}$, as shown above, collide on a frictionless, horizontal surface. Disk 1 , having an initial counterclockwise angular velocity $\omega_{0}$ and a center-of-mass velocity $v_{0}=\frac{1}{2} \omega_{0} R$ to the right, makes a grazing collision with disk II initially at rest. If after the collision the two disks stick together, the magnitude of the total angular momentum about the point $P$ is

A. zero

B. $\frac{1}{2} M R^{2} \omega_{0}$

C. $\frac{1}{2} M R v_{0}$

D. $M R v_{0}$

E. dependent on the time of the collision


Input:
The long thin cylindrical glass rod shown above has length $\ell$ and is insulated from its surroundings. The rod has an excess charge $Q$ uniformly distributed along its length. Assume the electric potential to be zero at infinite distances from the rod. If $k$ is the constant in Coulomb's law, the electric potential at a point $P$ along the axis of the rod and a distance $\ell$ from one end is $\frac{k Q}{\ell}$
multiplied by

A. $\frac{4}{9}$

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

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

D. $\ln 2$

E. 1


Input:
The positronium "atom" consists of an electron and a positron bound together by their mutual Coulomb attraction and moving about their center of mass, which is located halfway between them. Thus the positronium "atom" is somewhat analogous to a hydrogen atom. The ground-state binding energy of hydrogen is 13.6 electron volts. What is the ground-state binding energy of positronium?

A. $\left(\frac{1}{2}\right)^{2} \times 13.6 \mathrm{eV}$

B. $\frac{1}{2} \times 13.6 \mathrm{eV}$

C. $13.6 \mathrm{eV}$

D. $2 \times 13.6 \mathrm{eV}$

E. $(2)^{2} \times 13.6 \mathrm{eV}$


Input:
The screen of a pinhole camera is at a distance $D$ from the pinhole, which has a diameter $d$. The light has an effective wavelength $\lambda .(\lambda<<D)$ For which of the following values of $d$ will the image be sharpest?
A. $\sqrt{\lambda D}$
B. $\lambda$
C. $\frac{\lambda}{10}$
D. $\frac{\lambda^{2}}{D}$
E. $\frac{D^{2}}{\lambda}$


Input:
Which of the following best illustrates the acceleration of a pendulum bob at points $a$ through $e$ ?

A.


Input:
The coefficient of static friction between a small coin and the surface of a turntable is 0.30 . The turntable rotates at 33.3 revolutions per minute. What is the maximum distance from the center of the turntable at which the coin will not slide?

A. $0.024 \mathrm{~m}$

B. $0.048 \mathrm{~m}$

C. $0.121 \mathrm{~m}$

D. $0.242 \mathrm{~m}$

E. $0.484 \mathrm{~m}$


Input:
A satellite of mass $m$ orbits a planet of mass $M$ in a circular orbit of radius $R$. The time required for one revolution is

A. independent of $M$

B. proportional to $\sqrt{m}$

C. linear in $R$

D. proportional to $R^{3 / 2}$

E. proportional to $R^{2}$


Input:
In a nonrelativistic, one-dimensional collision, a particle of mass $2 m$ collides with a particle of mass $m$ at rest. If the particles stick together after the collision, what fraction of the initial kinetic energy is lost in the collision?

A. 0

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

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

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

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

SCRATCHWORK


Input:
A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature $T$. The average total energy of the oscillator is

A. $\frac{1}{2} k T$

B. $k T$

C. $\frac{3}{2} k T$

D. $3 k T$

E. $6 k T$


Input:
An ideal monatomic gas expands quasi-statically to twice its volume. If the process is isothermal, the work done by the gas is $W_{i}$. If the process is adiabatic, the work done by the gas is $W_{a}$. Which of the following is true?

A. $W_{i}=W_{a}$

B. $0=W_{i}<W_{a}$

C. $0<W_{i}<W_{a}$

D. $0=W_{a}<W_{i}$

E. $0<W_{a}<W_{i}$


Input:
Two long, identical bar magnets are placed under a horizontal piece of paper, as shown in the figure above. The paper is covered with iron filings. When the two north poles are a small distance apart and touching the paper, the iron filings move into a pattern that shows the magnetic field lines. Which of the following best illustrates the pattern that results?

A.


Input:
A positive charge $Q$ is located at a distance $L$ above an infinite grounded conducting plane, as shown in the figure above. What is the total charge induced on the plane?

A. $2 Q$

B. $Q$

C. 0

D. $-Q$

E. $-2 Q$


Input:
Five positive charges of magnitude $q$ are arranged symmetrically around the circumference of a circle of radius $r$. What is the magnitude of the electric field at the center of the circle? $\left(k=1 / 4 \pi \epsilon_{0}\right)$

A. 0

B. $k q / r^{2}$

C. $5 k q / r^{2}$

D. $\left(k q / r^{2}\right) \cos (2 \pi / 5)$

E. $\left(5 k q / r^{2}\right) \cos (2 \pi / 5)$


Input:
A 3-microfarad capacitor is connected in series with a 6 -microfarad capacitor. When a 300 -volt potential difference is applied across this combination, the total energy stored in the two capacitors is

A. $0.09 \mathrm{~J}$

B. $0.18 \mathrm{~J}$

C. $0.27 \mathrm{~J}$

D. $0.41 \mathrm{~J}$

E. $0.81 \mathrm{~J}$


Input:
An object is located 40 centimeters from the first of two thin converging lenses of focal lengths 20 centimeters and 10 centimeters, respectively, as shown in the figure above. The lenses are separated by 30 centimeters. The final image formed by the two-lens system is located

A. $5.0 \mathrm{~cm}$ to the right of the second lens

B. $13.3 \mathrm{~cm}$ to the right of the second lens

C. infinitely far to the right of the second lens

D. $13.3 \mathrm{~cm}$ to the left of the second lens

E. $100 \mathrm{~cm}$ to the left of the second lens


Input:
A spherical, concave mirror is shown in the figure above. The focal point $F$ and the location of the object $O$ are indicated. At what point will the image be located?

A. I

B. II

C. III

D. IV

E. $\mathrm{V}$

SCRATCHWORK


Input:
Two stars are separated by an angle of $3 \times 10^{-5}$ radians. What is the diameter of the smallest telescope that can resolve the two stars using visible light ( $\lambda \cong 600$ nanometers) ? (Ignore any effects due to Earth's atmosphere.)

A. $1 \mathrm{~mm}$

B. $2.5 \mathrm{~cm}$

C. $10 \mathrm{~cm}$

D. $2.5 \mathrm{~m}$

E. $10 \mathrm{~m}$


Input:
An 8-centimeter-diameter by 8 -centimeter-long $\mathrm{NaI}(\mathrm{Tl})$ detector detects gamma rays of a specific energy from a point source of radioactivity. When the source is placed just next to the detector at the center of the circular face, 50 percent of all emitted gamma rays at that energy are detected. If the detector is moved to 1 meter away, the fraction of detected gamma rays drops to

A. $10^{-4}$

B. $2 \times 10^{-4}$

C. $4 \times 10^{-4}$

D. $8 \pi \times 10^{-4}$

E. $16 \pi \times 10^{-4}$


Input:
Five classes of students measure the height of a building. Each class uses a different method and each measures the height many different times. The data for each class are plotted below. Which class made the most precise measurement?

A.


Input:
A student makes 10 one-second measurements of the disintegration of a sample of a long-lived radioactive isotope and obtains the following values.

$$
3,0,2,1,2,4,0,1,2,5
$$

How long should the student count to establish the rate to an uncertainty of 1 percent?

A. $80 \mathrm{~s}$

B. $160 \mathrm{~s}$

C. $2,000 \mathrm{~s}$

D. $5,000 \mathrm{~s}$

E. $6,400 \mathrm{~s}$


Input:
The ground state electron configuration for phosphorus, which has 15 electrons, is

A. $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{1} 3 p^{4}$

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

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

D. $1 s^{2} 2 s^{2} 2 p^{6} 3 s^{1} 3 d^{4}$

E. $1 s^{2} 2 s^{2} 2 p^{6} 3 p^{2} 3 d^{3}$


Input:
The energy required to remove both electrons from the helium atom in its ground state is $79.0 \mathrm{eV}$. How much energy is required to ionize helium (i.e., to remove one electron)?

A. $24.6 \mathrm{eV}$

B. $39.5 \mathrm{eV}$

C. $51.8 \mathrm{eV}$

D. $54.4 \mathrm{eV}$

E. $65.4 \mathrm{eV}$


Input:
The primary source of the Sun's energy is a series of thermonuclear reactions in which the energy produced is $c^{2}$ times the mass difference between

A. two hydrogen atoms and one helium atom

B. four hydrogen atoms and one helium atom

C. six hydrogen atoms and two helium atoms

D. three helium atoms and one carbon atom

E. two hydrogen atoms plus two helium atoms and one carbon atom


Input:
In the production of $X$ rays, the term "bremsstrahlung" refers to which of the following?

A. The cut-off wavelength, $\lambda_{\min }$, of the $X$-ray tube

B. The discrete $\mathrm{X}$-ray lines emitted when an electron in an outer orbit fills a vacancy in an inner orbit of the atoms in the target metal of the X-ray tube

C. The discrete X-ray lines absorbed when an electron in an inner orbit fills a vacancy in an outer orbit of the atoms in the target metal of the X-ray tube

D. The smooth, continuous X-ray spectra produced by high-energy blackbody radiation from the $\mathrm{X}$-ray tube

E. The smooth, continuous X-ray spectra produced by rapidly decelerating electrons in the target metal of the X-ray tube


Input:
In the hydrogen spectrum, the ratio of the wavelengths for Lyman- $\alpha$ radiation ( $n=2$ to $n=1)$ to Balmer- $\alpha$ radiation $(n=3$ to $n=2)$ is

A. $5 / 48$

B. $5 / 27$

C. $1 / 3$

D. 3

E. $27 / 5$

SCRATCHWORK


Input:
An astronomer observes a very small moon orbiting a planet and measures the moon's minimum and maximum distances from the planet's center and the moon's maximum orbital speed. Which of the following CANNOT be calculated from these measurements?

A. Mass of the moon

B. Mass of the planet

C. Minimum speed of the moon

D. Period of the orbit

E. Semimajor axis of the orbit


Input:
A particle is constrained to move in a circle with a 10 -meter radius. At one instant, the particle's speed is 10 meters per second and is increasing at a rate of 10 meters per second squared. The angle between the particle's velocity and acceleration vectors is

A. $0^{\circ}$

B. $30^{\circ}$

C. $45^{\circ}$

D. $60^{\circ}$

E. $90^{\circ}$

SCRATCHWORK


Input:
A stone is thrown at an angle of $45^{\circ}$ above the horizontal $x$-axis in the $+x$-direction. If air resistance is ignored, which of the velocity versus time graphs shown above best represents $v_{x}$ versus $t$ and $v_{y}$ versus $t$, respectively?

\begin{tabular}{|c|c|c|}
\hline & $v_{x}$ vs. $t$ & $v_{y} \mathrm{vs}$

A. & I & IV

B. & II &

C. & II & III

D. & II & V

E. & IV & V \\
\hline
\end{tabular}


Input:
Seven pennies are arranged in a hexagonal, planar pattern so as to touch each neighbor, as shown in the figure above. Each penny is a uniform disk of mass $m$ and radius $r$. What is the moment of inertia of the system of seven pennies about an axis that passes through the center of the central penny and is normal to the plane of the pennies?

A. $(7 / 2) m r^{2}$

B. $(13 / 2) m r^{2}$

C. $(29 / 2) m r^{2}$

D. $(49 / 2) m r^{2}$

E. $(55 / 2) m r^{2}$


Input:
A thin uniform rod of mass $M$ and length $L$ is positioned vertically above an anchored frictionless pivot point, as shown above, and then allowed to fall to the ground. With what speed does the free end of the rod strike the ground?

A. $\sqrt{\frac{1}{3} g L}$

B. $\sqrt{g L}$

C. $\sqrt{3 g L}$

D. $\sqrt{12 g L}$

E. $12 \sqrt{g L}$


Input:
The eigenvalues of a Hermitian operator are always

A. real

B. imaginary

C. degenerate

D. linear

E. positive

$$
\begin{gathered}
\left|\psi_{1}\right\rangle=5|1\rangle-3|2\rangle+2|3\rangle \\
\left|\psi_{2}\right\rangle=|1\rangle-5|2\rangle+x|3\rangle
\end{gathered}
$$


Input:
The states $|1\rangle,|2\rangle$, and $|3\rangle$ are orthonormal. For what value of $x$ are the states $\left|\psi_{1}\right\rangle$ and $\left|\psi_{2}\right\rangle$ given above orthogonal?

A. 10

B. 5

C. 0

D. -5

E. -10


Input:
The state $\psi=\frac{1}{\sqrt{6}} \psi_{-1}+\frac{1}{\sqrt{2}} \psi_{1}+\frac{1}{\sqrt{3}} \psi_{2}$ is a linear combination of three orthonormal eigenstates of the operator $\hat{O}$ corresponding to eigenvalues $-1,1$, and 2 . What is the expectation value of $\hat{O}$ for this state?

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

B. $\sqrt{\frac{7}{6}}$

C. 1

D. $\frac{4}{3}$

E. $\frac{(\sqrt{3}+2 \sqrt{2}-1)}{\sqrt{6}}$

SCRATCHWORK


Input:
Which of the following functions could represent the radial wave function for an electron in an atom? ( $r$ is the distance of the electron from the nucleus; $A$ and $b$ are constants.)
I. $A \mathrm{e}^{-b r}$

II. $A \sin (b r)$

III. $A / r$

A. I only

B. II only

C. I and II only

D. I and III only

E. I, II, and III


Input:
Positronium is an atom formed by an electron and a positron (antielectron). It is similar to the hydrogen atom, with the positron replacing the proton. If a positronium atom makes a transition from the state with $n=3$ to a state with $n=1$, the energy of the photon emitted in this transition is closest to

A. $6.0 \mathrm{eV}$

B. $6.8 \mathrm{eV}$

C. $12.2 \mathrm{eV}$

D. $13.6 \mathrm{eV}$

E. $24.2 \mathrm{eV}$


Input:
If the total energy of a particle of mass $m$ is equal to twice its rest energy, then the magnitude of the particle's relativistic momentum is

A. $m c / 2$

B. $m c / \sqrt{2}$

C. $m c$

D. $\sqrt{3} m c$

E. $2 m c$


Input:
If a charged pion that decays in $10^{-8}$ second in its own rest frame is to travel 30 meters in the laboratory before decaying, the pion's speed must be most nearly

A. $0.43 \times 10^{8} \mathrm{~m} / \mathrm{s}$

B. $2.84 \times 10^{8} \mathrm{~m} / \mathrm{s}$

C. $2.90 \times 10^{8} \mathrm{~m} / \mathrm{s}$

D. $2.98 \times 10^{8} \mathrm{~m} / \mathrm{s}$

E. $3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}$


Input:
In an inertial reference frame $S$, two events occur on the $x$-axis separated in time by $\Delta t$ and in space by $\Delta x$. In another inertial reference frame $S^{\prime}$, moving in the $x$-direction relative to $S$, the two events could occur at the same time under which, if any, of the following conditions?

A. For any values of $\Delta x$ and $\Delta t$

B. Only if $|\Delta x / \Delta t|<c$

C. Only if $|\Delta x / \Delta t|>c$

D. Only if $|\Delta x / \Delta t|=c$

E. Under no condition


Input:
If the absolute temperature of a blackbody is increased by a factor of 3 , the energy radiated per second per unit area does which of the following?

A. Decreases by a factor of 81 .

B. Decreases by a factor of 9 .

C. Increases by a factor of 9 .

D. Increases by a factor of 27.

E. Increases by a factor of 81 .

SCRATCHWORK


Input:
Consider the quasi-static adiabatic expansion of an ideal gas from an initial state $i$ to a final state $f$. Which of the following statements is NOT true?

A. No heat flows into or out of the gas.

B. The entropy of state $i$ equals the entropy of state $f$.

C. The change of internal energy of the gas is $-\int P d V$.

D. The mechanical work done by the gas is $\int P d V$.

E. The temperature of the gas remains constant.


Input:
A constant amount of an ideal gas undergoes the cyclic process $A B C A$ in the $P V$ diagram shown above. The path $B C$ is isothermal. The work done by the gas during one complete cycle, beginning and ending at $A$, is most nearly

A. $600 \mathrm{~kJ}$

B. $300 \mathrm{~kJ}$

C. 0

D. $-300 \mathrm{~kJ}$

E. $-600 \mathrm{~kJ}$


Input:
An AC circuit consists of the elements shown above, with $R=10,000 \mathrm{ohms}, L=25$ millihenries, and $C$ an adjustable capacitance. The $\mathrm{AC}$ voltage generator supplies a signal with an amplitude of 40 volts and angular frequency of 1,000 radians per second. For what value of $C$ is the amplitude of the current maximized?

A. $4 \mathrm{nF}$

B. $40 \mathrm{nF}$

C. $4 \mu \mathrm{F}$

D. $40 \mu \mathrm{F}$

E. $400 \mu \mathrm{F}$


Input:
Which two of the following circuits are high-pass filters? [FIGURE]

A. I and II

B. I and III

C. I and IV

D. II and III

E. II and IV


Input:
In the circuit shown above, the switch $S$ is closed at $t=0$. Which of the following best represents the voltage across the inductor, as seen on an oscilloscope?

A. Voltage


Input:
Maxwell's equations can be written in the form shown below. If magnetic charge exists and if it is conserved, which of these equations will have to be changed?
I. $\nabla \cdot \mathbf{E}=\rho / \epsilon_{\mathrm{o}}$

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

III. $\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

IV. $\nabla \times \mathbf{B}=\mu_{\mathrm{o}} \mathbf{J}+\mu_{\mathrm{o}} \epsilon_{\mathrm{o}} \frac{\partial \mathbf{E}}{\partial t}$

A. I only

B. II only

C. III only

D. I and IV

E. II and III


Input:
Three wire loops and an observer are positioned as shown in the figure above. From the observer's point of view, a current $I$ flows counterclockwise in the middle loop, which is moving towards the observer with a velocity $v$. Loops $A$ and $B$ are stationary. This same observer would notice that

A. clockwise currents are induced in loops $A$ and $B$

B. counterclockwise currents are induced in loops $A$ and $B$

C. a clockwise current is induced in loop $A$, but a counterclockwise current is induced in loop $B$

D. a counterclockwise current is induced in loop $A$, but a clockwise current is induced in loop $B$

E. a counterclockwise current is induced in loop $A$, but no current is induced in loop $B$

SCRATCHWORK


Input:
The components of the orbital angular momentum operator $\mathbf{L}=\left(L_{x}, L_{y}, L_{z}\right)$ satisfy the following commutation relations.

$\left[L_{x}, L_{y}\right]=i \hbar L_{z}$,

$\left[L_{y}, L_{z}\right]=i \hbar L_{x}$,

$\left[L_{z}, L_{x}\right]=i \hbar L_{y}$.

What is the value of the commutator $\left[L_{x} L_{y}, L_{z}\right]$ ?

A. $2 i \hbar L_{x} L_{y}$

B. $i \hbar\left(L_{x}^{2}+L_{y}^{2}\right)$

C. $-i \hbar\left(L_{x}^{2}+L_{y}^{2}\right)$

D. $i \hbar\left(L_{x}^{2}-L_{y}^{2}\right)$

E. $-i \hbar\left(L_{x}^{2}-L_{y}^{2}\right)$


Input:
The energy eigenstates for a particle of mass $m$ in a box of length $L$ have wave functions

$\phi_{n}(x)=\sqrt{2 / L} \sin (n \pi x / L)$ and energies $E_{n}=n^{2} \pi^{2} \hbar^{2} / 2 m L^{2}$, where $n=1,2,3, \ldots$

At time $t=0$, the particle is in a state described as follows.

$$
\Psi(t=0)=\frac{1}{\sqrt{14}}\left[\phi_{1}+2 \phi_{2}+3 \phi_{3}\right]
$$

Which of the following is a possible result of a measurement of energy for the state $\Psi$ ?

A. $2 E_{1}$

B. $5 E_{1}$

C. $7 E_{1}$

D. $9 E_{1}$

E. $14 E_{1}$


Input:
Let $|n\rangle$ represent the normalized $n^{\text {th }}$ energy eigenstate of the one-dimensional harmonic oscillator, $H|n\rangle=\hbar \omega\left(n+\frac{1}{2}\right)|n\rangle$. If $|\psi\rangle$ is a normalized ensemble state that can be expanded as a linear combination

$|\psi\rangle=\frac{1}{\sqrt{14}}|1\rangle-\frac{2}{\sqrt{14}}|2\rangle+\frac{3}{\sqrt{14}}|3\rangle$ of the eigenstates, what is the expectation value of the energy operator in this ensemble state?

A. $\frac{102}{14} \hbar \omega$

B. $\frac{43}{14} \hbar \omega$

C. $\frac{23}{14} \hbar \omega$

D. $\frac{17}{\sqrt{14}} \hbar \omega$

E. $\frac{7}{\sqrt{14}} \hbar \omega$


Input:
A free particle with initial kinetic energy $E$ and de Broglie wavelength $\lambda$ enters a region in which it has potential energy $V$. What is the particle's new de Broglie wavelength?

A. $\lambda(1+E / V)$

B. $\lambda(1-V / E)$

C. $\lambda(1-E / V)^{-1}$

D. $\lambda(1+V / E)^{1 / 2}$

E. $\lambda(1-V / E)^{-1 / 2}$

SCRATCHWORK


Input:
A sealed and thermally insulated container of total volume $V$ is divided into two equal volumes by an impermeable wall. The left half of the container is initially occupied by $n$ moles of an ideal gas at temperature $T$. Which of the following gives the change in entropy of the system when the wall is suddenly removed and the gas expands to fill the entire volume?

A. $2 n R \ln 2$

B. $n R \ln 2$

C. $\frac{1}{2} n R \ln 2$

D. $-n R \ln 2$

E. $-2 n R \ln 2$


Input:
A gaseous mixture of $\mathrm{O}_{2}$ (molecular mass $32 \mathrm{u}$ ) and $\mathrm{N}_{2}$ (molecular mass $28 \mathrm{u}$ ) is maintained at constant temperature. What is the ratio $\frac{v_{r m s}\left(\mathrm{~N}_{2}\right)}{v_{r m s}\left(\mathrm{O}_{2}\right)}$ of the root-mean-square speeds of the molecules?

A. $\frac{7}{8}$

B. $\sqrt{\frac{7}{8}}$

C. $\sqrt{\frac{8}{7}}$

D. $\left(\frac{8}{7}\right)^{2}$

E. $\ln \left(\frac{8}{7}\right)$


Input:
In a Maxwell-Boltzmann system with two states of energies $\epsilon$ and $2 \epsilon$, respectively, and a degeneracy of 2 for each state, the partition function is

A. $\mathrm{e}^{-\epsilon / k T}$

B. $2 \mathrm{e}^{-2 \epsilon / k T}$

C. $2 \mathrm{e}^{-3 \epsilon / k T}$

D. $\mathrm{e}^{-\epsilon / k T}+\mathrm{e}^{-2 \epsilon / k T}$

E. $2\left[\mathrm{e}^{-\epsilon / k T}+\mathrm{e}^{-2 \epsilon / k T}\right]$


Input:
At $20^{\circ} \mathrm{C}$, a pipe open at both ends resonates at a frequency of 440 hertz. At what frequency does the same pipe resonate on a particularly cold day when the speed of sound is 3 percent lower than it would be at $20^{\circ} \mathrm{C}$ ?

A. $414 \mathrm{~Hz}$

B. $427 \mathrm{~Hz}$

C. $433 \mathrm{~Hz}$

D. $440 \mathrm{~Hz}$

E. $453 \mathrm{~Hz}$


Input:
Unpolarized light of intensity $I_{0}$ is incident on a series of three polarizing filters. The axis of the second filter is oriented at $45^{\circ}$ to that of the first filter, while the axis of the third filter is oriented at $90^{\circ}$ to that of the first filter. What is the intensity of the light transmitted through the third filter?

A. 0

B. $I_{0} / 8$

C. $I_{0} / 4$

D. $I_{0} / 2$

E. $I_{0} / \sqrt{2}$


Input:
The conventional unit cell of a body-centered cubic Bravais lattice is shown in the figure above. The conventional cell has volume $a^{3}$. What is the volume of the primitive unit cell?

A. $a^{3} / 8$

B. $a^{3} / 4$

C. $a^{3} / 2$

D. $a^{3}$

E. $2 a^{3}$

SCRATCHWORK


Input:
Which of the following best represents the temperature dependence of the resistivity of an undoped semiconductor?

A.


Input:
The figure above shows a plot of the timedependent force $F_{x}(t)$ acting on a particle in motion along the $x$-axis. What is the total impulse delivered to the particle?

A. 0

B. $1 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$

C. $2 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$

D. $3 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$

E. $4 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$


Input:
A particle of mass $m$ is moving along the $x$-axis with speed $v$ when it collides with a particle of mass $2 m$ initially at rest. After the collision, the first particle has come to rest, and the second particle has split into two equal-mass pieces that move at equal angles $\theta>0$ with the $x$-axis, as shown in the figure above. Which of the following statements correctly describes the speeds of the two pieces?

A. Each piece moves with speed $v$.

B. One of the pieces moves with speed $v$, the other moves with speed less than $v$.

C. Each piece moves with speed $v / 2$.

D. One of the pieces moves with speed $v / 2$, the other moves with speed greater than $v / 2$.

E. Each piece moves with speed greater than $v / 2$.

SCRATCHWORK


Input:
A balloon is to be filled with helium and used to suspend a mass of 300 kilograms in air. If the mass of the balloon is neglected, which of the following gives the approximate volume of helium required? (The density of air is 1.29 kilograms per cubic meter and the density of helium is 0.18 kilogram per cubic meter.)

A. $50 \mathrm{~m}^{3}$

B. $95 \mathrm{~m}^{3}$

C. $135 \mathrm{~m}^{3}$

D. $270 \mathrm{~m}^{3}$

E. $540 \mathrm{~m}^{3}$


Input:
A stream of water of density $\rho$, cross-sectional area $A$, and speed $v$ strikes a wall that is perpendicular to the direction of the stream, as shown in the figure above. The water then flows sideways across the wall. The force exerted by the stream on the wall is

A. $\rho v^{2} A$

B. $\rho v A / 2$

C. $\rho g h A$

D. $v^{2} A / \rho$

E. $v^{2} A / 2 \rho$


Input:
A proton moves in the $+z$-direction after being accelerated from rest through a potential difference $V$. The proton then passes through a region with a uniform electric field $E$ in the $+x$-direction and a uniform magnetic field $B$ in the $+y$-direction, but the proton's trajectory is not affected. If the experiment were repeated using a potential difference of $2 \mathrm{~V}$, the proton would then be

A. deflected in the $+x$-direction

B. deflected in the $-x$-direction

C. deflected in the $+y$-direction

D. deflected in the $-y$-direction

E. undeflected


Input:
For an inductor and capacitor connected in series, the equation describing the motion of charge is

$$
L \frac{d^{2} Q}{d t^{2}}+\frac{1}{C} Q=0
$$

where $L$ is the inductance, $C$ is the capacitance, and $Q$ is the charge. An analogous equation can be written for a simple harmonic oscillator with position $x$, mass $m$, and spring constant $k$. Which of the following correctly lists the mechanical analogs of $L, C$, and $Q$ ?

\begin{tabular}{cccc}
& $\underline{L}$ & $\underline{C}$ & $\underline{Q}$ \\
A. & $m$ & $k$ & $x$ \\
B. & $m$ & $1 / k$ & $x$ \\
C. & $k$ & $x$ & $m$ \\
D. & $1 / k$ & $1 / m$ & $x$ \\
E. & $x$ & $1 / k$ & $1 / m$
\end{tabular}

SCRATCHWORK


Input:
An infinite, uniformly charged sheet with surfacecharge density $\sigma$ cuts through a spherical Gaussian surface of radius $R$ at a distance $x$ from its center, as shown in the figure above. The electric flux $\Phi$ through the Gaussian surface is

A. $\frac{\pi R^{2} \sigma}{\epsilon_{0}}$

B. $\frac{2 \pi R^{2} \sigma}{\epsilon_{0}}$

C. $\frac{\pi(R-x)^{2} \sigma}{\epsilon_{0}}$

D. $\frac{\pi\left(R^{2}-x^{2}\right) \sigma}{\epsilon_{0}}$

E. $\frac{2 \pi\left(R^{2}-x^{2}\right) \sigma}{\epsilon_{0}}$


Input:
An electromagnetic plane wave, propagating in vacuum, has an electric field given by $E=E_{0} \cos (k x-\omega t)$ and is normally incident on a perfect conductor at $x=0$, as shown in the figure above. Immediately to the left of the conductor, the total electric field $E$ and the total magnetic field $B$ are given by which of the following?
$\underline{E}$
$\underline{B}$
A. 0
0
B. $2 E_{0} \cos \omega t$
0
C. 0 $\left(2 E_{0} / c\right) \cos \omega t$
D. $2 E_{0} \cos \omega t$ $\left(2 E_{0} / c\right) \cos \omega t$
E. $2 E_{0} \cos \omega t$
$\left(2 E_{0} / c\right) \sin \omega t$


Input:
A nonrelativistic particle with a charge twice that of an electron moves through a uniform magnetic field. The field has a strength of $\pi / 4$ tesla and is perpendicular to the velocity of the particle. What is the particle's mass if it has a cyclotron frequency of 1,600 hertz?

A. $2.5 \times 10^{-23} \mathrm{~kg}$

B. $1.2 \times 10^{-22} \mathrm{~kg}$

C. $3.3 \times 10^{-22} \mathrm{~kg}$

D. $5.0 \times 10^{-21} \mathrm{~kg}$

E. $7.5 \times 10^{-21} \mathrm{~kg}$

SCRATCHWORK


Input:
The distribution of relative intensity $I(\lambda)$ of blackbody radiation from a solid object versus the wavelength $\lambda$ is shown in the figure above. If the Wien displacement law constant is $2.9 \times 10^{-3} \mathrm{~m} \cdot \mathrm{K}$, what is the approximate temperature of the object?

A. $10 \mathrm{~K}$

B. $50 \mathrm{~K}$

C. $250 \mathrm{~K}$

D. $1,500 \mathrm{~K}$

E. $6,250 \mathrm{~K}$


Input:
Electromagnetic radiation provides a means to probe aspects of the physical universe. Which of the following statements regarding radiation spectra is NOT correct?

A. Lines in the infrared, visible, and ultraviolet regions of the spectrum reveal primarily the nuclear structure of the sample.

B. The wavelengths identified in an absorption spectrum of an element are among those in its emission spectrum.

C. Absorption spectra can be used to determine which elements are present in distant stars.

D. Spectral analysis can be used to identify the composition of galactic dust.

E. Band spectra are due to molecules.

$$
C=3 k N_{A}\left(\frac{h v}{k T}\right)^{2} \frac{\mathrm{e}^{h v / k T}}{\left(\mathrm{e}^{h v / k T}-1\right)^{2}}
$$


Input:
Einstein's formula for the molar heat capacity $C$ of solids is given above. At high temperatures, $C$ approaches which of the following?

A. 0

B. $3 k N_{A}\left(\frac{h v}{k T}\right)$

C. $3 k N_{A} h v$

D. $3 k N_{A}$

E. $N_{A} h v$


Input:
A sample of radioactive nuclei of a certain element can decay only by $\gamma$-emission and $\beta$-emission. If the half-life for $\gamma$-emission is 24 minutes and that for $\beta$-emission is 36 minutes, the half-life for the sample is

A. 30 minutes

B. 24 minutes

C. 20.8 minutes

D. 14.4 minutes

E. 6 minutes


Input:
The ${ }^{238} \mathrm{U}$ nucleus has a binding energy of about 7.6 $\mathrm{MeV}$ per nucleon. If the nucleus were to fission into two equal fragments, each would have a kinetic energy of just over $100 \mathrm{MeV}$. From this, it can be concluded that

A. ${ }^{238} \mathrm{U}$ cannot fission spontaneously

B. ${ }^{238} \mathrm{U}$ has a large neutron excess

C. nuclei near $A=120$ have masses greater than half that of ${ }^{238} \mathrm{U}$

D. nuclei near $A=120$ must be bound by about 6.7 $\mathrm{MeV} /$ nucleon

E. nuclei near $A=120$ must be bound by about 8.5 MeV/nucleon

SCRATCHWORK


Input:
When ${ }_{4}^{7} \mathrm{Be}$ transforms into ${ }_{3}^{7} \mathrm{Li}$, it does so by

A. emitting an alpha particle only

B. emitting an electron only

C. emitting a neutron only

D. emitting a positron only

E. electron capture by the nucleus with emission of a neutrino


Input:
Blue light of wavelength 480 nanometers is most strongly reflected off a thin film of oil on a glass slide when viewed near normal incidence. Assuming that the index of refraction of the oil is 1.2 and that of the glass is 1.6 , what is the minimum thickness of the oil film (other than zero)?

A. $150 \mathrm{~nm}$

B. $200 \mathrm{~nm}$

C. $300 \mathrm{~nm}$

D. $400 \mathrm{~nm}$

E. $480 \mathrm{~nm}$


Input:
Light from a laser falls on a pair of very narrow slits separated by 0.5 micrometer, and bright fringes separated by 1.0 millimeter are observed on a distant screen. If the frequency of the laser light is doubled, what will be the separation of the bright fringes?

A. $0.25 \mathrm{~mm}$

B. $0.5 \mathrm{~mm}$

C. $1.0 \mathrm{~mm}$

D. $2.0 \mathrm{~mm}$

E. $2.5 \mathrm{~mm}$


Input:
The ultraviolet Lyman alpha line of hydrogen with wavelength 121.5 nanometers is emitted by an astronomical object. An observer on Earth measures the wavelength of the light received from the object to be 607.5 nanometers. The observer can conclude that the object is moving with a radial velocity of

A. $2.4 \times 10^{8} \mathrm{~m} / \mathrm{s}$ toward Earth

B. $2.8 \times 10^{8} \mathrm{~m} / \mathrm{s}$ toward Earth

C. $2.4 \times 10^{8} \mathrm{~m} / \mathrm{s}$ away from Earth

D. $2.8 \times 10^{8} \mathrm{~m} / \mathrm{s}$ away from Earth

E. $12 \times 10^{8} \mathrm{~m} / \mathrm{s}$ away from Earth


Input:
Two identical blocks are connected by a spring. The combination is suspended, at rest, from a string attached to the ceiling, as shown in the figure above. The string breaks suddenly. Immediately after the string breaks, what is the downward acceleration of the upper block?

A. 0

B. $g / 2$

C. $g$

D. $\sqrt{2} g$

E. $2 g$


Input:
For the system consisting of the two blocks shown in the figure above, the minimum horizontal force $F$ is applied so that block $B$ does not fall under the influence of gravity. The masses of $A$ and $B$ are 16.0 kilograms and 4.00 kilograms, respectively. The horizontal surface is frictionless and the coefficient of friction between the two blocks is 0.50 . The magnitude of $F$ is most nearly

A. $50 \mathrm{~N}$

B. $100 \mathrm{~N}$

C. $200 \mathrm{~N}$

D. $400 \mathrm{~N}$

E. $1,600 \mathrm{~N}$

SCRATCHWORK


Input:
The Lagrangian for a mechanical system is

$$
L=a \dot{q}^{2}+b q^{4}
$$

where $q$ is a generalized coordinate and $a$ and $b$ are constants. The equation of motion for this system is

A. $\dot{q}=\sqrt{\frac{b}{a}} q^{2}$

B. $\dot{q}=\frac{2 b}{a} q^{3}$

C. $\ddot{q}=-\frac{2 b}{a} q^{3}$

D. $\ddot{q}=+\frac{2 b}{a} q^{3}$

E. $\ddot{q}=\frac{b}{a} q^{3}$

$$
\left(\begin{array}{l}
a_{x}^{\prime} \\
a_{y}^{\prime} \\
a_{z}^{\prime}
\end{array}\right)=\left[\begin{array}{ccc}
1 / 2 & \sqrt{3} / 2 & 0 \\
-\sqrt{3} / 2 & 1 / 2 & 0 \\
0 & 0 & 1
\end{array}\right]\left(\begin{array}{l}
a_{x} \\
a_{y} \\
a_{z}
\end{array}\right)
$$


Input:
The matrix shown above transforms the components of a vector in one coordinate frame $S$ to the components of the same vector in a second coordinate frame $S^{\prime}$. This matrix represents a rotation of the reference frame $S$ by

A. $30^{\circ}$ clockwise about the $x$-axis

B. $30^{\circ}$ counterclockwise about the $z$-axis

C. $45^{\circ}$ clockwise about the $z$-axis

D. $60^{\circ}$ clockwise about the $y$-axis

E. $60^{\circ}$ counterclockwise about the $z$-axis


Input:
The mean kinetic energy of the conduction electrons in metals is ordinarily much higher than $k T$ because

A. electrons have many more degrees of freedom than atoms do

B. the electrons and the lattice are not in thermal equilibrium

C. the electrons form a degenerate Fermi gas

D. electrons in metals are highly relativistic

E. electrons interact strongly with phonons


Input:
An ensemble of systems is in thermal equilibrium with a reservoir for which $k T=0.025 \mathrm{eV}$.

State $A$ has an energy that is $0.1 \mathrm{eV}$ above that of state $B$. If it is assumed the systems obey Maxwell-Boltzmann statistics and that the degeneracies of the two states are the same, then the ratio of the number of systems in state $A$ to the number in state $B$ is

A. $\mathrm{e}^{+4}$

B. $\mathrm{e}^{+0.25}$

C. 1

D. $\mathrm{e}^{-0.25}$

E. $\mathrm{e}^{-4}$


Input:
The muon decays with a characteristic lifetime of about $10^{-6}$ second into an electron, a muon neutrino, and an electron antineutrino. The muon is forbidden from decaying into an electron and just a single neutrino by the law of conservation of

A. charge

B. mass

C. energy and momentum

D. baryon number

E. lepton number


Input:
A particle leaving a cyclotron has a total relativistic energy of $10 \mathrm{GeV}$ and a relativistic momentum of $8 \mathrm{GeV} / c$. What is the rest mass of this particle?

A. $0.25 \mathrm{GeV} / c^{2}$

B. $1.20 \mathrm{GeV} / c^{2}$

C. $2.00 \mathrm{GeV} / c^{2}$

D. $6.00 \mathrm{GeV} / c^{2}$

E. $16.0 \mathrm{GeV} / c^{2}$

SCRATCHWORK


Input:
A tube of water is traveling at $1 / 2 c$ relative to the lab frame when a beam of light traveling in the same direction as the tube enters it. What is the speed of light in the water relative to the lab frame? (The index of refraction of water is $4 / 3$.)

A. $1 / 2 c$

B. $2 / 3 c$

C. $5 / 6 c$

D. $10 / 11 c$

E. $c$


Input:
Which of the following is the orbital angular momentum eigenfunction $Y_{\ell}^{m}(\theta, \phi)$ in a state for which the operators $\mathbf{L}^{2}$ and $L_{z}$ have eigenvalues $6 \hbar^{2}$ and $-\hbar$, respectively?

A. $Y_{2}^{1}(\theta, \phi)$

B. $Y_{2}^{-1}(\theta, \phi)$

C. $\frac{1}{\sqrt{2}}\left[Y_{2}^{1}(\theta, \phi)+Y_{2}^{-1}(\theta, \phi)\right]$

D. $Y_{3}^{2}(\theta, \phi)$

E. $Y_{3}^{-1}(\theta, \phi)$


Input:
Let $|\alpha\rangle$ represent the state of an electron with spin up and $|\beta\rangle$ the state of an electron with spin down. Valid spin eigenfunctions for a triplet state $\left({ }^{3} S\right)$ of a two-electron atom include which of the following?
I. $|\alpha\rangle_{1}|\alpha\rangle_{2}$

II. $\frac{1}{\sqrt{2}}\left(|\alpha\rangle_{1}|\beta\rangle_{2}-|\alpha\rangle_{2}|\beta\rangle_{1}\right)$

III. $\frac{1}{\sqrt{2}}\left(|\alpha\rangle_{1}|\beta\rangle_{2}+|\alpha\rangle_{2}|\beta\rangle_{1}\right)$

A. I only

B. II only

C. III only

D. I and III

E. II and III


Input:
The state of a spin- $\frac{1}{2}$ particle can be represented using the eigenstates $|\uparrow\rangle$ and $|\downarrow\rangle$ of the $S_{z}$ operator.

$$
\begin{aligned}
& S_{z}|\uparrow\rangle=\frac{1}{2} \hbar|\uparrow\rangle \\
& S_{z}|\downarrow\rangle=-\frac{1}{2} \hbar|\downarrow\rangle
\end{aligned}
$$

Given the Pauli matrix $\sigma_{x}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$, which of the following is an eigenstate of $S_{x}$ with eigenvalue $-\frac{1}{2} \hbar$ ?

A. $|\downarrow\rangle$

B. $\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)$

C. $\frac{1}{\sqrt{2}}(|\uparrow\rangle-|\downarrow\rangle)$

D. $\frac{1}{\sqrt{2}}(|\uparrow\rangle+i|\downarrow\rangle)$

E. $\frac{1}{\sqrt{2}}(|\uparrow\rangle-i|\downarrow\rangle)$


Input:
An energy-level diagram of the $n=1$ and $n=2$ levels of atomic hydrogen (including the effects of spin-orbit coupling and relativity) is shown in the figure above. Three transitions are labeled $A, B$, and $C$. Which of the transitions will be possible electric-dipole transitions?

A. $B$ only

B. $C$ only

C. $A$ and $C$ only

D. $B$ and $C$ only

E. $A, B$, and $C$

SCRATCHWORK


Input:
One end of a Nichrome wire of length $2 L$ and cross-sectional area $A$ is attached to an end of another Nichrome wire of length $L$ and crosssectional area $2 A$. If the free end of the longer wire is at an electric potential of 8.0 volts, and the free end of the shorter wire is at an electric potential of 1.0 volt, the potential at the junction of the two wires is most nearly equal to

A. $2.4 \mathrm{~V}$

B. $3.3 \mathrm{~V}$

C. $4.5 \mathrm{~V}$

D. $5.7 \mathrm{~V}$

E. $6.6 \mathrm{~V}$


Input:
A coil of 15 turns, each of radius 1 centimeter, is rotating at a constant angular velocity $\omega=300$ radians per second in a uniform magnetic field of 0.5 tesla, as shown in the figure above. Assume at time $t=0$ that the normal $\hat{\mathbf{n}}$ to the coil plane is along the $y$-direction and that the selfinductance of the coil can be neglected. If the coil resistance is $9 \mathrm{ohms}$, what will be the magnitude of the induced current in milliamperes?

A. $225 \pi \sin \omega t$

B. $250 \pi \sin \omega t$

C. $0.08 \pi \cos \omega t$

D. $1.7 \pi \cos \omega t$

E. $25 \pi \cos \omega t$

SCRATCHWORK


Input:
Two spherical, nonconducting, and very thin shells of uniformly distributed positive charge $Q$ and radius $d$ are located a distance $10 d$ from each other. A positive point charge $q$ is placed inside one of the shells at a distance $d / 2$ from the center, on the line connecting the centers of the two shells, as shown in the figure above. What is the net force on the charge $q$ ?

A. $\frac{q Q}{361 \pi \epsilon_{0} d^{2}}$ to the left

B. $\frac{q Q}{361 \pi \epsilon_{0} d^{2}}$ to the right

C. $\frac{q Q}{441 \pi \epsilon_{0} d^{2}}$ to the left

D. $\frac{q Q}{441 \pi \epsilon_{0} d^{2}}$ to the right

E. $\frac{360 q Q}{361 \pi \epsilon_{0} d^{2}}$ to the left

SCRATCHWORK


Input:
A segment of wire is bent into an arc of radius $R$ and subtended angle $\theta$, as shown in the figure above. Point $P$ is at the center of the circular segment. The wire carries current $I$. What is the magnitude of the magnetic field at $P$ ?

A. 0

B. $\frac{\mu_{0} I \theta}{(2 \pi)^{2} R}$

C. $\frac{\mu_{0} I \theta}{4 \pi R}$

D. $\frac{\mu_{0} I \theta}{4 \pi R^{2}}$

E. $\frac{\mu_{0} I}{2 \theta R^{2}}$


Input:
A child is standing on the edge of a merry-goround that has the shape of a solid disk, as shown in the figure above. The mass of the child is 40 kilograms. The merry-go-round has a mass of 200 kilograms and a radius of 2.5 meters, and it is rotating with an angular velocity of $\omega=2.0$ radians per second. The child then walks slowly toward the center of the-merry-goround. What will be the final angular velocity of the merry-go-round when the child reaches the center? (The size of the child can be neglected.)

A. $2.0 \mathrm{rad} / \mathrm{s}$

B. $2.2 \mathrm{rad} / \mathrm{s}$

C. $2.4 \mathrm{rad} / \mathrm{s}$

D. $2.6 \mathrm{rad} / \mathrm{s}$

E. $2.8 \mathrm{rad} / \mathrm{s}$

SCRATCHWORK


Input:
Two identical springs with spring constant $k$ are connected to identical masses of mass $M$, as shown in the figures above. The ratio of the period for the springs connected in parallel (Figure 1) to the period for the springs connected in series (Figure 2) is

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

B. $\frac{1}{\sqrt{2}}$

C. 1

D. $\sqrt{2}$

E. 2

SCRATCHWORK


Input:
The cylinder shown above, with mass $M$ and radius $R$, has a radially dependent density. The cylinder starts from rest and rolls without slipping down an inclined plane of height $H$. At the bottom of the plane its translational speed is $(8 g H / 7)^{1 / 2}$. Which of the following is the rotational inertia of the cylinder?

A. $\frac{1}{2} M R^{2}$

B. $\frac{3}{4} M R^{2}$

C. $\frac{7}{8} M R^{2}$

D. $M R^{2}$

E. $\frac{7}{4} M R^{2}$


Input:
Two small equal masses $m$ are connected by an ideal massless spring that has equilibrium length $\ell_{0}$ and force constant $k$, as shown in the figure above. The system is free to move without friction in the plane of the page. If $p_{1}$ and $p_{2}$ represent the magnitudes of the momenta of the two masses, a Hamiltonian for this system is

A. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}-2 k\left(\ell-\ell_{0}\right)\right\}$

B. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}+2 k\left(\ell-\ell_{0}\right)^{2}\right\}$

C. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}-k\left(\ell-\ell_{0}\right)\right\}$

D. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}-k\left(\ell-\ell_{0}\right)^{2}\right\}$

E. $\frac{1}{2}\left\{\frac{p_{1}^{2}}{m}+\frac{p_{2}^{2}}{m}+k\left(\ell-\ell_{0}\right)^{2}\right\}$

SCRATCHWORK


Input:
The solution to the Schrödinger equation for the ground state of hydrogen is

$$
\psi_{0}=\frac{1}{\sqrt{\pi a_{0}^{3}}} e^{-r / a_{0}}
$$

where $a_{0}$ is the Bohr radius and $r$ is the distance from the origin. Which of the following is the most probable value for $r$ ?

A. 0

B. $a_{0} / 2$

C. $a_{0}$

D. $2 a_{0}$

E. $\infty$


Input:
The raising and lowering operators for the quantum harmonic oscillator satisfy

$a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle, a|n\rangle=\sqrt{n}|n-1\rangle$

for energy eigenstates $|n\rangle$ with energy $E_{n}$.

Which of the following gives the first-order shift in the $n=2$ energy level due to the perturbation

$$
\Delta H=V\left(a+a^{\dagger}\right)^{2}
$$

where $V$ is a constant?

A. 0

B. $V$

C. $\sqrt{2} V$

D. $2 \sqrt{2} V$

E. $5 \mathrm{~V}$


Input:
An infinite slab of insulating material with dielectric constant $K$ and permittivity $\epsilon=K \epsilon_{0}$ is placed in a uniform electric field of magnitude $E_{0}$. The field is perpendicular to the surface of the material, as shown in the figure above. The magnitude of the electric field inside the material is

A. $\frac{E_{0}}{K}$

B. $\frac{E_{0}}{K \epsilon_{0}}$

C. $E_{0}$

D. $K \epsilon_{0} E_{0}$

E. $K E_{0}$


Input:
A uniformly charged sphere of total charge $Q$ expands and contracts between radii $R_{1}$ and $R_{2}$ at a frequency $f$. The total power radiated by the sphere is

A. proportional to $Q$

B. proportional to $f^{2}$

C. proportional to $f^{4}$

D. proportional to $\left(R_{2} / R_{1}\right)$

E. zero

SCRATCHWORK


Input:
A beam of light has a small wavelength spread $\delta \lambda$ about a central wavelength $\lambda$. The beam travels in vacuum until it enters a glass plate at an angle $\theta$ relative to the normal to the plate, as shown in the figure above. The index of refraction of the glass is given by $n(\lambda)$. The angular spread $\delta \theta^{\prime}$ of the refracted beam is given by

A. $\delta \theta^{\prime}=\left|\frac{1}{n} \delta \lambda\right|$

B. $\delta \theta^{\prime}=\left|\frac{d n(\lambda)}{d \lambda} \delta \lambda\right|$

C. $\delta \theta^{\prime}=\left|\frac{1}{\lambda} \frac{d \lambda}{d n} \delta \lambda\right|$

D. $\delta \theta^{\prime}=\left|\frac{\sin \theta}{\sin \theta^{\prime}} \frac{\delta \lambda}{\lambda}\right|$

E. $\delta \theta^{\prime}=\left|\frac{\tan \theta^{\prime}}{n} \frac{d n(\lambda)}{d \lambda} \delta \lambda\right|$


Input:
Suppose that a system in quantum state $i$ has energy $E_{i}$. In thermal equilibrium, the expression

$$
\frac{\sum_{i} E_{i} e^{-E_{i} / k T}}{\sum_{i} e^{-E_{i} / k T}}
$$

represents which of the following?

A. The average energy of the system

B. The partition function

C. Unity

D. The probability to find the system with energy $E_{i}$

E. The entropy of the system


Input:
A photon strikes an electron of mass $m$ that is initially at rest, creating an electron-positron pair. The photon is destroyed and the positron and two electrons move off at equal speeds along the initial direction of the photon. The energy of the photon was

A. $m c^{2}$

B. $2 m c^{2}$

C. $3 m c^{2}$

D. $4 m c^{2}$

E. $5 m c^{2}$

SCRATCHWORK


Input:
A Michelson interferometer is configured as a wavemeter, as shown in the figure above, so that a ratio of fringe counts may be used to compare the wavelengths of two lasers with high precision. When the mirror in the right arm of the interferometer is translated through a distance $d, 100,000$ interference fringes pass across the detector for green light and 85,865 fringes pass across the detector for red $(\lambda=632.82$ nanometers $)$ light. The wavelength of the green laser light is

A. $500.33 \mathrm{~nm}$

B. $543.37 \mathrm{~nm}$

C. $590.19 \mathrm{~nm}$

D. $736.99 \mathrm{~nm}$

E. $858.65 \mathrm{~nm}$

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Input:
The wave function of a particle is $e^{i(k x-\omega t)}$, where $x$ is distance, $t$ is time, and $k$ and $\omega$ are positive real numbers. The $x$-component of the momentum of the particle is

A. 0

B. $\hbar \omega$

C. $\hbar k$

D. $\frac{\hbar \omega}{c}$

E. $\frac{\hbar k}{\omega}$


Input:
The longest wavelength $x$-ray that can undergo Bragg diffraction in a crystal for a given family of planes of spacing $d$ is

A. $\frac{d}{4}$

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

C. $d$

D. $2 d$

E. $4 d$


Input:
The ratio of the energies of the $K$ characteristic $\mathrm{x}$-rays of carbon $(Z=6)$ to those of magnesium $(Z=12)$ is most nearly

A. $\frac{1}{4}$

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

C. 1

D. 2

E. 4


Input:
What is $\frac{F(R)}{F(2 R)}$ ?

A. 32

B. 8

C. 4

D. 2

E. 1


Input:
Suppose there is a very small shaft in the Earth such that the point mass can be placed at a radius of $R / 2$. What is $\frac{F(R)}{F\left(\frac{R}{2}\right)}$ ?

A. 8

B. 4

C. 2

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

E. $\frac{1}{4}$


Input:
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.


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


Input:
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 $(-b, 0,-c)$


Input:
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. $\frac{\mu_{0} i}{2 \pi r} \frac{r^{2}-b^{2}}{c^{2}-b^{2}}$


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


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


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


Input:
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


Input:
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}


Input:
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. $\frac{11}{2} R$


Input:
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. $2760 \mathrm{~J}$


Input:
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.


Input:
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


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


Input:
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. $3 d=2 w$


Input:
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


Input:
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. $10 \mathrm{~m}$


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


Input:
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


Input:
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


Input:
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. $23 \mathrm{~min}$


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