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

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

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

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

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

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

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

Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
The nuclear decay above is an example of a process induced by the

A. Mössbauer effect

B. Casimir effect

C. photoelectric effect

D. weak interaction

E. strong interaction

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
Consider a single electron atom with orbital angular momentum $L=\sqrt{2} \hbar$. Which of the following gives the possible values of a measurement of $L_{z}$, the $z$-component of $L$ ?

A. 0

B. $0, \hbar$

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

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

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

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
Characteristics of the quantum harmonic oscillator include which of the following?

I. A spectrum of evenly spaced energy states

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

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

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

A. I only

B. IV only

C. I and IV only

D. II and III only

E. I, II, III, and IV
Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
A muon can be considered to be a heavy electron with a mass $m_{\mu}=207 m_{e}$. Imagine replacing the electron in a hydrogen atom with a muon. What are the energy levels $E_{n}$ for this new form of hydrogen in terms of the binding energy of ordinary hydrogen $E_{0}$, the mass of the proton $m_{p}$, and the principal quantum number $n$ ?

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

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

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

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

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

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A large, parallel-plate capacitor consists of two square plates that measure $0.5 \mathrm{~m}$ on each side. A charging current of $9 \mathrm{~A}$ is applied to the capacitor. Which of the following gives the approximate rate of change of the electric field between the plates?

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

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

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

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

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


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
The circuit shown in the figure above consists of eight resistors, each with resistance $R$, and a battery with terminal voltage $V$ and negligible internal resistance. What is the current flowing through the battery?

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

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

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

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

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


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
In the AC circuit above, $V_{i}$ is the amplitude of the input voltage and $V_{o}$ is the amplitude of the output voltage. If the angular frequency $\omega$ of the input voltage is varied, which of the following gives the ratio $V_{o} / V_{i}=G$ as a function of $\omega$ ?

A.


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A wire loop that encloses an area of $10 \mathrm{~cm}^{2}$ has a resistance of $5 \Omega$. The loop is placed in a magnetic field of $0.5 \mathrm{~T}$ with its plane perpendicular to the field. The loop is suddenly removed from the field. How much charge flows past a given point in the wire?

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

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

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

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

E. $1 \mathrm{C}$


Answer scores:
A: 1
B: 0
C: 0
D: 0
E: 0
Two nonrelativistic electrons move in circles under the influence of a uniform magnetic field $\mathbf{B}$, as shown in the figure above. The ratio $r_{1} / r_{2}$ of the orbital radii is equal to $1 / 3$. Which of the following is equal to the ratio $v_{1} / v_{2}$ of the speeds?

A. $1 / 9$

B. $1 / 3$

C. 1

D. 3

E. 9
Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
Which of the following statements about bosons and/or fermions is true?

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

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

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

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

E. Bosons and fermions obey the Pauli exclusion principle.

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
The discovery of the $J / \psi$ particle was especially significant because it provided evidence for which of the following?

A. Parity violation in weak interactions

B. Massive neutrinos

C. Higgs bosons

D. Charmed quarks

E. Strange quarks


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
The figure above shows an object $O$ placed at a distance $R$ to the left of a convex spherical mirror that has a radius of curvature $R$. Point $C$ is the center of curvature of the mirror. The image formed by the mirror is at

A. infinity

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

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

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

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

Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
A uniform thin film of soapy water with index of refraction $n=1.33$ is viewed in air via reflected light. The film appears dark for long wavelengths and first appears bright for $\lambda=540 \mathrm{~nm}$. What is the next shorter wavelength at which the film will appear bright on reflection?

A. $135 \mathrm{~nm}$

B. $180 \mathrm{~nm}$

C. $270 \mathrm{~nm}$

D. $320 \mathrm{~nm}$

E. $405 \mathrm{~nm}$


Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A model of an optical fiber is shown in the figure above. The optical fiber has an index of refraction, $n$, and is surrounded by free space. What angles of incidence, $\theta$, will result in the light staying in the optical fiber?

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

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

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

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

E. $\sin ^{-1}\left(\sqrt{n^{2}-1}\right)<\theta<\sin ^{-1}\left(\sqrt{n^{2}+1}\right)$
Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A gas at temperature $T$ is composed of molecules of mass $m$. Which of the following describes how the average time between intermolecular collisions varies with $m$ ?

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

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

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

D. It is proportional to $m$.

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

Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
A particle can occupy two possible states with energies $E_{1}$ and $E_{2}$, where $E_{2}>E_{1}$. At temperature $T$, the probability of finding the particle in state 2 is given by which of the following?

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

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

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

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

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

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

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

A. 0

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

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

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

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

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A $1 \mathrm{~kg}$ block attached to a spring vibrates with a frequency of $1 \mathrm{~Hz}$ on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an $8 \mathrm{~kg}$ block placed on the same table. Which of the following gives the frequency of vibration of the $8 \mathrm{~kg}$ block?

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

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

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

D. $1 \mathrm{~Hz}$

E. $2 \mathrm{~Hz}$


Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
A uniform disk with a mass of $m$ and a radius of $r$ rolls without slipping along a horizontal surface and ramp, as shown above. The disk has an initial velocity of $v$. What is the maximum height $h$ to which the center of mass of the disk rises?

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

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

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

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

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

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
A mass, $m$, is attached to a massless spring fixed at one end. The mass is confined to move in a horizontal plane, and its position is given by the polar coordinates $r$ and $\theta$. Both $r$ and $\theta$ can vary. If the relaxed length of the spring is $s$ and the force constant is $k$, what is the Lagrangian, $L$, for the system?

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

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

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

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

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


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A mass $m$ attached to the end of a massless rod of length $L$ is free to swing below the plane of support, as shown in the figure above. The Hamiltonian for this system is given by

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

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

A. $\theta$

B. $\phi$

C. $\dot{\theta}$

D. $p_{\theta}$

E. $p_{\phi}$


Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
A rod of length $L$ and mass $M$ is placed along the $x$-axis with one end at the origin, as shown in the figure above. The rod has linear mass density $\lambda=\frac{2 M}{L^{2}} x$, where $x$ is the distance from the origin. Which of the following gives the $x$-coordinate of the rod's center of mass?

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

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

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

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

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

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

A. 0

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

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

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

E. 1
Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
Which of the following are the eigenvalues of the Hermitian matrix $\left(\begin{array}{cc}2 & i \\ -i & 2\end{array}\right)$ ?

A. 1,0

B. 1,3

C. 2,2

D. $i,-i$

E. $1+i, 1-i$

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

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
Consider the Pauli spin matrices $\sigma_{x}, \sigma_{y}$, and $\sigma_{z}$ and the identity matrix $I$ given above. The commutator $\left[\sigma_{x}, \sigma_{y}\right] \equiv \sigma_{x} \sigma_{y}-\sigma_{y} \sigma_{x}$ is equal to which of the following?

A. $I$

B. $2 i \sigma_{x}$

C. $2 i \sigma_{y}$

D. $2 i \sigma_{z}$

E. 0

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A spin- $\frac{1}{2}$ particle is in a state described by the spinor

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

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

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

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

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

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

E. 1


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
An electron with total energy $E$ in the region $x<0$ is moving in the $+x$-direction. It encounters a step potential at $x=0$. The wave function for $x \leq 0$ is given by

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

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

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

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

A. $R=0$

B. $R=1$

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

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

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


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
Two thin, concentric, spherical conducting shells are arranged as shown in the figure above. The inner shell has radius $a$, charge $+Q$, and is at zero electric potential. The outer shell has radius $b$ and charge $-Q$. If $r$ is the radial distance from the center of the spheres, what is the electric potential in region I $(a<r<b)$ and in region II $(r>b)$ ?

\section*{Region I}

\section*{Region II}

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

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

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

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

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

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

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

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
In static electromagnetism, let $\mathbf{E}, \mathbf{B}, \mathbf{J}$, and $\rho$ be the electric field, magnetic field, current density, and charge density, respectively. Which of the following conditions allows the electric field to be written in the form $\mathbf{E}=-\nabla \phi$, where $\phi$ is the electrostatic potential?

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

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

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

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

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

Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
A long, straight, hollow cylindrical wire with an inner radius $R$ and an outer radius $2 R$ carries a uniform current density. Which of the following graphs best represents the magnitude of the magnetic field as a function of the distance from the center of the wire?

A.


Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
A parallel-plate capacitor has plate separation $d$. The space between the plates is empty. A battery supplying voltage $V_{0}$ is connected across the capacitor, resulting in electromagnetic energy $U_{0}$ stored in the capacitor. A dielectric, of dielectric constant $\kappa$, is inserted so that it just fills the space between the plates. If the battery is still connected, what are the electric field $E$ and the energy $U$ stored in the dielectric, in terms of $V_{0}$ and $U_{0}$ ?
$\underline{E}$
$\underline{U}$
A. $\frac{V_{0}}{d}$
$U_{0}$
B. $\frac{V_{0}}{d}$
$\kappa U_{0}$
C. $\frac{V_{0}}{d}$
$\kappa^{2} U_{0}$
D. $\frac{V_{0}}{\kappa d}$
$U_{0}$
E. $\frac{V_{0}}{\kappa d}$
$\kappa U_{0}$

Answer scores:
A: 0
B: 1
C: 0
D: 0
E: 0
An observer $O$ at rest midway between two sources of light at $x=0$ and $x=10 \mathrm{~m}$ observes the two sources to flash simultaneously. According to a second observer $O^{\prime}$, moving at a constant speed parallel to the $x$-axis, one source of light flashes $13 \mathrm{~ns}$ before the other. Which of the following gives the speed of $O^{\prime}$ relative to $O$ ?

A. $0.13 c$

B. $0.15 c$

C. $0.36 c$

D. $0.53 c$

E. $0.62 c$

Answer scores:
A: 0
B: 0
C: 1
D: 0
E: 0
Let $\hat{\mathbf{J}}$ be a quantum mechanical angular momentum operator. The commutator $\left[\hat{J}_{x} \hat{J}_{y}, \hat{J}_{x}\right]$ is equivalent to which of the following?

A. 0

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

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

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

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

Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
Which of the following ions CANNOT be used as a dopant in germanium to make an $n$-type semiconductor?

A. As

B. $\mathrm{P}$

C. $\mathrm{Sb}$

D. $\mathrm{B}$

E. $\mathrm{N}$
Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
In the Compton effect, a photon with energy $E$ scatters through a $90^{\circ}$ angle from a stationary electron of mass $m$. The energy of the scattered photon is

A. $E$

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

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

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

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

Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
Which of the following is the principal decay mode of the positive muon $\mu^{+}$?

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

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

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

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

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


Answer scores:
A: 0
B: 0
C: 0
D: 1
E: 0
A small particle of mass $m$ is at rest on a horizontal circular platform that is free to rotate about a vertical axis through its center. The particle is located at a radius $r$ from the axis, as shown in the figure above. The platform begins to rotate with constant angular acceleration $\alpha$. Because of friction between the particle and the platform, the particle remains at rest with respect to the platform. When the platform has reached angular speed $\omega$, the angle $\theta$ between the static frictional force $\mathbf{f}_{\mathrm{s}}$ and the inward radial direction is given by which of the following?

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

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

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

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

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

Answer scores:
A: 0
B: 0
C: 0
D: 0
E: 1
The partition function $Z$ in statistical mechanics can be written as

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

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

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

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

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

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

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

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

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


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