shayekh/physics_gre huggingface dataset:
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 (1 = correct):
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 (1 = correct):
A: 0
B: 0
C: 0
D: 0
E: 1