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GATE 2020 PH – Question 25

Thermodynamics and Statistical Mechanics · Laws of thermodynamics · 1 mark · Multiple choice

A small disk rotates about its suspending fibre. Its small-angle Hamiltonian is $H=p_\theta^2/(2I)+\alpha\theta^2/2$, with inertia I and restoring torque per unit angle $\alpha$. In thermal equilibrium at T, its mean and RMS angular deflections are

Diagram for GATE 2020 PH question 25
  1. $\theta_{avg}=0,\quad\theta_{rms}=(k_BT/\alpha)^{3/2}$
  2. $\theta_{avg}=0,\quad\theta_{rms}=(k_BT/\alpha)^{1/2}$
  3. $\theta_{avg}\ne0,\quad\theta_{rms}=(k_BT/\alpha)^{1/2}$
  4. $\theta_{avg}\ne0,\quad\theta_{rms}=(k_BT/\alpha)^{3/2}$

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Correct answer: (B) $\theta_{avg}=0,\quad\theta_{rms}=(k_BT/\alpha)^{1/2}$

Explanation

The equilibrium distribution is even in $\theta$ because the potential is quadratic and symmetric, so $\langle\theta\rangle=0$. Classical equipartition assigns $k_BT/2$ to the quadratic potential term: $\alpha\langle\theta^2\rangle/2=k_BT/2$. Hence $\theta_{rms}=\sqrt{k_BT/\alpha}$, B.

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