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GATE 2022 PH – Question 46

Classical Mechanics · small oscillations: coupled oscillations and normal modes · 2 marks · Multiple choice

A piston of mass m is fitted to an airtight horizontal cylindrical jar. The cylinder and piston have identical unit area of cross-section. The gas inside the jar has volume V and is held at pressure P= P௔௧௠௢௦௣௛௘௥௘. The piston is pushed inside the jar very slowly over a small distance. On releasing, the piston performs an undamped simple harmonic motion of low frequency. Assuming that the gas is ideal and no heat is exchanged with the atmosphere, the frequency of the small oscillations is proportional to

  1. $\sqrt{P/(\gamma mV)}$
  2. $\sqrt{\gamma P/(mV)}$
  3. $\sqrt{P/(mV^{\gamma-1})}$
  4. $\sqrt{\gamma P/(mV^{\gamma-1})}$

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Correct answer: (B) $\sqrt{\gamma P/(mV)}$

Explanation

**Small adiabatic compression.** The gas obeys $PV^\gamma=\text{constant}$ (adiabatic, ideal gas). If the piston (unit area) is displaced by $x$ into the jar, the volume changes by $\delta V=-x$ (area = 1), and the pressure changes by
$$\delta P=-\gamma\frac{P}{V}\,\delta V=\frac{\gamma P}{V}\,x .$$

**Restoring force** on the piston (unit area): $F=-\delta P\times1=-\dfrac{\gamma P}{V}x$. This is a spring force with spring constant
$$k=\frac{\gamma P}{V}.$$

**Frequency:**
$$f=\frac1{2\pi}\sqrt{\frac km}=\frac1{2\pi}\sqrt{\frac{\gamma P}{mV}}\;\propto\;\sqrt{\frac{\gamma P}{mV}}\quad(\text{option B}).$$