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 = P௧௦. 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
Options:
(A) ඨ P
𝛾mV
(B) ඨ P𝛾
Vm
(C) ඨ P
mVఊିଵ
(D) ඨ 𝛾P
mVఊିଵ
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.46)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Nuclear & Particle Physics** within *Section 9: Nuclear and Particle Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Nuclear & Particle Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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