An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M. The piston and the cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas is V₀ and its pressure is P₀. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency (a) 1/2π(Aγ P₀)/(V₀ M) (b) 1/2π(V₀ M P₀)/(A²γ) (c) 1/2π√((A²γ P₀)/(V₀ M)) (d) 1/2π√((M V₀)/(Aγ P₀))

Answer: (c). Isolated system: adiabatic, PVγ = const, so Δ P = -(γ P₀)/(V₀)Δ V = -(γ P₀ A x)/(V₀). Restoring force F = AΔ P = -(γ P₀ A²)/(V₀)x. f = 1/2π√((γ P₀ A²)/(M V₀)), answer (c).

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An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M.$ The piston and the cylinder have equal cross sectional area $A$ . When the piston is in equilibrium, the volume of the gas is $V_0$ and its pressure is $P_0.$ The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency

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Answer: (c)

Isolated system: adiabatic, $PV^\gamma$ = const, so $\Delta P = -\frac{\gamma P_0}{V_0}\Delta V = -\frac{\gamma P_0 A x}{V_0}$ .
Restoring force $F = A\Delta P = -\frac{\gamma P_0 A^2}{V_0}x$ .
$f = \frac{1}{2\pi}\sqrt{\frac{\gamma P_0 A^2}{M V_0}}$ , answer (c).

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