An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M . The piston and cylinder have equal cross sectional area A . When the piston is in equilibrium, the volume of the gas is $\mathrm{V}_{0}$ and its pressure is $\mathrm{P}_{0}$. 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:

(B) $\frac{1}{2 \pi} \sqrt{\frac{A^{2} \gamma P_{0}}{M V_{0}}}$

(A) $\frac{1}{2 \pi} \frac{V_{0} M P_{0}}{A^{2} \gamma}$

(C) $\frac{1}{2 \pi} \sqrt{\frac{M V_{0}}{A \gamma P_{0}}}$

(D) $\frac{1}{2 \pi} \frac{A \gamma P_{0}}{V_{0} M}$

JEE Main · 2013 · 07 Apr 2013 · Offline · Elasticity & SHM

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass M . The piston and 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:

(A) $\frac{1}{2 π} \frac{V _{0} M P _{0}}{A ^{2} γ}$
(B) $\frac{1}{2 π} \frac{A ^{2} γ P _{0}}{M V _{0}}$ ✓ Correct
(C) $\frac{1}{2 π} \frac{M V _{0}}{A γ P _{0}}$
(D) $\frac{1}{2 π} \frac{A γ P _{0}}{V _{0} M}$

Answer: (B)

Solution

The piston rests where the gas pressure holds up its weight, and a small downward push

x

changes the volume by

A x

. Because the system is isolated the compression is adiabatic,

P V^{γ} = const

, so differentiating gives a pressure change

d P = - \frac{γ P _{0}}{V _{0}} A x

. That produces a restoring force

F = - A d P = - \frac{γ P _{0} A ^{2}}{V _{0}} x

, the signature of simple harmonic motion with

ω^{2} = \frac{γ P _{0} A ^{2}}{M V _{0}}

. The frequency is therefore

f = \frac{1}{2 π} \frac{A ^{2} γ P _{0}}{M V _{0}}

JEE Main · 2013 · 07 Apr 2013 · Offline · Single correct option

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JEE Main · 2013 · 07 Apr 2013 · Offline · Elasticity & SHM

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