An ideal gas enclosed in a 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

Options

A. `(1)/(2pi) (V_(0)MP_(0))/(A^(2)gamma)`
B. `(1)/(2pi)sqrt(A^(2)gammaP_(0))/(MV_(0))`
C. `(1)/(2pi)sqrt(MV_(0))/(AgammaP_(0))`
D. `(1)/(2pi)(AgammaP_(0))/(V_(0)M)`

Answer

B

Explanation

For adiabatic process, `PV = K`
`In (PV) = in (K)`
`(dP)/(P) + gamma (dV)/(V) 0`
`dP = (Pgamma)/(V)dV`
Let the small displacement traversed by pistion of mass M be y
`dP = (Pgamma)/(V) (Ay)`
Restoring force F = dP
`F = (-P_(0)gammaA^(2))/(V_(0)) y`
`a = ((P_(0)gammaA^(2))/(V_(0)M))y to` it executes S.H.M.
`v = (1)/(2pi) sqrt((P_(0)gammaA^(2))/(V_(0)M))`