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 posittion and released. assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency

Explanation

Suppose the piston is displaced through a
distance x above the equilibrium positon. The volume
of the gas increases by `(Delta V = Ax)`. As the system is
completely isolated from its surrounding, the process is adiabatic . Thsu,
`(pV^gamma) = constant `
or, `In (p + gamma In V = constant)`
`(Delta p / p)` + gamma(Delta V) / V) = 0`
or, `Delta p = - (gamma p) / V (Delta V)`.
As the piston is only slightly pushed, we can write
` Delta p = - (gamma p_0) / (V_0) Delta V`.
The resultant force acting on the piston in this positon is
`F = (A Delta p) = - A (gamma P_0) / (V_0) Delta V`
= - A^(2) (gamma P_0) / (V_0) x = - kx`
where `k = A^(2) (gamma P_0) / (V_0) .
Thus the motion of the piston is simple harmonic . The angualar frequency omega is given by
`omega = sqrt (k / M )= sqrt(( A^2gamma P_0) / (MV_0)) `
and the freuency is `v = (omega / 2 pi) = (1 / 2 pi) sqrt(( A^2gamma P_0) / (MV_0)) `.