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

Options

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

Answer

C

Explanation

`(Mg)/(A) = P_(0) , P_(0)V_(0^(gamma)) = Pv^(gamma)`
`Mg = P_(0)A………(1) , P_(0)Ax_(0^(gamma)) = PA(x_(0)) - x)^(gamma)`
let piston is displaced by `x , P = (P_(0)x_(0^(gamma)))/((x_(0) - x)^(gamma))`
`Mg - ((P_(0)x_(0^(gamma)
)))/((x_(0) - x)^(gamma))A = F_("rastoring")`
`P_(0)A(1 - x_(0^(gamma))/((x_(0) - x)^(gamma))) = F_("restoring")`
`F = -(gammaP_(0)Ax)/(x_(0)) :. f = (1)/(2pi)sqrt((gammaP_(0)A)/(x_(0)M)) = (1)/(2pi)sqrt((gammaP_(0)A^(2))/(x_(0)M))`