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)(AgammaP_(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))/(AgammaP_(0)))`

Answer

C

Explanation

FBD of piston at equilibrium

`rArr P_(atm) A + mg = P_(0)A"……(i)`
FBD of piston when piston is pushed down a distance x

`P_(atm) + mg - (P_(0) + dP) A = m(d^(2))/(dt^(2))"....."(ii)`
Process is adiabatic
`rArr PV' = C rArr -dP = (gammaPdV)/(V)`
Using `1,2,3` we get `f = (1)/(2pi)sqrt((A^(2)gammaP_(0))/(MV_(0)))`