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 volum of the gas is `V_(0)` and its pressur is `P_(0)`. The pistong 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)(gammaAP_(0))/(V_(0)M)`
B. `(1)/(2pi)(V_(0)MP_(0))/(gammaA^(2))`
C. `(1)/(2pi)sqrt((gamma A^(2)P_(0))/(MV_(0)))`
D. `(1)/(2pi)sqrt((MV_(0))/(gamma^(2)P_(0)))`

Answer

C

Explanation

For sudden changes in pressure and volume of the gas `, PV^(gamma=a const ant)`
or `V^(gamma)dP+gammaPV^(gamma)dV=0 ordP =-(gammaP)/(V)dV`
Net force on piston, `F_(n et)=(dP)xxA=-(gammaP_(0))/(V_(0))dVxxA`
`["Given , inititally" P=P_(0), V=V_(0)]`
Let the piston be moved inwards through a small distance `x `, then decrease in the volume of the gas,
`dV=Ax`
`:. F_(n e t)=-(gammaP_(0))/(V_(0))Ax xxA=-(gamma P_(0)A^(2))/(V_(0))x` ...(i)
As `F prop x` and it is directed towards equilibrium position, hence, if piston is left free, it will execute linear SHM.
Comparing (i) with ,`F=-kx,` we have
spring factor, `k=(gammaP_(0)A^(2))/(V_(0))`
Inertia factor, `m=M(` mass of piston `)`
Frequency, `v=(1)/(2pi)sqrt((spri ng fact o r)/(i n ertial fact o r))`
`=(1)/(2pi)sqrt((gammaP_(0)A^(2)//V_(0))/(M))=(1)/(2pi)sqrt((gamma P_(0)A^(2))/(MV_(0)))`