A cylinder piston of mass `M` sides smoothly inside a long cylinder closed at and enclosing a certain mass of gas The cylinder is kept with its axis horizontal if the piston is distanced from its equations positions it oscillation simple harmonically .The period of oscillation will be

Options

A. `T = 2pi sqrt(((Mh)/(PA)))`
B. `T = 2pi sqrt(((MA)/(Pb)))`
C. `T = 2pi sqrt(((M)/(PAh)))`
D. `T = 2pi sqrt(MPhA)`

Answer

A

Explanation

Let the piston be displace through distance `x` toward left then velocity drcrease pressed increase if pressure and `DeltaV` is decrerase in volume then considring the pracess as take place gravity (i.e.sothermal)
`P_(1)V_(1) = P_(2)V_(2) rArr PV (P + Deltap)(V - DeltaV)`
`rArr PV = PV + Delta PV - DeltaP DeltaV`
`rArr Delta P.V - P (Ah) = P(Ax) rArr DeltaP(Ax) rArr DeltaP = (P.x)/(h)`
This execute pressed is responsible the reastroing force `(F)` to the pistom of mass `M`
Hence `F = Deltap A = (pks)/(h)`
comparing a with `(F) = kx rArr k = M omega^(2) = (pA)/(h)`
`rArr omega = sqrt((pA)/(Mh)) rArr T = 2pi sqrt((Mh)/(pA))`