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An ideal gas is trapped in a vertical cylinder of cross-sectional area A by a frictionless piston of mass M. When the gas volume is V0 and its pressure is P0, the piston is in equilibrium. If the piston is given a small displacement from this position and then let go, and the entire setup is thermally isolated, the piston performs SHM with frequency
- (1/2π)√(AγP0/(V0M))
- (1/2π)√(V0MP0/A²γ)
- (1/2π)√(A²γP0/(MV0))
- (1/2π)√(MV0/(AP0γ))
Correct answer: (1/2π)√(A²γP0/(MV0))
Solution
For an adiabatic small displacement x, dP = -(gamma*P0/V0)*A*x, so restoring force = -(gamma*P0*A^2/V0)*x. Then omega^2 = gamma*P0*A^2/(M*V0) and f = (1/2pi)*sqrt(A^2*gamma*P0/(M*V0)).
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