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A cylindrical piston of mass M and cross-sectional area A slides smoothly inside a long horizontal cylinder (closed at one end) enclosing a gas. The piston is displaced from its equilibrium position and undergoes simple harmonic motion under isothermal conditions. What is the period of oscillation? (h = equilibrium length of gas column)

1. 2*pi*sqrt(M*h/(P*A))
2. 2*pi*sqrt(M*A/(P*h))
3. 2*pi*sqrt(M/(P*A*h))
4. 2*pi*sqrt(M*P*h*A)

Correct answer: 2*pi*sqrt(M*h/(P*A))

Solution

Equilibrium: pressure P, gas length h, volume V0 = Ah. Displace piston by +x: new volume = A(h+x). Isothermal: P*Ah = P'*A(h+x) => P' = Ph/(h+x). Restoring force F = -(P' - P)*A = -A[Ph/(h+x) - P] = -A*P[h/(h+x) - 1] = -A*P*(-x)/(h+x) ≈ PA*x/h (inward, restoring) for small x. So F = -(PA/h)*x. Spring constant k = PA/h. T = 2*pi*sqrt(M/k) = 2*pi*sqrt(Mh/(PA)).

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