A frictionless piston of mass m and cross-sectional area A closes an ideal gas of equilibrium pressure P and equilibrium volume V. The other side of the piston is maintained at constant pressure. For small rapid oscillations, the gas compression and expansion may be treated as adiabatic with ratio of specific heats γ. What is the angular frequency?
Options

Question

A frictionless piston of mass m and cross-sectional area A closes an ideal gas of equilibrium pressure P and equilibrium volume V. The other side of the piston is maintained at constant pressure. For small rapid oscillations, the gas compression and expansion may be treated as adiabatic with ratio of specific heats γ. What is the angular frequency?

Options

G1 Physics · Accepted Answer

For a small piston displacement x, ΔV = Ax. For an adiabatic change, PVᵞ = constant. Therefore, for small changes, ΔP = −γP(ΔV/V) = −γPAx/V. The restoring force on the piston is F = AΔP = −γPA²x/V. Hence, kₑ = γPA²/V. Therefore, ω = √[kₑ/m] = √[γPA²/(mV)]. Correct option: B.