A vertical cylinder is closed at the bottom. Gas (assumed to be ideal) is enclosed in the cylinder by a close-fitting but frictionless piston above which there is an evacuated space. The piston is displaced slightly from its equilibrium position and released. Show that the period of the resulting oscillations is 2̃̀∑h/(̃̀g), where ̃̀ = cp/cV, h is the height of the piston above the closed end when it is in equilibrium. Hints: The ideal gas effectively acts as a spring. The spring constant is k = -df/dh, where the force f can be related to the pressure p and the vertical displacement h can be related to the volume V. The angular frequency of a harmonic oscillator is ̃̀ = ̃̀k/m, where m is the mass of the piston in the present case. The process is assumed to be adiabatic and reversible.