An ideal gas is enclosed in a vertical cylindrical container and supports a freely moving piston of mass M . The piston and the cylinder have an equal cross-sectional area A . Atmospheric pressure is p 0 and when the piston is in equilibrium, the volume of the gas is V 0 . The piston is now displaced slightly from its equilibrium position. Assuming that the system, is completely isolated from, its surroundings, what

Step-by-step solution

In equilibrium, pressure inside the cylinder = pressure just outside it or p = p 0 + Mg A When piston is displaced slightly by an amount x, change in volume, dV = - Ax Since, the cylinder is isolated from the surroundings, nature of the process is adiabatic. In adiabatic process, dp dV = - γ p V or increase in pressure inside the cylinder, dp = - γ p V dV = γ p 0 + Mg A V 0 Ax ​ This increase in pressure when multiplied with area of cross-section A will give a net upward force (or the restoring force). Hence, F = -