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An ideal gas is enclosed in a vertical cylindrical container fitted with a frictionless piston of mass m and area A. Initially the gas occupies volume V0 at equilibrium. The piston is pushed down until the gas volume becomes V0/8 and then released from rest. Assuming the compression and expansion are adiabatic (gamma = 5/3), the piston will:

1. Remain stationary at V0/8 after release
2. Rise and execute simple harmonic motion about the equilibrium position
3. Rise and come to rest at the original equilibrium position V0
4. Rise beyond V0 and oscillate about a position above V0

Correct answer: Rise and execute simple harmonic motion about the equilibrium position

Solution

Initially the system is in equilibrium at volume V0. The piston is compressed adiabatically to V0/8 — a large compression. When released from rest, the gas pressure (much higher than equilibrium) pushes the piston upward. In an adiabatic process, when the piston passes back through V0, the gas is at the original equilibrium pressure and the piston has kinetic energy (gained from the work done by the gas). The piston overshoots, the gas expands further (below equilibrium pressure), decelerating the piston until it stops, then returns. Since the process is reversible and adiabatic (no energy loss), the piston oscillates between V0/8 and a symmetric upper volume. For small oscillations about the equilibrium, this motion is SHM. The piston executes oscillatory (approximately SHM for small amplitudes) motion about the equilibrium position.

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