# Gas-Driven Piston Undergoing Simple Harmonic Oscillation

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This Demonstration shows simple harmonic oscillation of an isothermal ideal gas in a piston being driven by a pressure gradient. The piston is assumed to be frictionless and thermal effects of successive expansion and compression of the gas are neglected. This idealized system is a perpetual motion machine!

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Contributed by: T. Kirkpatrick (April 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Givens for the problem: 1. Initial displacement of piston = α 2. Initial velocity of piston = β 3. Initial pressure of gas = 4. Initial (equilibrium) length of gas = 5. Mass of piston = 6. Spring constant = 7. Cross-sectional area of piston =

The sum of all forces acting on the piston are:

The pressure as a function of position is determined by the ideal gas law, under isothermal conditions

.

Therefore, the differential equation describing the oscillations of the piston is:

.

In the limit that , this can be accurately approximated as

.

The solution to this differential equation governs the motion of the piston, as shown in the graphics of this Demonstration.

## Permanent Citation