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
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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.
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