- #1

vladimir69

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## Homework Statement

A horizontal piston-cylinder system containing n mole of ideal gas is surrounded by air at temperature [itex]T_{0}[/itex] and pressure [itex]P_{0}[/itex]. If the piston is displaced slightly from equilibrium, show that it executes simple harmonic motion with angular frequency [itex]\omega=\frac{AP_{0}}{\sqrt{MnRT_{0}}}[/itex], where A and M are the piston area and mass, respectively. Assume the gas temperature remains constant.

## Homework Equations

PV=nRT

F=Ma

P=F/A

x=amount by which piston moves

L=length of cylinder

F = force on piston

## The Attempt at a Solution

[tex]V_{0}=AL[/tex]

[tex]V=A(L-x)[/tex]

[tex]F_{net}=A(P_{0}-P)[/tex]

[tex]L=\frac{nRT_{0}}{P_{0}A}[/tex]

[tex]P=\frac{ALP_{0}}{V}[/tex]

Popping all this into the mix gives

[tex]F_{net}=\frac{-P_{0}^2A^2x}{nRT_{0}-P_{0}Ax}[/tex]

[tex]M\frac{d^2x}{dt^2}=\frac{-P_{0}^2A^2x}{nRT_{0}-P_{0}Ax}[/tex]

If I get rid of the [itex]P_{0}Ax}[/itex] term then we arrive at the correct answer but not sure why one should throw away this term

Thanks