Thermodynamics and simple harmonic motion

[vladimir69]

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

 

[ehild]

The denominator is equal to [tex]
nRT_0 (1-x/L)
[/tex]

It is assumed that the piston is only slightly displaced from equilibrium, so that x/L<<1. That means that its second or higher power can be neglected with respect to it.
We often apply this method in Physics when working with small quantities: use Taylor expansion and keep linear terms.
Your expression of force can be written in terms of (x/L),

[tex]F=AP_0(1-\frac{1}{1-x/L})
[/tex]

Expanding the the fraction with respect to x/L :

[tex]F=AP_0(1-(1+x/L+(x/L)^2+(x/L)^3+...))

[/tex]

Omitting all terms but linear, we get:

[tex]F=-AP_0(x/L)

[/tex]

ehild

 

[vladimir69]

ok
thanks for that