A piston of mass m divides a cylinder containing gas into two equal parts. Suppose the piston is displaced to the left to a distance x and let go Fig. Assuming the process to take place at a constant temperature, find the frequency of the piston's oscillations.

Answer

`omega_(0)^(iost)=sqrt(2pV//md^(2))`

Explanation

When the volume of a gas is changed at constant temperature we have, according to the Boyle-Maniotte Jaw
`p_(1)(d-x)S=p_(2)(d+x)S=pdS`
The force acting on the pistion is
`F=(p_(1)-p_(2))S=(2pxSd)/(d^(2)-x^(2))=(2pVx)/(d^(2)-x^(2))`
where V is the volume of one half of the vessel. As can be seen, the force does not conform to Hooke.s law, and the oscillations are not harmonic. But for small deflections of the piston (when `xltltd` ), the force will be quasi-elastic: `F=2pVx//d^(2)` and the oscillations of the piston will be harmonic. The rigidity of the system is`k=F//x=2pV//d^(2)`.