Question

An ideal gas is contained in a piston cylinder arrangement as shown in the figure. The area of the piston is $A$ and its mass is $m$. The surrounding atmospheric pressure is $P_0$. Under equilibrium condition, the volume of the gas is $v_0$. Find the angular frequency of small vertical oscillations of the piston assuming the expansion and compression of the gas to be adiabatic (adiabatic constant $=\gamma$)

Answer 1

Adaibatic process

$P{V}^r=const$

$\frac{dp}{p}\ne \frac{d{V}_0}{V_0}=0$

$dp=\frac{-ph}{V_0}=dV$

$F=\left(dP\right)A=\frac{P_{0}V{A}^2}{V_0}x$

$a=\frac{-P_{0}r{A}^2}{m{V}_0}x$

$\omega =\sqrt{\frac{P_{0}V{A}^2}{m{V}_0}}$.