c
\({\frac{M g}{A}=P_{0}}\)                            \(  {P_{0} V_{0}^{\gamma}=P V^{\gamma}}\)

\({\mathrm{Mg}=\mathrm{P}_{0} \mathrm{A}}{\ldots(1)}\)          \(  {P_{0} A x_{0}^{\gamma}=P A\left(x_{0}-x\right)^{\gamma}}\)

                                          \(P=\frac{P_{0} x_{0}^{\gamma}}{\left(x_{0}-x\right)^{\gamma}}\)

Let piston is displaced by distance \(x\)

\(M g-\left(\frac{P_{0} x_{0}^{\gamma}}{\left(x_{0}-x\right)^{\gamma}}\right) A=F_{\text {restoring }}\)

\(P_{0} A\left(1-\frac{x_{0}^{\gamma}}{\left(x_{0}-x\right)^{\gamma}}\right)=F_{\text {restoring }} \quad\left[x_{0}-x \approx x_{0}\right]\)

\(F=-\frac{\gamma P_{0} A x}{x_{0}}\)

Frequency with which piston executes \(SHM.\)

\(f=\frac{1}{2 \pi} \sqrt{\frac{\gamma P_{0} A}{x_{0} M}}=\frac{1}{2 \pi} \sqrt{\frac{\gamma P_{0} A^{2}}{M V_{0}}}\)