An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M$. The piston and the cylinder have equal cross sectional area $A$. When the piston is in equilibrium, the volume of the gas is $V_0$ and its pressure is $P_ 0$. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency
JEE MAIN 2013, Diffcult
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${\frac{M g}{A}=P_{0}}$ $ {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 $SHM.$
$f=\frac{1}{2 \pi} \sqrt{\frac{\gamma P_{0} A}{x_{0} M}}=\frac{1}{2 \pi} \sqrt{\frac{\gamma P_{0} A^{2}}{M V_{0}}}$
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