Question

An ideal gas having adiabatic index $\gamma$, enclosed in a vertical cylindrical container supports a freely moving piston of mass $M$. The piston and the cylinder have almost 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 and the process is adiabatic, the piston executes a simple harmonic motion with frequency

• A. $\frac{1}{2\pi }\frac{A\gamma p_0}{V_{0}M}$
• B. $\frac{1}{2\pi }\frac{V_{0}M{p}_0}{A^2\gamma }$
• C. $\frac{1}{2\pi }\sqrt{\frac{A^2\gamma p_0}{M{V}_0}}$
• D. $\frac{1}{2\pi }\sqrt{\frac{M{V}_0}{A\gamma p_0}}$

Answer 1

The correct option is C $\frac{1}{2\pi }\sqrt{\frac{A^2\gamma p_0}{M{V}_0}}$

At the equilbrium position of piston, pressure of gas in the cylinder is $p_0$, cross -sectional area of piston is $A$


$\Rightarrow F=Mg$

or $p_{0}A=Mg$

When the piston is displaced slightly by $dx$ downwards, a restoring force $dF$ will act on the piston in upward direction due to compression of gas.

On applying the bulk modulus relation for adiabatic process.

$\frac{-dp}{\left(\frac{dV}{V}\right)}=\gamma p$

At equilibrium position of piston

$dp=-\gamma \left(\frac{p_0}{V_0}\right)dV=-\gamma \left(\frac{p_0}{V_0}\right)Adx$

$\therefore$ Restoring force on piston is

$dF=\left(dp\right)A$

$\Rightarrow dF=-\gamma \left(\frac{p_0}{V_0}\right)Adx\times A$

or, $dF=-\left(\frac{\gamma p_0{A}^2}{V_0}\right)dx$

Comparing it with simple harmonic motion equation,

Since $dF\propto -dx$

$dF=-kdx$

Thus we get,

$k=\frac{\gamma p_0{A}^2}{V_0}$

Hence frequency of SHM will be

$f=\frac{1}{2\pi }\sqrt{\frac{k}{M}}$

$\therefore f=\frac{1}{2\pi }\sqrt{\frac{\gamma p_0{A}^2}{M{V}_0}}$

$\begin{array}{c}\text{Why this question}?\\ \text{It intends to challenge your basic understanding of SHM.}\\ \text{Tip: When the piston is displaced slightly, the displacement}\\ \text{will be very small and system would not get sufficient time}\\ \text{for heat exchange. Thus we can relate it with bulk modulus}\\ \text{for adiabatic process.}\end{array}$