Question

A piston of mass $M_0$ can slide along axis of a cylinder of cross-section area 'A'. Cylinder is kept horizontal and contains n moles of monoatomic gas at pressure $P_0$. Assume piston is made of conducting material. When piston is slightly displaced it performs SHM with angular frequency $\omega$. Then mass of piston is

Answer 1

M_0=\frac{A^2P_0^2}{nRT,\omega^2}

Answer 2

1. For an isothermal process, $PV=nRT$ implies

   $$V_0=\frac{nRT}{P_0}.$$

2. A small displacement $x$ causes

   $$P\approx P_0\left(1-\frac{Ax}{V_0}\right).$$

3. Restoring force:

   $$F\approx -\frac{A^2P_0}{V_0}\,x.$$

4. With $\omega^2=\frac{k}{M_0}$, solve for $M_0$ to get:

   $$M_0=\frac{A^2P_0^2}{nRT\,\omega^2}.$$