A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of gas. The cylinder is kept with its axis horizontal. If the piston is disturbed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will be

detailed solution

Correct option is A

Let the piston be displaced through distance x towards left, then volume decreases, pressure increases. If ΔP  is increase in pressure and ΔV  is decrease in volume, then considering the process to take place gradually (i.e. isothermal)P1V1=P2V2⇒PV=(P+ΔP)(V−ΔV)   ⇒PV=PV+ΔPV−PΔV−ΔPΔV  ⇒ΔP.V−P.ΔV=0   (neglecting  ΔP.ΔV)ΔP(Ah)=P(Ax)⇒ΔP=P.xh    This excess pressure is responsible for providing the restoring force (F) to the piston of mass M. Hence  F=ΔP.A=PAxh Comparing it with  |F|=kx⇒k=Mω2=PAh  ⇒ω=PAMh⇒T=2πMhPA Short trick : by checking the options dimensionally. Option (a) is correct.