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. Pressure inside the gas in equilibrium is P. Consider the piston to be diathermic so that process is isothermal the period of oscillation will be:

Options

A. `T=2pisqrt(((Mh)/(PA)))`
B. `T=2pisqrt(((MA)/(Ph)))`
C. `T=2pisqrt(((M)/(PAh)))`
D. `T=2pisqrt(MPhA)`

Answer

A

Explanation

pV=constant
`Rightarrow pdV+Vdp=0 Rightarrow dp=-(pdV)/(V)=-(Pax)/(Ah)=-(Px)/(h)`
`Rightarrow x=`Dosplacement from Mean position `Rightarrow F=(Pax)/(h), Rightarrow M(d^(2)x)/(dt^(2))=-(PA)/(h)x` `Rightarrow (d^(2)x)/(dt^(2)=-(PA)/(Mh)x`, `Rightarrow omega=sqrt((PA)/(Mh))`,`Rightarrow T=2pisqrt((Mh)/(PA))`