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 in distrubed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will (assume T = constant):

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))`