A fixed thermally conducting cylinder has a radius R and height `L_0`. The cylinder is open at ita bottom and has a small hole at its top. A piston of mass M is held at a distance L from the top surface, as shown in the figure. The atmospheric pressure is `P_0`.

While the piston is at a distance 2L from the top, the hole at the top is sealed. The piston is then released, to a position where it can stay in equilibrium. In this condition, the distance of the piston from the top is

Options

A. `((2P_0piR^2)/(piR^2P_0+Mg))(2L)`
B. `((P_0piR^2-Mg)/(piR^2P_0))(2L)`
C. `((P_0piR^2+Mg)/(piR^2P_0))(2L)`
D. `((P_0piR^2)/(piR^2P_0-Mg))(2L)`

Answer

D

Explanation

(d) The condition for equilibrium of the
piston is
`Mg=(P_0-p)piR^2`
`rArr p=(-Mg)/(piR^2)+P_0`

Since the cylinder is thermally conducting, the
temperature remains the same.
Therefore
`P_0xx(2LxxpiR^2)=pyxxpiR^2 rArr y=(P_0)/pxx2L`
`=(P_0)/[P_0-(Mg)/(piR^2)]xx2L=[(P_0xxpiR^2)/(P_0piR^2-Mg)]xx2L`