A fixed thermally conducting cylinder has a radius R and height `L_0`. The cylinder is open at its 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_(0)piR^(2))/(piR^(2)P_(0)+Mg))(2L)`
B. `((P_(0)piR^(2)-Mg)/(piR^(2)P_(0)))(2L)`
C. `((P_0piR^2+Mg)/(piR^2P_0))(2L)`
D. `((P_(0)piR^(2))/(piR^(2)P_(0)-Mg))(2L)`

Answer

D

Explanation

The condition for equlibrium of the piston is
`Mg=(P_(0)-p)piR^(2)`

`p=(-Mg)/(piR^(2))+P_(0)`
Since the cylinder is thermally conducting the temperature remain the same.
Therefore `P_(0)xx(2LxxpiR^(2))=pyxxpiR^(2)`
`y=(P_(0))/xx(2L)=(P_(0))/([P_(0)=(Mg)/(pir^(2))])xx2L`
`=(P_(0)xxpir^(2))/(p_(0)piR^(2)-Mg)xx2L`