In the arrangement shown in Fig. gas is thermally insulated. An ideal gas is filled in the cylinder having pressure `P_(0)` (gt atmospheric pressure `P_(0)`). The spring of force constant `K` is initially unstretched. The piston of mass `m` and area `S` is frictionless. In equilibrium, the piston rises up by distance `x_(0)`, then

Options

A. final pressure of the gas is `P_(0)+(Kx_(0))/(S)+(mg)/( S)`
B. work done by the gas is `(1)/(2) Kx_(0)^(2)+mgx_(0)`
C. decrease in internal energy of the gas is `(1)/(2) Kx_(0)^(2)+mgx_(0)+ P_(0)Sx_(0)`
D. all of the above

Answer

A, C

Explanation

Equilibrium of pistone gives
`PS=P_(a)S+ mg+ Kx_(0)`
`P=P_(a)+(mg)/(S)+(Kx_(0))/(S)`
`(P=` final pressure of gas `)`
Work done by the gas
`=` work done against atmospheric pressure `+` elastic potential energy stored in the spring `+` increase in gravitational potential energy of the piston
`=P_(a)DeltaV+(1)/(2) Kx_(0)^(2)+mgx_(0)=PSx_(0)+(1)/(2)Kx_(0)^(2)+mgx_(0)`
There occurs decrease in internal energy of the gas, because the gas is thermally insulated and hence, work is done at the expense of internal energy of the gas.