Question

A vertical cylinder of cross-sectional
area S contains one mole of an ideal monatomic gas under a piston of mass M. At a
certain instant, a heater which transmits to
a gas an amount of heat q per unit time is
switched on under the piston.
Determine the established velocity v
of the piston under the condition that the
gas pressure under the piston is constant
and equal to $p_0$, and the gas under the piston is thermally insulated.

Answer 1

Gas pressure $=P_0+\frac{Mg}{S}$, where M is mass of the piston.
As $C_v=\frac{3}{2}R$
$\therefore \Delta U=\frac{3}{2}R\Delta T=\frac{3}{2}P\Delta V$
$Q=P\Delta V+\Delta U=P\Delta V+\frac{3}{2}P\Delta V=\frac{5}{2}P\Delta V$
$\Delta V=Sdx$
$Q=q.dt=\frac{5}{2}PSdx$
$\frac{dx}{dt}=\frac{2q}{5PS}=\frac{2q}{5(P_0+\frac{Mg}{S})S}$