A piston travels between two stops and is restrained by a linear spring as shown. The cross$-$

sectional area of the piston is $0\mathrm{.}05$ m $2$ and the initial volume of the gas in the cylinder is

$0\mathrm{.}01$ m $3.$ At the top stops, the volume is $0\mathrm{.}03$ m $3.$ The spring constant is $50\mathrm{.}0$ kN$/$m$.$ The

gas inside the cylinder is initially at $0\mathrm{.}50$ MPa and the pressure required to just raise the

piston from the stops is $1\mathrm{.}00$ MPa. The final gas pressure in the cylinder is $2\mathrm{.}00$ MPa.

$($a$)$ Draw a P$-$V diagram showing the process.

$($b$)$ Calculate the work done by the gas, in kJ$.$