$10)$ In the operation of a single$-$cylinder internal combustion piston engine, one charge of fuel explodes to drive the piston outward in the power stroke. Part of its energy output is stored in a turning flywheel. This energy is then used to push the piston inward to compress the next charge of fuel and air. In this compression process, assume an original volume of V $[$L$]$ of diatomic ideal gas at atmospheric pressure is compressed adiabatically to V$\text{'}[$L$].($a$)$ Find the work input required to compress the gas. $($b$)$ Assume the flywheel is a solid disk of mass m $[$kg$]$ and radius r $[$m$],$ turning freely without friction between the power stroke and the compression stroke. How fast must the flywheel turn immediately after the power stroke? This situation represents the minimum angular speed at which the engine can operate without stalling.