Problem $10\mathrm{.}81)$ The flywheel $($Disk $\backslash ($ A $\backslash ))$ is made of a lighter material than the clutch plate $($disk $\backslash ($ B $\backslash )),$ and the moment of inertia of $\backslash (\backslash$operatorname$\{$disk$\}$ A $\backslash )$ about the shaft is one$-$third that of disk $\backslash ($ B $\backslash ).$ The moment of inertia of the shaft is negligible. With the clutch disconnected, $\backslash ($ A $\backslash )$ is brought up to an angular speed $\backslash (\backslash$omega$\_\{$A$\}$ ; B $\backslash )$ is initially at rest. The accelerating torque is then removed from $\backslash ($ A $\backslash ),$ and $\backslash ($ A $\backslash )$ is coupled to $\backslash ($ B $\backslash )$ by pushing the disks together with forces acting along the axis, so as not to apply any torque on either disk. The disks rub against each other and eventually reach a common angular speed $($Ignore bearing friction.$)$

The design specifications allow for a maximum of $2400$ J of thermal energy to be developed when the connection is made.

$($a$)$ Find the common angular speed $\backslash (\backslash$omega $\backslash )$ after the coupling.

$($b$)$ What can be the maximum value of the original kinetic energy of $\backslash (\backslash$operatorname$\{$disk$\}$ A $\backslash )$ so as not to exceed the maximum allowed value of the thermal energy?