In a damping system, a circular disk immersed in oil $($viscosity$,$ density$)$ is used as a damper with adjustable gaps a and b$.$

$($a$)$ Show that the damping torque, T is proportional to the angular rotation speed, $1,$ in accordance with the relationship $T=Cw.$ Hint: de ne the torque dT contributed by an in nitesimal element on the surface of the disk and integrated over the disk.

$($b$)$ Identify C in terms of given variables.

$($c$)$ Show that C is a minimum when $a=b$ for fixed total gap, $g=a+$