pts$.)$ A disk with radius $2R$ rotates about a vertical, frictionless axle with angular speed ${}_i.$ A second disk with radius $R,$ having the same density with the first one, is at rest. The second one drops onto the first disk. Because of the friction between the surfaces, the two eventually reach the same angular speed. Assume that the work done by rotational friction between the disks is negligible

$($a$)(10$ pts$.)$ Calculate the final angular speed $(f)$ of the two disk system.

$($b$)(10$ pts$.)$ Calculate the ratio of the final rotational kinetic energy to the initial rotational kinetic energy.