$(25$ pts $)$ A flywheel rotates without friction at an angular velocity ${}_0=500re\frac{v}{m}in$ on a frictionless, vertical shaft of negligible rotational inertia. A second flywheel, which is at rest and has a moment of inertia four times that of the rotating flywheel, is dropped onto it$.$ Because friction exists between the surfaces, the flywheels very quickly reach the same rotational velocity, after which they spin together. $($a$)$ Use the law of conservation of angular momentum to determine the angular velocity ${}_i$ of the combination. $($b$)$ What fraction of the initial kinetic energy is lost in the coupling of the flywheels?