d$)$ In class we did a problem involving a stationary disc dropping onto a rotating disc. The discs eventually reached the same angular velocity. This in the same problem, but with both discs initially rotating!

A cylinder with moment of inertia I$1$ rotates CCW $($Counter ClockWise$)$ with angular velocity $+$w$0$ about a frictionles vertical axle. A second cylinder, with moment of inertia I$2,$ initially rotating CW $($clockwise$)$ with angular velocity $-$w$0,$ drops onto the first cylinder $($See the figure below$).$ Since the surfaces are rough, the two cylinders eventually reach the same angular speed $.$

$($i$)$ Find an expression for $.$

$($ii$)$ Show that kinetic energy is lost in this situation, and calculate the ratio of the final to the initial kinetic energy. Your answer must NOT contain ${}_0$ or $.$