Problem $8$: A cylinder climbs a step $(8$ pts$)$

A uniform cylinder of mass $\backslash ($ M $\backslash )$ and radius $\backslash ($ R $\backslash )$ initially at rest is released from a height $\backslash ($ h $\backslash )$ and rolls without slipping down a guide $(\backslash ($ h $\backslash )$ is the height of the center of the cylinder relative to the bottom of the guide$).$ In the horizontal section of the guide, the cylinder hits a step of height $\backslash ($ d $\backslash )($less than $\backslash ($ R $\backslash )).$ The situation is illustrated above.

$($a$)(2$ pt$)$ Determine the angular velocity and velocity of the center of mass of the cylinder right before the collision. You can assume pure rolling motion: the change in potential energy of the cylinder between its initial position and the point of collision is converted into the total kinetic energy of the cylinder.

$($b$)(2$ pt$)$ Assume that the cylinder does not bounce back but instead climbs the step, having the point of contact $\backslash ($ P $\backslash )$ between the cylinder and the step $(\backslash ($ P $\backslash )$ is always stationary$).$ Write down the total angular momentum of the system immediately before the collision with respect to the point of contact $\backslash ($ P $\backslash ).$

$($c$)(2$ pt$)$ After the collision and during the climb the ball is purely rotating around the fixed point $\backslash ($ P $\backslash ).$ Find the angular velocity of the system around $\backslash ($ P $\backslash )$ during the climb.

$($d$)(2$ pt$)$ Determine the minimum value for $\backslash ($ h $\backslash )$ required for the cylinder to climb the step.