A cylinder rotating on a plane with friction and then moving across a frictionless plane to a collision

[Jason Ko]

Homework Statement

A cylinder of radius R, mass M and moment of inertia I
about its symmetry axis starts at x = -d. It travels with initial speed Vo towards the
positive x direction, and rotates with angular velocity Vo /R clockwise, as shown in the
figure. For x < 0, the friction is non-zero. For 0 < x < d, the surface is frictionless.
Assume that the cylinder makes an elastic collision with a frictionless immovable wall at
x = d. Find the final translational velocity (magnitude and direction, when it is no longer
changing) of the center of mass of the cylinder, in terms of R, M, I and Vo. Assume that
surface extends to negative infinity.

Relevant Equations

V=rω

I think the angular velocity keep increasing on the plane with friction and the translational velocity keep decreasing due to friction while the total kinetic energy is conserved. When it moves to the frictionless plane, all energy converts to translational kinetic energy and it stop rolling. When it collides with the wall, all energy goes to the cylinder itself since the wall is unmovable. After it moves to the plane with friction, its translational velocity decreases till 0. I know something’s wrong there but where exactly?

 

[haruspex]

I think the angular velocity keep increasing on the plane with friction and the translational velocity keep decreasing due to friction

Why?
Remember, friction acts to oppose relative motion of surfaces in contact.
What surfaces in contact are moving relatively to each other?

When it moves to the frictionless plane, all energy converts to translational kinetic energy and it stop rolling.

Why? What torque stops the rotation?