A simple problem just for fun

[wrobel]

A uniform massive cylinder is free to rotate about its fixed horizontal axis. A uniform disk of smaller radius rolls without slipping along the inner surface of the cylinder. The system is in the uniform gravitational field g. Prove that any motion of the system starting from rest is periodic or an equilibrium.

 

[.Scott]

So the fixed horizontal axis supports the outer cylinder against the "uniform gravitational field". The inner cylinder must be supported by a "sticky" contact with the large cylinder?
The system is put into motion by pushing the outer cylinder, the inner cylinder, or both?

It seems to me that the gravitational field serves no real purpose. As long as there is a way to "fix" the axis, the experiment could be done in free fall.

 

[Baluncore]

A uniform disk of smaller radius rolls without slipping along the inner surface of the cylinder.

Does the disk have a finite thickness, volume, mass and inertia?

Is the disk constrained to remain in a plane orthogonal to the cylinder axis, or can it fall onto one side and so stop rolling, then just flip heads and tails?