How Can I Calculate the Angular Speed of a Ring Rolling Without Slipping?

[TSny]

It seems that it is supposed that the ring remains in a horizontal plane. Strange

Yes, strange. I don't see how it could move in a truly horizontal plane.

 

[zwierz]

it would be a good task to write honest equations of stationary motion that is when an angle of ring's incline is constant and the speed of ring's center is constant and ring's center moves along a horizontal cirle