Question

A small body a is fixed to the inner side of a thin rigid hoop at what values will the hoop move without bouncing

Answer 1

Clearly, the tendency of bouncing of the hoop will be maximum when the small body A, will be at the highest point of the hoop during its rolling motion. Let the velocity of C.M. of the hoop equal v at this position. The static friction does no work on the hoop, so from conservation of mechanical energy;

E1 = E2

please refer attachments Figure 1

As the hoop has no acceleration in vertical direction, so for the hoop

please refer attachments Figure 2