Values of Velocity at which the body will move without bouncing

[sodaboy7]

Homework Statement

A small point mass is fixed to the inside of a thin rigid ring(hoop) of radius R and mass equal to that of the point mass. The hoop rolls without slipping over on a horizontal plane; at the moments when the point mass gets into lower position, the center of the hoop moves with velocity V₀. At what values of V₀ the hoop will move without bouncing ?

2. The attempt at a solution

As the point mass m attached to the hoop moves up it gains potential energy. Since total energy must be conserved, the kinetic energy of the (hoop+point mass system) reduces and thus the rotating hoop slows down. the hoop will have its maximum tendency to jump off the ground when the point mass is located at the highest point due to centrifugal force, that's all I can think about.

 

[Spinnor]

Does this make sense?

Edit, in my sketch my v = 2V_o

 

[brettjor]

In a perfect world, no bouncing, just diminished G's. In the real world, imperfections in the hoop and "road" surface could easily create bouncing. Isn't this why we balance rotational bodies?

 

[sodaboy7]

Does this make sense?

Edit, in my sketch my v = 2V_o

It does make much sense but the answer is √(8gR). how?
I found its solution on following thread:
http://irodovsolutionsmechanics.blogspot.com/2009_02_01_archive.html

but the solution is very fancy and i don't think its a proper one, take look at it.