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.

 

[asteriatic]

I would like to first clarify that the term "bouncing" may not be the most appropriate term to use in this scenario. Instead, the phenomenon being described is the maximum velocity at which the hoop can move without losing contact with the ground. This can be better understood by considering the forces acting on the hoop and point mass system.

At the moment when the point mass is at its lowest position, the hoop is experiencing a downward force due to gravity and an upward force from the ground. This upward force must be equal in magnitude to the weight of the hoop and point mass in order for the hoop to remain in contact with the ground. This can be expressed as:

Fground = mg + ma = mg + mV0^2/R

where m is the mass of the hoop and point mass system, a is the acceleration of the hoop, V0 is the velocity of the center of the hoop, and R is the radius of the hoop.

In order for the hoop to remain in contact with the ground, the acceleration a must be zero. This means that the maximum velocity at which the hoop can move without losing contact with the ground is when the centripetal force (mV0^2/R) is equal to the weight of the hoop and point mass. This can be expressed as:

mg = mV0^2/R

Solving for V0, we get:

V0 = √(gR)

Therefore, the values of V0 at which the hoop will move without bouncing are all velocities less than or equal to √(gR). Any velocity greater than this will result in the hoop losing contact with the ground.