Ring rolling inside a cone

1. The problem statement, all variables and given/known data
(See attachment)

Assume that the surface has friction and a small ring of radius ##r## rolls on the surface without slipping.

Assume conditions have been set up so that (1) point of contact between the ring and the cone moves in a circle at height ##h## above the tip and (2) the plane of ring is at all times perpendicular to the line joining the point of contact and the tip of the cone.

What is the frequency of this circular motion?

You may work in the approximation where ##r## is much smaller than radius of circular motion.

2. Relevant equations

3. The attempt at a solution
The attachment 2 shows the side view snapshot of the motion.

The CM of ring rotates in a circle of radius ##r_e=h\tan\theta-r\cos\theta##. The force acting on it are shown in the attachment. The frictional force can be resolved into two components. One along the slant height of cone (f) and the other along the tangent to the point of contact (not shown).

Now I am confused at this point. If I take torque about the CM of ring, the only contributing torque is due to the two components of friction mentioned above. The torque due to friction along the slant height would tend to rotate the ring and not allow the circular motion to take place, doe this mean I have to assume that component to be zero? But doing that doesn’t make sense to me. :confused:

Any help is appreciated. Thanks!

Attached Images
File Type: png ringincone.png (143.3 KB)
File Type: png ringincone1.png (7.2 KB)


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