Difficult problem in Rotational Mechanics JEE 2017 Comprehension paper

AI Thread Summary
The discussion revolves around a challenging problem from the JEE Advanced 2017 exam related to rotational mechanics, specifically calculating the kinetic energy of a ring. The no-slip condition is applied to relate the angular velocity and linear velocity, leading to the formulation of equations involving normal force and friction. The user expresses confusion over the role of the instantaneous axis of rotation (IAOR) and questions why it is assumed to be at the center of the small ring, despite their derived equations not yielding clarity. There is a desire for a deeper understanding of the physics involved, particularly regarding the relationship between the ring's center motion and its rotation. The inquiry highlights the complexity of applying theoretical concepts to solve practical problems in rotational dynamics.
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Homework Statement
One twirls a circular ring (of mass M and radius R) near the tip of one's finger as shown in Figure 1 In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity omega_o. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure-2). The coefficient of friction between the ring and the finger is u and the acceleration due to gravity is g
Relevant Equations
its down
This question is from the Jee Advanced 2017. This question stumped me because it was very difficult to do, the first part of the comprehension was asking for the kinetic energy of the ring.

This is what I mean.
Using the no-slip condition we can write ##\omega R-v_c=\omega_0r##. This is the no-slip condition. the relative velocity between the finger and the point of contact of the ring is ##0##. I assumed the angular velocity vector to be pointing out of the page and the finger to be moving in the counterclockwise direction, this is obvious through intuition.

Now if you draw fbd of the ring, you can come up with the equation ##N=\frac{mv^2}{x}##. Where ##N## is the normal force and ##x## is the distance between the center of mass and the IAOR(instantaneous axis of Rotation). the other equation is ##f=mg## and ##f=\mu N## where ##f## is the frictional force.

I don't know what to do with these how do they help solve the questions. This is the actual physics way to do it. All the other solutions on youtube and else where just assume by intuition(not obvious though) that the IAOR is at the center of the small circle, which basically sovles the question. Now I want proof of that, why is IAOR of this system the center of small ring. I came up with equations as shown as above but for some reason they aren't helping why?

I used the defining feature of the system to create equations, what did I miss.
Screenshot 2024-01-26 212303.png
 
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I presume ##v_c## is the instantaneous velocity of the centre of the ring. I find it more natural to express the ring's motion as the sum of a circular motion of its centre and its motion, rate ##\omega##, about that centre.
What are the radius and rate of the former and the radius of the latter?
 
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