Rolling without slipping on a curved surface

[NTesla]

I am saying that you cannot apply it to find the moment of inertia of the ball's motion about C.

Yes, i agree that that is what you were saying. I'm not sure why you thought I was disagreeing with you on this point. In the statement: "He is saying that parallel axis theorem can't be applied as you and I have done so far." "as" is the operative word. Meaning that you are saying that PAT can't be applied "as" I and kuruman had applied to calculate the MoI about point C.

The effective MoI, what I called its virtual MoI in post #46, about C results from the kinematic relationship between the motion of the sphere’s centre and the motion of the rest of the sphere about that centre. The MoI found by the parallel axis theorem assumes a particular kinematic relationship, namely, that the sphere moves as though it is fixed to a pendulum pivoted at C. Since that is not how it moves here, the PAT gives the wrong answer.

I agree and have understood this part already. But when I'm urging on to try to figure out a way to find the MoI about point C, which takes into account the unusual circumstance of the sphere's movement in the original question, you said that it's pointless. My disagreement is on that point. I don't think it's pointless.

I've reversed engineered the moment of Inertia of the sphere about point C. It's coming to be = $\frac{2}{5}mr^{2} + m\left (R-r\right)^{2} - \frac{2}{5}mrR$. The first two terms are expected and are easy to understand. However, the calculative origin of the 3rd term needs to be understood.

Humour me: calculate the relationship between the angular velocity of the ball's centre about C and the angular momentum of the ball about C, as described in post #28. We can then see if this gives the book answer.

Maybe it will give the book's answer, maybe it won't. There could be and are more than 1 way to reach the answer. My need is to understand how to reach answer using my own approach, and not to abandon it, in favour of other methods. I'll give time to solve it using your suggested approach also, once I understand and solve the question using the approach that I've been on.

 

[haruspex]

My need is to understand how to reach answer using my own approach

But I am arguing that you cannot arrive at it using your approach rather than mine because the steps you have to take to find the effective MoI involve my approach.

 

[haruspex]

I've reversed engineered the moment of Inertia of the sphere about point C. It's coming to be = $\frac{2}{5}mr^{2} + m\left (R-r\right)^{2} - \frac{2}{5}mrR$.

I get the same. Curiously, that gives 0 for $5R=7r$.