You are correct. I was confused on *exactly* how the string will apply torque. I now understand that the string won't pass through COM of sphere when extended and will be making some angle with the contact point so as to provide net torque. So my methods 1 and 3 are wrong and second one is...
As shown in figure there's a homogeneous solid sphere. It is rotating about axis which is passing through point P directed perpendicular to the plane of paper. (In short like a pendulum).
I'm neglecting gravity and assuming a force F which is directed perpendicular to the string. (The string...