How do I proof:

Textbook solution :

1 vote

I think the question is stating that the situation shown is a position of equilibrium for the centre of the small sphere. Which simply means that the resultant force on the small sphere is zero. However, the resultant torque about the CM is not zero. We must also assume that there is no slipping between sphere and cylinder, so the sphere accelerates about its COM.

Please explain the solution given by them: https://pasteboard.co/HXsICHVJ.png

I do not understand that solution either. Have you tried solving the problem yourself using my suggestion above?

No, but I derived that for a ball in the pure rotation along incline with inclination $\theta$, possess acceleration along incline given by $a=\dfrac{F/m}{1+\frac{k^2}{R^2}}$, where $F$ is a force along incline acting at CM (excluding frictional), $k$ is its radius of gyration. Using this I understand that $g\sin\theta$ term but why $\alpha r$?

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