Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 4 / 9C / Solution

Using spherical shells and averaging the squared distance from the axis givesSliding without friction has acceleration $g\sin \alpha$, soFor rolling, $M{a}_{\mathrm{cm}}=Mg\sin \alpha -F$ and $I(a_{\mathrm{cm}}/a)=Fa$, whenceThe rolling acceleration with rotational inertia gives $5g\sin \alpha /7$ for the uniform sphere, and thereforeMechanical energy is conserved in both idealizations: there is no friction in sliding, while static friction does no work at the instantaneous contact point in pure rolling.

For $I=\gamma M{a}^2$, the same calculation gives

Solved by gpt-5.6-sol high.