Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 4 / 11B / Solution

For axis unit vector $\hat{\omega }$,For a body, replace the sum by $I=\int \rho (x)r_{\perp }^{2}dV$.

For the hollow ball,Energy conservation and $v=R_{+}\Omega$ giveThe factor $\beta$ increases with $\mu$, so if ${\mu }_A>{\mu }_B$, ball B accelerates more and arrives first. Moving the cavity off centre removes material with a larger lever arm and lowers the moment of inertia about the centre of mass; despite the resulting wobble, ball C therefore rolls down faster than A.