Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 4 / 9A (Dynamics and Relativity)

For rolling without slipping, $\dot{s}=a\Omega$. Conservation of energy givesand differentiation yields

For a sphere with density proportional to $r^c$,so $I/(M{a}^2)=2(c+3)/(3(c+5))$. A uniform disc has $I/(M{a}^2)=1/2$. Equal launch speeds give equal return times exactly when the accelerations, and hence these ratios, agree. Solving gives $c=3$.

Let $M$ be the mass of a complete uniform sphere of radius $a$. The intact marble has $I_1=2M{a}^2/5$. Each removed bubble has mass $M/8$, radius $a/2$, and centre a distance $a/2$ from the rolling axis. By the parallel axis theorem, each removesThus $M_2=3M/4$, $I_2=2M{a}^2/5-7M{a}^2/80=5M{a}^2/16$, and $I_2/(M_2{a}^2)=5/12$. The two accelerations are therefore $(5/7)g\sin \alpha$ and $(12/17)g\sin \alpha$. From rest, distance is proportional to acceleration, so