Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 4 / 9B / Solution

Let the full cube mass be $M=8\rho a^3$ and the removed cylinder mass $m_h=2\pi \rho a{r}^2$. About the rod, the remaining moment isThe remaining centre-of-mass distance $\ell$ obeys $(M-m_h)\ell =M\gamma a$, so the small-angle equation is $I\ddot{\theta }+Mg\gamma a\theta =0$. Hence