Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 4 / 11B (Dynamics and Relativity)

Differentiating a vector using $(dA/dt{)}_S=(dA/dt{)}_{S\prime }+\omega \times A$ twice yields
$m\ddot{x}=F-2m\omega \times \dot{x}-m\dot{\omega }\times x-m\omega \times (\omega \times x)$.

Along $BC$, write $x=(2a,s,0)$. With $\omega =-\alpha \hat{z}/t$, the tangential equation isA stationary point is $s_0=2a/\alpha$, lying on the side iff $\alpha \ge 1$. The general perturbed motion isExcept on the decaying-mode fine tuning, the growing term carries the bead to $B$ or $C$, according to its sign.