Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 14A (Classical Dynamics)

Let $r$ be the cylindrical radius of the point of contact and $\varphi$ its azimuth. Rolling without slip or twist gives $|\omega |=v/a$ and thereforeup to an irrelevant additive constant. The factor $7/5$ combines translational and rotational inertia.

Time and azimuthal invariance give energyand angular momentum $J=(7m/5)r^2\dot{\varphi }$. They represent mechanical energy and vertical angular momentum.

The radial effective potential is $V_{\mathrm{eff}}=mgr\cot \alpha +5J^2/(14m{r}^2)$. Initially $r=8a$, $J=(7m/5)(8a)v_0$. Comparing the initial energy with $V_{\mathrm{eff}}(2a)$ giveswhich is exactly $v_0^2\ge (4/7)ga\cot \alpha$. Larger speed raises the centrifugal barrier at the neck faster than it raises the initial kinetic energy.

A holonomic constraint is $F(q,t)=0$. In the multiplier formulation its generalized constraint force is $Q_i=\lambda {\partial }_{i}F$. Along an allowed motion, its power is ${\sum }_i{Q}_i{\dot{q}}_i=\lambda (\dot{F}-{\partial }_{t}F)$, which vanishes when the constraint has no explicit time dependence.

At the circular minimum, $v^2=(5/7)gr\cot \alpha$ and $r=8a$. Projecting the centre-of-mass equation on the cone normal givesThis is the magnitude of the normal constraint force.