Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 11C (Dynamics and Relativity)

The moment of inertia about an axis iswhere $r_{\perp }$ is the perpendicular distance to the axis. The disc mass isTo leading order in $\delta /r$, the distance from the $x$-axis is $|y|$. Using polar coordinates in the disc,

The axis $y=0,z=h$ is parallel to the $x$-axis and lies a distance $|h|$ from it. The parallel axis theorem therefore givesto leading order in the thickness.

Measure $z$ from the apex along the cone's symmetry axis. A thin disc at height $z$ has radiusand mass $dM=\rho \pi s^{2}dz$. Its moment about a parallel diameter through its centre is $s^{2}dM/4$, while its centre is distance $z$ from the required axis. Part (a)(ii) givesHenceSince $M=\rho \pi R^{2}H/3$,

Without friction, rotational kinetic energy is conserved:The constant angular speed is $\omega =\sqrt{2K_0/I}$, so one full rotation takes

The work law means that friction exerts a constant opposing torque of magnitude $W$. Until the cone stops,Its initial angular speed is ${\omega }_0=\sqrt{2K_0/I}$, and uniform angular deceleration givesTherefore