 Solution

Let $t=0$ be the start of deposition and $I_0=M{R}^2/2$ be the original moment of inertia of a uniform disc. Assume that all deposited sand sticks and remains at radius $r$, with negligible thickness and no losses. The sand's incoming angular momentum about the vertical axis is zero. Neither the bearing nor gravity supplies an axial torque, so conservation of angular momentum givesConsequently the angular velocity and angular acceleration areThe rotational kinetic energy isThus, for $\alpha r^2>0$,This is mass-loaded rotation: the sticking process dissipates mechanical energy even though angular momentum is conserved. Indeed $\dot{K}=-\alpha r^2{\omega }^2/2$. If $r=0$ or $\alpha =0$, no additional axial moment of inertia is deposited and the nonzero initial rotational kinetic energy never halves.