A 50 kg flywheel with radius of gyration $k_G = 0.4$ m is rotating at 20 rad/s. A brake is applied, creating a constant moment of $M = 20$ N$\cdot$m opposing the motion. How long does it take for the flywheel to stop?

A uniform slender rod of mass $M=4$ kg and length $L=1$ m hangs vertically from a smooth pivot at its upper end $O$. A bullet of mass $m=0.01$ kg traveling horizontally at $v=400$ m/s strikes the rod at its center ($L/2$ from pivot) and becomes embedded in it. Determine the angular velocity of the rod immediately after the impact.

A uniform rectangular plate of mass $m=5 \text{ kg}$ and dimensions $a=0.4 \text{ m}$ (width) and $b=0.6 \text{ m}$ (height) rests on a smooth horizontal surface. A horizontal impulse $\int F \, dt = 10 \text{ N}\cdot\text{s}$ is applied at the top right corner of the plate, perpendicular to the height $b$. Determine the initial linear velocity of the mass center and the initial angular velocity of the plate immediately after the impact.

During a dismount from the high bar or a springboard dive, a gymnast initiates a somersault in a "layout" (straight) position and then rapidly pulls their arms and legs into a "tuck" (cannonball) position to increase their spin rate. Before landing, they "open up" back to a layout position. Analyze this maneuver using the principle of conservation of angular momentum.