A 10 kg slender rod of length $L=2$ m is released from rest from a horizontal position. It rotates about a pin at one end $O$. Find its angular velocity when it is vertical.

A solid sphere of mass $m=5$ kg and radius $r=0.2$ m is released from rest at the top of an incline of height $h=2$ m. It rolls without slipping. Determine the velocity of its center of mass at the bottom of the incline.

A constant couple moment $M = 50 \text{ N}\cdot\text{m}$ is applied to a flywheel of mass $m=20 \text{ kg}$ and radius of gyration $k=0.3 \text{ m}$. The flywheel starts from rest. Using the principle of work and energy, determine the angular velocity of the flywheel after it has completed 5 full revolutions.

Disc brakes on a car utilize friction pads to slow down the rotation of a wheel hub. Analyze the operation of disc brakes using the work-energy principle for a rigid body. Discuss the transformation of kinetic energy and the thermal implications of non-conservative work during emergency braking.

Flywheel energy storage systems (FESS) are used to store energy electrically by converting it into rotational kinetic energy. They are often used for grid load-leveling or uninterrupted power supplies (UPS). Analyze the design parameters that dictate a flywheel's energy capacity and the primary limitations based on rigid body kinetics and material strength.