A motor drives a flywheel with an initial angular velocity of $20 \, \text{rad/s}$. When power is cut, it experiences a constant angular deceleration of $-2 \, \text{rad/s}^2$. Determine the time required to stop and the number of revolutions it makes before stopping.

A $10 \, \text{m}$ ladder rests against a vertical wall and a horizontal floor. The bottom of the ladder is pulled away from the wall at $2 \, \text{m/s}$. At the instant the ladder is $6 \, \text{m}$ from the wall, determine the velocity of the top of the ladder using the IC method.

A four-bar linkage mechanism consists of a crank AB ($L_1 = 0.2 \text{ m}$), a connecting rod BC ($L_2 = 0.5 \text{ m}$), and a rocker CD ($L_3 = 0.4 \text{ m}$). At a given instant, crank AB has an angular velocity of $\omega_{AB} = 10 \text{ rad/s}$ (counter-clockwise) and an angular acceleration of $\alpha_{AB} = -5 \text{ rad/s}^2$ (clockwise). Analyze the relative velocity and relative acceleration of point B with respect to point A.

A planetary gear train consists of a central "sun" gear (fixed), a rotating "carrier" arm, and one or more "planet" gears that mesh with the sun gear and revolve around it. Analyze the complex general plane motion of a planet gear as the carrier arm rotates at a constant angular velocity $\omega_c$, and identify the Instantaneous Center of Zero Velocity (IC) for the planet gear.

A common problem involves a ladder sliding down a vertical wall and across a horizontal floor. The top of the ladder moves downwards, and the bottom moves outwards. Analyze the velocity of the midpoint of the ladder as the top approaches the floor. What is the trajectory of the midpoint, and what happens to its velocity vector at the instant the ladder goes flat?