14.11 : 角冲量和动量原理：解决问题

A heavy ball with a known mass is attached to a rod of known length and negligible mass. The system is subjected to a particular torque. Knowing the torque expression and the initial velocity, determine the speed of the ball when t equals 2 seconds.

Initially, a free-body diagram of the system is drawn, and to solve, the angular impulse and momentum principle is recalled.

The principle states that the initial angular momentum of the particle, added to the sum of all the angular impulses applied to the particle during a specific period, equals the final angular momentum of the particle.

The initial and final momenta are calculated as the product of the ball's mass, the moment arm, and the initial and final velocities, respectively.

While the angular impulse is the integral of the torque over time, the integration is solved by substituting the limits.

By substituting the known quantities into the initial and final momentum equations and rearranging them, the velocity of the ball can be calculated.