14.11:
Principle of Angular Impulse and Momentum: Problem Solving
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.
14.11:
Principle of Angular Impulse and Momentum: Problem Solving
Consider a ball of mass m, attached to a massless rod of known length, subjected to a time-dependent torque. If the initial velocity of the mass is known, then the final velocity of the mass for time t can be determined using the principle of angular impulse and momentum.
Initially, a free-body diagram of the system is drawn to illustrate all the forces acting upon the system, providing a crucial understanding of the dynamics at play. Then, the principle of angular impulse and momentum is applied to the system. This principle dictates that the initial angular momentum of an object, added to the sum of all angular impulses exerted on it over a specific period, equates to its final angular momentum.
As a result, the initial and final angular momenta of the sphere are determined by multiplying the sphere's mass, the moment arm (the perpendicular distance from the axis of rotation to the line of action of the force), and the initial and final velocities, respectively.
Next, the angular impulse is calculated by taking the integral of the torque over the given time interval. The integral can be solved by substituting the limits of the time interval into the equation. Finally, all necessary values are substituted into the initial and final angular momenta equations. By rearranging these equations, the final velocity of the sphere can be calculated.