Couples and angular acceleration

In just the same way that a body's linear velocity can be changed by the application of a force for a time t its angular velocity can be changed by a couple (Fd) acting on it for a time t.

If a couple is applied for a time t as shown in the diagram, the angular velocity of the body will be changed from w_o to w₁ this means the body experiences an angular acceleration (a).

Suppose that the body rotates through an angle q during the acceleration. We can then write down three equations for the motion:
q = w_ot + ½ at²
w₁² = w_o² + 2aq
w = [w₁ – w_o]/t

The work done by the couple is given by the equation:


where r is the distance of one of the forces from the axis of rotation and n is the number of rotations. The angle through which the body has rotated during the acceleration is 2pn.

If a continuous input of energy is required to maintain a constant angular velocity w against a frictional couple T then:


So applying a torque to a body of a certain moment of inertia will change its angular acceleration. The torque T, the moment of inertia I and the angular acceleration a are related by the equation: