4.4 Rotational Kinematics

As we study rotational kinematics, we will use the example of a rigid body, one in which all parts rotate at the same speed (the angle through which all parts of the body rotate is equal for each unit of time). Figure 4.6 represents such a body.

At a point in time, t_i, one element of mass, represented by m_i in Figure 4.6, is located some angle q from a reference line. The body rotates and, after some amount of time, dt, the mass is located at an angle dq from its position at time t_i. The distance traveled, ds_i, is equal to the product of the radius of its circular path of travel, r_i, and dq , which is also equal to the product of the linear speed of m_i, v_i, and the elapsed time, dt.

The angular displacement, dq , is equal to the quotient of the distance traveled (an arc length), ds_i, and the radius, r_i.

Thus, the angular velocity (or rotational speed) of the body, represented by w (omega), is

The units of w are radians per second (rad/s). Since we are dealing with a rigid body, w has the same value for all parts of the body. Angular velocity is considered positive if rotation occurs in a counterclockwise direction and negative if rotation occurs clockwise.

Angular acceleration, a (alpha), is change in angular velocity per unit time:

The units of angular acceleration are radians per second per second, or radians per second squared (rad/s²).

As was the case with translational motion, angular velocity is equal to the sum of initial angular velocity and the product of angular acceleration and time:

Integrating once, we find that position is equal to the sum of initial position, the product of initial velocity and time, and one half the product of acceleration and time squared:

The analog between translation and rotation has now been defined. We can complete it with the equation for the square of angular velocity:

which is derived from Equations 4.8 and 4.9.