Angular Motion

Angular motion is the motion of an object around a fixed axis point, such as a swinging pendulum or an orbiting planet. By convention, measurements of angular motion are considered positive when the motion is counter-clockwise and negative when the motion is clockwise.

Angular Displacement

Angular displacement ( $\theta$ ) is the change ( $\Delta$ ) in angle in radians ( $rad$ ) of the arc length ( $S$ ) traveled by an object along a circular path. The angular displacement of an arc length equal to the radius ( $r$ ) of a circle is, by definition, $1\ rad$ . Angular displacement can therefore be calculated as $\theta = \Delta S/r$ , though the $\Delta$ is often omitted for brevity.

Angular displacement is the basis for angular motion because it allows all points on a rotating line to be measured, rather than just a single point by solving for the missing variable in the angular displacement calculation.

Variable	Solution
Angular displacement	$\theta = S/r$
Arc length	$S = r\theta$
Radius	$r = S/\theta$

Angular Velocity

Angular velocity ( $\omega$ ) is the change in angular displacement over the change in time ( $t$ ), or $\omega = \Delta\theta/\Delta t$ (again, the $\Delta$ is often omitted). Essentially, "radians per second".

Angular velocity can be converted to speed (note that speed is not the same as velocity) by substituting arc length for angular displacement. Given $S = r\theta$ and $\omega = \theta/t$ , therefore $r\omega = r\theta/ t$ which equals $S/t$ and $S/t$ is distance over time, which is speed.

Angular Acceleration

Angular acceleration ( $\alpha$ ) is the change in angular velocity over time, or $\alpha=\Delta\omega/\Delta t$ . Essentially, "radians per second per second" or "radians per second squared".

Angular acceleration can be converted to tangential acceleration ( $A_{tan}$ ), which is acceleration which doesn't account for the change in direction in circular motion, by substituting arc length for angular displacement. Given $S = r\theta$ and $\alpha=\omega/t$ , therefore $r\alpha=r\omega/t$ , which is the change in angular velocity over time.