To describe the motion of rotating or spinning objects, we need a more natural set of variables than the x's, v's, and a's we have been using so far.

Angular variables

Angular displacement (angle):

θ

=

s r

Angular velocity:

ω

=

v r

Angular acceleration:

α

=

at r

The angular acceleration is connected to the tangential acceleration, not the centripetal acceleration. There is an angular acceleration only when the rotation rate changes.

If the angular velocity is constant, then the speed of a point on the rotating object is:

v

=

2πr T

where T is the period, the time to go around once.

The angular speed is:

ω

=

v r

=

2π T

These angular variables are vectors, just like their straight-line motion cousins. Which way do they point? Take the fingers of your right hand and curl them the way an object is spinning. Stick your thumb out and you get the direction of the angular velocity. The angular acceleration is in the same direction if the object is speeding up its spin, and in the opposite direction if it's slowing down.

We'll often use clockwise and counterclockwise to indicate direction. As with straight-line motion, we can define the positive direction based on what's convenient in a particular case.