The Radian:
Suppose you watch an ant moving on the rim of a 13 inch diameter car wheel (radius = 6.5 inches). And suppose the ant travels a distance of 6.5 inches along the circumference. That's the definition of the radian! The angle that the ant has moved through is called one radian.
Let's put this into symbols. I'll let s be the distance moved along the rim, r the radius, and theta the angle:
Setting up a proportion...if the ant travels all the way around, s is the circumference and the angle is 360o (except I'll write it in radians - 2 Pi). So the ratio of the partial distance to the circumference equals the ratio of the partial angle to the total angle:
(Please - for you purists among us - I know I used the thing to "prove" itself!)
The angle must be measured in radians, since the total angle above was expressed in radians. (Try using 360o instead of 2-Pi. The expression is a lot messier!)
Angular Velocity and Acceleration:
Taking derivatives, and assuming that the circle's radius r is held constant,
where
(The assumption of constant r is inapplicable for motion on a spiral, for example. More general equations must be derived.)
Please remember! Since the derivatives were taken of a "displacement" along the arc, we obtained tangential velocity vT and acceleration aT , as opposed to radial quantities. A reminder -the radial (or centripetal) acceleration aR is:
These equations "connect" the linear quantities (displacement along the arc, tangential velocity, and tangential acceleration) to the angular quantities (angular displacement, angular velocity, and angular acceleration).
Example 1: An ant on a wagon wheel
Example 2: Using calculus to find angular quantities