Angular and Tangential Quantities

When a wheel of radius r rotates about an axis whose direction is fixed, a point on the rim of the wheel is described in terms of the circumferential distance l it has moved, its tangential speed v, and its tangential acceleration a_T. These quantities are related to the angular quantities θ , ω , and α , which describe the rotation of the wheel, through the relations: l = r θ v = r ω a_r = α provided radian measure is used for θ, ω and α

By simple reasoning, l can be shown to be the length of belt wound on the wheel or the distance the wheel would roll (without slipping) if free to do so.

In such cases, v and a_T refer to the tangential speed and acceleration of a point on the belt or of the center of the wheel.

Radians are worth using because they make it very simple to convert between angular and tangential quantities. If one uses degrees, one must constantly go back to "360 degrees equals one revolution, which is a distance two pi times the radius ..."

Given the angular velocity omega of an object, and its angular acceleration alpha, one can calculate tangential displacement s = R * omega * t

tangential speed v = R * omega T tangential acceleration a = R * alpha

If an object moves along a circular path, but changes its speed as it moves (slowing down or speeding up), then it is NOT in uniform circular motion.

It retains a centripetal acceleration (towards center of circle) because, at any particular moment, it has some tangential velocity, and  centripetal acceleration a = V ^2/c R But it also has a tangential acceleration due to its change in tangential speed; one way to express it is tangential accleration a = R * alpha