Individual points on a

rotating object undergo

circular motion. How fast is each point moving? Each point has an

instantaneous linear velocity **v** with which it would continue if the forces holding the object together suddenly disappeared. We now relate the

magnitude *v* - i.e., the

linear speed - to the angular speed ω of a rotating object. The definitiion of angular measure in

radians gives

**θ** = s / r

Differentiating this expression with respect to time we have

**dθ / dt** = ( 1 / r ) * ( ds / dt )

because the radius *r* is constant. But dθ / dt is the angular velocity, and ds / dt is the linear speed, v, so ω = v / r, or

v = ω · r

Thus the linear speed of any point on a rotating object is proportional both in the angular speed of the object and to the distance from that point to the axis of rotation.