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.