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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.

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