Gears are common in mechanical equipment and are used to transfer load and to change velocities.

First, consider a point on a circular rotating body. The body rotates by dθ over time dt. The change in displacement is

     dr = r dθ

By dividing by dt, the velocity of the point is

     dr/dt = r dθ/dt

This can be written in vector form as

     v = ω × r

The velocity direction is tangent to the circular path of point P.

Since the velocity is always tangential to the radius, two touching objects will have identical velocities at the point of contact.

Similarly, tangent accelerations are equal, but the normal accelerations are not.