When an object is moving in a circular path and
its velocity's magnitude is constant, then its acceleration is a vector
that points towards the center of the circular path and its magnitude is
given by:
where v is the magnitude of the object's velocity, r is the radius and
ac is the magnitude of the acceleration. Note that this only holds
when the object's motion is uniformly circular -- it is not speeding up
and slowing down, like a child on a swing, but is moving with the same
rotational speed, like a scratch on a CD.
If the magnitude of the velocity (speed) of the body moving in a circle
is not constant, there is still instantaneous centripetal acceleration,
ac, given by the expression above. However, in this case there is
also tangential acceleration, atan, which accounts for the body speeding
up or slowing down. If the tangential acceleration is not zero, then
the magnitude of the total acceleration is given by