Now I have a little quiz for you. We have learned that
acceleration is the rate of change of the velocity vector.
My question is can a particle with constant speed
have a finite acceleration. (show of hands). If anyone says yes have
them explain how this is possible.
Indeed this is possible because a vector can change direction
without changing magnitude and when this occurs we have
acceleration with no resulting change in speed. Of course in general
motion in two or three dimensions involves changes in speed as well as
changes in the direction of 
. There is however one
type of motion in which only
the direction of the velocity vector changes not its magnitude and this is
uniform circular motion.
We consider a particle moving at constant speed around a circle.
Because we are dealing with kinematics here we do not ask why it does
so, we simply wish to determine the velocity and acceleration
associated with this type of motion. We denote by T the period of
the motion: the time taken to complete one orbit. The corresponding
frequency is
Units for f are s
also called Hertz. In motors the
frequency of rotation is measured in Revolutions Per Minute:
To describe the location of the particle it is convenient
to use polar coordinates 
. Because we have uniform circular
motion the polar angle 
increases in proportion to time ie
Here we introduced the angular velocity,
With these equations we can easily derive an expression for the speed
in uniform circular motion. The length of the arc spanned by the angle
is
We notice the well known expression for the circumference of a circle
when we insert 
. The speed in orbit is
