So if you have a rigid body say a wheel, how do you describe
the position of various points? The wheel rotates about
some fixed axis. Points on the axis don't move but points
off axis do. A point on the wheel can be described as
being some distance r from the axis. If you watch
it as it rotates, you see that when this point moves along
the arc of a circle. If the wheel rotates by an angle
(measured in radians), the arc is of length
Is this right? Think about a few case. If 
,
you get 
which is the circumference of a circle.
If 
, you get half the circumference. So it
makes sense.
So we now know how to relate the distance traveled by a point on the wheel, to how much it rotated.
Now let's introduce another concept, the angular velocity.
This is the rate the angle changes with time. In other
words, how fast the wheel is spinning. Suppose the wheel
rotates one revolution in a second. That means it rotates
radians in a second. The average angular velocity
over this time interval is 
. So in general,
we can define the average angular velocity as
This is similar to the definition of average velocity, except in that case you were concerned with distance traveled, not angle. In analogy with regular velocity, we can write the instantaneous angular velocity as
We can define angular acceleration in analogy to acceleration, the average acceleration being
and the instantaneous angular acceleration as
Now how to we relate the the velocity of a point to its
angular velocity? Well the distance traveled is 
,
so the speed of the point
or
The direction of velocity is along a circular arc. That was pretty straighforward, and we've talked about this before. What's a little trickier is the acceleration.
Remember the centripedal acceleration. Even when the wheel is rotating
at constant speed, a point at radius r feels a radial acceleration
. But what if the angular velocity
of the wheel is changing? Then there's a tangential acceleration
in the direction of the velocity
This is perpendicular to the centripedal acceleration, and the two accelerations when combined (in vector form) give the total acceleration of a point on the wheel.
