We can dynamically describe the process of rolling without slipping by
first drawing a figure and showing the relative velocities of various
points on a wheel:

Because the part of the wheel in contact with the ground is not moving,
it becomes the axis of rotation of the ball. This concept is difficult
to grasp: it seems more logical to state that the axis of rotation of
the ball is simply the center of the ball. The distinction that must
be made is that the axis of rotation of the ball is constantly
changing: each instant a new part of the ball comes in contact with the
floor and the axis of rotation changes.

Given that we define the axis of rotation in this way, we can relate
the velocity of the center of mass to the angular velocity of the ball.
We know that the center of mass is a distance r away from the axis of
rotation (the ground). Thus, by our equation for relating v and
σ, we see that:

v_{cm} = σr

Recall also that our equation for total kinetic energy involved two
variables: v_{cm} and σ. In the special case of rolling
without slipping, these variables are not independent, and through the
above relation we can generate expressions for the total kinetic energy
of an object in terms of one or the other:

K

=

Mv_{cm}^{2} + I

K

=

Mσ^{2}r^{2} + Iσ^{2}

As the equations show, in the special case of rolling without slippage,
we can uniquely determine the motion of the object by simply knowing
either its linear or angular velocity.

Conclusion

In combining our study of combined motion with our study of rotational
dynamics, we gain the ability to predict the motion of an object in a
variety of situations. The next step in the development of our
understanding of rotational motion is the introduction of the concept
of angular momentum. (Note: the next section in this SparkNote
is actually a calculus-based section describing the derivation of
inertial momentum.
This is
not a topic covered in courses such as AP Physics. If you would like
to skip the topic and go on to Angular
Momentum, it's fairly
obvious where you should click.)