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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:

Figure %: A wheel, rolling without slipping, with the velocity of three points shown
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:

vcm = σr    

Recall also that our equation for total kinetic energy involved two variables: vcm 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=Mvcm2 + I  
K=Mσ2r2 + Iσ2  

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.)

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