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Rotational Dynamics

Combined Rotational and Translational Motion

Problems

Problems

We have studied rotation on its own, and translation on its own, but what happens when the two are combined? In this section we study the case in which an object moves linearly, but in such a manner so that the object's axis of rotation remains unchanged. If the axis of rotation is changed, then our equations of rotation no longer apply. Here, we will only study cases in which our equations of rotation work.

The most familiar example of combined rotational and translational motion is a rolling wheel. While it is rolling, the axis of the wheel remains the axis of rotation, and our equations apply.

Kinetic Energy of Combined Motion

One important principle of combined motion is that the kinetic energies of translation and rotation are additive. In other words, we can get the total kinetic energy of a body by simply adding its rotational and translational kinetic energy. We must be careful, however, because we never truly defined translational kinetic energy for a rigid body (we only had a definition for a single particle). We solve this problem by simply using the velocity of the Center of Mass of the object, which provides the velocity of the rigid body. Thus the total kinetic energy of a particle is given by:

K = Mv cm 2 + 2    

This equation can be quite useful. Say a rolling ball ascends a hill until it stops. We can calculate the maximum height the ball will reach by using the above equation and relating total kinetic energy to potential energy.

Rolling Without Slippage

Many times we will know the velocity of an object, or its angular velocity, but not both. Usually if this is the case the problem is unsolvable. In the special case of rolling without slipping, however, we can generate a solution.

Rolling without slipping is defined as the special case of combined rotational and translational motion in which there is no relative motion between the object and the surface with which it is in contact. Examples of rolling without slipping include a car driving on a dry road and a pool ball rolling across the table. In each case, the surface can apply only static friction, since the object does not move relative to the surface. Also, this frictional force does no work and dissipates no energy. Thus an object rolling without slipping will continue with the same linear and angular velocity, unless acted on by another force.

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:

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 = 2 r 2 + 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.)

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