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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.
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} + Iσ ^{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.
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.
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