Rotation
 The work of a curious fellow

When everything that goes around, comes around...

 This section of the course deals with a different way of representing the laws of motion, based on the motion about some fixed line. In the next section, called rotational dynamics, we will build a model based on rotation, allowing you to predict the future of rotating objects. In this section we will introduce the parameters that describe rotation and cover enough of the basics to help you develop some intuition about general motion of many particle systems. In the translational motion which we have been studying, every particle undergoes the same motion. Now we will consider a kind of motion which is fundamentally different. We will give the name "rotation" to the motion of a solid object in which every particle follows a circular path such that the center of the circle that each particle traces lies on a straight line. That line which contains the centers of each particle's circular trajectory we will call the "axis of rotation". The axis of rotation may pass through the object or not. The obvious difference between rotation and translation is that in translation the object as a whole goes somewhere, with no guarantee that it will ever come back. In pure rotation it remains in the vicinity of the axis of rotation and periodically returns to its original position. I use the term "pure" rotation because it is possible that an object undergo both translation and rotation simultaneously. This is illustrated in the Combined Motion display. In this display we have a solid object made up of ten visible particles. Remember that being a solid means that the particles maintain their original separations. Because this is a computer model rather than a real object, we can also show the center of mass of the object. It is the larger white spot.
 What we will do next is develop the tools necessary to predict the future of an object undergoing this sort of motion. A different set of variables is used to describe rotation. The displacement in translation was measured along the coordinate axes, or as a distance and direction. In rotation we will measure the angle from a reference line to the position of interest and call it the angular displacement theta(the Greek letter q). Angular displacement is evidently a scalar quantity since one number completely defines it. If the reference line is the x axis, the angular displacement is the angular position. The units of angular measure will be the "radian". A radian is the ratio or the length of the arc cut by an angle, to the length of the radius of that arc. See the Angular Displacement display for an illustration. Since it is the ratio of two lengths, the radian is unitless.
 We also have measures in rotation analogous to velocity and acceleration. Velocity in angular measure is the rate of change of the angular position with respect to time, Dq/Dt. Since the angular position as we have defined it is a scalar quantity, so is average the angular velocity, being a scalar divided by another scalar, the difference in time. The instantaneous angular velocity is the limit as Dt approaches zero. We will use the Greek letter omega, w, to symbolize the instantaneous angular velocity. In thinking about angular velocity, we are faced with one of the things that makes introductory physics courses skip over much of the study of rotation. I argued convincingly a while back that average angular velocity, Dq/Dt, was a scalar quantity. When we let Dt approach zero to get the instantaneous angular velocity, the Dq becomes a vector. In fact it becomes a vector pointing off in a direction not even in the plane in which the rotation is taking place. It boggles the mind. Rather than interrupt our life at this point to ponder this perversion, consider the following experiment to help you believe that finite rotations behave differently than infinitesimal ones. Take a beer stein and hold it with the handle to your right and the opening facing upward. We are going to consider two kinds of rotation of the stein. An "A" rotation will move the opening directly away from you. A "B" rotation will rotate the stein clockwise as you look down on it. Now perform a 90-degree A rotation, followed by a 90 degree B rotation. You should end up with the stein's handle toward you and the opening to your right. Return the stein to its original position and then give it a 90 degree B rotation followed by a 90 degree A rotation. Now you see the handle on top and the opening away from you. Evidently for large rotations the order is important. For the mathematicians among you, the finite rotation operators A and B do not commute. Now repeat the experiment with 10-degree rotations. The difference in the final positions in the two cases when the rotation is small is well nigh undetectable. In the limit of infinitely small rotations, the operators do in fact commute. This does not prove that instantaneous angular velocity and acceleration are vectors, but lends credibility that the instantaneous values might be qualitatively different than the average ones.
 The average angular acceleration is the rate of change of Dq/Dt over some time difference so it too is a scalar. The instantaneous angular acceleration is a vector symbolized by the Greek letter alpha, a. The Angular Velocity and Acceleration display shows the successive angular displacement of an object undergoing constant angular acceleration where a = 2 radians per second squared, r/s2. The display shows 125 steps of .02 seconds each, about enough to complete one revolution starting from rest. Notice the angular displacement is larger in each step as w increases.