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Unformatted text preview: Chapter 1 Rotation of an Object About a Fixed Axis 1.1 The Important Stuff 1.1.1 Rigid Bodies; Rotation So far in our study of physics we have (with few exceptions) dealt with particles , objects whose spatial dimensions were unimportant for the questions we were asking. We now deal with the (elementary!) aspects of the motion of extended objects , objects whose dimensions are important. The objects that we deal with are those which maintain a rigid shape (the mass points maintain their relative positions) but which can change their orientation in space. They can have translational motion , in which their center of mass moves but also rotational motion , in which we can observe the changes in direction of a set of axes that is “glued to” the object. Such an object is known as a rigid body . We need only a small set of angles to describe the rotation of a rigid body. Still, the general motion of such an object can be quite complicated. Since this is such a complicated subject, we specialize further to the case where a line of points of the object is fixed and the object spins about a rotation axis fixed in space. When this happens, every individual point of the object will have a circular path, although the radius of that circle will depend on which mass point we are talking about. And the orientation of the object is completely specified by one variable , an angle θ which we can take to be the angle between some reference line “painted” on the object and the x axis (measured counter-clockwise, as usual). Because of the nice mathematical properties of expressing the measure of an angle in radians , we will usually express angles in radians all through our study of rotations; on occasion, though, we may have to convert to or from degrees or revolutions. Revolutions, degrees and radians are related by: 1revolution = 360degrees = 2 π radians 1 2 CHAPTER 1. ROTATION OF AN OBJECT ABOUT A FIXED AXIS MT113 MT114 MT115 Figure 1.1: A point on the rotating object is located a distance r from the axis; as the object rotates through an angle θ it moves a distance s . [Later, because of its importance, we will deal with the motion of a (round) object which rolls along a surface without slipping. This motion involves rotation and translation, but it is not much more complicated than rotation about a fixed axis.] 1.1.2 Angular Displacement As a rotating object moves through an angle θ from the starting position, a mass point on the object at radius r will move a distance s ; s length of arc of a circle of radius r , subtended by the angle θ . When θ is in radians , these are related by θ = s r θ in radians (1.1) If we think about the consistency of the units in this equation, we see that since s and r both have units of length, θ is really dimensionless ; but since we are assuming radian measure, we will often write “rad” next to our angles to keep this in mind....
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- Summer '09