object in which the forces between the atoms are so strong, and of such character that the little forces that are needed to move it do not bend it. Its shape stays essentially the same as it moves about. If we wish to study the motion of such a body, and agree to ignore the motion of its center of mass, there is only one thing g left for it to do, and that is to turn. We have to describe that. How? Suppose there is some line in the body which stays put (perhaps it includes the center of mass and perhaps not), and the body is rotating about this particular line as an axis. How do we define the rotation? That is easy enough, for if we mark a point somewhere on the object, anywhere except on the axis, we can always tell exactly here the object is, if we only know where this point has gone to. The only thing needed to describe the position of that point is an angle. So rotation consists of a study of the variations of the angle with time In order to study rotation, we observe the angle through which a body has ar angle inside the obje self; it is not that we draw some angle on the object. We are talking about the angular change of the position of the whole thing, from one time to another First, let us study the kinematics of rotations. the angle will change with time, and just as we talked about position and velocity in one dimension, we may talk about angular position and angular velocity in plane rotation In fact, there is a very interesting relationship between rotation in two dimensions and one-dimen- sional displacement, in which almost every quantity has its analog. First, we have the angle 8 which defines how far the body has gone around; this replaces the distance y, which defines how far it has gone along In the same manner, we have a velocity of turning, w= de/dt, which tells us how much the angle changes in a econd, just as u ds/ dt describes how fast a thing moves, or how far it moves in a second. If the angle is measured in radians, then the angular velocity w will be so and so many radians per second. The greater the angular velocity, the faster the object is turning, the faster the angle changes. We can go on: we can differ- to d20/dt2 the angular acceleration. That would be the analog of the ordinary accel- Now of course we shall have to relate the dynamics of rotation to the laws of dynamics of the particles of which the object is made so we must find out how a articula when the velocity is such and such. To do this let us take a certain particle which is located at a distance r from the axis and say P(x,y it is in a certain location P(x, y)at a given instant, in the usual manner(Fig. 18-1) If at a moment At later the angle of the whole object has turned through 4e, then this particle is carried with it. It is at the same radius away from O as it was before but is carried to e. The first thing we would like to know is how much the distance x changes and how much the distance y changes. If oP is called [, then the length PQ is r Ae, because of the way angles are defined. The change in x, then, is simply Fig 18-1. Kinematics of two-dimen the projection of r Ao in the x-direction sional rotation x=- PQ sin 8=-r△0·(y/) (186) Similarly, x△θ (187) If the object is turning with a given angular velocity a, we find, by dividing both sides of (18.6)and(18.7)by Af, that the velocity of the particle is x=- y and v=+ωx (188) course if we want to find the magnitude of the velocity, we just write D=√好+√②y2+x2=Vx2+p2=w.(89) It should not be mysterious that the value of the magnitude of this velocity is wr; in fact, it should be self-evident because the distance that it moves is r 40 and the distance it moves per second is r△/△r,orra