19 Center of Mass;Moment of Inertia 19-1 Properties of the center of mass In the previous chapter we found that if a great many forces are acting on a 19-1 Properties of the center of mass complicated mass of particles,whether the particles comprise a rigid or a nonrigid body,or a cloud of stars,or anything else,and we find the sum of all the forces 19-2 Locating the center of mass (that is,of course,the external forces,because the internal forces balance out),then 19-3 Finding the moment of inertia if we consider the body as a whole,and say it has a total mass M,there is a certain point“inside'"the body,called the center of mass,such that the net resulting 194 Rotational kinetic energy external force produces an acceleration of this point,just as though the whole mass were concentrated there.Let us now discuss the center of mass in a little more detail. The location of the center of mass (abbreviated CM)is given by the equation ∑mr RCM ∑m (19.1) This is,of course,a vector equation which is really three equations,one for each of the three directions.We shall consider only the x-direction,because if we can understand that one,we can understand the other two.What does XcM= ∑mx:/∑m:mean?Suppose for a moment that the object is divided into little pieces,all of which have the same mass m;then the total mass is simply the number N of pieces times the mass of one piece,say one gram,or any unit.Then this equation simply says that we add all the x's,and then divide by the number of things that we have added:XcM =mxi/mN x:/N.In other words, XcM is the average of all the x's,if the masses are equal.But suppose one of them were twice as heavy as the others.Then in the sum,that x would come in twice. This is easy to understand,for we can think of this double mass as being split into two equal ones,just like the others;then in taking the average,of course,we have to count that x twice because there are two masses there.Thus X is the average position,in the x-direction,of all the masses,every mass being counted a number of times proportional to the mass,as though it were divided into "little grams."From this it is easy to prove that X must be somewhere between the largest and the smallest x,and,therefore lies inside the envelope including the entire body.It does not have to be in the material of the body,for the body could be a circle,like a hoop,and the center of mass is in the center of the hoop,not in the hoop itself. CM Of course,if an object is symmetrical in some way,for instance,a rectangle, so that it has a plane of symmetry,the center of mass lies somewhere on the plane B of symmetry.In the case of a rectangle there are two planes,and that locates it uniquely.But if it is just any symmetrical object,then the center of gravity lies Fig.19-1.The CM of a compound somewhere on the axis of symmetry,because in those circumstances there are as body lies on the line joining the CM's of many positive as negative x's. the two composite parts. Another interesting proposition is the following very curious one.Suppose that we imagine an object to be made of two pieces,A and B(Fig.19-1).Then the center of mass of the whole object can be calculated as follows.First,find the center of mass of piece A,and then of piece B.Also,find the total mass of each piece,MA and MB.Then consider a new problem,in which a point mass MA is at the center of mass of object A,and another point mass Ma is at the center of mass of object B.The center of mass of these two point masses is then the center of mass of the whole object.In other words,if the centers of mass of various parts 19-1