out that the fundamental atomic laws, which we call quantum mechanics, are quite different from Newton s laws and are difficult to understand because all our direct experiences are with large-scale objects and the small-scale atoms behave like nothing we see on a large scale. So we cannot say, "An atom is just like a planet going around the sun, or anything like that. It is like nothing we are familiar with because there is nothing like it. as we apply quantum mechanics to larger and larger things, the laws about the behavior of many atoms together do not reproduce themselves, but produce new laws, which are Newton's laws, which then continue to reproduce themselves from, say, micro-microgram size, which still is billions and billions of atoms, on up to the size of the earth, and above Let us now return to the center of mass The center of mass is sometimes called the center of gravity, for the reason that, in many cases, gravity may be considered uniform. Let us suppose that we have small enough dimensions that the gravitational force is not only proportional to the mass, but is everywher parallel to some fixed line. Then consider an object in which there are gravitational forces on each of its constituent masses. Let mi be the mass of one part. Then the gravitational force on that part is m: times g. Now the question is, where can we apply a single force to balance the gravitational force on the whole thing, so that the entire object, if it is a rigid body, will not turn? The answer is that this force must go through the center of mass, and we show this in the following way. In order that the body will not turn, the torque produced by all the forces must add up to zero because if there is a torque, there is a change of angular momentum, and thus a rotation. So we must calculate the total of all the torques on all the particles, and see how much torque there is about any given axis; it should be zero if this axis is at the center of mass. Now, measuring x horizontally and y vertically, we know that the torques are the forces in the y-direction, times the lever arm x(that is to say,the force times the lever arm around which we want to measure the torque) Now the total torque is the sum T mixi=82 mixis (193) so if the total torque is to be zero, the sum 2mi x; must be zero. But mixi=MX, the total mass times the distance of the center of mass from the axis Thus the x-distance of the center of mass from the axis is zero Of course, we have checked the result only for the x-distance, but if we use the true center of mass the object will balance in any position, because if we turned it 90 degrees, we would have y's instead of xs. In other words, when an object is supported at its center of mass, there is no torque on it because of a parallel gravitational field. In case the object is so large that the nonparallelism of the gravitational forces is significant, then the center where one must apply the balanc ing force is not simple to describe, and it departs slightly from the center of mass That is why one must distinguish between the center of mass and the center of gravity. The fact that an object supported exactly at the center of mass will balance in all positions has another interesting consequence. If, instead of gravitation, we have a pseudoforce due to acceleration, we may use exactly the same mathe matical procedure to find the position to support it so that there are no torques produced by the inertial force of acceleration. Suppose that the object is held in inside a box, and that the box, and everything contained in it, accelerating. We know that, from the point of view of someone at rest relative to this accelerating box, there will be an effective force due to inertia. That is, to make the object go along with the box, we have to push on it to accelerate it, and this force is"balanced"by the force of inertia, which is a pseudoforce equal to the mass times the acceleration of the box To the man in the box this is the same situation as if the object were in a uniform gravitational field whose"g " value is equal to the acceleration a. Thus the inertial force due to accelerating an object has no torque about the This fact has a very interesting consequence. In an inertial frame that is not accelerating, the torque is always equal to the rate of change of the angular mo- mentum. However, about an axis through the center of mass of an object which