CONSERVATION OF ANGULAR MOMENTUM

We usually think of the theory of Special Relativity as being about motion in straight lines and that thought is usually true to the teaching of the fundamentals of the subject. Why, then, do I want to introduce you to the physics of rotary motion? It turns out that the conservation law pertaining to rotary motion has an astonishing relation to Relativity and in the next chapter I will show you what that relation is.

Let's recall our proposition that the Universe exists vis-a-vis Absolute Nothingness. Nothing, including space, exists outside the Universe, so the Universe as a whole object can have no motion. In the last chapter I considered that rule to apply to motion in straight lines, but it also applies to rotary motion. With no outside space to define the direction of an axis, the Universe cannot spin. But we know that inside the Universe objects do go round and round, so we infer, as before, that all rotary motions in the Universe must always add up to a net zero and from that theorem derive two rules:

1. The angular motion of a body or system of bodies remains unchanged unless that body or system of bodies interacts with another body or system of bodies; and

2. Any change in the angular motion of a body or system of bodies must be accompanied by an equal and oppositely directed change in the angular motion of another body or system of bodies.

Before we can see whether the angular motion in those rules is the same as the angular momentum that physicists use, we must devise some aids to the imagination, ones that will help us to analyze rotary motion as our dynamically equivalent bodies enabled us to infer the form of linear momentum. Because rotary motion involves bodies revolving about some axis or axes and because any rotating body can be regarded as comprising a collection of such bodies, we suspect that the size of a body will influence the amount of rotary motion it possesses. Thus, we will use in our analysis what physicists call "point masses", imaginary bodies that possess mass but, like mathematical points, no size. We can use these fictions legitimately only so long as the sizes of the bodies they represent are irrelevant to the solution of the problem on which we focus our attention. In this case we legitimize their use by the understanding that we can represent any large body by a suitable array of point bodies: it's not much different from representing a body as comprising an extended collection of atoms.

Imagine that two dynamically equivalent point masses move at the same speed in opposite directions and imagine further that they are connected to each other by a massless string (another useful fiction employed by physicists). That string, being unable to stretch, causes the bodies to exert equal and oppositely directed forces upon each other and those forces make the bodies move in circles about a common center (which, in this case, is the midpoint of the string). We can see that the linear momenta of the two bodies are equal and oppositely directed, so we know that they cancel each other and, consequently, the system as a whole has zero linear momentum. But because the bodies are going round and round, we can say that the system's rotary motion is certainly not zero.

Bring into the laboratory of your mind a second system of point masses, one identical to the first one except that it is not rotating. The two systems come together in such a way that the point masses of one collide with and stick to the point masses of the other. Based on what we learned in the last chapter, we can say that the resulting system will be twice as massive as either of the ones that comprise it and that the speed at which the component point masses move will be half that of the first system. We follow the obvious chain of experiments and so obtain the equally obvious result: the rotary motion of one of these systems is proportional to the mass of the point masses that comprise it and to the speed (not the velocity) with which they move.

But that's not the only experiment that we can conceive. Recall the original state of the first system described above and alter the description slightly. We now have two identical point masses connected each to the other by two massless strings of different lengths and revolving about the midpoint of the shorter string. What happens when we cut the shorter string? Freed from the constraint, the point masses will move in straight lines until the longer string goes taut, then they will resume moving in circles. We assume that no force was exerted upon the system when the string was cut (it might have been burned by laser beams, for example) and we notice that the force that the longer string exerted upon each of the bodies when it went taut was exerted along a line that passes through the center of the system, so we claim that the rotary motion of the system has not changed; that is, we state that no force has been exerted that would make the system move faster or slower around its center. Can we now determine how fast the point masses are moving in their new circles?

We know right away that the new speed at which each point mass moves will be smaller than the point mass's original speed. We know that's true to Reality because we can follow one of the point masses in our imaginations. What we see is that we can draw three straight lines representing velocities that we associate with the point mass:

Line 1; the velocity that the point mass has just after the shorter string breaks,

Line 2; the velocity that the point mass has just after the longer string goes taut, and

Line 3; the velocity that the longer string subtracts from the point mass's original velocity when it goes taut.

Because the string is limp, it can only exert a force in a direction parallel to itself, so Line 3 is parallel to the longer string at the instant it goes taut. Because the string does not stretch, Line 2 must be perpendicular to the longer string at the instant that string goes taut. And because of the way in which velocities add and subtract, Line 1 joins with Line 2 and Line 3 to form a perfect right triangle whose hypotenuse is Line 1. We know that either side of a right triangle is always shorter than the hypotenuse, so we know that the velocity that the point body has after the longer string goes taut (represented by Line 2) must be smaller than the velocity that the body had before the longer string goes taut.

How much smaller? We can work out an answer to that question by drawing another triangle, one that's similar to our velocity triangle. Again we specify the three sides:

Line A; half the length of the shorter string at the instant the shorter string breaks,

Line B; half the length of the longer string at the instant the longer string goes taut, and

Line C; the line that the point body traces between the breaking of the shorter string and the tautening of the longer string.

That length triangle is similar to the velocity triangle in the geometric sense that the internal angles of the two triangles are the same; thus, the ratio of two sides of one triangle is equal to the ratio of the corresponding sides of the other triangle. In this case Line 1 corresponds to Line B because both lines are the hypotenuses of their respective triangles. Line 2 corresponds to Line A, a fact that can be discerned by making the length of the longer string much greater than 1.414 times the length of the shorter string: in that case Line A will be much shorter than Line B and Line 2 will also be the short side of its triangle. The proportionality theorem of plane geometry now tells us that the length of Line 1 is to the length of Line B as the length of Line 2 is to the length of Line A; that is, the speed that one of the bodies has when the shorter string breaks is to the length of the longer string as the speed that the body has after the longer string goes taut is to the length of the shorter string.

Given that proportion, we can then say that the quantity that remained unchanged in that experiment was the product of each body's speed of revolution about the center of the system and that body's distance from the center of the system. Though that distance is constrained by the strings, there was a short time when it was changing, when the bodies were moving in straight lines. At any given instant during that time the calculation had to be the product of the distance from the center of the system and that part of the body's velocity that was oriented at a right angle to the line defining that distance. Even though the bodies were moving in straight lines, they could still be regarded as displaying rotary motion relative to the center of their system.

We can go on and devise other imaginary experiments along these lines, but the result to which they will point us will be the same as the result we obtain from combining the results of the two experiments that we have just performed. That result can be expressed as a simple rule:

3. For any extremely small body moving with a specific velocity at some distance from a defined axis, the amount of rotary motion that body possesses relative to that given axis is equal to the product of the body's mass, its distance from the axis, and that part of the velocity that's perpendicular to the line drawn between the body and the axis.

The product described in that rule is, in fact, the angular momentum of the body, so Rules 1 and 2 are the rotary equivalents of Newton's first and third laws of motion. That rule also applies to larger bodies because we can represent any such body as a collection of point bodies revolving about a common axis.

We now recognize Rule 2 as the law of conservation of angular momentum. That law states that in the absence of any torques exerted upon it by any other bodies, a given body will retain an unchanging angular momentum. If that body does interact with another body, the torques that the bodies exert each upon the other must be equal and oppositely directed, the direction of a torque, like the direction assigned to an angular momentum, being regarded as pointing along the axis of rotation.

That conservation law gives us an important clue to the fundamental structure of Reality. Noether's theorem correlates it with the isotropy of space; that is, because angular momentum is conserved, the Universe is so structured that the laws of physics are the same in all directions, just as conservation of linear momentum necessitates that the Universe be so structured that the laws of physics are the same at all positions in space. Thus scientists performing identical experiments in laboratories facing in different directions will obtain identical results from those experiments. But that conservation law does something more: it tells us something very important about the size and shape of space. In the next chapter I'll show you how that works out.

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