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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