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FINISHING AN EXAMPLE: NEWTON'S LAW OF GRAVITATION

These two laws, of inertia and of exactly opposite action- and reaction-forces, will be very useful as we turn now to extend Galilleo's law of gravity for fallers near Earth to Newton's increadibly more universal one, which first convinced the human race decisively that it was possible to understand nature.

Recall that from dimensional analysis, from Keplar's observations of the motions of the planets, we had arrived at the observation that their orbital diameters (lengths) and orbital periods (times) were related as

Then from Galilleo we had the for objects falling near the earth,

Newton did the following steps one way; we will do them in a simpler way that is really still guessing at this point, but is well motivated in the context of dimensional analysis, and which arrives at the same result. We follow Newton in taking the giant leap of supposing that the gravitation causing both these actions could be encoded in a single law. Since the constant g has dimensions , and the one thing that is special for all objects falling very near the surface of the earth is that they all have about the same distance from its center, namely the radius of the earth, the next natural supposition is that the constant we see as g is related to that other constant in the fashion

because the moon is also ``falling'' toward the earth; it just keeps missing because it is also flying to the side as it falls. (A not-too-hard estimate of the radius of the earth shows that these numbers do have the right sizes to fulfill the relation, which is left for the problems) More generally, then, we would expect that for objects not near the surface of the earth, the relevant radius is not , but whatever radius they have at a given moment. So the equation of motion would be expected to be

But then we have learned that this very useful object-external world distinction can be made in a general way if we relate not a, but ma, to the state of the external world, so we re-write Galilleo's relation with our new assumptions as

This is reasonable, because the same ma that is felt by a spring scale is the ma to which the body reacts when falling, which is why it seems reasonable to regard m as the coupling strength of the object to the earth's gravity. Now, how much more might it be possible to say about the dimensionful constant that describes the common body at the center, about which our fallers of interest are oribiting?

Here we make use of the action-reaction relation for forces. Apart from the fact that the earth is bigger, there is no obvious reason to treat it as the ``external world'' and the moon as the ``object'' in any description of the earth-moon system. There is no evidence that anyone holds the earth like a popsicle on a stick and somehow ``keeps it from falling'' any more than that is true of the moon. Thus we could just as well have regarded the moon as the ``force-contributing external world', and the earth as the object. But in this case, we know what force the moon would have contributed, specifically,

(it is convenient to use capital M and A for the earth, to keep the two distinct). But we still, on completely independent grounds, have all of our previous reasons to expect that the ``F = ma'' of the earth can be described as

So now we have the same quantity expressed in terms of two different points of view. Specifically, getting rid of the , which is just the same multiplicative factor in both of these equations,

Since, by our previous experience that, since all things fall at the same rate, and the factor of m could even have been left out of the original equation, the only way the mass m of a small faller (even the moon) appears on one side is the way it appeared when we added it to ma, as the single factor outside multiplying the orbital constant of the earth. Conversely, since there seems nothing obviously special about which point of view we choose to make the description, we expect the same thing to be true on the other side, so M(earth) only appears as we have explicitly related it.

So the only way these two equations can make sense vis-a-vis one another is if:

  1. the (orbital constant of the Earth) contains a factor M(earth) to match the one on the other side of the equation, and
  2. the complementary same is true for the orbital constant of the moon.
Thus we expect that the only way the mass of the earth appears in the orbital constant of the earth is as:

This places our original guessed extension of Galilleo's gravitational law in the form:

and of course all the expectations apply in just the same way to the earth, so the law takes exactly the same form. This is very nice, because taking account of the opposite directions of the ``pull', this gives ma = -MA, because they are both given by just the same expression.

At this stage, we have exhausted our dimensional analysis. There are no other obvious dimensional quantities available, and through reasoning extrapolated from the motion of the planets and the simple falling of objects near our own earth, we have shown that our present use of them exhausts the roles of the dimensionful properties m,M, and r that we have.

Thus, it seems, whatever the simple constant is that is left, it does not depend on any of the masses of the earth, moon, or anything else, nor on their motions. At this point Newton's conjecture, which has proven true in ways he could never have expected, is that this simple constant remaining must be a dimensionful constant that describes whatever is special about gravitational interactions. We give that constant a symbol G because it promises to be useful; note that its dimensions are

and name it Newton's constant in honor of its discoverer.



Next: FREE-BODY DIAGRAMS Up: NEWTON'S REVOLUTION: THE Previous: CONSEQUENCES AND CAVEATS


desmith@
Thu Aug 31 12:01:42 CDT 1995