We start with an already familiar example that happens to take almost the form of one of Newton's relations. It is Galilleo's discovery of the law of gravitation, as it is relates to objects near the earth. Recall that, by trial and error or however, we eventually conclude through measurement that for objects falling near the earth, the thing that is always the same about their motion is something called the acceleration, and that if we measure it we find that it always takes the value a = g, where g is some number with the units of acceleration
(or ), which can simply be
measured and recorded, after which it can be used to predict the
motion of all such objects considered in the future.
This is an interesting relation because we don't really ``understand'' where this number g comes from. Presumably it depends on properties of the earth, and on the fact that the earth is ``affecting the faller gravitationally', but these words do not amount to a prescription for predicting the value of g from other things that we may already know. Yet in spite of this ignorance, of all the complexity of the state of the earth, and even of the deeper rules that describe gravitation, everything that is special about the fact that it is the earth affecting the faller and not some other body is contained in this one dimensionful number g. On the other side of the equation lies the only relevant quantity that the motion of the object has which we are not free to change at will, namely its acceleration. (Recall that we could change particular distances, speeds or times by dropping it in different ways.) But because the acceleration, as has been noted, tells how the velocity is changing, and in its turn the velocity tells how the position is changing, the fact that we know the accelleration at every moment of the objects'' motion allows us, if we measure the velocity at some starting moment, to extrapolate, an instant at a time forward, and so to predict the velocity at any later time from our known acceleration and its starting value. (An important homework problem at the end of this chapter works through the details of hos this extrapolation is carried out.) Then similarly, if we check the starting position, knowing the velocity from our previous extrapolation, we can predict all of the later positions as well.
This kind of relation (or in this case, this use of the relation
we already had) is called an equation of motion.
There is
an important difference in what we do with it from what we did
with typical relations obtained from dimensional analysis.
There, we looked for any property of all possible allowed
motions that was repeatable and common to all of them, and
sought a way to predict it. Such a number, common to all the
motions that an object might have, independently of how they
were started, could obviously therefore not distinguish by
itself which particular motion the object had in a particular
case.
With our new use of Galilleo's law (the same as a question/answer pair, but now used as an equation of motion), we have added something to the dimensional analysis result, because in this case the number that was constant, the acceleration, happens to be enough to uniquely enable us to extrapolate what the motion will be at all later times, once we have made a complete description of that motion at some starting time (in this case, what we needed was a starting position and a starting velocity). The equation of motion is special because it tells us enough about each moment along the path of the object to enable us to predict that particular motion entirely from just the two starting numbers. The important thing to note about the equation of motion is that, unlike a generic relation from dimensional analysis, which will only give one number to describe all realistically possible paths, an equation of motion has to provide a whole sequence of numbers for each moment along any actual path, in order to make it possible to reconstruct that path. Not all relations we obtain from dimensional analysis will be enough, by themselves, to produce such a description, and it will often not take such an obvious form when they do. That happened in this case (with malice of forthought) because the quantity that happens to be common to all such falling motions (the downward acceleration) is also constant and so the same at each moment along any actual path, and also happens to be sufficient to reconstruct the whole path.
To take the next step in refinement beyond dimensional analysis, and be able to predict not only properties of all allowed paths, but also the particular details of any given individual path, given a knowledge of how it starts, we would like to be able to produce not only constants of the motion, but full equations of motion for the path.
While this is already done, in a sense, for the falling object near the earth, it is not done for another case we have considered, that of the mass bobbing on the end of a spring. How then, shall we proceed? For this case, the acceleration is not at all constant (the object bobs first one way and then its opposite), and the quantity predicted as constant by dimensional analysis (the characterstic frequency of the motion), does nothing to identify any particular path chosen by starting the object in a particular manner, since it is common to all such paths.
Unlike the free faller, none of whose properties seemed to
affect its motion at all, and for which the actual observed property of the motion, the acceleration, was the only
one that entered as a necessary description of the object, the
motion of the object on the spring is not measureable in
isolation, but depends on how massive the object itself is, as
well as on characteristics of the spring that holds it. On a
given spring, more massive objects oscillate more slowly, and
lighter ones more quickly, and for a single object tested on
several springs, ``stiffer'' springs will cause faster
oscillations, and less stiff springs slower ones. Newton
observed that, despite all this interplay of characteristics, it
was possible to achieve a separation into properties of the
object alone, and properties of the spring alone. Specifically,
he found that if the spring is left to hang motionless, the
object will have some ``resting''
position, as shown in fig. 5.3.
When it is then set in motion, it is possible to measure its position at each instant of time relative to that resting position, and that remarkably at each instant its acceleration is related only to that ``distance from rest'' and to properties of the spring and the object, through a relation
where again k is a single number with dimensions that is sufficient to describe quantitatively the stiffness of the particular spring being considered. A quick check of the dimensions of k (left for the homework) reveals that this number has the same dimensions as the constant of proportionality obtained two chapters ago. In fact, it turns out to be the very same number, as could be checked by direct measurements of masses, accelerations, positions and overall frequencies as these were described previously.
This new relation is clearly an equation of motion for the
object on the spring, because at each instant, starting from the
beginning, if we have known the velocity coming into that
instant, and we check the position of the object at that
instant, the position can be used to predict the acceleration,
and thus to predict the change of velocity, and from the
velocity the change in position leading into the next brief
moment, and so on indefinitely. (How to do this precisely
required a branch of mathematics which did not exist prior to
Newton, and which (he?) named the calculus
(after a small
Roman stone used for keeping count) when he invented the subject
to provide a concrete method for calculating the solutions to
such problems.)
The fact that such an equation of motion can be obtained in such a form for springs might have been an isolated lucky case, except that in re-examining Galilleo's law for fallers near the earth, it is possible to notice that, by adding a somewhat artificial factor of the mass to both sides of the equation, this law can be written in an equally correct and useful form as
Here again, m is an intrinsic property of the object itself, and a the characteristic of its motion that we would like to be able to predict, while g characterizes all of the complexity of the fact that it is the earth that is affecting the motion, and in some sense one may think of the other m on the ``mg'' side of the equation as representing a ``strength of attachment'' of the falling object to the gravitational influence of the earth. (The strangeness of placing this equation in such a form will be addressed later, in the section on general relativity. It becomes even stranger when one does what we will do shorty, in taking the notion of the quantity represented by ``mg'' seriously as physically meaningful, at which point it becomes very puzzling indeed why this ``strength of attachment to the earth's gravity'' should have just such a relation to the mass of the object (that characterizes its resistance to being moved) that the two could have been left out of the equation altogether, as Galilleo did in its first version. This self-created puzzle has been the subject of several remarkable experiments continuing into the present, and was not presented with a satisfactory ``explanation'' until Einstein created the description of Gravity as a geometric phenomenon. But that is another story, and in the mean time there is much to be learned by placing the equation in this form, as we shall therefore proceed to do.)
We now have two equations of motion for two quite different
systems, which both have the same remarkable form. The fact
that they predict the correct motion for their respective
objects means that it is possible, in these cases at least, to
divide the entire problem of the motion of a mass under either
of these two circumstances into two parts, which we could call
the object
and its external world.
The properties
of the object are very simple, the acceleration which
characterizes its motion, which is the quantity we would like to
be able to predict, and the mass of the object itself, which as
we have noted is that property of the object which characterizes
its resistance to ``being accelerated'. While the other side of
the equation must therefore contain a description of the entire
rest of the world, we find that that description may be put into
a very simple form: in the case of the faller near the earth,
there is some single number, g, which characterizes an
acceleration, and another reference to m, though this time as a
``strength of coupling'' of the object to the earth's gravity,
and in the case of the spring, all of its internal structure is
characterized by a single number representing the stiffness, and
the position of the spring away from its resting place.
As we have noted, once the acceleration of the object is given by the equations of motion, that knowledge can be combined piecewise to extrapolate the entire motion from an initial knowledge of the position and velocity. Certainly, the same could have been remarked about the velocity, in which case only the original position would have to be known to perform the extrapolation. The thing that is special about the acceleration, and the first important part of Newton's discovery, is that, had we tried to find an equation of motion for the body which made use of the velocity, or even the position itself at each moment, on the object-side of the equation, there would in general have been no similarly simple form for such an equation. Yet Newton discovered that, not only for these two cases, but for a very broad collection of motions, if we look for an equation that relates the accleration of the object, together with its resistance to being accelerated, in the combination ``ma', we can find that this combination can be neatly related at each instant to the properties of the external world within which the object moves at that instant.
This is our first example of what is called a instantaneous, or ``local in time'' description of the dynamics of some object, which means that by having chosen to ask about the right characteristics of the motion, and thus having chosen the correct description of the object, we are able to find an answer that depends only on the characteristics of the external world at that moment, without specific reference to what happened before, except as that is responsible for creating the velocity and position with which the object ``lives into'' the new moment in question.
It turns out that this description of the ``influence'' of the
external world on the object, which we see as being responsible
for a particular value of ``ma'' determined by what is the state
of the external world at that moment, is very useful as well as
very intuitive, so it has been given a special name,
the Force,
which
has become so universal that even the dimensions
of any quantity that are the same as
are now known
as the dimensions of Force. So just as a special name has
been given to velocity and acceleration because they are
frequently encountered and easily intuited quantities, a name
has been given to this other combination of dimensions, the
force. Hence we arive at the famous statement that
which is in fact a way of describing what about the object's motion (ma) takes the form it does, and how that relation to the state of the object's external world can be encoded in some simple number, F.