Around this discovery that for a large class of physical systems, such a distinction can be made, Newton framed a set of consequences that follow from the fact that it is the force and not some other quantity that makes possible such a local description.
First, he noticed that, since it is the acceleration and
not some more primitive quantity (such as velocity, which
requires less extrapolation to obtain the sequence of positions
that make up the actual path) which is constrained to have a
particular value because of the presence of the rest of the
world, if there is no force in a given instance, the
acceleration will then be zero, and the velocity will be
constant. This consequence is the origin
of Newton's Law of Inertia,
that
there is not anything more special about one
speed than about any other speed, but that constant speed
is the characteristic of an object that is not ``acted upon'' by
an external world. Hence the famous saying is not only that
``an object at rest tends to remain at rest
', but rather
that ``an object at rest or in a state of uniform motion
[i.e., constant velocity] tends to remain in that state unless
acted upon by some outside force'. This brings us to
recognize that, in the absense of the influence of an external
world, there is something about the motion of the object that
stays the same, which should naturally be of interest to
us in describing the system, because things that are the same at
every moment along the path can be simpler to describe than
things that are constantly changing, and so have to be
recalculated at each new moment.
It is not obvious yet which thing we should consider as staying
the same, because in Newton's formulation, in the absence of a
force, both the acceleration a alone, and the combination ma,
remain zero, so we could take either the velocity of which
the acceleration is the rate of change, recall, or some
combination like (of which
then ma is the rate of change), or even anything else that
depends only on the velocity and material properties of the
object, as being the useful or ``physically important'' quantity
that stays the same. We can suspect, though, that, since the
``Force'' exerted by the external world has no direct reference
to producing only an a, but rather always determines the
value of ma in combination, so that two forces can produce
very different accelerations in different objects if they have
different masses and so different resistances to being moved,
the relevant quantity ``produced'' by the force from the
external world should be taken to be the combination ma. In
that case, the thing which the force of the external world acts
to change, and which thus stays unchanged in the absence
of that force, should be taken to be ``mv'. This quantity,
which corresponds to Newton's concept of ``inertia'' has since
come to be known more widely by the name momentum,
and the
units
to be called the units of momentum.
Momentum turns out to be a very important concept, about which we will have a great deal to say in the next chapter.
For now, we remark some cautions. Newton's ``law of intertia',
which we will see in the next chapter has a much more general
statement called the principle of conservation of
momentum, is a very important and very basic principle from
which a great many particular phenomena can be understood. And
in a way, Newton recognized this importance, which is why in his
presentation, and in many of the exegeses that have succeeded
it, the ``law of inertia'' is given primacy as a ``more
fundamental'' principle, or as being closer to a ``starting
point'' from which to understand the laws. Its rate of change,
ma, is then only related to characteristics of the external
world as a later refinement. There is a certain logic to this,
but it can lead to dangerously wrong prejudices.
Newton may have been able to verify in a large collection of cases that this quantity, mv, which he called inertia, was conserved, and that when it changed by interaction with an external world, it did so in such a way that F = ma. But it should seem to the cautious reader that the cases demonstrate the rule. It is through the cases that people come to notice that this rule is true. There is no matter of ``principle'' that we have seen so far that tells us that this quantity ``mv'' stays the same in the absence of influences from an external world. Though a great genious may have guessed at the importance of this conservation statement, as a more basic principle it does not yet come from anything more than the observations which have already given us F=ma. It may be viewed up to this point as merely a consequence of the discovery that the equation works for this set of cases.
This is important because we do not know how far beyond this set of discovered cases the same result should be expected to hold. Is it always ma that equals F? Is it always mv that is the important unchanged object in the absense of interactions, and which is changed proportionally to F when they are present? So far, we have no grounds to say. This point is mentioned here, because even physicists have all-too-often succumbed to an unfortunate ``reverence'' for the great men of the past, and led themselves into strange confusions and awkward and confounding use of words, because they assume they know some principle more broadly than they actually do. We will see later that, while the principle of conservation of momentum is indeed in an important sense ``more fundamental'' than F = ma, once we find out the real observational origin that gives it its fundamentality (the observable property of summetry, as it turns out), that principle will also tell us what the momentum is that is conserved. And it will not always be mv. Correspondingly F will not always equal ma, though it will always equal something with the dimensions of an ma.
This warning is offered for protection against people who for some reason think that, because Newton chose to present a ``law of inertia'' as closer to a starting point than F = ma, and because the inertia that follows from F = ma involves the quantity mv, somehow it is given as a principle that momentum is necessarily the mv of any object. When, in relativity, we have a need to understand what is conserved, and how it changes, we will find that the momentum is not at all mv. If, at that time, we understand where the observational basis of our more fundamental momentum principles are (namely, in symmetry), this will come as no particular surprise, because we will expect that until deductions from the observations tell us what the momentum is, we have no reason to expect anything in particular. Had we chosen to hang dogmatically on a rhetoric that momentum equals mv, somehow handed down on stone tablets from Newton, we would be forced to invent all sorts of strange notions about a ``mass that depends on how fast the object is moving relative to the person who is seeing it, at which point the mass would lose all of its utility as a simple measureable property of the object alone.
All that confusion is unnecessary, even though it is very common. The way to avoid it at the start is to recognize that our ``philosophical impressions'' of momentum have no weight except what is given them by the equations that we have directly checked by observation to be true, in which case it is perfecly sensible and appropriate to regard F=ma as a discovered relation which we can apply in a wide collection of cases, from which constancy of mv follows as a mere consequence until we can observe it in something more basic and far-reaching.
Finally we turn to one last consequence of our law, which is very
useful and is the third of Newton's fundamental propositions about how
man can explain the workings of nature. Consider the case of the
object on the spring. Suppose we place a speck of dust between the
end of the spring and the object, and decide that we want to describe
the speck of dust, as in fig. 5.4.
In this new case the speck of dust is our object of interest, and both the spring and the large thing attached to its end are parts of that speck of dust's ``external world'. Now if we choose that speck of dust to be very light, compared to every other aspect of the spring-mass system, it becomes clear that for practical purposes it has ignorable resistance to being accelerated. Yet the speck of dust, in spite of having a mass that we can approximate as zero, does not move with infinite acceleration. It moves, sandwiched between the spring and the heavy mass, in precisely whatever way they would move if it were not there. Thus, in any correct description of this speck of dust, the force on the F side of its equation cannot only be the force from the end of the spring that we mentioned in the last section. In fact, to the extent that we really mean to approximate the mass as zero, for F to be the same as (``equal to'') ma of the dust, for any finite acceleration, the F must be zero, or as close to it as we like to consider.
This means that, for the speck of dust sandwiched between the spring and the heavy object, there must be two forces ``pushing'' on opposite sides of it, of equal magnitude but in exactly opposite directions, so that in addition to the F from the spring that we had before, there is another ``-F'' that cancels it to zero in the dust's external world. This -F, of course, must be coming from the ``intertia'' of the heavy object. So we see that, in those cases when it is appropriate to regard a heavy object as a part of an external world, its resistance to being moved causes it to effectively exert a -F against anything that pushes on it, in response to its resisting that push.
At this stage, of course, we can dispense with considering the
speck of dust, and recognize that precisely that -F which the
mass was exerting on the speck of dust is the -F that it is
``exerting back'' on the end of the spring, whether by means of
an interposed speck of dust or just directly. We thus arrive at
what is sometimes called Newton's law of action and
reaction. It usually has a somewhat confusing-sounding
statement, until one thinks of it in context of the way one
tries to form equations of motion by dividing nature into an
object and an external world. We state it like this: Whenever
it is possible to describe the motion of some system as F = ma,
where ma is the ``response'' of what we want to consider as the
object and F is the ``force exerted on it'' by its ``external
world', we can also choose to to regard the same equation the
other way around, by considering a description in which the
original ``object'' is now treated as a part of a newly divided
``external world'. In this latter case, the role of the
previously-object as incorporated in this newly-specified
external world is to contribute a ``force'' exactly opposite to
that which accounted for its motion before. By example:
When we had ,
we could also have considered
where the latter is effectively regarded as zero becase the mass of the speck is so small. But when we turn to considering the speck itself, we change the roles of the big object, and include it on the ``force'' side of the equation, which then becomes:
at which point we are considering the term as just another force, which happens to be exactly
opposite to
when the dust speck
mass is chosen to be zero. There is a whole formal structure
for extracting great use from this way that the same term can be
treated either as the response of an object of interest, or
alternatively as just another part of something else's
``external world''. It will be explored below, after we have
seen a particularly remarkable and historically significant
example of what this kind of relatiion can be used to deduce.