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about the properties of force. |
For example, in dealing with force the tacit assumption is always made that |
the force is equal to zero unless some physical body is present, that If we fnd a |
force that is not equal to zero we also find something ¡in the neighborhood that is |
a source of the force. 'PThis assumption is entirely diferent from the case of the |
“gorce” that we introduced above. Ône of the most important characteristics of |
force is that it has a material origin, and this is nø£ just a defñnition. |
Newton also gave one rule about the force: that the forces between interacting |
bodies are equal and opposite—action equals reaction; that rule, it turns out, |
--- Trang 232 --- |
1s not exactly true. In fact, the law = ma is not exactly true; iÝ iE were a |
defnition we should have to say that 1t is øløas exactly true; but ït is not. |
The student may object, “[ do not like this imprecision, I should like to have |
everything defned exactly; in fact, it says in some books that any science 1s |
an exact subject, in which cuerwthing is deñned.” TỶ you insist upon a precise |
defnition of force, you will never get itl Pirst, because Newton”s Second Law is |
not exact, and second, because in order to understand physical laws you must |
understand that they are all some kind of approximation. |
Any simple idea is approximate; as an illustration, consider an object,... what |
is an object? Philosophers are always saying, “Well, Just take a chair for example.” |
"The moment they say that, you know that they do not know what they are talking |
about any more. What ¿s a chair? Well, a chair is a certain thing over there.... |
certain?, how certain? 'Phe atoms are evaporating from ït from time to tỉme——=not |
many atoms, but a few—dirt falls on it and gets dissolved in the paint; so to |
defne a chaïr precisely, to say exactly which atoms are chaïir, and which atoms are |
air, or which atoms are dirt, or which atoms are paint that belongs to the chaïr is |
impossible. So the mass of a chair can be defned only approximately. In the same |
way, to delñne the mass of a single object is impossible, because there are not any |
single, left-alone objects in the world—every object is a mixture of a lot of things, |
so we can deal with it only as a series Of approximations and idealizations. |
'The trick is the idealizations. 'To an excellent approximation of perhaps one |
part in 1010, the number of atoms in the chair does not change in a minute, and |
1Í we are not too precise we may idealize the chair as a defñnite thing; in the same |
way we shall learn about the characteristics of force, in an ideal fashion, if we |
are not too precise. Ône may be dissatisied with the approximate view of nature |
that physics tries to obtain (the attempt is always to increase the accuracy of the |
approximation), and may prefer a mathematical definition; but mathematical |
defnitions can never work in the real world. A mathematical definition will be |
good for mathematics, in which all the logic can be followed out completely, but |
the physical world is complex, as we have indicated in a number of examples, such |
as those of the ocean waves and a glass of wine. When we try to isolate pieces of it, |
to talk about one mass, the wine and the glass, how can we know which is which, |
when one dissolves in the other? 'Phe forces on a single thing already involve |
approximation, and if we have a system of discourse about the real world, then that |
system, at least for the present day, must involve approximations of some kind. |
'This system ïs quite unlike the case of mathematics, in which everything can |
be defñned, and then we do not knou what we are talking about. In fact, the glory |
--- Trang 233 --- |
of mathematies is that 0e do no‡ hœue to sa that tue are talking abou‡. The gÌory |
1s that the laws, the arguments, and the logic are independent of what “it” is. lÝ |
we have any other set of objects that obey the same system of axioms as Buclid”s |
geometry, then if we make new defñnitions and follow them out with correct logic, |
all the consequences will be correct, and it makes no diference what the subject |
was. In nature, however, when we draw a line or establish a line by using a light |
beam and a theodolite, as we do in surveying, are we measuring a line in the sense |
of Euclid? No, we are making an approximation; the cross hair has some width, |
but a geometrical line has no width, and so, whether Buclidean geometry can be |
used for surveying or not is a physical question, not a mathematical question. |
However, from an experimental standpoint, not a mathematical standpoint, we |
need to know whether the laws of Euelid apply to the kind of geometry that we |
use in measuring land; so we make a hypothesis that it does, and it works pretty |
well; but it is not precise, because our surveying lines are not really geometrical |
lines. Whether or not those lines of Euclid, which are really abstract, apply to |
the lines of experience is a question for experienece; it is not a question that can |
be answered by sheer reason. |
In the same way, we cannot just call ?' = rmma a defnition, deduce everything |
purely mathematically, and make mechanics a mathematical theory, when me- |
chanics is a description of nature. By establishing suitable postulates it is always |
possible to make a system of mathematics, just as Euclid did, but we cannot make |
a mathematics of the world, because sooner or later we have to fnd out whether |
the axioms are valid for the objects of nature. Thus we immediately get involved |
with these complicated and “dirty” objects of nature, but with approximations |
©V€T lnCreasing In accuracy. |
12-2 Eriction |
The foregoing considerations show that a true understanding of NÑewton'°s laws |
requires a discussion of forces, and ït is the purpose of this chapter to introduce |
such a discussion, as a kind of completion of Newton's laws. We have already |
studied the defnitions of acceleration and related ideas, but now we have to |
study the properties of force, and this chapter, unlike the previous chapters, will |
not be very precise, because forces are quite complicated. |
To begin with a particular force, let us consider the drag on an airplane fying |
through the air. What is the law for that force? (Surely there is a law Íor every |
force, we rmus‡ have a lawl) One can hardly think that the law for that force will |
--- Trang 234 --- |
be simple. 'Iry to imagine what makes a drag on an airplane flying through the |
air—the air rushing over the wings, the swirling in the back, the changes going |
on around the fuselage, and many other complications, and you see that there is |
not going to be a simple law. Ôn the other hand, it is a remarkable fact that the |
drag force on an airplane is approximately a constant times the square of the |
velocity, or F` cu. |
Now what is the status of such a law, is i9 analogous to F' = ma? Not at |
all, because in the first place this law is an empirical thing that is obtained |
roughly by tests in a wind tunnel. You say, “Well ' = rmø might be empirical |
too.” That is not the reason that there is a diference. The difference is not that |
1t is empirical, but that, as we understand nature, this law is the result of an |
enormous complexity of events and is not, fundamentally, a simple thíng. ITf |
we continue to study it more and more, measuring more and more accurately, |
the law will continue to become more complicated, not /ess. In other words, as |
we study this law of the drag on an airplane more and more closely, we find |
out that it is “falser” and “falser,” and the more deeply we study it, and the |
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