All input welcome. I did give a quote in a separate post from
<A History of Mechanics> by Rene Dugas which seemed to support
my view. Did you see it?
I also took another look at a "history of science" type book
that I read some time ago, <The Birth of a New Physics> by
I. Bernard Cohen. This is a great little book dealing primarily
with Kepler, Galileo and Newton. In discussing Newton's laws
Cohen suggests the following experiment:
Measure the acceleration a1 and a2 produced by two different
forces F1 and F2 and then compare the ratios to see if
F1/F2 = a1/a2
Cohen doesn't mention whether Newton actually did an experiment
of this type but clearly this is not a "test" of Newton's law
if the force is merely defined as some scaler times the acceleration.
Cohen also discusses one of the first critical tests of Newtonian
mechanics which goes something like this: First one uses Newtons
law of universal gravitation to compute the force exerted by the
Earth on the Moon [this answers Feynman's question: "what is the
force"]. Next one uses Newtons laws of motion to predict the
acceleration from which one can calculate how far the Moon will
deviate from its inertial (straight line) path in one second.
Newton found this to be 0.0536 inches. One can also determine
this value purely from geometry if one knows the orbital path
of the moon. Making a simplifying assumption of circular motion at
constant speed one can calculate a value of 0.0539 inches.
It is predictions such as this that won the world over to the
Newtonian view. Suppose Newton had just used the geometry
mentioned above to compute the acceleration and then said:
"well, there must be a force acting on the moon equal to the
mass of the moon times this acceleration" the Aristotelian
would say "why" and Newton would say "because there's this
acceleration and my law says ...". The Aristotelian would
then use the same geometry to compute the velocity and then
say: "well, there must be a force acting on the moon proportional
to this velocity". Newton says "why" and the Aristotelian replies:
"that's what the laws of physics say".
DR:===
>"Definition IV. An impressed force is an action exerted upon a body in
>order to change its state, either of rest or of uniform motion in a
>right line." (Definitions and Scholium, _Principia_, Book I)
>
>The second law, it seems to me, is a stipulation of how he wants to
>quantify/measure force - the definition of a metric. In your original
>message (the one I initially responded to) you said that "For Newton's
>second law to have any significance it is necessary for all three
>quantities (force, mass and acceleration) to be defined independently of
>the law itself." Did you mean to claim that the concepts had to be
>defined independently (even if the metric is stipulative) or did you
>mean that all three had to be independently measurable, the equation
>then being simply an empirical generalization? In any case, do you take
>the definition of force above to be in terms independent of
>acceleration?
>
Let me follow my usual tactic of providing an example
rather than answering your question directly :).
Suppose I place a small block of mass m on a surface
inclined at some angle theta from the horizontal and
observe that the block immediately begins accelerating
down the incline. If I understood you correctly, your
approach would be to measure the acceleration and
then use that together with the mass as a measure of
the force. I'll come back to this in a minute. First
let's consider a different problem of first calculating
the forces and then using Newton's law to calculate the
acceleration. First we identify what physical forces are
acting on the block. This we can do without any knowledge
of Newton's law. What we have physically acting on the
block are the weight and a really complicated distribution
of forces on the surface of the block in contact with
the inclined plane. This distribution is usually replaced
by its resultant which is then resolved into two components,
N and F where N is the component normal to the plane and
F (the friction force) is the component parallel to the
plane. It turns out that in most cases the friction force
depends only on the normal force and can often be reasonably
approximated as being some constant factor multiplied by the
normal force:
F = u*N where u is a constant which depends on the contact
surfaces.
Sorry for going into so much detail, the intent is to illustrate
what Feynman meant about getting involved with the "dirty"
objects of nature. People knew how to determine the weight
of an object long before Newton. They also knew about friction
forces but I'm not sure to what extent they could be calculated.
I do know that Stevin (1548-1620) solved the problem of
a block held in equilibrium on an inclined plane by a
counter weight and was able to determine the tension in
the connecting cable (another type of force). Stevin also
established the parallelogram law for adding forces.
Also, Galileo correctly deduced that the
drag force on a body moving through a fluid is proportional
to the velocity squared. Two points here, (1) Galileo did not
know Newton's law and (2) this force is proportional to
v^2 not the acceleration.
Getting back to our example, the point here is that the forces
(normal force, friction force and weight) have a physical
significance independent of Newton's law and would in fact be
present on the block even if the angle theta were small enough so
that the block were in equilibrium (i.e. the forces are
present even if the acceleration is zero).
OK, let's go on. At this point we know the weight W and the
relation between the friction force and the normal force
(F = uN). Now we can apply Newtons law to determine N and
the acceleration a. Now if we wish we can measure the
acceleration. Chances are the measured value will differ
from the calculated value due either to experimental errors
or innacuracies in the friction law.
Now, let's turn this around and suppose we take your suggested
approach (as I understand it). We measure the acceleration
and then compute a force m*a. Does this quantity (m*a) correspond
to any of the physically identifiable forces mentioned above?
No, it doesn't.
So, in answer to your question, in order for Newton's law
to be useful (in order to predict something like the
acceleration in the problem above or the deviation of the
moon from its inertial path discussed earlier), I have to have
some concept of force that is independent of the law itself.
In the above example I need to know what the weight is
and I have to have some way of estimating what the contact
forces are (the friction law F = uN).
DR:====
>Second, I'm not sure why you take the Feynman stuff to support your
>general contention.
>
>a. Feynman says (in your quote):
>"That is what Newton's laws say, so
>the most precise and beautiful definition of force imaginable might
>simply be to say that force is the mass of an object times the
>acceleration." I take it that you would claim that he is not using
>'definition' in any precise sense here?
>
You may want to read the quote more carefully. Feynman is saying
that it is very tempting to define force in this way, but that
such a definition is useless.
DR:======
>b. Feynman says (in your quote):
> "If we have discovered a fundamental law, which
>asserts that the force is equal to the mass times the acceleration, and
>then _define_ the force to be the mass times the acceleration, we have
>found out nothing."
>I'm not sure that that makes good sense. If we
>*discovered* that F = ma, we would (presumably on your view) have to
>already have some independent concept and measure of F. What would it
>mean to then subsequently *define* F as ma? Wouldn't we already
>*have* a definition?
>
What Feynman is saying is that it *doesn't* make sense to define
F as ma. It just *seems* to be a good idea which turns out to be
useless. People understood the concept of force long before Newton.
Remember that Newton had still to deal with the Aristotelian view
of physics according to which no body could move unless a force
were acting on it. Further the force was thought to be proportional
to the velocity rather than the acceleration. This emphasizes the
truly radical nature of the first law, which says that a body
can move at constant velocity in a straight line *forever* with *no*
force acting on it. "Say What!!!" say the Aristotelians. I don't
think it would bode well for Newton to just define what it is that
he needs to show :).
DR:==
>c. Feynman says (in your quote):
> "The real content of Newton's laws is this: that the
>force is supposed to have some _independent properties_, in addition to
>the law _F = ma_; but the _specific independent properties that the
>force has wer not completely described by Newton or by anyvbody else,
>and therefore the physical law _F = ma_ is an incomplete law. It
>implies that if we study the mass times the acceleration and call the
>product the force ... then we shall find that forces have some
>simplicity;... it is a suggestion that the forces will be simple."
>
>One way of reading that is that "F=ma" is a gross definition, but that
>the intricacies of its specific applications and extensions will have to
>be worked out case by case (empirically). If that is what Feynman
>intends, that would not necessarily change the definitional character of
>the equation - only its detailed completeness. And saying that "if we
>study the mass times the acceleration and *call* the product the force
>..." (my emphasis) certainly sounds very like defining force in the
>relevant terms.
>
>The thrust of the Feynman passage, it seems to me, is that Newton begins
>with some definitions, but that since confining ourselves to mere
>definitions will never generate any actual *science*, we have to find a
>content for those definitions. So, says Feynman, "the real
>[*scientific*] content of Newton's laws" involves pursuing the F=ma
>framework/structure into nature empirically, and in adopting that
>framework/structure as foundamental guiding principle in doing one's
>science, one is thereby commited to a picture of nature as ultimately
>reducing to a simple underlying pattern/motif (specifically, that the
>world is F=ma-ish). That sounds to me very like the Rothman approach I
>mentioned last time, and very different from what I understood you to be
>saying.
>
This seems to coincide more or less with the way I view things,
perhaps I've been unclear. I guess where we disagree is that,
according to my view, the "F=ma framework/structure" is that
there is a general law of nature (Newton called it a
law of motion) which states that the motion of an object can
be determined from the forces that are acting on it. Of course,
we can also solve the inverse problem, but since its given
as a law of *motion* the most direct interpretation is that
it is a law that predicts motion given forces. Now we can
go into the real world and see if the knowledge of the forces
acting on a body allow us to determine the motion. If the forces
are themselves just a scaler quantity times the acceleration then
there is nothing to check, no empirical content. Newton's law
is true by definition.
This is the point of Feynman's "gorce" example, from which we
get Feynman's Second Law:
"everything stands still except when a gorce is acting"
Now if we physically identify this "gorce" with force then
we actually have a law from Aristotelian physics. I'm not
quite sure if Feynman chose the example for this reason or
not. Anyway, if one merely defines "gorce" as the rate of
change of position (i.e. velocity) then the "law" is certainly
true, but also useless since it merely says that things
either stand still or they don't stand still. Note that Feynman
says that this example is analogous to defining force as
mass time acceleration. In this case the "law" merely states
that things either accelerate or they don't accelerate.
DR:======
>That impression is strengthened by the Feynman addendum you included. n
>Feynman says:
>"...we cannot just call F=ma a definition, deduce everything purely
>mathematically, and make mechanics a mathematical theory, when mechanics
>is a description of nature. [snip] [S]ooner or later we have to find out
>whether the axioms are valid for the objects of nature. Thus we
>immediately get involved with these compolicated and 'dirty' objects of
>nature..."
>
>That does not imply that F=ma is not definitional - only that we cannot
>*stop* with a definition and its purely abstract derivations if we are
>going to do real science. In particular, we have to see *if the system
>applies* - i.e., as Feynman says, "whether the axioms are valid for the
>objects of nature." But that is just the Rothman scheme.
>
>So again, I'm not sure that Feynman really helps you here, unless I'm
>misreading either you or him. I will ignore the possible charge that I
>am misreading me.
>
I must say that I was rather surprised by this as I thought that
Feynman had sealed the case for me :). After all, he does say
specifically that defining the force by means of Newton's law
is useless.
Now, I would tend to agree that Rothman and Feynman seem to be
saying similar things. Do you have a reference for the "Rothman
scheme" so I can look at it in more detail? BTW, how can we
possibly check whether "the system applies" unless we have some
means of determining a force independently of Newton's law?
phew, sorry for going on and on. An example of the principle
of inertia perhaps?
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"If cucumbers had anti-gravity,
sunsets would be more interesting"
-- Wesley Elsberry