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Re: [Phys-l] Equations (causal relationship)



John Denker skribis:

Ken Caviness wrote:

1. Why do you have one spring attached to a stage, the other two
attached to posts attached to the table?

I needed to attach the particle to the stack-of-stages somehow.
I treat this attachment as one of the springs, and account for
the associated force as one of the forces.

Ok. There are 3 springs (3 forces) and 3 stages (allowing us to talk
about 3 relative acceleration vectors). Here you are treating the final
spring as the attachment to the top stage, yet if the particle stays
aligned with the top stage and both are at rest relative to the table it
doesn't matter if this spring is attached to the table or to the top
stage. My sense of the "rightness of things" was that all three 3
springs should be attached to poles attached to the table, but it
doesn't matter in your original setup. (More on this below.)

That appears to be an
essential asymmetry in your setup.

Life can be so asymmetric sometimes.

I had understood that all forces
were measured with respect to the lab frame, therefore we want all
springs attached to fixed posts in the lab frame.

The force is the force. You do not need a reference frame in order
to measure the force (non-relativistically speaking). All observers
in all frames agree what the force is.

All observers in all _inertial_ frames agree. Each of your intermediate
stages is accelerating relative to the table (which is assumed to be
inertial), and thus is non-inertial. But indeed, if your last spring is
attached to the top stage (which we agree is at rest relative to the
table) then everything is ok. It is only because I wanted to extend the
experiment to allow swapping the order of stages that this asymmetry is
problematic. If any spring were attached to any stage which is
accelerating (or even moving, for that matter) relative to the table and
particle, the force would vary, i.e., it would be a different problem
from the one described in the experiment.

So what I have done may
look ugly to some eyes, but it doesn't cause any problems with the
physics.

Due to my ineffective communication, no doubt, you have misunderstood my
concern. I was not concerned here with aesthetics but with getting the
right answer. Vector addition is commutative, the answer is unaffected
by a symmetric swap of the vectors. Reflecting this, your stages can be
stacked in any order without affecting the result. There must be no
asymmetry in how the 3 stages are treated, or you will not get the
result you are attempting to show. In fact, it is _only_ because the
top stage is in fact at rest relative to the table that you can get away
with the asymmetry of attaching the last spring to the top stage instead
of attaching it to the table like the other two. This asymmetry weakens
the thought experiment, and indeed was a contributing factor in _my_
misunderstanding of your explanation -- I thought, "One end of a spring
attached to a stage? But the stages are accelerating!"

Ugliness is indeed in the eye of the beholder, but as an educator I'm
frequently concerned with more effectively communicating the physics,
not merely ensuring there are "no problems" with it.

Anyway, I think I've understood you now. ;-) Allow me to paraphrase,
to check:

The particle stays lined up with the top stage. The (zero) acceleration
of the top stage is the vector sum of the relative accelerations of the
various stages, each with respect to the stage/table below them. The
"components" of the acceleration of the top stage (i.e., the individual
relative accelerations of each stage with respect to the stage/table
below it) are physically meaningful, thus there is some meaning to these
"components" of the acceleration of the particle itself.

Is this a correct restatement of the argument?

It's clear that the experiment can be generalized to N springs and N
stages, to springs and stages operating in 3 dimensions, even to nonzero
particle acceleration (at least as an approximation for a short period
of time, until the particle moves and the springs are no longer parallel
to the axes of the stages). Yes, a very nice thought experiment!

2. .... The particle is subjected to _multiple_ forces. We can
measure them. The particle experiences _one_ acceleration, we can
measure it.

I still see that as a matter of opinion, having nothing to do with
the laws of physics.

The laws of physics say that only the net force matters.
The laws of physics say that only the net acceleration matters.

The laws of physics say we can add force vectors to find the net
force.
The laws of physics say we can add acceleration vectors to find the
net
acceleration.

Maybe you *want* to decompose the force into a set of forces, and
maybe you *don't* want to decompose the acceleration in the same
way ... but what you "want" isn't physics.

I hear you, and I think that you are right (mostly). I do still see a
difference between the proposed decompositions: a physical difference, a
difference in meaning, but no mathematical difference. Again, it's not
a matter of my *wanting* to decompose the net force [*], I just read the
meters on the springs. That's data.

[* What I actually *want* is to decompose all the forces and
accelerations into x- and y- (and z-) components using some convenient
choice of coordinates, but you're right, that isn't physics, that's just
a convenience.]

Clearly, the accelerations of the stages are also data. But when you
apply your decomposition of the particle's zero acceleration into what I
was pleased to call "fictitious components" (that is, the [real]
relative accelerations of the stages), you are singling out one
decomposition as physically meaningful, giving it undeserved importance
(for the particle). That is (IMO) what some other posters on this
thread meant in referring to the underdefined system

F1 + F2 + F3 = m(a1 + a2 + a3)

So the upshot of my musings is that I'm becoming convinced that although
you have indeed succeeded in showing three nonzero accelerations that
add to zero, I would argue that the _particle_ still only has one
physically measurable and meaningful acceleration.

Hmmm, a related question: we use the net force all the time, but can
you measure it? No, you calculate it from the measured actual forces.
Or you calculate it as the opposite of an extra force with exactly
balances all the original forces. So in my book, the original forces
are real (in some sense), the net force a convenient mathematical
device. In the same way, I would call your decomposition of the
particle's acceleration into components a convenient mathematical
device. But at least the net force is unique for any given set of real
forces! Your decomposition of the zero acceleration is arbitrary (to a
great degree), but you *wanted* to decompose it that way, right?
[Sorry, I couldn't resist needling you just a little on that one.]

......... But notice
here neither F nor m need be considered causative, we simply have a
correlation between them.

That's the general case. Why not just accept it as the general case,
and move on?

Agreed. [I actually meant "neither F nor a", as I'm sure everyone
realized.] So, I will stipulate that I see no reason in classical
physics to assign a causative value to F as opposed to a. What we have
is a correlation, not evidence of causation.

But you've given short shrift to my musings on the implications of the
special and general relativistic pictures. I would still be interested
in any comments on either or both.

We can still add the forces as vectors in
special relativity, but we can't add relative velocities with our
non-relativistic formulas and although I haven't done it I would
think
that relative accelerations would be even worse. Does this mean
force
is more "basic" in some way than acceleration?

No.

Force is not much used in general relativity (or even special
relativity).
There are even different schools of thought as to whether force should
be defined as d(p)/d(t) or d(p)/d(tau). Most people don't care enough
to even have an opinion on the matter.

Yes, quite right. The dp/dtau seems to be more mathematically motivated
(by the 4-vector notation), the dp/dt by a direct extension of the
classical concepts.

However, under special relativity, forces retain some semblance of the
behavior we expect: they add vectorially. Take your thought experiment,
treat it using the formalism of special relativity. All forces are
measured in the lab frame, no transformations needed, no problems there.
But your stages' accelerations are each measured relative to the
stage/table below it, so we need to use relativistic formulas for the
addition of relative accelerations. I re-quote here your previous
statements:

The laws of physics say we can add force vectors to find the net
force.
The laws of physics say we can add acceleration vectors to find the
net
acceleration.

Under special relativity, only the first is actually correct, the second
is an approximation of strictly limited validity. Perhaps this
distinction seems not very important to you, but much of the elaboration
of your argument has been based on an assumed *equivalence* of the two
techniques: vector addition of forces, vector addition of
accelerations. But in fact, vector addition of forces is valid, vector
addition of relative velocities is an approximation -- convenient that
we can use the same method for each case, but they are intrinsically
different. That's what I was attempting to point out, together with the
fleeting thought that the simpler case (forces) might be more
fundamental than the more complicated case (accelerations). [But why
assume simple = fundamental? How to prove the validity of Occam's
Razor?]

Instead, they just keep track of the momentum. Any force problem can
trivially be transformed into a momentum problem.

Of course. And in general relativity we can have relative accelerations
without forces at all. So a force doesn't cause an acceleration (or a
rate of change in momentum), and in fact the only reason we see a close
correlation is that other factors are being ignored. There, that
answers the original question: force doesn't cause acceleration.

[No wait, that still leaves a loophole: a force could still be _one_
cause of acceleration, another being spacetime curvature. Oh, well!]

Back to grading,

Ken Caviness