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



Nice picture. Thanks.

But the picture and description below raise two new questions in my
mind:

1. Why do you have one spring attached to a stage, the other two
attached to posts attached to the table? That appears to be an
essential asymmetry in your setup. 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 other option is
to attach one spring each to the table, the blue stage, and the
green/striped stage (working from the bottom up), ensuring that the
force given by the spring extension is relative to the same frame of
reference as the corresponding acceleration. But I would think this
would not work, since the stages are accelerating and over time would
move, resulting in varying forces, not what was desired. That would put
us back to needing all springs attached to poles on the table, but
somehow keeping the alignment of the spring with the corresponding
stage. Or have I still misunderstood the significance of the thought
experiment?

2. I still see an essential difference between these accelerations
which happen to add up to zero and the forces. As has been pointed out,
we could program the stages in an infinite number of ways such that the
acceleration of the top stage relative to the table is zero. Of course,
I agree that a_i is the acceleration which the particle _would_have_
experienced had only F_i been applied. But it wasn't the only force
applied, and in fact the particle experiences only one acceleration,
here zero. Your stages' individual relative accelerations carefully
mimic these fictitious "might-have-been" non-linearly-independent
"components" of the acceleration, but that doesn't make the fiction
reality. The particle is subjected to _multiple_ forces. We can
measure them. The particle experiences _one_ acceleration, we can
measure it. Your experiment imputes multiple "component" accelerations
to the particle, but please explain why there is any meaning at all to
these, why they are more meaningful than 0+0+0, or any other choice.
Whereas the individual forces have clear physical meaning: I can read
off the magnitudes and measure the angles. The exist individually, I
need to use their sum. The separate accelerations you have found also
exist individually for the stages, but not (I would argue) for the
particle itself. Mathematically both of these sets of vectors have
meaning, and we know how to work with them, but only the set of forces
describes anything in the physical world.

Having argued that position, let me back off slightly: I certainly can
see that from one point of view, force and acceleration simply accompany
one another and it merely depends on the experimenter's preferences
which is treated as the independent variable. We do a fun lab in
Introductory Physics with a PASCO cart, force sensor and motion sensor /
range finder. Just pull/push at will on the force sensor attached to
the cart, let the computer record force & distance data, calculate the
acceleration from the distances, and graph F vs. a on a scatter plot.
The points jump all over, but clearly delineate (ha ha!) a line (through
the origin) whose slope we then calculate. Surprise! It's m. We
measure the mass of the cart and compare to our slope. Fun. But notice
here neither F nor m need be considered causative, we simply have a
correlation between them.

That makes me think again of general relativity. Rather than a
gravitation force F which depends linearly on the mass m, and an
acceleration a = F/m which oh so conveniently gives us the same
acceleration for all masses in the gravitational field, whatever their
mass might be -- rather than that, we have all objects moving on
geodesics (the straightest line they can find) in a given gravitational
field. So of course they end up with the same acceleration! So rather
than saying that a force "causes" an acceleration, there is no force at
all. :-) (At least not a gravitational force.) The acceleration is
caused by the spacetime curvature, which is a result of the mass-energy
distribution.

Lastly, no one has jumped in on my previous comment about the special
relativistic version of F = m a. I did misspeak in my haste, the
relativistic formula is _not_ F = gamma m a. What I was thinking of is
the extra factor of gamma = 1/sqrt(1-v^2/c^2) in the relativistic
momentum: p = gamma m v. The relativistic version of N2L is: F =
d(gamma m v)/dt, the right hand side ends up being a function of both v
and a. But my query stands: 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? Don't know, but it does
mean that the relative accelerations of the different stages (in the
thought experiment) with respect to each other will not be treated in
the same as the forces (in the lab frame).

My last two paragraphs seem to be leaning opposite directions on the
issue of the relative importance of force or acceleration, but both
highlight _differences_ in their handling. Time for me to get back to
grading papers....

Ken

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu
[mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John
Denker
Sent: Wednesday, May 03, 2006 12:11 AM
To: Forum for Physics Educators
Subject: Re: [Phys-l] Equations (causal relationship)

Michael Edmiston wrote:

Can you draw a picture?

http://www.av8n.com/physics/img48/f3a3.png

The rectangles represent the translation stages. Each stage moves in
the
direction given by the long axis of the rectangle.

The red stage moves relative to the green/striped stage.
The green/striped stage moves relative to the blue stage.
The blue stage moves relative to the table.

The particle of interest is shown as a gray disk. It is not attached to
anything except the three springs.

The springs represent the three forces. That is, by observing the
elongation
of the springs, we can ascertain the forces being applied to the
particle.

One spring is anchored to the red stage. The other two springs are
anchored
to posts attached to the table. The other (non-anchored) end of each
spring
attaches to the particle.

The springs produce three nonzero forces F1 F2 F3 that sum to zero.

The stages produce three nonzero accelerations a1 a2 a3 that sum to
zero.

The particle is attached only indirectly to the top stage, but we can
observe that the particle does not move relative to the top stage, so
the acceleration (zero) of the top stage is also the acceleration of
the particle. You can reach the same conclusion in several other ways,
including by using a verrrry stiff spring to connect the particle to
the top stage.

He said to align the moving stages with F1, F2, and F3 respectively,
and
adjust their operation so that they accelerate with a1 = F1/m, a2 =
F2/m, and a3 = F3/m. Okay, but they could also be adjusted to F1/k,
F2/k, and F3/k where k is any number.

Yeah, so what?

The decomposition of zero net acceleration can be done many ways; my
chosen values a1 a2 a3 are not unique ... but in the same breath I
point out that the decomposition of zero net force can be done in many
ways; the values F1 F2 F3 are not unique.

I hate to sound like a stuck recording, but the point remains:
anything you can say about the force vectors I can say about the
acceleration vectors. They're vectors.

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