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[Phys-l] accelerated frames (was: pseudo force)



On 10/26/2006 10:18 AM, Bob LaMontagne wrote:

??? The original question was WHAT IS a pseudo-force? From the responses so
far, it appears that the terms pseudo-force and non-inertial force are being
used interchangeably. Is that correct?


That's a good question. I don't have an answer, but that doesn't
make it any less of a good question. :-)

I think most people would answer "yes", but I think we might
want to step back and seriously consider other options.

In particular, I think the important issue is more sharply
focussed if we ask
** What are the laws of motion in accelerated frames? [1]

IMHO it is not a foregone conclusion that "pseudo forces" are the
key to answering question [1]. I apologize if I got the discussion
started on the wrong foot by asking about pseudo forces.

Surely question [1] has a good answer. We believe that motion
is lawful, even motion relative to an accelerated frame.

========

By way of terminology, I would like to distinguish the following:
-- Newton's laws, by which I mean Newton's laws in their conventional
Newtonian form.
-- the laws of motion, by which I mean the correct laws of motion,
in whatever form applies to the situation at hand.

Just to save words, I don't want to keep saying "Newton's laws in
their conventional Newtonian form". We all agree that Newton's
laws do not apply in accelerated frames. So the key question
reduces, again, to asking what laws /do/ apply. Optionally we
can compare and contrast the actual laws to Newton's laws.

N2 asserts that in a Newtonian frame,
a = Fu/m [1]
where Fu is the force exerted /upon/ the body by its surroundings.

Now, just to state the obvious, we *do* know how to generalize N2
to rotating frames. Assuming nonzero mass, theanswer can be written
as:

a = Fu/m + centrifugal acceleration + Coriolis acceleration [2]

or equivalently

a = (1/m) (Fu + centrifugal force + Coriolis force) [3]

To make contact with David Bowman's analysis, I don't think of
equation [2] as a violation of N2, nor as an attempt to "repair"
N2 by adding artificial forces. It's just a different equation,
valid in a different regime ... with the special property that
it reduces to N2 in the limit of zero rotation-rate.

So it appears that at least part of the discussion revolves around
philosophical and metaphysical questions about how to /interpret/
the terms in the equations of motion. If you write things in the
form of equation [3], we have a couple of frame-dependent terms
with dimensions of force, and we face the question of whether
they are "really" forces. I don't know the answer, and I'm not
sure the answer has any physical significance. The equation of
motion is what it is, no matter what interpretation we place
upon the terms. The arbitrariness is particularly clear when
we look at equation [2], where the frame-related terms have
dimensions of acceleration, and could perfectly well be moved
to the other side of the equation, where they wouldn't seem
force-like at all.

==========================

Returning to the more-general topic of the full laws of motion in
contrast to Newton's laws, I would like to particularly emphasize
the distinction between:
-- violation of N3, and
-- violation of conservation of momentum.

Momentum is well defined, as a physical entity unto itself, independent
of any reference frame. The momentum doesn't care whether you have a
reference frame that is freely falling, uniformly accelerated, rotating,
... or no reference frame whatsovever.

Momentum is conserved. If you find some laws of motion that make it
look like momentum is not conserved, the laws are wrong. Try again.

The true momentum may not be equal to MV in such-and-such reference
frame, and it may be /inconvenient/ to evaluate the true momentum in
that frame ... but that's just the cost of doing business in that
frame.

====================================================

Now, if you want to hear something entertaining that I just noticed:
Newton's laws are not strictly valid in the usual Newtonian laboratory
frame, even if we neglect the earth's rotation.

This is related to the idea of gravity being a pseudo force, and the
point that N3 is *sometimes* violated by pseudo forces. We must ask,
are there any conditions where N3 is /not/ violated by pseudo forces?

Obviously there is the trivial case where the pseudo forces are zero,
i.e. when we are using a freely-falling reference frame. We exclude
this trivial case from further consideration.

It turns out that there are nontrivial solutions, namely in cases
where our local frame is moving parallel to the CM of the system.
In an isolated system, the CM (considered as an abstraction) will
undergo unaccelerated motion, but that does not mean that a local
coordinate system located at the CM will be freely falling, since
there could be humongous local gravitational fields at that point.
The same logic applies even more strongly to locales distant from
the CM but moving parallel to it; there could be humongous gravitational
fields at the location of our local frame.

Four versions of this are depicted here:
http://av8n.com/physics/img48/pseudo-force.png

In the upper left corner, we have the world lines of two massive
particles A and B. They are initially at rest with respect to
one another, but they accelerate toward each other (due to gravity)
as time goes on. The observer O is moving parallel to the CM
and observes that momentum is conserved, using the convenient
MV definition of momentum.

In the lower left corner, we have the same scenario, except
that the observer chooses to move along an accelerated path,
pacing object B. In this case the observer finds that the
MV of object B remains zero, while the MV of object A starts
out zero but changes with time. This observer concludes that
MV is not conserved. (In my opinion, momentum is conserved,
but MV is not the right way to calculate MV.)

The upper right corner is very similar to the upper left
corner, except that one object (E for earth) is much more
massive than the other object (A for anvil). When we drop
the anvil, it accelerates toward the earth ... *and* the
earth accelerates ever so slightly toward the anvil. The
observer in this case is moving parallel to the CM, and
observes that MV is conserved. It may not be entirely
obvious how and why MV is conserved. The observer says
"hypotheses non fingo".

In the lower right corner, we have the same scenario except
that the observer attaches himself to the laboratory frame,
which is attached to the earth. This is ever so slightly
different from the CM frame, because the earth accelerates
toward the anvil. This observer finds that the MV of the
earth remains zero, while the MV of the anvil starts out
zero but changes with time. This observer concludes that
MV is not conserved.

This lower-right case is a trap for the unwary, because in
terms of position and velocity, the observer is "almost"
paralleling the CM, since the earth's path and the CM's
path are "almost" the same ... but in terms of MV, this
observer does not "almost" get the right answer. His
answer is wrong by 100%. That is, 100% of the anvil's MV
is unaccounted-for.

To summarize: In all four of these scenarios, the observer
is using a bona-fide Newtonian reference frame, by which I
mean a frame that differs from a freely-falling frame by
some overall acceleration of the frame. However, in only
two of these four Newtonian frames is MV conserved.