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[Phys-L] point particles puzzle

• From: John Denker <jsd@av8n.com>
• Date: Sun, 29 Jun 2014 10:56:21 -0700

Hi Folks --

Here's a tricky little puzzle.

I call it a "puzzle" not a "paradox", because I
believe that the laws of physics, when correctly
stated, are free of paradoxes. In my experience,
anything that looks like a paradox results from
misstating the laws of physics. So, then, the
challenge is to figure out in what way the laws
are being misstated or misused here:

1) Newton's laws, in their most fundamental form, apply
to point particles. We know this because the laws,
as usually stated, refer to "the" motion without any
allowance for rotation. Also the laws speak of "the"
force, whereas an extended object will in general be
subject to multiple different forces.

This is usually not too much of a problem, because
an extended object can be treated as a /collection/
of point particles.

2) Let's apply this to a spring being stretched at a
steady rate. One end is fixed, while the other is
being pulled along at a steady rate.

We consider the spring as a collection of point
particles. Each such point is moving at a steady
rate, although the rate will differ from point to
point. Note the contrast:
For each point, the rate is the same from time to time.
At each time, the rate is different from point to point.

3) There is no net force on any of the points. That
should be obvious from the fact that each point
exhibits zero acceleration.

4) Since there is no net force, there is no F•dx.
Therefore no work is being done on any of the points.

5) Since we are treating the spring as a collection
of points, no work is being done on the spring.

This is a problem, because it violates conservation
of energy. The guy who is pulling the spring is
doing some F•dx work, and if the energy is not
going into the spring, where is it going?

I need to think about it some more, but here are
my current thoughts about how to untangle this mess:

Gerng gur fcevat nf n pbyyrpgvba bs fznyy cnepryf.
Gur fvmr bs rnpu cnepry pna or fznyy, ohg vg pna'g
or mreb. Gur cnepry pna'g or cbvagyvxr. Vg unf
gb or fgergpunoyr.

Gur obhaqnel bs rnpu cnepry pna unir mreb guvpxarff,
ohg gur yrsg obhaqnel unf gb or noyr gb zbir
eryngvir gb gur evtug obhaqnel. Gur Sqk jbex ba
bar obhaqnel pna or qvssrerag sebz gur bgure.

Nyy bs guvf vf rnfvre vs jr sbphf nggragvba ba
zbzraghz (naq zbzraghz sybj) engure guna sbepr.

Does anybody have a better way to think about
this? The goal is to come up with something
that is
a) Simple enough to be used in the introductory
course, and also