This thought experiment looks flawed--it is a contradiction
within itself, just like the early theory of atoms (which
lets one law of physics, namely Coulomb's law, but ignores
another law that says an accelerating charge radiates EM
radiation).

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THOUGHT-EXPERIMENT:
Consider an old-fashioned battering ram. This is
basically a big tree trunk with a lot of handles
attached. Suspended from a framework, a gang of guys
divides into two groups, one on each side of the tree
trunk, then grabs it up by the handles, and moves
it to a door that needs to be battered. OK now,
let's imagine a modest experiment involving a very
special battering ram, not too thick, but so long
that it has two thousand handles, needing a thousand
guys on each side of the thing, and so tough that not
even Superman can break it. --Oh, and we also need
Superman for this thought-experiment.

So, this is one unrealistic assumption--a battering ram
that is so tough that it can't be broken--and since you
didn't give anything like Young's modulus, I suppose it's
"rigid," too, isn't it? By itself, it's not too
unreasonable--it can be a good assumption, just like
massless pulleys and frictionless surface, as well as so
called "free fall" on earth.

Due to the length of this battering ram, and the
speed of sound, we first need to rig up a speaker
system, say, every three meters. Now all the men
can hear the commands to move the battering ram at
pretty much the same time, and can apply their
efforts pretty much in unison. For purposes of
this thought-experiment, we assume they DO apply
their efforts in unison.

OK, Superman, please stand near that end of the
battering ram. All the rest of you guys, here's
what we want you to do: When you hear the word
"Forth", you will each apply a short, sharp, jerk
to your battering ram handle, toward Superman, and
then let it coast. Superman, let it coast toward
you until you hear the word "Back" -- then apply a
powerful force to that end of the battering ram,
to push it back. That's all. We just want a
simple oscillation, with equal-magnitude forces
being applied, pushing it forth and back. Only
the WAY those forces are applied will be different
...get ready!

For the sake of simplicity, let's assume that when
Superman applies his Super force to the end of the
battering ram, it takes exactly one whole second
for that mechanical force, moving at the speed of
sound inside the battering ram, to reach the far
end. Also, we are going to say, "Forth! Back!
Forth! Back!..." at the rate of four words per
second. Physicists, what IS the Equation of
Motion for the battering ram?
-------------------------------------------

Moving at the speed of sound inside the battering ram,
eh? Now, tell me, do you have slightest idea of what
the speed of sound inside the battering ram is?

By your assumption (or the way I interpreted it--
please give the required information (i.e. property
of the material) if my interpretation is incorrect)
the ram is very rigid. That lets us assume one
important constant.

Well, my physics book has that the speed of
longitudinal wave (which is what sound wave is...
well... I think--I know that it's the case in air,
but I'm not too sure about a solid) in a solid rod
is, v = sqrt(Y/rho), where Y is Young's modulus and
rho is the density of the material. Since we are
not dealing with a blackhole, we can assume that
the density is some reasonable, finite number.
Well, Young's modulus, as we all know, is
stress/strain. Stress is proportional to the
force applied on the rod (er, normal to cross
section) and strain is proportional to the change
in length of the rod... do you see the problem you
just ran into?

Since the rod is rigid, strain approaches 0, and
Young's modulus approaches infinity. So, since rho
is finite, v_sound approaches infinity, and we
know that it cannot be true--because of the theory
of special relativity. So, we need relativistic
corrections on the speed of sound.

But... as soon as we start considering relativistic
effect, your whole scenario breaks apart--even if
you put speakers everywhere possible, the fastest
speed sound signal can travel through the wires is
the speed of light--so, in that approximation
limit, your man will not be pushing the rod in
unison (and you should know, that, in theory of
relativity, "unison" has not much meaning... as
there is nothing that is "simultaneous").

So, here is my answer for "equation of motion
for the battering ram": For the given scenario,
there can't be an equation of motion for the
battering ram, as (if I understood correctly)
the author intends no upper limit on the amount
of impulse allowed on the battering ram (as
is clear in his assertion that the battering ram
cannot be broken) while stipulating that the
mechanical force can "only" travel at the speed
of sound, which, following from his assertions,
is infinite. (And... if we are going to start
approximating the battering ram as some sort of
"super spring," a spring (so... non-rigid
assumption in effect now.. allowing speed of
sound to be finite in the "rod") with infinite
elastic range, then I'm afraid it's outside my
expertise--as I do not believe it will be a SHM
any longer--I'll need a few more classes on
advanced mechanics before I tackle that.) So,
this is a poorly thought out scenario
(specifically, it is a scenario where the
theories, assumptions, and approximations are
taken out of their valid range), and whoever
worries too much over this scenario is, well, not
spending enough time in studying physics and
wasting too much time on some sort of modern
alchemy (that is, trying to break the conservation
law of momentum).

Best wishes,
Andrew