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Re: dropped slinky



John Mallinckrodt wrote:

> I've put an animation built with Interactive Physics online
> at http://www.csupomona.edu/~ajm/special/slinky_drop.mov

Server: Apache/1.3.26 (Unix) mod_ssl/2.8.9 OpenSSL/0.9.6b
Last-Modified: Fri, 22 Nov 2002 19:38:27 GMT
Content-Length: 1033314
Content-Type: video/quicktime

That movie displays what one might call non-ideal
behavior.

In particular, the movie shows 9 red beads that
come together and _stick_. That implies there must
be some sort of dissipation mechanism; otherwise
the beads wouldn't stick. I'm not sure exactly
what sort of dissipation is being assumed, or why.
No such dissipation was specified in the original
statement of the problem.

It is edificational to consider another brand of
slinky, i.e. one that doesn't have this "sticky"
property. The slinky is, after all, just a medium
for the propagation of waves, and most media do
not have the property that they compress once and
get stuck in the compressed state. A slinky does
weird things when the turns slap together, but it
doesn't just stick.

It is important to document what assumptions are
being made. I am assuming:
1) the medium has no internal dissipation.
2) the medium has no nonlinearities.
3) the medium is a continuum, i.e. infinitely
many tiny beads connected by infinitely many
tiny springs. In the unstressed state it has
a uniform mass per unit length and a uniform
compliance (inverse spring constant) per unit
length.

A brand-new slinky probably doesn't satisfy
assumption (2), because it goes nonlinear when
the turns slap against each other, but if you
get an old, abused slinky such that at rest the
pitch of the helix is sufficiently open, then it
should pretty much conform to these assumptions.

Most importantly, note that waves in other media
(sound in air, currents in coax cables, etc.)
usually satisfy these assumptions quite nicely.

You can pretty much figure out in your head what
the solution looks like. (For those who are
allergic to theory, the experimental answer is
given below.)

It's just a wave propagation problem. You're
applying a driving force that is a step function:
the force is mg for t<0 and the force is zero for
t>0.

You know how the medium responds to such a driving
force. If you feed in a step function, a step
propagates down the medium at the speed c. It
then reflects off the far end (which has a 100%
impedance mismatch) and comes back. It then
reflects off the near end, et cetera.

This is super-obvious if you know anything about
the Green function for the wave equation, but
you don't even need that, so long as you have
a good feel for how the wave equation responds
to a driving force (a source term).

You might think that things get messed up by the
fact that the slinky is significantly distorted
by the initial conditions. But you can make that
go away by a change of coordinates: the natural
unit of measure along the medium isn't distance
(in units of dx) but rather mass (in units of dm).
If you write the wave equation in the new units,
everything is perfectly uniform. You have "cells"
with constant mass per cell and a constant amount
of spring per cell. This approach is pretty
obvious if you start from the discrete case (N
discrete masses connected by N-1 discrete springs)
and then pass to the continuum limit (N going to
infinity).

The solution is plotted in this figure:
http://www.monmouth.com/~jsd/physics/gif48/slinky.gif

I plot the position (as a function of time) for 8
points on the slinky. This does not mean there
are 8 beads. Rather, my slinky is continuous; I've
just put a tiny dot of paint on 8 points. All
points are stationary for all times t<0. At t=0,
I let go of the slinky. The top of the slinky (shown
in blue) immediately drops away, travelling at a
pretty good clip. Points a distance X away don't
even know that I've let go, and they don't find
out until time t=X/c

Note that the speed at which the top of the slinky
moves (the slope of the blue line) is essentially
the amplitude of the excitation. It is _not_ the
speed of sound. Remember, it would be a trivial
modification of the problem to consider a transverse
wave, in which case the amplitude would be related
to the transverse velocity, clearly a different
animal from the speed of propagation of the wave
front, which is what we call c, the speed of sound.

You can eyeball the speed of sound in this figure
by looking at the locus of the "corners" where the
wavefront reaches the 8 different points.

When the wave hits the far end (the corner in the
red curve), it propagates back toward the near
end. As this reflected wave reaches each of the
various points, it gives them another step-
function increase in velocity.

By the time the reflected wave reaches the near
end (having completed one round-trip) the slinky
is the same size and shape as when it started.
But all elements have picked up some downward
velocity.

And yes, every time the wavefront passes a point,
that point is subjected to a reeeeally large
acceleration. There's a step change in velocity
spread over an infinitesimal time. Presumably
the continuum approximation breaks down at the
smallest length scales, so the acceleration is not
quite infinite, but it is going to be reeeeally
large. Large accelerations on small length-scales
and small time-scales are not _ipso facto_ unphysical.
12 Gees applied to you would make you very unhappy,
but 1200 Gees applied to a bacterium wouldn't be
a big deal.

Note that the overall shape of the figure is parabolic.
Every point on the slinky is picking up the right
amount of velocity per unit time (1 g) but it picks
it up in discrete steps.

To answer Tina's question: Students in an intro
course who haven't done waves yet don't have a ghost
of a chance of understanding this problem. But it's
not a graduate-level problem either. Juniors in
physics or EE after they've seen the wave equation
for the second time should be able to figure out
most of this, especially if you give 'em a couple
of hints.

_/\_ _/\_ _/\_

This posting is the position of the writer, not that of Mrs. Hu, Mrs.
Wye, or Mrs. Howe.

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.