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Re: physics of bouncing



At 02:11 PM 3/13/01 -0800, John Mallinckrodt wrote:

I would expect the superball to swing freely
far longer than it would bounce against *any* surface. And, of
course, this would be precisely because internal friction
dominates.

JM has a good point here. I guess I was imagining smallish superballs, for
which air drag does dominate, but in the limit of large superballs, JM is
certainly right. (That's because the air drag should scale like the area
of the superball, while the energy lost to internal friction, per bounce,
should be a more-or-less constant fraction of the total energy, which
scales like the volume of the superball. The drag due to the string makes
this effect even stronger.)


Whether or not air drag dominates internal friction, I think it is safe to
conclude that the two of them, collectively, account for the lion's share
of the dissipation.

===========

Here is another interesting collision scenario, in addition to the ones
discussed yesterday:

Let the ball be made of steel, and let there be a thick layer of hard
rubber on the face of the anvil. A diagram of the general situation can be
found at
http://www.monmouth.com/~jsd/physics/ballistic-pendulum.gif

In this case it seems that internal friction in the rubber is the dominant
contribution to the overall dissipation, over a wide range of
parameters. This dissipation occurs mainly in the bulk of the rubber, not
just at the surface.

What might we learn by placing a temperature-sensitive card between the
ball and the target surface? As far as I can tell:

*) The card doesn't tell us anything about the magnitude of momentum transfer.

*) The card doesn't tell us anything about the magnitude of energy transfer.

*) The card doesn't tell us anything about the dominant dissipative processes.

*) The card mainly demonstrates a process that would not occur in the
absence of the card, namely temperature rise due to irreversible crushing
of the card.

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

Meanwhile.......

The observed behavior of the card *can* be used to motivate a thoughtful
analysis of the physics of the situation.

One *correct* piece of physics that we could learn is the following: A
steel hammer falling on a steel anvil creates very high local forces -- as
a consequence of Newton's laws.

In particular, the force at impact (F_i) is larger than the muscle-force
used to drive the hammer (F_d) by the ratio of
-- the time-duration (t_d) over which the drive is applied, relative to
-- the duration of impact.
The latter scales roughly like sqrt(m/k), where m is the mass of the hammer
and k is the effective spring constant. This expression we recognize as
the acoustic resonance time (t_res) of the hammer-head. (We are assuming
the hammer is light compared to the anvil). So:
F_i / F_d scales like t_d / t_res

You can easily derive this result for yourself.

This can be a pretty big number. This is why it is easy to damage the card
during a steel-on-steel collision, whereas you can't damage it simply by
pushing on it with static muscular forces applied to the same-sized area.