pool table physics

[Ben Crowell <crowell01@LIGHTANDMATTER.COM>]

Brian Whatcott wrote:

> I expect that there is some upward precession of the horizontal spin
> axis nearer the cushion towards the vertical, which effects a
> forward roll along the compressed cushion during the rebound.
>
> This is a rather different effect from the "cancelling" frictional
> effect Ben has in mind.

Ahh...interesting idea. Of course in the case of a ball rolling in
perpendicularly to the cushion, it requires a spontan -- e.g. a pencil
balanced on its tip breaks reflection symmetry spontaneously.
All I mean by "spontaneously breaking the symmetry" is that
arbitrarily similar initial conditions can lead to very
different motion later on.

There are other ways to break the assumptions I made about the
motion. I was just now thinking about the assumption that the
ball's center of mass only moves horizontally. Although I pooh-
poohed John Denker's point about hopping, he does have a point
in that it's possible that the center of mass deviates /invisibly/
from horizontal motion. It's then possible that the ball rolls
without slipping on the /cushion/: either it can make a tiny,
invisible upward hop, or it can slide against the table top
and push down into the table. There could also be multiple
microscopic bounces.

I wonder if some of this could be tested via video capture?
It might happen too fast, though.

John Denker wrote:
> ... and I think it was appropriate to comment on these inconsistencies and
> omissions.
It's difficult to present this kind of thing via ascii.
Maybe the pdf file I posted on my web site
(http://www.lightandmatter.com/pool/pool.pdf) will be helpful in
this regard.

Re the issues John Denker raises about terminology, here's the
terminology (fairly standard, I think?) I've been using:
static friction = friction in which the surfaces are not
       slipping over each other; there can be motion, but
       not relative motion at the point of contact; the
       surface has no memory
kinetic friction = friction without memory, excluding static
       friction
rolling friction: cannot be understood if the surface has no
       memory -- neither static nor kinetic friction can
       slow down a ball that's initially rolling without
       slipping, since both are zero

> I assume that "static friction" in this context means rolling friction
Static friction is zero for a ball that's rolling without
slipping.


John Denker wrote:
> > Sliding friction is dissipative.  I don't see how it could possibly
> > cancel.  You lose energy on the way in, and you lose energy on the way out.

I wrote:

> Impulse is defined as momentum transfer, not energy transfer.

John Denker wrote:
> Hmmmmmmmm.  Consider the following proof-by-contradiction.  We adopt
> (temporarily!) Ben's hypothesis that the situation is symmetric, with
> momentum being "loaned" to the cushion on the way in, and "recovered" from
> the cushion on the way out, by means of sliding friction.  That implies the
> (linear and angular) momentum should be restored to their symmetric
> values.  We assume the mass is unchanging.  So.... that would rather imply
> that the energy is restored, wouldn't it?  But we know that energy is lost,
> because sliding friction dissipates energy as I stated.  That's a
> contradiction.  Therefore the hypothesis (that the momentum is restored) is
> not just questionable, it is untenable.

I didn't say that the situation was exactly symmetric, only approximately
so. It's not valid to talk about "loaning" momentum, and I never did;
the momentum transfer to the ball is always in the same direction,
which is why the ball's momentum reverses its sign. Again, you need
to distinguish clearly between energy and momentum. Energy is loaned
in an elastic collision; momentum isn't.

John Denker wrote:
> The way I do it, I find that for a cushion 1/5th of a radius above the
> midline, it suffices to have a force inclined upwards at 11.5
> degrees.  This does not strike me as impossible.
It's impossible based on the assumptions given in the pdf
file I posted on my site (http://www.lightandmatter.com/pool/pool.pdf).
You probably posted this before I posted that link. Of course at
least one of my assumptions has to be wrong, since the result
disagrees with reality. The angle you give could, for example,
occur if the usual textbook model of friction was incorrect.
(Actually it's not even possible for the angle to stay constant
throughout the collision, if there's an abrupt reversal of the
direction of the kinetic friction force at the cushion.)