Re: pool table physics

["John S. Denker" <jsd@MONMOUTH.COM>]

At 09:43 AM 4/22/01 -0700, Ben Crowell wrote:
> John, I think what you're missing here is that I've actually calculated
> all this stuff.

Well...
 -- I saw some assumptions stated.
 -- I saw some calculations reported.
 -- I saw some experiments reported.
 -- I saw some summary conclusions.

... and on the other hand...
 -- I detected some seemingly useful experiments that were either not done
or not reported.
 -- I detected some seemingly useful calculations that were either not
done or not reported.
 -- I detected some inconsistencies between the assumptions, calculations,
experiments and conclusions that were reported.

... and I think it was appropriate to comment on these inconsistencies and
omissions.


> > 3) It should be obvious from theoretical considerations that a purely
> > normal force would be highly unsatisfactory.  A purely normal force has no
> > lever arm about the center of the ball.  That makes it kinda hard to exert
> > a torque.

I stand by that.

> Well, nature doesn't really care what's "satisfactory."

But pool-table designers do care.  I thought a major goal here was to
understand the observed characteristics of the table.

> On a real pool
> table, we observe that the collision is highly elastic, so the kinetic
> friction forces are evidently fairly small,
> ... I assume static friction between the ball and
> the tabletop, and kinetic friction between the ball and the cushion.

And I tried to point out that such an assumption doesn't make sense.  It
certainly doesn't cover the general case.

Also a request:  Let's henceforth be careful to distinguish between
 -- sliding friction
 -- rolling friction
 -- strictly static friction (of strictly non-moving objects)

I can't tell whether the term "kinetic friction" is supposed to cover
sliding, or a combination of sliding and rolling, or whatever.

> (They can't both be static friction forces, since then the ball couldn't
> be moving!

I assume that "static friction" in this context means rolling friction.  I
assume that's what was intended, since the magnitude of rolling friction is
comparable to truly static friction, and very unlike sliding friction.  And
the alternative doesn't make sense.

But no matter what repairs I make, I don't see any way that statement can
be true.  A ball can roll cleanly against two constraints.  No problem.

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

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.

So my remark about the energy should not have been so lightly dismissed.

> The calculation is only about a page of algebra, and I'd encourage
> anyone interested in discussing the topic to just go ahead and
> do it before posting. You have six impulses:
>  impulse due to the tabletop's normal force
>  impulse due to the tabletop's static friction force
>  impulse due to the cushion's normal force before the ball stops
>  impulse due to the cushion's normal force after the ball stops
>  impulse due to the cushion's kinetic friction force before the ball stops
>  impulse due to the cushion's kinetic friction force after the ball stops

Setting up the problem that way is neither necessary nor sufficient for
obtaining a solution.

> ... you can easily verify numerically that it's not
> possible to get a physically reasonable solution using the value
> of b from an actual pool table.

Really?????

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.  See note below.

More generally, if the cushion acts a distance A above the midline, then
the force should be inclined upwards at an angle
                           .4 R - A
        theta = arctan(-----------------)
                        sqrt(R^2 - A^2)

Or equivalently
        tan(theta) = .4/cos(alpha) - tan(alpha)
        where alpha = A/R

Deriving this requires a diagram plus half a page of algebra.

All this assumes we want the property that natural roll inbound results in
natural roll outbound.

==========

Note:  For any A less than 0.4 R, the force will be inclined upwards.  If
it is inclined too much upwards, the ball may "hop" when it hits the
cushion, depending on the speed of the ball and the hardness of the cushion.