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Re: pool table physics (long)



There seem to be quite a few assumptions about the purpose of
positioning the cushion height, and indignation or puzzlement
that the table construction details do not agree with the
given assumptions.

I am fairly sure that the folks who put together the early billiards
table were more interested in three visible features:
1) that a ball, hit no matter how hard, would not jump the cushion
nor yet bounce off the baize.
2) That the approach angle to the normal and the departure angle
were fairly similar.
3) that there was maximal speed conservation.

So these are my assumptions for the pool table too.
And I assume that these constraints can be met by the usual
cushion height ratio.

Sincerely

Brian

At 12:37 4/21/01 -0700, Ben Crowell wrote:
I recently came across a classical mechanics problem that seems to be
a good example of the adage that "problems worthy of attack prove
their worth by hitting back." I thought I'd post about it here,
because (a) I'm curious to see if anyone can give a convincing
solution, (b) an incorrect analysis seems to turn up in a lot of
textbooks, and (c) the experimental side of the problem might be an
interesting research project for a talented student at the earliest
stages of her/his physics education.

Here's a paraphrase of the way the problem was stated when I came
across it on an old CalTech entrance exam. A pool ball is rolling
without slipping, and collides perpendicularly with the cushion. The
cushions on a pool table are built so that their point of contact with
a ball is above the ball's center by a certain height b. Assume that
the cushion's force on the ball is horizontal, and determine the
optimal value of b if the ball is to be rolling without slipping again
when it rebounds. By "optimal," I assume they mean the value of b that
will work without requiring any static frictional force at the ball's
point of contact with the tabletop beneath it.

If you solve this problem, you get b/r=2/5. This is a purely
geometrical result arising from the moment of inertia of a uniform
sphere. There are two difficulties. First, if you measure b/r on a
real pool table, it's closer to 1/5. Second, the assumption of a
purely horizontal force from the cushion is unrealistic. It's true
that any vertical component will be canceled out by a normal force
from the tabletop, but that doesn't mean that the torques from these
two force components cancel.

Actually, if you had to make an approximation about the cushion's
force, it would seem more reasonable to assume that it was purely
normal. The cushion's kinetic friction force reverses its direction at
the moment when the ball is instantaneously at rest. On a real pool
table, we observe that the collision is highly elastic, so the kinetic
friction forces are evidently fairly small, and more importantly, the
kinetic frictional impulses during the incoming and outgoing portions
of the collisions should very nearly cancel, due to the approximate
similarity between the incoming and outgoing motion. Under this
assumption, the only torque is from static friction with the tabletop,
and the result is that the minimum value of b/r is 1/sqrt(1+7mus/2),
where mus is the coefficient of static friction. The result is no
longer purely geometrical. I've offered one of my classes extra credit
for measuring mus and muk, but it's certainly not possible to
reproduce the real-life value of b/r with any reasonable value of mus.

I next tried relaxing the assumption of zero impulse due to the
cushion's kinetic friction force. I defined a parameter
alpha=(nc1-nc2)/nc1, where nc1 is the impulse due to the cushion's
normal force when the ball is on the way in, and nc2 is the
corresponding outgoing impulse. For a perfectly elastic collision,
alpha is 0, and for a completely inelastic collision, it's 1. The
kinetic friction impulses are of course proportional to the impulses
nc1 and nc2. In this model, it's no longer possible to get a
closed-form expression for b, but to reproduce the observed value of
b, you have to use alpha~1, which is obviously unrealistic.

I corresponded about this problem with Ron Shepard, author of an
excellent free online book called Amateur Physics for the Amateur Pool
Player (http://www.playpool.com/apapp/). Shepard treats the collision
of the cue stick with the cue ball in his book, but hasn't explicitly
calculated the case of the collision with a cushion. I feel a bit
awkward here because I'm grateful to Ron for taking the time to
discuss this problem with me, but we didn't end up agreeing on the
analysis, and I'm afraid if I try to relay his ideas here I'll end up
misrepresenting them. Ron points out that the people who originally
did the (presumably empirical) work of designing the cushions probably
looked for the maximum rebound distance. I did some calculations, and
found that, counterintuitively, the maximum rebound distance is not
necessarily achieved by making the ball rebound without slipping.
However, the effect is on the order of mur/muk, where mur~1/100 is the
coefficient of rolling resistance; since mur/muk is presumably pretty
small, I doubt that this is a significant issue. It would be
interesting, however, to capture some video on a computer and find out
whether the ball really does slip a little on the rebound.

What else could be going on? Maybe gravity is important, despite the
short duration of the collision. Maybe the felt displays memory
effects, so the usual assumptions about kinetic friction are wrong -
after all, rolling resistance doesn't exist unless you assume that the
felt has memory. But any more complicated model is probably going to
require a lot of empirical data from real pool tables.


brian whatcott <inet@intellisys.net> Altus OK
Eureka!