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Re: pool table physics

At 12:37 PM 4/21/01 -0700, Ben Crowell wrote:

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.

OK. That's optimal in the sense that natural roll inbound results in
natural roll outbound.

the assumption of a
purely horizontal force from the cushion is unrealistic.

Right.

It's true
that any vertical component will be canceled out by a normal force
from the tabletop,

I wouldn't assume that in all cases!

1) You can investigate this experimentally. You can get various sizes of
balls. If you get ones that are too large for the cushions on your table,
they will "hop" if they hit the cushion with enough momentum. Apparently
there is enough friction with the cushion that they can "climb" up it.

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.

I wouldn't assume that either!

2) The importance of friction can also be observed even with standard-sized
balls by experimenting with side-spin. (Detailed analysis of this is
beyond the grasp of average freshmen. But it wouldn't hurt them to at
least make the observation, so that they know that the assumption of a
_horizontal_ axis of rotation doesn't cover all of reality.)

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.

On a real pool
table, we observe that the collision is highly elastic, so the kinetic
friction forces are evidently fairly small,

I assume "kinetic friction" means sliding friction.

What about static friction?

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.

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.

Once you start considering friction, you need to account for the fact that
the cushion is not perfectly rigid, and not at all perpendicular to the
ball at the contact-point. It can store energy by flexing as well as by
compressing.

It would be fortuitous if all these complicated interactions resulted in a
force that was purely horizontal. Therefore it would be fortuitous if a
cushion height of 7/10ths of the ball diameter turned out to be the right
answer.


... Ron Shepard, author of an
excellent free online book called Amateur Physics for the Amateur Pool
Player (http://www.playpool.com/apapp/).

An interesting little treatise.

"problems worthy of attack prove
their worth by hitting back."

That's from _Grooks_ by Piet Hein, MIT Press (1966).