Re: pool table physics

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

Regarding
> >   http://www.monmouth.com/~jsd/physics/pool.gif

at 07:35 AM 4/24/01 -0700, Ben Crowell wrote:

> There's nothing wrong with the diagram, but the direction of the force
> vector can't be chosen arbitrarily. It's determined (up to a sign) by the
> standard model of kinetic friction.

Friction is one contribution among many;  it cannot be singled out as
"determining" the direction of the force.  In some cases it's not even the
most interesting contribution.  And kinetic friction (which I take to mean
"sliding friction") is not the only contribution to the friction, or
necessarily the most interesting contribution.  In particular:

a) One way to create a force of the required type is for the contact
between the cushion and the ball to be locally quasi-static.  Then the
direction and magnitude of the force is determined by the engineered
elastic properties of the cushion.  There will be some combination of
compression and flexion.  In particular, the effective center of the
flexing motion is a critical adjustable parameter.

The only requirement on the friction in this case is that there be _enough_
friction.  In this limit, the details of the friction model (standard model
or otherwise) do not enter the calculation.

b) More generally, there could be nontrivial sliding in addition to the
above-mentioned contributions.  In this case the details of the friction
law do matter, but they still cannot be singled out as "determining" the
direction of the force.  The engineered elastic properties remain important.

c) In practice, the design probably proceeded the other way around.  For
any given frictional and elastic properties, the cushion height can be
adjusted appropriately.  So it is not necessary that "the direction of the
force vector ... be chosen arbitrarily."

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

I thought the major goal here was to understand the observed cushion
height.  In the presence of nonzero friction, it seems inevitable that the
force of the cushion on the ball will include an upward component, as the
rolling ball tries to "climb" the cushion.  Therefore it seems inevitable
that the optimal height must be less than 0.7 times the ball diameter, in
accordance with the formulas I presented previously.

An almost-inescapable corollary is that if you roll a ball (with natural
spin) against the cushion hard enough, it will hop.