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Re: pool table physics



At 06:04 PM 4/22/01 -0700, Ben Crowell wrote:

... 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

Real objects rolling without slipping on real surfaces *do* exhibit rolling
friction. You can attribute this to "memory" if you like. Sometimes the
surface deforms; sometimes the object deforms.

I renew my recommendation that we speak of
-- sliding friction,
-- rolling friction, and
-- static friction (for strictly static objects only).

Ron Shepard uses this terminology at
http://www.playpool.com/anonftp/pub/APAPP/apappTotl.pdf
and I'm perfectly happy with the way he uses it.

Real rolling friction may be locally quasi-static, but it deviates slightly
from being exactly static, and the deviations are crucial. Strictly static
friction is non-dissipative. Rolling friction is dissipative.

Static friction is zero for a ball that's rolling without
slipping.

Well, there's *some* kind of friction that is nonzero during rolling. I
call it rolling friction. Let's not focus on what it isn't. Let's focus
on what it is. It has friction while it rolls. We have to deal with it.

I [jsd] 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.

Ben wrote:
the momentum transfer to the ball is always in the same direction,
which is why the ball's momentum reverses its sign.

Right. Good. We can use that to clarify the energy argument:

... didn't say that the situation was exactly symmetric, only approximately
so. ...

I still say that the hypothesis that the sliding friction makes an even
"approximately" symmetric contribution is totally untenable.

Friction dissipates energy. In general,
Power = Force dot velocity
Let's say that the velocity is in the +X direction on the way in, and
in the -X direction on the way out. The force of sliding friction must be
opposite to the velocity. Therefore it must be in the -X direction on the
way in, and in the +X direction on the way out. It is a much better
approximation to call it ANTI-symmetric than symmetric.

I [jsd] 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.

I made a diagram that may help:
http://www.monmouth.com/~jsd/physics/pool.gif
You can calculate the horizontal component of the force. You can calculate
the torque. Plug in and turn the crank.

It's impossible based on the assumptions given in the pdf
file I posted on my site (http://www.lightandmatter.com/pool/pool.pdf).
.... Of course at
least one of my assumptions has to be wrong, since the result
disagrees with reality.

I have been unable to identify a clearly-wrong assumption. I suspect there
are some unstated assumptions floating around.

The angle you give could, for example,
occur if the usual textbook model of friction was incorrect.

I don't think I'm using any heretical or unphysical models for
friction. I'm assuming the table manufacturer contrives _enough_ friction
... and also provides the right sort of elastic properties for the cushion.

(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.)

The assumption of "abruptness" may be causing trouble. An infinitely
abrupt collision would involve infinitely large forces. I assume a finite
force and a nonzero interaction time in all my calculations.

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.

For non-abrupt collisions, there's no need to assume that the ball actually
hops. In normal play, hitting the cushion could cause a modest reduction
(nowhere near a 100% reduction) in the force with which the ball rests on
the table. If it's rolling without slipping, this reduction would have
only the tiniest of consequences, and would be hard to detect by casual
observation.

Possible experiment: Roll a ball with natural spin _really hard_ against a
cushion. If it hops, it provides qualitative support for the explanation
I've suggested.

Possible experiment: Cover the cushion with something that provides
relatively low friction. A piece of smooth bond paper might be a good
thing to try. If this significantly impairs the bounce, by sending the
ball off with too little spin, it provides qualitative support for the
explanation I've suggested.

All this assumes we are dealing with a table where the cushion contacts the
ball about 1/5th not 2/5ths of a radius above the midline.