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pool table physics (long)



Brian Whatcott wrote:

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.
#3 would be extremely difficult to measure with the technology of
centuries ago, nor would there be any easy way to measure the
amount of kinetic energy that was retained (which is a not
the same thing). The easy thing to measure is the rebound
_distance_, which, as stated in my post, is optimized if
the ball rebounds with an angular velocity that is very
close to the rolling-without-slipping value.

I wrote:
It's true
that any vertical component will be canceled out by a normal force
from the tabletop,
John S. Denker wrote:
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.
There are various types of motion that can occur, including motion
in which the ball hops. But you can verify experimentally that
hops don't occur under typical conditions. I'm
only working on one possible type of solution to the equations of motion.

John S. Denker wrote:
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.)
John, I think what you're missing here is that I've actually calculated
all this stuff. The third calculation I referred to in my original post
took into account both frictional forces.

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.
Well, nature doesn't really care what's "satisfactory." I gave a
physical argument in my post as to why the impulse due to kinetic
friction from the cushion should approximately cancel out (this
is not the same as saying the force is zero), and then I calculated
the results both with and without this assumption.

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?
As stated in my post, I assume static friction between the ball and
the tabletop, and kinetic friction between the ball and the cushion.
(They can't both be static friction forces, since then the ball couldn't
be moving! It's conceivable that at some point in the motion they're
both kinetic, but this seems unlikely, since the ball seems to come
off the cushion rolling without slipping.)

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 i 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.

Once you start considering friction, you need to account for the fact that
the cushion is not perfectly rigid,
Of course it's impossible to have either object remain perfectly
rigid during the collision. That's elementary mechanics.
and not at all perpendicular to the
ball at the contact-point.
I'm not sure what you mean here. You might want to look at a real
cushion's geometry. What do you mean by "perpendicular to the ball"?
It can store energy by flexing as well as by
compressing.
Of course. My calculation used conservation of momentum and
angular momentum.
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
As discussed in my post, I then calculated the results if the force
isn't purely horizontal, and the result is in even worse disagreement
with experiment.

To sum up, I calculated the results in a model in which:
- Both the cushion and the tabletop exert both normal and frictional
forces and normal forces on the ball.
- The usual textbook model of static and kinetic friction is
assumed.
- Gravity is neglected.
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
The last four impulses have to be split into before and after parts
because the kinetic friction force reverses signs after the ball turns around.
It's easier if, instead of b, you work in terms of an angle theta,
defined as the angle between the surface of contact and the vertical.
Write down conservation of momentum and angular momentum.
You end up with an equation that relates b to the quantity alpha
defined in my original post. The equation can't be solved for b in
closed form, but 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.