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I am fairly sure that the folks who put together the early billiards#3 would be extremely difficult to measure with the technology of
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.
It's trueJohn S. Denker wrote:
that any vertical component will be canceled out by a normal force
from the tabletop,
I wouldn't assume that in all cases!There are various types of motion that can occur, including motion
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.
2) The importance of friction can also be observed even with standard-sizedJohn, I think what you're missing here is that I've actually calculated
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 purelyWell, nature doesn't really care what's "satisfactory." I gave a
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 poolAs stated in my post, I assume static friction between the ball and
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, theImpulse is defined as momentum transfer, not energy transfer.
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.
Once you start considering friction, you need to account for the fact thatOf course it's impossible to have either object remain perfectly
the cushion is not perfectly rigid,
and not at all perpendicular to theI'm not sure what you mean here. You might want to look at a real
ball at the contact-point.
It can store energy by flexing as well as byOf course. My calculation used conservation of momentum and
compressing.
It would be fortuitous if all these complicated interactions resulted in aAs discussed in my post, I then calculated the results if the force
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