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International Conference on Teaching Physics Innovatively17th August 2015
Funny motions of billiard balls
Máté VIGH
ELTE, Department of Physics of Complex Systems
Introduction - aims of my talk
•Show that the physics of billiard balls can be understood on
high school level
•Show that billiards provide a great opportunity to illustrate the
conservation of angular momentum
•The variety of the possible motions and the difficulty of the
physical ideas behind them make this topic interesting for
students from classroom physics to the level of International
Phsyics Olympiad
Physics of billiards in literature
Gustave-Gaspard Coriolis:
(1792-1843)
Theorie mathematique des effets
du jeu de billard (“Mathematical
Theory of the Game of Billiards”)
Arnold Sommerfeld:
(1868-1951)
Mechanics
And many more…
1st example: motion along a line
Problem 1. A ball resting on a table is struck by a cue tip at point T.
The point T, the center C, the touching point P and the line of action
of the resulting impulse lie in the same vertical plane.
Construct the direction in which the cue should be aligned in order that
after the shot, the ball’s subsequent rotational and translational motion
terminate at the same instant (and the ball comes to a halt). (Air drag
is negligilbe.)
1. Initially the angular momentum is zero
with respect to point P
2. Finally the angular momentum is zero
with respect to point P
3. All of the forces (except the striking
impulse) have zero net torque on
point P.
4. The line of action of the impulse
should go through point P.
Solution
2nd example: Coriolis massé shot
Problem 2. In the previous problem, if the line of action of the resulting
impulse doesn’t lie in the vertical plane defined by the point T, the
center C, the touching point P, the motion will be more complicated.
(Coriolis-massé shot).
Find:
•Trajectory of the COM of ball
•Final direction of motion of COM
Trajectory of the COM of the ball
Velocity of the lowermost point of the ball:
Equations of motion:
From these 3 equations:
Frictional force:
The direction of velocity of the lowermost
point is constant.
The direction and magnitude of the
frictional force (the only horizontal force) is
constant.
The COM of the ball moves on a
parabolic path until the pure
rolling occurs (at point B).
Using the conservation of angular momentum it can be shown that the ball
will move parallel with the PA line when the pure rolling occurs.
VIDEO
Books about funny problems
P. Gnädig – G. Honyek – M. Vigh:
333 Furfangos Feladat Fizikából
(Hungarian)
P. Gnädig – G. Honyek – K. Riley:
200 Puzzling Physics Problems