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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

Thank you for your attention! Thanks to my colleagues:

József CSERTI

professor, ELTE

László OROSZLÁNY

assistant professor

Tamás TASNÁDI

BME Math. Institute

Péter GNÄDIG

ELTE

Gyula HONYEK

Radnóti High School

Péter VANKÓ

BME Phys. Institute