The kinematics of a sphere of radius R and mass m bouncing elastically in a horizontal channel is studied in detail. Explicit expressions for the position and linear and angular velocities of the sphere at any collision are obtained. It is shown that if the moment of inertia I=γmR2 is such that (1γ)(1+γ)=cos2πk with k an integer and k4, the motion is periodic, with repetition after k collisions for k even, and after 2k collisions for k odd. The motion of a homogeneous sphere (which is not periodic) is analyzed by investigating the mean properties of its velocities and positions. The analogy between this motion and that of a periodic oscillator is discussed.

1.
R. L.
Garwin
, “
Kinematics of the ultraelastic rough ball
,”
Am. J. Phys.
37
,
88
92
(
1969
).
2.
G. L.
Strobel
, “
Matrices and superballs
,”
Am. J. Phys.
36
,
834
837
(
1968
).
3.
B. T.
Hefner
, “
The kinematics of a superball bouncing between two vertical surfaces
,”
Am. J. Phys.
72
,
875
883
(
2004
).
4.

Within this simplification, the developments that follow may be applied to other objects with cylindrical symmetry (for example, rings and cylinders) whose center of mass coincides with their geometrical center, whose cross section in the Oxy plane is a circle or a circumference, and that rotate around the z axis.

5.
I. S.
Gradshteyn
and
I. M.
Ryszhik
,
Table of Integrals, Series, and Products
, 5th ed. (
Academic
, London,
1994
), p.
36
.
6.

Because cosθ=(1γ)(1+γ), and 0<γ1, we must have 0<θπ2, and therefore k4.

7.
H. J.
Pain
,
The Physics of Vibrations and Waves
, 2nd ed. (
Wiley
, New York,
1979
), p.
42
.
8.
Two examples of such animations can be found in pwp.net.ipl.pt/dem.isel/jtavares/simulations.htm.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.