Details

The physics of a ball rolling inside a cylinder is worked out in the paper by Gualtieri et al; they analyze the various forces involved, notably a Coriolis torque that acts around the -axis. A physical demonstration of the process is described in the paper by Matsuura. We enhanced the work of Gualtieri et al by allowing an initial spinning around the -axis. This matches the behavior of a golf ball, which would be spinning at a rate proportional to its speed as it falls into the cup (see second snapshot). Solving the differential equations yields a formula of the form . But in order to learn where points are at time requires the numerical solution of a differential equation so that the angular velocity at each instant is as it should be. The angular velocity is computed from by the following equation, where is the radius of the ball, is the radius of the cylinder, and is a constant that represents the (negative of) the angular speed of the ball around the central axis of the cylinder: . Note that is a linear function of . The program for the soccer ball was taken from a Demonstration by Greg Wilhelm.

References:

M. Gualtieri, T. Tokieda, L. Advis-Gaete, B. Carry, E. Reffet, and C. Guthman, "Golfer's Dilemma," American Journal of Physics, 74(6), 2006 pp. 497–501.

A. Matsuura, "Strange Physical Motion of Balls in a Cylinder," Proc. of Bridges Conference, Banff, 2005.