Rolling along a square path: The dynamics of biased balls

Abstract

A biased ball rolled and spun on a horizontal surface exhibits interesting dynamics. We investigate the motion of a truncated billiard ball, via experiments, analytical methods, and numerical solutions of the equations of motion for a biased sphere rolling without slipping. Solutions are identified where the center of mass moves in a circular or a square path, and we investigate other quasi-periodic motions of the ball.

Curious behaviour of spinning coffee cups and other spinning objects

“…Other experiments indicate that a cone shape is not a significant factor since retrograde motion can be observed with both straight cylinders as well as a tapered coffee cup. Furthermore, retrograde motion can also be observed with a truncated or a biassed sphere [7,8] indicating that the dominant effect involves the direction of the angular momentum of the spinning object and the direction of the torque acting on the object, as it does with many other spinning objects. Supplementary video Croquet.mov shows an example of retrograde motion of a croquet ball with its top cut off.…”

“…The total torque is equal to the rate of change of angular momentum, with the result that the average precession frequency about the remote axis is given to a good approximation by [7][8][9]]…”

“…where D is the distance from Q to G and I 0 is the moment of inertia for rotation about an axis parallel to the spin axis and passing through P. The precession frequency in equation ( 1) is an average frequency during each precession cycle since the frequency varies with time due to the periodic variation in the tilt angle and the torque as the object leans towards or away from the remote precession axis [7]. The sign of D in (c) and (d) is opposite that in (a) and (b) so Ω also changes sign.…”