Two-dimensional Newton Cradles

Live Script simulating two-dimensional arrays of colliding pendulums.

Newton's cradle is a one-dimensional array of identical pendulums, the suspended balls nearly in contact in equilibrium, that exhibit surprising motions.[1] If one ball is displaced and released, it will collide with its neighbor and come to rest. A rapid series of collisions results in the ball at the opposite end magically launched into motion. If you have played with such a toy, you may have noticed it is unstable to transverse motions. Often double-string suspensions are used to limit the instability.

This Live Script simulates two-dimensional motions of arrays of colliding pendulums. It can be used to study the simple Newton's cradle instability and more complex behavior. When an edge ball or cluster of balls of a lattice of pendulums is displaced at some angle and released, a rapid sequence of two dimensional collisions will generally devolve into chaotic motion, and then resolve into a coordinated collision-free motion without collision energy loss.

This script motivates and demonstrates that if the suspension lengths are identical, throughout the evolving motion, the center-of mass oscillates as if it were a single undamped simple pendulum and hence the final motion remarkably remembers the direction of the released ball. This result applies also to the linear Newton's cradle. It also demonstrates through simulation that the total energy decays approximately exponentially towards the coordinated motion final state.

The script supports triangular and rectangular suspension point arrays with or without slight randomization of the support point locations. The user may choose an arbitrary number of balls, variable masses, variable suspension lengths, variable ball radii, and a wide of initial conditions.

This script will interest students and instructors in physics and related fields. It requires only a background in elementary and impact mechanics. 'Try this' suggestions, coding challenges, hyperlinks, and references are included for further exploration. Additional educational Live scripts by the author are here. Closely related Live Scripts are enumerated in the references [1-5].