Newton's Cradle Lab (2D)

What a Newton's cradle shows

Lift one ball and release it: one ball leaves the far end at the same speed. Lift two: two leave. Momentum p = sum m v and kinetic energy E = sum 1/2 m v^2 are both conserved in the collision, and for equal masses the only outcome that keeps both is "k balls in, k balls out". With unequal masses the outcome follows from the same two equations (a heavy ball hitting a light chain pushes out several balls).

Pendulums

Each ball of mass m_i hangs on strings of length L = 0.20 m, so the angle obeys theta'' = -(g/L) sin(theta) + F_t / (m L) where F_t is the contact force component along the swing direction. A ball lifted by theta0 arrives at the bottom with v = sqrt(2 g L (1 - cos theta0)). The period is not 2 pi sqrt(L/g) at large angles but 4 sqrt(L/g) K(sin(theta0/2)) with the complete elliptic integral K.

Hertz contact between the balls

Two touching spheres squeeze each other with a force F = k d^(3/2) (d = overlap). A collision lasts t_c = 2.943 d_max / v0 with d_max = (5 m* v0^2 / (4 k))^(2/5), so softer contact means a longer pulse. The pulse travels through the chain ball by ball (graph 4). To lose energy, a damping term is added (Hunt and Crossley), F = k d^(3/2) (1 + 3/2 alpha d'), and the page tunes it so the restitution at 1 m/s equals the slider value e.

Numerical checks on this page

(1) The simulated swing period agrees with the exact elliptic-integral period. (2) A lossless two-ball impact lasts as long as the Hertz formula predicts and the balls exchange velocities. (3) Two of five balls lifted, no loss: about all the kinetic energy ends up in the two balls that leave.

Honest simplifications

The motion is planar and the strings are rigid and massless; there is no friction between the balls and no rotation. Contact is made softer than real steel (a few tenths of a millisecond in reality, 1.5 ms by default) so it can be integrated in real time with a leapfrog step of at most 20 microseconds; the behaviour in the cradle is the same. The momentum shown is the horizontal momentum of the balls: string tension also acts, so it is only conserved inside a collision, not over a whole swing.

• Classic: one ball, then Two balls swing in and Three balls: the same number of balls leaves as arrived.
• Slow the time speed to 0.02x and watch graph 4: the contact force pulse moves along the chain.
• Heavy ball hits light chain and Light ball hits heavy chain: unequal masses break the "k in, k out" rule.
• Soft putty balls: low restitution, the energy graph drops after each collision and the balls stick together.
• Increase the contact time slider: a softer ball makes the pulse longer and the transfer less clean.