Newton's Cradle Steel Ball Rig: Hertz Impacts, Contact Force and Momentum Hand-off

What is simulated

A row of identical steel balls (radius 12.7 mm, mass 67.4 g) hanging side by side on strings. Each ball is a pendulum, m L2 θ'' = −m g L sinθ + L F·t, where F is the contact force from its neighbours and t = (cosθ, sinθ) is the direction of motion along the arc. There is no rule such as "k balls in, k balls out": whatever leaves the row is whatever the contact forces produce.

Contact law

Two touching steel spheres press together with the Hertz force F = k δ3/2, where δ is the overlap of the two surfaces, k = (4/3) E* √R*, E* = E / (2(1 − ν2)) and R* = R/2. With steel (E = 200 GPa, ν = 0.3) the overlap at 0.5 m/s is only about 15 micrometres and one impact lasts about 90 microseconds, so the integrator uses 0.5 microsecond steps while a pair is touching or closing and 100 microsecond steps otherwise. Real steel loses a little energy in each impact; this is represented by the Hunt-Crossley factor (1 + 3(1 − e) δ'/(2 vref)) with a reference speed of 0.5 m/s, so the restitution e is exact only near that speed.

Why several balls can leave

A compression pulse runs through the row of touching balls in a few tens of microseconds while the balls themselves hardly move. In a perfect Hertz chain with identical balls the pulse delivers almost all the momentum and energy to the same number of balls on the far side. Tiny leftover motion of the other balls is real physics (small gaps and the nonlinear contact), and the energy graph shows it.

Simplifications

• Balls move in one plane along arcs of equal effective length L. The real cradle hangs each ball from two strings in a V; that keeps the ball in plane and the page draws it, but the dynamics use one effective length.
• Strings are rigid and massless; frame flexing, ball spin, friction between balls and sound radiation are not modelled.
• Air drag is a simple linear rate per second. Contact loss uses the Hunt-Crossley form, which is a fit, not a derivation from material data.
• The time scale slider only slows the clock; the physics step does not change.

Checks

Five boxes: the peak overlap and duration of a single impact against the closed-form Hertz solution, the period of a single ball against the large-amplitude pendulum series, conservation of kinetic + gravitational + contact energy for lossless balls, the number of balls that carry the energy out after the first impact for k = 1, 2 and 3 lifted balls, and the restitution produced by the damping term.