M036 · Finite elastic contact

Newton’s Cradle & Contact Time

Does one ball in always mean exactly one ball out? One ball in gives an almost isolated outgoing ball when equal masses touch elastically, but the rule is approximate. Several contacts compress at once, leaving small motions in the other balls. Unequal masses, gaps and dissipation can change the outgoing pattern substantially.

Slow down a Newton’s cradle collision. Change individual masses, gaps and stiffness to see overlapping contacts and departures from one-in, one-out.

Current largest contact force $82.2\,\mathrm{N}$

Contacts pushing now $4$

First pair pulse FWHM $0.474\,\mathrm{ms}$

Last bob maximum swing $24.7\,\mathrm{{}^{\circ}}$

Model assumptions

Mass is concentrated at each sphere centre; massless rigid rods constrain motion in a vertical plane and can push or pull. Intrinsic sphere spin, rod and frame collisions, air drag and plasticity are omitted. All sphere pairs interact through unilateral Hertz contact. Equal radii and independent effective masses describe idealized coated or core-loaded spheres; effective modulus includes both surfaces. Optional clipped Hunt–Crossley damping prevents tensile contact force; its coefficient has units of seconds per metre and is not a calibrated restitution coefficient. Contact overlap represents indentation, not interpenetrating rigid solids. The record ends when indentation reaches five percent of radius. Contact patches are see-through annotations of the Hertz area. Gravity and fixed supports exchange momentum; constraint forces do no work. Energy plus accumulated dissipation is checked. The replay repeats one computed first-impact and first-swing trial, with extra slowing during contact; it does not simulate long-term synchronization.

Start with the essentials

Focus question: Does one ball in always mean exactly one ball out?
One-sentence intuition: The row carries a finite force pulse through several simultaneous contacts; separate two-body collisions are only an approximation.

Core mathematical model

Elastic compression of two spheres

F_{\mathrm H}=k\delta_+^{3/2},\qquad k=\frac43E^*\sqrt{R^*},\qquad R^*=\frac{R_1R_2}{R_1+R_2}

Compression is the overlap of the undeformed spherical surfaces. The positive part gives zero force after separation. Effective modulus includes both surfaces.

A constrained bob feels every contact

m_iL^2\ddot\theta_i=\mathbf F_i^{\mathrm{contact}}\cdot\frac{\partial\mathbf r_i}{\partial\theta_i}-m_igL\sin\theta_i

Every pair contributes to the net contact force. Rod constraints keep the centres on circles; no instantaneous velocity-swap rule is applied.

Dissipation and the external supports

F=F_{\mathrm H}\max(0,1+\alpha\dot\delta),\qquad E_{\mathrm{mech}}+Q=E_0,\qquad \Delta\mathbf P=\int\mathbf F_{\mathrm{external}}\,dt

Compression speed is positive while the surfaces approach. Dissipated work accumulates in heat. Gravity and the fixed pivots exchange momentum even though ideal rod forces do no work.

Common difficulties

Equal energy and momentum do not fix every velocity

Typical misconceptionConservation alone forces exactly one ball to leave.

Better mental modelA row has many outgoing velocities. The contact law and its finite duration select the motion; the default last ball receives most, but not exactly all, of the energy.

A replay is one finite trial

Typical misconceptionEach loop continues the previous oscillation.

Better mental modelThe same first-impact and first-swing record repeats. The contact stage has extra slow motion, and the clock reports physical time.

Run the experiment

01
Resolve the pulse
Start with One ball released. Pause during the gold contact glow and compare the rows below.
What to observe: Several contact rows light up together; the first-pair force has a resolved half-height width.
02
Change the outgoing motion
Try Two balls released, A heavy middle bob and Small gaps between balls. Click any bob to change its own mass.
What to observe: The outgoing energy split and contact overlap change. Mass edits stay with the selected bob.
03
Separate stiffness from loss
Increase Effective contact modulus, then try Dissipative contact.
What to observe: Stiffer elastic contact gives a shorter pulse; damping also lowers the outgoing swing by converting mechanical energy to heat.