Slinky Drop

What is modelled

A slinky hangs from one end, stretched by its own weight, and the top is let go. The top coils fall at once, but the bottom coil does not move at all until the whole slinky has collapsed onto it, and the centre of mass falls exactly like a stone. The page models the slinky as 60 equal masses joined by springs and lets you watch the collapsing front travel down.

Equations

• Each link between coil i and coil i+1 pulls with F = F0 + k (d − h) while the gap d is larger than the coil thickness h (2 mm). F0 is the prestress of a real slinky (zero for the ideal one). Coils cannot overlap: when two coils meet, they stick (perfectly inelastic) and a whole stack moves with its mean velocity, so momentum is conserved and energy is lost.
• Hanging at rest, the link below coil i carries the weight of the coils underneath it, Ti = (N−1−i) m g, so it is stretched by (Ti − F0)/k: the gaps are widest at the top. With F0 = 0 the hanging length is L = m g N(N−1) / (2k), which is how the slider sets the stiffness.
• After release the only outside force is gravity, so the centre of mass obeys ycm = y0 − g t²/2 exactly, whatever happens inside.
• Collapse front (continuum limit, ideal slinky): the stack of collapsed coils above the front has mass q = sM and moves with speed M g t / q (total momentum M g t, the part below is still at rest). The front is where the stack meets the first coil that has not moved yet, which gives s²/2 − s³/3 = g t² / (4L), the top has fallen L (2s − s²), and the front reaches the bottom at tc = √(2L / 3g). This time does not depend on the mass, the number of coils or the stiffness except through L.
• Integration: velocity Verlet with a step of 0.2 ms, followed by the contact projection described above.

Checks run in the page

• Damped relaxation of the chain to the closed-form hanging shape.
• Centre of mass against y0 − g t²/2 for your own parameters, including prestress.
• The simulated bottom-coil wait against √(2L/3g).
• The top-coil drop at 0.6 t_c against the momentum-balance solution, and that the bottom coil has not moved.

Honest simplifications

• The slinky is a one-dimensional vertical chain: the real coil is a spiral of flat wire, it can tilt, twist and swing sideways, and its coils touch with friction and slight bounce. Here all collisions are perfectly inelastic and exactly vertical.
• A real slinky is not a zero-length spring: the prestress slider approximates its tight coils with a constant extra force F0 and a hard stack. The closed-form front time is for F0 = 0 only; with prestress the page still conserves momentum and the centre of mass, but the front theory does not apply.
• No air drag, no wire mass distribution inside a coil, no loss of energy to sound or heat except in the collisions; the coil spacing at rest is fitted by the stiffness slider, not measured on a real toy.
• The floor is a rigid inelastic stop, used only after the collapse.

• Press Release in slow motion: the red stack grows downward while the bottom coil (blue ring) stays put, and the yellow centre-of-mass marker stays level with the stone.
• Pick the heavy slinky: the wait does not change. Pick the long soft one (2.1 m): it grows as the square root of the length.
• Raise the prestress: the lowest coils stay stacked while hanging, as in a real toy.