The spring whose bottom does not know it has been dropped

The point that keeps moving as if nothing had happened established that a body’s centre of mass responds only to external forces, whatever its parts do to each other. A shell bursting in flight leaves its centre of mass on the original parabola; a diver twisting in the air cannot change where the centre of mass lands. The rule is so general that it is easy to stop noticing what it does not say. It says nothing about where any particular part of the body goes.

A soft spring — the toy that walks down stairs — makes the omission vivid. Hold it by the top so that it hangs stretched under its own weight, and let go. The centre of mass begins to fall at once, at the acceleration of anything dropped. The bottom of the spring does not move. It hangs in the air, unsupported, for a quarter of a second, while the top races down, and it begins to fall only when the collapsing coils from above arrive and hit it. Slow-motion films of it are startling, and the explanation is three statements about momentum, tension and how fast news travels.

A spring stretched by its own weight #

An ideal soft spring has negligible length when unstretched, so its whole hanging length comes from the stretch its own weight produces. Each turn is pulled by the weight of everything below it: nothing at the bottom, the whole weight $Mg$ at the top. The turns near the top are stretched most, the turns near the bottom hardly at all, and the coils crowd together at the bottom.

The arithmetic is short. Label each coil by the fraction $\xi$ of the spring’s mass below it. The tension there is $Mg\xi$, and a slice of the spring holding a fraction $d\xi$ of the mass has a stiffness $k/d\xi$, so it stretches by $Mg\xi\,d\xi/k$. Adding the stretches from the bottom up puts the coil at fraction $\xi$ at height $Mg\xi^2/2k$, and the whole spring hangs $Mg/2k$. Half the mass sits in the bottom quarter of the length. A toy spring of a quarter of a kilogram that hangs a metre has a stiffness of about one and a quarter newtons per metre — soft enough that its own weight is the largest force it ever meets.

What the bottom can know #

At the moment the top is released, the force on the top coil changes from the holding hand’s pull to nothing. Every other coil feels exactly what it felt before: the tension above it and the tension below it are both set by the stretch of the neighbouring turns, and those turns have not moved yet. The bottom coil, in particular, has its weight balanced by the tension of the turn above it, and it will go on being balanced until that turn shortens.

So the question of when the bottom starts to fall is the question of how fast a disturbance travels down the spring. Nothing is allowed to be rigid made the same point about pushing one end of a rod: the far end cannot move until a wave of compression has travelled to it, at the speed of sound in the material, and in relativity that speed must be less than light’s. A soft spring makes the delay macroscopic, because its sound speed is slow — a soft spring of light coils is about as far as a solid can get from rigid.

The snapshots show what happens. The top coils collapse onto one another into a falling block. The block falls on the coils beneath it and gathers them up, one after another, and everything below the block hangs exactly as it did before release. The boundary between the collapsed block and the untouched, still-stretched spring is a front that moves down through the coils. When it reaches the bottom, the whole spring is a compact bundle, and only then does the bottom begin to fall.

A front that outruns sound #

The front is not an ordinary wave. A small disturbance — a gentle tap at the top of a hanging spring — travels down it at the spring’s speed of sound, and for an ideal spring of zero unstretched length that speed is uniform when measured in mass: a tap crosses equal fractions of the spring’s mass in equal times, taking $\sqrt{M/k}$ to get from top to bottom. For a spring that hangs a metre that is 0.452 seconds, exactly the time a stone would take to fall a metre — a coincidence of this particular model rather than a law.

The collapse takes 0.261 seconds, much less than the 0.452 a tap would take. The front outruns the spring’s own sound, as a shock wave outruns the sound speed of the gas ahead of it, and for the same reason: it is not a small disturbance riding on the material but a large one that changes the material as it passes, from stretched to compacted. A supersonic front cannot be anticipated by the material ahead of it, because no signal from the front can get ahead of the front. That is the deeper sense in which the bottom does not know: not merely that news travels at a finite speed, but that the news of this particular event travels faster than any warning could.

Momentum that is all in the block #

The moving part of the spring is the block, and the rest is at rest. That makes the momentum easy to follow. The whole spring’s momentum changes only through gravity, at the rate $Mg$, so at time $t$ after release it is $Mgt$ downward — and all of it is in the block. If the front has reached the coil at mass fraction $\xi_f$, the block holds the fraction $1 - \xi_f$ of the mass, and its velocity is

$$V = \frac{gt}{1 - \xi_f}.$$

The block must also stay at the static height of the coil it is about to hit, since the coils below the front have not moved. Combining that with the hanging profile gives a single equation for the front, $(M/k)\int_{\xi_f}^1 (1 - \xi)\,\xi\,d\xi = t^2/2$, and it reaches the bottom, $\xi_f = 0$, when

$$t_c = \sqrt{\frac{M}{3k}}.$$

For a spring that hangs a length $L = Mg/2k$, that is $\sqrt{2L/3g}$ — the time it would take to fall freely through a third of its own hanging length. A spring that hangs a metre collapses in 0.261 seconds.

The figure shows the whole of the argument in three curves. The centre of mass follows the parabola that every dropped object follows, as it must. The top falls far faster than a stone, reaching the bottom’s level in barely more than half the time a stone would take to fall that far. The bottom stays at its starting height, exactly, until the top arrives. The top’s lead and the bottom’s lag balance so that their average obeys the rule the centre of mass is bound by.

A top that is snapped down and then slows #

The top’s motion is odder than its fast fall suggests. At the instant of release the top coil has the full weight of the spring pulling it down through the tension of the stretched turn below it, and nothing pulling it up. It is a very light object under a large force, and it is flung downward almost instantly.

In the ideal spring the top reaches a speed of $g\sqrt{M/k}$ immediately — about four and a half metres a second for a spring that hangs a metre — and then slows down while it falls. The reason is in the block’s momentum. It grows steadily, as $Mgt$, but the block’s mass grows faster, because the front sweeps through the crowded lower coils quickly, and the velocity of a body whose momentum grows more slowly than its mass must fall. The raindrop that falls at a seventh of g met the same bookkeeping in a drop growing as it fell through mist, where the accumulated mass held the acceleration to a seventh of $g$; here the accumulation is fast enough to reverse the sign. By the time the block hits the bottom, its speed has dropped to $gt_c$, the speed of the centre of mass, and from then on the collapsed spring falls as one object.

Where the energy goes #

As the spring collapses, gravity does work on it and its stretched turns give up their elastic energy. Not all of that becomes motion.

Each coil the block meets is at rest, and the block is moving at metres a second. The meeting is a collision in which the two end up moving together, which is as inelastic as a collision can be, and a perfectly inelastic collision of a moving body with a stationary one always loses kinetic energy — the pile that lands heavier than it weighs found the same loss in a chain coming to rest link by link on a scale. Nearly half of the energy released goes into the coils’ collisions, which is the clatter a real spring makes as it collapses and the warmth it briefly carries. The collapse front is a shock in the precise sense used for waves in gases: a moving boundary across which the state of the material jumps — here from stretched and still to compacted and moving — and in which energy is dissipated however smooth the material is on either side. The front that steepens until it cannot followed how such a front forms in sound, and the spring’s front is the same structure made visible.

The bookkeeping is the same as for a rocket run backwards. The push that needs nothing to push against found a rocket gaining speed by throwing mass away; the block loses speed by gathering mass up, from a stack of coils at rest, and the loss of speed with each coil gathered is exactly what keeps the total momentum growing at the rate gravity sets and no faster.

Why a real spring is a little different #

The ideal spring has no unstretched length. A real toy spring does: its coils touch when it is relaxed, and when it hangs, the bottom few coils carry so little weight that they do not separate at all. The bottom of a real spring is therefore a small collapsed block from the start, and the collapse front arrives at the top of that block rather than at a single coil. The ideal formula then gives the collapse time slightly wrongly — Rod Cross and Michael Wheatland worked out the correction in 2012 and found measured collapse times of a few tenths of a second matching it — but nothing about the bottom’s stillness changes. The collapsed bottom block hangs motionless until the front arrives, as a single coil would.

The demonstration is easy to make more striking. Hang a tennis ball from the bottom of the spring and let go of the top: the ball hangs motionless while the spring collapses down onto it, and only then do ball and spring fall together. The ball stretches the spring further, but it also makes every turn carry nearly the same tension, and the same momentum argument then gives a collapse time of $\sqrt{Mm/k(M+m)}$ for a ball of mass $m$ — approaching $\sqrt{M/k}$, the time a tap takes to cross the spring, for a ball much heavier than the spring. However heavy the ball, it hangs still for a time set by the spring alone. The effect is not subtle and not small — it is a quarter of a second, longer than human reaction time — and it went largely unremarked for decades because nobody watched the bottom.

Air resistance is negligible over a quarter of a second for a metal spring. The coils of a real spring also oscillate sideways and twist as they collapse, which the one-dimensional model ignores, and the collisions are not perfectly inelastic: the bounces that add up to a stop followed how a coefficient of restitution below one makes repeated bounces finish in finite time, and coils bounce a little, so the block is not quite solid, and some of the energy the model counts as lost comes back as rattling. The model is the cleanest one that has the right momentum and the right signal speed, and those two are all the bottom’s behaviour depends on.

A delay set by the slowest news #

The general lesson is about the difference between what a body does as a whole and what each part does. The centre of mass’s motion is fixed by external forces and responds instantly; the motion of any particular part depends on the forces at that part, and those change only when something has travelled there to change them. In a stiff body the travel time is so short that the distinction never shows. In a soft spring it is a quarter of a second, and the bottom hangs in the air for all of it.

A steel bar a metre long, hung from one end and released, does the same thing on a scale nobody could see: sound crosses it in about two hundred microseconds, during which a free body falls a fifth of a micrometre, and the bottom’s wait is lost in the bar’s own vibration. The spring makes the wait visible by carrying its news a few thousand times more slowly. Between the two lie the cases where the wait matters in practice. A mooring line or a crane cable under heavy tension, cut at one end, sends a front of release along its length at the speed of its own longitudinal waves; the far end feels nothing until the front arrives and then is jerked violently, which is why snapping lines are lethal at distances where nobody expects them to move. The same separation governs any long, soft, loaded structure that is suddenly released: a slack chain lifted from a table, a tether cut at one end, a bungee cord let go. It also governs the opposite case, a long object pushed at one end, where the far end cannot know of the push until a compression wave arrives. Collisions are easier than forces showed that momentum accounting answers what happens to a system as a whole without asking about the forces inside; the falling spring is the case where the inside matters, and where the answer to “what does the bottom do?” is “nothing, until told.”

What the drawings leave out #

The figures follow two hundred point coils joined by ideal springs of zero unstretched length, merging them inelastically when they meet, in one dimension, with gravity the only external force. The analytic collapse time and the simulated one agree to a part in a hundred; the centre of mass’s acceleration comes out as $g$ to the precision of the integration; and the bottom coil’s velocity stays at zero until the front arrives, which is a check on the computation rather than a discovery. The real spring’s unstretched length, sideways motion, bouncing coils and air are absent. The domain of the drawings is a spring hanging a metre, released from rest, until shortly after its collapse.

Still open: how the front behaves in a real, bouncing spring #

The ideal model makes the collapsed block perfectly solid. In a real spring the coils that meet bounce slightly, so the block is a dense, rattling zone rather than a solid, and high-speed films show waves running back up through it as it falls. How those waves affect the front’s speed, how much energy they carry away and return, and whether the collapse of a real spring can be described by a single effective coefficient of restitution for its coils are questions answered, so far, by fitting particular springs. The same problem — a front of collapse running into a loaded, stretched medium — appears in the failure of tethers and cables and in the collapse of granular columns, where the details of the contacts matter much more.

The headline does not depend on any of that. A spring hanging Mg/2k under its own weight, released at the top, keeps its bottom exactly still until a front of collapsing coils arrives, $\sqrt{M/3k}$ later — 0.261 s for a spring that hangs a metre — while the centre of mass falls at g throughout, the top is snapped to $g\sqrt{M/k}$ and slows as it gathers coils, and 46 per cent of the energy released is lost in the coils’ collisions. The bottom is not held up; it simply has not yet been told.