The push that pays only for the hill

The hill that gives it back introduced potential energy as a landscape a body moves in, and the rest of these arguments have used that picture to ask how much energy is where. The energy that depends on the observer found that the amount depends on who is measuring; the speed at which grip hands over to power asked how fast it can be delivered; the floor that does no work asked who supplies it; and the wall that moves while the ball is in flight asked what happens when the landscape itself changes.

All of those arguments treat energy as the thing that decides what happens. There is a large class of processes where it does not — where plenty of energy is available and nothing happens, until a small disturbance lets it go, and then it goes all at once. A standing domino is the cleanest example there is. Its landscape has a small hill in front of a long drop, and the difference between the size of the hill and the size of the drop is what lets a line of dominoes, each larger than the last, carry a fall from a flick of a fingernail to a slab of wood as tall as a person.

A hill in front of a drop #

A domino stands on its narrow face. To knock it over, its centre of mass must be raised: tipping forward about its front bottom edge lifts the centre of mass until it is directly over that edge, after which the domino falls on its own.

The figure draws the landscape for a domino of the standard proportions, about six and a half times as tall as it is thick. As it tips, its centre of mass rises from half its height to half its diagonal. For a thin domino that is a tiny rise — 0.006 of its weight times its height, at a tilt of nine degrees — and then the landscape falls away steeply. If nothing stopped it, the domino would drop through 0.42 of its weight times its height, seventy times the hill. A neighbour standing half its height away stops it earlier, at thirty degrees, by which point it has dropped 0.028, still 4.6 times the hill.

The domino is metastable: resting at the bottom of a shallow dip with a deep valley beside it. Slide or topple asked when a push on a block tips it rather than sliding it, which is the question of whether the push can raise the centre of mass over the edge; here the question is what happens after it has. The answer is that the domino gives back far more than it was given, and the difference was put in by whoever stood it up. Lifting a domino from lying flat to standing on its end raises its centre of mass by nearly half its height; tipping it over again takes one per cent of that.

Shape decides the ratio #

How much a domino returns for each unit of push depends only on its shape and on how far away its neighbour is.

The figure plots the ratio of released energy to barrier against thickness. For a thin domino the hill is tiny, because the centre of mass barely rises before it passes over the edge, while the drop is almost half the height. The barrier grows as the square of the thickness-to-height ratio, and the ratio collapses as dominoes thicken: a thin card returns nearly ninety times its push, a standard domino about five, and a thick block spaced at half its height returns nothing at all, because it strikes its neighbour while its centre of mass is still higher than where it began. A wider spacing lets the domino fall further before contact and releases more, which is why closely set thick blocks cannot sustain a chain and widely set thin ones can.

The number that matters is a ratio of two energies set entirely by geometry. It does not depend on the domino’s mass, its material or its absolute size, and it is the whole of what lets a chain do more than pass along a disturbance of fixed size.

Why the next one can be bigger #

Scale a domino up by a factor $k$ in every dimension. Its mass grows as $k^3$ and the heights through which its centre of mass is raised and lowered grow as $k$, so both its barrier and the energy it releases grow as $k^4$. A domino whose released energy exceeds its own barrier by a factor $R$ can therefore tip one whose barrier is up to $R$ times its own — one larger by a factor of up to $R^{1/4}$.

For a standard domino the ratio is 4.6 and its fourth root is 1.47: a domino can tip one nearly half as large again, if all the energy it releases before striking goes into its neighbour. In practice some is lost in the collision and some goes into sliding, and the figure shows the ceiling for half and a quarter of the energy passed on. In 1983 Lorne Whitehead built a chain in which each domino was 1.5 times the size of the one before, close to the ceiling, and set it off with a push on a domino five millimetres tall. The fourth-root dependence is what makes the ceiling so insensitive: to double the allowed growth, the energy ratio would have to be sixteen times larger.

The fourth power has a consequence that makes the chain dramatic. Each step multiplies the energy by $k^4$ — by five, for $k = 1.5$.

The figure follows fourteen wooden dominoes growing by 1.5 a step. The first, five millimetres tall and weighing a tenth of a gram, needs two nanojoules to tip — less than the energy of a grain of sand falling its own length. The fourteenth is almost a metre tall, weighs fifty kilograms and releases thirteen joules as it falls, more than six thousand million times the first push. At every step the filled point, the energy released, lies above the next open point, the energy needed; the margin at each step is the same factor, set by the shape.

A chain drawn to scale #

It helps to see what growth by half as much again looks like.

The figure draws the first six of such a chain. The growth looks modest from one domino to the next — each is only half as tall again as its predecessor — and after six steps the last stands seven and a half times taller than the first. Its barrier, going as the fourth power, is more than three thousand times the first’s. A person pushing the sixth domino with the force that tipped the first would not notice having touched it, and yet the first, falling, tips the second, the second tips the third, and the sixth goes over as surely as the first.

What is passed along the chain is therefore neither a force nor an energy of fixed size. It is a trigger, adequate at each step because each domino arrives at its neighbour carrying the energy of its own fall, and each fall is larger than the last. The spacing matters for the same reason: each domino must be placed within reach of its predecessor’s top, and the gap in the figure grows with the dominoes, at half the height of each one, so that every domino falls through the same angle before it strikes.

A chain with a multiplication factor #

The same arithmetic, with a different store, governs the most consequential chain reactions there are. A uranium-235 nucleus holds energy in the arrangement of its protons and neutrons, which the mass that is missing found as a difference in mass between a heavy nucleus and the lighter ones it could split into — about two hundred million electronvolts for each fission. What stops the nucleus releasing it spontaneously is a barrier: to split, it must first deform, and the deformation costs energy, about six million electronvolts. A slow neutron absorbed by the nucleus supplies that, and the nucleus splits, releasing its store and two or three further neutrons.

The ratio of release to barrier is some thirty, like a very thin domino, and the fall passes on as neutrons. Each fission’s neutrons go on to cause, on average, some number $k$ of further fissions, and that number plays the part of the domino chain’s growth per step: below one the chain dies out, above one it grows exponentially, and at exactly one it runs steadily, which is what a reactor’s control system holds it to. The energy of the first neutron is a few hundredths of an electronvolt; the energy released by the chain is limited only by the store of nuclei that were placed within reach of one another, which is the reason the arrangement of the fuel — its size, its shape, and what slows the neutrons between fissions — is the whole of reactor design.

Here too nothing is amplified. The nuclei were assembled, their energy stored, in the explosions of stars before the Solar System formed, and a reactor spends that store a little at a time by holding the multiplication factor at one.

Permission, not energy #

Where did the thirteen joules come from? Not from the first push, which supplied a two-thousand-millionth of it, and not from any mechanism in the chain. Every one of the fourteen dominoes was lifted onto its end by whoever set up the chain, and each held its own store of energy from that moment. The chain did not amplify energy. It passed along a signal — fall now — from each store to the next, and each store paid for the signal’s next step out of its own reserve.

That distinction is easy to state and routinely confused. The first figure’s landscape has two numbers in it, the height of the hill and the depth of the drop, and energy conservation constrains only the second: it says how much energy a domino can release, and nothing about whether it will. The barrier a new phase has to climb found the same structure in a supersaturated vapour, which could release a great deal of energy by condensing and sits indefinitely because the first droplet costs energy to form. The exponential that decides everything found it in chemistry, where a mixture of fuel and oxygen can sit for centuries with an enormous release available, because the barrier to reaction is large compared with the thermal energy. In each case the release is enormous, the trigger is small, and what determines the outcome is the trigger.

A wave that does not fade #

The chain has a further property that makes it more than a curiosity. Set up a line of identical dominoes and knock over the first, and the fall propagates down the line at a steady speed, with each domino arriving at its neighbour with the same energy as the last. It does not die away with distance, as a push passed along a line of springs would, because each domino tops up the wave from its own store.

That is the defining property of what physicists call an excitable medium: a system in which each part sits in a metastable state, can be triggered by a disturbance from its neighbour, releases stored energy, and then passes the disturbance on. A forest fire is one: each tree holds chemical energy that its neighbour’s heat can release. So is a nerve fibre. A nerve impulse travels along an axon a metre long without growing weaker because each patch of membrane holds energy in the concentration differences of sodium and potassium ions across it, pumped there beforehand at the cost of the cell’s chemical fuel, and a small change in voltage from the neighbouring patch opens channels that let the ions flow. The signal is regenerated at every step, and like the domino wave it cannot run backwards, because the patch behind has spent its store and must be reset — for a nerve, over a millisecond or two; for a domino, by hand.

The cost of a hair trigger #

If a thinner domino returns more for each unit of push, why are dominoes not made as thin as cards? Because the hill is also what keeps them standing. A barrier is protection as well as an obstacle, and a store of energy guarded by too small a hill is released by whatever disturbance comes along first — a draught, a vibration of the table, a neighbouring domino settled slightly out of line.

The numbers make the difficulty concrete. The first domino in the cascade, five millimetres tall, is held upright by a barrier of two nanojoules. A breath of air moving at a few centimetres per second past a face a few square millimetres in area delivers forces comparable with its weight over the time it takes to tip it, which is why the smallest dominoes of such a chain are the hardest to set up without starting it. Every system of this kind faces the same trade: the lower the barrier compared with the store, the larger the growth the chain can sustain and the more easily it goes off by accident. Every minimum is a parabola found that the bottom of any smooth well is approximately a parabola, with a curvature that sets how stiffly the system resists small disturbances; a standing domino’s well is not smooth — it is a corner, the edge it stands on — which is why it resists small disturbances firmly and then gives way completely once the centre of mass passes over the edge.

Chemistry uses the same trade deliberately. Explosives used in mining are chosen to have high barriers, so that they can be handled and struck without going off, and are set off by a small charge of a far more sensitive compound, a detonator, whose low barrier makes it dangerous to handle and which is therefore used in tiny quantities. It is a two-domino chain: a small, sensitive store triggering a large, stable one.

What the drawing leaves out #

The energy argument gives a ceiling, not a prediction. It assumes that all, or a stated fraction, of the energy released before contact goes into lifting the next domino, but the transfer happens in a collision, and what matters in a collision is momentum as well as energy. A small domino striking a much heavier one near the top pushes it with a brief impulse; whether that tips the heavier one depends on where it strikes and on friction at both feet, and the heavy one can slide rather than tip, as slide or topple found for a push applied too low. Detailed studies of domino chains, since the 1980s, have followed the angular momentum through each collision and the speed of the wave that results, and find it close to but below the energy ceiling, which is why Whitehead’s 1.5 is at the edge of what works.

The figures also treat the domino as a rigid block with a sharp edge. Real dominoes have rounded edges, which lower the barrier slightly, and bounce and slip at their feet. None of that changes the scaling, which comes only from the size: barrier and release both go as the fourth power, so their ratio is set by shape.

Still open: how fast the fall travels #

The speed at which a toppling wave moves down a line of identical dominoes, as a function of their spacing and shape, has been measured carefully and modelled in several ways, and it is not fully settled. Simple models that treat each collision as instantaneous and perfectly inelastic predict the right trends and speeds within about ten per cent, but the wave also depends on how many dominoes are leaning on one another at once — a toppled domino continues to push as it falls and slides down its neighbour — and on friction between their faces, which is hard to measure in the relevant conditions. Whether a single formula can give the wave speed for any spacing, or whether the detailed contact mechanics always has to be included, is argued in a small literature that is, unusually, contributed to by both physicists and people who build very large domino displays.

The habit worth carrying away is to ask of any process whether energy is its limit or only its supply. A standing domino holds seventy times the energy needed to tip it, stored when it was stood up, and a push pays only for the hill in front of that store — so a chain can grow by the fourth root of the ratio at each step, and thirteen steps turn a flick into a fall six thousand million times larger without any energy being amplified. Whenever a small cause produces a large effect, there was a store, and somebody filled it.