Balance

The Line That Decides

Anything resting on the ground stands as long as a line straight down from its balance point lands inside its footprint, the outline of everything it stands on. For a table that is the rectangle of its legs. For a person it is the outline round both feet.

The reason is that the ground can push but never pull, and only where it touches. Tip a block onto one edge and the ground’s whole push gathers at that edge, while the weight still pulls down at the balance point. A push and a pull that are not in one line set a body turning. While the line from the balance point lands inside the edge, they turn the block back onto its base. Once the line passes the edge, they turn it over, and nothing the ground can do will stop it.

So the angle at which a thing tips does not depend on its weight, only on the width of its base against the height of its balance point. A block twice as tall as it is wide tips at about 27 degrees, and a cube at 45. Lay the tall block on its side and it can lean to 63 degrees and still fall back. Wide and low stands; tall and narrow goes over.

Turning a corner is like tilting the floor. The road pushes the tires sideways, and that push turns the car; but it acts down at the ground, below the balance point, so it also tips the car toward the outside of the turn. Tipping a car over on a flat road would take a sideways push of more than its own weight, and its tires let go first, at about its weight. So a car slides rather than rolls, unless it trips on a curb. A loaded truck, its balance point high above its wheels, can roll over in a turn that pushes it sideways with only a quarter to a half of its weight.

Every Resting Thing Sits in a Valley

There is a second way to see the same rule. To tip the block, its balance point had to be lifted: tilting raised it, until it stood right over the edge. Past there, any further tilt lowers it, and the block falls of its own accord. Resting on its base, the block sat in a valley. Every small tilt lifted its balance point, and so every small tilt was undone.

That is what stable means. A body resting on firm ground is stable if every nudge lifts its balance point, like a marble in a bowl; unstable if some nudge lowers it, like a marble on a dome; and neutral if a nudge changes nothing, like a log on its side. How stable it is depends on how deep the valley is: how far the balance point must rise before the body falls. A wide, low block sits in a deep, wide valley. It tips only at a large angle, and only after a large lift.

A rounded bottom with weight low inside makes a valley in every direction, which is why a toy with a weighted round base always rocks back upright. Can a shape do it with no weight inside, made of one material throughout? Not a flat one: every flat shape of even thickness, with no dents in its outline, can be proved to have at least two ways to rest on its rim. A solid can, and one has been found: the Gömböc, a curved body with one way to rest and one way to balance, which rocks back to its one valley from any position.

High-domed tortoises, the Indian star tortoise among them, have shells close to that shape, and right themselves mostly by it. The first such body with flat faces and sharp corners, made of one material, was described recently, in a paper posted online. It has 56,946 faces and was checked by computer in exact arithmetic, but other mathematicians have yet to review it.

i A tortoise and a Gömböc: shapes that right themselves ×

Balanced on a Point

A pencil standing on its point is the opposite: a peak. Its balance point is as high as it can be, any lean lowers it, and so any lean grows, faster the further over it is. For an ordinary pencil the lean doubles about every twelfth of a second.

That doubling is why no amount of care is enough. Set a pencil upright to within a millimeter and it is down in about two thirds of a second. Set it a thousand times more exactly, to a thousandth of a millimeter, and it stands only about three quarters of a second longer. Set it a million times more exactly, to the width of a few atoms, and it still falls in about two seconds. Every thousandfold of care buys the same three quarters of a second.

A broom can be balanced on a palm because it falls more slowly. The time to fall grows with the square root of the length, so a broom eight times as long as a pencil falls about three times more slowly, slowly enough for the eye to see the lean and the hand to get under it. Even skilled balancers need almost a quarter of a second to respond. In one study the shortest stick anyone kept up on a fingertip for four minutes was 32 centimeters long.

Could a perfect hand on a perfectly still table do better? It is often said that even then the uncertainty principle of quantum mechanics, which lets no pencil be both exactly upright and perfectly still, would topple it within about four seconds. Worked through with care, the uncertainty would decide only for a pencil set upright to within a millionth of an atom’s width, which no real point or table allows. A recent study finds that warmth gets there first: the jiggle of the pencil’s own atoms and of the air around it shakes it millions of times more than quantum uncertainty does. It is still down within a few seconds, because the lean grows so fast that even such different starts end almost alike.

Two tricks hold a point up without catching it. A spinning top turns its fall into a slow circling. And an upside-down pendulum stands if its support is shaken up and down fast enough, its peak turned into a valley by the shaking. A pendulum a meter long, shaken ten centimeters each way, needs about seven shakes a second.

Standing Is Falling, Caught

A standing person is a broom: a balance point about a meter above a footprint the size of two shoes, and a body that is a peak. Left to itself it would be on the floor in a second or two. It stays up by catching itself. The springiness of the calf muscles and their tendons does much of the work, but not all of it. One measurement put it at about nine tenths of what it would take, and later studies find less.

The brain does the rest. From what the eyes, the inner ears and the soles of the feet report, it nudges the calf muscles between two and three times a second. All the while the body sways forward and back within a couple of centimeters, without its owner noticing. Standing still is a fall caught again and again. Walking stretches the act out: at each step the body falls forward past its footprint, and the swinging leg lands in time to make a new one.

A bicycle’s footprint is a line, the line between where its two tires touch the road. Lean, and the balance point is no longer over it. A rider balances mainly by steering into the fall: turning the front wheel toward the lean runs the tires back under the balance point. A bicycle can even do it alone. The standard model of a bicycle’s motion shows one of ordinary build, its rider sitting perfectly still with hands off the bars, straightening itself after a shove whenever it coasts at about 15 to 22 kilometers an hour.

The spinning wheels were long said to hold a bicycle up, like a top. Yet a bicycle whose front wheel had its spin cancelled, by a second wheel spun the other way, was ridden with ease. And a riderless machine was built on tiny wheels, with almost no spin and its front wheel touching the road just ahead of the steering axis instead of behind it. It still steered itself into every lean above about eight kilometers an hour, because of where its weights sat. No single effect is the answer; several together turn the wheel toward the fall.

When the Ground Gives Way

The rule of the base assumes ground that does not give. The Leaning Tower of Pisa shows what happens when it does. Before its rescue, the tower leaned five and a half degrees, and the line from its balance point, 22.6 meters up, landed 2.3 meters from the middle of a base nearly 10 meters in radius: well inside. On hard rock it could have leaned about 23 degrees before tipping. Yet it was close to falling. The clay under it is soft, and each small extra lean pressed the south side further down, adding more turning than the ground could push back. Engineers call this leaning instability.

After a trial, engineers spent most of two years drawing 38 cubic meters of soil, a little at a time, out from beneath the tower’s north side. It settled back by half a degree, its top by 45 centimeters, to the lean it had in 1838. It leans about five degrees today, and has barely moved since.

A ship has no ground at all, and floats with its balance point above the push of the water, which looks as if it should roll straight over. The water pushes up as hard as the hull weighs, through the middle of the water the hull has pushed aside. Heel the ship, and the part under water changes shape, gaining on the low side and losing on the high side, so the push moves out toward the low side. If it moves out past the balance point, the two turn the ship back upright. So builders make a hull wide, which carries the push further out, and keep the balance point low, with heavy stone or iron deep inside.

Under the Swedish warship Vasa, the push could barely get past. She carried 64 guns, 48 of them heavy bronze cannon, on two gun decks above 120 tonnes of stone ballast, and her upper hull was too tall and heavy for the hull under the water. Weeks before she sailed, 30 men running back and forth across her deck set her rolling so hard that the test was stopped after three runs. She set out on her first voyage, in Stockholm. About 1,300 meters out, a gust heeled her over, water poured in through her open lower gunports, and she sank within minutes, with about 30 people. She has since been raised, and stands in a museum in Stockholm.

Stones That Only Press

An arch of stone needs no glue. Each stone is pressed by its neighbors and passes the push on, and stone takes pushing well and pulling hardly at all. So an arch stands as long as the push can run through it within the stones, crossing every joint inside the stone: the rule of the base, applied at every joint. It can be stated as a theorem: if the stones cannot slide, and any one line of push that carries the arch’s weight can be found inside the stones everywhere, the arch stands.

The best curve for that line is a hanging chain turned upside down. A chain can only pull, so it hangs in the one curve in which every link pulls straight along its neighbors. Turn the curve over and every pull becomes a push of the same size. The rule fits in a sentence: as the flexible line hangs, so the rigid arch will stand, turned upside down.

Hang weights on the chain and it bends to carry them, and the line of push in the arch bends the same way. When the dome of St Peter’s in Rome cracked, a chain was loaded with 32 weights, each in proportion to a slice of the dome. Its curve turned over fitted inside the dome’s thickness: the dome was safe. An architect spent ten years designing a church near Barcelona upside down, in strings hung with small sacks of lead shot. Only its crypt was ever built.

The same rule lets blocks lean out with no arch at all. Stack identical blocks at a table’s edge, each pushed as far out as it will go, and the top one can hang half its length past the block below it, that block a quarter past the next, the next a sixth, and so on. Four blocks put the top one wholly past the table; two lengths take 31 blocks, and three take 227. Lay some of them side by side as counterweights, like bricks in a wall, and the reach grows far faster: 111 blocks can reach out three lengths.

Where a Valley Stops Holding

All of this assumes that a thing at rest stays in its valley until something pushes it out. For anything you can see, it does. Every object is jiggled by the random shaking of atoms that we feel as warmth, but the jiggle is tiny beside the lift it takes to tip over a book. Shrink the object, though, and the lift falls fast, while the energy of the jiggle stays the same: halve a block’s size and the lift it needs falls sixteenfold. For a speck of glass a thousandth of a millimeter across, lying in water so that it does not stick to the bottom, the two are about equal, and the speck tips back and forth by itself. Smaller things never settle into one valley. They wander between them, in the restless jiggling called Brownian motion.

Smaller still, quantum mechanics lets a thing leave a valley without climbing out at all. The nitrogen atom of an ammonia molecule can sit in either of two valleys, one on each side of the triangle of its three hydrogen atoms. Started in one, it crosses to the other and back about 24 billion times a second, through a ridge it lacks the energy to climb: it travels as a wave, and its wave reaches through. The first maser, the forerunner of the laser, ran on that flip.

And the largest valley of all may not be the lowest. Empty space has a resting state too. If the particles known today are all there are, whether that is the very bottom or a valley with a deeper one beyond depends mostly on the masses of two of them, the Higgs boson and the top quark. The best measurements lean, though not by much, toward a valley that is not the lowest, but one that would hold for unimaginably longer than universe has existed. A more exact mass for the top quark is the main thing needed to settle it.