Rotation
Why a Spin Keeps Itself, and Swerves When Pushed
The Push That Misses the Middle
Throw a hammer end over end and watch it closely. One point of it, its balance point, sails along a smooth arc, the same arc a stone would follow. The rest of the hammer wheels round that point at a steady rate, all the way. Every motion of a solid body is these two at once: the travel of its balance point, which the laws of motion already describe, and a turning about that point. Gravity cannot change the hammer’s turning. Near the ground it pulls every bit of the hammer the same way, each in proportion to its mass, so its pulls add up to one pull at the balance point, and a pull through the balance point moves a body without turning it.
What sets a body turning is a push whose line misses its balance point, or misses the pivot it is held by. How hard the push turns it depends on two things: how hard it pushes, and how far its line passes from the pivot. Their product is the twist, which physicists call torque. Push a door near its handle and it swings open; push as hard right beside the hinges and it barely moves. A wrench twice as long turns a stuck bolt twice as hard for the same pull.
A lever runs on the same rule. Two weights on a beam balance when their distances from the pivot are in inverse proportion to them, so a weight three times as far out balances three times its weight. Give me a place to stand, Archimedes is said to have boasted, and I will move the Earth. The lever gives nothing for free. Its long end moves three times as far as its short one, so what it gains in push it pays for in distance, and the work, the push times the distance, comes out the same at both ends.
Where the Mass Sits
The same twist can set one body spinning fast and another hardly at all. For motion in a straight line, a body’s reluctance to change its speed is its mass. For turning, what counts is where the mass sits. Mass near the axis hardly moves when the body turns, so it costs little to set going. Mass far out has to travel all the way round a wide circle. Its share of the reluctance grows with the square of its distance from the axis: the same mass twice as far out is four times as hard to set turning, and four times as hard to stop. This reluctance to change its turning is called the moment of inertia.
A race down a slope shows it. Let a ball, a solid cylinder and a ring roll down side by side. Each turns as it rolls, so some of the energy it gains in falling goes into its spinning, and less is left to carry it along. The ball, its mass gathered toward its middle, puts only two sevenths of its energy into spin. The cylinder puts in a third, and the ring, its mass out at the rim, nearly half. So the ball wins and the ring comes last, every time.
Neither size nor weight changes the order. A big heavy ring and a small light one keep level all the way down, and any solid ball beats any ring. Only the shape counts: how the mass is spread, compared with the radius the body rolls on.
A tightrope walker’s long pole uses the same rule. It adds little weight, but its ends are far from the walker. In a model of a walker on a wire, a pole 12 meters long and weighing 13 kilograms, like those of professional walkers, nearly triples the walker’s reluctance to tip. A lean then grows about half again as slowly, which gives the walker time to catch it. The pole helps in a second way: turning it one way pushes the walker the other way, a real correction and not just a slower fall.
A Spin That Keeps Itself
Once a body is turning, it goes on turning until a twist from outside stops it, just as a moving body goes on moving until a push stops it. What it keeps is its angular momentum: its reluctance to change its turning, times how fast it turns. Earth has spun for four and a half billion years with nothing to drive it. Only the drag of the tides slows it.
The rule has a surprising side. A body that changes its shape changes that reluctance, and then its rate of turning must change to keep the product the same. Sit on a stool that turns freely, a weight held out in each hand, and set it turning slowly. Pull the weights in and the spin speeds up, though nothing outside has pushed you. Push them out again and it slows.
If all the mass were in the weights, pulling them in to half the distance would make the spin four times as fast. Most of the body cannot be pulled in, so the gain is smaller: a sitter holding four kilograms in each hand spins a little more than twice as fast. A figure skater spins up the same way, starting with her arms and a leg stretched out and drawing them in.
Spinning faster carries more energy, and that does not come free. Left to itself, each weight would go straight on, off its circle, so the arm must pull it inward just to keep it going round; the sitter feels that as the weight tugging outward. Drawing the weight in means pulling it inward while it moves inward, and that takes work. The work of the arms goes into the spin, and letting the weights back out takes it back. The angular momentum stays the same throughout. The energy of the spin does not, but every bit of it is accounted for.
Can a body with no spin at all turn itself over? A cat dropped upside down lands on its feet. When one was filmed doing it, at twelve pictures a second, scientists argued over whether the rule had been broken. It had not. A body with no spin can still end up turned round, by changing its shape on the way. The cat bends and twists at its middle and draws in and stretches out its legs, so that its front and back halves turn against each other and their spins cancel at every instant. How much each of those moves does is still argued over in physics journals. Astronauts turn themselves round in midair the same way.
A Spin Has a Direction
Angular momentum has a direction, along the axis of the spin, and the direction is kept too. A spinning wheel left alone goes on pointing the same way. So would one that was not spinning, if nothing twisted it. What spin adds is that a twist turns its direction only slowly, and sideways. A heavy wheel spun at up to 200 turns a second, in rings that let it point any way, once showed Earth turning beneath its steady axis. It gave the gyroscope its name, from the Greek for watching the turning.
Hang a spinning bicycle wheel by a cord tied to one end of its axle, and let go of the other end. A wheel that is not spinning swings down and hangs from the cord. The spinning one swings its axle round the cord instead, in a level circle. Each moment, the twist of its weight adds a little spin about a level line across the axle, the line it would turn about if it fell. Added to the big spin the wheel already has, that little spin does not tip the axle down. It swings it sideways. This circling is called precession.
That is the account in the textbooks, and it leaves something out. Let go, the wheel does not simply start to circle. It dips a little first, and bobs, and the bobbing dies away as it settles into a level circle a little lower down. The circling is a spin of its own, about the upright cord, and nothing gave the wheel any: the twist of its weight is always level, with no share about the upright. The only place that spin can come from is the wheel’s own, tipped down a little. To go around, it has to go down a little. The dip of a teaching gyroscope has been measured, and it is the size the rule predicts.
All the while, the cord holds up the whole weight of the wheel. Nothing about it defies gravity. The faster the wheel spins, the more slowly it circles, since the same twist then has a bigger spin to turn: spun at half the rate, it circles twice as fast and dips four times as far. A rifle bullet is spun by the grooves of its barrel for the same reason. The air’s twist on its nose swings it round its path in a small circle, instead of tipping it over.
Earth is a spinning wheel too. Its spin makes it bulge at the equator, and the pulls of the Sun and the Moon on the bulge try to stand its tilted axis upright. Instead the axis circles, once in about 26,000 years. Comparing the places of stars with those written down some 150 years earlier is enough to show the drift. The north pole of the sky circles with it. Five thousand years ago the pole star was Thuban, in the constellation of the Dragon; in about twelve thousand years it will be Vega.
Why a Top Stands Where a Pencil Cannot
A pencil set on its point falls, because any lean lets its weight lever it further over. A spinning top is a pencil whose lean is turned sideways. When it leans, the twist of its weight swings its spin round instead of tipping it, so the lean circles instead of growing. Spinning fast enough, a top stands upright and seems not to move at all; it is said to sleep.
Fast enough is a definite rate. It depends on the top’s size and on how its mass is spread, not on how heavy it is. The higher its balance point, the faster it must spin; the more of its mass lies out at its rim, the slower it may go. The tops in the scene, each 18 grams of beech on a steel point, need about 13 turns a second. The air and the friction at its point slowly take its spin away. Below that rate it starts to lean and to circle, and the slower it spins, the further it leans and the faster it circles, since a smaller spin is swung round more quickly by the same twist. Then the edge of its body strikes the floor, and it falls like the pencil it always was.
A rolling wheel does the same. Lean a rolling coin and it curves toward the side it leans to, because the twist of its weight swings its spin round. A bicycle’s spinning front wheel is one of the things that steer a riderless bicycle into a lean and so catch its fall. It is not what holds up a bicycle with a rider. A bicycle whose front wheel had its spin canceled, by a second wheel spun the other way, was still easy to ride. Launched on its own, it fell over.
Why Everything Out There Spins
Look anywhere in the sky and things are turning: planets, stars, the discs of gas round young stars, whole galaxies. They spin because the clouds of gas they formed from were never perfectly still. A cloud is stirred by the motions of its own gas and tugged by its neighbors, so it carries some spin, however slight. As gravity pulls it in, its mass comes closer to the axis while its spin is kept, so it turns faster and faster.
A spinning cloud also flattens. Along its axis nothing holds it, and it falls straight in. Across the axis its spin holds it out. Its parcels of gas run into one another, and the energy of their motion warms the gas, which shines it away as light. But no collision can take away the cloud’s spin. So the cloud settles into a flat disc, each parcel on a circle as wide as its spin allows, and the gas with almost no spin falls to the middle. That is why the planets go round the Sun in nearly one plane, and why spiral galaxies are flat.
The young star cannot keep the spin. A typical cloud core that makes a star carries, for its mass, hundreds to thousands of times more spin than a star could hold even turning fast enough to tear itself apart. Most of it is handed outward: to the disc and the planets that form in it, to companion stars, and away on magnetized winds, some of them squeezed into the two narrow jets a young star shoots out along its axis. The Sun holds 99.8 percent of the solar system’s mass but less than one percent of its spin. Jupiter alone carries about three fifths.
Collapse spins up dying stars too. When the core of a big star falls in on itself, it becomes a neutron star the size of a city, spinning up to tens of times a second: the Crab pulsar, born in a supernova seen almost a thousand years ago, turns about thirty times a second. The fastest spinners are older. A neutron star with a close companion can draw in its gas, and each bit that lands brings the swing of its orbit, spinning the star up. One has been caught switching between feeding on its companion and shining as a pulsar. The fastest known, in the star cluster Terzan 5, turns 716 times a second.
Where Turning Stops Being Simple
How fast can anything spin? Holding each part of a spinning body on its circle takes an inward pull, which the body’s strength or its gravity has to supply, and a flywheel spun too fast bursts. Strength limits how fast the rim can move, so the smaller the body, the faster it can turn: a speck of glass held by laser light in a vacuum has been spun about six billion times a second. A neutron star is held together by its gravity instead. At 716 turns a second, the equator of the fastest is moving at about a fifth of the speed of light. Theory puts the rate at which a neutron star’s gravity could no longer hold on to its equator at around a thousand turns a second, the exact figure depending on how wide neutron stars really are, which is still being measured. None has been found spinning faster than 716. Whether something brakes them first, such as the grip of the star’s magnetism on the gas that feeds it, or gravitational waves carrying spin away, is not settled. Catching those waves from a fast neutron star, or finding one that spins faster, would tell.
At the other end of the scale, spin comes in steps. Quantum mechanics lets the angular momentum of anything, measured along any line, take only values one fixed step apart, the same step for everything. A spinning bicycle wheel holds some 10³⁴ steps, far too many for the steps to show. A molecule holds a few. A molecule of carbon monoxide cannot turn slowly: it either does not turn at all, or turns with at least one whole step, and when it gives up that step it sends out radio waves 2.6 millimeters long, which carry the step away. Radio telescopes map the cold gas between the stars by that glow.
And some angular momentum has nothing going round at all. Hang an iron rod on a fine thread inside a coil, and reverse the iron’s magnetism: the rod twists on its thread. The magnetism of iron comes from its electrons, and the angular momentum they carry flips with it, so the rod turns the other way to keep the total. The first measurement found the turn expected if the magnetism came from the turning of the electrons’ clouds round their nuclei, but later ones found the rod turning only half as far. The puzzle was cleared up once the electron’s own spin was discovered. That spin is real angular momentum, as the twisting rod shows. Yet nothing in the electron is going round: it has no size that anyone has found, and no parts.
