Module 10 · rotational motion
Almost everything in Modules 1 to 6 treated objects as points; Module 2's spinning stool was the exception, added because angular momentum belonged with the other conservation laws. This module goes back and rebuilds mechanics for things that turn — force becomes torque, mass becomes moment of inertia, momentum becomes angular momentum — a real translation with visible seams, and a skater who speeds up by folding her arms and a bicycle that will not fall over as the payoff.
Push a door in the middle of its face and it swings. Push it just as hard right next to the hinge and almost nothing happens. Same force, same direction, same door — completely different result. So force alone cannot be the whole story when an object can turn. What matters is force and where you apply it, and the combination has its own name.
You need to undo a very tight bolt with a spanner, and you have one spare hand. What helps most?
Notice what happens as you swing the angle toward 0° or 180°: the torque dies, and at exactly 0° a force of any size does nothing at all. A push straight along the spanner, aimed at the bolt, cannot turn it. Everything that acts through the pivot is torque-free, and that is more useful than it sounds: for a body that is not turning, the torques balance about any point you care to name, not only about the real hinge. Choose the hinge, and the hinge's own reaction force drops out of the sum completely — it acts through your chosen point, so it has no torque about it. One unknown, gone for free, which is why balance problems (engineers call it "taking moments") so often collapse to one line.
An object is in rotational equilibrium when the torques trying to turn it one way exactly match those trying to turn it the other. That is why a child can balance an adult on a seesaw by sitting further out. It is also the principle behind every lever you have ever used: crowbars, wheelbarrows, scissors, nutcrackers, cranes, and your own forearm.
Newton's second law says F = ma: force, resisted by mass, produces acceleration. The rotational version is identical in shape:
But here is the interesting part, and the reason this is not just relabelling. Mass is a single number for an object. Moment of inertia is not — it depends on where the mass sits relative to the axis. Mass far from the axis has to travel a bigger circle to achieve the same spin rate, so it resists far more. How much more? A bit of mass m at distance r moves at speed v = ωr, so its kinetic energy is ½m(ωr)² = ½(mr²)ω². The r has been squared twice over: once because a wider circle means more speed, once again because energy goes as speed squared. So each bit contributes mr², and I is those contributions added up over the whole object — move a bit of mass twice as far out and it resists four times as hard. In the table below M is the object's total mass and R is the radius.
| Shape, spun about its centre | Moment of inertia | Why |
|---|---|---|
| Hoop or thin ring | MR² | every gram is at the maximum radius. Worst possible case. |
| Solid disc or cylinder | ½MR² | mass spread from centre to rim, so the average r² is half. |
| Solid sphere | ⅖MR² | a lot of the mass is packed near the middle, doing very little work. |
| Rod about its centre | 1/12 ML² | here L is the rod's length, not angular momentum; four times more, 1/3 ML², about its end. |
One thought to hold in mind before the next question. A rolling object carries kinetic energy in two places at once — ½mv² for the forward motion of its centre, and ½Iω² for the spin about that centre. Both come out of the same store on the way down.
A hoop, a solid disc and a solid sphere — same mass, same radius — are released together at the top of a ramp. They all roll without slipping. Which reaches the bottom first?
This is a genuinely startling result and worth stating plainly: a marble beats a bicycle wheel down a ramp, and a heavy marble ties exactly with a light one. Size and weight are irrelevant; only the shape's distribution of mass counts. Most people guess wrong the first time, which is why the experiment is worth running rather than just reading about.
A flywheel is a hoop on purpose: engineers put the mass at the rim precisely to maximise I, so it stores as much rotational energy as possible and smooths out the jerks of a piston engine. A tightrope walker's long pole uses the same trick for a different purpose — not to store energy but simply to be hard to turn: it makes her enormously harder to rotate, so she has seconds rather than fractions of a second to correct a wobble. And a figure skater's spin, in the next bench, is this table read backwards.
Module 2 found two conserved quantities: momentum mv, unchangeable for an isolated system, and angular momentum — the skater who pulls her arms in, and Kepler's equal areas. Here it gets a name and a formula: angular momentum, L = Iω. For a single object of mass m moving at speed v at distance r from the axis, I = mr² and ω = v/r, so L = mvr — momentum times lever arm, just as torque is force times lever arm. It is the form a planet obeys (Module 2's equal areas) and the form Module 22 will quantise. And it is conserved for the same deep reason momentum is. Module 2 showed that momentum is conserved because space has no special place; angular momentum is conserved because space has no special direction — turn an isolated system round to face a different way and nothing about it changes, so nothing can twist it, so L cannot change. (The "internal forces come in third-law pairs" story also gives the right answer here, but only if the paired forces act along the line joining the two bodies — the strong form of the third law — and Module 2 showed that this is the argument that stops working.) If nothing outside twists you, your angular momentum is fixed, whatever you do inside.
This produces the most famous trick in physics — you have already met it in Module 2 as a preview. A skater spinning with her arms out has a large I and a modest ω. She pulls her arms in. Her mass has not changed, and nothing outside has twisted her — but I has dropped, so ω must rise to keep the product constant. She speeds up dramatically, having done nothing but fold her arms.
When the skater pulls her arms in, her rotational kinetic energy goes up — sometimes by a factor of two or three. Where did that energy come from?
Two conservation laws, both true at once, and they do not say the same thing. Angular momentum Iω is fixed by the absence of external torque. Kinetic energy ½Iω² is not fixed, because she is doing work. Halve I and ω doubles, so ½Iω² doubles too — and every joule of that comes out of her arms. Let them fly back out and she gets it back, slowing down as she does.
| Where angular momentum shows its hand | What is going on |
|---|---|
| A diver tucks to somersault | tuck for a small I and a fast tumble, straighten to slow down and enter the water cleanly. The number of somersaults is decided at the moment she leaves the board. |
| A cat lands feet-first | dropped upside down with zero angular momentum, it still turns over — by twisting its front and back halves in opposite directions with different moments of inertia, so the total stays zero throughout. It ends the right way up having never broken the law. |
| A helicopter needs a tail rotor | spinning the main rotor one way would spin the helicopter's body the other. The tail rotor supplies the external torque that stops it. |
| A collapsing star becomes a pulsar — a tiny, furiously spinning star whose beam sweeps past us like a lighthouse | a star the size of the Sun collapses to a ball 20 km across. I falls by a factor of about 5 × 10⁹, so ω rises by the same factor — a star that turned once a month would end up turning nearly two thousand times a second. Real pulsars turn between once a second and 700 times a second, because the supernova throws most of the star's mass, and most of its angular momentum, away. |
| The Earth's day is getting longer | tides drag on the ocean floor and steal angular momentum from the Earth's spin. It is not lost: the Moon receives it and drifts 3.8 cm further away each year. The day was about 23 hours long when the dinosaurs lived; you would have to go back some 600 million years to find a 22-hour day. |
The usual explanation is gyroscopic, and it needs one idea we have not used yet. Angular momentum has a direction as well as a size — the direction is along the axle (curl the fingers of your right hand the way the wheel is spinning, and your thumb points the way L points). Tipping the axle does not change how fast the wheel spins; it swings that whole arrow round to point somewhere new, and swinging L takes a torque. So a spinning wheel does not simply topple when you tip it — the gravity torque, instead of toppling it, pushes its axis sideways and it precesses in a slow circle. This is real, and it is why a spinning top stays up.
But it is not really why bicycles work. Riderless bicycles built with counter-rotating wheels that cancel all gyroscopic effect still balance themselves perfectly well. What actually keeps a bicycle up is steering into the fall. When the bike leans left, the front wheel turns left — partly because the rider turns it, and partly on its own: the front wheel touches the ground a little behind the point where the steering axis would meet it, and that offset (bicycle designers call it trail) makes the wheel swivel into a lean by itself. That is how a riderless bicycle manages without anybody. The bike then curves left, and the ground has to supply the mv²/r of Module 6 through the tyre contact patch, which is below your centre of mass. An inward push applied below the centre of mass torques the bike back upright. Nothing pushes you outward; the inward push is simply applied in the wrong place to keep you leaning. A rider adds to this constantly and without thinking, which is why balancing is far easier to do than to describe.
Sit on a swivel chair holding something heavy in each hand, arms out, and have someone start you turning gently. Pull your arms in. You will speed up sharply, and you will feel yourself doing the work. Then, for the better experiment: hold a spinning bicycle wheel by its axle, arms out, and tip the axle sideways. The chair turns underneath you. You have not pushed against anything — you tilted a vector, and the rest of the system had to compensate.
1. Two children sit on a seesaw. One weighs 30 kg and sits 2 m from the pivot. The other weighs 40 kg. For balance, she must sit:
2. A tin of soup and a tin of the same size full of frozen solid soup are rolled down a ramp together. The liquid one wins easily. Why?
3. A skater spinning at 1 turn per second pulls her arms in and halves her moment of inertia. Her new spin rate and her new kinetic energy are: