Why does a spinning skater speed up when she pulls her arms in?

The everyday picture, and why it is wrong

Watching a skater accelerate into a blur, the natural reading is effort. She looks like she is straining, so she must be spinning herself faster, the way you would pedal harder on a bike. On that picture, speed comes from work, and a better athlete simply pushes more.

That is not what is happening. Once she is up on one skate with her blade tracing a tight circle, almost nothing outside her is twisting her. She has no way to push herself around faster. The acceleration is not effort applied to turning. It is the consequence of changing her shape.

Spin is a quantity, and it is conserved

Physics tracks rotation with a quantity called angular momentum. It depends on two things: how fast something is turning, and how far its mass sits from the axis it turns around. Mass held far out counts for much more than the same mass held close in, which is why a wide stance is hard to turn and a tight one is easy.

The rule is that this quantity only changes when something outside applies a twist. Physicists call an outside twist a torque: a foot planted on the ground, a hand on a wall, friction. With no net outside torque, the total is locked.

So when the skater pulls her arms and her free leg in toward her body, one of the two factors drops sharply. Her mass is now much closer to the axis, and she is far easier to turn. The total cannot change, so the other factor has to climb. She spins faster, whether she wants to or not.

The trade is not free. Her rotational energy genuinely rises during the pull, and that energy comes from her muscles working to drag her limbs inward against the outward pull of the rotation. Angular momentum is conserved; energy is supplied. Try it seated on a swivel chair holding two heavy books, and you can feel the effort the moment you fold your arms in.

The same rule, at absurd scales

A gymnast's dismount is the textbook version with numbers on it. She releases the bar turning about once a second, curls into a tuck, and by the time she opens to land she is turning about four times a second. Nothing touched her in flight. Halving how far her mass sits from her axis was enough.

Push the same trade to its extreme and you get a pulsar. When a massive star's core collapses to a ball roughly 20 kilometres across, all of that mass moves in toward the axis at once, and the rate has to climb to match. In 2006, Hessels and colleagues reported PSR J1748-2446ad in the cluster Terzan 5, turning 716 times every second, the fastest star known.

The rule also runs in the other direction, as a bill that has to be paid. Tidal friction moves angular momentum from Earth's rotation into the Moon's orbit. Laser ranging off the mirrored panels Apollo 11, 14 and 15 left on the lunar surface shows the Moon receding about 3.8 centimetres a year, and eclipse records reaching back to 720 BC show our day lengthening by roughly 1.8 milliseconds per century, slightly less than tidal friction alone predicts, with other processes inside Earth making up the difference.

Why it matters for machines

Because spin can only be traded and never created, a spacecraft in vacuum with nothing to push against can still steer. It carries heavy flywheels called reaction wheels: spin a wheel one way and the whole spacecraft rotates the other way, leaving the total unchanged. Stop the wheel and the spacecraft stops, now aimed somewhere new.

That also makes those wheels a single point of failure. NASA's Kepler telescope carried four and needed three; the first failed in July 2012 and the second in May 2013, and on 15 August 2013 NASA ended attempts to recover them, closing the original planet survey. Closer to home, the same rule is why every single-rotor helicopter needs a tail rotor: the engine turning the main rotor one way pushes the fuselage the other, and the tail rotor is what holds the heading.