Watch a figure skater start a spin with her arms out wide. It looks slow and graceful. Then she pulls her arms in and, with no push from anything, she blurs into a spinning column. Nobody gave her a shove. She did it by changing her shape alone.
This is a skater seen from above. Drag anywhere on the picture to move her hands in and out, or use the buttons. Watch the numbers as you go.
Model: torso 0.8 kg·m² plus two arms of 3 kg each, treated as points at the hand distance. Start: 1 turn per second, arms out.
Spinning things have angular momentum, written L. For a rigid spinner it is the product of two things: how hard the object is to spin up, called the moment of inertia I, and how fast it turns, ω. So L = I × ω.
On smooth ice the blade barely grips, so almost nothing pushes on her to change L. In the real world friction slowly drains it, but over a few seconds L is close to constant. If I × ω stays fixed and she makes I smaller, ω has to get bigger. There is no other place for the change to go.
The moment of inertia adds up every bit of mass multiplied by its distance from the axis squared: I = Σ m r². The square is the whole trick. Move a hand to half its distance and that hand's contribution drops to a quarter, not a half.
In the toy model above, the arms out at 0.8 m give I of 4.64, and arms in at 0.15 m give about 0.94. That is a drop of nearly five times, so the spin rises nearly five times: from one turn a second to nearly five. Real skaters are lumpier than two points on sticks, but the same idea applies, and they also pull their free leg in and stand tall and narrow.
The fastest spin on ice skates in the Guinness records is 342 revolutions per minute, which is 5.7 turns every second, set by Olivia Oliver of Canada in 2015.
Look at the energy box. It goes up as the arms come in. Spinning energy is ½Iω², and with L fixed that equals L² / 2I, so shrinking I raises it. Halve I and the energy doubles.
Energy cannot appear from nowhere. It comes from her muscles. As she spins, her arms want to fly outward, so pulling them in means pulling against that outward tendency, the way you tug on a string with a ball whirling on its end. That pull does work, and the work becomes spin energy. Throw the arms back out and the energy is handed back to the arms as they swing outward; she slows down again.
The same rule shows up wherever something rotates without a push from outside. Divers tuck into a tight ball to spin quickly through the air, then stretch out to slow down for the entry. A cat can twist itself around in mid-air by changing its shape. Even the toy ball on a string you swing around your finger speeds up as the string winds in.
Pull the arms in and out several times. Notice that the product of inertia and spin never changes: when the inertia number halves, the rpm doubles. Then try moving your hands in slowly versus quickly. The result is the same, because only the shape at that moment matters. The skater's spin is a trade between size and speed, and the ice keeps the books balanced.