Angular momentum: why a spinning skater speeds up

Guide · Updated 2026-11-10

Understand conservation of angular momentum, moment of inertia and rotational energy with a skater who pulls her arms in, and a worked example.

Open the Spinning skater Free, in your browser, nothing uploaded.

The spinning skater tool shows why a figure skater turns faster when she pulls her arms in, and why that is not magic: a quantity called angular momentum has to stay the same. It suits students in Class 11 and anyone curious about rotation. To find the point a body balances on, see centre of mass. To see rotation mixed with rolling, try the rolling race.

What it is

Pushing a trolley needs force and gives it momentum. Turning something needs a twist and gives it angular momentum. For a body spinning about a fixed line (its axis):

L = I × ω

• L is the angular momentum, in kg m²/s.
• I is the moment of inertia, in kg m². It tells you how hard the body is to spin.
• ω (omega) is the spin speed, in radians per second (rad/s). One full turn is about 6.28 rad.

Moment of inertia depends on where the mass is. For small weights, I = Σ m r², where m is each mass and r is its distance from the axis. Because r is squared, mass far out counts a lot more than mass near the middle.

If no outside twist acts on the body, L stays the same. So when the skater pulls her arms in, r gets smaller, I gets smaller, and ω must get bigger to keep L the same.

The energy of the spin is E = ½ I ω², which can also be written E = L² ÷ (2I). With L fixed, a smaller I means more energy. The extra energy is the work her muscles do pulling the arms in.

How to use it

1. Set the weight in each hand, from 0 to 4 kg. Heavier weights give a bigger change when the arms move.
2. Set the spin with arms out ω₀, the speed when the hands are 1 m from the axis.
3. Press Pull arms in, or drag the r slider, which runs from 0.2 m to 1 m.
4. Watch ω, I, L and the kinetic energy. L stays put while the others change.
5. Press Arms out to go back, and Pause to freeze the picture.

A worked example

Pick our own numbers: 1.5 kg in each hand and a starting spin of 3 rad/s. The model body alone has I = 1 kg m², and the two weights add 2 × m × r².

Arms out, r = 1 m.

• I = 1 + 2 × 1.5 × 1² = 4 kg m²
• L = 4 × 3 = 12 kg m²/s
• E = ½ × 4 × 3² = 18 J

Arms in, r = 0.4 m.

• I = 1 + 2 × 1.5 × 0.4² = 1.48 kg m²
• L is still 12 kg m²/s, so ω = 12 ÷ 1.48 = 8.1 rad/s
• E = ½ × 1.48 × 8.108² = 48.6 J

The spin is 2.7 times faster, and the energy has gone from 18 J to 48.6 J. That is a gain of 30.6 J, which the skater supplied by pulling.

Where it is used

• Sport. Divers tuck to spin faster and open out to slow down before they enter the water. Gymnasts do the same.
• Space. Satellites use spinning wheels. Changing a wheel’s speed turns the satellite the other way, with no fuel.
• Astronomy. A collapsing cloud of gas spins faster as it shrinks. This is part of why young stars and the discs around them rotate.
• Everyday. Swivel chairs let you feel it: sit, hold weights out, get a spin, then pull them in.

Common mistakes

• Saying energy is conserved too. In this case it is not. The skater does work, so the energy rises. Only L is fixed.
• Forgetting the square. Halving r makes the weights’ share of I four times smaller, not two times.
• Mixing units. Use kg, m and rad/s. If a spin is given in turns per minute, change it to rad/s first.
• Ignoring the body. The body adds its own I. Treating the arms as the only mass gives the wrong ω.

Questions people ask

Why does the skater slow down when she opens her arms? I goes up, L stays the same, so ω must fall.

What happens on a real ice rink? Ice has very little friction, so L stays nearly constant and the model works well. On rough ground the floor would twist her and L would change.

Does a heavier skater spin faster? Not by itself. What changes the spin is how much I changes when she moves her arms.