The Spinning Skater: Angular Momentum's Free Speed-Up

The conservation law behind the trick

A figure skater begins a spin with her arms extended, spinning at a modest rate. She pulls her arms in against her body, and without pushing against anything, without any external force at all, she visibly speeds up - sometimes dramatically. The explanation is conservation of angular momentum: on frictionless ice, with no external torque acting about the skater's spin axis (gravity and the normal force from the ice both act through, or parallel to, that axis and contribute no twist), the quantity L = I·ω must stay exactly constant throughout the spin.

Moment of inertia: mass, but weighted by distance

Angular momentum L = I·ω pairs two quantities: angular velocity ω, and moment of inertia I, the rotational analogue of mass. For a body made of many small mass elements, I = Σ m·r² sums each element's mass weighted by the square of its distance r from the rotation axis - mass close to the axis barely counts, mass far from the axis counts a great deal, because of that squared term. Extending an arm moves a meaningful fraction of body mass much farther from the spin axis, which increases I substantially even though total body mass never changes; pulling it back in reverses that.

L = I · ω              (angular momentum, conserved with zero net torque)
I = Σ m_i · r_i²        (moment of inertia — mass weighted by r²)

arms out:  I_out  (large)   →  ω_out  (small)
arms in:   I_in   (small)   →  ω_in   (large)

I_out · ω_out  =  I_in · ω_in      (L unchanged)

Rotational kinetic energy is not conserved

Here is the part that trips people up: while angular momentum L stays fixed, rotational kinetic energy KE = ½Iω² does not. Substituting L = Iω lets you rewrite KE = L²/(2I) - with L constant, shrinking I necessarily increases KE. The skater genuinely gains rotational kinetic energy by pulling her arms in, and that energy is not free; it comes from real mechanical work her muscles do pulling her arms inward against the outward-flinging tendency (in the rotating, non-inertial frame this is often described loosely as working "against the centrifugal effect"). Extending the arms back out does the reverse, converting rotational kinetic energy back into work done by the skater's muscles resisting the arms being flung outward.

The same law, everywhere

This is not a special "figure skating" law - it is a completely general consequence of rotational symmetry (formally, Noether's theorem links angular momentum conservation to the laws of physics being the same in every direction), and it shows up wherever an object's mass redistributes while spinning freely. A platform diver pulls into a tight tuck to spin rapidly through multiple somersaults, then extends into a layout to slow the rotation for a clean entry. A cat, dropped upside down, uses differential tucking and extending of its front and back halves - conserving zero total angular momentum while still managing to rotate itself right-side up, a genuinely clever bit of biomechanics rather than a violation of the law. On a cosmic scale, a star's collapsing core at the end of its life shrinks from roughly the size of the Sun to a neutron star only about 20 km across; conserving its angular momentum over that enormous change in radius spins it up from a leisurely rotation to hundreds of revolutions per second, which is exactly what a pulsar is observed to do.

What changes it, what doesn't

It is worth being precise about what breaks the conservation. Any external torque about the spin axis - friction with the ice at the skate blade, a push from another skater, air resistance on outstretched arms - changes L over time, exactly as an external force changes ordinary linear momentum. In an idealized frictionless simulation, none of those act, so the only way L changes is if you apply one deliberately; everything else the skater does, redistributing her own mass by moving her arms, legs or torso, conserves L exactly while trading kinetic energy for the muscular work of that redistribution.

Frequently asked questions

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Everything above runs in your browser - open Angular Momentum: Spinning Skater and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.