Angular Momentum Calculator
Angular momentum of a spinning body, and the new speed when its moment of inertia changes — the skater-pulling-in-arms calculation.
Why a spinning skater speeds up when they pull their arms in.
How the angular momentum calculator works
Angular momentum L = Iω is conserved when no external torque acts. So if the moment of inertia falls, the angular velocity must rise in exact proportion.
The energy does not stay constant, though: halving the inertia doubles the speed and *doubles* the kinetic energy. The extra comes from the work the skater does pulling their arms in against the outward pull.
Formula: L = Iω; I₁ω₁ = I₂ω₂
Worked examples
| Inputs | Angular momentum | Note |
|---|---|---|
| A skater pulling in | 25.13274 kg·m²/s | 60 rpm becomes 200 rpm |
| No change | 25.13274 kg·m²/s | speed unchanged, no work done |
| Arms out instead | 25.13274 kg·m²/s | the reverse — speed falls, energy is given back |
FAQFrequently asked questions
Why does a skater speed up when they pull their arms in?
Because angular momentum is conserved. Reducing the moment of inertia forces the angular velocity up in exact proportion.
Where does the extra energy come from?
From the skater. Pulling the arms inward against the outward pull is real work, and it goes into the rotation.
Is angular momentum always conserved?
Only when no external torque acts. Friction at the ice or air resistance will slow any real spin.
Why does a cat always land on its feet?
By changing its moment of inertia in two halves of its body and rotating them differently — turning over with zero net angular momentum.
Where else does this matter?
Planetary orbits, neutron stars spinning at hundreds of turns a second after collapse, helicopter rotors, and gyroscopes.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Measurement Institute — Australia's measurement authority
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.