By Zencalcy Editorial TeamLast reviewed About us
When no external torque acts, total angular momentum stays constant. If the moment of inertia changes — arms pulled in, a rotor folded, a star collapsing — the angular velocity must change in the opposite direction so that Iω is unchanged.
Final spin after the moment of inertia changes.
I₁ ω₁ = I₂ ω₂
I₁ and ω₁ are the initial inertia and angular velocity; I₂ and ω₂ the final values. The relation assumes no net external torque, so friction and air drag must be negligible.
| Symbol | Meaning |
|---|---|
| I₁, I₂ | Initial and final moment of inertia, in kg·m². |
| ω₁, ω₂ | Initial and final angular velocity, in rad/s. |
Starting at 10 rad/s with I = 4 kg·m² gives L = 40 kg·m²/s. Halving the inertia to 2 kg·m² raises the spin to 20 rad/s, keeping L at 40.
Continue with the Moment of Inertia Calculator, Physical Pendulum Calculator, or the Rolling Kinetic Energy Calculator — or browse more rotational and periodic motion tools and all physics calculators.
Pulling the arms in moves mass closer to the axis, reducing I. With L constant, ω must rise by the same factor that I falls.
Results are computed from the standard equations of classical mechanics. Unit conversions use the SI definitions published by NIST. Air resistance is ignored unless the calculator names a drag model.
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Last reviewed: September 2026. See how we build and check every tool on the about page.
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Content updated: September 2026