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Angular Momentum Conservation Calculator

Find the new rotational speed and kinetic energy change when a spinning object's moment of inertia changes with no external torque, like a figure skater pulling in their arms.

Final angular velocity
150 RPM
Angular momentum (conserved)
31.4159 kg·m²/s
Initial kinetic energy
98.696 J
Final kinetic energy
246.7401 J

How it works

With no external torque acting on a spinning object, its angular momentum L = I×ω is conserved — so if its moment of inertia changes from I1 to I2 (mass redistributing closer to or farther from the spin axis), its angular velocity must change to match: ω2 = I1×ω1 ÷ I2. This is the classic figure-skater-pulling-their-arms-in scenario: pulling mass inward shrinks I, so ω rises to keep L fixed, the same reasoning that speeds up a diver's spin when they tuck or a star's spin when it collapses.

Rotational kinetic energy, KE = ½×I×ω², behaves differently — it is NOT conserved here even though momentum is. Substituting the conservation relation shows KE = L²/(2I), so shrinking I while L stays fixed necessarily raises KE. That extra energy isn't free: it's exactly the work the skater's own muscles do pulling their arms in against the spin's outward pull, reported here as the gap between initial and final kinetic energy.

Moment of inertia (I) is taken as a direct input rather than derived from a specific shape, since the underlying conservation law applies to any rigid body regardless of its shape — the defaults (5 kg·m² shrinking to 2 kg·m² at 60 RPM) reproduce the standard skater example, multiplying the spin rate's rotational kinetic energy by 2.5x while conserving angular momentum exactly.

FAQ

Why does kinetic energy increase if no external torque is doing work?

No external torque means no external agent is twisting the object, but internal work is still possible — the skater's arm muscles do real mechanical work pulling the arms in against the outward tension the spin creates, exactly like winding a spring. That internal work is where the kinetic energy gain comes from; angular momentum, not energy, is what stays conserved when there's no external torque.

What if I want to shrink I by changing a shape's radius instead of entering I directly?

For a specific simple shape (solid disk, hoop, solid or hollow sphere) with a known mass, this site's Flywheel Kinetic Energy calculator computes I = k×m×r² for you — compute I1 and I2 there for the before/after radius, then enter those two results into this calculator's I1/I2 fields to find the resulting spin-rate and energy change.

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