SKATER
Pull in: I₁ω₁ = I₂ω₂ holds, KE jumps. L conserved, energy not — two different books.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A skater changes the radius of two spinning masses from r₁ to r₂. Pulling inward produces the familiar spin-up. External torque is zero.
No torque
External torque is zero, so L = Iω is fixed. Moving the masses inward shrinks I and raises ω.
Solve
The new spin is ω₂ = L/I₂. Because KE = L²/(2I), moving inward raises kinetic energy through positive muscle work; moving outward lowers it and returns mechanical energy. The displayed sign of ΔKE follows the chosen direction.
Change radius
Angular momentum stays fixed throughout. As the masses move in, the spin and energy rise. The animation and energy bar follow that direction continuously.
Audit
At every arm length along the pull, the product of moment of inertia and angular velocity holds the same value to a part in a trillion — L is genuinely constant, not merely constant at the endpoints. The spin ratio equals the inertia ratio, the energy ratio equals it too, and the energy gain matches one-half L-squared times the change in one-over-I. Two different books, both balanced.