⛸️ Angular Momentum Conservation — Spinning Skater

No — the total mass (body plus both arms) is fixed throughout. Only the distribution of that mass relative to the spin axis changes as the arm-extension radius r varies. Moment of inertia depends on both mass and how far each piece of mass sits from the axis (I = Σm·d²), so moving the same mass closer to the axis reduces I without removing any mass at all.

A diver or gymnast tucks into a tight ball to spin or somersault faster mid-air, then extends to slow the rotation before landing. A falling cat twists its front and back halves in opposite senses with different limb configurations to right itself without any external torque. On a much larger scale, a collapsing stellar core conserves its angular momentum as it shrinks from thousands of kilometers to a few kilometers across, spinning up to hundreds of rotations per second to form a pulsar.

It idealizes the skater as a rigid torso of fixed moment of inertia plus two point masses at an adjustable radius, ignoring the true distributed mass of arms, legs and torso, any friction with the ice, air resistance, and the small torque a real skater's blade edge can exert while gliding. It also treats the extension change as instantaneous rather than modeling the muscular force and finite time actually needed to move the arms.

The arm-extension slider sets r, the live radius of the point-mass arms from the spin axis (0.15 m fully tucked to 0.9 m fully extended); moving it while the figure spins changes the moment of inertia instantly and the angular velocity responds live to keep L constant. The initial spin-rate slider (ω₀) resets the conserved angular momentum L using the current extension and the chosen spin rate. The Reset button returns the arms to full extension and re-establishes L from the ω₀ slider.

Each frame recomputes ω(r) = L / I(r) from the live extension r and the fixed conserved angular momentum L, then integrates the spin angle forward with φ += ω(r)·dt using the actual elapsed time between frames. Because ω is recalculated from scratch every frame rather than cached, dragging the extension slider mid-spin produces an immediate, continuous change in spin rate rather than a delayed or stepped one.

The conservation law itself — L = Iω constant with zero external torque — is exact and is exactly what the simulator enforces at every frame. The specific numeric values (torso moment of inertia 0.8 kg·m², two 3 kg point-mass arms, extension range 0.15–0.9 m) are reasonable order-of-magnitude approximations for an adult figure skater rather than a biomechanically precise model, chosen so the demonstrated physics matches real skating intuition.