🎾 Spin axis

The body is a "book" with three unequal sides. Pick which principal axis to spin it about:

Spin rate ω: 6.0 rad/s Initial wobble: 3% Playback speed: 1.0×

📐 Conserved & state

‖L‖ ang. momentum—

Rot. energy E—

Flips so far0

ω₁ / ω₂ / ω₃—

I₁ : I₂ : I₃—

Tip. Spin about the middle axis ② and wait — the book will suddenly flip end-over-end, again and again, even though nothing touches it. The other two axes spin smoothly forever. Drag in the scene to orbit the camera.

axis ① (long)

axis ② (middle)

axis ③ (flat)

L (angular momentum)

The intermediate axis theorem. A free rigid body has three principal axes with moments of inertia I₁ < I₂ < I₃. Rotation about the axis of smallest (I₁) or largest (I₃) moment of inertia is stable; rotation about the intermediate axis (I₂) is unstable. The motion obeys Euler's equations for torque-free rotation:

I₁ ω̇₁ = (I₂ − I₃) ω₂ ω₃
I₂ ω̇₂ = (I₃ − I₁) ω₃ ω₁
I₃ ω̇₃ = (I₁ − I₂) ω₁ ω₂

Cosmonaut Vladimir Dzhanibekov noticed this in 1985 aboard the Salyut 7 station when a spinning wing-nut floated off and repeatedly flipped. Astronauts call it the tennis racket effect — toss a racket and try to make it spin about its handle-to-edge axis; it twists half a turn every time. Throughout the motion angular momentum L (the gold vector, fixed in space) and the rotational energy E are both conserved — the flip is not a glitch, it is the only way the intermediate-axis trajectory can stay on both conservation surfaces (the polhode).