Intermediate-Axis Spin Really Does Flip - Rotational Dynamics and Orbital Mechanics Step by Step

Every value in this lesson is rounded to four decimal places from a real physim numerical integration of a rigid body spinning about its intermediate principal axis (scenario examples/rigid-body-rotation-intermediate-axis.json, integrator dp45, rtol 1e-10, atol 1e-12), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The spin starts almost purely on the middle axis

At t = 0.0000 seconds the spin vector is omega one = 0.0010, omega two = 5.0000, omega three = 0.0010 radians per second — the two small-axis components are essentially at rest next to the middle-axis spin.

\omega=(0.0010,\ 5.0000,\ 0.0010)\ \mathrm{rad/s}

The middle-axis spin flips while the ledgers stay pinned

At the mid-flip checkpoint t = 2.9800 seconds omega two has already fallen to 4.6179 radians per second, and by t = 4.0000 seconds it has flipped sign to -4.3482 while omega one and omega three have grown to -2.4684 and 1.4251 radians per second — a genuine instability, not a rounding artifact. The energy stays 25.0000 joules and the squared angular momentum stays 100.0000 (in squared kilogram-meter-per-second units) at every sampled row.

\omega_{2}:\ 5.0000\ \to\ 4.6179\ \to\ -4.3482\ \mathrm{rad/s},\quad E=25.0000\ \mathrm{J},\quad L^{2}=100.0000\ \mathrm{kg}^{2}\,\mathrm{m}^{4}/\mathrm{s}^{2}