🌀 Gyroscope Precession Simulation

Gravity pulling on the tilted disk's centre of mass creates a torque τ = mgr·sinθ about the pivot, and the simulation integrates the rigid-body law dL/dt = τ. Because this torque is always perpendicular to the angular momentum vector L = Iω, it can only rotate L's direction, never shrink it — so instead of toppling, the spin axis sweeps a steady cone around the vertical, which is precession.

The simulator computes precession rate as Ω = mgr / (Iω), where m is the disk's mass, g is 9.81 m/s², r is the pivot-to-centre-of-mass distance, I is the moment of inertia, and ω is the spin rate. Both the torque τ = mgr·sinθ and the horizontal component of angular momentum that must rotate scale with sinθ in the same way, so the sinθ factors cancel and the resulting precession rate is independent of tilt angle.

The spinning disk has a fixed radius of 0.12 m and mass 0.3 kg, and the code computes its moment of inertia about the spin axis with the standard solid-disk formula I = ½mR². This fixed I, together with the spin rate slider, sets the angular momentum L = Iω shown in the fact chips, and it is this same I that appears in the denominator of the precession-rate formula.

Earth is itself a giant, slightly tilted spinning top: the combined gravitational pull of the Sun and Moon on its equatorial bulge creates a torque analogous to mgr·sinθ here, and Earth's rotation axis responds exactly the way this simulator's disk does — by precessing around a cone instead of tipping over. Earth's enormous moment of inertia and comparatively slow torque stretch that same Ω = torque/(Iω) relationship out to a period of about 26,000 years, compared with the few seconds to minutes seen on screen.

The dashed circle is the theoretical precession path: the fixed circle of radius 2r·sinθ that the axle tip would follow under perfectly steady precession. The coloured line is the actual simulated tip path, recorded frame by frame as the physics integrates φ += Ω·dt and spin phase += ω·dt. At the fast-spin, small-tilt settings used in most presets the two curves overlap almost exactly, which is a visual check that the numerical integration tracks the analytic precession formula correctly.