Gyroscopic precession — 2D heavy-top lab

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What to try
  1. Start with "Steady precession": the tilt stays constant and the tip traces a perfect circle in the top view.
  2. Select "Released from rest": the axis first falls, then nutates in small cycloid loops while drifting around (precession).
  3. Double the spin rate: the steady precession rate halves (mgl / I3w) and the nutation frequency doubles.
  4. Lower the spin to 60 rad/s: the fast-top approximation breaks down, the loops grow, and for very slow spin no steady precession exists.
  5. Make the arm longer or the disc heavier: gravity torque mgl rises, so precession is faster.
The science

A symmetric top with fixed pivot has Euler angles θ (tilt), φ (precession) and ψ (spin). With moments of inertia I1 (transverse, about the pivot) and I3 (spin axis) its Lagrangian is L = ½ I1(θ̇² + φ̇² sin²θ) + ½ I3(ψ̇ + φ̇ cosθ)² − m g l cosθ.

pψ = I3 ω3,   pφ = I1 φ̇ sin²θ + pψ cosθ (both conserved) I1 θ̈ = I1 φ̇² sinθ cosθ − pψ φ̇ sinθ + m g l sinθ Steady precession: I1 cosθ Ω² − pψ Ω + m g l = 0 Fast top: Ω ≈ m g l / (I3 ω3), nutation frequency ≈ I3 ω3 / I1

The page integrates the θ equation with fourth-order Runge-Kutta (0.5 ms steps) and obtains φ, ψ from the conserved momenta. The disc is a uniform cylinder-thin disc on a massless rod: I3 = ½ m r², I1 = ¼ m r² + m l².

Simplifications: frictionless pivot, rigid massless rod, no air drag, so the spin never decays. The "fast-top" checks only appear when pψ² > 100 I1 m g l cosθ (the approximation's validity).