Spinning Top

A spinning top exhibits two of the most beautiful phenomena in classical mechanics: precession (the slow rotation of the spin axis around the vertical) and nutation (a wobbling superimposed on the precession). These arise from the interplay of angular momentum and gravitational torque, governed by Euler's equations for rigid body rotation.

I dω/dt + ω × (Iω) = τ — Euler's equation for rigid body rotation

Spin Rate: 30

Tilt (°): 25

Gravity: 9.8

Precession Rate—

Tilt Angle—

Spin (rad/s)—

Energy—

How it works: The top is modeled as an axially symmetric rigid body with principal moments of inertia I₁ = I₂ (transverse) and I₃ (spin axis). Gravity exerts a torque about the contact point. Rather than directly solving Euler angles' ODEs, we integrate the angular momentum vector L in the lab frame. The precession rate is approximately Ω = Mgl / (I₃ω₃), faster for heavier, slower-spinning, or more tilted tops. Nutation appears as a rapid oscillation superimposed on the steady precession when the initial conditions do not perfectly match steady precession.

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