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The science
Why a top does not fall over
A top of mass m whose centre of mass is a distance l from the pivot feels a gravity torque τ = m g l sinθ horizontal and perpendicular to its angular momentum L = I3ω. A torque changes L by dL/dt = τ, so L turns around the vertical instead of falling: the axis precesses at Ω = m g l / (I3 ω) in the fast-top limit. Faster spin means slower precession.
Exact equations
With Euler angles (θ tilt, φ azimuth) the Lagrangian gives I1θ'' = I1φ'² sinθ cosθ - I3ω3φ' sinθ + m g l sinθ and the conserved azimuthal momentum pφ = I1φ' sin²θ + I3ω3 cosθ, with I1 = ¼mR² + m l² about the pivot and I3 = ½mR². Steady precession requires I3ω3φ' - I1φ'² cosθ = m g l, which has a real solution only above a minimum spin. Released from rest the axis nods (nutation) while it precesses; its mean precession rate still follows the fast-top formula. The equations are integrated with a fixed-step fourth-order Runge-Kutta method.
Honest simplifications: the tip is a perfect frictionless pivot; the top is a uniform disc on a massless rod; friction only slows the spin (it does not move the pivot or create sliding), and the top is stopped when it reaches the table at about 89 degrees.
What to try
- Choose Steady precession: the tilt graph is flat and the top view is a circle.
- Choose Released from rest and see the nutation wiggles in the tilt graph and the looping path in the top view.
- Choose Fast spin and check that the precession rate drops and stays equal to the dashed fast-top value.
- Choose Spinning down: as the spin decays the precession speeds up and the tilt grows until the top falls.