Gyroscopic Precession

A rapidly spinning rotor under gravitational torque can steadily turn about the vertical. This module uses the slow-precession approximation, rather than a full solution of rigid-body dynamics.

Let M be the assembly mass, r the pivot-to-center-of-mass distance, I the spin inertia, ω the spin angular speed and θ the tilt from vertical. Spin angular momentum is L=Iω. The torque magnitude is τ=Mgr sin θ. Steady precession rotates L at rate Ω, so |dL/dt|=Ω L sin θ.

Equating torque and angular-momentum change gives Ω=τ/(L sin θ)=Mgr/(Iω). The sin θ factor cancels for the supported nonzero tilts. The period is T=2π/Ω, and Ω/ω measures the separation between precession and spin rates.

Worked example with selectable controls

Set Rotor assembly mass M=1 kg, Center-of-mass lever r=10 cm, Spin inertia I=0.02 kg·m², Spin angular speed omega=100 rad/s and Tilt from vertical theta=90 degrees. The model fixes g=10 m/s².

Convert r=10/100=0.1 m.
L=Iω=0.02×100=2 kg·m²/s.
τ=Mgr sin θ=1×10×0.1×1=1 N·m.
Ω=Mgr/L=1/2=0.5 rad/s.
T=2π/0.5=4π≈12.5663706144 s.
Ω/ω=0.5/100=0.005, which is small compared with1.

Experiment: tilt versus spin

Open the Gyroscopic Precession module and enter the worked example using its labelled controls.

Change Tilt from vertical theta from90 to30. Torque halves to0.5 N·m while Ω stays0.5 rad/s. The horizontal component L sin θ also halves, so the same precession rate suffices.
Set Spin angular speed omega to200. L doubles to4 kg·m²/s, Ω halves to0.25 rad/s and the period doubles to8π≈25.1327412287 s.
Inspect the vertical axis, tilted L axis, angle and projected precession cone. Drag to pan and use Reset view to recover the original framing.

Approximation and limits

The listed grid has positive spin and Ω/ω≤0.02. The model neglects angular momentum associated with precession itself, nutation, pivot friction and spin losses; it assumes an already established steady precession. No zero-spin or vertical-tilt singular case is selectable. It does not predict a zero-spin top falling or solve the full Euler equations. The diagram is a static schematic projection, with dimensions scaled for readability.

Primary reference

OpenStax University Physics Volume1, section11.4: Precession of a Gyroscope derives torque, spin angular momentum and slow precession under Ω much smaller than ω.

Gyroscopic Precession