Gyroscopic Precession — 2D Rigid-Body Lab

Spin & Inertia

Spin rate (ω) 25 rad/s

Moment of inertia (I) 1.00 kg·m²

Torque (τ = mgr sinθ)

Mass (m) 2.00 kg

Pivot–CM distance (r) 0.60 m

Tilt from vertical (θ) 35°

Live Readout

Angular momentum L=Iω

0.0 kg·m²/s

Torque τ=mgr·sinθ

0.0 N·m

Precession rate Ω

0.0 °/s

Precession period

0.0 s

How it works

A rigid disk spins about its own axis at ω, giving it spin angular momentum L = Iω pointing along that axis. Gravity pulls down on its centre of mass, a distance r from the pivot, producing a horizontal torque τ = mgr sinθ about the pivot (θ measured from vertical). Because torque is the rate of change of angular momentum, and only the horizontal component L sinθ of the spin vector can be swung around by a horizontal torque, the axis sweeps a cone at a steady precession rate:

L  = I·ω
τ  = m·g·r·sinθ
Ω  = τ / (I·ω·sinθ)   =   m·g·r / (I·ω)

The sinθ cancels — steady precession rate does not depend on how far the gyroscope is tilted, only on torque and spin. What it does depend on is ω: double the spin rate and Ω is cut in half. That is the counter-intuitive gyroscopic effect this lab is built to show — spin faster and the axis wanders more slowly around the vertical, not faster.

Side view (left) — an oblique projection of the real 3D axis vector, spin marks rotating at ω, tracing the precession cone as a glowing loop.
Top-down orbit (top right) — looking straight down the vertical: the true circular path the axis tip sweeps at rate Ω, radius set by sinθ.
Ω vs ω curve (bottom right) — the inverse relationship Ω=mgr/(Iω) plotted live, with a marker at the current spin rate.

Spin axis / disk Precession cone trace Ω(ω) curve, current point