Gyroscope and precession - 3D heavy top

Why a spinning wheel does not fall: Lagrange top with nutation, steady precession and spin-down on a Blender-built bench.

Switch to 2D

Presets

Playback

What to try

  1. "Steady precession": the tilt stays put and the tip draws a perfect circle on the protractor.
  2. "Released from rest": the top first dips, then loops - that is nutation; the average sideways speed is still mgl/L.
  3. Halve the spin: the precession rate doubles. Double the mass or lever arm: it doubles too.
  4. "Too slow: it falls" and "Spin-down to the table": with too little angular momentum gravity wins.
  5. "On the Moon": six times weaker gravity, six times slower precession.
  6. Use "Top-down" and watch the green arrow (precession) against the red torque arrow.

Parameters

Camera

Layers

Quality

Model check

The science

Model

A rotor of mass m spins about an axle that is supported at a fixed pivot. With Euler angles theta (tilt from the vertical), phi (precession) and psi (spin) the Lagrangian of this heavy symmetric top is L = ½ I1 (theta'² + phi'² sin²theta) + ½ I3 (psi' + phi' cos theta)² − m g l cos theta. For a uniform disc: I3 = ½ m r² and, about the pivot, I1 = ¼ m r² + m l².

Conserved quantities

p_psi = I3 w3 and p_phi = I1 phi' sin²theta + p_psi cos theta are conserved, and so is the energy. That leaves one equation: I1 theta'' = I1 phi'² sin theta cos theta − p_psi phi' sin theta + m g l sin theta, with phi' = (p_phi − p_psi cos theta)/(I1 sin²theta). It is integrated with 4th-order Runge-Kutta at 1 ms.

Steady precession and nutation

Constant tilt needs I1 cos theta W² − p_psi W + m g l = 0. The slow root is W = 2mgl / (p_psi + sqrt(p_psi² − 4 I1 m g l cos theta)). If the discriminant is negative (too little spin) no steady precession exists and the top tumbles. A fast top gives the textbook W ≈ m g l / L with L = I3 w3, i.e. the torque tau = m g l sin theta turns the angular momentum sideways instead of toppling it. Releasing the top with no initial precession produces the cycloid-like wobble called nutation, with frequency about L / I1.

Bearing drag and the table

The optional drag torque acts on the spin axis only: dp_psi/dt = −c p_psi, dp_phi/dt = −c p_psi cos theta. As the spin decays the precession speeds up, the wobble grows and finally the rotor reaches the table (the run stops there). Real bearings have more complicated friction; this is a simple viscous model.

Numerical checks run in the page

  • Energy conserved for a top released from rest (no drag), tolerance 1e-6.
  • Integrated steady-precession rate against the analytic root above, tolerance 0.5 %.
  • Integrated nutation period against 2 pi / sqrt(−dF/dtheta), with the derivative taken numerically from the equation of motion.

Simplifications

Rigid rotor, massless frictionless axle, point pivot with no slip, no air drag, uniform gravity. Real rotors also show a slightly different I1 because of the hub and axle mass. The shown spin is capped at about 3.5 revolutions per second (and a translucent blur disc fades in) because a screen cannot display 40 revolutions per second; all physics uses the true spin.

Loading model...
Loading Blender model...
Tilt theta(t): nutation
Precession rate (green: exact steady value)
Top view of the axle tip

šŸŒ€ Gyroscope & Precession

Interactive gyroscope simulator showing angular momentum, torque and precession. Spin up the rotor to see how L = IĻ‰ resists tilting ā€” and when gravity applies torque, the gyroscope precesses instead of falling.

šŸ”¬ What It Demonstrates

A spinning gyroscope resists changes to its orientation because angular momentum L is a conserved vector. Gravity creates a torque Ļ„ = r Ɨ mg perpendicular to L, causing the spin axis to precess at rate Ī© = Ļ„/L rather than topple.

šŸŽ® How to Use

Increase the spin speed and observe how the gyroscope holds steady. Reduce spin to see it wobble and eventually topple. The vectors L, Ļ„, and Ī© are drawn live to show their perpendicular relationships.

šŸ’” Did You Know?

Gyroscopic precession explains why a thrown boomerang returns, why a bicycle stays upright at speed, and how spacecraft attitude control uses reaction wheels ā€” all based on conservation of angular momentum.

About the Gyroscope & Precession Simulation

This is a real 3D rigid-body gyroscope rendered with Three.js. A spinning rotor sits on a gimballed axle pivoted at one end of a stand; gravity pulls on its centre of mass and applies a torque about the pivot. Instead of toppling, the spin axis sweeps around the vertical ā€” precession ā€” while bobbing slightly up and down ā€” nutation. The simulation integrates the heavy-top equations each frame, drawing the angular-momentum vector L, the torque vector Ļ„, and a trail traced by the axis tip so you can orbit the camera and inspect the motion from any angle.

The physics follows from L = IĻ‰ and dL/dt = Ļ„. Because gravity's torque Ļ„ = r Ɨ mg is perpendicular to the spin angular momentum, it rotates L sideways rather than speeding the fall, giving a steady precession rate Ī© = mgr/(IĻ‰). This conserved-angular-momentum behaviour stabilises spinning tops and bicycles, steadies spacecraft via reaction wheels and control-moment gyros, makes a thrown boomerang curve back, and drives the slow ~26,000-year precession of Earth's own rotation axis.

Frequently Asked Questions

What is gyroscopic precession?

Precession is the slow circular sweep of a spinning gyroscope's axis around the vertical. When gravity applies a sideways torque, conserved angular momentum makes the spin axis rotate around instead of simply falling over.

Why doesn't the spinning gyroscope just fall down?

Gravity's torque acts perpendicular to the spin angular momentum L. Per dL/dt = Ļ„, this changes the direction of L rather than adding downward motion, so the axis turns sideways (precesses) and the gyroscope stays up while it spins fast enough.

How is the precession rate calculated?

For a steadily precessing gyroscope it is Ī© = mgr/(IĻ‰), where m is mass, g gravity, r the pivot-to-centre-of-mass distance, I the moment of inertia, and Ļ‰ the spin rate. Faster spin or larger I gives a slower precession.

What is nutation?

Nutation is a small up-and-down nodding of the spin axis superimposed on the smooth precession. It appears as little looping cusps in the tip trail and is excited whenever the gyroscope is disturbed, for example by the "Give it a nudge" button.

Why does faster spin make the gyroscope more stable?

Faster spin gives a larger angular momentum L = IĻ‰. A bigger L resists the same gravitational torque more strongly, producing slower precession and a steadier, more upright axis that is harder to tip over.

What happens if I set the spin to zero?

With little or no spin there is almost no angular momentum to redirect the torque, so gravity simply topples the rotor. The "No spin (falls)" preset demonstrates this ā€” the axle leans over and drops instead of precessing.

What do the L and Ļ„ arrows represent?

The yellow arrow is the angular-momentum vector L pointing along the spin axis; its length grows with spin. The red arrow is the gravitational torque Ļ„, drawn horizontally and perpendicular to L, showing the direction in which L is pushed.

Does this explain why a bicycle stays upright?

Partly. The spinning wheels add gyroscopic stability, and when a moving bike leans, gyroscopic and steering effects produce a corrective turn. Rider steering and trail geometry also matter, but conserved angular momentum is a key ingredient.

How do spacecraft use gyroscopes?

Satellites and telescopes use reaction wheels and control-moment gyroscopes to point and stabilise without expending fuel. Spinning these internal rotors exchanges angular momentum with the spacecraft body to rotate or hold its orientation.

Is Earth itself a gyroscope?

Effectively yes. The Earth spins like a giant top, and the gravitational pull of the Sun and Moon on its equatorial bulge applies a torque that makes its rotation axis precess once roughly every 26,000 years ā€” the precession of the equinoxes.

Made with šŸŸ  Blender Ā· Three.js