# Gyroscope & Spinning Top Simulator - Precession, Nutation, 3D

> 3D heavy symmetric top simulator: RK4 rigid-body motion with precession and nutation (cusps, waves, loops), sleeping-top threshold and a bicycle-wheel gyro.

- URL: https://qilingtools.com/en/physics/gyroscope
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Set the spin rate, initial tilt, initial precession rate, mass and moments of inertia, and watch a top precess and nutate in a 3D view you can rotate by dragging. The tip of the axis traces cusps, waves or loops on a sphere, and the angular momentum L and gravity torque τ are drawn live. Alongside the animation the tool gives the exact steady-precession rates, the fast-top approximation and the sleeping-top threshold. It also reports the simulated mean precession, the nutation period and checks that energy, L_z and L₃ stay conserved.

## How to use

1. Pick a preset: toy-top nutation patterns, a sleeping top, a gyroscope on a stand or a bicycle wheel hanging from a string
2. Adjust the spin ω₃, initial tilt θ₀ and initial precession φ̇₀, or edit the mass, ℓ, I₁ and I₃
3. Press Play and drag on the canvas to change the view while watching L, τ and the axis trace
4. Compare the steady-precession solution, fast-top approximation and simulated values, and read the θ(t) and φ̇(t) charts

## Features

- Full rigid-body dynamics of a heavy symmetric top, RK4 integration with no singularity near upright
- Rotatable perspective 3D view, a rim marker that shows the spin and an axis-tip trace for precession and nutation
- Live angular momentum, torque, angular velocity and gravity vectors
- Both steady-precession roots and the fast-top approximation Mgℓ/(I₃ω₃) compared with the simulation
- Sleeping-top condition ω₃² > 4MgℓI₁/I₃² with demonstrations above and below it
- Conservation drift of energy, L_z and L₃, plus θ(t) and φ̇(t) charts

## FAQ

### Why doesn't a spinning top fall over?

Gravity's torque about the pivot, τ = ℓê₃ × (−Mg ẑ), is always horizontal and perpendicular to the axis. In a fast top the angular momentum L points almost along the axis, and the torque changes only its direction, not its size. The tip of L therefore circles horizontally, which is precession. For a fast top the precession rate is about Ω = Mgℓ/(I₃ω₃), so the faster it spins, the slower it precesses.

### What is nutation, and why does the trace show cusps, waves or loops?

If the initial precession rate is not exactly the steady value, the axis also nods up and down while it precesses. That nodding is nutation, with θ oscillating between two limits. Released from rest (φ̇₀ = 0), the tip momentarily stops at the top of each nod and draws cusps. A smaller precession in the same direction gives waves, and one in the opposite direction gives small loops. The faster the spin, the faster and smaller the nutation, until it is hard to see at all.

### What is a sleeping top?

A top spinning upright looks perfectly still, as if asleep. Linear stability analysis gives the condition ω₃² > 4MgℓI₁/I₃². Above that critical spin, small disturbances only cause small wobbles. Once friction slows it below the threshold, the top starts to wobble widely and falls. The presets show the same top just above and just below the critical spin.

### How is the motion computed, and how accurate is it?

The model is a heavy symmetric top with a fixed pivot, Lagrangian L = ½I₁(θ̇² + φ̇²sin²θ) + ½I₃(ψ̇ + φ̇cosθ)² − Mgℓcosθ. To avoid the Euler-angle singularity at θ = 0, the tool integrates the equivalent vector equations dL/dt = τ and dê₃/dt = ω × ê₃ with fourth-order Runge–Kutta and a small step. It agrees point by point with the Euler-angle equations. Relative drift of energy, L_z and L₃ is typically below 10⁻⁹, as the results table shows. Pivot friction and air drag are ignored.

### How can a bicycle wheel hang sideways from a string?

One end of the axle hangs from the string. The fast-spinning wheel has a large angular momentum, so gravity's torque only makes it precess slowly in the horizontal plane instead of dropping. With the axle horizontal (θ = 90°), the exact steady precession equals Mgℓ/(I₃ω₃). For the preset 2 kg wheel, 0.15 m lever arm and 8 revolutions per second, one precession circle takes about 20 seconds; halve the spin and it precesses twice as fast.

## Related tools

- [Unit Converter](https://qilingtools.com/en/math/unit): Free online unit converter: length, area, volume, weight, temperature, time, speed, data, pressure, energy and angle — one value converts every unit at once.
- [Vector Calculator](https://qilingtools.com/en/math/vector): Free online vector calculator for 2D and 3D: addition, scalar multiples, dot and cross products, magnitude, angle, unit vector and projections, with checks.
- [Significant Figures Calculator](https://qilingtools.com/en/math/significant-figures): Free online sig fig calculator: count significant figures digit by digit, round to N sig figs or decimals with six rounding modes, and compute by sig fig rules.
- [Tone Generator](https://qilingtools.com/en/audio/tone-generator): Free online tone generator: sine, square, sawtooth and triangle waves from 1 to 20,000 Hz, a note picker, sweeps, white/pink/brown noise and a WAV download.
- [Audio Spectrogram](https://qilingtools.com/en/audio/spectrogram): Free online spectrogram generator: see time, frequency and level from a Hann-windowed STFT, with FFT size, log axis, dB range and colour maps. Exports PNG.
- [Decibel Meter](https://qilingtools.com/en/audio/decibel): Free online decibel meter: measure noise with your mic in real time, A/C/Z weighting, fast/slow response, min/max/average/Leq, 60-second graph and calibration.
- [Physics Formula Calculator](https://qilingtools.com/en/physics/physics): Physics formulas solved with steps: SUVAT, projectiles, Newton’s laws, inclines, energy, Ohm’s law, resistors, RC, buoyancy, waves, lenses, gravity.
- [Fourier Series & FFT Visualizer](https://qilingtools.com/en/math/fourier): Fourier series coefficients and animated partial sums for square, sawtooth, triangle or any function (Gibbs), draw-your-own epicycles and FFT spectra.