---
title: "Gyroscope Precession Bench Interactive Lab | Sandbox Physics"
description: "Why does a spinning wheel precess instead of simply falling? Because gravity's torque on the wheel points sideways, not down. A torque changes angular momentum in its own direction, so the wheel's large spin angular momentum swings around the vertical instead of tipping over. Energy, the angular momentum about the vertical and the spin along the axle all stay constant during the motion."
canonical_url: "https://sandboxphysics.com/labs/gyroscope-precession/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/gyroscope-precession/"
last_modified: "2026-10-22"
---

M055 · Torque on a spinning rigid body

# Gyroscope Precession Bench

**Why does a spinning wheel precess instead of simply falling?** Because gravity's torque on the wheel points sideways, not down. A torque changes angular momentum in its own direction, so the wheel's large spin angular momentum swings around the vertical instead of tipping over. Energy, the angular momentum about the vertical and the spin along the axle all stay constant during the motion.

Release a spinning flywheel held at one pivot, start it on either steady-precession branch, reverse its spin or switch gravity off. Follow the angular-momentum and torque arrows in three dimensions, measure the nutation and compare the full motion with the slow-precession estimate.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The wheel is a uniform thin disk on a massless shaft, with the distance measured from the fixed pivot to the disk's centre. The drawn disk thickness, support and hub are only illustrations and carry no mass. The model is an ideal symmetric rigid body with one point fixed, in uniform gravity, and the full equations are integrated without assuming a small tilt or fast spin. Spin here means the component of angular velocity along the wheel's axis, which differs from the rate of the Euler spin angle. The bearing can pull as well as push, so this is not a tip that slides or bounces freely on a table. Friction, air drag, deformation, collisions with the floor or support, and spin decay are left out. The full-record plots include samples calculated ahead of the current time. At high spin the screen frame rate can make the rotation look wrong (aliasing); pause and use the fine-step button to follow the phase. The upright stability threshold applies only to infinitesimal tilts away from vertical spinning.

Physics tutorial

## Gyroscope Precession Bench

**Background** A spinning flywheel is held at one fixed pivot, and you can turn the view around it in three dimensions. Release it freely, start it on either steady-precession branch, reverse the spin or switch gravity off. Follow the angular-momentum and torque arrows, measure the nutation, and compare the full motion with the slow-precession estimate.

**Why it matters** The same rigid-body law explains the flywheel in an instrument and a top that wobbles as it spins. If you watch only the spinning surface, you miss what matters: the axis itself is moving.

Start with the essentials

- **Focus question**: Why does a spinning wheel precess instead of simply falling?
- **One-sentence intuition**: Gravity's sideways torque changes the angular momentum in the fixed frame, yet the energy, the vertical part of the angular momentum and its part along the axis all stay constant.

### Core mathematical model

#### Moments of inertia about the pivot

$$
I_3=\frac12mR^2,\qquad I_\perp=\frac14mR^2+m\ell^2
$$

The wheel is a uniform thin disk on a massless shaft, and the distance is measured from the fixed pivot to the disk's centre. The drawn disk thickness, support and hub are only illustrations and carry no mass.

#### Equations of motion in vector form

$$
\dot{\mathbf L}=mg\ell\,\mathbf e_z\times\mathbf n,\qquad\dot{\mathbf n}=\frac{\mathbf L\times\mathbf n}{I_\perp}
$$

Every quantity is taken about the fixed pivot. The symmetry axis and the angular momentum in the fixed frame change together, and the spin angle is tracked with a quaternion.

#### Angular velocity and spin about the axis

$$
\boldsymbol\omega=\frac{\mathbf L}{I_\perp}+\left(\frac1{I_3}-\frac1{I_\perp}\right)(\mathbf L\cdot\mathbf n)\mathbf n,\qquad\omega_3=\dot\psi+\dot\phi\cos\theta
$$

The physical spin about the axis is not the rate of the Euler spin angle. Mixing the two up gives the wrong condition for exact steady precession.

#### Two steady-precession branches

$$
I_\perp\cos\theta\,\Omega^2-I_3\omega_3\Omega+mg\ell=0
$$

A real root gives a constant tilt only if the wheel starts with no nutation. A kick disturbs that solution. When the axis is horizontal, the equation becomes linear.

#### Slow-precession approximation

$$
\Omega_{\mathrm{slow}}\simeq\frac{mg\ell}{I_3\omega_3}
$$

This estimate drops the term with the transverse moment of inertia. It is accurate when the precession is slow compared with the spin about the axis, and it is undefined at zero spin.

#### Three conserved quantities

$$
E=\frac{I_\perp}{2}(\dot\theta^2+\dot\phi^2\sin^2\theta)+\frac{I_3\omega_3^2}{2}+mg\ell\cos\theta,\qquad L_z=\mathrm{const},\quad\mathbf L\cdot\mathbf n=\mathrm{const}
$$

Potential energy is zero when the disk's centre is level with the pivot. The largest errors are divided by nonzero energy and angular-momentum scales of the system, so they still mean something when a conserved quantity is itself zero.

#### Upright stability threshold

$$
|I_3\omega_3|>2\sqrt{I_\perp mg\ell}
$$

This is a linear stability condition for small tilts near vertical. Below the threshold, perfectly vertical spinning is still an exact solution; add a small tilt to see that it is unstable.

#### Force from the pivot

$$
\mathbf R=m\ell\ddot{\mathbf n}+mg\mathbf e_z
$$

The ideal bearing can pull down as well as push up. A tip resting freely on a surface would need a separate model for one-sided contact and collisions.

### Common difficulties

#### Angular momentum always points along the shaft

**Typical misconception** The shaft and the angular momentum point the same way at every spin rate.

**Better mental model** Whenever the axis moves, the transverse moment of inertia adds to the angular momentum. Compare the blue arrow with the shaft at low spin or on the fast branch.

#### A top that stays upright must be stable

**Typical misconception** If a vertical top stays vertical, it must be stable.

**Better mental model** Below the threshold, the undisturbed vertical solution still exists. Compare runs that start from a small tilt to see how the top responds to a disturbance.

#### The pivot works like a tip on a table

**Typical misconception** An ordinary table could supply the downward pull that the pivot provides.

**Better mental model** Here a bearing holds the tip in place. A table cannot pull the tip down, and sliding, lifting off and collisions are outside the model.

### Run the experiment

1. 01

   #### Set up and look around

   Select free release, look at the overview, top and side views, then drag the orange tilt handle.

   **What to observe:** The new starting axis changes both the scene and the recorded numbers.

2. 02

   #### Compare the steady branches

   Select the slow and fast branches with the nutation kick left at zero. Then reverse the spin.

   **What to observe:** The tilt stays constant only when the wheel starts exactly on that solution. Released without the right precession, it usually nutates.

3. 03

   #### Test the approximation and the upright top

   Reduce the spin and check the approximation. For the conical top, compare the stable and unstable small-tilt presets.

   **What to observe:** The slow-precession approximation can fail, and the upright stability threshold shows up only when you disturb the top.

4. 04

   #### Check the numerical accuracy

   Review the full record and look at the conservation errors and the difference between two step sizes. Use fine phase steps to follow the rotor markings.

   **What to observe:** Small numerical errors show that the model is solved accurately. They say nothing about friction or table contact, which the model leaves out.

Physical conclusion

Set up the body's shape and its angular momentum first, then keep the exact rigid-body motion separate from the approximate precession rule.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)