---
title: "Spinning Top: Precession & Nutation Interactive Lab | Sandbox Physics"
description: "When does a nearly upright spinning top stay upright? When it spins faster than a threshold set by its mass, shape and moments of inertia. A heavy top with its mass far from the tip needs more spin. Above the threshold, a small tilt just makes the top nod around upright; below it, the same tilt grows and the top falls away from vertical."
canonical_url: "https://sandboxphysics.com/labs/spinning-top/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/spinning-top/"
last_modified: "2026-10-22"
---

M057 · Precession and the sleeping top

# Spinning Top: Precession & Nutation

**When does a nearly upright spinning top stay upright?** When it spins faster than a threshold set by its mass, shape and moments of inertia. A heavy top with its mass far from the tip needs more spin. Above the threshold, a small tilt just makes the top nod around upright; below it, the same tilt grows and the top falls away from vertical.

Spin a uniform solid cone in three dimensions with its tip held in a bearing. Tell apart its spin, precession and nutation, compare the two steady precession rates, and nudge an upright 'sleeping' top on either side of its stability threshold. The force at the pivot and an independent check of the numerical error update alongside.

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

The top is a uniform solid cone: an ideal axisymmetric rigid body in uniform gravity, with its apex fixed by an ideal bearing. The same radius and height set the centre of mass and the moments of inertia about the pivot; the handle, marker and support have no mass. The equations of motion are integrated without any small-tilt or fast-spin approximation. Spin means the component of angular velocity along the body 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. Plots of the full run include samples already computed for later times. Fast rotation can alias at the screen's frame rate; pause and use the fine-step button to check the phase. The upright stability threshold applies only to infinitesimal disturbances of vertical spinning.

Physics tutorial

## Spinning Top: Precession & Nutation

**Background** A uniform solid cone spins in three dimensions with its tip held in a bearing. You can tell apart its spin, precession and nutation, compare the two steady precession rates, and nudge an upright 'sleeping' top on either side of its stability threshold. The force at the pivot and an independent check of the numerical error are shown alongside.

**Why it matters** The same rigid-body law explains a flywheel in an instrument and a nutating top. If you watch only the spinning surface pattern, you miss what matters: how the axis moves.

Start with the essentials

- **Focus question**: When does a nearly upright spinning top stay upright?
- **One-sentence intuition**: Gravity's sideways torque changes the angular momentum, but the energy, the vertical component of angular momentum and its component along the body axis all stay constant.

### Core mathematical model

#### Cone geometry and inertia about the pivot

$$
\ell=\frac34h,\qquad I_3=\frac3{10}mR^2,\qquad I_\perp=\frac3{20}mR^2+\frac35mh^2
$$

The top is a uniform solid cone whose apex is held by an ideal bearing. The same radius and height set the centre of mass and the moments of inertia about the pivot. The handle, marker and support have no mass.

#### Full vector equations of motion

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

All quantities are taken about the fixed pivot. The symmetry axis and the angular momentum evolve together, and a quaternion keeps track of the spin angle.

#### Angular velocity and axial spin

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

The actual 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 rates of steady precession

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

Each real root gives a constant tilt, but only if the top starts with no nutation. A kick disturbs that motion. 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 leaves out the transverse-inertia term. It is accurate when the precession is slow compared with the spin, and it is undefined at zero spin.

#### Three conserved quantities

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

The potential energy is zero when the centre of mass is level with the pivot. The largest errors are divided by nonzero physical scales of energy and angular momentum, 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 local, linear stability condition. Below the threshold, perfectly vertical spinning is still an exact solution; tilt the top slightly to reveal the instability.

#### Reaction force at the held pivot

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

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

### Common difficulties

#### The angular momentum always points along the shaft

**Typical misconception** At any spin rate, the angular momentum points exactly along the shaft.

**Better mental model** Whenever the axis moves, the transverse inertia adds a part that points off the axis. Compare the blue arrow with the shaft in the low-spin or fast-branch preset.

#### Staying vertical proves stability

**Typical misconception** A vertical top that remains vertical must be stable.

**Better mental model** Below the threshold, the exact undisturbed solution still exists. Compare the small-tilt presets to see how the top responds when it is disturbed.

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

**Typical misconception** An ordinary table surface can pull the tip down too.

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

### Run the experiment

1. 01

   #### Set the tilt and look around

   Select a 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 three-dimensional scene and the recorded data.

2. 02

   #### Compare the two steady precessions

   Select the slow and fast branches with the nutation kick at zero, then reverse the spin.

   **What to observe:** The tilt stays constant only because the top starts with exactly the right precession. Released without it, the top generally nutates.

3. 03

   #### Test the approximation and the sleeping top

   Lower the spin and compare with the approximation. Then compare the stable and unstable small-tilt presets for the conical top.

   **What to observe:** The slow-precession approximation can fail. The local stability threshold shows up only when the top is disturbed.

4. 04

   #### Check the numerical accuracy

   Replay the whole run and look at the conservation errors and the difference between two step sizes. Use the fine phase step to follow the rotor markings.

   **What to observe:** Small numerical errors show that the model's own equations are solved accurately. They cannot vouch for friction or tabletop contact, which the model leaves out.

Physical conclusion

Fix the body's shape first, then set its angular momentum. Keep the exact rigid-body motion apart from approximate rules for precession.

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)