---
title: "Tennis-Racket Theorem Interactive Lab | Sandbox Physics"
description: "Why does a free object spinning about its middle axis keep flipping over? Because spin about the middle axis is unstable. A tiny wobble about the other axes grows exponentially at first, and the body flips over, again and again, while its angular momentum and energy stay exactly the same. Spin about the axis with the largest or smallest moment of inertia only wobbles a little."
canonical_url: "https://sandboxphysics.com/labs/tennis-racket/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/tennis-racket/"
last_modified: "2026-10-22"
---

M056 · Intermediate-axis instability of a free rigid body

# Tennis-Racket Theorem

**Why does a free object spinning about its middle axis keep flipping over?** Because spin about the middle axis is unstable. A tiny wobble about the other axes grows exponentially at first, and the body flips over, again and again, while its angular momentum and energy stay exactly the same. Spin about the axis with the largest or smallest moment of inertia only wobbles a little.

Spin an asymmetric rigid body about each principal axis and add a tiny sideways spin. Turn the three-dimensional view and compare the half-turn flips with steady rotation. Follow the body angular velocity and the curve where the angular-momentum sphere meets the energy ellipsoid.

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

The body is a uniform rectangular block in free space, with no external torque, damping, deformation or collisions. The theorem is named after a tennis racket, but the Lab models a block, not a calibrated racket. The stripes on its faces are massless markings. The camera follows the centre of mass, and translation is decoupled from rotation. Axes are numbered by the block dimensions, so the numbers need not follow the order of the moments of inertia. The linear growth and oscillation rates are local estimates; large flips are computed from the full nonlinear equations. Spin exactly about a principal axis with no perturbation remains an exact solution, even about the unstable axis. The sphere and ellipsoid are drawn in the same body-frame angular-momentum coordinates. Curves for the full record include samples already calculated for later times. Fast spin can alias at the screen frame rate; pause and use fine steps.

Physics tutorial

## Tennis-Racket Theorem

**Background** The same rigid body is released three times, once about each principal axis, with the same shape and the same small perturbation.

**Why it matters** A free object can flip over even though its energy and its angular momentum in the fixed frame are both conserved.

Start with the essentials

- **Focus question**: About which principal axis does a tiny sideways spin grow?
- **One-sentence intuition**: About the axis of intermediate inertia, a small wobble first grows exponentially; the nonlinear motion then flips a body axis over again and again, while the conserved angular momentum vector stays fixed in space.

### Core mathematical model

#### Principal moments of inertia of a block

$$
I_1=\frac{m(b^2+c^2)}{12},\quad I_2=\frac{m(a^2+c^2)}{12},\quad I_3=\frac{m(a^2+b^2)}{12}
$$

Changing a dimension can change the order of the moments of inertia, but the axis numbers stay attached to the body.

#### Euler equations

$$
I_1\dot\omega_1=(I_2-I_3)\omega_2\omega_3\quad\text{and cyclic permutations}
$$

Even with no torque, the angular velocity measured in the body frame can change, because the body frame itself rotates.

#### Angular momentum in the fixed frame

$$
\mathbf L=R I\boldsymbol\omega_b,\qquad\dot{\mathbf L}=\mathbf 0
$$

The vector stays fixed in the inertial frame, while its components along the moving body axes change.

#### Momentum sphere and energy ellipsoid

$$
\sum_i L_{b,i}^2=|\mathbf L|^2,\qquad\sum_i\frac{L_{b,i}^2}{I_i}=2E
$$

Both surfaces are written in terms of the body-frame angular momentum, not the angular velocity. The teal trace lies on the curve where they meet.

#### Growth rate near the intermediate axis

$$
\lambda=|\Omega|\sqrt{\frac{(I_j-I_{\min})(I_{\max}-I_j)}{I_{\min}I_{\max}}}
$$

This positive growth rate applies when the chosen axis has a moment of inertia strictly between the other two. It describes how an infinitesimal wobble starts to grow; it is not the flip frequency.

#### Oscillation about a stable axis

$$
\nu=|\Omega|\sqrt{\frac{(I_j-I_k)(I_j-I_\ell)}{I_k I_\ell}}
$$

The product under the square root is positive for the smallest and largest inertias. When two inertias are equal, the exponential prediction no longer applies, but that does not mean the motion settles down by losing energy.

### Common difficulties

#### A flip with no torque

**Typical misconception** If nothing twists the object, its angular velocity should stay the same.

**Better mental model** What stays fixed is the angular momentum in the fixed frame. The body has different moments of inertia along different axes, and as it turns, the same angular momentum corresponds to a changing angular velocity.

#### Axis numbers do not decide stability

**Typical misconception** The second axis is always the intermediate one, so its number tells you whether it is stable.

**Better mental model** The numbers stay attached to the body. After you change the dimensions, read the three inertias to see which axis is in the middle.

### Run the experiment

1. 01

   #### Compare the axes

   Run the Minimum-inertia axis, Intermediate-axis flip and Maximum-inertia axis presets, then use Review full record.

   **What to observe:** Spun about the intermediate axis, the body flips that axis over. About the other two axes, the deviation stays small.

2. 02

   #### Perturb the exact solution

   Compare the Exactly unperturbed and Tiny perturbation presets, then drag the orange handle to change the perturbation size.

   **What to observe:** With no perturbation, even the unstable spin stays exact. The smaller the initial perturbation, the later the body departs from it.

3. 03

   #### Follow the intersection curve

   Pause near a flip and look at the body angular velocity and the angular-momentum sphere and ellipsoid.

   **What to observe:** The body-frame components move along both conserved surfaces at once, while the angular momentum in the fixed frame stays put.

4. 04

   #### Change the shape

   Make two or three dimensions equal, then compare what changing the mass does and how the defects depend on step size.

   **What to observe:** Equal inertias change how small deviations grow. Changing the mass rescales the energy and angular momentum but leaves the attitude unchanged.

Physical conclusion

Before you call a spin unstable, add a tiny perturbation, find the order of the moments of inertia and check that the motion stays on both conserved surfaces.

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)