Spinning a Book in Mid-Air

Teach me

Speed

Torque-free 3:2:1 book — spin it about a principal axis · drag to orbit

L (angular momentum)

ω (angular velocity)

covers ⊥ e₃

sides ⊥ e₂

ends ⊥ e₁

body frame · tip of L̂ on the sphere of fixed |L|

Guided Inquiry

1 · The book in empty space

Let’s meet our book. A 3:2:1 box floats free — no force, no torque. You’ll see three coloured face-pairs, three labelled principal axes e₁ (length), e₂ (width), e₃ (thickness), an arrow for L and an arrow for ω. To get started, choose an axis of spin in the sidebar.

2 · Spun about its thickness

Choose the axis through the thickness of the box, the axis of the largest principal moment, and keep the disturbance small. Before pressing play, predict what the box will do.

3 · Spun about its length

Now choose the axis along the length of the box, the axis of the smallest principal moment, with the same small disturbance. Before pressing play, predict what the box will do.

4 · Spun about its width

Now choose the axis along the width of the box, the axis of the intermediate principal moment, with the same small disturbance. Before pressing play, predict what the box will do.

5 · The arrow for L

Keep the same set-up and watch the arrow for L. Before the next turn, predict what the arrow does while the box turns over.

6 · A ten times smaller disturbance

Reduce the size of the disturbance by a factor of ten. Before pressing play, predict how the time before the first turn changes.

7 · The tennis-racket theorem

So here’s the whole story. Rotation about the smallest and largest principal moments is stable, but rotation about the intermediate one is not — and the turning over happens with L and T never changing. As a final check, run the width axis at two different values of ε and compare the measured first turn against the λ⁻¹ln(1/ε) row each time.

Listen

Controls

Disturbance ε (fraction of spin rate)0.020

Axis of spinthickness e₃

Readouts

t since release

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|L|

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The Physics

Rotation about principal axes · what this sim shows, the physics behind it, and every control

At a glance

This sim spins a free box, with no gravity and no torque, about its length, width or thickness, adds a small disturbance, and follows what the spin does next. The box's shape decides the outcome, which matters for a thrown book, a tennis racket and a spacecraft.

What you'll learn

What the three principal axes and principal moments of a box are.
How the moments of inertia decide what a small disturbance does about each axis.
How the time before the first turn depends on the size of the disturbance.
Which quantities stay constant while the box moves, and which ones change.

The sim's box. e₁ runs along the length, e₂ the width, e₃ the thickness. Ends ⊥ e₁ green, sides ⊥ e₂ violet, covers ⊥ e₃ cyan.

What's on screen

The box, with the face colours above and a line through each axis (the chosen spin axis is thicker).
L (amber arrow): the angular momentum.
ω (rose arrow): the angular velocity.
Body-frame inset (bottom right; in the Guided Inquiry it appears at step 4): the direction of L seen from the box, with a trail.
Readouts: time since release, kinetic energy T and |L|, plus (in Free Exploration) the measured first turn and an estimate of it.

See itSpin the box about its thickness. Watch for: the box spinning on the stage while t counts up in the Readouts.

The controls

Press ⌖ on a row to see where that control sits in the sim. The disturbance and the axis are the two controls that set up a run, and both restart it. The rest change only what you see or when it plays.

Control	What it changes	Range
Disturbance ε	The size of the starting disturbance: a small extra spin component εω₀ added to the main spin ω₀. Changing it restarts the run.	0.002–0.1 opens at 0.020
Axis of spin	Which principal axis carries the main spin: length e₁ (smallest moment), width e₂ (middle), thickness e₃ (largest). Choosing one restarts the run.	e₁, e₂ or e₃ opens at e₂ in Free Exploration
Readouts	Time since release, T and \|L\|. Free Exploration also shows the first turn (measured) and the estimate λ⁻¹ln(1/ε) (section 05).	panel
Body-frame inset	The tip of L̂, the unit vector along L, drawn on a sphere fixed to the box, with a blue trail.	stage, bottom right
The stage	Drag it to orbit the view. This changes the viewpoint only, not the motion.	drag
▶ Play / ⏸ Pause	Starts or freezes the run.	top bar
Speed	Plays the motion faster or slower on screen. The physics stays the same.	0.25×–4× opens at 1×
↻ Reset	Restarts from t = 0 with ε back at 0.020 and the axis back at thickness e₃.	top bar

Fixed: m = 1 kg; box 0.30 × 0.20 × 0.10 m; main spin ω₀ = 2π rad/s, one turn per second of the t readout. At 1× the sim plays 0.4 s of motion per real second.

Three axes, three moments

Every rigid body has three principal axes, at right angles through its centre of mass. Spin the body exactly about one of them and ω and L both lie along it, so the spin does not change. For a uniform box they run through the centres of the faces. Their moments of inertia are:

The box's principal moments. Here I₁ = 0.0042, I₂ = 0.0083 and I₃ = 0.0108 kg·m².

With no torque, the angular velocity components obey Euler's equations:

Euler's equations for a free body. These are all the sim integrates.

They keep two quantities constant, whatever the starting spin: the kinetic energy T and the size |L| of the angular momentum.

See itSpin the box about its thickness. Watch for: T and |L| keeping the same digits while t counts up.

One sign decides the stability

Spin about axis k and add a small disturbance. To first order, a small tilt q of the spin away from axis k obeys the equation below, where i and j are the other two axes:

The sign of C decides whether the tilt oscillates or grows.

Spin axis	C (s⁻²)	A small disturbance
e₁, length (smallest I)	−12.1	oscillates, period 1.80 s
e₂, width (middle I)	+9.11	grows exponentially
e₃, thickness (largest I)	−18.9	oscillates, period 1.44 s

The axes with the smallest and the largest moment are stable. The one in between is not.
The sign of C comes from the ordering of the moments alone. The size of ε and the spin rate set only the sizes and the timing.
The growth lasts only while the tilt is small. Sections 05 and 06 follow it further.

See itSpin about the length, then about the width. Watch for: the length run showing "no turn" in the card, and the width run filling in a first turn within a few seconds.

How long before the first turn

About the width, the growth rate comes from the box's own moments:

The tilt multiplies by e every 1/λ = 0.33 s.

A tilt that starts at a fraction ε of the spin grows until it is of order one, and the estimate for that time is:

The wait depends on the logarithm of 1/ε.

This calculator starts at the sim's ε. Moving it does not change the sim.

Making ε ten times smaller adds a fixed ln 10/λ = 0.76 s to the wait, rather than multiplying it by ten. The sim's measured first turn comes about 0.4 s after the estimate at every ε, because the estimate is built from the linearised equations and the sim integrates the full ones. At ε = 0.02 the estimate is 1.30 s and the measured turn is 1.71 s.

See itRun the width spin at ε = 0.1, then at ε = 0.01. Watch for: the card listing both measured first turns, less than a second apart although ε changed tenfold.

Why the turn repeats, and nothing is lost

Both are exactly conserved for a free body.

Because T and |L| are fixed, the tip of L̂ in the box's frame can only move along a closed curve. For a width spin with a small ε, that curve passes close to +e₂ and to −e₂. L̂ lingers near one, swings across to the other (a turn of the box), lingers there, and swings back.

The turns are regular. At ε = 0.02 they come at 1.71 s, 5.12 s and 8.54 s: the first after half a gap, then every 3.42 s.
ε sets the timing. A smaller ε makes L̂ linger longer near each of +e₂ and −e₂, but the pattern of turns is the same. Only ε = 0 never turns.
L does not move. No torque acts, so L is fixed in space. The box moves around it, and what changes is which face points along L.
Nothing is lost. T and |L| keep the same six decimals through every turn.

See itRun the width spin at 4× speed through three turns. Watch for: the card listing each turn time, with equal gaps after the first.

See itRun the width spin and watch the dashed line along L. Watch for: the dashed line staying put while the box turns, and T and |L| keeping their digits.

Where it matters

The Dzhanibekov effect. In 1985, aboard Salyut 7, cosmonaut Vladimir Dzhanibekov released a wing nut from its bolt. It spun steadily, then turned over, again and again at regular intervals. The nut was spinning about its middle-moment axis, the unstable one.
Thrown objects. A tennis racket tossed with a spin about the axis across the head, in the plane of the strings, tends to turn over. The same goes for a phone or a book tossed about its width. Spins about the other two axes stay steady.
Spacecraft. Spin-stabilised satellites depend on it. Explorer 1 (1958) was spun about its long, smallest-moment axis, and soon after launch its spin shifted to end-over-end rotation.

Go deeper: why Explorer 1 changed its spin

The sim's body is perfectly rigid. A real body flexes, and the flexing turns rotational energy into heat. No torque acts, so |L| cannot change, but T slowly falls.
For a fixed |L|, T = |L|²/2I along a principal axis, which is lowest for the largest I. Energy loss therefore drives the spin towards the largest-moment axis.
So with energy loss the smallest-moment axis is not permanently stable. The sim does not model this.

Summary

In one line. A free box spins steadily about its smallest-moment and largest-moment axes. A small disturbance about the middle axis grows exponentially until the box turns over, again and again, with T and |L| unchanged.

Key points

The sign of C = (I_j − I_k)(I_k − I_i)ω₀²/I_iI_j decides: negative gives an oscillation, positive gives exponential growth.
The first turn comes after about ln(1/ε)/λ (the sim measures about 0.4 s more), and each tenfold cut in ε adds a fixed ln10/λ = 0.76 s.
T and |L| are constant, which confines L̂ to a closed curve in the box, so the turns repeat.

Common mix-ups

"Something must be pushing the box over." No torque acts. The turn comes from Euler's equations alone.
"The smallest-moment axis must be the least stable." For a rigid body it is as stable as the largest. Only the middle axis is unstable.
"A tiny enough disturbance means no turn." Any ε above zero turns. Smaller ε only delays it.

Things to try on your own

Run the width spin at ε = 0.1 and at ε = 0.002, and compare each measured first turn with its estimate.
Run the length and the thickness at ε = 0.1 and compare the loops the inset's trail makes around each axis.

Predict before reveal	Each axis is predicted before play; the separatrix and the λ⁻¹ln(1/ε) row appear only after the relevant card.
Cognitive conflict	Distractors carry real student intuitions (smallest moment = least stable, turning over = dissipation) and the feedback names them.
Invariants made visible	T and \|L\| are printed to six decimals from the sim's own integrator, and the L arrow is drawn from the conserved initial vector.
Operational numbers	Every number on screen — t, T, \|L\|, the measured first turn — is read or compared by a card.
Honest depiction	Real SI values and one stable metres-per-pixel scale.
Body-frame companion	The inset shows the tip of L̂ riding the sphere–ellipsoid intersection, so "it repeats" becomes something to see, not to believe.