Three Axes, One of Them Treacherous

Last timeGoing Round, and Staying Round

A freely spinning body keeps two quantities at once, and the two are not enough to keep it steady. About its middle axis it will always turn over.

Pick up a book, hold it closed, and throw it spinning. There are three sensible

ways to do it, and one of them does something the other two do not.

The reason is that a body does not have a single resistance to being spun. It has

three, one for each of three special axes, and for a book they are plainly

different: easiest about the long axis down the spine, harder about the axis

across the width, hardest of all about the axis through the covers.

FIG 1The three resistances of a book, and which spin survives

The three numbers used throughout this lesson. Only their order matters for what follows, and the middle one is the problem. Nothing about the book is unusual: any object whose three resistances are all different has exactly one treacherous axis.

Two conserved quantities, and still a surprise

Thrown freely, with nothing touching it, the book keeps two things.

FIG 2What a freely spinning body keeps

: the three resistances, in increasing order, which are properties of the body and never change
: how fast it is spinning about each of the three axes, which is what changes
: twice the energy of rotation, conserved because nothing is doing work on the body in flight
: the squared size of the angular momentum, conserved because nothing is twisting the body in flight

Two different weighted sums of the same three squares. The resistances appear once in the first and twice in the second, which is the entire reason the two expressions are independent and therefore worth having both.

Two conditions on three numbers leaves a curve, exactly as the two conditions on

the colliding blocks left a circle. And here is where the analogy stops being

comforting. A curve is a strong restriction, and it is not a promise that the

body stays near where it started. The curve through a state near the middle axis

runs all the way across to the far side.

The lesson stops here

5 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

Three Axes, One of Them Treacherous

Two conserved quantities, and still a surprise

5 more paragraphs to go

This is the reading half

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