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*From*: John Denker <jsd@av8n.com>*Date*: Wed, 26 Dec 2012 11:17:54 -0700

On 12/26/2012 07:26 AM, Carl Mungan wrote:

Recall that the IAT says that if you throw a book (taped closed) or

tennis racquet into the air with spin, it will start tumbling about

its intermediate principal axis, in contrast to the stable rotation

observed about its large and small axes.

Well, I wouldn't have stated the theorem quite that way,

but presumably we all know what the general topic is.

a) Spinning in the direction associated with the large

eigenvector is stable in the sense that it has the least

energy per unit angular momentum. This is true and quite

important in some situations, for example for a spacecraft

in orbit where it is subject to internal forces but no

external forces. Internal damping cannot change the

angular momentum but can (and often does) dissipate the

rotational energy.

b) The argument for spinning in the direction associated

with the small eigenvector is much sketchier. The usual

hand-wavy argument says such motion is not stable, but

it can be steady in the short term, in the same way that

a pencil can be balanced on its point.

I think the actual book-tossing experiment is more properly

explained in terms of what is *noticeable* rather than what

is stable.

It *is* possible to toss a book such that it spins around its

intermediate axis. It requires more skill and more bother

than the other axes, but it is definitely doable.

Consider the following hypothesis:

a) The states "near" the large-eigenvalue rotation "look"

similar to it.

b) The states "near" the small-eigenvalue rotation "look"

similar to it.

c) The states "near" the intermediate-eigenvalue rotation

"look" different.

Whether this hypothesis is even true (let alone intuitive)

depends on how you define "near" and "look".

I never put much stock in this theorem anyway. In practice,

it is perfectly possible for an airplane to exhibit a steady

spin around the intermediate axis, aka a steep spin. In this

case the force-terms in the equation of motion are more relevant

than the inertial terms.

The alternative to a steep spin is a flat spin, i.e. a spin

in the large-eigenvalue direction. This is more stable than

a steep spin, and correspondingly more obnoxious ... but it

is certainly not the only spin mode.

**Follow-Ups**:**Re: [Phys-L] intermediate axis theorem***From:*Bernard Cleyet <bernardcleyet@redshift.com>

**References**:**[Phys-L] intermediate axis theorem***From:*Carl Mungan <mungan@usna.edu>

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