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Re: [Phys-L] intermediate axis theorem

• 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.