Re: [Phys-L] intermediate axis theorem

[John Denker <jsd@av8n.com>]

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.