Re: The Olympics (more nit picking)

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding:

>                        Divers actually start rotating their entire
> bodies rather rapidly around a new (body-centered) axis.  If you watch
> the slow-motion replays of the olympic diving, it is clear that this
> rotation can be started in the air.
>
>    Divers are taking advantage of the fact that the conservation of
> angular momentum does not imply the conservation of angular velocity if
> the body's moments of inertia change. As physicists, we are used to
> this fact with respect to the magnitude of the angular velocity.  It is
> also true that the direction of angular velocity can change even while
> angular momentum is conserved, if the moments of inertia change.

Angular velocity isn't even conserved if the inertia tensor *doesn't
change* (by internal mass redistributions rather than by rigid rotation)
whenever the conserved angular momentum vector is not parallel to an
eigenvector (i.e. principal axis) of that tensor.  Whenever the angular
momentum is not a principal axis direction of the inertia tensor the
angular velocity changes according to Euler's equations.

...
> The moment of inertia  I  of a body is a tensor, not a scalar.

I think the usual nomenclature for these terms is to reserve the term
'moment of inertia' to mean the *scalar* projection of the inertia
tensor along a given axis through the point about which the inertia
tensor is evaluated.  The inertia tensor's components depend on the
central (origin) *point* about which the mass distribution is taken.
Usually for free unrestricted rotation this point is taken as the
center of mass of the mass distribution.  A moment of inertia (scalar) is
only defined for a given *axis* (not for a given reference *point* like
the full inertia tensor is).

> ...  If only the diagonal components are
> treated, then each component of of angular velocity can change in
> magnitude individually, then the x-component of angular velocity cannot
> be turned into a y-component or vice versa.

Not quite.  Look up Euler's equations (but be careful; they involve
motion referred to the rotating body axes).  Rotation is only steady
about a given axis if that axis is a principal axis of the inertia
tensor.  This means that if the x-component of angular velocity is
nonzero, then it will not change for free rotation *only* if *both* the y
and z components of angular velocity are precisely zero in a coordinate
system in which the inertia tensor is diagonal (so that only its 3
diagonal components are nonzero as Maurice suggested).

> When the off-diagonal components of
> I  are non-zero, then you can increase one component of
> angular velocity at the expense of another.

As viewed from an external inertial frame of reference the inertia tensor
for a *rigid* body is *itself* changing in time (as the body rotates) in
such a way that if it was initially diagonal at some time, it would
always very quickly change to having nonzero off-diagonal elements which
would induce changes in the angular velocity components *unless* the
angular momentum vector was along a principal axis direction of that
initially diagonal inertia tensor (and even then, some nonzero
off-diagonal tensor elements would appear, but they would not cause any
changes in the components of the angular velocity because those angular
velocity components which could have been changed by them would always
have stay zeroed out from the beginning).

> You can always, of course, find a set of axes in which a constant  I  is
> diagonal.

But it will only stay constant if the axes to which it refers are
rotating with the rigid body.  Otherwise I changes with time (whether it
is diagonal or not) in the absense of simplifying symmetries.

David Bowman
David_Bowman@georgetowncollege.edu