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Re: The Olympics



Justin Parke wrote:

I just want to clarify since it was my post that started this whole discussion in the first place.

I understand that angular position can be changed without violating >conservation of angular momentum (i.e. the cat landing on its feet), >however, it seemed to me as I watched the diving that the divers were >terminating an existing rotation and it was THAT which seemed to >violate the law. I thought perhaps they were merely slowing the >rotation by means of changing their moment of inertia but it really >seemed that the rotation stopped.

True, but irrelevant to the question of diving. The cat's trick is
to rotate the front and back of its body separately, changing the
position of its legs so as to get a net rotation after going through a
cycle of different twists. 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.
Twisting divers are manipulating their bodies to take advantage of this
possibility as well. The details are naturally very complicated and
therefore difficult to follow. It is especially difficult since our
eyes tend follow the diver's body, putting our conceptialization in a
rotating frame instead of a stationary one.

The moment of inertia I of a body is a tensor, not a scalar. For
simplicity in elementary texts, it is treated as if it had only three
components, but in fact I is a 9-component object in general. The
components used in elementary discussions are just the diagonal
components of the full tensor. 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. When the off-diagonal
components of I are non-zero, then you can increase one component of
angular velocity at the expense of another.

You can always, of course, find a set of axes in which a constant I is
diagonal. However, divers can and do change their moments of inertia,
and the axes in which the tensor is diagonal after the change may not be
the same as those before the change, so you can't ignore the
off-diagonal elements.

Fortunately for divers, you don't have to know all this in order to do a
twisting dive. Unfortunately for physicists, knowing the theory doesn't
really help you do a twisting dive. I taught myself to do every type of
dive but twists, but I could never figure out how to move myself so that
the twisting would start. Since I didn't like to land in the water
lengthwise, I was unable to force myself to experiment enough to figure
out the dives.

Note for nit-picking expert physicists: there is a constraint on the
components of I so that there are only 8 independent components. I
haven't included enough detail for the constraint to make a difference
in the discussion above.

I also seemed to have missed a few posts so perhaps someone already addressed this. Any diver/physicists out there?

Justin Parke
Oakland Mills High Schools

--
Maurice Barnhill (mvb@udel.edu)
Department of Physics and Astronomy
University of Delaware
Newark, DE 19716