I heard that one of the first satellites we launched
spun out of control after the radio antennae were deployed. The reason
given was that the spin axis of a body will change if the mass is
redistributed. In two dimensions, if a skater pulls in her arms she will
spin faster because angular momentum is conserved. In three dimensions
will something different happen?
No. I think the analogy that fits the satellite is simpler: if the skater
thinks she's going to pull in both arms but somehow only one comes in, she'll
probably fall over.
Is this related to the fact that if you hold a hammer near the end
of the handle and give it a toss upward while causing it to spin on an
axis perpendicular to the long dimension of the handle, the hammer will
begin to spin with an axis parallel to the long dimension of the handle?
Rotations don't mix unless there is some external force.
You've got a couple of different issues here, which are related, but not
identical. When a skater pulls her arms in, she's changing her "moment of
inertia," not her rotational axis. Now, if she pulls just one arm in,
she'll change her rotation axis. Unless the axis of a rotating object is
fixed, say, by an axle, the axis of any rotation will be its center of
mass. This is the case no matter how many dimensions you happen to be
living in. The same is true for the conservation of momentum and angular
As far as the rotation of your tossed hammer is concerned, the moment of
inertia about the long axis of the handle is less than about an axis
perpendicular to the handle. Thus, a smaller amount of angular momentum
will be manifested as a greater angular velocity. When you toss the
hammer, even though you are trying to apply torque only about the
perpendicular axis, so that the hammer will flip end-over-end, you also
will give it some torque about the long axis. It's the same as throwing a
ball; although you try to throw it in one direction, you'll miss just a
little (or in the case of a klutz like me, a lot), and give it some force
in a sideways direction. Since the hammer has angular momentum about both
axes, it will spin about both axes.
Richard E. Barrans Jr.
Argonne National Laboratory
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Update: June 2012