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The gyrogenerator



On Monday Chuck Britton sent me a query on my posting. He wanted a good
explanation of what happened to the angular momentum lost by the Earth
in this problem. With my brain disengaged I sent him an incorrect
answer. Of course when I reengaged my brain I recognized that it was
wrong, and I apologize to Chuck for my sloppy thinking. Unfortunately
for me, the reengagement of my brain occurred while I was trying to
sleep. Fortunately for me, the thinking clarified my Gedankenapparat
and showed me the flaw in the implicit hypothesis. I believe I now
understand "where the angular momentum of Earth goes" when energy is
extracted from the Earth-gyroscope system.

I do believe my Gedankenapparat, the gyrogenerator, is a good way to
look at Fred Bucheit's conundrum, so I will continue to examine that
apparatus. This feeble medium does not permit me to draw the vector
diagrams that, I am sure, would clarify this for more of the group. If
you are interested, however, try to follow what I say, draw your own
diagrams, and please do clarify and correct it where necessary.

Here's my original posting, annotated with inspirations acquired
between 4 am and 6 am Tuesday.

Consider an electric generator to be mounted solidly to the Earth with
its rotor axis parallel to Earth's axis of rotation. This can be done
anywhere on Earth's surface or even within the Earth; that doesn't
matter. A gyroscope with a large rotor is now attached by its cage to
the shaft of the generator, its rotational axis being perpendicular to
that of the generator, and no gravitational torques apply due to
imbalances. Assume everything is ideal, frictionless, and the generator
is initially unloaded.

So far, so good. I then made a small slip that is founded in what you
will soon see is a misconceived hypothesis.

The gyroscope is set spinning.

I should have asked at this point "Where did the angular momentum
necessary to set the gyroscope spinning come from?" For the time being
let me assume that the gyro was already spinning when I attached it
perpendicular to the generator shaft.

It will be observed that the gyroscope rotates (precesses) once per
day* on the generator's axis.

The observer is mistaken, of course. The gyro maintains its
orientation; it is the observer who rotates (the generator rotates too,
of course). For ease of conception imagine that this is all set up at
the north pole, though that really doesn't matter.

This is normal behavior for a gyroscope. Energy and angular momentum
of the Earth-gyroscope system are both conserved.

No swindle there. I will point out, however, that the angular momentum
of the system is the (vector) sum of the Earth's angular momentum and
that of the gyro. It is important to recognize that the angular
momentum of this system is not parallel to the angular momentum of the
Earth. That means it is not parallel to the axis of the generator. As
seen from an inertial frame, the Earth's angular momentum remains
constant, unaffected by the gyro.

To extract work from this system one must merely load the generator. A
dead short will (ideally) clamp the generator rotor. The gyroscope
will no longer rotate in the stationary Earth frame. No work is done;
the angular momentum and energy of the Earth-gyroscope system are both
conserved.

Again, I wasn't wrong, but I was glib. The motion of the clamped system
as described from an inertial frame is different from that of the
unclamped system. In the clamped case both the angular momentum of the
Earth and that of the gyro precess about an axis parallel to the total
angular momentum of the system at a rate of one rotation per day.

In the clamped case the gyro exerts a small torque on the Earth, and
the Earth exerts an equal and opposite torque on the gyro. These
torques act in directions perpendicular to the plane determined by the
two individual angular momenta and thus do not affect the magnitude of
either. Energy is conserved.

The next question is "How does one get useful work out of this
device?" The answer is that one must load the generator with some
electrical device - a motor, heater, guitar amp... Another question is
"How can one get the maximum power out of this device?" The answer is
that somewhere between open and shorted generator outputs there is an
optimum load impedance at which the gyroscope will

appear to

precess at somewhat less than one rotation per day. Energy is now
being extracted from the Earth-gyroscope system, but it reappears on
the generator output. There is no problem with angular momentum
conservation since there is no external torque acting on the
Earth-gyroscope system.

At this load there will be components of the torques in the plane of
the two angular momenta, and perpendicular to the gyro axis. The
magnitude of the gyro's angular momentum will not change, but that of
the Earth will change slightly, as will the angle it makes with respect
to the gyro's axis.

(The comment on the tides is somewhat misleading in this sense because
the tidal forces do exert external torque on Earth.)

I will leave the question of optimization (impedance matching) as an
exercise for the reader. I haven't worked it out myself, but I would
be unsurprised to find that at match the gyroscope would precess at
one half rotation per day.

Let's consider my conjectured impedance matched load. At this load the
gyro will seem to have made one rotation about the generator axis after
two days. At that time the system will appear to be unchanged, but in
fact the angular momentum of Earth will have changed slightly in its
direction and magnitude, and the sum of the angular momenta will still
be the same. Note, however, that the two angular momenta are no longer
precisely perpendicular to one another. The question of where the
Earth's "lost" angular momentum goes is answered as simply as that!

I can use this model to determine quantitatively the power output of a
gyrogenerator, but we have company coming for dinner, so again, I leave
an exercise for the reader.

Merry Solstice and Happy Perihelion!

Leigh

* That's a sidereal day, of course.

Our oldest son (the gamma ray astronomer) has brought me a shiny new
telescope for Christmas, even though neither of us is Christian. I
picked up our youngest son from the airport this afternoon. My brother
just showed up from Washington, and tomorrow Our daughter, son-in-law,
and granddaughter will fly in for a week. I hope that your Christmas is
as wonderful as it is for us atheists.

Merry Christmas all,

Leigh