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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.
The gyroscope is set spinning.
It will be observed that the gyroscope rotates (precesses) once per
day* on the generator's axis.
This is normal behavior for a gyroscope. Energy and angular momentum
of the Earth-gyroscope system are both conserved.
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.
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
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.
(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.
Merry Solstice and Happy Perihelion!
Leigh
* That's a sidereal day, of course.