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Of course it will work, if, perhaps, not economically. Let's modify the
Gedankenapparat somewhat to make the point a bit more clearly:
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.