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The gyrogenerator (concluded)



We derived the following relationship for the gyrogenerator in the
previous posting:

d KE B dB
------ = - A k (K - k) = --- ----
dt I dt

KE = Earth's rotational kinetic energy
A = angular momentum of the gyroscope
k = angular speed of the gyro's precession
K = angular rotation speed of Earth
B = magnitude of Earth's angular momentum
I = Earth's moment of inertia

(All these quantities are defined with respect to some frame or point,
of course, but I have suppressed that information here. I won't let my
students get away with that.)

First let us see what power, P, we might expect to extract from this
apparatus. At the impedance matched load we will have:

K = 2 pi radians per day

k = K/2

2
P = A K /4

I tried to think of an appropriate gyro for this purpose, but my
experience is limited. Consider a midsize automobile tire on one wheel
of a car moving at 100 km/hr. Using my best guesses I conclude that

2
A = 50 kg m /s

It will come as no surprise that given this gyro, the gyrogenerator can
produce a maximum output power of 66 nW. I haven't checked this
thoroughly, however, and I defer to John Mallinckrodt's calculation.

We now look at the question that started this analysis: what do the
individual angular momenta do as power is drawn from the generator? The
first thing to note is that both A and B precess about the direction of
the total angular momentum of the system, L. For what follows, picture
A, B, and L in their plane.

L is the vector sum of A and B. Initially the angle between A and B is
90 degrees. As energy is dissipated the magnitude of B diminishes, but
the vector sum remains constant. Thus both A and B must change in
direction, the angle between them becoming smaller. That is, the two
vectors become more nearly aligned with each other and with the their
sum. For any practical case we choose the angle will remain very nearly
90 degrees for a very long time, of course.

This tendency for individual angular momenta in a system with
dissipative coupling between the individual angular momenta to approach
alignment is a very general one. Its results can be seen in the
relative directions of spins and orbits in the solar system and in the
tendency of satellite orbits to become equatorial. As tidal forces
dissipate energy the angular momenta align as they must to conserve
total angular momentum. Deviations from this tendency are usually
attributed to catastrophic encounters in the early history of the solar
system.

I seem to be a bit ill, so I'll send this off without further
elaboration. I really enjoyed this topic. I now understand it better,
and I hope others do, too.

Leigh