As we saw last time, there is an implicit fallacy in the
Gedankenapparat as I originally conceived it. While it is possible to
align the generator shaft initially parallel to Earth's rotational
axis, it will begin to depart from that alignment as soon as the
generator is loaded. Nevertheless, this setup can be used to calculate
the power capacity of the gyrogenerator with only a very small
handwave.
Consider what happens in the first small time interval, dt, after the
generator is loaded. The mutual torques begin to act. (For conceptual
ease I will refer to the gyrogenerator used in Santa's workshop at, of
course, the North Pole.)
The torque exerted on the gyro by Earth is directed upward. This torque
is perpendicular to the horizontal axis of the gyro and thus it
produces a change in the direction, but no change in the magnitude, of
the gyro's angular momentum, A.
The torque exerted on Earth by the gyro is equal in magnitude but
opposite in direction, that is, downward. This torque will produce a
small change in the magnitude of Earth's angular momentum, B. B is
initially directed upward, so its magnitude will decrease. (There will
also be a change in the direction of Earth's angular momentum, but that
is both small and of second order in time.)
We now observe the angular speed of precession, k, of the gyro with
respect to an inertial frame. From this rate of precession we infer the
magnitude of the torque, T, exerted on the gyro by Earth.
T = A k
The gyro does work* on the Earth-generator at a rate, P, equal to the
product of the torque and the relative angular velocity (that is, the
angular velocity with which the generator shaft turns). Let K be the
angular velocity of Earth and the stator of the generator.
P = T (K - k) = A k (K - k)
It is immediately apparent that I conjected correctly; the system is
impedance matched at k = K/2. Hooray for physical intuition!)
The rotational kinetic energy of Earth, KE, is the source of this
power, of course. Let I be Earth's moment of inertia, so
2
B
KE = ----
2I
and
d KE B dB
------ = - A k (K - k) = --- ----
dt I dt
where B means the magnitude of Earth's angular momentum.
Note that I slipped a small handwave into this argument. I implicitly
claim that the calculation is insensitive to first order to departures
from exact alignment of the generator axis with Earth's rotational
axis. I am confident that this is true, but I have not proved it.
It's Christmas Day here in Canada. I have to go to the airport to pick
up our Edmonton family members, including our only grandchild. I'll
leave the numerical calculation 'til later, but I expect it will take a
beaudacious (Canadian spelling) gyro to power a string of my pretty
blue LED Christmas lights.
Looking forward to the warmth of perihelion,
Leigh
(This would be so much easier if we had graphic capability. Dan?)