Foucault's Pendulum is used to determine the latitude of the location of the Pendulum. Such pendulum oscilates through the same plane so, as the Earth rotate around its axis, it seems, to an observer, that the pendulum is rotating the plane it oscilates in as time goes on. By measuring the period of this apparent rotation of the plane, one is able to determine the latitude of the pendulum's location, as the period is T = 24h / sinα, with α being the latitude. So a pendulum like this seems to be useless to determine the position of the Island. But here is what I think: the center of mass is a particular point of a system of particles defined as the average of their position weighted by their masses. Moreover, the center of mass (CM) makes possible to consider the Earth as a particle with all the mass concentrated in the said point. This way, considering an object on Earth's surface, we can consider the Gravitational Force acting on that object to be directed to the CM (not considering any gravimetric anomaly).
Now, let's suppose the Island is moving through Space. This can hypothetically cause the center of mass to change it's position.
A pendulum will oscilates only through one plane, that means the plane NEVER changes. But, if the CM moves, the Pendulum will probably correct its oscillation following the new gravitational force coming from the new CM. So the pendulum, while oscilating in the very same plane, can have a correction in its direction.
So as the CM changes its position we would see the Pendulum rotating around a different axis (as seen by an observer on Earth). If we position the pendulum on a map that takes into account such aspects, the pendulum could hypothetically indicate the actual position of the Island.
Of course this theory has some unclear aspects, as any other theory:
- With the Island moving, the CM of the Earth would change in an imperceptible way, because the mass of the Island is negligible in relation to Earth's mass.
- Even if the CM moves considerably, why nobody on Earth seems to notice this too (as there is more than Mrs Hawking's pendulum in the world)?
Anyway, as seen in episode 502, Mrs Hawking seems to have found the actual position of the Island. I tried to enhance a screencap of the map she used.
As you can see, the lines converge in a point just above the Equator: we can see French Polynesia and Kiribati to the South and Hawaii to the North. So every grid of that map is 30° x 30°. Considering this piece of information, I think we could approximate the location of the Island to the coordinates:
0° - 5° N, 160° - 165° W
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