# Further background information on Navigation based around Gravity and Gyros by Antonio Nafarrate

Let us take a fun ride to the North Pole and set up a pendulum. Once we are done we should launch it to swing taking care that we do not push it left or right so it swings in a nice vertical plane. After an hour has passed we notice that the plane of oscillation has rotated about 15 degrees. If we could see the stars we will find that the plane of oscillation is always aligned with the same stars. This results because it is the Earth rotating under the pendulum and this is exactly what the French Physicist Leon Foucault did but not at the North pole but at the Pantheon in Paris in front of a large audience. Some gentlemen assured that they could feel the Earth move and several ladies fainted according to reports of the time. It must have been a great show. See: http://en.wikipedia.org/wiki/Foucault_pendulum

One revolution of the Earth with reference to the stars is called a Sidereal day and it is equal to 23 hrs. 56 min, 4.0996 sec. Let us call it TP.  If the pendulum is set up at some other location X and we measure the time that takes for a complete revolution, say we measure TX. Then the Latitude of X is equal to the angle with a sine function equal to TP divided by TX.  More mathematically expressed : Latitude X = arcsine TP/TX .  So this is the formula that measures Latitude.

Many places have Foucault pendulums available to visitors. Foucault also coined the word “Gyroscope”. See http://en.wikipedia.org/wiki/Gyroscope

Foucault noticed that a long brass rod in the chuck of his lathe if set up to oscillate say in a vertical plane it will continue doing so even when he turned the lathe on.

Now let us go to the Equator and set up a spinning top. Instead of the ones that are toys like the ones that children have played since thousands of years let us make one out of a small electric motor so it will spin for a long time.

Before we play lets refresh some terminology from the Physics of Mechanics. The product of mass times velocity is called momentum and usually the letter “p” is the choice of the experts so we have p=mv (mass multiplied by velocity -simple enough), In dynamics of rotations there is a similar formula that momentum is called Angular momentum represented by L , the mass is the moment of inertia I and the velocity is the angular velocity omega. Then for the dynamics of rotations the formula is L=Iomega.

Now for fun we can launch our top, it may wobble as it develops full angular velocity but soon it calms down as it aligns with the direction of Gravity and appears to stay upright in a condition sometimes called “asleep”, if we give it a little push it reacts with some wobbliness until calms down again. The wobbliness before calming is called “precession”. Because it happens at a certain speed we can measure a “precessional frequency” in revolutions per seconds.

Our electric top has been made to look like those disk shaped ones with a stem in the upper part and pointy lower end to sit firmly on the ground and it may even be mounted in a frame as some tops sold as Gyroscopes but they are not unless the frame has three axis of freedom or a so called Cardanic suspension. After say one hour of spinning we can see that the stem that initially was pointing at some star now is pointing 15 degrees East of the star. What happens ?. Angular momentum is always conserved, what went wrong ?. Well actually the rotation of the Earth is sort of “pulling the rug” from under the top and forcing it to point East of the original position.

Now we will take our stationary spinning top onto Earth and we will carry it in our hands. We start by going East. We move our top smoothly avoiding sudden strong shakes as these will produce large transitory precessional motions. Precessional motions are when the top starts wobbling as it spins.

We will proceed along a parallel of latitude so the changes in the top’s position will be only in Longitude (360 degrees around the world North-South). Say that in the stationary condition the precessional frequency of the top is copied by a biological oscillator that is not affected by motions. Let us call this frequency f. The frequency will now increase to f+df . If now we move it West the frequency will become f-df . We have here some mechanism that in some way is a mechanical analog of the Doppler frequency shift (beat) so familiar in acoustics and optics.  (to understand the  Doppler effect see: http://en.wikipedia.org/wiki/Doppler_effect).

By comparing these frequencies by the usual “beat” or interference method, a beat period related to the reciprocal of df (1/df) will be observed, assuming linear conditions and noting that for most cases f>>df (f is much greater than df), meaning that the tangential velocity of a point on the earth surface (except near the polar regions) is considerably larger than the velocity of motion of organisms under their own power, the zeros and maximum values of the amplitude of the beat will happen at equidistant points along the parallel independent of the velocity of motion. The integration of the velocity to obtain the distance becomes as simple as counting steps, the organism only has to count the beats to know how many units of distance it has moved from a “home” or reference meridian (longitude). It is needed to point  out here that the accelerations encountered during the top displacement do not count because they get fully cancelled by decelerations in the start to stop motion. Only the initial and final position determines the total integrated effect again if simple linearity is assumed.

We described how to measure the N-S displacements by measuring the rotation of the plane of oscillation of a Foucault pendulum but the beat method is also applicable for the N-S displacements. And again the displacements will be reduced to units of time.  It is possible that animals pick up these beats and integrate them unconsciously to deliver bearing and position.

The fact that Latitude and Longitude can both be measured in the same units, time, constitutes a very satisfactory result from a logical standpoint.