Measuring the effect

The average Coriolis force Fc = 2 m weV exerted on the pendulum bob is of the order of 10-6 times the gravitational force Fg = m g. Yet we would like not only
to detect it but to measure its effect, namely measure we' to an accuracy of say 0.5% or better. That is, we want we to be larger than any spurious precession rate
caused by the gravitational force by a factor of more than 102. How this remarkable enhancement of 108 to 109 is achieved is outlined below.

The numerical value of the ratio of Coriolis to gravitational effects is given on the left. Begin with

10-6 = average Coriolis force / gravitational force.

     But note that the pendulum bob is supported by a wire and most of the gravitational force is cancelled by the tension force in the wire. The remaining
     gravitational force is less by a factor of the maximum displacement b = 4º (= 0.07 rad) of the pendulum. That is 10-6/0.07 or
1.4 x 10-5 = we'/w, The ratio of  the  direct effect of the Coriolis force to the gravitational force.

     The pendulum is approximately an harmonic oscillator. Only the residual anharmonic forces can give rise to a precession competing with the Coriolis
     precession. The ratio anharmonic / harmonic force is = ½b2 so that

5x10-3 = we'/w,  Maximum possible precession rate from the anharmonic gravitational force.

     The anharmonic force produces no precession if the pendulum motion is strictly planar. The maximum possible precession is reduced to the actual
     precession by the ratio b/a, the measure of the ellipticity of the motion. For our support anisotropy and Charron ring the maximum b/a = 1/50. The
     result is

0.25 = we'/w, Spurious precession rate. This, being of the order unity, shows that the earth's rotation is now just barely observable.

     The precession caused by the anharmonic residual of the gravitational force is reduced by a compensating anharmonic magnetic force. The cancellation
     cannot be made perfect because of uncontrollable fluctuations in various drive and damping mechanisms. There are also corrections of order (b/a)2 in
     the preceding ratio that are not affected by the magnetic force. With reduction by another factor of 50 there remains

10 = we'/w, Compensated spurious precession rate.

     The spurious contributions to the precession are still mostly of an oscillatory or random nature. The mean of these spurious contributions is more than
     an order of magnitude smaller so that by measuring over the period of a week, say, we get

2 x 102 = we'/w, Mean compensated spurious precession rate.

     An accuracy of 5 x 10-3 or 0.5% on we' means we can determine the latitude of our Foucault pendulum with an uncertainty of about 30 km.
 

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