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
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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