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It turns out that any circle is possible when I try to find
it numerically rather than analytically. I still do not see
what is wrong with the derivation of the cos(TET) formula.
The numerical results shown below were obtained by
choosing an arbitrary TET and finding (by trial and error)
the circular speed which makes the restoring force and the
tangential component of the centrifugal force equal. Note
that for a large TET, such as 40 degrees, the numerical result
is the same as that obtained from the cos(TET) formula. Was
the generality of the formula lost when the sin(TET) was
canceled on both sides of the force=force equation?
for TET=40 degr --> f=1.272 rot/second
for TET=20 degr --> f=1.149
for TET=10 degr --> f=1.121
for TET= 1 degr --> f=1.114
for TET=0.1 degr --> f=1.112
As expected, the frequency (or period=1/f) does not
depend on the amplitude (TET) when TET is small.
Nothing different from what one finds from the well
known pendulum formula.
Speculations from my previous message should be
ignored. But what is wrong with the derivation?