Re: "simple" pendulum

["JACK L. URETSKY (C)1998; HEP DIVISION, ARGONNE NATIONAL LAB ARGONNE, IL 60439" <JLU@HEP.ANL.GOV>]

Hi all-
        The conical pendulum ain't simple either.  Since the string length
is fixed ("l"), we can use spherical coordinates with angle theta and phi.
Te "phi" angular momentum is a constant of the  motion that couples wiith
the theta motion.  The coupled equations are highly non-linear.  I think
they say the following:
        Start the motion tangent to the theta direction (so as to start
the string in a direction to generate a cone), but with two low an
angular velocity about the verttical axis.  The string will rapidly
align itself with a new angle with a higher rotational frequency about the
vertical axis, and then oscillate in theta about this new angle.  I think,
but am not sure, that the frequency of oscillations in theta about this
new angle is sqrt(g/l).
        I almost flunked a mechanics exam in grad school because I tried
to solve this problem instead of the one intended.  Fortunately, Kerson
Huang, the post-doc who gave the exam, was merciful.
                                        Regards,
                                                Jack
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> The typical equation given for the period of a simple pendulum is
> (2*pi)sqrt(g/L)(1 + (1/4)sin^2(theta/2) + (9/64)sin^4(theta/2) ...)
> i.e. an infinite series.
>
> When I solve the conical pendulum I get that the period is (2*pi)sqr
> t(g/L)sqrt(sec(theta)).
>
> they don't look quite the same to me, and I suspect they are not the
> same.  If they are the same, why would anyone report the simple
> pendulum period in the infinite series form if they could equally well
> have written it as the square-root of the secant?
>
> Has anyone done this?  Are the periods of the simple pendulum and
> conical pendulum truly the same, or are they, in fact, different?

They're different.  Different physics, different math.  The conical
pendulum just goes around and around at constant speed.  The simple
pendulum speeds up as it goes through the origin.  This affects the period.
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        "I scored the next great triumph for science myself,
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had marveled over that mystery a long time.  We had
followed the cows around for years - that is, in the
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