Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: "simple" pendulum



At 06:08 PM 8/4/99 -0400, Michael Edmiston wrote:

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.

The expression for the period of the simple pendulum involves an elliptic
integral. I guarantee you won't find a closed-form expression for it in
terms of sines, cosines, and/or simple algebra. You will find all sorts of
expansions for it in places like Abramowitz and Stegun.

The expansion for theta near zero you already know. A more challenging one
is the expansion for theta near 180 degrees -- i.e. a pendulum almost
exactly upside down. To make it into a physics puzzle, ask somebody to
estimate the maximum time a pencil could possibly stay balanced on its
point, in the presence of a given amount of thermal agitation. That's a
hard, hard, hard puzzle for most people.

It's called a "simple" pendulum to distinguish it from a compound pendulum.
That doesn't mean it's anywhere near as simple as a simple harmonic
oscillator.