One of the most rewarding aspects of this list is how often something I
read sends me to pencil and paper, or into the lab, and I discover a
new twist about something for which I thought I "knew it all." Such is
the case of the simple pendulum versus conical pendulum.
Since my last posting on this subject I have revisited the subject both
on paper and in the lab. I have found a few things I find most
interesting. Here they are:
(1) I committed several reciprocal errors in my first derivation of the
period of the conical pendulum.
The correct period for the conical pendulum is: T =
2*pi*sqrt(L/g)*sqrt(cos(theta)).
The correct period for the simple pendulum is: T =
2*pi*sqrt(L/g)*(infinite series).
The infinite series for the simple pendulum has a leading term of one
followed by a power series containing even powers of sin(theta/2).
(2) These are not the same. The most striking difference is that the
period increases with theta for the simple pendulum, but the period
decreases with theta for the conical pendulum. This struck me as so
interesting that I went into the lab to confirm it. As I reported
earlier, my students experimentally demonstrate that the period of a
simple pendulum gets longer as the amplitude increases, hence that
apparatus was already set up in the lab. Putting the pendulum bob into
circular orbits at various amplitudes quickly confirmed that the period
indeed decreases as the radius of the circular orbit (conical pendulum
amplitude) increases.
(3) At small amplitudes these pendulums indeed have identical periods
of 2*pi*sqrt(L/g).
(4) Even though I originally goofed it up, the period for the conical
pendulum is easy to derive without using calculus. The motion of a
simple pendulum can only be described by a differential equation having
no simple solution, and the expression for the period is an infinite
series.
Summary: It is not correct that the math for these two systems is the
same, even though the small-angle results for the periods are
identical. Isn't it interesting that, on one hand, the very different
motions have identical periods for small angles, yet on the other hand,
their deviations from this are in opposite directions as the amplitude
increases. And, of course, neither pendulum is an example of simple
harmonic motion. And, as I point out to my students (and as other
writers have pointed out), even though the "simple pendulum"
differential equation is simple to derive, what happens after that is
far from simple.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817