simple pendulum

[Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>]

This is about the formula for T of a simple pendulum.
In an elementary course we say T=2*Pi*sqr(L/g)
happens to be a correct "mathematical model"; it agrees
with experimental data at small amplitudes. In a more
advanced course we try to derive the formula.

Approaches differ but, as far as I can remember, the
derivations always refer to a particle moving with a
constant speed along the "reference circle". Students
are probably puzzled. What does a linear motion (of
a mass attached to a spring, or of the pendulum bob
at small amplitudes) have to do with circular motion?
We say: "nothing, except that the mathematical
descriptions are similar. Imagine a circle and do the
analysis."

The "ball in a dish" thread showed a new way of
deriving the pendulum period formula.

Instead of dealing with a simple pendulum we
consider a conical pendulum. Here trajectories
are circular and T=2*Pi*r/v is easier to introduce.
The rest is only a matter of calculating v for a
given angle. Students can again be puzzled. Why
do we use a conical pendulum approach to find
the period of a simple pendulum? "Because the
motion of a bob of a conical pendulum can be
viewed as a superposition of two simple pendular
motions."

Why is this approach better than the one based on
the idea of a circle of reference? Because the circular
trajectory is more real (less abstract) and because the
analysis can be used to calculate T for any amplitude,
not for small amplitudes only.

Solving two problems at the same time (and ignoring
one of the solutions) is often simpler than dealing with
problems on the one-by-one basis. Think about complex
amplitudes in optics; we use them and ignore imaginary
solutions at the end. The alternative, long trigonometric
relations, is more cumbersome.

Inspired by Bob Sciamanda's new signature file, and
by the current energy thread, let me add:

If you think that energy is not real then represent
it by an imaginary number.

> If you require a real number, please rotate your
> telephone by 90 degrees, and try again."

Aha, now I know why they are eliminating rotary
dials. A conspiracy to confuse us. Anonymous.

Ludwik Kowalski