Re: simple pendulum

[Bea or Rick Tarara <rbtarara@SPRYNET.COM>]

Hi from San Antonio...the local access number for mindspring seems to work.

Just a few notes here to refresh Ludwik's memory on the normal derivation of
the simple pendulum:

-----Original Message-----
From: Ludwik Kowalski <KowalskiL@MAIL.MONTCLAIR.EDU>
To: PHYS-L@LISTS.NAU.EDU <PHYS-L@LISTS.NAU.EDU>
Date: Wednesday, August 04, 1999 8:38 AM
Subject: simple pendulum


> 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."
Actually the derivation from circular motion goes along with a mass and
spring in SHM--not the pendulum.  Here there is a strong physical
connection, namely that the one-dimensional (either vertical or horizontal)
projection of the circular motion matches that of the mass on the spring.

Derivation of the simple pendulum then uses 2pi*sqrt(m/k) and the notion
that SHM requires a restoring force pointed back at the equilibrium position
that is proportional to the displacement from that position.  For the
pendulum the restoring force is the projection of the weight of the mass
pointed back at the equilibrium position (mgsin(theta)) and the displacement
is arc length or L*theta.  Using the SHM equation we get
2pi*sqrt(mL*theta/mg*sin(theta)). In the small angle approximation then
sin(theta) = theta and we get the expected 2pi*sqrt(L/g).  Outside the small
angle approximation the motion is not SHM.

Rick