Notice: there is a question at the end of this long e-mail. You can
skip to there if you don't want to read the stuff between here and
there.
I agree there is more than one approach to the simple pendulum. But I
think some approaches are more appropriate than others depending upon
the mathematical level of the students. The conical pendulum might be
a good way to approach the simple pendulum without using calculus,
assuming the answer to my question is affirmative. If the answer is
negative, then the conical pendulum and simple pendulum are not the
same.
In any event, if the students have studied calculus and rotational
motion, I prefer to analyze the simple pendulum that way.
I introduce the simple pendulum after having examined the spring
pendulum and torsional pendulum. First the students see how the SHM
differential equation arises from Newton's laws and Hooke's law for the
spring pendulum. Then they analyze the torsional pendulum using
rotational equivalents of Newton's and Hooke's laws (torque, angular
acceleration, torsional spring constant) and they see the resulting
differential equation is identical to that for the spring pendulum.
Hence, if this differential equation is used as the definition of SHM,
then the spring pendulum and torsional pendulum are both SHM.
Once the torsional pendulum has been analyzed from a rotational
viewpoint (torque, angular acceleration) it is very easy to analyze the
simple pendulum the same way. In fact, I never approach the simple
pendulum any other way. If the textbook analyzes the simple pendulum
using Cartesian coordinates and linear equations, I tell the students
to skip it, and I give them a supplemental handout.
For the simple pendulum, the resulting differential equation is not the
SHM differential equation. Hence, the simple pendulum is not SHM. But
upon noticing that sin(theta) is about equal to theta for small angles,
we see that the simple pendulum differential equation is approximately
the same as the SHM equation to the extent that we can replace
sin(theta) with theta.
Finally I have the students test this in lab. Using photogate timers
accurate to 0.1-millisecond they can determine that the spring pendulum
and torsional pendulum have periods that are amplitude independent.
This is characteristic of SHM. Then they can easily verify that the
simple pendulum period is amplitude dependent; the period increases
with increasing amplitude. This is opposite of what most students
originally assume. This means, for example, that a pendulum clock will
"speed up" (gain time) as it "runs down" if running down implies the
pendulum loses amplitude as the clock spring unwinds. This is
counter-intuitive, and students get a kick out of it. It can lead to a
discussion about how clockmakers tried to make escapements that
produced constant pendulum amplitude as the spring unwound, otherwise
the clocks would gain time as they unwound.
There is something I like about the conical approach that I did not
realize until reading Ludwick Kowalski's posting. Using the
differential equation approach, even though the differential equation
is easy to obtain, it is not easy to arrive at the formula for the
period. Most calculus-based textbooks simply print the formula for the
period and it can be seen that the period is amplitude dependent. But
the formula is typically not derived or proved. My students
experimentally confirm the period dependence follows the equation given
in the textbook, but they don't know how to derive that equation (nor
do I). However, when solving the conical pendulum, the period's
dependence upon amplitude is easily seen.
Unfortunately I am having trouble equating the two.
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)).
I admit I currently do not have sufficient energy to work through the
series to see if the square-root of the secant is the same as the
infinite series in sin(theta/2) given for the simple pendulum. But
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?
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