Physics 103: Fall 2005

**Pendulums**

With the invention of the pendulum clock, civilization
suddenly had an accurate timekeeper, independent of the heavenly motions.
Before the pendulum, the best clocks were accurate to perhaps ˝ hour per day.
After the pendulum, clocks were routinely accurate to within a few *seconds*
per day, and were rapidly improved after that. How was it that such a simple
and revolutionary invention escaped notice for so many centuries? I have never
heard a plausible answer.

A simple pendulum is just a mass *m* on a string. Its
length L is the length from the support to the center of the mass. (Notice
this means that if you try to change *m* holding *L* fixed, you might
have to make little adjustments to allow for a different size *m*.)

About the only thing to measure for a given pendulum is its
period T, or equivalently its frequency f=1/T, or equivalently its angular
frequency _{}=2_{}f=2_{}/T. For many
purposes the simplest quantity is the angular frequency. (Try not to make your
pendulums too short: short ones can swing in a more complicated way, with the
mass pivoting on the knot that holds it. That is a more complicated pendulum,
with internal motion in addition to the normal pendulum swing.)

(1) The most important things about pendulums can be seen
without measuring anything! A priori, we might expect the frequency of a
pendulum to depend on both *m* and *L*. Find out how the pendulum
frequency depends on mass by constructing two pendulums, using two different
masses, one about twice the mass of the other. Let the pendulums run at the
same time, and adjust their lengths to make them have the same frequency.
What really matters, *m* or *L*?

(2) To find what role the length *L* plays, adjust the
lengths of two pendulums so that one has twice the frequency of the other
(i.e., makes two swings while the other makes only one). You will probably
find that when the frequency goes up by a factor of 2, the length goes down by
a factor of 4.

(3) Another variable, rather subtle, is the amplitude of
the pendulum’s swing. Galileo stated in writing that the frequency doesn’t
depend on the amplitude, but this is not true! In the decades following
Galileo’s death, Christian Huyghens went to great effort to make a
pendulum-like mechanism that would be truly *isochronic*, i.e., would have
a frequency independent of amplitude. See if you can tell just the sign of the
small amplitude effect: make two pendulums that have the same frequency for
small swings, as nearly as you can arrange it, and then observe them if one of
them has a larger amplitude. Is it slower or faster? Try reversing their
roles.

(4) Finally take some real data! Measure T vs L, for some
pendulum, and make a neat data table in Excel. Use Excel also to find the
corresponding f and _{}. Use the log-log idea to test for a
power law relationship between _{} and L (find the power!). Also,
supposedly,

_{}

Here the constant of proportionality is supposed to
be *g*! Use your data to test this idea graphically, and if these things
seem indeed to be proportional, determine *g* as the slope.

Data table:

Sketches of graphs, labeled:

Use your results from (1)-(4) to write a paragraph explaining everything one has to know about the pendulum:

OPTIONAL: please comment on the lab, if you wish, -- suggested improvements?