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Re: Coupled Pendulums



Look at this mostly from an energy viewpoint, but also a little bit from a
frequency/phase viewpoint.

I'll use "energy transfer" even though we have just had a lengthy discussion
about the merits of this language.

When one pendulum starts with energy and the other has none, it is clear
energy is being transferred from the higher energy pendulum to the lower
one. When the pendulums are identical frequency, the surprising thing seems
to be that all the energy transfers so the original pendulum stops and the
second pendulum has all the energy. This is contrary to the typical thought
that energy will transfer until the energies are matched, and then the two
pendulums stay at that amplitude.

Here is why. When energy transfers from the first pendulum to the second,
the second pendulum slightly lags the first pendulum in phase. If the
frequencies are matched this lag persists to the end... the second pendulum
always lags the first. So the second pendulum is always "pulled" by the
first even when the second one has essentially gained all the energy and the
first one has just about lost it. Transfer is essentially complete.

Once the original swinging pendulum has stopped, the second one tugs on the
first, and energy transfer starts going the other direction. Now the first
pendulum always lags the second one in phase, so the energy all comes back.
And so on, and so on.

When the two pendulums have different lengths they have different
frequencies. Initially the first pendulum leads the second, and energy
transfers from the first to second. But the frequencies differ, so the lead
of the first eventually switches around to being a lag. Since energy passes
from the leader to the lagger, once the phase relationship switches, the
energy flow switches direction also.

The more different the frequencies are, the quicker the phase switches and
the quicker the energy-flow direction switches. Watch the period of time
that the second pendulum starts, stops, starts, etc. This period is short
if the pendulums are quite different, and longer if the pendulums are not so
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