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Re: Coupled Pendulums

On Mon, 24 Sep 2001, Michael Edmiston wrote:
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.

The coupled-pendula demonstration hides a more subtle demonstration:
spectral line splitting. The higher the coupling between the two pendula,
the faster the energy will "slosh" from one to the other. Suppose the
energy of the whole system could be displayed on a spectrum analyzer. If
the two pendula were not coupled at all, then both would swing at exactly
the same frequency, and the spectrum analyzer would display a single sharp
frequency peak. If the pendula are coupled together and the energy starts
"sloshing" slowly from one to the other, then the spectrum analyzer
displays two peaks, and the distance between the two peaks (the difference
frequency) is the same as the slow frequency of "sloshing". If the two
pendula are coupled more and more solidly together, the two peaks on the
spectrum analyzer will move farther and farther apart.

In the real world, when a group of atoms are assembled into a molecule or
a crystal, their emission/absorbtion lines are split into a bunch of
separate lines. When enough atoms are bound (coupled) together, we say
that they have energy "bands" instead of lines. In semiconductor physics,
the conduction band and valence band can be crudely explained by looking
at strings of coupled pendula. The same applies to chemistry, where
molecules have wide emission bands while single atoms have narrow emission
peaks. The "bands" are actually a densely packed collection of narrow
peaks, and the spacing between the peaks is the same as the slow "sloshing
frequency" between all those coupled resonators.


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