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Re: [Phys-l] neat video of synchronizing metronomes



On 07/21/2011 09:48 AM, Brian Blais wrote:
Interesting synchronization of metronomes. Any good physics lessons
here?

As for the basic physics, see my note from yesterday.

Perhaps some lab activity?

There are lots of activities ... none of them ideal. You can
google for the relevant PIRA numbers:

http://www.google.com/search?q=3a70+coupled
http://www.google.com/search?q=3a70.20+OR+3a70.25+OR+3a70.30

Note that all versions of the actual apparatus (that I know of) are
non-ideal and hard to analyze quantitatively. Typically the coupling
is nonlinear and/or its magnitude is not easily determined and/or
there are annoying additional degrees of freedom.

This stands in contrast to the highly idealized system that everybody
analyzes:
http://de.wikipedia.org/wiki/Gekoppelte_Pendel

Note the lack of sag in the idealized spring ... in contrast to
the extreme sag in reality:
http://www.physics.uci.edu/~demos/pdf/waves/3a70.25-spring_coupled_pendula.pdf

If you are building your own apparatus:
*) For spring-coupled pendulums:
-- Use rod-type pendulums ... no strings.
-- Any spring will sag under its own weight. Sag is bad,
because it makes the coupling nonlinear and hard to
quantify. You can remove /some/ of the sag by adding some
pre-tension. To do this without changing the equilibrium
position of the pendulums (which would ruin the symmetry)
you can connect *three* springs to two pendulums. The
two outer springs attach to the frame. Other options are
proposed below.
-- Attach the springs with clamps, so that you can reposition
them. Positions nearer the pivot demagnify the effect of the
spring.
-- Use big heavy pendulums, to minimize the effect of the
weight of the springs.

*) For dowel-coupled pendulums, the analysis is messier, since
there is all the complexity of a compound pendulum plus the
complexity of coupled pendulums. Three variables. OTOH once
you have done the analysis, it is a reasonably faithful model
of the actual apparatus.
http://www.physics.uci.edu/~demos/pdf/waves/3a70.20-dowel_coupled_pendula.pdf

*) In all cases, you want the supports to be immovable, lest
they contribute additional hard-to-quantify couplings.

=============

I have some ideas for improving the apparatus. I haven't actually
tried them, so this must be considered preliminary and hypothetical.
Maybe it will get folks thinking in a good direction.

Consider this:
http://www.av8n.com/physics/img48/coupled-pendulums.png
or perhaps even better:
http://www.av8n.com/physics/img48/coupled-oscillators.png

Among other features, note that the springs are vertical, so
that sag isn't a factor.