Mark Sylvester suggests changing my spring pendulum analogy to a double
spring. That's a closer analogy, and I like it. However, all I wanted
was an analogy to show two things: (1) In the absence of processes that
cause a change in thermal energy, the pendulum (either one) oscillates
forever. (2) If there are processes that cause a change in thermal
energy, the pendulum (either one) will end up in its equilibrium
position, and all energy it had been given (when placed into
oscillation) will have become thermal energy.
In light of item (2), I do not understand Mark's (and others')
viewpoint that it is interesting that the dissipated energy depends on
the initial and final states rather than the "dissipation factor"
involved. Isn't that the way it always works? The energy difference
between two states is the energy "dissipated" regardless of how (or how
fast) the energy is dissipated. If a ball drops straight down from a
certain height, or rolls down an incline (same height), or falls
through a viscous medium (through same height), or ricochets down a
maze of pegs (from same height), the energy lost is the same in all
cases. If that does not amaze us, the capacitor problem should not
amaze us.
The two separate capacitors describe the initial state. When hooked
together (and oscillations have ceased) they are in the final state.
There is a fixed energy difference between those states. That amount
of energy must be dissipated into whatever modes are available, and at
whatever rate is appropriate for that system. I think the only thing
interesting about this is it is basically an example of conservation of
energy, but it has the ability to confuse the heck out of people. I
first met this problem on a graduate-school prelim-exam. Many grad
students had fits with it for a while (including me). Afterwards I
wondered what all the hullabaloo was about.
Mark is correct that I erred when I wrote "RC circuit." I had actually
meant to write RLC circuit (assuming there is resistance); LC circuit
is appropriate if we can imagine zero resistance.
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