An example with a person periodically bending on a swing is also somewhat
tricky. The person can indeed increase (or decrease) the pendulum's energy by
appropriate bending and unbending, but this conclusion depends on what we mean
by "pendulum's energy". If this is just the mechanical energy of
(pendulum+person), the conclusion is true. But this is only a certain fraction
of system's energy. If we include also the internal energy of the person, then
the swings will increase at the cost of his/her thermodynamic free energy (or
decrease with the gain in the corresponding thermal energy). In this broader
context the total energy of the pendulum will conserve, as it should for an
isolated system.
OTOH, we can indeed change the pendulum's energy in a more simple way by just
hitting it periodically with the same frequency and appropriately chosen phase.
This would be a trivial example of resonance with a possible huge change of
pendulum's energy at fixed frequency. The pendulum's energy now is not conserved
since, being subject to an external force, it is no longer an isolated system.
It is also worth noting that energy conservation is not the same as
invariance.
Moses Fayngold,
NJIT
John Denker wrote on Wed, June 8, 2011 4:46:32 PM
... I insist that if you perturb the system, all bets are off.
In a complicated system, if you project out some little
one-dimensional piece of the phase space, that piece will
not in general be conserved.
This is important. Otherwise it would be impossible to build
things like refrigerators and heat engines and quantum non-
demolition measuring devices and zillions of other useful
things.
In particular, let's do the experiment: Go to the playground.
Select the longest (tallest) swingset you can find. Stand on
the seat of the swingset. Modulate the effective length of the
pendulum by bending you knees at the end of each half cycle,
and straightening your knees at the middle of each half cycle
(so as to do work against centrifugal force). The variation
in length is indicated by the light-gray figure 8 in the figure: http://www.av8n.com/physics/img48/dpa.gif
This leaves ω substantially unchanged. I guarantee that you
can make very large changes in E and therefore in E/ω.
Indeed, this works so well that it is quite possible for E
to grow quite large, sooner than you might have expected,
so please be careful.
This is moderately interesting as a Gedankenexperiment, but
it is even more interesting as a real experiment. I beseech
to actually do the experiment. It's well worth it.
I've done this experiment many many times over the years.
You are going to have a very hard time convincing me that
E/ω is invariant under the given conditions.
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