BELIEVE ME NOT! - - A SKEPTIC's GUIDE
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Why is SHM characteristic of such an enormous variety of phenomena?
Because for sufficiently small displacements from equilibrium,
every system with an equilibrium configuration satisfies
the first condition for SHM: the linear restoring force.
Here is the simple argument:
a linear restoring force is equivalent to a potential energy
of the form
-- i.e. a
"quadratic minimum" of the potential energy at the
equilibrium configuration q = 0.
But if we "blow up" a graph of V(q) near q = 0,
every minimum looks quadratic under sufficient magnification!
That means any system that has
an equilibrium configuration also has some analogue of a
"potential energy" which is a minimum there;
if it also has some form of inertia so that
it tends to stay at rest (or in motion) until acted upon
by the analogue of a force, then it will automatically
exhibit SHM for small-amplitude displacements.
This makes SHM an extremely powerful paradigm.
Jess H. Brewer -
Last modified: Sun Nov 15 13:51:11 PST 2015