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# Re: Harmonic Motion and Simple Harmonic Motion

I think the "textbook" definition of simple harmonic motion is that it
follows the differential equation: d^2x/dt^2 + w^2x = 0 where w = omega =
angular frequency. For example, for a Hooke's law spring, w^2 = k/m.

Another way to say it is that the restoring force responsible for the
oscillation must be a linear function of displacement. For a Hooke's law
spring F = -kx.

The general solution to this differential equation can be a sine or a cosine
or any linear combination of the two. But since the angular frequency
appears the same way in the either the sine or cosine or combination, the
graph of x(t) for this differential equation will be a simple sinusoid, i.e.
will not be complex.

Although it's true, I don't typically hear of simple harmonic motion
described as a simple sinusoid. I usually see or hear SHM expressed as
coming from a linear restoring force which coupled with Newton's 2nd law
gives the differential equation above.

A problem my students often have is getting it straight that a "simple
pendulum" follows harmonic motion, but not simple harmonic motion. I have
to keep telling them the word simple does not mean the same thing in these
two phrases. One way I stress this is via lab. A spring pendulum and
torsional pendulum follow simple harmonic motion, one part of which is that
the period is not dependent on amplitude. Students verify this is lab. On
the other hand, the simple pendulum, not being SHM, indeed has an amplitude
dependent period; its period increases as amplitude increases. We measure
this in lab, and also explain how this necessitates a constant amplitude
pendulum swing if a pendulum clock is going to be accurate. In fact, it is
surprising for students to realize that a pendulum clock will speed up as it
unwinds if the amplitude decreases as a result of unwinding. The typical
assumption is the clock will slow down as it unwinds.

Anyway, SHM requires linear restoring force and no other forces.

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