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*From*: Michael Edmiston <edmiston@BLUFFTON.EDU>*Date*: Thu, 28 Jun 2001 17:07:12 -0400

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

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