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*From*: Bernard Cleyet <anngeorg@PACBELL.NET>*Date*: Sat, 30 Jun 2001 17:28:34 -0700

I thought a) was eliminated by the restriction "in one cycle."

I thought I was defining 2 times the arc length.

Which is another example of "our" inability to use definitions.

c.f. Tim F.

bc

I was attempting eccentricity -- the mathematicians have succeeded.

"John S. Denker" wrote:

At 01:29 AM 6/30/01 -0700, Bernard Cleyet wrote:

And I suppose your math prof. would say the period of a pendulum is the

distance traveled by the bob in one cycle?

The word "distance" here could mean a couple of different things:

a) arc length (S), i.e.

-- the distance along the trajectory, i.e.

-- the integral of the speed (|V|) dt,

-- which grows without bound as the pendulum continues to swing.

b) the displacement (X),

-- which also is a distance, i.e.

-- the integral of the velocity (V) dt,

-- which oscillates within fixed bounds.

Mathematicians and physicists would agree that the displacement (X) is a

periodic function of the arc length (S). This periodic function obviously

has a period, which has dimensions of distance.

However I'm not sure anyone should be so bold as to call this !the! period,

because that would imply uniqueness, and the arc-length parameterization is

not !the! only possible parameterization. It is more conventional to use

time as the parameter. Indeed it would be hard to analyze a pendulum

without introducing time. Using arc length would be a bit eccentric.

Meanwhile... one should not become too dogmatic about the primacy of the

time variable. There are other very similar problems where something more

akin to the arc-length variable is primary. Example: consider the

camshaft in a car engine. The position of each cam-follower is a periodic

function of the angle of rotation of the camshaft (or, equivalently, the

arc-length of motion of a given point on the circumference of the

camshaft). It may on occasion be a periodic function of time, if the

engine happens to be running at steady RPM, but more generally it is not.

====================

The idea of periodicity is clear:

A steady change in a certain variable returns

something to its initial state.

Examples:

1) A pendulum is periodic in time. (It is also periodic in arc-length,

but time is the more natural variable.)

2) A cam is periodic in angle.

3) A crystal is periodic in space.

4) A monochromatic wave is periodic in space and periodic in time.

Case (4) clearly permits the word "period" to be correctly used as the

answer to two different questions.

=======================

General philosophical & pedagogical remarks:

When teachers ask open-ended questions, they ought to tolerate unexpected

or even eccentric answers.

This is clearly necessary when it comes to open-ended questions about names

and terminology. If I show somebody a picture such as

http://www.airliners.net/photos/small/o/oy-vkf.gif

and as "what is this?" there are many possible answers:

-- it's an airplane

-- it's an airliner

-- it's an Airbus

-- it's an A320

... and if I don't ask a more-specific question I shouldn't expect a

more-specific answer. The same applies to open-ended questions about

physics terminology. The terminology is highly irregular, always has been,

and probably always will be.

More generally, I am appalled when teachers take the fascist attitude that

every question has one and only one correct answer.

Many questions have a solution set containing more than one element.

I remember a quiz in high school where we were asked to construct

such-and-such using straightedge and compass. I turned in a construction

that was correct and concise -- but it wasn't what the teacher was

expecting, so it was marked wrong. Phooey! I'm still rankled by it.

If the teacher was trying to teach me to stick to the conventional answers,

the lesson had no effect :-). Now, I get paid a lot of money to do

research, and to hire researchers. Outsiders think this involves solving

problems where no conventional answer exists... but more often it involves

re-examining old problems and finding that the conventional answer is in

need of improvement -- because it is obsolete, incomplete, or just plain wrong.

I think the world needs more unconventional thinkers, not less. I think

students should be rewarded, not punished, for unconventional-but-correct

answers.

It is amazing how many students finish their schooling having no idea what

a brainstorming session is, and being almost unable to imagine generating

multitudinous different answers to the same question.

http://classweb.gmu.edu/mgabel/nclc110_1997/groupcol.htm

I am reminded of Calandra's parable about the barometer:

http://homepages.paradise.net.nz/ianman/research/barometer.html

This parable could be used as the lead-in to a classroom exercise in

brainstorming: how many different solutions can the group come up

with? If done right, this could be highly entertaining as well as educational.

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