Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: unconventional answers (was: period vs. wavelength)

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.


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.

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
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

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.

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

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.