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Re: misconception bu omission





On Mon, 16 Feb 1998, Bob Sciamanda wrote:

The exponential models of radioactive decay and capacitor discharge predict
infinite lifetimes. There is always charge left; there is always un-decayed
matter left. We know that is not so! I cannot imagine presenting this model
without a full discussion of the logical origins, implications, etc of this
absurd mathematical statement. Can we (should we) use a better model? . . .
etc.

-Bob

Why is this a *special* difficulty? All mathematical models are
approximations, and students should be aware of the particular limits of
validity of the approximation. What "better" model do you propose for the
usual situations in which we use the exponential model?

Is this in any way different in character from the general fact that when
we apply calculus processes, like limits, differentiation, and
integration, the model implicitly assumes continuous, infinitely
subdividable matter? This causes no problem if you understand that in
physics the we don't take the limits all the way to infinitesimals or
infinities. We take the limits only so far as necessary for the result to
be meaningful and useful, but not so far that the atomic nature of matter,
or quantum effects, show up. When students are told of this early on, many
conceptual difficulties just don't arise later.

If we threw away models because they fail when pushed to absurd limits,
we'd not have many models left in physics. Every analogy fails when pushed
far enough. Math is the most powerful analogy because we can choose
appropriate mathematical analogies so that their failure point can be
isolated to regions which don't matter for the problems at hand. Physics
*needs* mathematics, but physics *isn't* mathematics. We use what math we
need to get a job done. We don't use "pure" math, but "good-enough" math.
We need to help students grasp this fact of life.

I'm not advocating that we teach elementary courses which promote
misconceptions which aren't easy to correct later. We probably shouldn't
even discuss things which can't be presented at a given level in a
reasonably honest manner, with models which give results which can be
checked in the lab. When necessary we need to remind students that a
particular presentation isn't the final or complete treatment, and if it
doesn't consume too much class time, indicate in a general way what a
better treatment looks like.

I'm watching the "weight" thread with amusement, noting how often people
fuss about fine details which are of no consequence to a wide variety of
situations. Next some will be advocating that, because relativity is
"better" than Newtonian mechanics, and "correctly" deals with things
Newtonian mechanics doesn't, then we should use relativity all the time,
in every problem, even those where there's no practical difference in the
results. Some people would use a sledgehammer to crack a walnut.

Why, I wonder, don't we adopt the method we used in Freshman physics back
in the 50's. The weight of an object was defined as the size of the
external force required to support that object at rest with respect to its
immediate environment. Thus, standing on the floor, your weight is
*approximately* mg, the size of the normal force of floor on your feet
(realizing, of course, that there's a centripetal force in this
accelerating frame of reference, small enough to neglect in many
situations, but an important matter which will be taken up later). The
astronaut is therefore weightless, using this definition of weight. The
person in a free-falling elevator is weightless in that frame of
reference. And when our textbook spoke of a body immersed in liquid, and
told us that the buoyant force was equal to the "loss of weight in the
liquid" we realized that the loss of weight was due to the fact that the
force required to support it (the tension in the string suspending it) is
less when it's in liquid than when it's hanging from the string in air.
Hence, the weight being equal to the string tension, the weight decreased
when taken from air to liquid. We knew that mg hadn't changed.

This was quite consistent with the language used in describing these
things, and the language which still persists today. It worked, in the
context of the situations described in the Freshman course, and didn't
require much traumatic unlearning later. Even the slower students 'got
it'. I submit that that approach to presenting the idea of weight is as
"good" for introductory courses, and no worse, than that used in most
textbooks, or some of the other proposals seen here. And it won't cause
students difficulties on any standardized exams which I have seen, a
matter of some importance when we discuss innovative ways to present
material.

Oh, I know why people enjoy discussing esoteric details of mathematical
physics which you never encounter in daily life. That's where the *fun* is!

-- Donald

......................................................................
Dr. Donald E. Simanek Office: 717-893-2079
Professor of Physics FAX: 717-893-2048
Lock Haven University, Lock Haven, PA. 17745
dsimanek@eagle.lhup.edu http://www.lhup.edu/~dsimanek
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