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RE: supplementary S.I. units




A. John Mallinckrodt wrote:

BTW, despite my own preference for the idea of an angular dimension
because of the consistency that it confers on calculations, I'm not
sure that I would want to try to teach it this way.


Nor would I. Granted, nature puts certain potholes in the way of
developing a consistent scheme for describing nature. Granted that we have
options about how we want to formulate this description. But I have read
the articles by those who try to circumvent the "problem" of the radian,
and I say "where's the problem?"

I also have real problems with teachers, especially in the introductory
courses, who devise their own schema for dealing with standard topics,
with a terminology and logic unknown to the larger community of
physicists. Certain things have become standard, sometimes for very good
reasons, sometimes because of habit. But before we destroy those standards
by "teching" non-standard approaches, we ought to understand them
thoroughly and attempt to convince our peers in the physics community,
publish our method for all to see, and gain the approval of a significant
segment of the community of physicists (not just physics teachers). I
don't think we want to go to a brave new world where each physics teacher
adopts his or her own private system of physics, unknown to all the
others. Physics is difficult enough inherently, and some standardization
of units, nomenclature and methodology makes life easier for all of us.
Granted, no one standardization scheme will be perfect, nor will it
please everyone, but it facilitates communication, and the alternatives
are far worse.

Textbooks often treat dimensions and units shabbily. Some even say
or imply that the terms are synonyms! Students' problem with radians often
stem from failure to distinguish dimensions from units. Many teachers have
never read the document "Symbols, Units and Nomenclature in physics" of
the International Union of Pure and Applied Physics (reprinted in the
Handbook of Chemistry and Physics).

There are measurable quantities which are dimensionless. Some are given
unit names, some are not. Specific gravity, being a ratio of two
densities, is dimensionless, and has no unit name. Pi, the ratio of
circumference to diameter of a circle, is dimensionless and has no unit
name. Index of refraction is another example.

The ratio of arc length to radius is a measure of angle, and is given the
name radian to distinguish it from other widely used measures: degree,
grad, etc. Of these three, the radian measure is the most "natural",
especially in the mathematical sense. This has some nice consequences:
sin(theta) and tan(theta) approach theta radians in the limit as theta
goes to zero. Not so if theta is measured in degrees.

Some measurables of physically different quantities have the same
dimensions. Work and torque are examples. Luminous flux and source
strength have the same dimensions, but different unit names.

These are not that difficult to deal with.

All equations in physics are tyrannical with respect to
dimensional consistency. With only a few exceptions, however,
they are completely neutral about units. ANY units carrying
appropriate dimensions are perfectly acceptable. For instance, in
the equation

Neutral about units? You mean all that effort physicsts have put in to
devise *coherent* unit *systems* has been in vain? Consider this equation
from a highway engineering book:

h
d = 67.39 - 0.33

Here d represents the distance in feet at which a road sign is generally
legible to an automobile driver, h is the height of the lettering in
inches.

The equation could be made to work for any choice of units by rewriting it

ch
d = K - d

where K, c, and d are constants. Each choice of units would require a
different set of values for the constants. Do we want to deal with this
sort of thing?!

To avoid such ugliness, physicists devised coherent unit systems, such as
cgs, MKS, English, etc. If you keep everything within one such system, you
don't have to deal with variable constants in equations. F = ma works in
any coherent system, so long as you express each quantity only in the
designated units appropriate for that system. You don't have to rewrite
the equations for each of the unit systems.

The radian happens to be the appropriate angular measure for *any*
coherent system. I consider that a simplification, not a 'problem'.

The definition of radian involves only two measurables, radius and arc
length. It requires no numeric constants in its definition. I consider
that a simplification, not a 'problem'.

I'd be interested to see John's method for dealing with the steradian, the
standard (natural) measure of solid angle.

I notice John's discussion of radian introduced the unit "cycle". What is
its precise definition, please? Put the definition in words, or an
equation. Not so simple, is it? John seems to be using "cycle" as a
"unit". Is it also dimensionless? What is gained by throwing out one
dimensionless unit and introducing another, even more problematic?

When I taught the introductory course, I gave students a checklist of info
about dimensions and units. All of it is obvious, but surprisingly, many
students don't notice these things.

1. Dimensions combine by the ordinary rules of algebra. Units do also.

2. Terms which are added or subtracted must have the same dimensions and
the same units.

3. Quantities on either side of the equal sign must have the same
dimensions and the same units.

4. Powers are dimensionless and unitless. However, if the power is a
mathematical expression, quantities within that expression may have
dimensions and units.

5. dy/dx and partial(y)/partial(x) have the dimensions and the units of
y/x (look at the formula for the definition of the derivative).

6. Integral (y dx) has the simensions and the units of yx.

7. Arguments of sin, cos, tan, log, etc. must be dimensionless, but may
have units.

8. The values of sin, cos, tan, log, etc. are dimensionless and unitless.

9. The mathematical constants pi and e are dimensionless and unitless.

I also note that to write 1 kilogram = 2.2 pounds is improper, because the
equation is inhomogenous and is not coherent with respect to units and
dimensions. We should avoid equating apples and oranges unless we are only
interested in "the number of pieces of fruit". We should say, "A one
kiolgram *mass* at the earth's surface *weighs* 2.2 pounds." Even that
needs more precision of language.

-- Donald

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