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RE: supplementary S.I. units



John: if I ever get back to writing "my" text, you can expect to have your
post stolen! Nice, Karl

The following is an excerpt from a post that I made to phys-l a
couple of years ago when a similar thread was evolving. It says
many of the same things as the recent AJP articles, but it
emphasizes the idea that the calculation of angular quantities
should depend on "tangential distance per unit angle" rather than
"radius."

I hope some will find it useful in clarifying some of these ideas.

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.

John

*************************

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

x = x_o + v_o * t + 1/2 a * t^2

we can specify x_o in furlongs, v_o in parsecs/millisec, t in
weeks, and a in inches/(hour fortnight) and get a perfectly
unambiguous (if unwieldy and uninterpretable!) answer like

x = 73.2 furlong + 1.65 x 10^-16 parsec week/millisec
+ 3.65 x 10^6 inch week^2/(hour fortnight)

(Of course a few unit conversions would be VERY nice at this point.)

Such is NOT (or at least APPEARS not to be) the case when it comes
to rotational equations. We tell students that some equations
like KE = 1/2 I w^2 ONLY work if we express w in rad/sec or at
least convert all angular units to radians (and then mysteriously
throw them out!) before presenting the final answer.

It needn't be this way. Suppose we take seriously the idea of an
angular dimension (call it "A") and a quantity r' which is the
tangential distance per unit angle at some specific distance from
an origin (to be carefully distinguished from the quantity r which
would still be the distance from that origin.) Then r' would have
dimensions L/A and, for a point 1.00 meter away from the origin,
we would have

r' = 1.00 m/rad = .0175 m/degree = 6.28 m/cycle

One can now go back through all of rotational kinematics and
dynamics rewriting equations using r' wherever appropriate and get
things like the following

QUANTITY "FORMULA" DIMENSIONS
angular momentum r' m v M L^2/(T A)
torque r' F M L^2/(T^2 A)
rotational inertia r'^2 m M L^2/A^2
angle s/r' A

Now you can calculate, for instance, the rotational kinetic energy
using the usual equation KE = 1/2 I w^2 with no requirements on
the units for I or w. For instance, if I were specified in
kg m^2/deg^2 (e.g., because one used the tangential distance "per degree"
instead of the more conventional "per radian" in its calculation)
and the angular velocity were specified in cycles/sec we would get
a perfectly unambiguous energy specified in Joule (cycle/degree)^2.
Of course one may well WANT to convert that quantity to Joules using
the simple conversion factor cycle/degree = 360, but it is NOT
necessary in order to carry unambiguous meaning. And the radian is
no longer any more special or mysterious than any other unit.

-----------------------------------------------------------------
A. John Mallinckrodt http://www.csupomona.edu/~ajm
Professor of Physics mailto:ajmallinckro@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223

Dr. Karl I. Trappe Desk Phone: (512) 471-4152
Physics Dept, Mail Stop C-1600 Demo Office: (512) 471-5411
The University of Texas at Austin Home Phone: (512) 264-1616
Austin, Texas 78712-1081