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Units & Dimensions.




This is a continuation of the discussion items mentioned by Dewey and in
particular by John Mallinckrodt, where I hope to play a little of a devil's
advocate:

Dewey wrote:
I still have not heard how the dot product of a force and a displacement
can have the _same_ dimensions as the cross product of the two. We
differentiate between the multiplicative product and the quotient, why not
the dot and cross products?

I would like to turn the question around and say that I don't understand why
one would think that they should have different dimensions. So my short
answer is: the units are the same for F*dot*r and F*cross*r, and this is a
consequence of the definition of how to compute dot and cross products. The
units factor out of the dot product and/or the cross product and are seen to
be same.

Now, for some discussion on the idea of adding angle as a dimension.

First let me say that from what I saw John's comments are correct with no
error in math or conception.

At one point he wrote:
______________________________
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 correct (if unwieldy)
answer like

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

A few unit conversions would be VERY nice at this point but are NOT
necessary.
__________________________________

At this point I would argue the necessity issue. (This may be what John
means by VERY nice). One can write those terms in all those mixed
(inconsistant) units, but I would argue that to actually add them up and get
a number for x, you must as a matter of necessity convert to a consistant
set of units.

In the equation 1/2*I*w^2 we tell students that they must calculate using
angular velocty as measured in radians. I don't think it is too mysterious
when you see that you derived the formula using the radian definition of
angular measure, so it is only valid if you use radian measure.

So why do we not say the units of kinetic energy (calculated from the above)
are kg-m^2-rad^2/s^2; here is the mystery! Apparently throwing away the
radian units. My answer is that the radians are dimensionless and therefore
don't have to be there; (they are somewhat like an intermediary idea used to
calculate the final result and not needed in the end). Besides we calculated
a kinetice energy, the formula was derived by adding up 1/2*M*v^2 's where
radians didn't appear. I.e. they weren't there in the beginning, so why
should they be there in the end. This gets to the difference between
dimensions and units.

John's example of introducing an angular dimension is a perfect example of
what I said about the number of dimensions being arbitrary and it is
perfectly valid to set up a system of mechanics where we choose 4 dimensions
(as John did) or 3 dimensions (what is done in the standard SI system of
units) or 2 dimensions (those guys that like to set c=1, where velocities
are dimensionless quantities).

I leave it to the readers to decide whether or not the mental gymnastics
needed to understand John's r' quantity are more difficult or awkward than
those mental gymnastics needed to understand the usual idea of the radian.

BTW the statement that d/dx (sin theta) = cos (theta) is valid only for
theta measured in radians; look in your calculus book and you'll see that
they derive it for angles measured in radians. And hence, this is a
situation analgous to the rotational kinetic energy example John mentioned.
(This is mentioned in Arons' book "A Guide to Introductory Physics Teaching,
page 100).

Also, there is a good discussion of radian measure in TPT February 1993
issue. The authors critisize the approach that John suggested and I quote
in part:

The comment is discussing the formula for tangential speed, namely
v_tan=r*w. They say (parenthetical comments are mine)

"The scheme that he (an article that is being critisized) advocates avoids
the radian in the units for tangential speed by using units m/rad for r;
this is based on the proposition that if arclength s divided by radius r is
radians, and if s is in meters, then r is in meters per radian. (I think
this r is equivalent to John's r' ) Multiplying r in m/rad by w in rad/s
then gives v_tan in m/s as desired. Hoever, in this scheme one now has
units for r that containg the radian, which would imply that the numerical
measure of the distance of the moving point from the axis of rotation
changes when one changes the units of angular measurement! Any such
approach is totally unacceptable if one takes seriously the fact that the
radian is only one of many possible units of angle measurement and must be
subject to the rules of unit conversion that apply to all other units"

I don't think I would make as strong a statement as found in the last
sentence of the paragraph I'm quoting, since I think one can make a
consistant system out of what John proposes; but I think the observation is
related (not sure though, since I haven't thought this comment through) to
the fact that John's r' quantity doesn't stand alone but is implicitly a
function of how far you are from the center.

Joel Rauber
rauberj@mg.sdstate.edu