"What to do about the radian" is one of phys-l's recurring themes.
Some newer readers might be interested in perusing the following
old threads in the phys-l archives:
December 1998: Radians, dimensions, & explanations
September 1998: supplementary S.I. units
September 1996: Units and dimensions in rotational dynamics
There have also been a couple of recent and relevant AJP articles
"Angles--Let's treat them squarely," K. R. Brownstein
AJP, vol. 65, No. 7, pages 605-614, (1997)
I have an elaborate method for rationalizing angular units that is
absurdly pedantic, that I would never burden students with, and
that I personally like very much. It is included in the 1996
thread and is similar to the approach presented by Brownstein in
his article. I also have a far simpler approach (included in the
December 1998 thread) that I find works pretty well with students.
It is based on the same kind of idea that David Bowman mentioned
in demystifying Michael Edmiston's "150 %kg" example via the
explicit recognition that
% = 0.01
To recap (and revise) my December 1998 posting:
1) I show students that the definition of the "radian measure" of
an angle demonstrates that
radian = 1 (exactly)
2) I point out that, because of this fact (i.e., the radian
exactly equaling one), we can always insert the radian or any
power of the radian into the units of *any* quantity and that,
similarly, we can remove it from the units of *any* quantity with
no effect whatsoever on the value of the quantity,
3) I further point out that this can *not* be done with other
angular units. Indeed, we find that
degree = (pi radian)/180 = pi/180 = 0.01745...
cycle = 2 pi radian = 2 pi = 6.283...
grad = (90 degree)/100 = [90 (pi/180)]/100 = pi/200 = 0.01570...
and, so on. In other words, although other angular units *are*,
like the radian, pure numbers, they are *not* equal to one and,
therefore, must be treated with more respect. (It is *not*, for
instance, O.K. to say that line frequency in the U.S. is 60/s.)
4) Finally, I show that, with these understandings, we can use
*any* angular units we like when we employ equations like KE =
(1/2)Iw^2. For instance if I = 10 kg m^2 and w = 300 degree/s, we
have
KE = (1/2) (10 kg m^2) (300 degree/s)^2
= 4.50 x 10^5 J degree^2
This is an unusual, but completely unambiguous unit for energy.
If we want to know the value in Joules, we first convert the
degree^2 to radian^2 by multiplying twice by the easily remembered
conversion factor
1 = (pi rad/180 degree)
obtaining
KE = 137 J rad^2
and then we simply throw away the rad^2 (remembering why we could
*not* do that with degree^2.)
Alternatively, of course, I could simply have used the fact that