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Re: Radians, dimensions, & explanations



Oh, oh! The radian thing again. I will refrain from repeating here my
own (wornout?) position that we ought to take seriously the notion of
angular dimension. From a purely practical and, I hope, pedagogically
useful standpoint, what I do is

1) show students that the definition of the "radian measure" of an angle
clearly demonstrates that there are, in fact, no units for the radian
measure of an angle,

2) say that we tack on the pseudounit "radian" simply as a courtesy to
people who want to be sure that when we say, properly, "the angle is 1.2"
we are not simply being sloppy and leaving off other possible units for
angle that do not share the radian's unique status,

3) go on to say that, because of the radian's unique status, 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,

4) point out 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 kg m^2 degree^2/s^2

Now, this is an unusual unit for energy; we might prefer Joules. So let's
first convert the degrees to radians by multiplying by
(pi rad/180 degree)^2 and obtaining

KE = 137 kg m^2 rad^2/s^2

and then throw away the radians (remembering why we *couldn't* do this
with the degrees!)

KE = 137 kg m^2/s^2 = 137 J

I won't say I like it, but it's the best I've got.

John

On Wed, 9 Dec 1998, Mike Wilson wrote:

Each year as I go through angular motion I realize
I do not have a simple explanation for why radians are a dimensionless
measurement.

I am always forced back into some discussion of the circumference of
a circle as 2 pi radi so the distance drops out.

My students routinely seem to have a rather blank expression at this
explanation. They are more than willing to just accept "because" as
an answer.

I hope I can provide a better answer.

Thanks for any ideas or feedback.

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