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Re: dimensionless units

On Wed, 29 Nov 2000, John Mallinckrodt wrote:

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

In my mind, I replace % with "per hundred" (or per cent). Unitwise, is %
equivalent to pph (parts per hundred)?

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,

Hmmm...something bothers me about this. Suppose the mixing ratio (mass of
water vapor to mass of dry air) is 0.005. I believe it is wise to
continue to include the units as 0.005 (kg water)/(kg dry air). By
keeping the units (even though they appear to just cancel) it becomes
clear that if you multiply the mixing ratio by the mass of dry air one
gets the mass of vapor (whereas if you multiply the mixing ratio by the
mass of vapor you get...?).

Shouldn't we treat the radian the same way? Isn't the radian equivalent
to (m of arc)/(m of radius)? [note that any unit of length could be used]

Saying it is equal to just "1" loses important information about what the
radian represents. When multiplying the angular velocity (rad/s) by the
radius, the "rad" converts the (m of radius) into a (m of arc) and one
gets a tangential velocity. It does not convert (m or arc) into a (m of
radius).

FWIW, I introduce this concept with my students by first having them cut a
piece of string 2-4 inches long from which they draw a circle with that
radius (using a compass). They then place the string along the arc of the
circle and draw a "pie slice" encompassing that arc. I ask them what the
ratio of s/r is. They say one. I then say that from now on, we will say
that is an angle of one. "One what?" they say. It is one meter of arc to
meter of radius or "radian" for short. They then place the string along
the arc and determine how many string lengths (radii) it takes to go all
the way around. This way they "discover" that there are 2*pi radians in
one full circle. I assume this is a popular way of introducing it - I'm
interested to hear if this produces the disappointing results everyone
seems to be complaining about (I seem to think it works pretty well).

----------------------------------------------------------
| Robert Cohen Department of Physics |
| East Stroudsburg University |
| bbq@esu.edu East Stroudsburg, PA 18301 |
| http://www.esu.edu/~bbq/ (570) 422-3428 |
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