Re: dimensionless units

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Michael's comments:

> I don't totally disagree with Denker and others who want to say that
> "radians" are units, but dimensionless units, but it sure causes problems
> for students who are trying to do dimensional analysis.

Not if they are careful.

> For example, if the angular velocity (w) of a rotating disk is given as 6.28
> radians per second, then you can calculate the frequency (f) of rotation by
> w/2pi.  But what shall we do with the units?

The frequency f is measured in Hz. The definition of a Hz is 1 cycle/s,
but w is measured in rad/s.  The conversion factor between cycles and
radians is that 2*pi rad = 1 cycle. So f (Hz) is given in terms of
w (rad/s) by f = w/(2*pi rad/cycle). Here the rad units cancel and one
is left with cycle/s or Hz in the result.  The dimensionless conversion
factor includes the quotient of the two units being converted between.

>                                             It looks to students like the
> frequency ought to be 1.00 radian per second.

Only if one forgets to include the quotient of units in the conversion
factor.

>                                               If we want it to be 1.00
> reciprocal second (or 1.00 Hz), how do we "get rid of" the radians?  I know
> there are games we can play to do it... but it confuses the students.
>
> I would rather say that the angular velocity is 6.28 reciprocal seconds and
> the frequency is 1.00 reciprocal second (or Hz).  That is, I just like to
> leave the word "radian" completely out of the discussion.

It is probably not a good idea to leave out the names of units (whether
they are dimensioned or not) from the discussion when there are multiple
different common unit measures of how to measure the quantity in question.
Since the dimensionless quantity "angle" has multiple units: rad, deg,
grad, and cycle are all in extant use one ignores their names at one's
peril.  This has nothing to do with angles being dimensionless.  That is
a different issue.

Also, converting units is a *different* activity than doing dimensional
analysis.  Dimensional analysis mostly involves power counting of a set
of base dimensions.  When doing dimensional analysis *none* of the
dimensionless quantities contribute to the dimension of the result of a
product or quotient of mutliple qunatities *no matter* which units any
dimensionless quantity is measured in.  But when converting units (or
when evaluating the composite units of a composite quantity) all the
factors of all the units count whether or not they happen to denominate
a dimensionless quantity.

David Bowman
David_Bowman@georgetowncollege.edu