Re: dimensionless units

[John Denker <jsd@MONMOUTH.COM>]

At 03:09 PM 11/29/00 -0500, Michael Edmiston wrote that using radians as units
> ... sure causes problems
> for students who are trying to do dimensional analysis.

The way I look at it, the existence of dimensionless units does not _cause_
a problem, it merely _exposes and clarifies_ a genuine fundamental
limitation of dimensional analysis.

Let's do an example:

Suppose you want to calculate one of the many forces that act on an
airplane.  Dimensional analysis suggests that it depends on the dynamic
pressure (one half rho vee squared) and on the area.  Now suppose it also
depends on the angle of attack.  Is it proportional to angle of attack to
the first power?  To the second power?  Some sort of exponential?  You will
never figure it out from dimensional analysis, because the angle is
dimensionless.

(In case you are wondering:  the lift goes like the first power, and the
induced drag goes like the second power, to a good approximation if the
angle isn't too large.)

The problem with radians per second versus degrees per second is a thinly
disguised variant of the previous problem.  The "per second" is helpful,
making a certain amount of dimensional analysis possible.  But don't let
your guard down:  dimensional analysis still cannot tell the difference
between radians and degrees.  That is, your formula might contain a factor
of (degrees per radian) raised to some arbitrary power.

In the formula for centrifugal force, there appears a factor involving the
rotation rate.  Dimensional analysis will not and can not tell you whether
you should evaluate that formula using radians per second, cycles per
second, degrees per second, or whatever.

Dimensional analysis can only produce proportionality relationships, not
equalities.

> how do we "get rid of" the radians?

For the purposes of dimensional analysis, you simply observe that they are
dimensionless and throw them away.  Throw away degrees and cycles,
too.  For the purposes of calculating equalities (not proportionalities)
you have to go back to the derivation of the formula you are using and
verify that it was derived using radians.

> I would rather say that the angular velocity is 6.28 reciprocal seconds and
> the frequency is 1.00 reciprocal second (or Hz).

Nitpick: As others have noted, according to convention, 1.00 reciprocal
second denotes one _radian_ per second.  A Hertz is something else, namely
one _cycle_ per second.

(Students, especially pre-calculus students, might argue that cycles are
more "natural" than radians, but the radian convention is very well
established and there's no point in fighting it.  You can calculate in
terms of cycles if you want, but if you mean Hz you have to say Hz, not
reciprocal seconds.)