Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: dimensionless units; j and jbar



At 09:24 PM 11/29/00 -0800, John Mallinckrodt posted a nice note about
dimensionless units.

One slight comment on this passage:

the definition of the "radian measure" of
an angle demonstrates that

radian = 1 (exactly)

One should keep in mind that this is a conventional definition, not a law
of physics.

1) I have a spreadsheet program running here. If you type in sin(1), you
get out 0.84. The machine is "calibrated" in radians. But there exist
calculators that can be configured to use degrees instead of radians, in
which case sin(1) = 0.017. I have on occasion found this to be a
convenient feature.

2) Electrical engineers (and others) use the word "frequency" in two
different ways. Sometimes they refer to "the frequency f". Other times
they refer to "the frequency omega". The relationship "omega = 2 pi f" is
taken for granted.

We could have a discussion of whether this constitutes an
abuse of notation, but let's not pursue that tangent.

Using EE notation in a mechanical formula, suppose we have a crank that
strikes a bell. The number of times (S) the bell is struck in a time
period (t) is
S = omega t / (2 pi)
where we have used "the frequency omega" (angular measure). But we could
equally well write
S = f t
where we have used "the frequency f" (circular measure).


==================

To summarize:
-- Sometimes degrees are more convenient than radians.
-- Sometimes cycles are more convenient than radians.
-- You must accept the fact that formulas like "S = f t" are quite common
in the literature. When you see a formula involving "the frequency", you
need to do some digging to ascertain whether is based on circular measure
or radian measure.

==================================================================

Another thought:

In typical physics formulas such as F=ma, one does not need to dig around
to find what units to use. When one switches, say, from English units to
SI units, the unit of force changes, the unit of mass changes, and the unit
of acceleration changes -- all in such a way that "F=ma" is still correct
as written.

This leads to a possibly constructive suggestion: We could define a new
symbol (let's call it "j" for now) which is defined to be the conversion
factor to radians from whatever units you are using. While we are at it,
we can define
jbar = j / (2 pi)
which is the conversion factor to cycles from whatever units you are using.

Then we can write the bell-ringing formula as
S = jbar t frequency
which is correct no matter whether the frequency is Hz or radians per
second or whatever.

As another example, the formula for the centrifugal field is
A = R j^2 frequency^2
and once again we don't care what units are used, so long as they are
consistent.

I realize that people would omit factors of j and jbar from typical
formulas, but it might be nice to have such conversion factors around, to
disambiguate things when necessary.