Re: [Phys-l] frequency: a modest proposal

[James McLean <mclean@geneseo.edu>]

So many responses so quickly!  I wish I had the time to be as attentive 
to this list. :-)

John Denker wrote in part:
> On 01/27/2010 01:33 AM, James McLean wrote in part:
>
> > ... why not define the unit 'cycle' = 2pi radians, and then make
> > 1 Hz = 1 cycle/s = 2pi radians/s = 2pi/s ?
>
> It's not a problem.  It's not even a change.  Radians
> and radians per second (not cycles per second) are 
> already traditional throughout mathematics, throughout
> electrical engineering, and in every physics book I
> can think of.
I think you and I are blinded by our deeper understanding ;-)

According to this system, the proper unit for wavelength (as commonly 
used) would be m/cycle.  If you were to measure the crest-to-crest 
distance to be 25cm, then writing lambda=25cm would be incorrect!
However, I doubt that you can find any book that would /not/ write 
lambda=25cm.

For a more advanced example, where is the book that gives different 
units for h and h-bar?

I have always attributed this to a sort of universal consensus to be 
sloppy in an organized way.  A conspiracy, if you will.  But I now 
realize that NIST and the textbooks are actually expressing a different 
structure.

> I cannot imagine any argument in favor of cycles/sec
> as equivalent to 1/sec.  I have to assume that those
> two NIST pages are just mistakes.
I have just come to understand what the argument is, myself.  It goes 
like this (much of this repeating things I'm sure we all know and agree on):

* Counting is a dimensionless activity.  Like all dimensionless 
quantities, units are permitted (e.g., dozen, gross, Avogadro's number), 
but there is one special unit which is equivalent to "one" and which may 
be omitted.  For counting, that unit could be the name of whatever you 
are counting ("8 widgets"), or more generically the word "unit" 
(awkwardly enough).

* If regularly spaced widgets are rolling off a conveyor belt, and you 
/count/ how many come out in each second, then an appropriate unit is 
"widgets/s" = Hz = 1/s.

* If instead a square wave is rolling off the conveyor belt, then the 
situation is directly analogous, where the "objects" being counted are 
now repetitions of the waveform shape, called cycles.
"cycles/s" = Hz = 1/s = widgets/s

* Now it is a very small jump to sinusoidal waves.  Thus we have two 
different concepts (counting frequency and angular frequency) which can 
be measured for the sinusoidal wave.  They share the same dimensions and 
units, but they are distinct. Just as torque and energy share the same 
dimensions and units, but are distinct.

* I believe there are a fair number of physicists who would additionally 
say that expressing counting frequency in 1/s (instead of Hz) is bad 
form, just as expressing torque in J (instead of Nm) is bad form.  Some 
might even say that those are "wrong."

If we push my "modest proposal" to an extreme, to entirely get rid of 
Hz=1/s, then I guess the base unit for counting objects would become the 
radian.  5 widgets = 31.415 radians.  I have to admit, that's absurd. 
But I'm not wholly satisfied the the "two distinct units" solution either.

Cheers,
-- James