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

[John Denker <jsd@av8n.com>]

On 02/02/2010 07:45 AM, Edmiston, Mike wrote:

> ... the usual problem with dimensionless units...
> when students are making sure "the units come right" the units in
> fact won't "come out right" because there is nothing to cancel the
> dimensionless unit that we explicitly wrote into the paperwork.
> Students have difficulty dealing with this, and I typically don't
> have a good solution other than telling them just to learn to live
> with it.

Agreed: they just need to learn to live with it.
In particular, they need to learn when to worry
about "left over" units, and when when not to.

Here's an example to prove that there will never 
be an easy "routine" solution, indeed no solution 
short of understanding the physics:
  Sometimes kg m^2 s^-2 is a unit of torque
  Sometimes kg m^2 s^-2 is a unit of energy

The interesting wrinkle is that a torque is also 
an energy per radian ... but my point is that we
should not restate the unit of torque as 
  kg m^2 s^-2 per radian
or anything like that ... even though from one
point of view it might be tempting to try that,
so that torque times (angle in radians) equals
energy.

Here's one counterargument: Torque is quite nicely
defined as r /\ F.  There is no need to mention
angles, or even mention motion, let alone circular 
motion, let alone uniform circular motion.

If you want to expand the wedge product in terms
if sines and if you want to expand the sine in
terms of the angle and if you measure the angle
in radians, that's OK ... but that is at least
three "if"s removed from requiring any changes
to the units of measurement.

As Jeffrey S. explained, a radian is a cubit of
arc divided by a cubit of radius, and the cubits
drop out.  The result is dimensionless.  The
units of the result -- radians -- are important,
but you have to stick them in by hand;  they
do not emerge and cannot emerge as a mechanical
consequence of the calculation.  In this example
they appeared out of thin air, and they can
disappear the same way.

Unit analysis is good as a check when you are
doing _analysis_ of an existing calculation,
but it is never more than a heuristic when used
to guide the _synthesis_ of a not-yet-finished 
calculation.

Sometimes you just need to understand the physics.

Similar words apply to dimensional analysis, which
is related to unit analysis but is not quite the 
same thing.  

To drive home the point that there is more to physics
than dimensional analysis, point out that there is 
such a thing as _non-dimensional_ scaling. There 
are lots of problems where non-experts might expect 
a trivial scaling result, but instead you get an 
interesting, completely nontrivial non-dimensional 
scaling result.
  http://www.av8n.com/physics/scaling.htm#sec-non-dimensional

Dimensional analysis is a tool.  Like any tool
it can be abused.  It allows experts to get the
right answer quickly.  It allows non-experts to
get the wrong answer quickly.