[Phys-l] units, dimensions, scaling

[John Denker <jsd@av8n.com>]

On 01/18/2010 02:16 PM, Jeff Loats wrote:

> On the first day of class I do a brief example to illustrate unit conversion
> (snore) and I usually spice it up ....

On a completely serious note, here are some points
that you might want to think about in connection with 
units.  There is a fairly natural segue from units
to dimensions, and from dimensions to scaling.

1) Units are *not* the same as dimensions.  The 
 existence of dimensionless units such as "degrees 
 of arc" should suffice to prove this point.
    http://www.av8n.com/physics/dimensionless-units.htm

 Units and dimensions are not entirely the same,
 but they're not entirely unrelated, either.  If
 you know the units you know the dimensions (but
 not conversely).

 A lot of students have misconceptions about this.

2) Sometimes dimensional analysis is presented as
 if it were a law of nature ... which it is not.

 Dimensional analysis should be considered a 
 heuristic for guessing the scaling behavior.
 Like all heuristics,
 -- In skilled hands, it is a way of getting
  the right answer quickly.
 -- In unskilled hands, it is a way of getting
  the wrong answer quickly.

===========

I mention this because scaling laws are tremendously
important.  Given any discussion of units, I would
be unable to resist the temptation to segue from
units to dimensions, and then from dimensions to
scaling.

Scaling laws have been central to physics since 
Day One of the modern era (1638) and remain so even 
now.  IMHO, they are grievously underemphasized in 
the typical curriculum.  They are IMHO more useful 
*and* easier and more age-appropriate than most of 
the stuff that is in the curriculum.

For more on all this, see
  http://www.av8n.com/physics/dimensional-analysis.htm
and especially
  http://www.av8n.com/physics/scaling.htm