When it comes to "dimensional" arguments and scaling arguments,
there seem to be several Piagetian stages
A) There comes a time when students figure out about units.
They can use the factor-label method to get the units come
out right.
B) There comes a time when students figure out about dimensions.
They realize that dimensions are not the same as units. There
exist dimensionless units. http://www.av8n.com/physics/dimensionless-units.htm
There are important results that can best be understood in terms
of dimensions, including some basic scaling laws: area scales
like the square of length, and volume scales like the cube of
length (independent of what units are used).
Many of the false positives are associated with non-dimensional
scaling situations.
Amusing pedagogical examples of non-dimensional scaling include:
-- The distance to the apparent horizon, which scales like the
square root of eye-height (subject to mild restrictions).
-- The mean free path as a function of molar volume and scattering
cross-section.
-- Chemical equilibrium density (or concentration) as a function
of system volume; in this case you can have three quantities all
with the same dimensions (and same units) such that one scales
like system volume to the -1 power, one scales like system volume
to the -½ power, and one scales like system volume to the 0 power.
This is a perfectly reasonable scaling law, but you could never
figure it out using dimensional analysis alone.
Here's the key idea: Suppose the answer contains a physically-relevant
dimensionless group raised to some power (x). Dimensional analysis
may help you identify such a group as being dimensionless ... but it
definitely will not tell you the exponent (x).
In such cases dimensional analysis won't give you the things you
need to know; instead it is more useful in helping you identify
the things you don't know, namely the exponents.
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Dimensional analysis is a classic example of how a little knowledge
can be a dangerous thing.
-- I've seen people who have a superficial knowledge of dimensional
analysis get into all sorts of trouble. They get bit by false
negatives and false positives, and go around and around in circles,
unable to figure out what the problem is.
-- The goal is to know how to do proper scaling, including non-
dimensional scaling. I'm not saying students can get to this
goal in one step ... but we should keep the goal clearly in
mind.