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Re: [Phys-L] dimensional analysis



On 07/25/2012 03:37 PM, Larry Smith wrote:
Seems to me that the term "dimensional analysis" is used in two
different ways. Some sources use it to mean simply converting from
one kind of units to another (how many square centimeters are there
in an acre?); this usage of the term is also called the factor label
method. Others use it to mean comparing the dimensions of the
physical quantities to find relationships between the quantities.

Do the two uses fall along disciplinary lines (chemistry vs.
physics)? Looking at many hits from a web search led me to that
tentative conclusion.

What do you mean when you say "dimensional analysis"?

Let's start with some basic facts:

a) Units are not the same thing as dimensions. This should be
obvious as soon as somebody mentions dimensionless units, such
as radians. Measuring an angle in radians is different from
measuring an angle in cycles or in degrees, different in some
very important ways (but not in all ways).

b) Obviously things that have different dimensions have different
units, but the converse does not hold.

To say the same thing the other way: Knowing the units tells you
the dimensions, but the converse does not hold.

c) Dimensional analysis is just one corner of a larger picture, namely
/scaling/. The rest of the picture is called nondimensional scaling.
That is, there are important laws that say how one dimensionless
quantity scales relative to another. No amount of dimensional analysis
will derive such laws for you.

Dimensional analysis is a heuristic for coming up with scaling laws.
It is not entirely reliable.

So, to return to the question that was asked: There is no doubt that
there is a food chain:
_Units_ tell you the _dimensions_ which might suggest a _scaling law_

However, there are lots of ways of knowing the dimensions with out fussing
with the units, and there are lots of ways of knowing the scaling laws
without fussing with the dimensions.

In increasing order of importance:
http://www.av8n.com/physics/dimensionless-units.htm
http://www.av8n.com/physics/dimensional-analysis.htm
http://www.av8n.com/physics/scaling.htm


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

As for the chemistry versus physics angle, my advice is don't go there.
I don't think there is any such angle, and even if there were, there
would be no advantage to pursuing it. My advice is to teach good science
based on good scientific explanations. In this case, as in most cases,
no sociological explanations are needed.

========

As a very minor point: "factor label" is not quite synonymous with unit
analysis. It is /one method/ -- not the only method -- for munging the
units.

By way of analogy, Gaussian elimination is one method -- not the only
method -- for solving systems of linear equations. There are other
methods such as Cramer's rule. For *most* applications Gaussian
elimination is treeeemendously better, although there are some niche
applications for Cramer's rule.

For /most/ applications, factor-label is so very powerful and convenient
that it might be hard for non-experts to imagine anything else, but
there are niche applications (such as computer programs that keep track
of units) where other approaches make sense.