The first steps in modeling any physical phenomena are the identification of
the relevant variables, and then relating these variables via known physical
laws. For sufficiently simple phenomena we can usually construct a
quantitative relationship among physical variables from first principles;
however, for many complex phenomena (which often occur in engineering
applications) such an ab initio theory is often difficult, if not
impossible. In these situations modeling methods are indispensable, and one
of the most powerful modeling methods is dimensional analysis.
You have probably encountered dimensional analysis
dimensional analysis. You have probably encountered dimensional analysis
in your previous physics courses when you were admonished to ``check your
units'' to ensure that the left and right hand sides of an equation had the
same units (so that your calculation of a force had the units of kg 
m 
). In a sense, this is all there is to dimensional analysis,
although ``checking units'' is certainly the most trivial example of
dimensional analysis (incidentally, if you aren't in the habit of checking
units, do it!). Here we will use dimensional analysis to actually solve
problems, or at least infer some information about the solution. Much of
this material is taken from Refs. [1] and [2];
Ref. [3] provides many interesting applications of dimensional
analysis and scaling to biological systems (the science of allometry.
The basic idea is the following: physical laws do not depend upon the
choice of the basic units of measurement. In other words, Newton's second
law, 
, is true whether we choose to measure mass in
kilograms, acceleration in meters per second squared, and force in newtons,
or whether we measure mass in slugs, acceleration in feet per second
squared, and force in pounds. As a concrete example, consider the angular
frequency of small oscillations of a point pendulum of length l and mass m
:
where g is the acceleration due to gravity, which is 
on earth (in the SI system of units; see below). To derive Eq. (1.1), one usually needs to solve the differential equation which results
from applying Newton's second law to the pendulum (do it!). Let's instead
deduce (1.1) from dimensional considerations alone. What can 
depend upon? It is reasonable to assume that the relevant variables are m,
l, and g (it is hard to imagine others, at least for a point pendulum).
Now suppose that we change the system of units so that the unit of mass is
changed by a factor of M, the unit of length is changed by a factor of L, and the unit of time is changed by a factor of T. With this change of
units, the units of frequency will change by a factor of 
, the units
of velocity will change by a factor of 
, and the units of
acceleration by a factor of 
. Therefore, the units of the quantity g/l will change by 
, and those of 
will change by
. Consequently, the ratio
is invariant under a change of units; 
is called a dimensionless
number.
Since it doesn't depend upon the
variables (m,g,l), it is in fact a constant. Therefore, from dimensional
considerations alone we find that
A few comments are in order: (1) the frequency is independent of the mass of the pendulum bob, a somewhat surprising conclusion to the uninitiated; (2) the constant cannot be determined from dimensional analysis alone. These results are typical of dimensional analysis--uncovering often unexpected relations among the variables, while at the same time failing to pin down numerical constants. Indeed, to fix the numerical constants we need a real theory of the phenomena in question, which goes beyond dimensional considerations.
