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# Dimensional Analysis

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.

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Notes by A. Dorsey, edited by V. Celli, Univ. of Virginia
Thu Jul 10 16:27:59 EDT 1997