Dimensional Analysis
نویسنده
چکیده
The first step in modeling any physical phenomena is 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 these 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 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/s). 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 arbitrariness in the choice of the basic units of measurement. In other words, Newton’s second law, F = ma, 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:
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تاریخ انتشار 2003