James F. Pricea)

Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543

Received 22 May 2002; accepted 4 November 2002 Dimensional analysis is a widely applicable and sometimes very powerful technique that is demonstrated here in a study of the simple, viscous pendulum. The ﬁrst and crucial step of dimensional analysis is to deﬁne a suitably idealized representation of a phenomenon by listing the relevant variables, called the physical model. The second step is to learn the consequences of the physical model and the general principle that complete equations are independent of the choice of units. The calculation that follows yields a basis set of nondimensional variables. The ﬁnal step is to interpret the nondimensional basis set in the light of observations or existing theory, and if necessary to modify the basis set to maximize its utility. One strategy is to nondimensionalize the dependent variable by a scaling estimate. The remaining nondimensional variables can then be formed in ways that deﬁne aspect ratios or that correspond to the ratio of terms in a governing equation. © 2003 American Association of Physics Teachers. DOI: 10.1119/1.1533057

I. ABOUT DIMENSIONAL ANALYSIS Dimensional analysis is a remarkable tool insofar as it can be applied to virtually all quantitative models and data sets. Topics in the recent literature include donuts, dinosaurs,1 and the most fundamental theories of physics.2 In some instances dimensional analysis is very powerful; results include the log-layer proﬁle of a turbulent boundary layer and the spectral slope in the inertial subrange of isotropic turbulence, both landmarks in ﬂuid mechanics.3 More often the result of dimensional analysis is a hint at the form of a solution or a more effective way to display or correlate a large data set. These kinds of results, though seldom complete if taken alone, are an essential element of many investigations. This paper is an introduction to dimensional analysis that aims to make the method and the results as accessible as possible.3,4 The plan is to investigate the motion of a simple pendulum while emphasizing the use of dimensional analysis as an adjunct to experimental, numerical, and theoretical methods.5 If the simple pendulum seems too familiar, skip ahead to Sec. III. If the use of nondimensional variables is also familiar, skip ahead to Sec. IV, where a general method of computing a basis set of nondimensional variables is presented. The effects of drag are considered in Sec. V, and concluding remarks are in Sec. VI. II. MODELS OF A SIMPLE PENDULUM Consider a pendulum that can be made and observed with simple tools; a small lead ﬁshing sinker having a mass of a few tens of grams suspended on a thin monoﬁlament line a few meters in length. The motion of such a pendulum will be only lightly damped by drag with the surrounding air and can be characterized by two distinct time scales—a regular, fast time-scale oscillation having a period, P, and a slow, more1 . Our goal or-less exponential decay with a time-scale, will be to learn how these time scales and some other variables, for example, tension in the line, vary with line length and the mass of the bob. A. A physical model To analyze the motion of the pendulum, we begin by listing the variables that are presumed to be relevant to the 437 Am. J. Phys. 71 5 , May 2003 http://ojps.aip.org/ajp/

aspect of the motion that is of interest. To start, consider the fast time-scale, oscillatory motion. The line will be idealized as rigid, so that the bob must swing along a constant radius. The motion of the bob is then deﬁned by the angle of the line to the vertical, (t), and its time derivatives; the angle is the dependent variable of this physical model and the time, t, is the only independent variable. Several properties of the pendulum would seem to be important—the mass of the bob, M, the length of the supporting line, L, and...