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.

1.
E. Thurairajasingam, E. Shayan, and S. Masood, “Modeling of a continuous food pressing process by dimensional analysis,” Comput. Ind. Eng. (in press);
J. R.
Hutchinson
and
M.
Garcia
, “
Tyrannosaurus was not a fast runner
,”
Nature (London)
415
,
1018
1022
(
2002
).
2.
F.
Wilczek
, “
Getting its from bits
,”
Nature (London)
397
,
303
306
(
1999
).
3.
This essay builds upon the introduction to dimensional analysis that can be found in most comprehensive fluid mechanics textbooks. Recent examples include P. K. Kundu and I. C. Cohen, Fluid Mechanics (Academic, New York, 2001);
B. R. Munson, D. F. Young, and T. H. Okiishi, Fundamentals of Fluid Mechanics (Wiley, New York, 1998), 3rd ed.;
D. C. Wilcox, Basic Fluid Mechanics (DCW Industries, La Canada, CA, 2000);
F. M. White, Fluid Mechanics (McGraw–Hill, New York, 1994), 3rd ed. An older but very useful reference is by H. Rouse, Elementary Mechanics of Fluids (Dover, New York, 1946).
A particularly good discussion of the relationship between dimensional analysis and other analysis methods is by C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (MacMillan, New York, 1974).
4.
An algorithm for computing nondimensional variables has been implemented in MATLAB and can be downloaded from the author’s web page, 〈http://www.whoi.edu/science/PO/people/jprice/misc/Danalysis.m〉 or from the MATLAB archive (the file name is Danalysis.m). Also available is a manuscript that treats aspects of dimensional analysis that are not touched on here, 〈http://www.whoi.edu/science/PO/people/jprice/class/ND.pdf〉.
5.
The simple pendulum is the starting point for most discussion of dimensional analysis including the classic text by P. W. Bridgman, Dimensional Analysis (Yale U.P., New Haven, CT, 1937), 2nd ed., which is an excellent introduction to the topic, and the more advanced treatment by L. I. Sedov, Similarity and Dimensional Methods in Mechanics (Academic, New York, 1959). Still more advanced is G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics (Cambridge U.P., Cambridge, 1996).
6.
An excellent discussion of physical measurement and much else that is relevant to dimensional analysis is given by A. A. Sonin, “The physical basis of dimensional analysis,” 2001. This unpublished manuscript is available from 〈http://me.mit.edu/people/sonin/html〉.
7.
What would be the result if the acceleration of gravity, g, was omitted, that is, what phenomenon would that entail? What if g were omitted, but an initial angular velocity dφ/dt was included? What if in place of g we used the acceleration due to the earth’s rotation, Ω2R?(Ω≐m0l0t−1 is the rotation rate of the earth and R is the distance normal to the rotation axis.)
8.
T. Szirtes, Applied Dimensional Analysis and Modeling (McGraw–Hill, Englewood Cliffs, NJ, 1997).
9.
S. Brückner and the University of Stuttgart Pi-Group 〈http://www.pigroup.de/〉, is an excellent resource for advanced applications of dimensional analysis.
10.
The calculation of a null space basis is, in effect, what all computational methods accomplish, and was noted by E. A. Bender, An Introduction to Mathematical Modelling (Dover, New York, 1977). We delegate the calculation to the computer, and emphasize those properties of the null space basis that are essential for the present purpose.
11.
For example, suppose that X5 is a speed in British engineering units, feet/second, and we wish to compute X5 in SI units, meters/second. This variable has dimensionality, D15=0(X5 does not have units of mass), D25=1 for length, and D35=−1 for time. The appropriate scale change factors are α1=0.435 (pounds to kilograms for nominal g), α2=0.3048 (feet to meters), and α3=1 (seconds to seconds). Thus X5=0.3048X5.
12.
G. Strang, Introduction to Linear Algebra (Wellesley–Cambridge Press, Wellesley, MA, 1998).
13.
There is no doubt that dimensional analysis has just added something significant to what we knew from numerical integrations (that is, the maximum tension is independent of L). Does this result from dimensional analysis constitute a satisfactory explanation? This is clearly a matter of degree and opinion, but my opinion is that it does not. On the one hand, dimensional analysis has deduced a very clear statement of the observation from a general principle (invariance to the choice of units) and a set of specific conditions (the physical model). This is a form of explanation, but one that seems shallow and unsatisfying; there is no connection to a physical principle, and not the slightest hint of quantitative limits. In this instance and frequently, we will have to look beyond the immediate problem at hand or use something more than dimensional analysis when we seek explanations with enough depth to confer a useful understanding. Consider the following: The period of a simple (inviscid) pendulum undergoing small amplitude motion is independent of the amplitude, and yet increases with the square root of the length. Can you explain these facts? One approach might be to use dimensional analysis to analyze oscillators that have a restoring force that is proportional to some arbitrary power of the displacement. A salient fact for the maximum tension shown in Fig. 3(b) is that the maximum value is exactly 5 (nondimensional units) and occurs at φ0=π. Is dimensional analysis of any further use for explaining this? Consider energy conservation.
14.
Detailed treatment of damping processes are by
P. T.
Squire
, “
Pendulum damping
,”
Am. J. Phys.
54
,
984
991
(
1986
) and
R. A.
Nelson
and
M. G.
Olsson
, “
The pendulum: Rich physics from a simple system
,”
Am. J. Phys.
54
,
112
121
(
1985
).
15.
One criterion is to follow conventions of the field. In this case Π1 is a drag coefficient, usually defined as Cd=H/12ρAU2, where A is the frontal area of the object. For the purpose of this essay we will consider other possible forms for Π1.
16.
More recent textbooks (Ref. 3), like this article, show only the curve that runs through the middle of a tight cloud of data points that have accumulated from many laboratory experiments, see for example, Rouse (Ref. 3). What is most important, but not evident from this kind of presentation, is that drag coefficients inferred from experiments made using a very wide range of spheres and cylinders moving at widely differing speeds and through many different viscous fluids (Newtonian fluids) do indeed collapse to a well-defined function of Reynolds number alone, just as dimensional analysis had indicated. This is a result, characteristic of dimensional analysis generally, that is at once profound and trivial. One might say trivial because, after all, dimensional analysis told us that the drag coefficient must depend upon Re alone. From this perspective, an effective collapse of the data verifies that carefully controlled laboratory conditions can indeed approximate the idealized physical model. It is profound in that dimensional analysis has shown the way to a useful result (Fig. 5), where there would otherwise have been be an unwieldy mass of highly specific data (as in going from Fig. 1 to Fig. 2). An open question of considerable practical importance is whether the steady drag laws are robust in the sense of giving useful estimates in practical problems, say our pendulum, in which the idealized conditions are inevitably violated. Other data sets have been developed to define the effects of idealized surface roughness, for example, but our pendulum has a long list of violations—time-dependence, a nearby solid boundary (the floor), slight surface roughness, etc., all present at once, so that we are on our own. About all that can be said is that it is important to understand the full set of assumptions under which a similarity law has been defined, and to be skeptical of applications outside of those bounds.
17.
Even at very large Re it does not follow that viscosity is entirely irrelevant. Significant changes in the drag coefficient occur at around Re≈2×105 due to changes in the viscous boundary layer and the width of the wake behind a moving sphere. This is the Re range of a well-hit golf ball or tennis ball, and is part of the reason that aerodynamic drag on these objects has a surprising sensitivity to surface roughness or spin. For much more detail on these phenomenon see S. Vogel, Life in Moving Fluids (Princeton U.P., New York, 1994), and
P.
Timmerman
and
J. P.
van der Weele
, “
On the rise and fall of a ball with linear and quadratic drag
,”
Am. J. Phys.
67
,
538
546
(
1999
).
18.
Can you calculate a Reynolds number for the bob and the line from the original six nondimensional variables of Eq. (39)? Which nondimensional variable is present in Eq. (39) but not in Eqs. (57) and (58)? How or why was it omitted? Under what conditions (what parameter range) would you expect to see a significant effect of the time-dependent motion? How could you test (in principle and in practice) that the steady drag formulations really are appropriate for modeling the damping of a simple pendulum? You might, for example, consider that the fluid medium was water in place of air (the approximate density and kinematic viscosity of water are ρ=1.0×103kg m−3 and ν=1.8×10−6m2/s at a temperature=0°C, and ρ=1.0×103kg m−3 and ν=0.7×10−6m2/s at a temperature=40°C). Given these results, can you think of a name more apt than “viscous” pendulum?
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