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.

## REFERENCES

*Fluid Mechanics*(Academic, New York, 2001);

*Fundamentals of Fluid Mechanics*(Wiley, New York, 1998), 3rd ed.;

*Basic Fluid Mechanics*(DCW Industries, La Canada, CA, 2000);

*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).

*Mathematics Applied to Deterministic Problems in the Natural Sciences*(MacMillan, New York, 1974).

*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).

*g*, was omitted, that is, what phenomenon would that entail? What if

*g*were omitted, but an initial angular velocity $d\phi /dt$ was included? What if in place of

*g*we used the acceleration due to the earth’s rotation, $\Omega 2R?$ $(\Omega \u2250m0l0t\u22121$ is the rotation rate of the earth and

*R*is the distance normal to the rotation axis.)

*Applied Dimensional Analysis and Modeling*(McGraw–Hill, Englewood Cliffs, NJ, 1997).

*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.

*g*), $\alpha 2=0.3048$ (feet to meters), and $\alpha 3=1$ (seconds to seconds). Thus $X5\u2032=0.3048X5.$

*Introduction to Linear Algebra*(Wellesley–Cambridge Press, Wellesley, MA, 1998).

*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 $\phi 0=\pi .$ Is dimensional analysis of any further use for explaining this? Consider energy conservation.

*A*is the frontal area of the object. For the purpose of this essay we will consider other possible forms for $\Pi 1.$

*Life in Moving Fluids*(Princeton U.P., New York, 1994), and

*American Journal of Physics*and

*The Physics Teacher*as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.