Don S. Lemons - A Student’s Guide to Dimensional Analysis

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10.1017/9781316676165

Don S.Lemons 2017

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I was charmed when as a young student I watched one of my physics professors, the late Harold Daw, work a problem with dimensional analysis. The result appeared as if by magic without the effort of constructing a model, solving a differential equation, or applying boundary conditions. But the inspiration of the moment did not, until many years later, bear fruit. In the meantime my acquaintance with this important tool remained partial and superficial. Dimensional analysis seemed to promise more than it could deliver.

Dimensional analysis has charmed and disappointed others as well. Yet there is no doubt that a deep understanding of its methods is useful to researchers in a number of fields. Attesting to this fact is that dimensional analysis is the subject of several good, graduate-level monographs. Even so, dimensional analysis is often ignored at the introductory level except when teachers admonish their students to check the units of their result and warn them against adding apples and oranges.

The problem for teachers and students is that dimensional analysis has no settled place in the physics curriculum. The mathematics required for its application is quite elementary of the kind one learns in a good high school course and its foundational principle is essentially a more precise version of the rule against adding apples and oranges. Yet the successful application of dimensional analysis requires physical intuition an intuition that develops only slowly with the experience of modeling and manipulating physical variables. But how much intuition is required?

I have written A Students Guide to Dimensional Analysis in the belief that the simple techniques of dimensional analysis can deepen our understanding and enhance our exploration of physical situations and processes at the introductory level. Thus, this text is designed for students who are taking or have taken an entry-level, mathematically oriented, university physics course. More experienced students and professionals may also find this text useful.

One elementary application of dimensional analysis is to a simple pendulum, that is, to a compact weight swinging from the end of a string. How long does the pendulum take to complete one cycle of its motion? It seems that this interval t might depend on the mass of the weight m , the acceleration of gravity g , the length of the string l , and the maximum angle of its swing . The adjective simple in simple pendulum means that the size of the weight is negligibly small compared to the length of the pendulum l . Therefore, we search for a function of m , l , g , and that produces t . We note that the International System or SI unit of the period t is the second, the SI unit of mass m is the kilogram, of length l is the meter, and of acceleration g is a meter divided by a second squared. The angle is dimensionless. We quickly discover that is the only combination of m , l , and g that produces a quantity whose SI unit is a second. Therefore, it must be that

where f () is an as yet undetermined, dimensionless function of .

This result is typical of dimensional analysis. Much is learned but something is left unlearned. The analysis teaches us that the period t of a simple pendulum is directly proportional to the square root of its length and does not at all depend on its mass m . We could experimentally verify these findings, but dimensional analysis suggests that limited time might be better spent exploring the undetermined function f () . However, this example misleads as well as informs. Not many applications of dimensional analysis are this simple, nor are we usually aware, as we probably are in this case, of the result beforehand.

Dimensional analysis is most revealing when we know something but not everything about a situation or process. Then dimensional analysis builds on that something we already know. We might, for instance, be familiar with the equations that describe a certain process but not have the skill or the time to solve them in the usual way. Or we might wish to expand our preliminary knowledge of a solution before completely determining its form. Then again, we might know only the category of the problem (that is, mechanics, thermodynamics, or electrodynamics) and the variable we want to determine (that is, an oscillation period, a pressure drop, or an energy loss rate). In all these cases, dimensional analysis constrains how the relevant physical variables and constants work together to produce the result sought.