Fifteen Eighty Four

Notes on Dimensional Analysis

Dimensional analysis has the dubious reputation of being both utterly trivial and, at the same time, ineffective.  Although both claims are understandable neither is well founded.  One can, indeed, learn the basic technique of dimensional analysis, what A Student’s Guide to Dimensional Analysis calls the “Rayleigh algorithm,” in ten minutes.  But its fruitful application requires experience and it is that experience that one who studies this text will cultivate.

Percy Bridgman, Nobel laureate and the author of the first textbook on the subject, called dimensional analysis “an analysis of an analysis.”  Bridgman’s phrase, at least, means that the Rayleigh algorithm is, by itself, not enough.  One must also have a prior analysis to which that algorithm is applied.  One cannot expect the algorithm to unlock the secrets of a natural phenomenon about which one knows nothing.  The Rayleigh algorithm is not magical.

A careless construction of the dimensional model can lead to a failed dimensional analysis.

According to A Student’s Guide to Dimensional Analysis a necessary first step is to construct what the text calls a “dimensional model.”  Such consists of determining which dimensional variables and constants enter into the phenomena — a task not often trivial or obvious.  A careless construction of the dimensional model can lead to a failed dimensional analysis.

There is another reason why dimensional analyses sometimes fail: expressing the dimensional variables and constants in terms of the wrong dimensions.  “What,” one may ask, “are not the right dimensions those of mass, length, time, and charge?”  The answer one discovers is “Not always.”  For in order to satisfy the pre-conditions of the Buckingham Pi Theorem (the fundamental theorem of dimensional analysis) one must choose the minimum set of dimensions in terms of which the dimensional variables and constants can be expressed.  Both the number and these identity of these minimum or “effective dimensions” may not be those of mass, length, time, and charge.

an intellectual activity that is not only challenging and effective but also delightful.

These effective dimensions must also be physically appropriate and recognized as such a priori, that is, before applying the Rayleigh algorithm.  One of my favorite examples is addressed in the worked problem “Cooking a Turkey.”  The physics of cooking involves the diffusion of heat into the cooked object just as if heat were a conserved fluid.  Never mind that the first law of thermodynamics allows us to identify a unit of heat with a unit of energy, whose dimensions are a combination of mass, length, and time.  In its diffusion heat does not produce work nor does work produce heat.  The first law of thermodynamics is never invoked nor is it needed.  For this reason heat has its own unique, effective dimension – a dimension imposed by the physics of the model.  Without embedding this physics in the dimensional model its analysis misleads.

A Student’s Guide to Dimensional Analysis emphasizes issues such as these.  Thus, the text presents dimensional analysis as an intellectual and not only an algorithmic activity – an intellectual activity that is not only challenging and effective but also delightful.

For a complete background to Dimensional Analysis including an introduction to the Raleigh algorithm please see the preface and introduction within our free sample chapter of A Student’s Guide to Dimensional Analysis