Dimensional analysis: an elegant technique for facilitating the teaching of mathematical modelling

Abstract

Dimension analysis is promoted as a technique that promotes better understanding of the role of units and dimensions in mathematical modelling problems. The authors' student base consists of undergraduate students from the Science and Engineering Faculties who generally have one or two semesters of calculus and some linear algebra as part of their curriculum. Because of ‘In Service Training’ which is an integral part of their education, they have a reasonable understanding of the link between theory and practice in their particular industry, but manipulating mathematical formulae is not necessarily a strong point. Dimensional analysis involves both dimensionless products and linear algebra and, because of the latter, this branch of mathematical modelling was, until recently, beyond the reach of most undergraduates. However, it has been found that the skills of a good technologist can be blended with the use of computer algebra systems to successfully teach dimensional analysis to these undergraduates. This note illustrates the concept of dimensional analysis by examining the simple pendulum problem and shows how dimensionless products can lead to the discovery of the connection between the period of the pendulum swing and its length. Dimensional analysis is shown to lead to interesting systems of linear equations to solve, and can point the way to more quantitative analysis, and two student problems are discussed. It is the authors' experience that dimensional analysis broadens a student's viewpoint to include units and dimensions as an integral part of any physical problem. With this approach coupled with a computer algebra systems such as DERIVE, students can concentrate on understanding the model and the modelling process rather than the solution technique. Finally, it has been observed that students find dimensional analysis fun to do.