Dimensional, or unit, analysis is a useful tool for finding relations between variables that describe a physical system. Although it has applications across all fields of physics, it is not a regular part of a typical undergraduate physics curriculum. Yes, students are instructed to include units in their homework answers and lab reports, and some students may remember that comparing the units on two sides of an equation can help determine errors in their solutions. But they are rarely taught how to use dimensional analysis to solve a problem.
The hydraulic jump of this drinking fountain in Venice, Italy, is one of the many values dimensional analysis can help calculate.
The hydraulic jump of this drinking fountain in Venice, Italy, is one of the many values dimensional analysis can help calculate.
A Student’s Guide to Dimensional Analysis by Don Lemons addresses that gap. An emeritus professor of physics at Bethel College in Kansas, Lemons has published books on variational principles, thermodynamics, entropy, and stochastic processes, all aimed at physics undergraduates. His latest book is written in a casual style, as if he were talking to his students and giving them step-by-step guidance. Lemons shares his personal experiences applying dimensional analysis to problems. For instance, he discusses the hydraulic jump, a phenomenon one can see in a kitchen sink. He then introduces the Reynolds number, a quantity indicating whether viscosity or inertia is dominant in a fluid, and reports the corresponding data found for his sink and faucet.
Later in the book, Lemons describes how the chimneys in an old building he once worked in encouraged natural air flow and kept the upper floors from becoming too hot in the summer. He explains that the air flow is due to air buoyancy caused by the temperature difference between the indoors and the outdoors, a transport phenomenon called the stack effect. He then shows the reader how to use dimensional analysis to calculate the air flow’s speed. Such anecdotes make dimensional analysis more accessible and less intimidating.
A Student’s Guide has seven chapters, each of which concludes with a list called “Essential Ideas” and several exercises. The book also has references and an appendix with answers to the end-of-chapter problems. Those resources will be helpful for students who want to learn the topic on their own.
The first chapter introduces basic concepts and theorems of dimensional analysis. The other six chapters present its applications to problems in different fields of physics: classical mechanics, hydrodynamics, thermal physics, electrodynamics, plasma physics, quantum physics, and cosmology.
The book’s examples demonstrate the potential power of dimensional analysis. For instance, students can find the oscillation frequency of a spring-and-mass system by analyzing the dimensions of physical variables and constants, without drawing force diagrams or writing equations of motion. Most examples discuss concepts that are covered in introductory physics courses, such as classical mechanics, heat, and electricity and magnetism. For more advanced topics, such as hydrodynamics and cosmology, Lemons helpfully explains the physics involved before showing how a given problem can be tackled with dimensional analysis.
As with any problem-solving skill, practice makes perfect. Inexperienced beginners can easily make mistakes and feel like they are not making progress. In the first chapter, the author points out a common error: forgetting to minimize the number of effective dimensions—that is, the number of dimensions or combinations of dimensions needed to express all the problem’s variables and constants. That gentle warning reminds the reader to approach the examples and problems in following chapters with care and attention to detail.
Although it is a useful and powerful tool, dimensional analysis has its limitations. One is that a result obtained with it necessarily includes an undetermined constant or function. Sometimes a relation derived via dimensional analysis seems quite complex. Lemons does not avoid addressing such limitations, but at the same time, he illustrates the ways in which dimensional analysis can help readers gain insight and direct them toward a complete solution.
Unlike many other books on the topic, A Student’s Guide to Dimensional Analysis does not include a thorough study of general theorems or detailed mathematical derivations. Mathematically inclined readers may want to choose another resource. However, Lemons’s book is a well-written entry-level text that will be of value to curious undergraduates in physics and engineering.
Hong Lin is a professor of physics at Bates College in Lewiston, Maine. She teaches introductory to upper-level physics courses, including modern physics, classical mechanics, electricity and magnetism, quantum mechanics, and mathematical methods of physics. Her research focuses on the nonlinear dynamics of optical systems and their applications.
