**Preface**

**My Goals For This Book**

Science and engineering students depend heavily on concepts of math-
ematical modeling. In an age where almost everything is done on a
*computer, it is my conviction that students of engineering and science are*
*better served if they understand and “own” the underlying mathematics*
*that the computers are doing on their behalf. Mathematics is a necessary*
language for doing engineering and science. This will remain true no mat-
ter how good computation becomes. I repeatedly tell students that it is
risky to accept computer calculations without having done some parallel
*closed-form modeling to benchmark the computer results. Without such*
*benchmarking and validation, how do we know that the computer isn’t talking*
*nonsense? Finally, I find it satisfying and fun to do mathematical manipula-*
tions that explain how or why something happens, and to use mathematics
to obtain corresponding numerical data or predictions.

*Thus, as it was for the first edition, my primary goal for this second edition*
*remains to engage the reader in developing a foundation for mathematical*
*modeling. Further, knowing that mathematical models are built in a range*
of disciplines—including physics, biology, ecology, economics, sociology,
military strategy, as well as all of the many branches of engineering—and
knowing that mathematical modeling is comprised of a very diverse set of
*skills and tools, I focused on techniques of particular interest to engineers,*
*scientists, and others who model continuous systems.*

**xiii**

**Features of This Edition**

Aided by a variety of reviewers’ comments and suggestions, this second edition features:

• A more formal statement of a principled approach to mathematical modeling (in Chapter 1). Ten principles are articulated and invoked as applications are developed, and each of them is identified by a key word (see below).

• Some 360 problems, many of which are designed to reinforce skills in mathematical manipulation. Many could be done with a computer algebra system (CAS), and there are others for which numerical pro- grams could be used. However, given my goals for this book, I would ask students do the problems in “the old-fashioned way.”

• A reordering and expansion of the applications chapters that reflects some sense of increasing complexity.

• Expanded figure captions that are intended to be more informative.

**How This Book Is Organized**

The book is organized into two parts: foundations and applications. The first part lays out the fundamental mathematical ideas of interest to the model builder: dimensional analysis, scaling, and elementary approxima- tions of curves and functions. The applications part of the book develops a series of models and discusses their origins, their validity, and their mean- ing. These models include a host of exponential models, traffic flow models, free and forced vibration of linear (and occasionally nonlinear) oscillat- ors, and optimization as done both with calculus and with elementary operations research techniques.

In the applications discussions, reference to the modeling principles is made by highlighting appropriate key words in the margin immediately adjacent to the appropriate text, as in:

“Lanchester wanted to describe the attrition of opposing forces at war. This

**Why?**

**Find?** required modeling the changes of two army populations whose respective
rates of attrition depend on the size of the opposing army.”

The foundations and applications parts of the book are connected only loosely. The following matrix indicates roughly how the chapters in each part relate to each other. In fact, the reader—and the teacher—can easily start with Chapter 5 and work through the applications models, referring back to corresponding discussions of the foundations as needed.

**Preface** **xv**
The problems distributed throughout and at the end of each chapter
(save Chapter 1) are an integral part of the book. Like bike riding and dan-
cing and designing, mathematical modeling cannot be learned simply by
reading. Skills are developed and honed by doing problems, both ele-
mentary and difficult. Thus, there are problems that provide drills in
basic skills, and there are problems that either develop new models or
expand on models developed earlier in the text. For example, in prob-
lems at the end of Chapter 3 we show how dimensional groups are used
to interpret experimental results. The problems in Chapter 5 demonstrate
how dimensional analysis interacts with other approaches to deriving the
governing equations for the oscillating pendulum, and the problems in
Chapter 7 include data on resonance and impedance for a variety of forced
oscillators.

Models

5 6 7 8 9

Tools

Exponential Growth and Decay

Traffic Flow Models

Modeling Free Vibration

Applying Vibration Models

What Is the Best?

2 Dimensional Analysis • • • •

3 Scaling • • •

4 Approximation • • • • •

As noted earlier, many of the problems could be done with a computer, whether a symbolic manipulator, a spreadsheet, or an algorithmic number cruncher. However, in order to learn to do mathematical modeling, the problems should be done in “closed form,” with pencil and paper, with access only to a simple electronic calculator. This will both reinforce skills and provide a basis for benchmarking future computer calculations.

Three appendices from the first edition have been moved closer to their
use in the book. A brief review of elementary transcendental functions is
now appended to Chapter 4; the mathematics of the first-order equation,
*dN/dt*−*λN* =0, is outlined in Section 5.2.2; and the mathematics of the
*second-order oscillator equation, md*^{2}*x/dt*^{2}+*kx* = *F(t), is detailed in*
Sections 7.2.2. and 8.6.

Lastly, the book can be used in several ways. The first edition was
developed for new courses in mathematical modeling that were offered
*to first-year engineering students at Carnegie Mellon University and at the*
University of Massachusetts at Amherst. The book could also serve as a
*first course in applied mathematics for mathematics majors, or as a “tech-*
nical elective” for various science and engineering majors, or conceivably
as a supplementary text in basic calculus courses. In hopes of extending

its audience, I have tried to enhance both the book’s accessibility and its flexibility.

**I Presume That You, the Reader, Have** **. . .**

**. . .**

*. . .*taken courses in elementary algebra, trigonometry, and first-year cal-
culus. I further presume that you recognize what a differential equation
*is and what it means for y(x)* *or y(x, t)*to be a solution of a differential
equation. While you won’t be asked to “solve” a differential equation, you
will be asked to confirm and manipulate some of the solutions that are
given. Finally, I do assume some basic understanding of first-year physics,
mainly mechanics.

**Acknowledgments**

This book is the second edition of a text originally published in 1980 and written by me and Elizabeth S. Ivey; Betty was then both a professor of physics at Smith College and an adjunct professor of mechanical engineer- ing at the University of Massachusetts at Amherst. When approached in the summer of 2000 by Academic Press to do a second edition, Betty and I decided that I would do the second edition alone, but I cannot view the second edition as complete without acknowledging the wonderful nature of our original collaboration and the long-standing friendship that resulted.

Many people deserve much credit for the good in this new version, although the responsibility for the bad (and the ugly) is entirely mine.

Professors Robert L. Borrelli (Harvey Mudd College), Edward A. Connors (University of Massachusetts at Amherst), Ricardo Diaz (University of North Colorado), Michael Kirby (Colorado State University), Mark S.

Korlie (Montclair State University), Thomas Seidman (University of Maryland Baltimore County), Caroline Smith (James Madison Univer- sity), and William H. Wood (University of Maryland Baltimore County) were kind enough to provide pre-publication reviews of this second edition that were very helpful and supportive.

Professor Ewart Carson (University of London) provided a solid and helpful review of the first edition that prompted several changes in the second.

The artist Miriam Dym once again agreed to design the cover for one of her father’s books, for which I am very grateful.

Drs. Ashkay Gupta and Ali Reza (Exponent FAA) provided a thought- ful review of the discussion of the vibration of a tall, slender building (Section 8.2).

**xvii**

Several Harvey Mudd colleagues provided helpful comments on vari- ous sections: Patrick Little (Engineering) of Chapter 9 on optimization;

R. Erik Spjut (Engineering) of Section 9.5.1 on nucleation; Lisa M. Sullivan (Humanities and Social Sciences) of Section 5.5 on exponential modeling of money matters; and Harry E. Williams (Engineering), who commented on much of the early material in painstaking detail.

Dr. John H. McMasters (The Boeing Company) allowed me to use parts of an unpublished manuscript on geometric programming in constructing Sections 9.5.2 and 9.5.3, and graciously reviewed the final product.

Professor David Powers (Clarkson University) served once again as a reviewer of this book. In fact, Dr. Powers has also reviewed the pre- publication prospectus of the first edition, provided a very useful re-review of that first edition that helped define the second edition, and then reviewed this manuscript during its preparation.

Professor Wilfred W. Recker (University of California, Irvine), a good friend for almost forty years, carefully reviewed the discussion of traffic flow modeling (Chapter 6) and provided some new materials.

Robert Ross, formerly of Academic Press, instigated this second edition, assisted ably by Mary Spencer. Subsequently, over the extended writing life of the project, Barbara Holland has been an encouraging and supportive editor, and she was assisted very ably by Tom Singer.

Professors Michael J. Scott (University of Illinois, Chicago) and William H. Wood (University of Maryland Baltimore County) commented insight- fully on my adaptation (for Section 9.4) of our joint paper on choosing among alternatives.

Gautam Thatte, Harvey Mudd College ’03, did a great job of researching and rendering illustrations.

Finally, Joan Dym has provided her usual friendship, love, and oft- needed distraction while this second edition was being written. It is a true pleasure to acknowledge what this has meant to me.

Clive L. Dym

Claremont, California May 2004

**1**

**What Is Mathematical** **Modeling?**

*We begin this book with a dictionary definition of the word model:*

**model** *(n): a miniature representation of something; a pattern of some-*
*thing to be made; an example for imitation or emulation; a description or*
*analogy used to help visualize something (e.g., an atom) that cannot be dir-*
*ectly observed; a system of postulates, data and inferences presented as a*
*mathematical description of an entity or state of affairs*

*This definition suggests that modeling is an activity, a cognitive activity in*
which we think about and make models to describe how devices or objects
of interest behave.

There are many ways in which devices and behaviors can be described.

We can use words, drawings or sketches, physical models, computer pro- grams, or mathematical formulas. In other words, the modeling activity can be done in several languages, often simultaneously. Since we are par- ticularly interested in using the language of mathematics to make models,

**3**

we will refine the definition just given:

**mathematical model***(n): a representation in mathematical terms of the*
*behavior of real devices and objects*

We want to know how to make or generate mathematical representations or models, how to validate them, how to use them, and how and when their use is limited. But before delving into these important issues, it is worth talking about why we do mathematical modeling.

**1.1 WhyDo We Do Mathematical Modeling?**

Since the modeling of devices and phenomena is essential to both engi- neering and science, engineers and scientists have very practical reasons for doing mathematical modeling. In addition, engineers, scientists, and mathematicians want to experience the sheer joy of formulating and solving mathematical problems.

**1.1.1 Mathematical Modeling and the** **Scientiﬁc Method**

*In an elementary picture of the scientific method (see Figure 1.1), we identify*
a “real world” and a “conceptual world.” The external world is the one
we call real; here we observe various phenomena and behaviors, whether
natural in origin or produced by artifacts. The conceptual world is the
world of the mind—where we live when we try to understand what is
going on in that real, external world. The conceptual world can be viewed
as having three stages: observation, modeling, and prediction.

*In the observation part of the scientific method we measure what is*
happening in the real world. Here we gather empirical evidence and “facts
on the ground.” Observations may be direct, as when we use our senses, or
indirect, in which case some measurements are taken to indicate through
some other reading that an event has taken place. For example, we often
know a chemical reaction has taken place only by measuring the product
of that reaction.

*In this elementary view of how science is done, the modeling part is*
concerned with analyzing the above observations for one of (at least) three
*reasons. These rationales are about developing: models that describe the*

**1.1 WhyDo We Do Mathematical Modeling?** **5**
The real world The conceptual world

Phenomena

Predictions Observations

Models (analyses)

Figure 1.1 An elementary
depiction ofthe*scientiﬁc method*
that shows how our conceptual
models ofthe world are related to
observations made within that real
world (Dym and Ivey, 1980).

*behavior or results observed; models that explain why that behavior and*
*results occurred as they did; or models that allow us to predict future*
behaviors or results that are as yet unseen or unmeasured.

*In the prediction part of the scientific method we exercise our models*
to tell us what will happen in a yet-to-be-conducted experiment or in
an anticipated set of events in the real world. These predictions are then
followed by observations that serve either to validate the model or to suggest
reasons that the model is inadequate.

The last point clearly points to the looping, iterative structure apparent in Figure 1.1. It also suggests that modeling is central to all of the conceptual phases in the elementary model of the scientific method. We build models and use them to predict events that can confirm or deny the models. In addition, we can also improve our gathering of empirical data when we use a model to obtain guidance about where to look.

**1.1.2 Mathematical Modeling and the** **Practice of Engineering**

*Engineers are interested in designing devices and processes and systems.*

That is, beyond observing how the world works, engineers are interested
in creating artifacts that have not yet come to life. As noted by Herbert
*A. Simon (in The Sciences of the Artificial), “Design is the distinguishing*
activity of engineering.” Thus, engineers must be able to describe and
analyze objects and devices into order to predict their behavior to see if

that behavior is what the engineers want. In short, engineers need to model devices and processes if they are going to design those devices and processes.

While the scientific method and engineering design have much in com- mon, there are differences in motivation and approach that are worth mentioning. In the practices of science and of engineering design, mod- els are often applied to predict what will happen in a future situation. In engineering design, however, the predictions are used in ways that have far different consequences than simply anticipating the outcome of an experiment. Every new building or airplane, for example, represents a model-based prediction that the building will stand or the airplane will fly without dire, unanticipated consequences. Thus, beyond simply validat- ing a model, prediction in engineering design assumes that resources of time, imagination, and money can be invested with confidence because the predicted outcome will be a good one.

**1.2 Principles of Mathematical Modeling**

*Mathematical modeling is a principled activity that has both principles*
behind it and methods that can be successfully applied. The principles are
*over-arching or meta-principles phrased as questions about the intentions*
and purposes of mathematical modeling. These meta-principles are almost
philosophical in nature. We will now outline the principles, and in the next
section we will briefly review some of the methods.

A visual portrayal of the basic philosophical approach is shown in Figure 1.2. These methodological modeling principles are also captured in the following list of questions and answers:

• **Why?**What are we looking for? Identify the need for the model.

• **Find?**What do we want to know? List the data we are seeking.

• **Given?**What do we know? Identify the available relevant data.

• **Assume?**What can we assume? Identify the circumstances that apply.

• **How?** How should we look at this model? Identify the governing
physical principles.

• **Predict?**What will our model predict? Identify the equations that will
be used, the calculations that will be made, and the answers that will
result.

• **Valid?** Are the predictions valid? Identify tests that can be made
*to validate the model, i.e., is it consistent with its principles and*
assumptions?

• **Veriﬁed?** Are the predictions good? Identify tests that can be made
*to verify the model, i.e., is it useful in terms of the initial reason it*
was done?

**1.2 Principles of Mathematical Modeling** **7**

**Why? **What are we looking for?

**Find? **What do we want to know?

**How? **How should we look at this model?

**Given? **What do we know?

**Assume? What can we assume?**

**Predict? What will**
our model predict?

**Valid? **Are the predictions valid?

**Improve? **Howcan we improve the model?

**Use? How will we exercise the model?**

OBJECT/SYSTEM

MODEL VARIABLES, PARAMETERS

**Verified? **Are the predictions good?

MODEL PREDICTIONS

VALID, ACCEPTED PREDICTIONS

TEST

Figure 1.2 A ﬁrst-order view of*mathematical modeling* that
shows how the questions asked in a principled approach to building
a model relate to the development ofthat model (inspired by
Carson and Cobelli, 2001).

• **Improve?**Can we improve the model? Identify parameter values that
are not adequately known, variables that should have been included,
and/or assumptions/restrictions that could be lifted. Implement the
iterative loop that we can call “model-validate-verify-improve-predict.”

• **Use?**How will we exercise the model? What will we do with the model?

*This list of questions and instructions is not an algorithm for building*
a good mathematical model. However, the underlying ideas are key to
mathematical modeling, as they are key to problem formulation generally.

Thus, we should expect the individual questions to recur often during the
modeling process, and we should regard this list as a fairly general approach
*to ways of thinking about mathematical modeling.*

Having a clear picture of why the model is wanted or needed is of prime importance to the model-building enterprise. Suppose we want to estimate how much power could be generated by a dam on a large river, say a dam located at The Three Gorges on the Yangtze River in Hubei Province in the People’s Republic of China. For a first estimate of the available power, we

wouldn’t need to model the dam’s thickness or the strength of its founda- tion. Its height, on the other hand, would be an essential parameter of a power model, as would some model and estimates of river flow quantities.

If, on the other hand, we want to design the actual dam, we would need a model that incorporates all of the dam’s physical characteristics (e.g., dimensions, materials, foundations) and relates them to the dam site and the river flow conditions. Thus, defining the task is the first essential step in model formulation.

We then should list what we know—for example, river flow quantities and desired power levels—as a basis for listing the variables or parameters that are as yet unknown. We should also list any relevant assumptions.

For example, levels of desired power may be linked to demographic or economic data, so any assumptions made about population and economic growth should be spelled out. Assumptions about the consistency of river flows and the statistics of flooding should also be spelled out.

Which physical principles apply to this model? The mass of the river’s water must be conserved, as must its momentum, as the river flows, and energy is both dissipated and redirected as water is allowed to flow through turbines in the dam (and hopefully not spill over the top!). And mass must be conserved, within some undefined system boundary, because dams do accumulate water mass from flowing rivers. There are well-known equa- tions that correspond to these physical principles. They could be used to develop an estimate of dam height as a function of power desired.

We can validate the model by ensuring that our equations and calcu- lated results have the proper dimensions, and we can exercise the model against data from existing hydroelectric dams to get empirical data and validation.

If we find that our model is inadequate or that it fails in some way, we
*then enter an iterative loop in which we cycle back to an earlier stage of the*
model building and re-examine our assumptions, our known parameter
values, the principles chosen, the equations used, the means of calculation,
and so on. This iterative process is essential because it is the only way that
models can be improved, corrected, and validated.

**1.3 Some Methods of Mathematical** **Modeling**

Now we will review some of the mathematical techniques we can use to help answer the philosophical questions posed in Section 1.2. These mathemati- cal principles include: dimensional homogeneity, abstraction and scaling,

**1.3 Some Methods of Mathematical Modeling** **9**
conservation and balance principles, and consequences of linearity. We will
expand these themes more extensively in the first part of this book.

**1.3.1 Dimensional Homogeneityand Consistency**

There is a basic, yet very powerful idea that is central to mathematical
*modeling, namely, that every equation we use must be dimensionally homo-*
*geneous or dimensionally consistent. It is quite logical that every term in an*
energy equation has total dimensions of energy, and that every term in
a balance of mass should have the dimensions of mass. This statement
*provides the basis for a technique called dimensional analysis that we will*
discuss in greater detail in Chapter 2.

In that discussion we will also review the important distinction between
*physical dimensions that relate a (derived) quantity to fundamental physi-*
*cal quantities and units that are numerical expressions of a quantity’s*
dimensions expressed in terms of a given physical standard.

**1.3.2 Abstraction and Scaling**

An important decision in modeling is choosing an appropriate level of
detail for the problem at hand, and thus knowing what level of detail is
*prescribed for the attendant model. This process is called abstraction and it*
typically requires a thoughtful approach to identifying those phenomena
on which we want to focus, that is, to answering the fundamental question
about why a model is being sought or developed.

For example, a linear elastic spring can be used to model more than just the relation between force and relative extension of a simple coiled spring, as in an old-fashioned butcher’s scale or an automobile spring. It can also be used to model the static and dynamic behavior of a tall building, perhaps to model wind loading, perhaps as part of analyzing how the building would respond to an earthquake. In these examples, we can use a very abstract model by subsuming various details within the parameters of that model.

We will explore these issues further in Chapter 3.

In addition, as we talk about finding the right level of abstraction or the
right level of detail, we are simultaneously talking about finding the right
*scale for the model we are developing. For example, the spring can be used*
*at a much smaller, micro scale to model atomic bonds, in contrast with*
*the macro level for buildings. The notion of scaling includes several ideas,*
including the effects of geometry on scale, the relationship of function to
scale, and the role of size in determining limits—all of which are needed
to choose the right scale for a model in relation to the “reality” we want to
capture.

**1.3.3 Conservation and Balance Principles**

When we develop mathematical models, we often start with statements that indicate that some property of an object or system is being conserved.

For example, we could analyze the motion of a body moving on an ideal,
*frictionless path by noting that its energy is conserved. Sometimes, as when*
we model the population of an animal colony or the volume of a river flow,
*we must balance quantities, of individual animals or water volumes, that*
*cross a defined boundary. We will apply balance or conservation principles*
to assess the effect of maintaining or conserving levels of important physi-
cal properties. Conservation and balance equations are related—in fact,
conservation laws are special cases of balance laws.

The mathematics of balance and conservation laws are straightforward
at this level of abstraction. Denoting the physical property being monitored
*as Q(t)and the independent variable time as t , we can write a balance law*
*for the temporal or time rate of change of that property within the system*
boundary depicted in Figure 1.3 as:

*dQ(t)*

*dt* =*q**in**(t)*+*g(t)*−*q**out**(t)*−*c(t),* (1.1)
*where q**in**(t)and q**out**(t)represent the flow rates of Q(t)into (the influx)*
*and out of (the efflux) the system boundary, g(t)* is the rate at which
*Q is generated within the boundary, and c(t)* *is the rate at which Q is*
*consumed within that boundary. Note that eq. (1.1) is also called a rate*
*equation because each term has both the meaning and dimensions of the*
*rate of change with time of the quantity Q(t).*

Efflux, q* _{out}*
System Boundary

Influx, q_{in}*Q(t)*

Consumption, c(t)

Generation, g(t)

Figure 1.3 A system boundary surrounding the object or
system being modeled. The inﬂux*q*_{in}*(t**), efﬂux**q**out**(t**),*
generation*g**(t**), and consumption**c(t**), affect the rate at*
which the property ofinterest,*Q(t**), accumulates within*
the boundary (after Cha, Rosenberg, and Dym, 2000).

**1.4 Summary** **11**
In those cases where there is no generation and no consumption within
*the system boundary (i.e., when g* =*c* = 0), the balance law in eq. (1.1)
becomes a conservation law:

*dQ(t)*

*dt* =*q**in**(t)*−*q**out**(t).* (1.2)
*Here, then, the rate at which Q(t)* accumulates within the boundary is
*equal to the difference between the influx, q**in**(t), and the efflux, q**out**(t).*

**1.3.4 Constructing Linear Models**

Linearity is one of the most important concepts in mathematical model-
*ing. Models of devices or systems are said to be linear when their basic*
equations—whether algebraic, differential, or integral—are such that the
*magnitude of their behavior or response produced is directly proportional*
to the excitation or input that drives them. Even when devices like the
pendulum discussed in Chapter 7 are more fully described by nonlinear
models, their behavior can often be approximated by linearized or per-
turbed models, in which cases the mathematics of linear systems can be
successfully applied.

We apply linearity when we model the behavior of a device or system
that is forced or pushed by a complex set of inputs or excitations. We obtain
the response of that device or system to the sum of the individual inputs
*by adding or superposing the separate responses of the system to each indi-*
*vidual input. This important result is called the principle of superposition.*

Engineers use this principle to predict the response of a system to a com- plicated input by decomposing or breaking down that input into a set of simpler inputs that produce known system responses or behaviors.

**1.4 Summary**

In this chapter we have provided an overview of the foundational material we will cover in this book. In so doing, we have defined mathematical modeling, provided motivation for its use in engineering and science, and set out a principled approach to doing mathematical modeling. We have also outlined some of the important tools that will be covered in greater detail later: dimensional analysis, abstraction and scaling, balance laws, and linearity.

It is most important to remember that mathematical models are repre-
*sentations or descriptions of reality—by their very nature they depict reality.*

Thus, we close with a quote from a noted linguist (and former senator from

California) to remind ourselves that we are dealing with models that, we hope, represent something that seems real and relevant to us. However, they are abstractions and models, they are themselves real only as models, and they should never be confused with the reality we are trying to model.

Thus, if the behavior predicted by our models does not reflect what we see or measure in the real world, it is the models that need to be fixed—and not the world:

*“The symbol is NOT the thing symbolized; the word is NOT the thing; the map is*
*NOT the territory it stands for.”*

*—S. I. Hayakawa, Language in Thought and Action*

**1.5 References**

*E. Carson and C. Cobelli (Eds.), Modelling Methodology for Physiology and*
*Medicine, Academic Press, San Diego, CA, 2001.*

*P. D. Cha, J. J. Rosenberg, and C. L. Dym, Fundamentals of Model-*
*ing and Analyzing Engineering Systems, Cambridge University Press,*
New York, 2000.

*C. L. Dym, Engineering Design: A Synthesis of Views, Cambridge University*
Press, New York, 1994.

*C. L. Dym and E. S. Ivey, Principles of Mathematical Modeling, 1st Edition,*
Academic Press, New York, 1980.

*S. I. Hayakawa, Language in Thought and Action, Harcourt, Brace,*
New York, 1949.

*G. Kemeny, A Philosopher Looks at Science, Van Nostrand-Reinhold,*
New York, 1959.

*H. J. Miser, “Introducing Operational Research,” Operational Research*
*Quarterly,***27(3), 665–670, 1976.**

*M. F. Rubinstein, Patterns of Problem Solving, Prentice-Hall, Englewood*
Cliffs, NJ, 1975.

*H. A. Simon, The Sciences of the Artificial, 3rd Edition, MIT Press,*
Cambridge, MA, 1999.

**2**

**Dimensional Analysis**

*We begin this chapter, the first of three dealing with the tools or techniques*
*for mathematical modeling, with H. L. Langhaar’s definition of dimensional*
*analysis:*

**Dimensional analysis***(n): a method by which we deduce information*
*about a phenomenon from the single premise that the phenomenon can be*
*described by a dimensionally correct equation among certain variables.*

This quote expresses the simple, yet powerful idea that we introduced
*in Section 1.3.1: all of the terms in our equations must be dimension-*
*ally consistent, that is, each separate term in those equations must have*
the same net physical dimensions. For example, when summing forces
to ensure equilibrium, every term in the equation must have the physi-
cal dimension of force. (Equations that are dimensionally consistent are
*sometimes called rational.) This idea is particularly useful for validat-*
ing newly developed mathematical models or for confirming formulas
and equations before doing calculations with them. However, it is also
a weak statement because the available tools of dimensional analysis are
rather limited, and applying them does not always produce desirable
results.

**13**

**2.1 Dimensions and Units**

*The physical quantities we use to model objects or systems represent con-*
*cepts, such as time, length, and mass, to which we also attach numerical*
values or measurements. Thus, we could describe the width of a soccer field
by saying that it is 60 meters wide. The concept or abstraction invoked is
length or distance, and the numerical measure is 60 meters. The numerical
measure implies a comparison with a standard that enables both commu-
nication about and comparison of objects or phenomena without their
being in the same place. In other words, common measures provide a
frame of reference for making comparisons. Thus, soccer fields are wider
than American football fields since the latter are only 49 meters wide.

The physical quantities used to describe or model a problem come in
*two varieties. They are either fundamental or primary quantities, or they*
*are derived quantities. Taking a quantity as fundamental means only that*
we can assign it a standard of measurement independent of that chosen for
the other fundamental quantities. In mechanical problems, for example,
mass, length, and time are generally taken as the fundamental mechanical
variables, while force is derived from Newton’s law of motion. It is equally
correct to take force, length, and time as fundamental, and to derive mass
from Newton’s law. For any given problem, of course, we need enough
fundamental quantities to express each derived quantity in terms of these
primary quantities.

While we relate primary quantities to standards, we also note that they
are chosen arbitrarily, while derived quantities are chosen to satisfy physical
laws or relevant definitions. For example, length and time are fundamental
quantities in mechanics problems, and speed is a derived quantity expressed
as length per unit time. If we chose time and speed as primary quantities,
the derived quantity of length would be (speed×time), and the derived
quantity of area would be (speed×time)^{2}.

*The word dimension is used to relate a derived quantity to the fun-*
damental quantities selected for a particular model. If mass, length, and
time are chosen as primary quantities, then the dimensions of area are
(length)^{2}, of mass density are mass/(length)^{3}, and of force are (mass ×
length)/(time)^{2}. We also introduce the notation of brackets [] to read as

“the dimensions of.” If M, L, and T stand for mass, length, and time, respectively, then:

[A=area] =*(*L*)*^{2}, (2.1a)

[ρ=density] =M*/(*L*)*^{3}, (2.1b)
[F =force] =*(*M×L*)/(*T*)*^{2}. (2.1c)

**2.2 Dimensional Homogeneity** **15**
*The units of a quantity are the numerical aspects of a quantity’s dimensions*
expressed in terms of a given physical standard. By definition, then, a unit
is an arbitrary multiple or fraction of that standard. The most widely
accepted international standard for measuring length is the meter (m), but
it can also be measured in units of centimeters (1 cm = 0.01 m) or of
feet (0.3048 m). The magnitude or size of the attached number obviously
depends on the unit chosen, and this dependence often suggests a choice
of units to facilitate calculation or communication. The soccer field width
can be said to be 60 m, 6000 cm, or (approximately) 197 feet.

Dimensions and units are related by the fact that identifying a quantity’s dimensions allows us to compute its numerical measures in different sets of units, as we just did for the soccer field width. Since the physical dimensions of a quantity are the same, there must exist numerical relationships between the different systems of units used to measure the amounts of that quantity.

For example,

1 foot (ft) ∼=30.48 centimeters (cm),
1 centimeter (cm) ∼=0.000006214 miles (mi),
1 hour (hr)=60 minutes (min)=*3600 seconds (sec or s).*

This equality of units for a given dimension allows us to change or convert units with a straightforward calculation. For a speed of 65 miles per hour (mph), for example, we can calculate the following equivalent:

65mi

hr =65mi

hr ×5280 ft

mi×0.3048m

ft ×0.001km

m ∼=104.6km
hr.
Each of the multipliers in this conversion equation has an effective value
of unity because of the equivalencies of the various units, that is, 1 mi=
5280 ft, and so on. This, in turn, follows from the fact that the numerator
and denominator of each of the above multipliers have the same physical
*dimensions. We will discuss systems of units and provide some conversion*
data in Section 2.4.

**2.2 Dimensional Homogeneity**

*We had previously defined a rational equation as an equation in which*
each independent term has the same net dimensions. Then, taken in its
*entirety, the equation is dimensionally homogeneous. Simply put, we cannot*
add length to area in the same equation, or mass to time, or charge to
stiffness—although we can add (and with great care) quantities having the
same dimensions but expressed in different units, e.g., length in meters

and length in feet. The fact that equations must be rational in terms of their dimensions is central to modeling because it is one of the best—and easiest—checks to make to determine whether a model makes sense, has been correctly derived, or even correctly copied!

We should remember that a dimensionally homogeneous equation is
independent of the units of measurement being used. However, we can
create unit-dependent versions of such equations because they may be
more convenient for doing repeated calculations or as a memory aid. In
*an example familiar from mechanics, the period (or cycle time), T*0, of a
pendulum undergoing small-angle oscillations can be written in terms of
*the pendulum’s length, l, and the acceleration of gravity, g :*

*T*0=2π

*l/g .* (2.2)

This dimensionally homogeneous equation is independent of the system of
units chosen to measure length and time. On the other hand, we may find
*it convenient to work in the metric system, in which case g* = 9.8 m/s^{2},
from which it follows that

*T*0*(s)*=2π

*l/9.8*∼=2√

*l.* (2.3)

*Equation (2.3) is valid only when the pendulum’s length is measured in*
meters. In the so-called British system,^{1}*where g* =32.17 ft/sec^{2},

*T*0*(sec)*=2π

*l/32.17*∼=1.1√

*l.* (2.4)

Equations (2.3) and (2.4) are not dimensionally homogeneous. So, while

**Why**

**not?** these formulas may be appealing or elegant, we have to remember their
limited ranges of validity, as we should whenever we use or create similar
formulas for whatever modeling we are doing.

**2.3 Why Do We Do Dimensional Analysis?**

We presented a definition of dimensional analysis at the beginning of this chapter, where we also noted that the “method” so defined has both power- ful implications—rational equations and dimensional consistency—and severe limitations—the limited nature of the available tools. Given this limitation, why has this method or technique developed, and why has it persisted?

1 One of my Harvey Mudd colleagues puckishly suggests that we should call this the American system of units as we are, apparently, the only country still so attached to feet and pounds.

**2.3 Why Do We Do Dimensional Analysis?** **17**

Figure 2.1 A picture of a “precision mix batch
mixer” that would be used to mix large quantities
of foods such as peanut butter and other mixes of
substances that have relatively high values of
density*ρ*and viscosity*µ*(courtesy of H. C. Davis
Sons Manufacturing Company, Inc.).

Dimensional analysis developed as an attempt to perform extended, costly experiments in a more organized, more efficient fashion. The under- lying idea was to see whether the number of variables could be grouped together so that fewer trial runs or fewer measurements would be needed.

Dimensional analysis produces a more compact set of outputs or data, with perhaps fewer charts and graphs, which in turn might better clarify what is being observed.

Imagine for a moment that we want to design a machine to make large quantities of peanut butter (and this author prefers creamy to crunchy!).

We can imagine a mixer that takes all of the ingredients (i.e., roasted peanuts, sugar, and “less than 2%” of molasses and partially hydrogen- ated vegetable oil) and mixes them into a smooth, creamy spread. Moving a knife through a jar of peanut butter requires a noticeably larger force than stirring a glass of water. Similarly, the forces in a vat-like mixer would be considerable, as would the power needed to run that mixer in an automated food assembly line, as illustrated in Figure 2.1. Thus, the electro-mechanical design of an industrial-strength peanut butter mixer depends on estimates of the forces required to mix the peanut butter. How can we get some idea of what those forces are?

It turns out, as you might expect, that the forces depend in large part on properties of the peanut butter, but on which properties, and how? We can

answer those questions by performing a series of experiments in which we
push a blade through a tub of peanut butter and measure the amount of
force required to move the blade at different speeds. We will call the force
*needed to move the blade through the peanut butter the drag force, F**D*,
because it is equal to the force exerted by the moving (relatively speaking)
peanut butter to retard the movement of the knife. We postulate that the
*force depends on the speed V with which the blade moves, on a character-*
*istic dimension of the blade, say the width d, and on two characteristics of*
the peanut butter. One of these characteristics is the mass density,*ρ, and*
*the second is a parameter called the viscosity,µ, which is a measure of its*

“stickiness.” If we think about our experiences with various fluids, includ- ing water, honey, motor oil, and peanut butter, these two characteristics seem intuitively reasonable because we do associate a difficulty in stirring (and cleaning up) with fluids that feel heavier and stickier.

Thus, the five quantities that we will take as derived quantities for this
initial investigation into the mixing properties of peanut butter are the
*drag force, F**D**, the speed with which the blade moves, V , the knife blade*
*width, d, the peanut butter’s mass density,ρ*, and its viscosity,*µ*. The fun-
damental physical quantities we would apply here are mass, length, and
time, which we denote asM,L, andT, respectively. The derived variables
are expressed in terms of the fundamental quantities in Table 2.1.

**Table 2.1** The ﬁve derived
quantities chosen to model the
peanut butter stirring

experiments.

**Derived quantities** **Dimensions**

*Speed (V )* L/T

*Blade width (d)* L
Density (*ρ*) M*/(*L*)*^{3}
Viscosity (*µ*) M*/(*L×T*)*
*Drag force (F** _{D}*)

*(*M×L

*)/(*T

*)*

^{2}

How did we get the fundamental dimensions of the viscosity? By a straightforward application of the principle of dimensional homogeneity to the assumptions used in modeling the mechanics of fluids: The drag force (or force required to pull the blade through the butter) is directly proportional both to the speed with which it moves and the area of the blade, and inversely proportional to a length that characterizes the spatial rate of change of the speed. Thus,

*F** _{D}* ∝

*VA*

*L* , (2.5a)

**2.4 How Do We Do Dimensional Analysis?** **19**
or

*F**D* =*µVA*

*L* . (2.5b)

If we apply the principle of dimensional homogeneity to eq. (2.5b), it follows that

[µ] =
*F**D*

*A* × *L*
*V*

. (2.6)

It is easy to show that eq. (2.6) leads to the corresponding entry in Table 2.1.

*Now we consider the fact that we want to know how F**D* *and V are*
*related, and yet they are also functions of the other variables, d,ρ, andµ,*
that is,

*F**D* =*F**D**(V ; d,ρ,µ).* (2.7)
Equation (2.7) suggests that we would have to do a lot of experiments and
plot a lot of curves to find out how drag force and speed relate to each
other while we are also varying the blade width and the butter density
and viscosity. If we wanted to look at only three different values of each
*of d,ρ, andµ, we would have nine (9) different graphs, each containing*
three (3) curves. This is a significant accumulation of data (and work!) for
a relatively simple problem, and it provides a very graphic illustration of
the need for dimensional analysis. We will soon show that this problem can
be “reduced” to considering two dimensionless groups that are related by
a single curve! Dimensional analysis is thus very useful for both designing
and conducting experiments.

**Problem 2.1.** Justify the assertion made just above that “nine (9)
different graphs, each containing three (3) curves” are
needed to relate force and speed.

**Problem 2.2.** Find and compare the mass density and viscosity of
peanut butter, honey, and water.

**2.4 How Do We Do Dimensional Analysis?**

Dimensional analysis is the process by which we ensure dimensional con- sistency. It ensures that we are using the proper dimensions to describe the problem being modeled, whether expressed in terms of the correct number of properly dimensioned variables and parameters or whether written in terms of appropriate dimensional groups. Remember, too, that we need consistent dimensions for logical consistency, and we need consistent units for arithmetic consistency.

How do we ensure dimensional consistency? First, we check the dimen-
sions of all derived quantities to see that they are properly represented
in terms of the chosen primary quantities and their dimensions. Then
*we identify the proper dimensionless groups of variables, that is, ratios*
and products of problem variables and parameters that are themselves
dimensionless. We will explain two different techniques for identifying such
*dimensionless groups, the basic method and the Buckingham Pi theorem.*

**2.4.1 The Basic Method of Dimensional Analysis**

The basic method of dimensional analysis is a rather informal, unstructured approach for determining dimensional groups. It depends on being able to construct a functional equation that contains all of the relevant variables, for which we know the dimensions. The proper dimensionless groups are then identified by the thoughtful elimination of dimensions.

For example, consider one of the classic problems of elementary mecha-
*nics, the free fall of a body in a vacuum. We recall that the speed, V , of such a*
*falling body is related to the gravitational acceleration, g , and the height, h,*
from which the body was released. Thus, the functional expression of this
knowledge is:

*V* =*V(g , h).* (2.8)

Note that the precise form of this functional equation is, at this point, entirely unknown—and we don’t need to know that form for what we’re doing now. The physical dimensions of the three variables are:

[V] =L*/*T,

[g] =L*/*T^{2}, (2.9)
[h] =L.

The time dimension,T, appears only in the speed and gravitational accele-
*ration, so that dividing the speed by the square root of g eliminates time*
and yields a quantity whose remaining dimension can be expressed entirely
in terms of length, that is:

*V*

√*g*

=√

L. (2.10)

If we repeat this thought process with regard to eliminating the length dimension, we would divide eq. (2.10) by√

*h, which means that*

*V*
*gh*

=1. (2.11)

**2.4 How Do We Do Dimensional Analysis?** **21**
Since we have but a single dimensionless group here, it follows that:

*V* =constant×
*gh*

(2.12) Thus, the speed of a falling body is proportional to

*gh, a result we*
should recall from physics—yet we have found it with dimensional analysis
alone, without invoking Newton’s law or any other principle of mechanics.

This elementary application of dimensional consistency tells us some- thing about the power of dimensional analysis. On the other hand, we do need some physics, either theory or experiment, to define the constant in eq. (2.12).

Someone seeing the result (2.12) might well wonder why the speed of a falling object is independent of mass (unless that person knew of Galileo Galilei’s famous experiment). In fact, we can use the basic method to build on eq. (2.12) and show why this is so. Simply put, we start with a functional equation that included mass, that is,

*V* =*V(g , h, m).* (2.13)

A straightforward inspection of the dimensions of the four variables in eq. (2.13), such as the list in eq. (2.9), would suggest that mass is not a variable in this problem because it only occurs once as a dimension, so it cannot be used to make eq. (2.13) dimensionless.

As a further illustration of the basic method, consider the mutual
revolution of two bodies in space that is caused by their mutual gravita-
tional attraction. We would like to find a dimensionless function that
*relates the period of oscillation, T**R*, to the two masses and the distance
*r between them:*

*T** _{R}* =

*T*

_{R}*(m*1

*, m*

_{2}

*, r).*(2.14) If we list the dimensions for the four variables in eq. (2.14) we find:

[m1],[m2] =M,

[T*R*] =T, (2.15)

[r] =L.

We now have the converse of the problem we had with the falling body.

Here none of the dimensions are repeated, save for the two masses. So, while we can expect that the masses will appear in a dimensionless ratio, how do we keep the period and distance in the problem? The answer is that we need to add a variable containing the dimensions heretofore missing to

*the functional equation (2.14). Newton’s gravitational constant, G, is such*
a variable, so we restate our functional equation (2.14) as

*T**R* =*T**R**(m*1*, m*2*, r, G),* (2.16)
*where the dimensions of G are*

[G] =L^{3}*/*MT^{2}. (2.17)

The complete list of variables for this problem, consisting of eqs. (2.15) and (2.17), includes enough variables to account for all of the dimensions.

Regarding eq. (2.16) as the correct functional equation for the two
revolving bodies, we apply the basic method first to eliminate the dimen-
*sion of time, which appears directly in the period T** _{R}* and as a reciprocal

*squared in the gravitational constant G. It follows dimensionally that*

*T**R*

√*G* =
L^{3}

M, (2.18a)

where the right-hand side of eq. (2.18a) is independent of time. Thus, the corresponding revised functional equation for the period would be:

*T**R*

√*G*=*T**R1**(m*1*, m*2*, r).* (2.18b)
We can eliminate the length dimension simply by noting that

*T** _{R}*√

√ *G*
*r*^{3}

= 1

M, (2.19a)

which leads to a further revised functional equation,
*T** _{R}*√

√ *G*

*r*^{3} =*T**R2**(m*1*, m*2*).* (2.19b)
We see from eq. (2.19a) that we can eliminate the mass dimension from
eq. (2.19b) by multiplying eq. (2.19b) by the square root of one of the two
masses. We choose the square root of the second mass (do Problem 2.6 to
find out what happens if the first mass is chosen),√

*m*_{2}, and we find from
eq. (2.19a) that

*T*√
*Gm*2

√*r*^{3}

=1. (2.20a)

This means that eq. (2.19b) becomes
*T** _{R}*√

*Gm*_{2}

√*r*^{3} =√

*m*2*T**R2**(m*1*, m*2*)*≡*T**R3*

*m*_{1}
*m*2

, (2.20b)

**2.4 How Do We Do Dimensional Analysis?** **23**
where a dimensionless mass ratio has been introduced in eq. (2.20b) to
*recognize that this is the only way that the function T**R3*can be both dimen-
*sionless and a function of the two masses. Thus, we can conclude from*
eq. (2.20b) that

*T**R*=

*r*^{3}
*Gm*_{2}*T**R3*

*m*_{1}
*m*_{2}

. (2.21)

This example shows that difficulties arise if we start a problem with an
incomplete set of variables. Recall that we did not include the gravitational
*constant G until it became clear that we were headed down a wrong path.*

*We then included G to rectify an incomplete analysis. With the benefit*
of hindsight, we might have argued that the attractive gravitational force
*must somehow be accounted for, and including G could have been a way*
to do that. This argument, however, demands insight and judgment whose
origins may have little to do with the particular problem at hand.

While our applications of the basic method of dimensional analysis show that it does not have a formal algorithmic structure, it can be described as a series of steps to take:

a. List all of the variables and parameters of the problem and their dimensions.

b. Anticipate how each variable qualitatively affects quantities of interest, that is, does an increase in a variable cause an increase or a decrease?

c. Identify one variable as depending on the remaining variables and parameters.

d. Express that dependence in a functional equation (i.e., analogs of eqs. (2.8) and (2.14)).

e. Choose and then eliminate one of the primary dimensions to obtain a revised functional equation.

*f. Repeat steps (e) until a revised, dimensionless functional equation is*
found.

*g. Review the final dimensionless functional equation to see whether the*
apparent behavior accords with the behavior anticipated in step “b”.

**Problem 2.3.** What is the constant in eq. (2.12)? How do you know
that?

**Problem 2.4.** Apply the basic method to eq. (2.2) for the period of
the pendulum.

**Problem 2.5.** Carry out the basic method for eq. (2.13) and show
that the mass of a falling body does not affect its speed
of descent.

**Problem 2.6.** Carry out the last step of the basic method for eqs.

(2.20) using the first mass and show it produces a form that is equivalent to eq. (2.21).

**2.4.2 The Buckingham Pi Theorem for** **Dimensional Analysis**

Buckingham’s Pi theorem, fundamental to dimensional analysis, can be stated as follows:

*A dimensionally homogeneous equation involving n variables in m primary*
*or fundamental dimensions can be reduced to a single relationship among*
n−*m independent dimensionless products.*

A dimensionally homogeneous (or rational) equation is one in which every independent, additive term in the equation has the same dimensions. This means that we can solve for any one term as a function of all the others. If we introduce Buckingham’snotation to represent a dimensionless term, his famous Pi theorem can be written as:

1=*(*2,3*. . . **n−m**).* (2.22a)
or, equivalently,

*(*1,2,3*. . . **n−m**)*=0. (2.22b)
*Equations (2.22) state that a problem with n derived variables and m*
*primary dimensions or variables requires n*−*m dimensionless groups*
to correlate all of its variables.

*We apply the Pi theorem by first identifying the n derived variables in a*
*problem: A*_{1}*, A*_{2},*. . .A*_{n}*. We choose m of these derived variables such that*
*they contain all of the m primary dimensions, say, A*1*, A*2*, A*3*for m* =3.

Dimensionless groups are then formed by permuting each of the remaining
*n*−*m variables (A*4*, A*5,*. . .A**n**for m* = *3) in turn with those m’s already*
chosen:

1=*A*_{1}^{a}^{1}*A*_{2}^{b}^{1}*A*_{3}^{c}^{1}*A*_{4},
2=*A*_{1}^{a}^{2}*A*_{2}^{b}^{2}*A*_{3}^{c}^{2}*A*5,

... (2.23)

*n−m*=*A*_{1}^{a}^{n}^{−}^{m}*A*^{b}_{2}^{n}^{−}^{m}*A*_{3}^{c}^{n}^{−}^{m}*A**n*.

**2.4 How Do We Do Dimensional Analysis?** **25**
*The a*_{i}*, b*_{i}*, and c** _{i}* are chosen to make each of the permuted groups

*i*

dimensionless.

For example, for the peanut butter mixer, there should be two dimen-
sionless groups correlating the five variables of the problem (listed in
Table 2.1). To apply the Pi theorem to this mixer we choose the blade
*speed V , its width d, and the butter densityρ*as the fundamental variables
*(m* = 3), which we then permute with the two remaining variables—the
viscosity*µand the drag force F**D*—to get two dimensionless groups:

1 =*V*^{a}^{1}*d*^{b}^{1}*ρ*^{c}^{1}*µ,*

2 =*V*^{a}^{2}*d*^{b}^{2}*ρ*^{c}^{2}*F**D*. (2.24)
Expressed in terms of primary dimensions, these groups are:

1= L

T
_{a}_{1}

L^{b}^{1}
M

L^{3}
_{c}_{1}

M LT

, 2=

L T

_{a}_{2}
L^{b}^{2}

M
L^{3}

_{c}_{2}
ML

T^{2}

. (2.25)

Now, in order for1and2 to be dimensionless, the net exponents for each of the three primary dimensions must vanish. Thus, for1,

L: *a*1+*b*1−*3c*1−1=0,
T: −a1−1=0,

M: *c*1+1=0,

(2.26a) and for2,

L: *a*2+*b*2−*3c*2+1=0,
T: −a2−2=0,

M: *c*_{2}+1=0.

(2.26b) Solving eqs. (2.26) for the two pairs of subscripts yields:

*a*1=*b*1 =*c*1= −1,

*a*2=*b*2 = −2, *c*2= −1. (2.27)
Then the two dimensionless groups for the peanut butter mixer are:

1 =
*µ*

*ρVd*

, 2 =

*F**D*

*ρV*^{2}*d*^{2}

. (2.28)

Thus, there are two dimensionless groups that should guide experiments with prototype peanut butter mixers. One clearly involves the viscosity of the peanut butter, while the other relates the drag force on the blade to

*T*

*mg*

Figure 2.2 The classical
pendulum oscillating through
angles*θ*due to gravitational
acceleration*g.*

the blade’s dimensions and speed, as well as to the density of the peanut butter.

In Chapter 7 we will explore one of the “golden oldies” of physics, mod- eling the small angle, free vibration of an ideal pendulum (viz. Figure 2.2).

There are six variables to consider in this problem, and they are listed
along with their fundamental dimensions in Table 2.2. In this case we
*have m*=*6 and n* =3, so that we can expect three dimensionless groups.

*We will choose l, g , and m as the variables around which we will per-*
*mute the remaining three variables (T*0,*θ, T ) to obtain the three groups.*

Thus,

1=*l*^{a}^{1}*g*^{b}^{1}*m*^{c}^{1}*T*_{0},

2=*l*^{a}^{2}*g*^{b}^{2}*m*^{c}^{2}*θ*, (2.29)
3=*l*^{a}^{3}*g*^{b}^{3}*m*^{c}^{3}*T .*

**Table 2.2** The six derived quantities
chosen to model the oscillating
pendulum.

**Derived quantities** **Dimensions**

*Length (l)* L

*Gravitational acceleration (g )* L*/*T^{2}

*Mass (m)* M

*Period (T*_{0}) T

Angle (θ) 1

*String tension (T )* *(M*×L)/T^{2}

**2.4 How Do We Do Dimensional Analysis?** **27**
The Pi theorem applied here then yields three dimensionless groups (see
Problem 2.9):

1= *T*0

*l/g*,

2=*θ*, (2.30)

3= *T*
*mg*.

*These groups show how the period depends on the pendulum length l*
*and the gravitational constant g (recall eq. (2.2)), and the string tension T*
*on the mass m and g . The second group also shows that the (dimensionless)*
angle of rotation stands alone, that is, it is apparently not related to any
of the other variables. This follows from the assumption of small angles,
which makes the problem linear, and makes the magnitude of the angle of
free vibration a quantity that cannot be determined.

*One of the “rules” of applying the Pi theorem is that the m chosen vari-*
*ables include all n of the fundamental dimensions, but no other restrictions*
are given. So, it is natural to ask how this analysis would change if we start
*with three different variables. For example, suppose we choose T*0*, g , and*
*m as the variables around which to permute the remaining three variables*
*(l,θ, T ) to obtain the three groups. In this case we would write:*

^{}_{1}=*T*_{0}^{a}^{1}*g*^{b}^{1}*m*^{c}^{1}*l,*

^{}_{2}=*T*_{0}^{a}^{2}*g*^{b}^{2}*m*^{c}^{2}*θ,* (2.31)
^{}_{3}=*T*_{0}^{a}^{3}*g*^{b}^{3}*m*^{c}^{3}*T .*

Applying the Pi theorem to eq. (2.31) yields the following three “new”

dimensionless groups (see Problem 2.10):

^{}_{1}= *l/g*
*T*_{0}^{2} = 1

^{2}_{1},

^{}_{2}=*θ* =2, (2.32)

^{}_{3}= *T*

*mg* =3.

We see that eq. (2.32) produce the same information as eq. (2.30), albeit
in a slightly different form. In particular, it is clear that1 and^{}_{1} con-
tain the same dimensionless group, which suggests that the number of
dimensionless groups is unique, but that the precise forms that these groups