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Useful Data
Earth data and gravity
Me Mass of the earth 5.98 * 1024
kg
Re Radius of the earth 6.37 * 106
m
g Free-fall acceleration 9.80 m/s2
G Gravitational constant 6.67 * 10-11
N # m2
/kg2
Thermodynamics
kB Boltzmann constant 1.38 * 10-23
J/K
R Gas constant 8.31 J/mol # K
NA Avogadro’s number 6.02 * 1023
particles/mol
T0 Absolute zero -273°C
patm Standard atmosphere 101,000 Pa
u Atomic mass unit (Dalton) 1.66 * 10-27
kg
Speeds of sound and light
vsound Speed of sound in air at 20°C 343 m/s
c Speed of light in vacuum 3.00 * 108
m/s
Electricity and magnetism
K Coulomb constant (1/4pP0) 8.99 * 109
N # m2
/C2
P0 Permittivity constant 8.85 * 10-12
C2
/N # m2
m0 Permeability constant 1.26 * 10-6
T # m/A
e Fundamental unit of charge 1.60 * 10-19
C
Quantum and atomic physics
h Planck constant 6.63 * 10-34
J # s 4.14 * 10-15
eV # s
U Planck constant 1.05 * 10-34
J # s 6.58 * 10-16
eV # s
aB Bohr radius 5.29 * 10-11
m
Particle masses
mp Mass of the proton (and the neutron) 1.67 * 10-27
kg
me Mass of the electron 9.11 * 10-31
kg
Common Prefixes
Prefix Meaning
femto- 10-15
pico- 10-12
nano- 10-9
micro- 10-6
milli- 10-3
centi- 10-2
kilo- 103
mega- 106
giga- 109
terra- 1012
Conversion Factors
Length
1 in = 2.54 cm
1 mi = 1.609 km
1 m = 39.37 in
1 km = 0.621 mi
Velocity
1 mph = 0.447 m/s
1 m/s = 2.24 mph = 3.28 ft/s
Mass and energy
1 u = 1.66 * 10-27
kg
1 cal = 4.19 J
1 eV = 1.60 * 10-19
J
Time
1 day = 86,400 s
1 year = 3.16 * 107
s
Force
1 lb = 4.45 N
Pressure
1 atm = 101 kPa = 760 mm Hg
1 atm = 14.7 lb/in2
Rotation
1 rad = 180°/p = 57.3°
1 rev = 360° = 2p rad
1 rev/s = 60 rpm
Greek Letters Used in Physics
Alpha a Nu n
Beta b Pi Π p
Gamma Γ g Rho r
Delta ∆ d Sigma g s
Epsilon P Tau t
Eta h Phi Φ f
Theta 𝚹 u Psi c
Lambda l Omega Ω v
Mu m
Mathematical Approximations
(1 + x)n
≈ 1 + nx if x V 1
sinu ≈ tanu ≈ u and cosu ≈ 1 if u V 1radian
ln11 + x2 ≈ x if x V 1

Brief Contents
PART I Force and Motion 2
Chapter 1 Physics for the Life Sciences 4
2 Describing Motion 25
3 Motion Along a Line 52
4 Force and Motion 90
5 Interacting Systems 124
6 Equilibrium and Elasticity 163
7 Circular and Rotational Motion 199
8 Momentum 230
9 Fluids 256
PART II Energy and Thermodynamics 308
Chapter 10 Work and Energy 310
11 Interactions and Potential Energy 337
12 Thermodynamics 378
13 Kinetic Theory 422
14 Entropy and Free Energy 459
PART III Oscillations and Waves 506
Chapter 15 Oscillations 508
16 Traveling Waves and Sound 543
17 Superposition and Standing Waves 578
PART IV Optics 620
Chapter 18 Wave Optics 622
19 Ray Optics 656
20 Optical Instruments 692
PART V Electricity and Magnetism 728
Chapter 21 Electric Forces and Fields 730
22 Electric Potential 770
23 Biological Applications of Electric Fields and Potentials 811
24 Current and Resistance 842
25 Circuits 871
26 Magnetic Fields and Forces 907
27 Electromagnetic Induction and Electromagnetic Waves 949
PART VI Modern Physics 990
Chapter 28 Quantum Physics 992
29 Atoms and Molecules 1025
30 Nuclear Physics 1058

University Physics
FOR THE LIFE SCIENCES

F
University Physics
FOR THE LIFE SCIENCES
RANDALL D. KNIGHT
California Polytechnic State University,
San Luis Obispo
BRIAN JONES
Colorado State University
STUART FIELD
With contributions by Catherine Crouch, Swarthmore College

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Library of Congress Cataloging-in-Publication Data
Names: Knight, Randall Dewey, author. | Jones, Brian, author. | Field, Stuart, author.
Title: University physics for the life sciences / Randall D. Knight,
California Polytechnic State University, San Luis Obispo, Brian Jones,
Colorado State University, Stuart Field, Colorado State University;
with contributions by Catherine Crouch, Swarthmore College.
Description: Hoboken: Pearson, 2021. | Includes index. | Summary:
“University Physics for the Life Sciences has been written in response
to the growing call for an introductory physics course explicitly
designed for the needs and interests of life science students
anticipating a career in biology, medicine, or a health-related field”—Provided by publisher.
Identifiers: LCCN 2020053801 | ISBN 9780135822180 (hardcover) | ISBN
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v
About the Authors
Randy Knight taught introductory physics for thirty-two years at Ohio State University and Cal-
ifornia Polytechnic State University, where he is Professor Emeritus of Physics. Professor Knight
received a PhD in physics from the University of California, Berkeley, and was a postdoctoral
fellow at the Harvard-Smithsonian Center for Astrophysics before joining the faculty at Ohio
State University. A growing awareness of the importance of research in physics education led
first to Physics for Scientists and Engineers: A Strategic Approach and later to College Physics:
A Strategic Approach. Professor Knight’s research interests are in the fields of laser spectroscopy
and environmental science. When he’s not in front of a computer, you can find Randy hiking,
traveling, playing the piano, or spending time with his wife Sally and their five cats.
Brian Jones has won several teaching awards at Colorado State University during his thirty
years teaching in the Department of Physics. His teaching focus in recent years has been the
College Physics class, including writing problems for the MCAT exam and helping students
review for this test. In 2011, Brian was awarded the Robert A. Millikan Medal of the American
Association of Physics Teachers for his work as director of the Little Shop of Physics, a hands-
on science outreach program. He is actively exploring the effectiveness of methods of informal
science education and how to extend these lessons to the college classroom. Brian has been
invited to give workshops on techniques of science instruction throughout the United States and
in Belize, Chile, Ethiopia, Azerbaijan, Mexico, Slovenia, Norway, Namibia, and Uganda. Brian
and his wife Carol have dozens of fruit trees and bushes in their yard, including an apple tree
that was propagated from a tree in Isaac Newton’s garden.
Stuart Field has been interested in science and technology his whole life. While in school he
built telescopes, electronic circuits, and computers. After attending Stanford University, he
earned a Ph.D. at the University of Chicago, where he studied the properties of materials at
ultralow temperatures. After completing a postdoctoral position at the Massachusetts Institute
of Technology, he held a faculty position at the University of Michigan. Currently at Colorado
State University, Stuart teaches a variety of physics courses, including algebra-based introduc-
tory physics, and was an early and enthusiastic adopter of Knight’s Physics for Scientists and
Engineers. Stuart maintains an active research program in the area of superconductivity. Stuart
enjoys Colorado’s great outdoors, where he is an avid mountain biker; he also plays in local ice
hockey leagues.
Contributing author Catherine Hirshfeld Crouch is Professor of Physics at Swarthmore
College, where she has taught since 2003. Dr. Crouch’s work developing and evaluating cur-
riculum for introductory physics for life science students has been used by faculty around the
country and has been supported by the National Science Foundation. She earned her PhD at
Harvard University in experimental condensed matter physics, and then remained at Harvard
in a dual postdoctoral fellowship in materials physics and physics education with Eric Mazur,
including developing and evaluating pedagogical best practices for undergraduate physics. She
has published numerous peer-reviewed research articles in physics education and experimental
physics, and has involved dozens of Swarthmore undergraduate students in her work. She is
married to Andy Crouch and they have two young adult children, Timothy and Amy.

vi
To the Student
If you’re taking a physics course that uses this text, chances
are that you intend a career in medicine or the life sciences.
What are you expected to learn in physics that’s relevant to
your future profession?
Understanding physics is essential to a mastery of the life
sciences for two key reasons:
■
■ Physics and physical laws underlie all physiological pro-
cesses, from the exchange of gases as you breathe to the
propagation of nerve impulses.
■
■ Many of the modern technologies used in biology and
medicine, from fluorescent microscopy to radiation ther-
apy, are based on physics concepts.
Because of this critical role, physics is a major component of
the MCAT.
Biological systems are also physical systems, and a deep
knowledge of biology requires understanding how the laws
of physics apply to and sometimes constrain biological pro-
cesses. One of our goals in this text is to build on the science
you’ve learned in biology and chemistry to provide a solid
understanding of the physical basis of biology and medicine.
Another important goal is to help you develop your quan-
titative reasoning skills. Quantitative reasoning is more than
simply doing calculations. It is important to be able to do cal-
culations, but our primary focus will be to discover and use
patterns and relationships that occur in nature. Right away,
in Chapter 1, we’ll present evidence showing that there’s a
quantitative relationship between a mammal’s mass and its
metabolic rate. That is, knowing the metabolic rate of a mouse
allows you to predict the metabolic rate of an elephant. Mak-
ing and testing predictions are at the heart of what science and
medicine are all about. Physics, the most quantitative of the
sciences, is a great place to practice these skills.
Physics and biology are both sciences. They share many
similarities, but learning physics requires a different approach
than learning biology. In physics, exams will rarely test your
ability to simply recall information. Instead, the emphasis will
be on learning procedures and skills that, on exams, you will
need to apply to new situations and new problems.
You may be nervous about the amount of mathematics
used in physics. This is common, but be reassured that you can
do it! The math we’ll use is overwhelmingly the algebra, ge-
ometry, and trigonometry you learned in high school.You may
be a bit rusty (see Appendix A for a review of the math we’ll
be using), and you almost certainly will understand this math
better after using it in physics, but our many years of teaching
experience find that nearly all students can handle the math.
This text does use some calculus, and your instructor
will decide how much or how little of that to include. Many
of the ideas of physics—how fast things happen, how things
accumulate—are expressed most naturally in the language of
calculus. We’ll introduce the ideas gently and show you how
calculus can be an important thinking and reasoning tool. In
fact, many students find they understand calculus best after
using it in physics. It’s important to become comfortable with
calculus because it is increasingly used as a quantitative tool
in the life sciences.
How To Learn Physics
There’s no single strategy for learning physics that works for
everyone, but we can make a few suggestions that will help
most students succeed:
■
■ Read all of each chapter! This might seem obvious,
but we know that many students focus their study on the
worked examples. The worked examples are important and
helpful, but to succeed on exams you will have to apply
these ideas to completely new problems. To do so, you
need to understand the underlying principles and logic that
are explained in the body of the chapter.
■
■ Use the chapter summaries. The chapter summaries are
designed to help you see the big picture of how the pieces
fit together. That said, the summaries are not a substitute
for reading the chapter; their purpose is to help you consol-
idate your knowledge after you’ve read the chapter. Notice
that there are also part summaries at the end of each of the
text’s six parts.
■
■ Actively participate in class. Take notes, answer ques-
tions, and participate in discussions. There is ample
evidence that active participation is far more effective for
learning science than passive listening.
■
■ Apply what you’ve learned. Give adequate time and
attention to the assigned homework questions and prob-
lems. Much of your learning occurs while wrestling with
problems. We encourage you to form a study group with
two or three classmates. At the same time, make sure you
fully understand how each problem is solved and are not
simply borrowing someone else’s solution.
■
■ Solve new problems as you study for exams. Questions
and problems on physics exams will be entirely new prob-
lems, not simply variations on problems you solved for
homework. Your instructor wants you to demonstrate that
you understand the physics by being able to apply it in new
situations. Do review the solutions to worked examples
and homework problems, focusing on the underlying rea-
soning rather than the calculations, but don’t stop there. A
much better use of time is to practice solving additional
end-of-chapter problems while, as much as possible, refer-
ring only to the chapter summaries.
Our sincere wish is that you’ll find your study of physics to be
a rewarding experience that helps you succeed in your chosen
field by enhancing your understanding of biology and medi-
cine. Many of our students report this was their experience!

vii
To the Instructor
University Physics for the Life Sciences has been written
in response to the growing call for an introductory physics
course explicitly designed for the needs and interests of life
science students anticipating a career in biology, medicine, or
a health-related field. The need for such a course has been
recognized within the physics education community as well
as by biological and medical professional societies. The Con-
ference on Introductory Physics for the Life Sciences Report
(American Association of Physics Teachers, 2014, available
at compadre.org/ipls/) provides background information and
makes many recommendations that have guided the develop-
ment of this text.
This new text is based on Knight Physics for Scientists
and Engineers (4th edition, 2017) and Knight, Jones, Field
College Physics (4th edition, 2019). As such, it is a research-
based text based on decades of studies into how students learn
physics and the challenges they face. It continues the engaging,
student-oriented writing style for which the earlier books are
known. At the same time, we have fully rethought the content,
ordering, examples, and end-of-chapter problems to ensure
that this text matches the needs of the intended audience.
Objectives
Our goals in writing this textbook have been:
■
■ To produce a textbook that recognizes and meets the needs
of students in the life sciences.
■
■ To integrate proven techniques from physics education re-
search and cognitive psychology into the classroom in a
way that accommodates a range of teaching and learning
styles.
■
■ To help students develop conceptual and quantitative rea-
soning skills that will be important in their professional
lives.
■
■ To prepare students to succeed on the Chemical and
Physical Foundations of Biological Systems portion of the
MCAT exam.
Content and Organization
Why develop a new textbook? What is needed to best meet the
needs of life science students? The purpose of this text is to
prepare students to grasp and apply physics content as need-
ed to their discipline of choice—biology, biochemistry, and/
or health sciences. However, the introductory physics course
taken by most life science students has for decades covered
pretty much the same topics as those taught in the course for
engineering and physics majors but with somewhat less math-
ematics. Few of the examples or end-of-chapter problems deal
with living systems. Such a course does not help life science
students see the relevance of physics to their discipline.
Many topics of biological importance are missing in a
standard introductory physics textbook. These include viscos-
ity, surface tension, diffusion, osmosis, and electrostatics in
salt water. Applications such as imaging, whether in the form
of fluorescence microscopy or scanning electron microscopy,
are barely touched on. A physics course designed for life sci-
ence students must be grounded in the fundamental laws of
physics, a goal to which this text remains firmly committed.
But how those laws are applied to the life sciences, and the
examples that are explored, differ significantly from their ap-
plication to engineering and physics.
To endeavor to connect physics to the life sciences, we
have added many topics that are important for biologists
and physicians. To make time for these, we’ve scaled back
some topics that are important for physicists and engineers
but much less so for students in the life sciences. There’s less
emphasis on standard force-and-motion problems; circular
and rotational motion has been de-emphasized (and the text
has been written to allow instructors to omit rotation entirely);
some aspects of electricity and magnetism have been reduced;
and relativity is omitted. After careful consideration, and con-
sultation with experts in biology and physics education, we’ve
made the choice that these topics are less relevant to the audi-
ence than the new content that needed to be added.
The most significant change is in the treatment of energy
and thermodynamics. Energy and entropy are crucial to all
living systems, and introductory physics could play a key role
to help students understand these ideas. However, physicists
and biologists approach these topics in very different ways.
The standard physics approach that emphasizes conserva-
tion of mechanical energy provides little insight into biological
systems, where mechanical energy is almost never conserved.
Further, biologists need to understand not how work is per-
formed by a heat engine but how useful work can be extracted
from a chemical reaction. Biologists describe energy use in
terms of enthalpy and Gibbs free energy—concepts from
chemistry—rather than heat and entropy. A presentation of
energy and entropy must connect to and elucidate reaction dy-
namics, enthalpy, and free energy if it is to help students see
the relevance of physics to biology.
Thus we’ve developed a new unit on energy and
thermodynamics that provides a coherent development of en-
ergy ideas, from work and kinetic energy through the laws of
thermodynamics. Students bring a knowledge of atoms and
molecules to the course, so a kinetic-theory perspective is
emphasized. Molecular energy diagrams and the Boltzmann
factor are used to understand what happens in a chemical re-
action, and ideas about randomness lead not only to entropy
but also to Gibbs free energy and what that tells us about the
energetics of reactions.
This text does use simple calculus, but more lightly than
in the calculus-based introductory course for physicists and

engineers. Calculus is now a required course for biology majors
at many universities, it is increasingly used as a quantitative
analysis tool in biological research, and many medical schools
expect at least a semester of calculus. Few results depend on
calculus, and it can easily be sidestepped if an instructor de-
sires an algebra-based course. Similarly, there are topics where
the instructor could supplement the text with a somewhat more
rigorous use of calculus if his or her students have the neces-
sary math background.
Although this text is oriented toward the life sciences, it
assumes no background in biology on the part of the instructor.
Examples and problems are self-contained. A basic familiarity
with chemistry and chemical reactions is assumed.
Key Features
Many of the key features of this textbook are grounded in
physics education research.
■
■ Annotated figures, now seen in many textbooks, were
introduced to physics in the first edition of Knight’s
Physics for Scientists and Engineers. Research shows that
the “instructor’s voice” greatly increases students’ ability
to understand the many figures and graphs used in physics.
■
■ Stop to Think Questions throughout the chapters are
based on documented student misconceptions.
■
■ NOTES throughout the chapters call students’ attention to
concepts or procedures known to cause difficulty.
■
■ Tactics Boxes and Problem-Solving Strategies help stu-
dents develop good problem-solving skills.
■
■ Chapter Summaries are explicitly hierarchical in design
to help students connect the ideas and see the big picture.
Instructor Resources
A variety of resources are available to help instructors teach
more effectively and efficiently. Most can be downloaded
from the Instructor Resources area of MasteringTM
Physics.
■
■ Ready-To-Go Teaching Modules are an online instruc-
tor’s guide. Each chapter contains background information
on what is known from physics education research about
student misconceptions and difficulties, suggested teaching
strategies, suggested lecture demonstrations, and suggested
pre- and post-class assignments.
■
■ Mastering Physics is Pearson’s online homework system
through which the instructor can assign pre-class reading
quizzes, tutorials that help students solve a problem with hints
and wrong-answer feedback, direct-measurement videos,
and end-of-chapter questions and problems. Instructors can
devise their own assignments or utilize pre-built assignments
that have been designed with a good balance of problem
types and difficulties.
■
■ PowerPoint Lecture Slides can be modified by the instructor
but provide an excellent starting point for class preparation.
The lecture slides include QuickCheck questions.
■
■ QuickCheck “Clicker Questions” are conceptual ques-
tions, based on known student misconceptions. They
are designed to be used as part of an active-learning
teaching strategy. The Ready-To-Go teaching modules
provide information on the effective use of QuickCheck
questions.
■
■ The Instructor’s Solution Manual is available in both
Word and PDF formats. We do require that solutions for
student use be posted only on a secure course website.
viii To the Instructor

ix
Instructional Package
University Physics for the Life Sciences provides an integrated teaching and learning package of support material for students
and instructors.
NOTE For convenience, instructor supplements can be downloaded from the Instructor Resources area of Mastering Physics.
Supplement Print Online
Instructor
or Student
Supplement Description
Mastering Physics
with Pearson eText
✓ Instructor
and Student
Supplement
This product features all of the resources of Mastering
Physics in addition to the new Pearson eText 2.0. Now
available on smartphones and tablets, Pearson eText 2.0
comprises the full text, including videos and other rich
media.
Instructor’s
Solutions Manual
✓ Instructor
Supplement
This comprehensive solutions manual contains com-
plete solutions to all end-of-chapter questions and
problems.
TestGen Test Bank ✓ Instructor
Supplement
The Test Bank contains more than 2,000 high-quality
problems, with a range of multiple-choice, true/false,
short answer, and regular homework-type questions.
Test files are provided in both TestGen®
and Word
format.
Instructor’s Resource
Materials
✓ ✓ Instructor
Supplement
All art, photos, and tables from the book are available
in JPEG format and as modifiable PowerPointsTM
. In
addition, instructors can access lecture outlines as well
as “clicker” questions in PowerPoint format, editable
content for key features, and all the instructor’s re-
sources listed above.
Ready-to-Go
Teaching Modules
✓ Instructor
Supplement
Ready-to-Go Teaching Modules provide instructors
with easy-to-use tools for teaching the toughest top-
ics in physics. Created by the authors and designed to
be used before, during, and after class, these modules
demonstrate how to effectively use all the book, media,
and assessment resources that accompany University
Physics for the Life Sciences.

x Instructional Package
Acknowledgments
We have relied on conversations with and the written pub-
lication of many members of the physics education com-
munity and those involved in the Introductory Physics for
Life Sciences movement. Those who may recognize their
influence include the late Lillian McDermott and members
of the Physics Education Research Group at the University
of Washington, Edward “Joe” Redish and members of the
Physics Education Research Group at the University of Mary-
land, Ben Dreyfus, Ben Geller, Bob Hilborn, Dawn Meredith,
and the late Steve Vogel. Ben Geller of Swarthmore College
provided useful insights into teaching thermodynamics and
contributed to the Ready-To-Go Teaching Modules.
We are especially grateful to our contributing author
Catherine Crouch at Swarthmore College for many new ideas and
examples, based on her experience developing a course for life
science majors, and for a detailed review of the entire manuscript.
Thanks to Christopher Porter, The Ohio State University,
for the difficult task of writing the Instructor’s Solutions Man-
ual; to Charlie Hibbard for accuracy checking every figure
and worked example in the text; to Elizabeth Holden, Univer-
sity of Wisconsin-Platteville, for putting together the lecture
slides; and to Jason Harlow, University of Toronto, for updat-
ing the QuickCheck “clicker” questions.
We especially want to thank Director HE Content
Management Science & Health Sciences, Jeanne Zalesky;
Product Manager Science–Physical Sciences, Darien Estes;
Senior Analyst HE Global Content Strategy–Physical Sci-
ences, Deborah Harden; Senior Development Editor, Alice
Houston; Development Editor, Edward Dodd; Senior Content
Producer, Martha Steele; Senior Associate Content Analyst
Physical Science, Pan-Science, Harry Misthos; and all the
other staff at Pearson for their enthusiasm and hard work on
this project. It has been very much a team effort.
Thanks to Margaret McConnell, Project Manager, and
the composition team at Integra for the production of the text;
Carol Reitz for her fastidious copyediting; Joanna Dinsmore
for her precise proofreading; and Jan Troutt and Tim Brum-
mett at Troutt Visual Services for their attention to detail in the
rendering and revising of the art.
And, last but not least, we each want to thank our wives
for their encouragement and patience.
Randy Knight
California Polytechnic State University.
Brian Jones
Colorado State University.
Stuart Field
Reviewers and Classroom Testers
Ward Beyermann, University of California–Riverside
Jim Buchholz, California Baptist University
David Buehrle, University of Maryland
Robert Clare, University of California–Riverside
Carl Covatto, Arizona State University
Nicholas Darnton, Georgia Tech
Jason Deibel, Wright State University
Deborah Hemingway, University of Maryland
David Joffe, Kennesaw State University
Lisa Lapidus, Michigan State University
Eric Rowley, Wright State University
Josh Samani, University of California–Los Angeles
Kazumi Tolich, University of Washington
Luc Wille, Florida Atlantic University
Xian Wu, University of Connecticut

xi
Detailed Contents
To the Student vi
To the Instructor vii
PART I Force and Motion 2
OVERVIEW The Science of Physics 3
Chapter 1 Physics for the Life Sciences 4
1.1 Why Physics? 5
1.2 Models and Modeling 7
1.3 Case Study: Modeling Diffusion 8
1.4 Proportional Reasoning: Scaling
Laws in Biology 14
1.5 Where Do We Go from Here? 20
SUMMARY 21
QUESTIONS AND PROBLEMS 21
Chapter 2 Describing Motion 25
2.1 Motion Diagrams 26
2.2 Position, Time, and
Displacement 27
2.3 Velocity 30
2.4 Acceleration 33
2.5 Motion Along a Straight Line 36
2.6 Units and Significant Figures 41
SUMMARY 46
Chapter 3 Motion Along a Line 52
3.1 Uniform Motion 53
3.2 Instantaneous Velocity 56
3.3 Finding Position from Velocity 60
3.4 Constant Acceleration 62
3.5 Solving Kinematics Problems 67
3.6 Free Fall 69
3.7 Projectile Motion 72
3.8 Modeling a Changing Acceleration 76
SUMMARY 80
Chapter 4 Force and Motion 90
4.1 Motion and Forces: Newton’s
First Law 91
4.2 Force 92
4.3 A Short Catalog of Forces 95
4.4 What Do Forces Do? 98
4.5 Newton’s Second Law 100
4.6 Free-Body Diagrams 102
4.7 Working with Vectors 106
4.8 Using Newton’s Laws 111
SUMMARY 116
Chapter 5 Interacting Systems 124
5.1 Systems and Interactions 125
5.2 Mass and Weight 125
5.3 Interactions with Surfaces 129
5.4 Drag 135
5.5 Springs and Elastic Forces 141
5.6 Newton’s Third Law 144
5.7 Newton’s Law of Gravity 150
SUMMARY 155
Chapter 6 Equilibrium and Elasticity 163
6.1 Extended Objects 164
6.2 Torque 164
6.3 Gravitational Torque and the Center
of Gravity 168
6.4 Static Equilibrium 172
6.5 Stability and Balance 176
6.6 Forces and Torques in the Body 179
6.7 Elasticity and Deformation 182
SUMMARY 190
Chapter 7 Circular and Rotational Motion 199
7.1 Uniform Circular Motion 200
7.2 Dynamics of Uniform Circular
Motion 202
7.3 The Rotation of a Rigid Body 208
7.4 Rotational Dynamics and Moment
of Inertia 212
7.5 Rolling Motion 219
SUMMARY 222

Chapter 8 Momentum 230
8.1 Momentum and Impulse 231
8.2 Conservation of Momentum 235
8.3 Collisions and Explosions 240
8.4 Angular Momentum 244
SUMMARY 248
Chapter 9 Fluids 256
9.1 Properties of Fluids 257
9.2 Pressure 258
9.3 Buoyancy 266
9.4 Surface Tension and Capillary Action 272
9.5 Fluids in Motion 279
9.6 Ideal Fluid Dynamics 281
9.7 Viscous Fluid Dynamics 284
9.8 The Circulatory System 289
SUMMARY 295
PART I SUMMARY Force and Motion 304
ONE STEP BEYOND Dark Matter and the Structure
of the Universe 305
PART I PROBLEMS 306
PART II Energy and
Thermodynamics 308
OVERVIEW Energy and Life 309
Chapter 10 Work and Energy 310
10.1 Energy Overview 311
10.2 Work and Kinetic Energy 313
10.3 Work Done by a Force That
Changes 319
10.4 Dissipative Forces and Thermal
Energy 322
10.5 Power and Efficiency 324
SUMMARY 330
Chapter 11 Interactions and
Potential Energy 337
11.1 Potential Energy 338
11.2 Conservation of Energy 340
11.3 Elastic Potential Energy 346
11.4 Energy Diagrams 350
11.5 Molecular Bonds and Chemical
Energy 354
11.6 Connecting Potential Energy
to Force 359
11.7 The Expanded Energy Model 360
11.8 Energy in the Body 362
SUMMARY 370
Chapter 12 Thermodynamics 378
12.1 Heat and the First Law
of Thermodynamics 379
12.2 Thermal Expansion 383
12.3 Specific Heat and Heat of
Transformation 385
12.4 Calorimetry 390
12.5 Heat Transfer 393
12.6 The Ideal Gas: A Model System 398
12.7 Thermodynamics of Ideal Gases 404
12.8 Enthalpy 409
SUMMARY 414
Chapter 13 Kinetic Theory 422
13.1 Connecting the Microscopic
and the Macroscopic 423
13.2 Molecular Speeds and Collisions 423
13.3 The Kinetic Theory of Gases 425
13.4 Thermal Energy and Specific Heat 431
13.5 kBT and the Boltzmann Factor 437
13.6 Reaction Kinetics and Catalysis 441
13.7 Diffusion 445
SUMMARY 452
xii Detailed Contents

Chapter 14 Entropy and Free Energy 459
14.1 Reversible and Irreversible
Processes 460
14.2 Microstates, Multiplicity, and
Entropy 463
14.3 Using Entropy 468
14.4 Spontaneity and Gibbs Free Energy 474
14.5 Doing Useful Work 479
14.6 Using Gibbs Free Energy 483
14.7 Mixing and Osmosis 485
SUMMARY 495
PART II SUMMARY Energy and Thermodynamics 502
ONE STEP BEYOND Entropy and the Living World 503
PART II PROBLEMS 504
PART III Oscillations and
Waves 506
OVERVIEW Motion That Repeats 507
Chapter 15 Oscillations 508
15.1 Simple Harmonic Motion 509
15.2 SHM and Circular Motion 513
15.3 Energy in SHM 516
15.4 Linear Restoring Forces 520
15.5 The Pendulum 525
15.6 Damped Oscillations 528
15.7 Driven Oscillations and Resonance 531
SUMMARY 534
Chapter 16 Traveling Waves and Sound 543
16.1 An Introduction to Waves 544
16.2 Visualizing Wave Motion 546
16.3 Sinusoidal Waves 550
16.4 Sound and Light 555
16.5 Circular and Spherical Waves 560
16.6 Power, Intensity, and Decibels 562
16.7 The Doppler Effect 565
SUMMARY 570
Chapter 17 Superposition and
Standing Waves 578
17.1 The Principle of Superposition 579
17.2 Standing Waves 580
17.3 Standing Waves on a String 583
17.4 Standing Sound Waves 587
17.5 The Physics of Speech 591
17.6 Interference Along a Line 593
17.7 Interference in Two and Three
Dimensions 599
17.8 Beats 603
SUMMARY 607
PART III SUMMARY Oscillations and Waves 616
ONE STEP BEYOND Waves in the Earth and the Ocean 617
PART III PROBLEMS 618
PART IV Optics 620
OVERVIEW The Wave Model of Light 621
Chapter 18 Wave Optics 622
18.1 Models of Light 623
18.2 Thin-Film Interference 625
18.3 Double-Slit Interference 630
18.4 The Diffraction Grating 634
18.5 Single-Slit Diffraction 638
18.6 Circular-Aperture Diffraction 642
18.7 X Rays and X-Ray Diffraction 644
SUMMARY 648
Detailed Contents xiii

21.6 The Motion of a Charged Particle in an
Electric Field 754
21.7 The Torque on a Dipole in an Electric
Field 756
SUMMARY 761
Chapter 22 Electric Potential 770
22.1 Electric Potential Energy 771
22.2 The Electric Potential 778
22.3 Calculating the Electric Potential 781
22.4 The Potential of a Continuous
Distribution of Charge 787
22.5 Sources of Electric Potential 791
22.6 Connecting Potential and Field 793
22.7 The Electrocardiogram 797
SUMMARY 802
Chapter 23 Biological Applications of
Electric Fields and Potentials 811
23.1 Capacitance and Capacitors 812
23.2 Combinations of Capacitors 816
23.3 Dielectrics 819
23.4 Electrostatics in Salt Water 824
23.5 The Membrane Potential of a Cell 829
SUMMARY 835
Chapter 24 Current and Resistance 842
24.1 A Model of Current 843
24.2 Defining Current 844
24.3 Batteries and emf 848
24.4 Resistance and Conductance 850
24.5 Ohm’s Law and Resistor Circuits 853
24.6 Energy and Power 856
24.7 Alternating Current 859
SUMMARY 864
Chapter 25 Circuits 871
25.1 Circuit Elements and Diagrams 872
25.2 Using Kirchhoff’s Laws 873
25.3 Series and Parallel Circuits 876
25.4 Measuring Voltage and Current 881
25.5 More Complex Circuits 882
25.6 Electric Safety 885
25.7 RC Circuits 888
25.8 Electricity in the Nervous System 891
SUMMARY 898
Chapter 19 Ray Optics 656
19.1 The Ray Model of Light 657
19.2 Reflection 659
19.3 Refraction 662
19.4 Image Formation by Refraction 667
19.5 Thin Lenses: Ray Tracing 668
19.6 The Thin-Lens Equation 674
19.7 Image Formation with Spherical
Mirrors 677
19.8 Color and Dispersion 680
SUMMARY 685
Chapter 20 Optical Instruments 692
20.1 Lenses in Combination 693
20.2 The Camera 694
20.3 The Human Eye 697
20.4 Magnifiers and Microscopes 702
20.5 The Resolution of Optical
Instruments 706
20.6 Microscopy 710
SUMMARY 717
PART IV SUMMARY Optics 724
ONE STEP BEYOND Phase-Contrast Microscopy 725
PART IV PROBLEMS 726
PART V Electricity and
Magnetism 728
OVERVIEW Charges, Currents, and Fields 729
Chapter 21 Electric Forces and Fields 730
21.1 The Charge Model 731
21.2 A Microscopic Model of Charge 737
21.3 Coulomb’s Law 740
21.4 The Electric Field 744
21.5 The Electric Field of Multiple
Charges 747
xiv Detailed Contents

28.3 Photons 1000
28.4 Matter Waves 1003
28.5 Energy Is Quantized 1005
28.6 Energy Levels and Quantum Jumps 1007
28.7 The Uncertainty Principle 1009
28.8 Applications of Quantum Physics 1012
SUMMARY 1017
Chapter 29 Atoms and Molecules 1025
29.1 Spectroscopy 1026
29.2 Atoms 1027
29.3 The Hydrogen Atom 1033
29.4 Multi-electron Atoms 1037
29.5 Excited States and Spectra 1040
29.6 Molecules 1043
29.7 Fluorescence and Bioluminescence 1046
29.8 Stimulated Emission and Lasers 1047
SUMMARY 1051
Chapter 30 Nuclear Physics 1058
30.1 Nuclear Structure 1059
30.2 Nuclear Stability 1063
30.3 The Strong Force 1066
30.4 Radiation and Radioactivity 1069
30.5 Types of Nuclear Decay 1074
30.6 The Interaction of Ionizing
Radiation with Matter 1079
30.7 Nuclear Medicine 1082
30.8 The Ultimate Building Blocks
of Matter 1086
SUMMARY 1091
PART VI SUMMARY Modern Physics 1098
ONE STEP BEYOND The Physics of Very Cold Atoms 1099
PART VI PROBLEMS 1100
Appendix A Mathematics Review A-1
Appendix B Periodic Table of Elements A-5
Appendix C Studying for and Taking
the MCAT Exam A-7
Appendix D Atomic and Nuclear Data A-11
Answers to Odd-Numbered Problems A-15
Credits C-1
Index I-1
Chapter 26 Magnetic Fields and Forces 907
26.1 Magnetism 908
26.2 The Magnetic Field of a Current 911
26.3 Magnetic Dipoles 915
26.4 The Magnetic Force on a Moving
Charge 920
26.5 Magnetic Forces on Current-
Carrying Wires 927
26.6 Forces and Torques on Magnetic
Dipoles 929
26.7 Magnetic Resonance Imaging 932
SUMMARY 940
Chapter 27 Electromagnetic Induction
and Electromagnetic Waves 949
27.1 Induced Currents 950
27.2 Motional emf 951
27.3 Magnetic Flux and Lenz’s Law 955
27.4 Faraday’s Law 960
27.5 Induced Fields 963
27.6 Electromagnetic Waves 965
27.7 Polarization 968
27.8 The Interaction of Electromagnetic
Waves with Matter 971
SUMMARY 977
PART V SUMMARY Electricity and Magnetism 986
ONE STEP BEYOND The Greenhouse Effect and Global
Warming 987
PART V PROBLEMS 988
PART VI Modern Physics 990
OVERVIEW New Ways of Looking at the World 991
Chapter 28 Quantum Physics 992
28.1 Physics at the Atomic Level 993
28.2 The Photoelectric Effect 994
Detailed Contents xv

2
Force and Motion
P A R T
I
Elite athletes push the human body’s physical limits.What forces act on
this sprinter as she accelerates? How much force can her muscles, tendons,
and bones endure? How much air can flow into her lungs? How rapidly can
blood be pumped through her veins?These are physics questions that help
us understand human performance, questions we’ll address in Part I.

The Science of Physics
Physics is the foundational science that underlies biology, chemistry, earth science,
and all other fields that attempt to understand our natural world. Physicists couple
careful experimentation with theoretical insights to build powerful and predictive
models of how the world works. A key aspect of physics is that it is a unifying dis-
cipline: A relatively small number of key concepts can explain a vast array of natural
phenomena. In this text, we have organized the chapters into parts according to six
of these unifying principles. Each of the six parts opens with an overview that gives
you a look ahead, a glimpse of where your journey will take you in the next few
chapters. It’s easy to lose sight of the big picture while you’re busy negotiating the
terrain of each chapter. In Part I, the big picture is, in a word, change.
Why Things Change
Simple observations of the world around you show that most things change. Some
changes, such as aging, are biological. Others, such as the burning of gasoline in
your car, are chemical. In Part I, we will look at changes that involve motion of one
form or another—from running and jumping to swimming microorganisms.
There are two big questions we must tackle to study how things change by moving:
■ How do we describe motion? How should we measure or characterize the mo-
tion if we want to analyze it quantitatively?
■ How do we explain motion? Why do objects have the particular motion they
do? When you toss a ball upward, why does it go up and then come back down
rather than keep going up? What are the “laws of nature” that allow us to predict
an object’s motion?
Two key concepts that will help answer these questions are force (the “cause”) and
acceleration (the “effect”). Our basic tools will be three laws of motion worked out
by Isaac Newton to relate force and motion. We will use Newton’s laws to explore
a wide range of problems—from how a sprinter accelerates to how blood flows
through the circulatory system. As you learn to solve problems dealing with motion,
you will be learning techniques that you can apply throughout this text.
Using Models
Another key aspect of physics is the importance of models. Suppose we want to
analyze a ball moving through the air. Is it necessary to analyze the way the atoms in
the ball are connected? Or the details of how the ball is spinning? Or the small drag
force it experiences as it moves? These are interesting questions, of course. But if
our task is to understand the motion of the ball, we need to simplify!
We can conduct a perfectly fine analysis of the ball’s motion by treating the
ball as a single particle moving through the air. This is a model of the situation. A
model is a simplified description of reality that is used to reduce the complexity of a
problem so it can be analyzed and understood. Both physicists and biologists make
extensive use of models to simplify complex situations, and in Part I you’ll begin to
learn where and how models are employed and assessed. Learning how to simplify
a situation is the essence of successful problem solving.
3
O V E R V I E W

LOOKING AHEAD
Biology Is Subject to the Laws of Physics
The rich diversity of life on our planet shares one thing in common: All living organisms are
subject to the laws of physics. Physical laws, such as energy conservation and the laws of
fluid flow, constrain what is possible for life, and life has responded to the challenge spec-
tacularly. Physics also provides powerful tools for the life sciences, enabling us to image
and measure cells and organisms ranging from viruses to humans. Biologists and physicians
emphasize the importance of understanding how these tools work—particularly, recognizing
when tools aren’t working correctly or won’t work effectively in a certain situation. Our goal
in this text is to help you understand living systems and biomedical tools more deeply by
exploring the physical mechanisms at work from large scales to small, from organisms down
to molecules. This first chapter will introduce you to several key ideas.
PHYSICS AND LIFE
This scanning electron microscope image
shows Salmonella bacteria (red) invading
immune cells.You’ll learn about electron
microscopes in Chapter 28.
GOAL To understand some of the ways that physics and quantitative reasoning shed light on biological processes.
Chapter Previews
Each chapter starts with a preview outlining
the major topics of the chapter and how they
are relevant to the life sciences.
Modeling
A dialysis machine serves as an artificial kid-
ney by having waste products diffuse through
a membrane from blood into a dialysis liquid.
Scaling
If you know the metabolic rate of a hamster,
you can calculate the metabolic rate of a
horse—or of any other mammal.
You’ll see how physicists model complex
situations as we construct a model of dif
fusion based on the idea of a random walk.
You’ll learn how scaling laws connect many
basic physiological processes to body size
or body mass.
Studies find that your understanding of a
chapter is improved by knowing what key
points to look for as you read.
Magnetic resonance imaging (MRI) is
just one of many ways that physics
has contributed to biology and
medicine.We’ll look at how images
like this are created in Chapter 26.
1 Physics for the Life
Sciences

1.1 Why Physics? 5
1.1 Why Physics?
Why take physics? Does knowing about projectiles, pendulums, or magnetic fields
help you understand biological systems? Do doctors or microbiologists or ecologists
think about or use the principles of physics?
Actually, the answer to these questions is Yes. The existence of separate biology,
chemistry, and physics departments may make it seem like these are distinct sci-
ences, but that couldn’t be further from the truth. Our overarching goal in this text is
to help you discover the central importance of physics to the life sciences. Biological
systems are part of the physical world, and biological processes are physical pro-
cesses, so the laws of physics can help you understand a great deal about biology,
biochemistry, and medicine. Let’s look at some examples:
Physics in biology
The circulatory system, from the heart to
the capillaries, is governed by fluid
dynamics. You’ll study the physics of
circulation in Chapter 9.
Neurons signal each other via electric
pulses that travel along axons. This is a
topic we’ll visit in Chapter 25.
The light-emitting properties of the green
fluorescent protein are used to visualize
cell structure. Fluorescence is covered in
Chapter 29.
Physics in biomechanics
The physics of locomotion affects how
fast an animal can run. Locomotion is the
topic of several chapters in Part I.
Physics helps us understand the limits of
athletic performance. Chapter 11 looks at
how the body uses energy.
And physics explains how sap rises in trees
through the xylem. This somewhat counter-
intuitive fluid flow is described in Chapter 9.
Physics in medicine
Pulse oximetry measures blood oxygen
with a clip-on device. Chapter 29 discusses
how blood absorbs different colors of light.
A patient prepares to have cancer treated
by proton irradiation. Radiation and radia-
tion therapy are topics in Chapter 30.
Lasers are used for high-precision eye surgery
of both the cornea and the retina. Chapter 20
covers the optics of the human eye.

6 c h a p t e r 1 Physics for the Life Sciences
These are among the many applications of physics to the life sciences that you’ll
learn about in this book. That said, this is a physics textbook, not a biology textbook.
We need to develop the underlying principles of physics before we can explore the ap-
plications, so we will often start with rolling balls, oscillating springs, and other simple
systems that illustrate the physical principles. Then, after laying the foundations, we will
move on to see how these ideas apply to biology and medicine. As authors, our sincere
hope is that this course will help you see why many things in biology happen as they do.
Physics and Biology
Physicists and biologists are scientists: They share many common views of what sci-
ence is and how science operates. At the same time, as TABLE 1.1 illustrates, physicists
and biologists often see the world in rather different ways. Your physics course will
be less stressful and more productive if you’re aware of these differences.
Physics tries to get at the essence of a process by identifying broad principles,
such as energy conservation, and applying them to systems that have been greatly
simplified by stripping away superfluous details. This approach is less common in
biology, where systems often can’t be simplified without throwing out details that
are essential to biological function. You may have taken biology courses in which
you were expected to memorize a great deal of information; there’s much less to
memorize in physics. Instead, most physicists agree that students demonstrate their
understanding of physics by using general principles to solve unfamiliar problems.
With that in mind, we can establish four large-scale goals for this text. By the end,
you should be able to:
• Recognize and use the principles of physics to explain physical phenomena.
• Understand the importance of models in physics.
• Reason quantitatively.
• Apply the principles of physics to biological systems.
The examples in this text will help you at each step along the way, and the end-of-
chapter problems will provide many opportunities for practice.
Mathematics
Physics is the most quantitative of all the sciences. As physics has developed, the
laws of physics have come to be stated as mathematical equations. These equations
can be used to make quantitative, testable predictions about nature, whether it’s the
orbit of a satellite or the pressure needed to pump blood through capillaries. This
approach to science has proven to be extremely powerful; much of modern bio-
medical technology—from electron microscopes to radiation therapy—depends on
the equations of physics.
But math is used in physics for more than simply doing calculations. The equa-
tions of physics tell a story; they’re a shorthand way to describe how different
concepts are related to one another. So, while we can use the ideal-gas law to calcu-
late the pressure in a container of gas, it’s more useful to recognize that the pressure
of a gas in a rigid container (one that has a constant volume) increases in exactly the
same proportion as the absolute temperature. Consequently, doubling the tempera-
ture causes the pressure to double. The ideal-gas law expresses a set of deep ideas
about how gases behave.
So, yes, we will often use equations to calculate values. But, more fundamentally,
physics is about using math to reason and to analyze; that is, math is a thinking tool
as much as it is a calculation tool. This may be a new way of using math for you,
but—with some practice and experience—we think you’ll come to recognize the
power of this way of thinking.
The math used in this book is mostly math you already know: algebra, geometry,
and some trigonometry. You might need some review (see Appendix A), but we’re
confident that you can handle the math. Physics does use many symbols to represent
TABLE 1.1 Characteristics of biology
and physics
Biology Physics
Irreducibly complex
systems
Simple models
More qualitative More quantitative
Focus on specific
examples
Focus on broad
principles
Seeing the details A scanning electron
microscope produces highly detailed images
of biological structures only a few nanome-
ters in size. The level of detail in this image
of a budding HIV virus can provide new
understanding of how the virus spreads and
how it might be stopped. Physics has been at
the forefront of developing a wide variety of
imaging technologies.

1.2 Models and Modeling 7
quantities, such as F for force, p for pressure, and E for energy, so many of our
equations—like the ideal-gas law—are algebraic equations that show how quanti-
ties are related to one another. It’s customary to use Greek letters to represent some
quantities, but we’ll let you know what those letters are when we introduce them.
We will, in some chapters, use a little bit of the calculus of derivatives and inte-
grals. Don’t panic! Our interest, once again, is not so much in doing calculations as in
understanding how different physical quantities are related to one another. And we’ll
remind you, at the appropriate times, what derivatives and integrals are all about. In
fact, most students feel that they come to understand calculus much better after study-
ing physics because physics provides a natural context for illustrating why calculus
is useful.
1.2 Models and Modeling
The real world is messy and complicated. A well-established procedure in phys-
ics is to brush aside many of the real-world details in order to discern patterns that
occur over and over. For example, an object oscillating back and forth on a spring,
a swinging pendulum, a vibrating guitar string, a sound wave, and an atom jiggling
in a crystal seem very different—yet perhaps they are not really so different. Each
is an example of a system oscillating around an equilibrium position. If we focus on
understanding the properties of a very simple oscillating system, we’ll find that we
understand quite a bit about the many real-world manifestations of oscillations.
Stripping away the details to focus on essential features is a process called model
ing. A model is a simplified picture of reality, but one that still captures the essence
of what we want to study. Thus “mass on a spring” is a model of almost all oscillat-
ing systems. Models allow us to make sense of complex situations by providing a
framework for thinking about them.
A memorable quote attributed to Albert Einstein is that physics “should be as
simple as possible—but not simpler.” We want to use the simplest model that al-
lows us to understand the phenomenon we’re studying, but we can’t make the model
so simple that essential features of the phenomenon are lost. That’s somewhat of a
problem in this text because understanding the physics of a biological system is not
the same as understanding the biology. Our models of biological systems may seem
to throw out much of the relevant biology, but keep in mind that our goal is to un-
derstand the ways in which the physical properties of the system have a meaningful
effect on aspects of the biology.
We’ll develop and use many models throughout this textbook; they’ll be one of
our most important thinking tools. These models will be of two types:
• Descriptive models: What are the essential characteristics and properties of a phe-
nomenon? How do we describe it in the simplest possible terms? For example,
we will often model a cell as a water-filled sphere. This omits all the details about
what’s inside, but for some purposes, such as estimating a cell’s mass or volume,
we don’t need to know what’s inside.
• Explanatory models: Why do things happen as they do? What causes what?
Explanatory models have predictive power, allowing us to test—against experi-
mental data—whether a model provides an adequate explanation of an observed
phenomenon. A spring-like model of molecular bonds will allow us to explain
many of the thermal properties of materials.
Biologists also use models. A biological model, often qualitative rather than
quantitative, is a simplified representation of the structure and function of a biologi-
cal system or a biological process. The cell model is a simplified presentation of an
immensely complex system, but it allows you to think logically about the key pieces
and processes of a cell. Mice, fruit flies, and nematodes are important model organ
isms that lend themselves to the study of particular biological questions without
unnecessary complications.
The eyes have it We study a model organ
ism not to learn details about the organism
but because the organism lends itself to the
understanding of broad biological principles.
Drosophila melanogaster, a common fruit
fly, has provided immeasurable insights into
genetics for more than a century.

At the same time, biology and medicine are becoming increasingly quantitative,
with models more and more like those constructed by physicists. For example:
• Mathematical models of enzyme kinetics provide quantitative predictions of
complex biochemical pathways.
• Neural network models improve our understanding of both real brains and artifi-
cial intelligence.
• Epidemiological models increase our knowledge of how disease spreads.
• Global climate models that illustrate the earth’s climate and how it is changing
depend on a complex interplay between living (e.g., photosynthesis) and nonliv-
ing (e.g., solar radiation) processes.
These models skip over many details in order to provide a big-picture understand-
ing of the system. That’s the purpose of a model. This is not to say that details are
unimportant—most scientists spend most of their careers studying the details—but
that we don’t want to let the trees obscure our view of the forest.
1.3 Case Study: Modeling Diffusion
Diffusion is one of the most important physical processes in biology. Oxygen diffuses
from your lungs to your blood, and neurotransmitters diffuse across the synapses that
connect one neuron to the next. We’ll study diffusion extensively in Chapter 13, but
as a case study let’s see how a physicist might model diffusion and conclude when
diffusion has biological significance. That is, our goal is to create a model that makes
testable predictions of how far molecules can diffuse and how long it takes them to
do so.
NOTE ▶ The analysis in this section is more complex than you are expected to
do on your own. However, it is expected that you will follow the reasoning and be
able to answer questions about the procedure and the results. ◀
On a microscopic scale, molecules are constantly jostling around and colliding
with one another, an atomic-level motion we’ll later associate with thermal energy.
Any one molecule moves only a short distance before a collision sends it off in a dif-
ferent direction. Its trajectory, if we could see it, might look something like FIGURE 1.1:
lots of short, straight segments of apparently random lengths in what appear to be
random directions.
This is a chaotic and complex motion. The essence of modeling is to make simpli-
fying assumptions, so how might we begin? The photographs in FIGURE 1.2 provide a
clue. The left photo shows blue dye carefully deposited as a thin layer in a test tube
of agar gel. The right photo is the same test tube one week later. It’s a slow process,
but the dye has diffused both up and down in a symmetrical pattern. This suggests
that we start not with the complex three-dimensional motion of a molecule but with
the simpler case of diffusion along a one-dimensional linear axis.
A sidewalk is a linear axis. Suppose you stand at one location on an east-west
sidewalk and flip a coin. If it’s heads, you take one step to the east; if tails, one step
to the west. Then you toss the coin again, and again, and again, each time randomly
taking a step to the east or to the west. You would be engaged in what physicists and
mathematicians call a one-dimensional random walk. Because collisions randomly
redirect molecules, let’s see what we can learn by modeling molecular motion as a
random walk.
NOTE ▶ Variations on the random-walk model are used in applications rang-
ing from protein folding and genetic drift to predicting share prices on the stock
market. ◀
In particular, let’s imagine a molecule that starts at the origin, x = 0, and then
takes a random walk along the x-axis. That is, the molecule, at regular intervals,
FIGURE 1.1 The possible motion of
one molecule as it collides with other
molecules.
FIGURE 1.2 Vertical diffusion of blue dye.

randomly takes a step whose length we’ll call d in either the +x@direction or the
-x@direction. At each step there’s a 50% chance of going either way. To help make
this clear, FIGURE 1.3 shows what might be the first 10 steps of the molecule. The first
two steps are to the right, the third back to the left, and so on. The molecule seems
to wander aimlessly—that’s the essence of random motion—and after 10 steps its
position is x10 = -2d.
Now diffusion involves not one molecule but many, so imagine that we have a very
large number of molecules that each move in one dimension along the x-axis. Assume
that each molecule starts at the origin and then undergoes a random walk. What can
we say about the collective behavior of a large number of random-walking molecules?
This is a problem that can be worked out exactly by using statistics, but the
mathematical manipulations are a bit tricky. Instead, we’ll explore the model as a
computer simulation, one that you could do yourself in a spreadsheet. Suppose you
put a zero in a spreadsheet cell to show a molecule’s starting position. In the cell to
the right, you use the spreadsheet’s random-number generator—a digital coin flip—
to either add d to the initial position (a step to the right) or subtract d from the initial
position (a step to the left). Then you do the same thing in the next cell to the right,
and then the next cell to the right of that, and so on, each time adding or subtracting
d from the previous cell with a 50% chance of each. The 101st cell will be the mol-
ecule’s position after taking 100 random steps.
Now you can add a second row of cells for a second molecule, a third row for
a third molecule, and so on. It might take a big spreadsheet, but you can have as
many molecules and as many steps as you wish. We’ve used exactly this procedure
to simulate the random walks of 1000 molecules for 100 steps each. FIGURE 1.4 gives
the result by using dots to show the final positions of each of these molecules. Note
that all molecules must be at an even multiple of d after an even number of steps, so
the possible positions after 100 steps are 0, {2d, {4d, and so on.You can see that the
first molecule—the top row—ends up at x100 = 4d while the 453rd molecule in
the 453rd row ends at x100 = 24d.
x
1
2
4
8
3
5
6
7
9
10
-3d -2d -d 0 d 2d 3d
Each step is
equally likely to
go right or left.
Step number
x10 is the molecule’s
position at the end
of step 10.
x10
x4
x0
FIGURE 1.3 The first 10 steps of one
possible random walk.
-30d
1
1000
-20d -10d 0
Position on the x-axis after 100 steps
Molecule
number
10d 20d 30d
Molecule 1 ends
up at x100 = 4d.
Molecule 453 ends
up at x100 = 24d.
FIGURE 1.4 The result of 1000 molecules following a random walk for 100 steps.
The motion may be random, but the collective motion of many molecules reveals
a pattern. Figure 1.4 shows that roughly half the molecules end up to the right of the
origin and half to the left. That’s exactly what we would expect from randomly tak-
ing steps to the right or left. Mathematically, we can say that the average position of
all 1000 molecules is xavg = 0. That is, the center of this array of dots is at the origin.
You can see this in the second photo of Figure 1.2: The dye has spread, but it has
done so symmetrically so that the center of the dye, its average position, is still in the
middle of the test tube.
The center may not have moved, but the molecules are spreading out, which is
exactly what diffusion is. However, the spreading is perhaps not as much as you
might have expected. After 100 steps, a molecule could have reached x = 100d, but

you can see from Figure 1.4 that, in fact, most are less than 10d from the origin.
Even the most adventuresome molecule—one in a thousand—has moved out a dis-
tance of only 30d. A moment’s thought tells why. Any molecule’s position after 100
steps is determined by 100 coin tosses. Although you don’t expect to get exactly
50 heads and 50 tails, you do expect the number of heads to be fairly similar to the
number of tails. In the same way, a molecule won’t move very far from the origin if
the number of steps it takes to the right is fairly similar to the number it takes to the
left. Reaching x = 100d would require tossing 100 heads in a row, an outcome that
is extraordinarily unlikely.
We can analyze the 1000 molecular positions in Figure 1.4 by creating the
histogram of FIGURE 1.5. A histogram is a bar chart that shows how many molecules
end up at each position. The histogram is a bell curve, showing that x100 = 0 is the
most likely final position, occurring 80 times out of the 1000 molecules, and that
final positions becomes less likely as you move away from the origin. The computer-
simulation bell curve is a bit ragged with only 1000 molecules, but, as you can
imagine, it would become a very smooth curve if we could run the simulation with
billions of molecules.
Measuring the Spread Due to Diffusion
Diffusing molecules spread out, and that’s exactly what the molecules in our model
are doing. How can we measure this? What quantity describes the amount of spread-
ing? Averaging the positions of all the molecules results in zero, as we’ve seen, so the
average position doesn’t tell us anything about spreading.You might think of averag-
ing the absolute values of the positions; those are all positive, and their average—
which is not zero—really would give some information about spreading. However, it
turns out that working with absolute values presents mathematical difficulties.
Rather than averaging absolute values, let’s average the squares of the positions
of each of the molecules. That is, we square the positions of all 1000 molecules,
add the squares, and then divide by 1000 to find the average. We’ll use the nota-
tion (x2
)avg to indicate this average of the squares of the positions. Squares are
positive, so (x2
)avg is not zero. But is it useful? To find out, FIGURE 1.6 graphs x2
during the 100 steps of our computer simulation both for one molecule and for
(x2
)avg when averaged over all 1000 molecules.
The graph for one molecule is always positive, as it must be, but not very useful.
It looks like the molecule managed to walk out to x = {8d in 33 steps, making
x2
= 64d2
, then returned to the origin after 43 steps, got as far as x = {9d after 87
and 89 steps, but ended back at x = {6d. All in all, it’s pretty much what you might
expect for one molecule engaged in a random walk.
But averaging the squares of the positions of a large number of molecules tells a
different story. You can see that (x2
)avg increases linearly with the number of steps,
and the average becomes a better and better straight line if we increase the number of
molecules in the average. Furthermore, it appears that (x2
)avg ≈ 50d2
after 50 steps
and (x2
)avg ≈ 100d2
after 100 steps. That is, our computer simulation suggests that
(x2
)avg = nd2
after n steps. And, indeed, a rigorous statistical analysis proves that this
is true.
The square root of (x2
)avg is called the root-mean-square distance xrms, often
called the rms distance. Taking the square root of (x2
)avg = nd2
gives
xrms = 2(x2
)avg = 1nd (1.1)
The root mean square—the square root of the mean (i.e., the average) of the
squares—is a useful way of describing the spread of a set of values that are sym-
metrical about zero. We’ll revisit this concept at several points throughout this text.
For our random-walking molecules, the root-mean-square distance xrms after n steps
is simply the step size d multiplied by the square root of n. Thus xrms = 116d = 4d
after 16 steps and xrms = 1100d = 10d after 100 steps, as our simulations showed.
0 20 40 60 80 100
100d2
80d2
60d2
40d2
20d2
0
Number of steps
(x2
)avg for 1000 molecules
x2
of one molecule
x2
FIGURE 1.6 A graph showing how x2
changes for one molecule and, on
average, for 1000 molecules during the
first 100 steps of a random walk.
-30d
80
60
40
20
0
-20d -10d 0
Position on the x-axis after 100 steps
Number
of
occurrences
10d 20d 30d
The most likely
final position is
x100 = 0.
xrms
FIGURE 1.5 A histogram of the 1000
molecular positions in Figure 1.4.

The root-mean-square distance is shown on the histogram of Figure 1.5, and you can
see that xrms is a reasonably good answer to the question: What is the typical distance
that the molecules have spread after n steps?
It can be shown, using the tools of probability and statistics, that xrms is the stan
dard deviation of the histogram, a term you may have encountered in statistics. One
can show that, on average, 68% of all the molecules have positions between -xrms
and +xrms, while only 5% have traveled farther than 2xrms. Of our 1000 molecules
that took 100 steps and have xrms = 10d, we would expect 680 to have traveled no
farther than 10d from the origin and only 50 to have exceeded 20d. This is a good
description of the results shown in Figures 1.4 and 1.5. Thus xrms appears to be quite
a good indicator of about how much spreading has occurred after n steps. We can say
that xrms is the diffusion distance.
What is the diffusion distance as a fraction of the maximum
possible travel distance after 102
, 106
, and 1012
steps?
PREPARE The maximum possible travel distance increases
linearly with the number of steps n, but the diffusion distance
increases more slowly, with the square root of n.
SOLVE If a molecule moves in the same direction for every
step, it will travel distance xmax = nd in n steps. This is the
maximum possible travel distance. The diffusion distance is
xdiff = xrms = 1nd. Thus
xdiff
xmax
=
1nd
nd
=
1
1n
Diffusive spreading
For the different values of n we can compute
xdiff
xmax
= •
10-1
for n = 102
10-3
for n = 106
10-6
for n = 1012
ASSESS Compared to how far a molecule could travel, the diffu-
sion distance rapidly decreases as the number of steps increases.
Diffusion is 10% of the maximum distance after 100 steps but
only one-millionth of the maximum distance after 1012
steps, a
number that is comparable to the number of collisions a molecule
undergoes each second. Having lots of collisions does not mean
rapid spreading.
EXAMPLE 1.1
NOTE ▶ Worked examples in this text will follow a three-part problem-solving
strategy: Prepare, Solve, and Assess. These examples are intended to illustrate
good problem-solving procedures, and we strongly encourage you to use the
same approach in your own work. We’ll look at this problem-solving strategy in
more detail in Chapter 3. ◀
Connecting to the Real World
Our analysis of the random walk has revealed some interesting similarities to what
is known about diffusion, but how can we judge whether this model is an accurate
description of diffusion? What quantitative prediction can we make to connect this
model to what happens in the physical world?
Two things that we can easily measure are distance and time, so let’s work out
the relationship between the diffusion distance and the length of time that the sys-
tem spends diffusing. We can then compare measurements to the predictions of this
model to see how good the model is. First, let’s represent the time interval between
steps of the random walk by the symbol ∆t, where ∆ is an uppercase Greek delta.
As you’ll see in Chapter 2, we use the symbol t to represent a specific instant of time
and ∆t to represent an interval of time. If the molecules start moving at time t = 0,
then at a later time t they will have taken n = t/∆t steps. If, for example, a step is
taken every ∆t = 1
10 s, there will have been 100 steps at t = 10 s. We can write xrms
in terms of t and ∆t as
xrms = 1nd =
A
t
∆t
d =
B
td2
∆t
= 1vdt
For the last step we used the fact that d/∆t = distance/time, as in miles per hour or
meters per second, is the molecule’s speed. Throughout this text we’ll use the symbol v
(from velocity) to represent speed, with v = d/∆t.

Let’s define the diffusion coefficient D to be
D =
1
2
d2
∆t
=
1
2
vd (1.2)
The diffusion coefficient connects our model to the physical world because it de-
pends on the speed v of molecules and the distance d traveled between collisions. A
factor of 1
2 is included in the definition to avoid a factor of 2 in an important equation
that you’ll meet in Chapter 13. Thus our model predicts that the root-mean-square
distance should increase with time as
xrms = 22Dt (1.3)
The real world is three dimensional, not a straight line. That turns out to be an easy
extension of our model. Suppose each step is {d along the x-axis and {d along
the y-axis and {d along the z-axis. In this 3D world, a molecule’s distance from the
starting point is r = 2x2
+ y2
+ z2
. Averaging the squares, as we did above, gives
(r2
)avg = (x2
)avg + (y2
)avg + (z2
)avg
The x-axis does not differ from the y-axis or the z-axis, so averages should be the
same along each axis. Thus
(r2
)avg = (x2
)avg + (y2
)avg + (z2
)avg = 2Dt + 2Dt + 2Dt = 6Dt (1.4)
If we define the three-dimensional root-mean-square distance as rrms = 2(r2
)avg ,
then
rrms = 26Dt (1.5)
This is our model’s prediction for how far molecules will diffuse in three dimensions
in time t.
Equation 1.5 is a quantitative, testable prediction of our model. It says that the
distance molecules diffuse should increase with the square root of the elapsed time.
This is not what your intuition suggests. If you walk or run or drive at a steady speed,
you go twice as far in twice the time. But because 22 = 1.41, our prediction is that
molecules will diffuse only 1.41 times as far in 20 s as they do in 10 s. It will take
40 s to diffuse twice as far as in 10 s. In fact, this prediction is confirmed by count-
less experiments.
To really test our model, we would also like to predict a numerical value for
the diffusion coefficient D. What do we know that will help us at least estimate D?
Estimating is a valuable scientific skill, one we’ll spend a fair amount of time on in
this text, so let’s look at how to approach an estimation.
Our random-walk model is a simplification of the idea that molecules are constantly
colliding and changing direction, so let’s think about what is known about molecular
motions and collisions. For one thing, we can infer that molecules in a liquid are pretty
much touching each other. Liquids, unlike gases, are essentially incompressible—the
molecules can’t get any closer together. The distance a molecule can move between
collisions is at most the diameter of a molecule, probably less. You learned in chem-
istry that the size of small molecules such as oxygen and water is about 0.1 nm,
where 1 nm = 1 nanometer = 10-9
m. So let’s estimate that the random-walk step
size, the distance between collisions, is d ≈ 0.05 nm = 5 * 10-11
m.
How fast do molecules move? We can estimate molecular speeds by thinking
about sound waves. Sound travels through a medium because the molecules run into
each other, propagating a sound wave forward. We’ll look at the details when we
study sound, but it seems reasonable that the sound speed is probably fairly similar to
the speed of a molecule. We could look up the speed of sound in a reference book or
on the Internet, but we can also estimate it from an experience that many of you have
had.You may have learned that the distance to a lightning strike can be determined by
counting the seconds from seeing the lightning to hearing the thunder, then dividing
by 5 to get the distance in miles. The implication is that the speed of sound is about
Crossing the gap Nerve impulses travel-
ing from neuron to neuron have to cross the
synaptic cleft between the axon terminal of
one neuron and the dendrite receptor of the
next. When the sending terminal on the axon
is electrically activated, it releases chemi-
cals called neurotransmitters that simply
diffuse across the gap. The gap is so narrow,
approximately 20 nm in width, that the
diffusing neurotransmitters reach the other
side and stimulate a response in less than a
microsecond.

1
5 mi/s. A mile is about 1600 m, so 1
5 mi/s = 320 m/s. Consequently, our second esti-
mate is that the speed with which molecules travel between collisions is v ≈ 320 m/s.
We have assumed that molecular speeds in liquids are not greatly different from
molecular speeds in air. You’ll learn when we study waves that the speed of sound in
water is about four times the speed in air. That’s a big difference for some applications,
such as whether ultrasound imaging works better in air or water, but it’s not a large
discrepancy when our goal is merely a rough estimate of the diffusion coefficient.
With that, we can now predict that the diffusion coefficient for small molecules
diffusing through a liquid like water should be approximately
D = 1
2 vd ≈ 1
2 (320 m/s)(5.0 * 10-11
m) ≈ 8 * 10-9
m2
/s (1.6)
Notice the rather unusual units for a diffusion coefficient; these arise because of the
square-root relationship between diffusion distance and time.
Also notice that the value is given to only one significant figure because the num-
bers that went into this calculation are only estimates; they are not precisely known.
This is called an order-of-magnitude estimate, meaning that we wouldn’t be sur-
prised to be off by a factor of three or four but that our estimate is almost certainly
within a factor of ten of the real value. That may seem a poor outcome, but initially
we had no idea what the value of a diffusion coefficient might be. To have deter-
mined by simple reasoning that D is approximately 10-8
m2
/s (rounding the value of
D in Equation 1.6 to the nearest power of 10), rather than 10-10
m2
/s or 10-6
m2
/s,
is an important increase in our understanding of diffusion, one that can be improved
only by careful laboratory measurements.
Estimate how long it takes small molecules such as oxygen to
diffuse 10 mm (about the diameter of a cell) and 10 cm (roughly the
size of a small animal) through water, the primary component of
intercellularfluid.Notethat 1 mm = 1 micrometer or 1 micron =
10-6
m.
PREPARE The root-mean-square distance rrms tells us how far
diffusion carries the molecules in time t. We’ll use our estimated
D ≈ 8 * 10-9
m2
/s as the diffusion coefficient.
SOLVE The root-mean-square distance is rrms = 26Dt. Solving
for the time needed for molecules to diffuse distance rrms gives
t =
(rrms)2
6D
Diffusing across a cell requires rrms = 10 mm = 1 * 10-5
m.
Thus
tcell =
(1.0 * 10-5
m)2
6(8 * 10-9
m2
/s)
≈ 2 * 10-3
s = 2 ms
Diffusion times
But diffusing through a small animal requires rrms = 10 cm =
0.1 m. In this case,
tcell =
(0.1 m)2
6(8 * 10-9
m2
/s)
≈ 2 * 105
s ≈ 2 days
ASSESS Diffusion of oxygen across a cellular distance of a few
mm is very rapid. Oxygen in your blood is easily transported
throughout cells by diffusion as long as no cell is more than a
couple of cells away from a capillary. Unicellular organisms
such as amoebas can use diffusion to passively harvest the oxy-
gen from their environment. But diffusion across larger distances
is impractically slow. A small animal, or one of your organs, is
roughly 104
times larger than a cell. Because diffusion times de-
pend on the square of the distance, diffusion across 10 cm takes
108
times as long as diffusion across a cell. The second photo in
Figure 1.2, the dye diffusion, was taken a week after the experi-
ment started, and the diffusion distance is only a few centimeters.
EXAMPLE 1.2
We should feel pretty satisfied. Starting with a very simplified idea about what hap-
pens in molecular collisions and using just a few basic ideas about the physical world,
we’ve put together a model of diffusion that has strong explanatory power. Our esti-
mated value of D for small molecules turns out to be about a factor of four too large, as
you’ll see in Chapter 13, but, as we might say colloquially, it’s certainly in the ballpark.
Furthermore, this result tells us why an amoeba doesn’t have lungs. A creature of
its size doesn’t need them! But you do. Diffusion is inadequate to transport oxygen
and other molecules throughout an organism of your size, so you need an actively
powered (i.e., using metabolic energy) circulatory system. The laws of physics have
real consequences for organisms.

Our goal in this section was to demonstrate how physicists model a complex situ-
ation and how physics can increase your understanding of biological systems. The
mathematics we used was not especially complex, but we certainly emphasized quan-
titative reasoning. Graphs were also a useful tool to shape our thinking. These are
the types of reasoning skills that we will help you develop throughout this textbook.
NOTE ▶ Each chapter will have several Stop to Think questions. They are
designed to see if you have processed and understood what you’ve read. The
answers are given at the end of the chapter, but you should make a serious effort
to think about these questions before turning to the answers. They are an impor-
tant part of learning. A wrong answer should spur you to review the section
before going on. ◀
1.4 Proportional Reasoning: Scaling Laws
in Biology
The smallest terrestrial mammal is the Etruscan shrew, with a length of 4 cm and
mass of a mere 2 g. The largest is the 7-m-long, 6000 kg African elephant. If we
were to enlarge a shrew by a factor of 175, giving it a 7 m length, would it look pretty
much like a furry elephant?
We can’t carry out the experiment, but we can simulate it with photographs.
FIGURE 1.7 shows a shrew and an elephant with, at least photographically, the same
body length. We could say that the shrew has been scaled to the size of the elephant.
In some regards, the answer to our question isYes. The scaled-up shrew is a bit chub-
bier than the elephant, but the height, the ear size, and other features are all about the
same. But what’s with the legs? Compared to the elephant, the legs of the scaled-up
shrew are twigs rather than tree trunks.
This is not an accident; there are underlying physical reasons why large animals
are not just scaled-up versions of small animals. It all has to do with scaling laws,
which are regularities in how the physical characteristics of organisms—from bone
size to heart rate—depend on the organism’s size.
Proportionality
You often hear it said that one quantity is proportional to another. What does this
mean? We say that y is linearly proportional to x if
y = Cx (1.7)
where C is a constant called the proportionality constant. This is sometimes written
y ∝ x, where the symbol ∝ means “is proportional to.” As FIGURE 1.8 shows, a graph of
y versus x is a straight line passing through the origin with slope C.
NOTE ▶ A graph of “a versus b” means that a is graphed on the vertical axis
and b on the horizontal axis. Saying “graph a versus b” is a shorthand way of say-
ing “graph a as a function of b,” indicating that b is the independent variable on
the horizontal axis. ◀
For example, the mass of an object is related to its volume by m = rV, where r
(the Greek letter rho—we use a lot of Greek letters in physics so as to have enough
The line passes
through the
origin.
The slope of
the line is C.
0 2 4
4C
2C
0
y
x
FIGURE 1.8 A graph showing linear
proportionality.
FIGURE 1.7 Comparing a shrew and an
elephant.
STOP TO THINK 1.1 Our estimate of the diffusion coefficient D is for the diffu-
sion of small molecules through water. The same type of reasoning predicts that the
diffusion coefficient for diffusion through air is _______ the diffusion coefficient for
diffusion through water.
A. Smaller than
B. About the same as
C. Larger than

1.4 Proportional Reasoning: Scaling Laws in Biology 15
symbols for our needs) is the object’s density. Thus mass is linearly proportional to
volume with, in this case, density being the proportionality constant.
Proportionality allows us to use ratios to draw conclusions without needing to
know the proportionality constant. Suppose x has an initial value x1, and thus y has
the initial value y1 = Cx1. If we change x to x2, then y changes to y2 = Cx2. The ratio
of y2 to y1 is
y2
y1
=
Cx2
Cx1
=
x2
x1
(1.8)
That is, the ratio of y2 to y1 is exactly the same as the ratio of x2 to x1, meaning that x
and y scale up or down by the same factor. If you double an object’s volume, its mass
doubles. If you decrease an object’s volume by a factor of three, its mass decreases
by a factor of three. This type of reasoning is called ratio reasoning.
NOTE ▶ Linear proportionality is more specific than a linear relationship. The
linear relationship y = Cx + B also has a straight-line graph, but the graph has a
y-intercept B; the line does not pass through the origin. Ratio reasoning does not
work with a linear relationship because the constants don’t cancel. A linear pro-
portionality is a special case of a linear relationship whose graph passes through
the origin (B = 0). ◀
Now consider the surface area of a sphere: A = 4pr2
. This is also a proportion-
ality, but in this case we would say, “The area is proportional to the square of the
radius” or A ∝ r2
. Proportionalities don’t have to be linear (i.e., dependent on the
first power of x); any relationship in which one quantity is a constant times another
quantity is a proportionality. And ratio reasoning works with any proportionality
because the constant cancels.
A spherical type A cell has a surface area of 2 mm2
. The diameter
of a type B cell is three times that of a type A cell. What is the
surface area of a type B cell?
PREPARE We could use the surface-area formula to find the radius
of a type A cell, multiple it by 3, and then calculate the surface
area of a type B cell. That’s a lot of calculation, though, with the
possibility for making mistakes. Instead, let’s use ratio reasoning.
SOLVE The ratio of the surface areas is
AB
AA
=
4pr 2
B
4pr 2
A
= a
rB
rA
b
2
The surface area of a cell
The proportionality constant 4p cancels, and we find that the ratio
of the areas is the square of the ratio of the radii. If rB /rA = 3,
then AB = 32
* AA = 9AA = 18 mm2
.
ASSESS This is what we mean by reasoning with math rather
than seeing math as simply a calculation tool. With practice,
this is a problem you can solve in your head! “Let’s see, area
is proportional to the square of the radius. If the radius in-
creases by a factor of 3, the area will increase by a factor of
32
= 9.”
EXAMPLE 1.3
For a person or object that moves at a steady speed, the time
needed to travel a specified distance is inversely proportional
to the speed; that is, ∆t ∝ 1/v, where v is the symbol for speed.
Angela gets to work in 15 minutes if she walks at a steady speed.
How long will it take if she rides her bicycle and cycles three
times as fast as she walks?
PREPARE We don’t know either the distance to work or Angela’s
walking speed, but we don’t need either if we use ratio reasoning.
SOLVE Because ∆t = C/v, with C being some unknown con-
stant, we have
∆tcycle
∆twalk
=
C/vcycle
C/vwalk
=
vwalk
vcycle
Time to work
The ratio of the times is the inverse of the ratio of the speeds. We
know that vcycle = 3vwalk, so
∆tcycle
∆twalk
=
vwalk
3vwalk
=
1
3
Thus ∆tcycle = 1
3 ∆twalk = 5 min.
ASSESS It makes sense that it takes one-third as long if Angela
goes three times as fast.
EXAMPLE 1.4

Proportionality helps us understand why the elephant’s legs are much thicker
than the legs of the scaled-up shrew. An object’s volume V depends on the
cube of some linear dimension l, such as a radius or an edge length. The exact
formula for volume depends on the specific shape, but for any object—including
animals—V ∝ l3
. For a given density, mass is proportional to volume and, because
all mammals have about the same density, mass also obeys M ∝ l3
. We say that
mass scales with the cube of the linear dimension. If we scale up an animal by a
factor of five, quintupling the size of the animal in all three dimensions, its mass
increases by a factor of 53
= 125.
An animal’s legs have to support its weight. An animal that’s 125 times more
massive needs bones that are 125 times stronger so as not to break under the load.
You will see in Chapter 6 that the strength of bones is proportional to their cross-
section area, but that’s a problem for large animals. Areas, whether those of circles,
squares, or any other shape, scale as the square of the linear dimension: A ∝ l2
. In
scaling up an animal by a factor of five, areas and bone strength increase by only a
factor of 52
= 25. In other words, bone strength is not keeping up with the animal’s
increase in weight. And to get from a shrew to an elephant we have to scale not by a
factor of five but by a factor of 175!
An animal’s mass scales as l3
, where l is its linear dimension, but its bone strength
scales as only l2
. The weight of an animal that is simply scaled up would quickly out-
strip its leg bones’ ability to support that weight, so a simple scaling up in size doesn’t
work. To support the weight, the cross-section area of the bones must also scale as l3
.
This means that the bone diameter has to scale as l3/2
, faster than the rate at which the
length of the animal increases. That’s why the diameter of the elephant’s tree-trunk-like
legs is a much larger fraction of its body length than the twig-like legs of the shrew.
Allometry
It may seem unlikely that there are mathematically expressible “laws of biology.”
The great core principles of biology—such as evolution, structure-function relation-
ships, and the connections between an organism and its genetic information—don’t
readily lend themselves to being expressed in equations. But consider FIGURE 1.9, a
graph of basal metabolic rate (the rate at which a resting animal uses energy, ab-
breviated BMR) versus mass m for a large number of mammals, ranging from the
smallest (the 2 g Etruscan shrew) to the largest (a 1.5 * 108
g blue whale). Recall
100
101
102
103
104
105
106
107
108
109
105
104
103
102
101
100
10-1
10-2
BMR (W)
m (g)
Blue whale
Moose
Human
Rabbit
Etruscan shrew
Trend line slope = 0.76
FIGURE 1.9 Basal metabolic rate as a function of mass for mammals.

1.4 Proportional Reasoning: Scaling Laws in Biology 17
that “BMR versus mass” graphs BMR on the vertical axis and mass—the indepen-
dent variable—on the horizontal axis.
NOTE ▶ Notice that graph axis labels tell us the units that are being used. Mass
is measured in grams (g), while BMR is measured in watts (W). Metabolic rate is
energy used per second, and the watt—just as used with your appliances to show
the rate of energy use—is the metric unit. ◀
The data points fall just about perfectly along a straight line, suggesting that there
is some kind of law relating a mammal’s BMR to its mass. The mass of the blue
whale is nearly 108
times larger than the mass of the shrew, so this apparent law
holds over an extremely wide range of masses. When two numbers differ by a factor
of L10, we say that they differ by an order of magnitude. Thus the range of masses
spans eight orders of magnitude. Similarly, the BMR values shown on the vertical
axis span about six orders of magnitude.
But you may have noticed that Figure 1.9 is not the type of graph you usually
work with. You’re familiar with graphs where the tick marks on the axis represent
constant intervals. That is, the tick marks might be labeled 0, 1, 2, 3, cwith an
interval of 1. Or 0, 20, 40, 60, cwith an interval of 20. But the horizontal-axis tick
marks in Figure 1.9 are labeled 100
, 101
, 102
, 103
, c. That is, each tick mark is a
constant factor of 10 larger than the preceding tick mark. The same is true on the
vertical axis.
Figure 1.9, called a log-log graph, graphs not BMR versus mass but the logarithm
of BMR versus the logarithm of mass—that is, log(BMR) versus log(m). Log-log
graphs are widely used in science, especially—as here—when the data span many
orders of magnitude. Data that fall on a straight line on a log-log graph are said to
follow a scaling law. We need to review some properties of logarithms to see what
this means.
Any positive number a can be written as a power of 10 in the form a = 10b
.
Simple cases are 1 = 100
, 10 = 101
, and 100 = 102
. You can use your calculator
to find that 210 = 3.16, so 3.16 = 101/2
= 100.50
. We define the logarithm of a as
follows:
If a = 10b
then log(a) = b (1.9)
Thus log(1) = 0, log(10) = 1, log(100) = 2, and log(3.16) = 0.50. You can use your
calculator to find that log(25) = 1.40, which means that 25 = 101.40
.
NOTE ▶ We’re using base10 logarithms, also called common logarithms. You
may also be familiar with natural logarithms. We’ll see those later, but data anal-
ysis is usually done with common logarithms. ◀
Because 10b # 10d
= 10b+d
, you should be able to convince yourself that
log(a # c) = log(a) + log(c). This is an important property of logarithms. And be-
cause an
is a # a # a # g # a, multiplied n times, the logarithm of an
is log(a) added
n times; that is, log(an
) = nlog(a).
Log-log graphs are widely used in science. It’s important to know how to read
them, but the axis labels can be confusing. FIGURE 1.10 shows a portion of the hori-
zontal axis from Figure 1.9 with two different ways of labeling the axis. The labels
above the axis—powers of 10—are what Figure 1.9 shows, but they are not what
is graphed. What’s actually graphed is the logarithms of these powers of 10, which
are the labels shown below the axis. That is, the tick mark labeled 103
actually rep-
resents the value 3.0, the logarithm of the label. So the axis really is increasing 0,
1, 2, 3, c, with a constant interval between the tick marks, but the tick marks are
labeled, instead, with the powers of 10. Notice how the midpoints between the tick
marks are about 3, 30, 300, c. This follows from the fact that their logarithms are
about 0.5, 1.5, 2.5,c.
This method of labeling, while potentially confusing until you get used to it, is
quite useful. The top labels in Figure 1.10 tell the values of the masses, which is what
These axis labels show the masses
whose logarithms are being graphed.
Because log(3) ≈ 0.5, the midpoint
between the tick marks is ≈ 3 * 10n
.
These axis labels show the
logarithms of the masses.
5.0
…
105
4.0
104
3.0
log[m (g)]
103
m (g)
2.0
102
1.0
101
3 30
0.0
100
FIGURE 1.10 Two ways to label a
logarithmic axis.

you really want to know. The bottom labels tell only the logarithms of the masses,
which are harder to interpret.
Returning to Figure 1.9, we see that the trend line—the straight line that best
“fits” the data—has a slope of 0.76. Slope, you will recall, is the rise-over-run ratio.
But the rise and run of what? Because this is a graph of log(BMR) versus log(m), the
slope is measured as the “rise” in log(BMR) divided by the “run” in log(m). That is,
the slope is determined from the unseen logarithm labels 0, 1, 2, 3, crather than
the power-of-ten labels used in the graph.
Let’s now imagine that some quantity Y (the BMR in this example) depends on
another quantity X (the mass) via the proportionality
Y = CXr
(1.10)
where C is a constant. We say that Y scales as the rth power of X.
We take the logarithm of both sides of Equation 1.10 and use the properties of
logarithms:
logY = log(CXr
) = logC + log(Xr
) = rlogX + logC (1.11)
To interpret this equation, recall that a linear equation of the form y = mx + b graphs
as a straight line with slope m and y-intercept b. By defining y = logY and x = logX,
we see that Equation 1.11 is actually a linear equation. That is, a graph of logY
versus logX is a straight line with slope r. Conversely, and importantly, if a graph
of logY versus logX is a straight line, then Y and X obey a scaling law like
Equation 1.10 and the value of the exponent r is given by the slope of the line.
NOTE ▶ Spreadsheets and other graphing software that you might use to ana-
lyze data usually have a “logarithmic axis” option. If you choose that option,
the computer calculates and plots the logarithms of the data for you (taking
logarithms is not something you have to do yourself) and then labels the axes
using the power-of-ten notation shown above the axis in Figure 1.10. Using the
logarithmic axis option allows you to look for functional relationships that are
scaling laws. ◀
Figure 1.9 graphed the logarithm of the basal metabolic rate against the logarithm
of the mass for a large number of mammals. We had no reason to suspect that these
two quantities are related in any particular way, but the graph turned out to be a
straight line with a slope of almost exactly 3
4. Consequently, we’ve discovered a scal-
ing law telling us that a mammal’s BMR scales as the 3
4th power of its mass m; that is,
BMR = Cm3/4
or, as we would usually write, BMR ∝ m3/4
. We could use the graph’s
y-intercept to determine the value of the constant C, but the constant is usually less
interesting than the exponent.
This result has broad generality across all mammals. It is interesting that birds
and insects also have metabolic rates that scale as the 3
4th power of mass. This scal-
ing law really seems to be a law of biology. Why? Scaling laws, especially ones that
span many orders of magnitude, tell us that there’s some underlying regularity in the
physics of the organism.
A simple model relating an organism’s metabolic rate to its size is based on
the fact that metabolism generates heat and that heat has to be dissipated. Heat
is dissipated through an organism’s surface via heat convection and radiation, so
this model predicts that BMR should be proportional to the organism’s surface
area. We’ve seen that surface area scales as A ∝ l2
, where l is the organism’s linear
size, and thus BMR should scale with the size of an organism as BMR ∝ l2
. The
organism’s mass is proportional to its volume, so mass scales as m ∝ l3
. Combining
these, by eliminating l, we see that this model predicts that BMR should scale with
the organism’s mass m as BMR ∝ m2/3
. That is, a geometric model in which the
organism’s BMR is determined by how quickly it can dissipate heat through its
surface predicts a scaling law, but one in which BMR should scale as the 2
3rd power
of mass.

eBook University Physics for the Life Sciences, 1e Randall Knight, Brian Jones, Stuart Field.pdf

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