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Biomechanics of Sport
and Exercise
Third Edition
Peter M. McGinnis
State University of New York, College at Cortland
Human Kinetics
Library of Congress Cataloging-in-Publication Data
McGinnis, Peter Merton, 1954-
Biomechanics of sport and exercise / Peter M. McGinnis. -- 3rd ed.
p. ; cm.
Includes bibliographical references and index.
I. Title.
[DNLM: 1. Biomechanics. 2. Exercise--physiology. 3. Sports--physiology. WE 103]
612.7'6--dc23
2012034730
ISBN-10: 0-7360-7966-1 (print)
ISBN-13: 978-0-7360-7966-2 (print)
Copyright � 2013, 2005, 1999 by Peter M. McGinnis
All rights reserved. Except for use in a review, the reproduction or utilization of this work in any form or by any elec-
tronic, mechanical, or other means, now known or hereafter invented, including xerography, photocopying, and record-
ing, and in any information storage and retrieval system, is forbidden without the written permission of the publisher.
The web addresses cited in this text were current as of August 2012, unless otherwise noted.
Acquisitions Editors: Amy N. Tocco and Loarn D. Robertson, PhD; Developmental Editor: Katherine Maurer;
Assistant Editors: Brendan Shea and Susan Huls; Copyeditor: Joyce Sexton; Indexer: Nancy Ball; Permissions
Manager: Dalene Reeder; Graphic Designer: Nancy Rasmus; Graphic Artist: Dawn Sills; Cover Designer: Keith
Blomberg; Photograph (cover): � Mike Kemp/Tetra Images/age fotostock; Photographs (interior): Courtesy Peter
M. McGinnis unless otherwise noted; photos on pages 5, 22, 51, 87, 115, 125, 150, 207, 239, 277, 315, and 361 �
Human Kinetics; Photo Asset Manager: Laura Fitch; Photo Production Manager: Jason Allen; Art Manager: Kelly
Hendren; Associate Art Manager: Alan L. Wilborn; Art Style Development: Joanne Brummett; Illustrations: �
Human Kinetics; Printer: Thomson-Shore, Inc.
Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
The paper in this book is certified under a sustainable forestry program.
Human Kinetics
Website: www.HumanKinetics.com
United States: Human Kinetics Australia: Human Kinetics
P.O. Box 5076 57A Price Avenue
Champaign, IL 61825-5076 Lower Mitcham, South Australia 5062
800-747-4457 08 8372 0999
e-mail: humank@hkusa.com e-mail: info@hkaustralia.com
Canada: Human Kinetics New Zealand: Human Kinetics
475 Devonshire Road Unit 100 P.O. Box 80
Windsor, ON N8Y 2L5 Torrens Park, South Australia 5062
800-465-7301 (in Canada only) 0800 222 062
e-mail: info@hkcanada.com e-mail: info@hknewzealand.com
Europe: Human Kinetics E4696
107 Bradford Road
Stanningley
Leeds LS28 6AT, United Kingdom
+44 (0) 113 255 5665
e-mail: hk@hkeurope.com
This third edition is dedicated to the memory of two strong,
outgoing, and adventurous women whose presence raised
the spirits of those around them--my mother, Doris Joye
McGinnis (1925-2009), and my friend, colleague, and
former student, Julianne Abendroth (1962-2011).
contents
preface vii
acknowledgments ix
student and instructor resources x
how to use MaxTRAQ xi
introduction Why Study Biomechanics? 1
What Is Biomechanics? 3 � What Are the Goals of Sport and Exercise
Biomechanics? 3 � The History of Sport Biomechanics 10 � The
Organization of Mechanics 12 � Basic Dimensions and Units of
Measurement Used in Mechanics 13 � Summary 15 � Learning
Aids 15
part I External Biomechanics 17
External Forces and Their Effects on the Body
and Its Movement
chapter 1 Forces 19
Maintaining Equilibrium or Changing Motion
What Are Forces? 20 � Classifying Forces 21 � Friction 23 �
Addition of Forces: Force Composition 26 � Resolution of
Forces 33 � Static Equilibrium 37 � Summary 44 � Learning
Aids 45
chapter 2 Linear Kinematics 51
Describing Objects in Linear Motion
Motion 52 � Linear Kinematics 53 � Uniform Acceleration and
Projectile Motion 68 � Summary 79 � Learning Aids 79 � Motion
Analysis Exercises Using MaxTRAQ 84
chapter 3 Linear Kinetics 87
Explaining the Causes of Linear Motion
Newton's First Law of Motion: Law of Inertia 88 � Conservation
of Momentum 90 � Newton's Second Law of Motion: Law of
Acceleration 98 � Impulse and Momentum 102 � Newton's
Third Law of Motion: Law of Action-Reaction 107 � Newton's
Law of Universal Gravitation 108 � Summary 108 � Learning
Aids 109 � Motion Analysis Exercises Using MaxTRAQ 112
chapter 4 Work, Power, and Energy 115
Explaining the Causes of Motion Without Newton
Work 116 � Energy 119 � The Work�Energy Relationship 121 �
Power 127 � Summary 129 � Learning Aids 129 � Motion Analysis
Exercises Using MaxTRAQ 132
iv
chapter 5 Torques and Moments of Force 133
chapter 6 167
chapter 7 Maintaining Equilibrium or Changing Angular Motion 195
chapter 8 217
What Are Torques? 134 � Forces and Torques in Equilibrium 141 �
What Is Center of Gravity? 145 � Summary 160 � Learning Aids 160
Angular Kinematics
Describing Objects in Angular Motion
Angular Position and Displacement 168 � Angular and Linear
Displacement 171 � Angular Velocity 173 � Angular and Linear
Velocity 173 � Angular Acceleration 176 � Angular and Linear
Acceleration 176 � Anatomical System for Describing Limb
Movements 178 � Summary 187 � Learning Aids 189 � Motion
Analysis Exercises Using MaxTRAQ 193
Angular Kinetics
Explaining the Causes of Angular Motion
Angular Inertia 196 � Angular Momentum 202 � Angular
Interpretation of Newton's First Law of Motion 205 � Angular
Interpretation of Newton's Second Law of Motion 208 � Angular
Impulse and Angular Momentum 209 � Angular Interpretation of
Newton's Third Law of Motion 209 � Summary 211 � Learning
Aids 211
Fluid Mechanics
The Effects of Water and Air
Buoyant Force: Force Due to Immersion 218 � Dynamic Fluid Force:
Force Due to Relative Motion 221 � Summary 233 � Learning
Aids 233
part II Internal Biomechanics 237
Internal Forces and Their Effects on the Body
and Its Movement
chapter 9 Mechanics of Biological Materials 239
263
Stresses and Strains on the Body 277
Stress 240 � Strain 250 � Mechanical Properties of Materials:
The Stress�Strain Relationship 252 � Mechanical Properties of the
Musculoskeletal System 256 � Summary 261 � Learning Aids 261
chapter 10 The Skeletal System
The Rigid Framework of the Body
Bones 265 � Joints 267 � Summary 274 � Learning Aids 274
chapter 11 The Muscular System
The Motors of the Body
The Structure of Skeletal Muscle 278 � Muscle Action 281 � Muscle
Contraction Force 285 � Summary 297 � Learning Aids 297
v
Contents
vi
chapter 12 The Nervous System 299
Control of the Musculoskeletal System
The Nervous System and the Neuron 300 � The Motor
Unit 301 � Receptors and Reflexes 304 � Summary 307 � Learning
Aids 308
part III Applying Biomechanical Principles 309
chapter 13 Qualitative Biomechanical Analysis 311
to Improve Technique 339
361
Types of Biomechanical Analysis 312 � Steps of a
Qualitative Biomechanical Analysis 313 � Sample 383
Analyses 322 � Summary 336 � Learning Aids 336
chapter 14 Qualitative Biomechanical Analysis
to Improve Training
Biomechanics and Training 340 � Qualitative Anatomical Analysis
Method 341 � Sample Analyses 344 � Summary 356 � Learning
Aids 360
chapter 15 Qualitative Biomechanical Analysis
to Understand Injury Development
Mechanical Stress and Injury 362 � Tissue Response to
Stress 364 � Mechanism of Overuse Injury 367 � Individual
Differences in Tissue Threshold 367 � Intrinsic and Extrinsic
Factors Affecting Injury 368 � Sample Analysis: Overuse Injuries
in Running 371 � Summary 380 � Learning Aids 380
chapter 16 Technology in Biomechanics
Quantitative Biomechanical Analysis 384 � Measurement
Issues 384 � Tools for Measuring Biomechanical
Variables 386 � Summary 392 � Learning Aids 392 � Motion
Analysis Exercises Using MaxTRAQ 393
appendix A Units of Measurement and Conversions 395
appendix B Answers to Selected Review Questions, Problems, and MaxTRAQ Exercises 399
glossary 415
references and suggested readings 425
web resources 429
index 431
about the author 443
preface
This textbook was written to intro- includes an exercise in which students can record and
analyze their own video clip.
duce undergraduate students to the field of sport and
exercise biomechanics. The text is primarily intended The organization of the book remains intact. The
for undergraduate students majoring in kinesiology, introductory chapter provides an introduction to biome-
exercise science, or physical education, but it is suitable chanics, which includes justifications for the study of
for students in other human movement fields as well. biomechanics. It also includes an overview of the organi-
Most of the examples and applications appearing in the zation of mechanics and an introduction to measurement
book are from sport or exercise, but examples from clini- systems. The rest of the book is divided into three parts.
cal and everyday human movement activities have also
been included. No matter what human movement field Part I concerns external biomechanics, or external
readers are interested in, knowledge of mechanics will be forces and their effects on the body and its movement.
valuable to them in their work. Many human movement Rigid-body mechanics with applications to human move-
professionals' only formal instruction in the mechanics of ment is the primary topic of this part of the text. Mechan-
human movement occurs during a single undergraduate ics is one of the most difficult topics for undergraduate
course in kinesiology or biomechanics. This book was students of human movement to understand, so this is the
developed with this constraint in mind. The goal of the most important and largest part of the book. The order
book and its ancillary materials is to present an introduc- of presentation of topics in this part differs from that in
tion to the biomechanics of human movement in a clear, most other biomechanics texts. Chapter 1 presents the
concise, user-friendly manner. concepts of force and static equilibrium. With forces as
the example, this chapter also introduces vector addition
This third edition is an improvement over the previous and resolution. The trigonometry used to add and resolve
edition in a number of ways. Photos and select figures forces is also explained. Chapter 2 discusses linear motion
have been updated. New material has been added, includ- and its description. This chapter includes equations that
ing new sport examples, new sample problems, and describe the motion of an object undergoing constant
discussion of new technologies used by researchers in acceleration and their application to describing projectile
quantitative biomechanical analysis. Review questions motion. Chapter 3 presents the causes of linear motion
and problems have been added to many chapters, and and introduces Newton's three laws of motions, as well
many problems have been enhanced with new diagrams as the conservation of momentum principle. Chapter 4
to help students visualize the mechanics of real-world discusses mechanical work and energy principles and also
scenarios. the mechanical work done by muscles. Torque, moment
of force, and center of gravity are introduced in chapter
Also new to this edition is the inclusion of video 5 preceding a discussion of angular kinematics in chapter
motion analysis software along with the book. With the 6. The causes of angular motion are presented in chapter
purchase of a new textbook, students receive a license 7, along with the angular analogs to Newton's three laws
and instructions to download an educational version of of motion. Part I concludes with a discussion of fluid
the motion analysis software MaxTRAQ, along with mechanics in chapter 8.
number of video clips. Chapters 2, 3, 4, and 6 include
exercises that require use of the MaxTRAQ software to Part II concerns internal biomechanics, or internal
measure kinematic variables in the various video clips forces and their effects on the body and its movement.
This part begins with a discussion of the mechanics of
that accompany the soft- biological materials in chapter 9. Stress and strain are
ware. The software can also introduced in this chapter, along with various concepts
be used to analyze human of material strength. Overviews of the skeletal system,
movement in video clips muscular system, and nervous system control are then
recorded and supplied by presented in chapters 10, 11, and 12.
the user, and chapter 16
vii
Preface
viii
Part III deals with the application of biomechanics. This is especially true if projectile motion and the equa-
Commonsense methods of applying biomechanics to the tions for projectile motion are discussed under linear
analysis of sport or human movement skills are presented kinematics. Since projectile motion is influenced by the
in the first three chapters of this part. The first of these force of gravity, an understanding of this force should
chapters, chapter 13, presents procedures for completing precede discussion of the effects of the force. Similarly,
qualitative biomechanical analyses to improve technique. torques are introduced before the discussion of angular
Chapter 14 presents a method of qualitative biomechani- kinematics.
cal analysis to improve training. A qualitative procedure
for identifying active muscle groups in phases or parts of Since mechanics uses equations to describe relation-
movements is emphasized in this chapter. Chapter 15 is ships or to define quantities, some knowledge of math-
an examination of how qualitative biomechanical analysis ematics (primarily algebra) is necessary. I have tried to
can be used to help understand the causes of injury. This write the book in such a way that even students with
chapter was written by Steven McCaw. Chapter 16 is the weak math skills can succeed in learning biomechanics.
concluding chapter of the book. It gives an overview of However, success in learning the material will come more
the technology used in conducting quantitative biome- easily to those who are better prepared mathematically.
chanical analyses.
Appendix A lists the principal units used for mechani-
Throughout the book, and especially in part I, the cal quantities in the International System of Units, as well
objective has been to allow students to discover the prin- as prefixes and the conversions to customary units used
ciples of mechanics for themselves. Common activities in the United States.
are observed, and explanations for these activities are then
developed. The resulting explanations reveal the underly- Each chapter of this book includes elements intended
ing mechanical concepts. This discovery process requires to help the reader learn the material. Each chapter begins
more active participation by the reader, but it results in a with a list of objectives and an opening scenario lead-
better understanding of the subject matter. ing to questions that readers can answer after reading
and understanding the material presented in the chapter.
What makes this book unique among biomechanics Practical examples of concepts are integrated into the text
textbooks is its order of presentation. In most under- throughout each chapter. Sample problems are presented,
graduate biomechanics textbooks, functional anatomy and the step-by-step procedures for solving them are
is presented before mechanics. This textbook presents illustrated. Problems and review questions appear at the
mechanics first. Bones and ligaments are the structural end of each chapter to test the reader's understanding
elements that support the human body. Muscles are the of the material presented. More problems and review
motors that move this structure. Understanding how the questions have been added to this edition. Answers to
forces exerted by bones and ligaments support the body, problems and most of the review questions appear in
and how the forces and torques produced by muscles do appendix B. Finally, the educational version of the video
work to move the body's limbs, requires a knowledge of motion analysis software MaxTRAQ and accompanying
forces and their effects. Mechanics is the study of forces video clips are included with this new edition. Exercises
and their effects. Thus mechanics should precede the that require the use of this software appear at the ends
study of the musculoskeletal system. of chapters 2, 3, 4, 6, and 16, and the solutions to these
exercises appear in appendix B. These video analysis
This book is also unique in its order of presentation problems may more clearly illustrate principles presented
of mechanical topics. The mechanics section of most in each chapter.
biomechanics textbooks begins with linear kinematics
and then deals with linear kinetics, angular kinematics, Throughout the text, I have tried to explain and illus-
and finally angular kinetics. This book introduces forces trate the concepts as clearly as possible and in such a way
before presenting linear kinematics. Because forces are that you, the reader, are actively involved in discovering
the causes of changes in motion and forces are in equi- them. Still, you might find that some of the material is
librium if no changes in motion occur, it makes sense to challenging. Occasionally, you might find yourself dis-
define and understand forces before discussing motion. tracted or confused while reading. Don't give up! Your
effort will be worthwhile.
acknowledgments
A book is not produced by one individual--a University, who was responsible for chapter 15; my
students and colleagues who provided suggestions for
team of people is involved. I thank the members of that improvement; and my extended family, whose support
team: the folks at Human Kinetics, including Amy Tocco, and encouragement is further acknowledged by the
Kate Maurer, and especially Loarn Robertson, whose appearance of their names in problems and questions
patience I tested; Victoria Berger of Motion Analysis throughout the book. Special thanks to my wife, Boodie,
Products, who was responsible for the inclusion of the for her love and support.
MaxTRAQ software; Steve McCaw of Illinois State
ix
student and
instructor resources
Student Resources Instructor Guide
Students, visit the free student web resource at www. Specifically developed for instructors who have adopted
HumanKinetics.com/BiomechanicsOfSportAndExercise. Biomechanics of Sport and Exercise, Third Edition, the
The web resource takes select end-chapter problems from instructor guide includes chapter summaries, objectives,
the textbook and provides a sequence of hints to guide and outlines, plus lecture ideas and sample outlines, stu-
you through the problems, helping you to develop your dent activities, and teaching tips. The instructor guide also
skills and gain confidence working through biomechani- includes PDFs that show the mathematical and graphic
cal problems. It also provides downloadable PDF copies work done to arrive at the correct answer for each of
of all end-chapter questions and problems. Problems that the end-chapter review problems. These can be used as
are worked through in the web resource are marked with teaching aids or for evaluating student work.
the web resource icon:
Test Package
?
The test package includes a bank of over 380 questions
In addition, with the purchase of a new textbook you created especially for Biomechanics of Sport and Exer-
will receive access to the MaxTRAQ motion analysis cise, Third Edition. Various question types are included,
software. To access and download this software, visit such as true-or-false, fill-in-the-blank, essay and short
www.motionanalysisproducts.com/Books/PM-BSE-R3. answer, and multiple choice. The test package is available
html. If you have purchased a used book or e-book, you for download in three different formats: Rich Text (.rtf),
may purchase the software separately at the MaxTRAQ Respondus, and Learning Management System (LMS).
site. The MaxTRAQ software is compatible with Win-
dows operating systems only. For more on MaxTRAQ, Image Bank
see the section "How to Use MaxTRAQ."
New for the third edition, the image bank includes most
Instructor Resources of the illustrations, photos, and tables from the text,
sorted by chapter. These are provided as separate files
The instructor guide, test package, and image bank for easy insertion into lecture presentations and other
are free to course adopters and are accessed at www. course materials, providing instructors with the flexibility
HumanKinetics.com/BiomechanicsOfSportAndExercise. to create customized resources.
x
how to use MaxTRAQ
Several chapters in this book links in the lower-right corner under Short Tutori-
als and Tours. The video shows several versions
include activities that use MaxTRAQ Educational 2D of MaxTRAQ. Follow along with the manual
software. MaxTRAQ Educational (ME) is a download- tracking version.
able software module that allows you to track and analyze
human movement, distance, and angles in video clips. � The video clips provided with the software
ME operates only under Windows operating systems require different deinterlacing settings. Follow
(Windows XP, Windows Vista, and Windows 7). the instructions at the beginning of each set of
exercises to set the deinterlacing options for the
To use ME to complete the activities in this text, you video clip used.
will need to download the software, install it on your
computer, and enter a key code to activate it. If you have � To view a video clip, open it using File--Open,
purchased a new print book, you may enter the code then use the player buttons in the center bottom
provided on the blue letter bound into the front of the of the screen. The buttons to the left and right
text. If you have purchased a used book or e-book, you of the center Stop button advance or rewind the
will need to purchase the MaxTRAQ Educational 2D video one frame at a time. You will use these a
software directly from the MaxTRAQ site and your key lot as you complete the exercises. Also note that
code will be sent to you via e-mail. you can adjust the brightness and contrast of a
video if needed to improve visibility.
To download the software, go to www.motionanalysis
products.com/Books/PM-BSE-R3.html. If you have � Note that the frame rate of the video clips varies.
purchased a new book, click on the New Books link The frame rate is shown to the lower right of the
under Book Downloads--MaxTRAQ Educational to MaxTRAQ screen in hertz (Hz). Hertz are cycles
download the software installer. If you have purchased per second, so a frame rate of 30 Hz equals 30
a used book or e-book, you can click on the Used Books frames per second (fps). For some of the video
link to purchase a copy of the software. clips, MaxTRAQ does not recognize the true
frame rate and the exercise will instruct you in
After you have saved the MaxTRAQ installer to a doing a conversion.
folder on your computer, double-click it to install the
software. See How to Install MaxTRAQ Educational at � The software functions you will use the most to
www.motionanalysisproducts.com/Books/PM-BSE-R3. complete the exercises in this text are setting the
html for more information. The first time you open the scale, digitizing points on the subject, and calcu-
program, a License window will appear. Click on Enter lating the changes in the x and y coordinates of
Key at the lower left. Carefully enter the key code that the points as the subject moves.
came with this book or the code you were sent via e-mail
(cut and paste the code) if you purchased the software � To set the scale, open a video, then go to
access separately. Once you have entered your key code, View--Tools and select Show Scale. Next,
access is provided for one year. click on Tools in the top menu bar and Se-
lect Scale. In the window that pops up, en-
Tips for Using MaxTRAQ Educational 2D ter the length of a known reference in the
video (for the video clips included with the
� The software download for this book includes 12 software, see the exercise instructions for
video clips created by the author. When you down- the length to enter). Click OK and then use
load the software, these video clips are saved to the mouse to left-click each end point of
your local disk at Program Files\Motion Analysis\ the known distance in the frame (for exam-
MaxTRAQP\VideoFiles. You may also use the ple, two points known to be 500 cm apart
MaxTRAQ software to analyze other video clips on a wall). When you click the rst point,
that are saved in .avi format. nothing will appear on the screen, but
when you click the second, a scale will ap-
� A tutorial video and guidance on using the pear. To hide the scale, go to View--Tools
video files are available at www.motionanalysis and uncheck Show Scale.
products.com/Books/PM-BSE-R3.html. See the
xi
How to Use MaxTRAQ then set the Point Zoom Magni cation un-
der the rst option in the right side of the
xii window. It is recommended that you use
Point Zoom when digitizing the ends of
� To digitize points on the subject, use the the reference measure.
arrows for Number of Points on the right
side of the MaxTRAQ screen to select the � For each digitized point, the coordinates
number of points you will need to digitize along the x (horizontal) and y (vertical)
in any single frame of video (often this axes will be displayed next to the point in
will be only one). Click the Digitize but- that order (x, y). To show or hide these co-
ton in the right-hand column, then posi- ordinates, go to View--Point Markers and
tion the cursor over the spot on the video check or uncheck Show Coordinates.
that you want to digitize and left-click the
mouse button. To digitize points in sub- � If you make a mistake when creating a
sequent frames of video, just advance the scale or digitizing a point, you can right-
video, position the cursor over the spot on click the element and select Delete Scale
the video, and left-click again. Repeat this or Delete Point to remove it.
in each frame and for each point that you
want to digitize. � If you are digitizing more than one point
in a frame, make sure that the point you
� To improve the accuracy of your digi- want to digitize next (point 1, point 2, and
tizing, you can enlarge the video image so on) is highlighted in the right column
around the point you wish to digitize. Po- before you click to set the location of that
sition the cursor near the point and right- point.
click the mouse. Select Zoom and then
Point Zoom in the windows that appear. � For many questions, it may be best to begin by
To zoom back out to regular view, right- working with your professor or a fellow student
click, select Zoom, and Point Zoom again. who has successfully used the software. If you
To adjust the magni cation of the Point need further advice, send questions or inquiries
Zoom, select Tools--Options on the top via e-mail to support@motionanalysisproducts.
menu bar. In the window that appears, se- com.
lect General from the choices on the left,
� Lukas Blazek | Dreamstime.comintroduction
Why Study Biomechanics?
objectives
When you finish reading this introduction, you
should be able to do the following:
� Define biomechanics
� Define sport and exercise biomechanics
� Identify the goals of sport and exercise
biomechanics
� Describe the methods used to achieve the
goals of sport and exercise biomechanics
� Be somewhat familiar with the history
and development of sport and exercise
biomechanics
� Define mechanics
� Outline the organization of mechanics
� Define length and the units of measure-
ment for length
� Define time and the units of measurement
for time
� Define mass and the units of measure-
ment for mass
1
Introduction
2
You are watching the Olympic Games on television when you see a high jumper
successfully jump over a crossbar set more than a foot above his head. The
technique he uses looks very awkward. He approaches the bar from the side,
and when he jumps off the ground he turns his back toward the bar. His head
and arms clear the bar first; he then arches his back and finally kicks his legs out
and up to get them over the bar. He lands in the pit in an ungainly position: on
his shoulders and back with his legs extended in the air above him. You think to
yourself, How can he jump so high using such an odd-looking technique? Certainly,
there must be another technique that is both more effective and more graceful
looking. Biomechanics might provide you with some insights to answer this and
other questions you have about human movement.
What is your motivation for studying or coach, whereas biomechanics tells you why those
techniques are best to teach or coach. A good knowledge
biomechanics? What can you gain from learning about of biomechanics will enable you to evaluate techniques
biomechanics? How will a working knowledge of biome- used in unfamiliar sport skills as well as to better evaluate
chanics assist you in your future endeavors? Will the time new techniques in sports familiar to you.
you spend learning about biomechanics be worthwhile?
You should consider these questions before investing a Perhaps the best outcome of study-
lot of time learning about biomechanics. ing and using biomechanics will be
improved performances by your
If you are like most readers of this book, you are prob- athletes or the accelerated learning
ably an undergraduate student majoring in kinesiology, of new skills by your students.
physical education, exercise science, or another human
movement science. If so, your answer to the question , ,
"Why study biomechanics?" may be that you are enrolled
in a required course in biomechanics for which this is the Figure I.1 Studying biomechanics will help you under-
required text. You are studying biomechanics to earn a stand why some sport techniques work and some don't.
required credit so that you can graduate. If this is your
answer, and for the majority of readers this may be true,
you are probably unable to answer the other questions
because you do not have enough prior knowledge of
biomechanics to know how it can benefit you. So let me
give you some reasons for studying biomechanics. This
may provide you with some intrinsic motivation to get
started on the task of learning about biomechanics.
You are probably planning a career as a physical
education teacher, coach, or some other physical activity
specialist; and you probably are or have been active as
a participant in one or more sports or fitness activities.
Suppose a student or athlete asks you, "Why do I have
to do this skill this way?" or "Why isn't this technique
better?" (see figure I.1). Perhaps you have even asked
such questions yourself. Was the coach or teacher able
to answer your questions? Were these questions asked
of you? Could you answer them? Traditional teaching
and coaching methods tell you what techniques to teach
E4696/McGinnis/Fig.0.1/410358/JG/R3-kh
Introduction
3
Athletic training and physical therapy students, as well it is not even limited to human activities; indeed, it is
as other students of sports medicine, will also benefit not even limited to animal activities! The range of titles
from a knowledge of biomechanics. A good knowledge indicates that biomechanics could include not only the
of biomechanics will help in diagnosing the causes of study of the movement of an athlete, but also the study
an injury. It can provide the mechanical basis for taping of the airflow in the athlete's lungs and the strength of
techniques, braces, and orthotic devices. An understand- the athlete's tissues.
ing of biomechanics may also guide therapists in their
prescriptions for rehabilitation and indicate to exercise Let's return to the word itself and examine it directly
specialists what exercises may be dangerous for certain for clues about its definition. The word biomechanics can
individuals. be divided into two parts: the prefix bio- and the root word
mechanics. The prefix bio- indicates that biomechanics
What Is Biomechanics? has something to do with living or biological systems.
The root word mechanics indicates that biomechanics
What is this science that promises so much? Before has something to do with the analysis of forces and their
going any further, we should agree on a definition of effects. So it appears that biomechanics is the study of
the word biomechanics. What is biomechanics? How forces and their effects on living systems. This comes
have you heard this word used by others? How have you very close to the definition of biomechanics presented
used this word? Your immediate response may be that by Herbert Hatze in 1974: "Biomechanics is the study
biomechanics has something to do with determining the of the structure and function of biological systems by
best techniques used by athletes in various sport skills. means of the methods of mechanics" (p. 189). This is a
Indeed, some biomechanists are involved in such work, much broader field of study than you may have at first
and we just highlighted technique evaluation as a primary thought. The study of the structure and function of plants
reason for studying biomechanics. But biomechanics as well as animals is encompassed by the definition of
encompasses much more than that. biomechanics.
Let's look in the library for clues to the definition. Biomechanics is the study of forces
Several journals have the word "biomechanics" or a and their effects on living systems.
derivation of it in their title. These include the Journal
of Biomechanics, the Journal of Biomechanical Engi- What Are the Goals
neering, the Journal of Applied Biomechanics, Sports of Sport and Exercise
Biomechanics, Clinical Biomechanics, Applied Bionics Biomechanics?
and Biomechanics, and Computer Methods in Biome-
chanics and Biomedical Engineering. Looking through Now let's focus on our specific topic of interest in bio-
the tables of contents of these journals, we find that the mechanics. Biomechanics includes the study of all living
Journal of Applied Biomechanics and Sports Biomechan- things, plant and animal; animal biomechanics includes
ics contain numerous articles about sport biomechanics, only animals as subjects of study; human biomechanics
such as "Upper-limb kinematics and coordination of includes only humans; and sport and exercise biome-
short grip and classic drives in field hockey," "Hydrody- chanics includes only humans involved in exercise and
namic drag during gliding in swimming," "Effects of bat sport. We might define sport and exercise biomechan-
grip on baseball hitting kinematics," "The influence of ics as the study of forces and their effects on humans in
the vaulting table on the handspring front somersault," exercise and sport.
and "Biomechanics of skateboarding: Kinetics of the
Ollie." These articles support our sport-related defini- Performance Improvement
tion of biomechanics. However, a look through the other
journals reveals a broad range of topics that at first may The ultimate goal of sport and exercise biomechanics is
appear unrelated, such as "Regional stiffening of the performance improvement in exercise or sport. A sec-
mitral valve anterior leaflet in the beating ovine heart," ondary goal is injury prevention and rehabilitation. This
"Biomechanical model of human cornea based on stromal secondary goal is closely related to the first and could
microstructure," "Simulation of pulmonary air flow with a almost be considered part of the primary goal, because
subject-specific boundary condition," and "Strain-energy an uninjured athlete will perform better than an injured
function and three-dimensional stress distribution in athlete. Well, how do biomechanists work toward achiev-
esophageal biomechanics." There's even an article titled ing these goals?
"Biomechanics of fruits and vegetables." From these titles
we can deduce that biomechanics is not limited to sport;
Introduction
4
The ultimate goal of sport and exer- belief that new and revolutionary techniques are regularly
cise biomechanics is performance developed by biomechanists, such developments are rare.
improvement in exercise or sport. Perhaps the reason is that biomechanics as a discipline
is a relatively new science. The much more common
Technique Improvement outcome of biomechanics research is the discovery of
small refinements in technique. One example of biome-
The most common method for improving performance chanics research that did greatly affect the technique and
in many sports is to improve an athlete's technique. This performances in a sport occurred in swimming in the late
is highlighted here as one motivation for studying bio- '60s and early '70s. Research done by Ronald Brown and
mechanics, and it is probably what you thought of when James "Doc" Counsilman (1971) indicated that the lift
asked how a biomechanist goes about trying to improve forces acting on the hand as it moved through the water
an athlete's performance. were much more important in propelling a swimmer
through the water than previously thought. This research
The application of biomechanics to improve technique indicated that rather than pulling the hand in a straight
may occur in two ways: Teachers and coaches may use line backward through the water to produce a propulsive
their knowledge of mechanics to correct actions of a drag force, the swimmer should move the hand back and
student or athlete in order to improve the execution of a forth in a sweeping action as it is pulled backward to
skill, or a biomechanics researcher may discover a new produce propulsive lift forces as well as propulsive drag
and more effective technique for performing a sport skill. forces (see figure I.2). This technique is now taught by
In the first instance, teachers and coaches use qualita- swimming teachers and coaches throughout the world.
tive biomechanical analysis methods in their everyday
teaching and coaching to effect changes in technique. Other examples of sports in which dramatic changes
In the second instance, a biomechanics researcher uses in technique produced dramatic improvement in per-
quantitative biomechanical analysis methods to discover formance include javelin throwing, high jumping, and
new techniques, which then must be communicated to the cross-country skiing. In 1956, before the Summer Olym-
teachers and coaches who will implement them. pic Games in Melbourne, Felix Erasquin, a 48-year-old
retired discus thrower from the Basque region of Spain,
Let's look at a simple example of the first case. As a experimented with an unconventional way of throwing
coach, suppose you observe that your gymnast is having the javelin. Erasquin had experience in barra vasca, a
difficulty completing a double somersault in the floor traditional Basque sport that involved throwing an iron
exercise. You might suggest three things to the gymnast bar called a palanka. A turn was used to throw the palanka,
to help her successfully complete the stunt: (1) jump and Erasquin incorporated this turn in his innovative jav-
higher, (2) tuck tighter, and (3) swing her arms more elin throwing technique. Rather than throwing using the
vigorously before takeoff. These suggestions may all conventional technique--over the shoulder with one hand
result in improved performance and are based on bio- from a run--Erasquin held the javelin with his right hand
mechanical principles. Jumping higher will give the just behind the grip. The tip of the javelin pointed down
gymnast more time in the air to complete the somersault. to his right, and the tail was behind his back and pointed
Tucking tighter will cause the gymnast to rotate faster upward. During the run-up, Erasquin spun around like a
due to conservation of angular momentum. Swinging discus thrower and slung the javelin from his right hand,
the arms more vigorously before takeoff will generate which guided the implement. To reduce the frictional
more angular momentum, thus also causing the gymnast forces acting on the javelin as it slid through his hand, it
to rotate faster. In general, this is the most common type had been dunked in soapy water to make it slippery. The
of situation in which biomechanics has an effect on the outstanding results achieved by Erasquin and others with
outcome of a skill. Coaches and teachers use biomechan- this technique attracted international attention. Several
ics to determine what actions may improve performance. throwers using this "revolutionary" technique recorded
throws that were more than 10 m beyond the existing
Coaches and teachers use biome- javelin world record. Officials at the International Ama-
chanics to determine what actions teur Athletic Federation (IAAF), the governing body for
may improve performance. track and field, became so alarmed that they altered the
rules for the event, and this unconventional technique
The second general situation in which biomechanics became illegal (see figure I.3). None of the records set
contributes to improved performance through improved with the Spanish technique were recognized as official
technique occurs when biomechanics researchers develop world records.
new and more effective techniques. Despite the common
Introduction
5
� Human Kinetics/J. Wiseman, reefpix.org
Figure I.2 Swimming techniques have been influenced by biomechanics.
Figure I.3 Current IAAF rules require athletes to throw the javelin over the shoulder with one hand.
In 1968, most world-class high jumpers used the strad- to jump 7 ft 4 1/4 in. (2.24 m). The technique became
dle technique (figure I.4a). But at the Olympics in Mexico known as the Fosbury Flop (figure I.4b). Its advantages
City, the gold medalist in the high jump used a technique over the straddle technique were its faster approach run
few had ever seen. Dick Fosbury, an American from and its ease of learning. No biomechanics researcher
Oregon State University, used a back layout technique had developed this technique. Fosbury achieved success
a� Zuma Press/Icon SMI
b
Figure I.4 Before 1968, most high jumpers used the straddle technique (a); but after 1968, many switched to the
Fosbury Flop technique (b), the technique used by almost all elite high jumpers today.
6
---
[Cuối tài liệu]
Index
441
overuse injuries and 366, 367, 367f as goal of biomechanics 4-7, 5f, 6f throwing technique
in running 372-375, 372f, 373f, 374f, ideal 313-317 fastball pitch bioanalysis of 322-326,
injury reduction and 10 324f, 325f
375f instruction on 321-322 impulse�momentum relationship and
shear 244-245, 244f, 245f, 364 observation of 317-320, 318f, 319f 103-104
stress continuum 365-366, 365f, 366f in sprint running 332-336, 334f, 335f work�energy relationship and 123, 124
tensile 241-243, 242f, 245f, 364 steps in qualitative biomechanical
threshold of 367-368 thrust reflex 306
tissue response to 364-367, 365f, 366f analysis of 313-322, 318f, 319f tightrope walkers 210-211
units of 240 technology in biomechanics time
stress fractures 373
stress relaxation 258, 258f, 259f, 260 accelerometers 389, 389f measurement of 14, 386-387
stress�strain relationship computer simulation and modeling 392 unit conversions 397t
elastic behavior 252-253, 253f, 254 data collection 385-386, 385f time of flight 77, 79
material strength and mechanical force platforms 341, 390, 391f timing devices 386-387
force transducers 390-391 tonic neck reflex 306, 307f
failure 254-256, 255f, 256f inertial measurement units 389-390 toppling force 153, 155, 155f, 156f, 157f
for muscles 260, 260f motion capture (mocap) systems 390 torque. See also center of gravity
plastic behavior 254 optical imaging systems 387-388, 388f center of gravity and 145-149
strain energy and 121 pressure sensors 391, 392f definition of 134
for tendons 260, 260f timing devices 386-387 examples of use of 136-137, 137f, 138f
stretching exercises 260 velocity-measuring systems 387 mathematical definition of 135-136,
stretch reflex 304-305 temporal phases, in anatomical analysis
striated (skeletal) muscle 278 136f
stride length 376 342 moment arm in 136-137, 136f, 137f,
strongman competitors 42-44, 42f, 44f tendon reflex 305-306
supination tendons 138f, 139, 139f
description of 187, 188f muscle contraction force and 296
injuries and 374-375, 374f, 375f Achilles tendinitis 375 muscle force estimates from
of the proximal radioulnar joint 269 adaptation to stress 365
supporter muscles 285 anisotropic behavior of 257 equilibrium equations 144
surface area, friction and 24-25, 25f Golgi tendon organ 305-306 muscular 137, 139-140, 139f
surface drag 224-225, 224f, 226 mechanical properties of 259-260, net 141-144, 143f
swimmers Newton's first law of motion and 205-
drag reduction by 227 259f, 260f
swimsuit design 7, 8, 8f muscle structure and 280-281, 280f, 207, 205f, 207f
technique improvement 4, 5f Newton's second law of motion and
swimsuit design 7, 8, 8f 281f
sympathetic nerves 300 tennis balls, coefficient of restitution of 98 208-209
synarthrodial joints 268, 269f tennis elbow (lateral epicondylitis) 10 sign conventions 136
synergy, in muscle function 285 tennis players in sport 137
synovial fluid 271, 271f strength training devices and 140-141,
synovial joints forehand drive analysis 326-331, 328f,
stability of 271-273, 272f, 273f 329f, 330f, 331f, 332f 140f, 141f
structure of 270-271, 271f units of 136
types of 268-270, 269f, 270f injury prevention in 10 torsion load 248, 249f, 364
synovial membrane 270-271, 271f purpose of serve by 314, 315 toughness 255, 255f, 256f
Syst�me International d'Unites (SI racket length and grip by 329-330, 330f trabecular bone 257
racket orientation 330-331, 331f traction dislocations 271, 272f
units) 14-15. See also units of spin by 231-232, 231f, 327, 331, 332f training
measurement tensile forces 21 football punt analysis for 346-349,
tensile stress
T analysis of 241-243, 242f 347f, 349t
takeoff distance 335, 335f from bending loads 245-248, 246f injuries and pace of 376
tangent 32 on bones 247-248 javelin throw analysis for 352-356,
tangential acceleration 176, 177f injuries and 364
technical training 340 tension. See also stress, mechanical 352f-354f, 357t-359t
technique active vs. passive 289-290, 292f physical 341
analysis of 241-243, 242f physiologic training zone 365-366,
characteristics of most effective 315- contraction velocity and 293-294, 294f
317 definition of 21 365f
fatigue and 295 principle of specificity 340
evaluation of 320-321 muscle fiber type and 295-296 qualitative anatomical analysis and
in fastball pitch 322-326, 324f, 325f stimulus duration and 295
in forehand drive in tennis 326-331, tetanic response 285, 286f, 303, 303f 341-344, 344t
third law of motion. See Newton's third sprint running analysis for 350-352,
328f, 329f, 330f, 331f, 332f
law of motion 351f, 353t
threshold potential 303 technical 340
technique deficiency analysis 8-9
vertical jump analysis for 345-346,
345f, 346t
training effect 366, 366f, 367
Index
442
translation 135, 135f. See also linear motion prefixes for 396t W
transverse abduction 187, 188f of speed 60 walking, center of gravity and 158
transverse axis of stress 240 water pressure 218-220, 219f
summary table of 395t watt (W) 128
identifying 180-183, 180f, 181t, 182f of time 14 Watt, James 128
joint movements around 183, 184, of torque 136 weight
of velocity 14, 62, 396t
184f, 187, 188f, 189t of work 116 calculation of 22
transverse (horizontal) plane 179-183, upward rotation 187 definition of 14
U.S. Golf Association (USGA) 97-98 Newton's law of universal gravitation
180f, 181t, 182f, 189t
traumatic injury 366 V and 108
triangles, right vectors. See also net force weightlifters
parts of 33 acceleration as 66 Newton's first law and 89-90, 89f
resolution of forces into 33-36, 34f, 35f addition of colinear forces by 26-28 power output by 129, 129f
resultant force from 31-33, 31f, 32f, addition of concurrent forces by 28-31, in static equilibrium 40-42, 41f, 42f
work done by 117-119, 117f
33f, 34 29f, 30f, 31f work�energy relationship in 121-122
similar 31-32, 32f angular momentum and 203 weight-lifting machines 140-141, 140f,
sum of angles of 33 angular velocity and 173
triceps brachii 281, 283 definition of 20 141f
trigonometric technique displacement as 57-60, 58f weight training, motor unit recruitment
of force triangle from graphic technique forces as 20, 26
momentum and 91 in 303
36-37, 37f resultant force from 26 Williams' model of repetitive stress 366,
resolution of horizontal and vertical sign conventions for 28
trigonometric technique with 31-34, 366f, 367
forces by 31-33, 31f, 32f, 33f, 34 windup position (pitcher's) 324-325, 325f
of resultant displacement 59 31f, 32f, 33f, 36-37, 37f withdrawal reflex 306-307, 307f
triple axel 314 velocity as 62-64 Wolff, Julius 364
trochanter 266, 267f velocity Wolff's law 364-365
trochoid joints 269, 270f absolute 222 work. See also work�energy relationship
trunk (intervertebral joints) angular 173-176, 174f, 176f
in a javelin throw 352-356, 352f-354f, angular and linear relationship 173- definition of 116
by muscles 119, 282, 342-344
358t 176, 174f, 176f negative 119
movements of 187, 188f average 62-63 power and 128-129
tubercle 266, 267f components of 62-63 in sport 117-119, 117f
tuberosity 266, 267f definition of 60 stability and 155
turbulent flow 225-227, 225f, 227f in dynamic fluid force 221-222, 222f units of 116
twist axis 180, 180f importance of 64, 65t work�energy relationship
twitch responses 285, 286f, 295-296, instantaneous 63-64, 68-69, 173 conservation of mechanical energy
moments of inertia and 202, 202f, 203f
303, 303f of muscle contraction 293-294, 294f 125-127
type I muscle fibers (SO) 295-296 in projectile motion 68-69 doing work to decrease energy 125
type IIA muscle fibers (FOG) 295-296 relative 221-222, 222f, 223f, 228 doing work to increase energy 123-125,
type IIB muscle fibers (FG) 295-296 units of 14, 62, 396t
vertical axis 180, 180f 125f
U vertical jump, qualitative anatomical examples of 121-125, 125f
ulnar deviation 185-186, 186f wrestlers
ultimate strength 255, 255f analysis of 345-346, 345f, 346t stability and 158-159, 159f
uniform acceleration 68 vestibular system 305-306 torque used by 137, 139f
unipennate muscles 281, 282f videotaping performances wrist joints, movements at 185-186, 186f
units of measurement
digitizing analog data from 341, 387- X
of acceleration 66 388 Xbox Kinect game system 390
of angles 168-169, 169f
of angular acceleration 176 kinematic data from 341, 387 Y
of angular velocity 173 to show performer 322 yellow marrow 265
conversions of 397t, 398t temporal phase information from 342 yield strength 255, 255f
of force 20, 22 viscous drag 224-225, 224f Young's modulus 253
of kinetic energy 120 volume, center of 220-221
of length 14 Z
of mass 15, 396t, 397t Z bands 279, 279f
of power 128
about the author
Peter M. McGinnis, PhD, is a professor in the depart- professor in the department of kinesiology at the Univer-
ment of kinesiology at the State University of New York, sity of Northern Colorado. During that time he served as
College at Cortland, where he has taught since 1990. He a sport biomechanist in the Sports Science Division of
is also the men's and women's pole vault coach at SUNY the U.S. Olympic Committee in Colorado Springs, where
Cortland. Before 1990, Dr. McGinnis was an assistant he conducted applied sport biomechanics research, tested
athletes, taught biomechanics courses to coaches, and
developed educational materials for coaches.
Dr. McGinnis is also the biomechanist for the pole
vault event for USA Track and Field. As a member of the
American Society of Testing Materials, he serves as chair
of the pole vault equipment subcommittee and the task
group on pole vault helmets. He has authored numerous
articles and technical reports about the biomechanics of
pole vaulting and has been a reviewer for Sports Biome-
chanics, the Journal of Applied Biomechanics, Research
Quarterly for Exercise and Sport, and the Journal of
Sports Sciences.
Dr. McGinnis is a member of numerous professional
organizations, including the American College of Sports
Medicine, American Society of Biomechanics, and the
International Society of Biomechanics in Sport. He
received a PhD in physical education from the University
of Illinois in 1984 and a BS in engineering from Swarth-
more College in 1976.
443
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HUMAN KINETICS
The Information Leader in Physical Activity & Health
P.O. Box 5076 � Champaign, IL 61825-5076 USA
Quick-Reference Equations
MATHEMATICAL FORMULAS Horizontal velocity: Perfectly inelastic collision of two objects
Pythagorean theorem v = vf = vi = constant
(2.22) m1u1 + m2u2 = (m1 + m2)v (3.19)
Horizontal acceleration: (2.23)
A2 + B2 = C2 (1.5) a=0 Coefficient of restitution
Trigonometric functions e = v1 � v2 = v2 � v1 (3.20)
u1 � u2 u1 � u2
sin = opposite side (1.6) Other equations governing projectile
hypotenuse (1.7) motion
(1.8) Time of flight: Newton's 2nd law: law of acceleration
cos = adjacent side (1.9)
hypotenuse (1.10) tup = tdown if yf = yi (2.20) F = ma (3.22)
(1.11)
tan = opposite side tflight = 2tup if yf = yi (2.21) Fx = max (3.23)
adjacent side (3.24)
(3.29)
= arcsin opposite side Parabolic equation: Fy = may
hypotenuse (3.30)
x 1 x 2 Impulse�momentum equation
vx 2 vx
yf = yi + vyi + g (2.27)
= arccos adjacent side F� t = m(vf - vi)
hypotenuse
LINEAR KINETICS Universal law of gravitation:
opposite side gravitational force
= arctan adjacent side Weight
LINEAR KINEMATICS W = mg (1.2) F = G m1m2
r2
Static and dynamic friction
Average speed Fs = sR WORK, POWER, AND ENERGY
Fd = dR
s = (2.5) Static equilibrium (1.3) Work
t (2.6) F = 0 (1.4)
(2.9) U = F�(d)
Average velocity (1.12) (4.2)
(2.14)
v = d (2.16) Kinetic energy
t (2.11)
(2.15) KE " 1 mv2 (4.4)
Average acceleration Fx = 0 (1.13) 2
Fy = 0 (1.14)
a = v f � vi Newton's 1st law: law of inertia Gravitational potential energy
t v = constant if F = 0 (3.1a)
PE = Wh (4.5)
PROJECTILE EQUATIONS
Strain energy
Vertical motion (y)
SE = 1 k x2
Vertical position: or 2 (4.7)
yf = yi + vit + 1 g(t)2 Work�energy principle
2 U = E
F = 0 if v = constant (3.1b)
1 g(t)2
yf = 2 if yi = 0 and vi = 0 Linear momentum (4.8)
(4.12)
Vertical velocity: L = mv (3.6) Power (4.13)
vf = vi + gt Conservation of momentum P = U
v2 = v2 + 2gy t
L = constant if F = 0 (3.7) P = F�v�
vpeak = 0 (2.19) Lx = constant if Fx = 0 (3.8) ANGULAR KINEMATICS
vf = gt if yi = 0 and vi = 0 (2.17) Angular position measured in radians
v2 = 2gy if vi = 0 (2.18) Ly = constant if Fy = 0 (3.9)
Vertical acceleration:
a = g = -9.81 m/s2 (2.10) Li = (mu) = m1u1 + m2u2 + m3u3 = arc length = (6.1)
+ . . . = m1v1 + m2v2 + m3v3 r r
+ . . . = (mv) = Lf = constant (3.11) Angular displacement and arc length
if F = 0
Perfectly elastic collision of two objects = r (6.4)
Horizontal motion (x) ( ) v1 Average angular velocity
Horizontal position: 2m2u2 + m1 - m2 u1
= m1 + m2 (3.17) f i (6.6)
x = vt (2.26) = t = t
Angular velocity and linear velocity Moment of inertia FLUID MECHANICS
Pressure
vT = r (6.8) Ia = miri2 (7.1)
P= F
Average angular acceleration Ia = mka2 (7.2) A
= = f �i (6.9) Moment of inertia: parallel axis theorem Density
t t
Ib = Icg + mr2 (7.3) = m (8.3)
Angular momentum V
Tangential acceleration
Drag force
aT = r (6.10) Ha = Iaa (7.4) FD = 1 CD Av2 (8.5)
(6.11) 2
Centripetal acceleration
Angular momentum of the human body Lift force
vT2
ar " r Ha = (Ii i + mir2i/cg i/cg) (7.5) FL = 1 CL Av2 (8.6)
2
ar = 2r (6.12) Conservation of angular momentum MECHANICS OF MATERIALS
Hi = Iii = Iff = Hf = constant if T = 0
ANGULAR KINETICS Stress
Torque (7.7)
=F (9.1)
Angular version of Newton's 2nd law A
T=F�r (5.1) Ta = Iaa (7.9) Shear stress
(7.10)
Static equilibrium ( ) Ta =F (9.2)
Ha H f � Hi
T = 0 (5.2) = t = t A
Strain
Center of gravity Angular impulse-momentum = C � Co (9.4)
T�at = (Hf � Hi)a Co
(W � r) = (W) � rcg (5.3) (7.11)
Elastic modulus
E = (9.5)
Abbreviations for Variables and Subscripts Used in Equations
Variables L = linear momentum = strain
m = mass = coefficient of friction
a = instantaneous linear acceleration P = power = density
a� = average linear acceleration P = pressure i = stress
P = force = sum of ...
A = area PE = gravitational potential energy = shear stress
r = radius = angular position
CD = coefficient of drag r = moment arm = instantaneous angular velocity
CL = coefficient of lift R = normal contact force � = average angular velocity
d = displacement s = instantaneous linear speed
�s = average linear speed Subscripts
e = coefficient of restitution t = time a = axis
T = torque b = axis
E = energy u = pre-impact velocity d = dynamic
U = work done cg = center of gravity
E = elastic modulus or Young's modu- v = instantaneous linear velocity D = drag
v = post-impact velocity f = final or ending
lus v� = average linear velocity i = initial or starting
V = volume i = one of a number of parts
FF� = force force W = weight L = lift
= average x = horizontal position o = original or undeformed
y = vertical position r = radial
Fd = dynamic friction force = instantaneous angular acceleration s = static
Fs = static friction force � = average angular acceleration T = tangential
F = net force = sum of forces = change in ... = final � initial x = horizontal
y = vertical
g = acceleration due to gravity
G = gravitational constant
h = height
H = angular momentum
I = moment of inertia
k = radius of gyration
k = stiffness or spring constant
KE = kinetic energy
= distance traveled or length