Introduction to Biomechanics for Human Motion Analysis Copyright© 1997 Waterloo Biomechanics All rights reserved. This book is protected by copyright. No part of this publication may be reproduced in any form, including photocopying, or utilized by any information storage and retrieval system without written permission from Waterloo Biomechanics. Critics or reviewers may quote brief passages in connection with a review or critical article in any medium.

ISBN 0-9699420-2-8

Waterloo Biomechanics 364 Warrington Drive Waterloo, Ontario, Canada N2L 2P6 Phone: +1-519-747-0077 FAX: +1-519-747-1894 Printed and bound at Graphic Services, University of Waterloo

Canadian Cataloguing in Publication Data Robertson, D. Gordon E., 1950Introduction to Biomechanics for Human Motion Analysis Bibliography: p. Includes index. 1. Human mechanics 2. Kinesiology I. Robertson, D. Gordon E.

10 9 8 7 6 5 4 3 2

Dedicated to my children, Heather and Andrea

TABLE OF CONTENTS

TABLE OF CONTENTS.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Photo credits.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 History of Biomechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Taxonomy of Biomechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Units of Measure and the System International. . . . . . . . . . . . . . . . . . . 4 Rules for Reporting Units in the Metric System. . . . . . . . . . . . . . . 5 1.4 Conversion of Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Numerical Accuracy and Significant Digits.. . . . . . . . . . . . . . . . . . . . 10 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2. FUNDAMENTAL CONCEPTS OF BIOMECHANICS. . . . . . . . . . . . . . . . . 13 2.1 Space, Mass and Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Frames of Reference, Planes and Axes. . . . . . . . . . . . . . . . . . . . . . . . 15 Cardinal, Relative or Anatomical Frame of Reference. . . . . . . . . 16 Anatomical position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Centre of mass and centre of gravity. . . . . . . . . . . . . . . . . 17 Newtonian or Absolute Frame of Reference. . . . . . . . . . . . . . . . . 19 2.4 Scalars and Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Scalars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Parallelogram Law for the Addition of Vectors.. . . . . . . . . . . . . . . . . 20 2.6 Coordinate Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Polar to Rectangular Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . 22 Rectangular to Polar coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . 23 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3. STATICS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Universal Law of Gravitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Resolution of Forces into Components. . . . . . . . . . . . . . . . . . . . . . . . 28 Rectangular Components and the Unit Vectors.. . . . . . . . . . . . . . 29 Addition of Forces by Summing Components.. . . . . . . . . . . . . . . 30 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Newton’s First Law of Motion: Law of Inertia. . . . . . . . . . . . . . . . . . 32 Equilibrium of a Particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 First Law of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

iv

Table of Contents Resultant Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Forces on Rigid Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Principle of Transmissibility of Forces. . . . . . . . . . . . . . . . . . . . . 35 Moment of a Force.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Levers of the Musculoskeletal System.. . . . . . . . . . . . . . . . . . . . . . . . 42 Classes of Levers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 First-class levers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Second-class levers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Third-class levers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Mechanical Advantage versus Speed Advantage. . . . . . . . . . . . . 47 3.5 Laws of Statics.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Moment of Force as a Vector Product. . . . . . . . . . . . . . . . . . . . . . 48 Laws of Statics.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Free-body Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Indeterminacy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Free-body diagrams: Rules. . . . . . . . . . . . . . . . . . . . . . . . 52 Steps for Solving Mechanics Problems. . . . . . . . . . . . . . . . . . . . . 53 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4. FRICTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.1 Laws of Dry Friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Empirically Determining the Coefficients of Friction. . . . . . . . . . . . . 65 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5. KINEMATICS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1 Linear Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Distance versus Displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Constant Linear Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Constant Linear Acceleration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2 Projectile Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 Angular Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Angular Displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Radian measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Angular Velocity and Acceleration. . . . . . . . . . . . . . . . . . . . . . . . 94 Constant Angular Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 Relationship between Linear and Angular Measures.. . . . . . . . . . . . . 97 Angular and Linear Displacement. . . . . . . . . . . . . . . . . . . . . . . . . 97 Angular and Linear Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Transverse velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Table of Contents

v

Angular and Linear Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . 99 Transverse acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Radial acceleration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Total linear acceleration.. . . . . . . . . . . . . . . . . . . . . . . . . . 99 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6. KINETICS: FORCES AND MOMENTS OF FORCE. . . . . . . . . . . . . . . . . . 103 6.1 Newton’s Second Law: Law of Acceleration. . . . . . . . . . . . . . . . . . 103 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.2 Moment of Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3 Moment of Inertia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4 Newton’s Third Law: Law of Reaction. . . . . . . . . . . . . . . . . . . . . . . 119 Centripetal and Centrifugal Forces. . . . . . . . . . . . . . . . . . . . . . . 119 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7. KINETICS: IMPULSE AND MOMENTUM. . . . . . . . . . . . . . . . . . . . . . . . . 125 7.1 Impulse-Momentum Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Linear Impulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Linear Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Application of the Impulse-Momentum Theorem. . . . . . . . . . . . 131 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 Angular Impulse and Momentum Theorem.. . . . . . . . . . . . . . . . . . . 136 Angular Impulse.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Angular Momentum.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.3 Conservation of Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Conservation of Linear Momentum.. . . . . . . . . . . . . . . . . . . . . . 141 Conservation of Angular Momentum. . . . . . . . . . . . . . . . . . . . . 142 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8. KINETICS: WORK, ENERGY AND POWER. . . . . . . . . . . . . . . . . . . . . . . 149 8.1 Work-Energy Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Work on a Particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Work on a Rigid Body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Mechanical Energy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Mechanical work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.2 Work of a Force or Moment of Force. . . . . . . . . . . . . . . . . . . . . . . . 157 Work of a Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Work of a Moment of Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.3 Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Average Power.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Instantaneous Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Power of a force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

vi

Table of Contents Power of a moment of force.. . . . . . . . . . . . . . . . . . . . . . 165 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.4 Conservation of Mechanical Energy. . . . . . . . . . . . . . . . . . . . . . . . . 168 Conservative Forces.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 8.5 Mechanical Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9. FLUID MECHANICS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9.1 Fluid Statics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Relative density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Pressure and depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Pascal’s Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Archimedes’ Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 9.2 Fluid Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Drag Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Terminal velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Streamline Flow and the Bernoulli Principle.. . . . . . . . . . . . . . . 181 Bernoulli’s principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Lift Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Magnus Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 APPENDIX A: TRIGONOMETRY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Trigonometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Basic Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 185 Tangent function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Sine function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Cosine function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Pythagorean Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Sine Laws.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Cosine Laws.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . 187 APPENDIX B: ALGEBRA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Algebra.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Order of operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Term.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Roots of a Quadratic Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . 189 APPENDIX C: PLANE GEOMETRY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Plane Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Table of Contents

vii

APPENDIX D: VECTOR ALGEBRA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Vector Algebra.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Magnitude of a Vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Negative of a Vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Changing the Magnitude of a Vector. . . . . . . . . . . . . . . . . . . . . . 193 Addition of Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Commutative Law for the addition of vectors. . . . . . . . . 194 Associative Law for the addition of vectors. . . . . . . . . . 194 Distributive Law of scalar multiplication over addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Multiplication of Vectors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 By a scalar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Scalar, dot or inner product. . . . . . . . . . . . . . . . . . . . . . . 194 Vector, cross or outer product. . . . . . . . . . . . . . . . . . . . . 195 Distributive property of vector product over addition. . . 195 APPENDIX E: CALCULUS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Differential Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Integral Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 APPENDIX F: TABLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 GLOSSARY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Textbooks.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 ANSWERS TO ODD NUMBERED QUESTIONS. . . . . . . . . . . . . . . . . . . . . . 222 EQUATIONS.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

PREFACE

This book was written to bridge the gap between the way mechanics is taught to engineering or physics students and to those students in kinesiology or physical education. In the past, students who wanted to study, in depth, the mechanics of human motion had to either study the mechanics of gears, pulleys and levers with the engineers or deal with the complexities of optics, wave motion and quantum electrodynamics with the physicists. In this text human motion is analyzed primarily in two dimensions with mathematical tools easily modified to handle three-dimensional motions in an advanced course. This book is intended to form the foundation upon which advanced analysis of mechanical human or animal motion can be undertaken. Chapters are arranged in a progression and should not be taught in a different order. The first two chapters are concerned with fundamental concepts that are required for all subsequent chapters. In particular, the metric system (System International) is defined and used almost exclusively throughout the text, although methods for converting between former measurement systems and the metric system are provided. Chapter three defines the laws of statics but is primarily concerned with developing understanding of vectors and the mathematical concepts associated with vectors. Vector algebra was selected as the optimal mathematical tool for handling the concept of forces acting on human bodies. Since this is an introductory text, however, analysis is centred around linear or planar motions on rigid bodies. The human body is therefore considered to be either a point mass or a single rigid body or parts of the body are considered in isolation to simplify their analysis. More complex analysis, such as, linked segment analysis or the elastic deformation of bodies is reserved for an advanced textbook. Chapter four analyzes the forces created by dry friction when two or more bodies are forced against each other. No analysis of the fluid friction or how uneven surfaces interact is provided. Chapter five introduces the concept of kinematics, that is, motion description, which is essential for quantifying the differences between one athletic technique and another or the differences among several individuals performing the same skill. Kinematic analysis is also the first step to performing inverse dynamical analysis which define the underlying causes of human movements. In chapters six, seven and eight, three different forms of kinetic analysis are presented. First, the force/mass/acceleration and its angular counterpart are defined, followed second by the impulse-momentum theorem and its angular counterpart and the third by the work-energy theorem and the concept of mechanical power are presented. Each chapter introduces these concepts with the laws from which they were derived. Sample problems are provided with the answers to odd numbered problems collected at the end of the text. Numerous example problems are given with each section. Chapter nine concludes the textual material with an introduction to the principles and concepts associated with fluid mechanics—both statics and dynamics—to enable understanding of the complex motions produced in a world

Preface

ix that includes a thick atmosphere and is plentiful with large and small bodies of water. I am indebted to Dr. Peter Stothart for writing this chapter. Peter has many years of teaching experience in biomechanics and structural anatomy and can put quite a curve on a volleyball. One of the most important features of this text is the use of example problems and sample questions within each chapter. Students have available many solved problems plus the answers to all the odd-numbered questions at the end of the book. Instructors may use the even-numbered problems for homework assignments or examination questions. Furthermore, there is a list of all the essential equations included at the back of the book for quick reference. Included in appendices are brief descriptions of some of the fundamental laws, rules and concepts associated with, trigonometry, algebra, planar geometry, vector algebra and calculus. In addition, tables on the System International, conversion factors and physical constants are provide as reference material. Finally, there is a glossary, bibliography and extensive index to assistant readers in their pursuit of biomechanical information in this text and others. Thanks are extended to many friends and colleagues who reviewed portions of this book, including Dr. Ton van den Bogert, Dr. Graham Caldwell, Dr. Joe Hamill, Ms. Sheila Purkiss, Mr. Li Li, Ms. Patti Turnbull and particularly Dr. Peter Stothart. I am also indebted to Dr. David Winter my mentor, friend and now editor, for supporting my work over the years. Special thanks are reserved for my wonderful wife, Dr. Lorna McLean, who offered advice and encouragement throughout the process of writing and editing. I would also like to thank my children for coping with my extended periods in front of the computer at all hours of the day and night. D. Gordon E. Robertson Ottawa, Ontario July 1997

Author, Dr. Gordon Robertson, and a former graduate student, Dr. Edward Lemaire, align a high-speed cine-camera in preparation for filming Canadian sprinters indoors. Two world records were set during the competition by Angella Issajenko and Ben Johnson.

x

Preface Photo and graphics credits. Page: x

Dan Curry

11, 83, 90, 110, 128, 134, 155, 160, 161, 170 146

Laura Fauquier

D. Gordon E. Robertson