Copyright © 2005 by F. A. Davis.
Copyright © 2005 by F. A. Davis. Joint Structure and Function: A Comprehensive Analysis Fourth Edition
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Copyright © 2005 by F. A. Davis. Joint Structure and Function: A Comprehensive Analysis Fourth Edition Pamela K. Levangie, PT, DSc Professor Physical Therapy Program Sacred Heart University Fairfield, CT Cynthia C. Norkin, PT, EdD Former Director and Associate Professor School of Physical Therapy Ohio University Athens, OH F. A. Davis Company • Philadelphia
Copyright © 2005 by F. A. Davis. F. A. Davis Company 1915 Arch Street Philadelphia, PA 19103 www.fadavis.com Copyright © 2005 by F. A. Davis Company Copyright © 2005 by F. A. Davis Company. All rights reserved. This book is protected by copyright. No part of it may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without written permission from the publisher. Printed in the United States of America Last digit indicates print number: 10 9 8 7 6 5 4 3 2 1 Acquisitions Editor: Margaret M. Biblis Development Editor: Jennifer Pine Design Manager: Carolyn O’Brien As new scientific information becomes available through basic and clinical research, recommended treatments and drug therapies undergo changes. The author(s) and publisher have done everything possible to make this book accurate, up to date, and in accord with accepted standards at the time of publication. The author(s), editors, and publisher are not responsible for errors or omissions or for consequences from application of the book, and make no warranty, expressed or implied, in regard to the contents of the book. Any practice described in this book should be applied by the reader in accordance with professional standards of care used in regard to the unique circum- stances that may apply in each situation. The reader is advised always to check product information (package inserts) for changes and new information regarding dose and contraindications before administering any drug. Caution is especially urged when using new or infrequently ordered drugs. Library of Congress Cataloging-in-Publication Data Levangie, Pamela K. Joint structure and function : a comprehensive analysis / Pamela K. Levangie, Cindy Norkin.— 4th ed. p. ; cm. Includes bibliographical references and index. ISBN 0–8036–1191–9 (hardcover : alk. paper) 1. Human mechanics. 2. Joints. [DNLM: 1. Joints—anatomy & histology. 2. Joints—physiology. WE 399 L655j 2005] I.Norkin, Cynthia C. II. Title. QP303.N59 2005 612.7′5—dc22 2004021449 Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by F. A. Davis Company for users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service, provided that the fee of $.10 per copy is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923. For those organizations that have been granted a photocopy license by CCC, a separate system of payment has been arranged. The fee code for users of the Transactional Reporting Service is: 8036–1191–9/05 0 ϩ $.10.
Copyright © 2005 by F. A. Davis. Dedication for the Fourth Edition For more than 20 years, we have been privileged to contribute to the professional development of students and practitioners . The four editions of Joint Structure and Function have been shaped as much by the faculty and students who use this text as by the changes in evidence and tech- nology. Therefore , we dedicate this 4th edition of Joint Structure and Function to the faculty, the students, and the health care professionals who are both our consumers and our partners.
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Copyright © 2005 by F. A. Davis. Preface to the Fourth Edition With the 4th edition of Joint Structure and Function, we chapters in best evidence and in clinical relevance. A continue a tradition of excellence in education that key change in our educational approach is in use of began more than 20 years ago. Although we entered patient cases, not as adjuncts to the text but as inte- the market when there were few resource options for grated elements within the text of each chapter. Patient our readers, we are now in an era of increasingly cases (in both highlighted Patient Case and Patient numerous choices in a variety of media. We continue Application boxes) substantially facilitate an under- with this edition to respond to the ever-accelerating standing of the continuum between normal and changes taking place in media and research technology impaired function, making use of emerging case-based as well as in the education of individuals who assess and problem-based learning educational strategies. We human function. In the move toward what many have maintained highlighted summary boxes (now believe will be a “paperless” society, the role of text- called Concept Cornerstones) while also adding high- books is evolving rapidly; learners demand changes but lighted Continuing Exploration boxes that provide the are not ready to give up the textbook as an educational reader or the instructor additional flexibility in setting modality. With the 4th edition, we attempt to meet the learning objectives. challenges before us and our learners by taking advan- tage of new technologies, current evidence, the exper- What is unchanged in this edition of Joint Structure tise of colleagues, and a more integrated approach to and Function is our commitment to maintaining a text preparing those who wish to understand human kinesi- that provides a strong foundation in the principles that ology and pathokinesiology. underlie an understanding of human structure and function while also being readable and as concise as Use of digital imaging technology allows us to sub- possible. We hope that our years of experience in con- stantially change the visual support for our readers. tributing to the education of health care professionals Line drawings (many taking advantage of our two-color allow us to strike a unique balance. We cannot fail to format) have been added or modified because these recognize the increased educational demands placed often work best to display complex concepts. However, on many entry-level health care professionals and hope we now include in this edition a greater variety of image that the changes to the 4th edition help students meet options, including photographs, medical imaging, and that demand. However, Joint Structure and Function, three-dimensional computer output that should better while growing with its readers, continues to recognize support learning and better prepare the reader for that the new reader requires elementary and inter- negotiating published research. Changes in size, layout, linked building blocks that lay a strong but flexible and two-color format provide a more reader-friendly foundation to best support continued learning and page and enhance the reader’s ability to move around growth in a complex and changing world. within each chapter. We continue to appreciate our opportunity to con- Recognizing the increasing challenge of remaining tribute to health care by assisting in the professional current in published research across many areas, we development of the students and practitioners who are now take advantage of the expertise of a greater num- our readers. ber of respected colleagues as chapter contributors. Our contributors straddle the environs largely of Pamela K. Levangie research, practice, and teaching—grounding their Cynthia C. Norkin vii
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Copyright © 2005 by F. A. Davis. Acknowledgments No endeavor as labor-intensive as updating a science Publisher, brought new energy and a contemporary and research-based textbook such as Joint Structure and vision to this project; Jennifer Pine, Developmental Function can be accomplished without the expertise and Editor, managed the project in a manner that merged support of many committed individuals. We appreciate Margaret’s vision with Jennifer’s own unique contribu- the very considerable investment of our continuing tions. We credit our artist, Anne Raines, with many new contributors, as well as the willingness of our new group clear images that appear in the book. We are grateful to of clinical and academic professionals to also lend their artists Joe Farnum and Timothy Malone, whose creative names and expertise to this project. Our thanks, there- contributions to previous editions also appear in the fore, to Drs. Borstad, Chleboun, Curwin, Hoover, 4th edition. Lewik, Ludewig, Mueller, Olney, Ritzline, and Snyder- Mackler as well as to Mss. Austin, Dalton, and Starr. All Of course, none of us would be able to would be brought from their various institutions, states, and able to make such large investment of time and energy countries their enthusiasm and a wealth of new knowl- to a project like this without the support of our col- edge and ideas. We also would like to thank the review- leagues and the ongoing loving support of families. We ers, listed on pages xiii and xiv, who provided us with can only thank them for giving up countless hours of many helpful suggestions for improving the text. our time and attention to yet another edition of Joint Structure and Function. We further extend our gratitude to FA Davis for their investment in this book’s future. Margaret Biblis, ix
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Copyright © 2005 by F. A. Davis. Contributors Noelle M. Austin, PT, MS, CHT Paula M. Ludewig, PT, PhD CJ Education and Consulting, LLC Associate Professor Woodbridge CT Program in Physical Therapy cj-education.com & University of Minnesota The Orthopaedic Group Minneapolis, Minnesota Hamden, Connecticut Michael J. Mueller, PT, PhD, FAPTA John D. Borstead, PT, PhD Associate Professor Assistant Professor Program in Physical Therapy Physical Therapy Division Washington University School of Medicine Ohio State University St. Louis, Missouri Columbus, Ohio Sandra J. Olney, PT, OT, PhD Gary Chleboun, PT, PhD Director, School of Rehabilitation Therapy Professor Associate Dean of Health Sciences School of Physical Therapy Queens University Ohio University Kingston, Ontario Athens, Ohio Canada Sandra Curwin, PT, PhD Pamela Ritzline, PT, EdD Associate Professor Associate Professor Department of Physical Therapy Krannert School of Physical Therapy University of Alberta University of Indianapolis Edmonton, Alberta Indianapolis, Indiana Canada Lynn Snyder-Macker, PT, ScD, SCS, ATC, FAPTA Diane Dalton, PT, MS, OCS Professor Clinical Assistant Professor of Physical Therapy Department of Physical Therapy Physical Therapy Program University of Delaware Boston University Newark, Delaware Boston, Massachusetts Julie Starr, PT, MS, CCS Don Hoover, PT, PhD Clinical Associate Professor of Physical Therapy Assistant Professor Physical Therapy Program Krannert School of Physical Therapy Boston University University of Indianapolis Boston, Massachusetts Indianapolis, Indiana Michael Lewek, PT, PhD Post Doctoral Fellow, Sensory Motor Performance Program Rehabilitation Institute of Chicago Northwestern University Chicago, Illinois xi
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Copyright © 2005 by F. A. Davis. Reviewers Thomas Abelew, PhD Ricardo Fernandez, PT, MHS, OCS, CSCS Assistant Professor Assistant Professor/Clinician Department of Rehabilitation Medicine Department of Physical Therapy and Human Emory University Movement Sciences Atlanta, Georgia Northwestern University Feinberg School of Medicine Chicago, Illinois Gordon Alderink, PT, PhD Assistant Professor Jason Gauvin, PT, SCS, ATC, CSCS Physical Therapy Department Physical Therapist Grand Valley State University Departments of Occupational Therapy and Physical Allendale, Michigan Therapy Duke University Mary Brown, PT, MEd Durham, North Carolina Physical Therapist Department of Rehabilitation Barbara Hahn, PT, MA Morristown Memorial Hospital, Atlantic Health System Director, Physical Therapist Assistant Program West Orange, New Jersey University of Evansville Evansville, Indiana John A. Buford, PT, PhD Assistant Professor of Physical Therapy John Hollman, PT, PhD Division of Physical Therapy Assistant Professor and Director School of Allied Medical Professions Program in Physical Therapy The Ohio State University Mayo School of Health Sciences Columbus, Ohio Rochester, MN Margaret Carton, MSPT Birgid Hopkins, MS, L.ATC Assistant Professor Director Allied Health, Nursing, and HPE Department Department of Sports Medicine Black Hawk College Merrimack College Moline, Illinois North Andover, Massachusetts Gary Chleboun, PT, PhD Edmund Kosmahl, PT, EdD Professor Professor School of Physical Therapy Department of Physical Therapy Ohio University University of Scranton Athens, Ohio Scranton, Pennsylvania Deborah Edmondson, PT, EdD Gary Lentell, PT, MS, DPT Assistant Professor/Academic Coordinator of Clinical Professor Education Department of Physical Therapy Department of Physical Therapy University of California, Fresno Tennessee State University Fresno, California Nashville, Tennessee xiii
Copyright © 2005 by F. A. Davis. Suzanne Reese, PT, MS Director, Physical Therapist Assistant Program xiv ■ Reviewers Allied Health Department Tulsa Community College Robin Marcus, PT, PhD, OCS Tulsa, Oklahoma Clinical Associate Professor Division of Physical Therapy Claire Safran-Norton, PT, PhD-ABD, MS, MS, OCS University of Utah Assistant Professor Salt Lake City, Utah Department of Physical Therapy Simmons College R. Daniel Martin, EdD, ATC Boston, Massachusetts Associate Professor and Director, Athletic Training Program Exercise Science, Sport, and Recreation Marshall University Huntingdon, West Virginia Matthew C. Morrissey, PT, ScD Department of Physiotherapy King’s College London, KCL London, England
Copyright © 2005 by F. A. Davis. Contents SECTION 1 Joint Structure and Angular Acceleration and Angular Equilibrium 37 Function: Foundational Concepts 2 Parallel Force Systems 38 Meeting the Three Conditions for Equilibrium 41 Chapter 1 Biomechanical Applications Muscle Forces 42 to Joint Structure and Function 3 Total Muscle Force Vector 42 Pamela K. Levangie, PT, DSc Torque Revisited 44 Changes to Moment Arm of a Force 45 Introduction 4 Angular Acceleration with Changing Torques 46 Moment Arm and Angle of Application Patient Case 4 of a Force 46 Part 1: Kinematics and Introduction Lever Systems, or Classes of Levers 48 to Kinetics 5 Muscles in Third-Class Lever Systems 50 Muscles in Second-Class Lever Systems 50 Descriptions of Motion 5 7 Muscles in First-Class Lever Systems 51 Types of Displacement 5 Mechanical Advantage 51 Location of Displacement in Space Trade-Offs of Mechanical Advantage 52 Direction of Displacement 9 Limitations to Analysis of Forces by Magnitude of Displacement 9 Lever Systems 53 Rate of Displacement 10 Introduction to Forces 10 Force Components 53 Definition of Forces 10 Resolving Forces into Perpendicular and Parallel Force Vectors 12 Components 54 Force of Gravity 15 Perpendicular and Parallel Force Effects 54 Introduction to Statics and Dynamics 19 Translatory Effects of Force Components 60 Newton’s Law of Inertia 19 Rotatory Effects of Force Components 61 Newton’s Law of Acceleration 20 Total Rotation Produced by a Force 62 Translatory Motion in Linear and Concurrent Multisegment (Closed-Chain) Force Force Systems 20 Analysis 63 Linear Force System 21 Determining Resultant Forces in a Linear Force Summary 66 System 21 Concurrent Force System 22 Chapter 2 Joint Structure and Newton’s Law of Reaction 24 Function 69 Additional Linear Force Considerations 25 Sandra Curwin, PT, PhD Tensile Forces 26 Joint Distraction 28 Introduction 70 Revisting Newton’s Law of Inertia 31 Joint Design 70 Shear and Friction Forces 32 Materials Used in Human Joints 71 Part 2: Kinetics – Considering Rotatory and Structure of Connective Tissue 72 Translatory Forces and Motion 35 Specific Connective Tissue Structures 77 Torque, or Moment of Force 35 General Properties of Connective Tissue 83 xv
Copyright © 2005 by F. A. Davis. xvi ■ Contents General Structure and Function 142 Structure 142 Mechanical Behavior 83 Function 150 Viscoelasticity 87 Time-Dependent and Rate-Dependent Regional Structure and Function 156 Structure of the Cervical Region 156 Properties 87 Function of the Cervical Region 161 Properties of Specific Tissues 89 Structure of the Thoracic Region 164 Function of the Thoracic Region 165 Complexities of Human Joint Design 91 Structure of the Lumbar Region 166 Synarthroses 91 Function of the Lumbar Region 170 Diarthroses 93 Structure of the Sacral Region 173 Function of the Sacral Region 174 Joint Function 98 Kinematic Chains 98 Muscles of the Vertebral Column 176 Joint Motion 99 The Craniocervical / Upper Thoracic Regions 176 General Changes with Disease, Injury, Lower Thoracic / Lumbopelvic Regions 180 Immobilization, Exercise, and Muscles of the Pelvic Floor 186 Overuse 102 Disease 102 Effects of Aging 187 Injury 102 Immobilization (Stress Deprivation) 103 Summary 188 Exercise 104 Overuse 106 Summary 107 Chapter 3 Muscle Structure and Chapter 5 The Thorax and Chest Function 113 Wall 193 Gary Chleboun, PT, PhD Julie Starr, PT, MS, CCS Introduction 113 Diane Dalton, PT, MS, OCS Patient Case 114 Introduction 193 Elements of Muscle Structure 114 Patient Case 193 Composition of a Muscle Fiber 114 The Contractile Unit 115 General Structure and Function 193 The Motor Unit 117 Rib Cage 193 Muscle Structure 119 Muscles Associated With the Rib Cage 200 Muscular Connective Tissue 121 Coordination and Integration of Ventilatory Muscle Function 123 132 Motions 208 Muscle Tension 123 Classification of Muscles 129 Developmental Aspects of Structure and Factors Affecting Muscle Function Function 209 Differences Associated with the Neonate 209 Effects of Immobilization, Injury, and Differences Associated with the Elderly 210 Aging 135 Immobilization 135 Pathological Changes in Structure and Injury 135 Function 210 Aging 136 Chronic Obstructive Pulmonary Disease 210 Summary 212 Summary 136 Chapter 6 The Temporomandibular Joint 215 SECTION 2 Axial Skeletal Joint Complexes 140 Don Hoover, PT, PhD Chapter 4 The Vertebral Column Pamela Ritzline, PT, EdD Diane Dalton, PT, MS, OCS 141 Patient Case 215 Introduction 142 Introduction 215 216 Patient Case 142 Structure 216 Articular Surfaces
Copyright © 2005 by F. A. Davis. Contents ■ xvii Articular Disk 217 219 Chapter 8 The Elbow Complex 273 Capsule and Ligaments 218 Upper and Lower Temporomandibular Joints Cynthia C. Norkin, PT, EdD Function 219 Introduction 273 Mandibular Motions 219 Muscular Control of the Temporomandibular Patient Case 274 Joint 222 Relationship with the Cervical Spine 223 Structure: Elbow Joint (Humeroulnar and Dentition 225 Humeroradial Articulations) 274 Articulating Surfaces on the Humerus 274 Age-Related Changes in the Articulating Surfaces on the Radius and Temporomandibular Joint 225 Ulna 275 Articulation 276 Dysfunctions 226 Joint Capsule 276 Inflammatory Conditions 226 Ligaments 278 Capsular Fibrosis 226 Muscles 280 Osseous Mobility Conditions 226 Articular Disk Displacement 227 Function: Elbow Joint (Humeroulnar and Degenerative Conditions 227 Humeroradial Articulations) 282 Axis of Motion 282 Summary 228 Range of Motion 284 Muscle Action 286 SECTION 3 Upper Extremity Joint Structure: Superior and Inferior Complexes 232 Articulations 289 Superior Radioulnar Joint 289 Chapter 7 The Shoulder Complex 233 Inferior Radioulnar Joint 289 Radioulnar Articulation 290 Paula M. Ludewig, PT, PhD Ligaments 290 John D. Borstead, PT, PhD Muscles 292 Introduction 233 Function: Radioulnar Joints 292 Axis of Motion 292 Patient Case 234 Range of Motion 293 Muscle Action 293 Components of the Shoulder Complex 234 Stability 294 Sternoclavicular Joint 234 Acromioclavicular Joint 237 Mobility and Stability: Elbow Complex 295 Scapulothoracic Joint 242 Functional Activities 295 Glenohumeral Joint 246 Relationship to the Hand and Wrist 295 Integrated Function of the Shoulder Effects of Age and Injury 296 Complex 259 Age 296 Scapulothoracic and Glenohumeral Injury 297 Contributions 259 Sternoclavicular and Acromioclavicular Summary 300 Contributions 260 Structural Dysfunction 262 Chapter 9 The Wrist and Hand Muscles of Elevation 263 Complex 305 Deltoid Muscle Function 263 Supraspinatus Muscle Function 264 Noelle M. Austin, PT, MS, CHT Infraspinatus, Teres Minor, and Subscapularis Muscle Function 264 Introduction 305 Upper and Lower Trapezius and Serratus Anterior Muscle Function 264 The Wrist Complex 305 Rhomboid Muscle Function 266 Radiocarpal Joint Structure 306 Muscles of Depression 266 Midcarpal Joint Structure 310 Latissimus Dorsi and Pectoral Muscle Function of the Wrist Complex 311 Function 266 Teres Major and Rhomboid Muscle Function 266 The Hand Complex 319 Carpometacarpal Joints of the Fingers 319 Summary 267 Metacarpophalangeal Joints of the Fingers 321 Interphalangeal Joints of the Fingers 324 Extrinsic Finger Flexors 325
Copyright © 2005 by F. A. Davis. xviii ■ Contents Extrinsic Finger Extensors 328 346 Tibiofemoral Alignment and Weight-Bearing Extensor Mechanism 329 Forces 395 Intrinsic Finger Musculature 333 Structure of the Thumb 337 Menisci 397 Thumb Musculature 339 Joint Capsule 399 Ligaments 402 Prehension 340 Iliotibial Band 407 Power Grip 341 Bursae 408 Precision Handling 344 Tibiofemoral Joint Function 409 Functional Position of the Wrist and Hand Joint Kinematics 409 Muscles 413 Summary 346 Stabilizers of the Knee 419 SECTION 4 Hip Joint 354 Patellofemoral Joint 420 Patellofemoral Articular Surfaces and Joint Chapter 10 The Hip Complex 355 Congruence 421 Motions of the Patella 422 Pamela K. Levangie, PT, DSc Patellofemoral Joint Stress 423 Frontal Plane Patellofemoral Joint Stability 425 Introduction 355 Weight-Bearing vs. Non–Weight-Bearing Exercises with Patellofemoral Pain 428 Patient Case 356 Effects of Injury and Disease 429 Structure of the Hip Joint 356 Tibiofemoral Joint 429 Proximal Articular Surface 356 Patellofemoral Joint 430 Distal Articular Surface 358 Articular Congruence 361 Summary 431 Hip Joint Capsule and Ligaments 362 Structural Adaptations to Weight-Bearing 365 Chapter 12 The Ankle and Foot Complex 437 Function of the Hip Joint 366 Motion of the Femur on the Acetabulum 366 Michael J. Mueller, PT, PhD, FAPTA Motion of the Pelvis on the Femur 368 Coordinated Motions of the Femur, Pelvis, and Introduction 437 Lumbar Spine 371 Hip Joint Musculature 373 Patient Case 438 Hip Joint Forces and Muscle Function in Definitions of Motions 438 Stance 378 Bilateral Stance 378 Ankle Joint 440 Unilateral Stance 379 Ankle Joint Structure 440 Reduction of Muscle Forces in Unilateral Ankle Joint Function 443 Stance 381 The Subtalar Joint 445 Hip Joint Pathology 385 Subtalar Joint Structure 445 Arthrosis 386 Subtalar Joint Function 447 Fracture 386 Bony Abnormalities of the Femur 387 Transverse Tarsal Joint 452 Transverse Tarsal Joint Structure 452 Transverse Tarsal Joint Function 454 Summary 388 Tarsometatarsal Joints 458 Tarsometatarsal Joint Structure 458 Chapter 11 The Knee Tarsometatarsal Joint Function 459 Lynn Snyder-Macker, PT, ScD, SCS, ATC, FAPTA Metatarsophalangeal Joints 460 Metatarsophalangeal Joint Structure 460 Michael Lewek, PT, PhD Metatarsophalangeal Joint Function 461 Introduction 393 Interphalangeal Joints 464 Patient Case 394 394 Plantar Arches 464 Structure of the Arches 464 Structure of the Tibiofemoral Joint Function of the Arches 465 468 Femur 394 Muscular Contribution to the Arches Tibia 395
Copyright © 2005 by F. A. Davis. Contents ■ xix Muscles of the Ankle and Foot 468 Chapter 14 Gait 517 Extrinsic Musculature 468 Intrinsic Musculature 472 Sandra J. Olney, PT, OT, PhD Deviations from Normal Structure and Introduction 517 Function 472 General Features 518 Summary 474 Patient Case 518 SECTION 5 Integrated Function 478 Kinematics 519 527 Phases of the Gait Cycle 519 Chapter 13 Posture 479 Gait Terminology 522 Joint Motion 524 Cynthia C. Norkin, PT, EdD Saunders’ “Determinants” of Gait Introduction 479 Kinetics 527 Ground Reaction Force 527 Patient Case 480 Center of Pressure 528 Kinetic Analysis 528 Static and Dynamic Postures 480 Internal and External Forces, Moments, and Postural Control 481 Conventions 530 Major Goals and Basic Elements of Control 481 Energy Requirements 534 Mechanical Energy of Walking 534 Kinetics and Kinematics of Posture 484 Mechanical Energy: Kinematic Approach 534 Inertial and Gravitational Forces 485 Mechanical Power and Work 537 Ground Reaction Forces 485 Muscle Activity 543 Coincident Action Lines 485 Ground Reaction Force: Sagittal Plane Sagittal Plane 486 Analysis 547 Optimal Posture 487 Kinematics and Kinetics of the Trunk and Upper Extremities 551 Analysis of Standing Posture 487 Trunk 551 Sagittal Plane Alignment and Analysis 488 Upper Extremities 553 Deviations from Optimal Alignment in the Sagittal Plane 493 Stair and Running Gaits 553 Frontal Plane Optimal Alignment and Analysis 498 Stair Gait 553 Deviations from Optimal Alignment in the Frontal Running Gait 555 Plane 498 Summary 558 Analysis of Sitting Postures 503 Effects of Age, Gender, Assistive Devices, and Muscle Activity 504 Orthoses 559 Interdiskal Pressures and Compressive Loads on Age 559 the Spine 505 Gender 560 Seat Interface Pressures 506 Assistive Devices 561 Orthoses 561 Analysis of Lying Postures 508 Interdiskal Pressures 508 Abnormal Gait 561 Surface Interface Pressures 508 Structural Impairment 562 Functional Impairment 562 Effects of Age, Pregnancy, Occupation, and Recreation on Posture 509 Summary 564 Age 509 Pregnancy 511 Index 569 Occupation and Recreation 511 Summary 512
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Copyright © 2005 by F. A. Davis. Joint Structure and Function: A Comprehensive Analysis Fourth Edition
Copyright © 2005 by F. A. Davis. Section 1 Joint Structure and Function: Foundational Concepts
Copyright © 2005 by F. A. Davis. 1 Chapter Biomechanical Applications to Joint Structure and Function Pamela K. Levangie, PT, DSc “HUMANS HAVE THE CAPACITY TO PRODUCE A NEARLY INFINITE VARIETY OF POSTURES AND MOVEMENTS THAT REQUIRE THE TISSUES OF THE BODY TO BOTH GENERATE AND RESPOND TO FORCES THAT PRODUCE AND CONTROL MOVEMENT.” Introduction Newton’s Law of Reaction Gravitational and Contact Forces PART 1: Kinematics and Introduction to Kinetics Additional Linear Force Considerations Tensile Forces Descriptions of Motion Tensile Forces and Their Reaction Forces Types of Displacement Joint Distraction Translatory Motion Distraction Forces Rotatory Motion Joint Compression and Joint Reaction Forces General Motion Revisiting Newton’s Law of Inertia Location of Displacement in Space Vertical and Horizontal Linear Force Systems Direction of Displacement Shear and Friction Forces Magnitude of Displacement Static Friction and Kinetic Friction Considering Vertical and Horizontal Linear Introduction to Forces Equilibrium Definition of Forces Force Vectors PART 2: Kinetics—Considering Rotatory and Force of Gravity Translatory Forces and Motion Segmental Centers of Mass and Composition of Gravitational Forces Torque, or Moment of Force Center of Mass of the Human Body Angular Acceleration and Angular Equilibrium Center of Mass, Line of Gravity, and Stability Parallel Force Systems Alterations in Mass of an Object or Segment Determining Resultant Forces in a Parallel Force System Bending Moments and Torsional Moments Introduction to Statics and Dynamics Identifying the Joint Axis about which Body Segments Newton’s Law of Inertia Rotate Newton’s Law of Acceleration Meeting the Three Conditions for Equilibrium Translatory Motion in Linear and Concurrent Force Muscle Forces Systems Total Muscle Force Vector Anatomic Pulleys Linear Force System Anatomic Pulleys, Action Lines, and Moment Arms Determining Resultant Forces in a Linear Force System Concurrent Force System Torque Revisited Changes to Moment Arm of a Force Determining Resultant Forces in a Concurrent Force System 3
Copyright © 2005 by F. A. Davis. 4 ■ Section 1: Joint Structure and Function: Foundational Concepts Angular Acceleration with Changing Torques Perpendicular and Parallel Force Effects Moment Arm and Angle of Application of a Force Determining Magnitudes of Component Forces Force Components and the Angle of Application of the Lever Systems, or Classes of Levers Force Muscles in Third-Class Lever System Muscles in Second-Class Lever System Translatory Effects of Force Components Muscles in First-Class Lever System Rotatory Effects of Force Components Mechanical Advantage Rotation Produced by Perpendicular Force Components Trade-Offs of Mechanical Advantage Rotation Produced by Parallel Force Components Limitations to Analysis of Forces by Lever Systems Rotatory Effects of Force Components Force Components Resolving Forces into Perpendicular and Parallel Total Rotation Produced by a Force Components Multisegment (Closed-Chain) Force Analysis Introduction of the body. The final two chapters integrate the func- tion of the joint complexes into the comprehensive Humans have the capacity to produce a nearly infinite tasks of posture and gait. variety of postures and movements that require the structures of the human body to both generate and In order to maintain our focus on clinically rele- respond to forces that produce and control movement vant applications of the biomechanical principles pre- at the body’s joints. Although it is impossible to capture sented in this chapter, the following case example will all the kinesiologic elements that contribute to human provide a framework within which to explore the rele- musculoskeletal function at a given point in time, a vant principles of biomechanics. knowledge of at least some of the physical principles that govern the body’s response to active and passive 1-1 Patient Case stresses on its segments is prerequisite to an under- standing of both human function and dysfunction. Sam Alexander is 20 years old, is 5 feet, 9 inches (1.75 m) in height, and weighs 165 pounds (~75 kg or 734 N). Sam is a We will examine some of the complexity of human member of the university’s golf team. He sustained an injury to his musculoskeletal function by examining the role of the right knee as he fell when his foot went through a gopher hole on bony segments, joint-related connective tissue struc- a slope. Physical examination and magnetic resonance imaging ture, muscles, and the external forces applied to those (MRI) resulted in a diagnosis of a tear of the medial collateral lig- structures. We will develop a conceptual framework ament, a partial tear of the anterior cruciate ligament (ACL), and a that provides a basis for understanding the stresses on partial tear of the medial meniscus. Sam agreed with the orthope- the body’s major joint complexes and the responses to dist’s recommendation that a program of knee muscle strengthen- those stresses. Case examples will be used to ground the ing was in order before moving to more aggressive options. The reader’s understanding in clinically relevant applica- initial focus will be on strengthening the quadriceps muscle. The tions of the presented principles. The objective is to fitness center at the university has a leg-press machine (Fig. 1-1A) provide comprehensive coverage of foundational kine- and a free weight boot (see Fig. 1-1B) that Sam can use. siologic principles necessary to understand individual joint complexes and their interdependent composite As we move through this chapter, we will consider functions in posture and locomotion. Although we the biomechanics of each of these rehabilitative options acknowledge the role of the neurological system in in relation to Sam’s injury and strengthening goals. motor control, we leave it to others to develop an [Side-bar: The case in this chapter provides a back- understanding of the theories that govern the role of ground for presentation of biomechanical principles. the controller and feedback mechanisms. The values and angles chosen for the forces in the vari- ous examples used in this case are representative but The goal of this first chapter is lay the biomechani- are not intended to correspond to values derived from cal foundation for the principles used in subsequent sophisticated instrumentation and mathematical mod- chapters. This chapter will explore the biomechanical eling, in which different experimental conditions, principles that must be considered to examine the instrumentation, and modeling can provide substan- internal and external forces that produce or control tially different and often contradictory findings.] movement. The focus will be largely on rigid body analysis; subsequent chapters explore how forces affect Human motion is inherently complex, involving deformable connective tissues (Chapter 2) and how multiple segments (bony levers) and forces that are muscles create and are affected by forces (Chapter 3). most often applied to two or more segments simultane- Subsequent chapters then examine the interactive ously. In order to develop a conceptual model that can nature of force, stress, tissue behaviors, and function be understood and applied clinically, the common through a regional exploration of the joint complexes
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 5 AB ᭣ Figure 1-1 ■ A. Leg- press exercise apparatus for strengthening hip and knee extensor muscles. B. Free weight boot for strengthening knee extensor muscles. strategy is to focus on one segment at a time. For the A purposes of analyzing Sam Alexander’s issues, the focus B will be on the leg-foot segment, treated as if it were one rigid unit acting at the knee joint. Figure 1-2A and B is a schematic representation of the leg-foot segment in the leg-press and free weight boot situations. The leg- foot segment is the focus of the figure, although the contiguous components (distal femur, footplate of the leg-press machine, and weight boot) are maintained to give context. In some subsequent figures, the femur, footplate, and weight boot are omitted for clarity, although the forces produced by these segments and objects will be shown. This limited visualization of a seg- ment (or a selected few segments) is referred to as a free body diagram or a space diagram. If proportional representation of all forces is maintained as the forces are added to the segment under consideration, it is known as a free body diagram. If the forces are shown but a simplified understanding rather than graphic accuracy is the goal, then the figure is referred to as a space diagram.1 We will use space diagrams in this chap- ter and text because the forces are generally not drawn in proportion to their magnitudes. As we begin to examine the leg-foot segment in either exercise situation, the first step is to describe the motion of the segment that is or will be occurring. This involves the area of biomechanics known as kinematics. Part 1: Kinematics and Introduction to Kinetics Descriptions of Motion Kinematics includes the set of concepts that allows us to ▲ Figure 1-2 ■ A. Schematic representation of the leg-foot seg- describe the motion (or displacement) of a segment ment in the leg-press exercise, with the leg-foot segment highlighted without regard to the forces that cause that movement. for emphasis. B. Schematic representation of the leg-foot segment in The human skeleton is, quite literally, a system of seg- the weight boot exercise, with the leg-foot segment highlighted for ments or levers. Although bones are not truly rigid, we emphasis.
Copyright © 2005 by F. A. Davis. 6 ■ Section 1: Joint Structure and Function: Foundational Concepts will assume that bones behave as rigid levers. There are five kinematic variables that fully describe motion or displacement of a segment: (1) the type of displace- ment (motion), (2) the location in space of the dis- placement, (3) the direction of displacement of the segment, (4) the magnitude of the displacement, and (5) the rate of displacement or rate of change of dis- placement (velocity or acceleration). Types of Displacement Translatory and rotatory motions are the two basic types ▲ Figure 1-4 ■ Rotatory motion. Each point in the forearm- of movement that can be attributed to any rigid seg- hand segment moves through the same angle, in the same time, at a ment. Additional types of movement are achieved by constant distance from the center of rotation or axis (A). combinations of these two. segments of the body are neither isolated nor uncon- ■ Translatory Motion strained. Every segment is linked to at least one other segment, and most human motion occurs as movement Translatory motion (linear displacement) is the move- of more than one segment at a time. The translation of ment of a segment in a straight line. In true translatory the forearm-hand segment in Figure 1-3 is actually pro- motion, each point on the segment moves through the duced by motion of the humerus, with rotation occur- same distance, at the same time, in parallel paths. In ring at both the shoulder and the elbow joints. In fact, human movement, translatory movements are gener- translation of a body segment rarely occurs in human ally approximations of this definition. An example of motion without some concomitant rotation of that seg- translatory motion of a body segment is the movement ment (even if the rotation is barely visible). of the combined forearm-hand segment as it moves for- ward to grasp an object (Fig. 1-3). This example as- ■ Rotatory Motion sumes, however, that the forearm-hand segment is free and unconstrained—that is, that the forearm-hand seg- Rotatory motion (angular displacement) is movement ment is not linked to the humerus. Although it is easi- of a segment around a fixed axis (center of rotation est to describe pure translatory motion by using the [CoR]) in a curved path. In true rotatory motion, each example of an isolated and unconstrained segment, point on the segment moves through the same angle, at the same time, at a constant distance from the CoR. True rotatory motion can occur only if the segment is prevented from translating and is forced to rotate about a fixed axis. This does not happen in human movement. In the example in Figure 1-4, all points on the forearm-hand segment appear to move through the same distance at the same time around what appears to be a fixed axis. In actuality, none of the body segments move around truly fixed axes; all joint axes shift at least slightly during motion because segments are not suffi- ciently constrained to produce pure rotation. ▲ Figure 1-3 ■ Translatory motion. Each point on the forearm- ■ General Motion hand segment moves through the same distance, at the same time, in parallel paths. When nonsegmented objects are moved, combinations of rotation and translation (general motion) are com- mon and can be very evident. If someone were to at- tempt to push a treatment table across the room by using one hand, it would be difficult to get the table to go straight (translatory motion); it would be more likely to both translate and rotate. When rotatory and trans- latory motions are combined, a number of terms can be used to describe the result. Curvilinear (plane or planar) motion designates a combination of translation and rotation of a segment in two dimensions (parallel to a plane with a maximum of three degrees of freedom).2–4 When this type of motion occurs, the axis about which the segment moves is not fixed but, rather, shifts in space as the object moves.
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 7 The axis around which the segment appears to move in y-axis any part of its path is referred to as the instantaneous center of rotation (ICoR), or instantaneous axis of x-axis rotation (IaR). An object or segment that travels in a z-axis curvilinear path may be considered to be undergoing rotatory motion around a fixed but quite distant CoR3,4; ▲ Figure 1-5 ■ Body in anatomic position showing the x-axis, that is, the curvilinear path can be considered a seg- y-axis, and z-axis of the Cartesian coordinate system (the coronal, ver- ment of a much larger circle with a distant center. tical, and anteroposterior axes, respectively). Three-dimensional motion is a general motion in which the segment moves across all three dimensions. Just as curvilinear motion can be considered to occur around a single distant CoR, three-dimensional motion can be considered to be occurring around a helical axis of motion (HaM), or screw axis of motion.3 As already noted, motion of a body segment is rarely sufficiently constrained by the ligamentous, mus- cular, or other bony forces acting on it to produce pure rotatory motion. Instead, there is typically at least a small amount of translation (and often a secondary rotation) that accompanies the primary rotatory mo- tion of a segment at a joint. Most joint rotations, there- fore, take place around a series of ICoRs. The “axis” that is generally ascribed to a given joint motion (e.g., knee flexion) is typically a midpoint among these ICoRs rather than the true CoR. Because most body segments actually follow a curvilinear path, the true CoR is the point around which true rotatory motion of the segment would occur and is generally quite distant from the joint.3,4 Location of Displacement in Space The rotatory or translatory displacement of a segment ▲ Figure 1-6 ■ The sagittal plane. is commonly located in space by using the three-dimen- sional Cartesian coordinate system, borrowed from moving in or parallel to one of three possible cardinal mathematics, as a useful frame of reference. The origin planes. As a segment rotates around a particular axis, of the x-axis, y-axis, and z-axis of the coordinate system the segment also moves in a plane that is both perpen- is traditionally located at the center of mass (CoM) of dicular to that axis of rotation and parallel to another the human body, assuming that the body is in anatomic axis. Rotation of a body segment around the x-axis or position (standing facing forward, with palms forward) coronal axis occurs in the sagittal plane (Fig. 1-6). (Fig. 1-5). According to the common system described Sagittal plane motions are most easily visualized as front- by Panjabi and White, the x-axis runs side to side in the to-back motions of a segment (e.g., flexion/extension body and is labeled in the body as the coronal axis; the of the upper extremity at the glenohumeral joint). y-axis runs up and down in the body and is labeled in the body as the vertical axis; the z-axis runs front to Rotation of a body segment around the y-axis or back in the body and is labeled in the body as the vertical axis occurs in the transverse plane (Fig. 1-7). anteroposterior (A-P) axis.3 Motion of a segment can occur either around an axis (rotation) or along an axis (translation). An unconstrained segment can either rotate or translate around each of the three axes, which results in six potential options for motion of that seg- ment. The options for movement of a segment are also referred to as degrees of freedom. A completely uncon- strained segment, therefore, always has six degrees of freedom. Segments of the body, of course, are not un- constrained. A segment may appear to be limited to only one degree of freedom (although, as already pointed out, this rarely is strictly true), or all six degrees of freedom may be available to it. Rotation of a body segment is described not only as occurring around one of three possible axes but also as
Copyright © 2005 by F. A. Davis. 8 ■ Section 1: Joint Structure and Function: Foundational Concepts of the segment (e.g., abduction/adduction of the upper extremity at the glenohumeral joint). Rotation and translation of body segments are not limited to motion along or around cardinal axes or within cardinal planes. In fact, cardinal plane motions are the exception rather than the rule and, although useful, are an oversimplification of human motion. If a motion (whether in or around a cardinal axis or plane) is limited to rotation around a single axis or translatory motion along a single axis, the motion is considered to have one degree of freedom. Much more commonly, a segment moves in three dimensions with two or more degrees of freedom. The following examples demon- strate three of the many different ways in which rota- tory and translatory motions along or around one or more axes can combine in human movement to pro- duce two- and three-dimensional segmental motion. Example 1-1 ▲ Figure 1-7 ■ The transverse plane. When the forearm-hand segment and a glass (all con- sidered as one rigid segment) are brought to the mouth Transverse plane motions are most easily visualized as (Fig. 1-9), rotation of the segment around an axis and motions of a segment parallel to the ground (medial/ translation of that segment through space occur simul- lateral rotation of the lower extremity at the hip joint). taneously. As the forearm-hand segment and glass ro- Transverse plane motions often occur around axes that tate around a coronal axis at the elbow joint (one pass through the length of long bones that are not truly degree of freedom), the shoulder joint also rotates to vertically oriented. Consequently, the term longitudi- translate the forearm-hand segment forward in space nal (or long) axis is often used instead of vertical axis. along the forearm-hand segment’s A-P axis (one degree Rotation of a body segment around the z-axis or A-P axis of freedom). By combining the two degrees of free- occurs in the frontal plane (Fig. 1-8). Frontal plane dom, the elbow joint axis (the ICoR for flexion of the motions are most easily visualized as side-to-side motions forearm-hand segment) does not remain fixed but moves in space; the glass attached to the forearm-hand segment moves through a curvilinear path. ▲ Figure 1-8 ■ The frontal plane. ▲ Figure 1-9 ■ The forearm-hand segment rotates around a coronal axis at the elbow joint and along A-P axis (through rotation at the shoulder joint), using two degrees of freedom that result in a moving axis of rotation and produce curvilinear motion of the fprearm-hand segment.
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 9 Example 1-2 (uniaxial or uniplanar) but in opposite directions. Flexion and extension generally occur in the sagittal With the forearm-hand and glass still being considered plane around a coronal axis, although exceptions exist as one rigid segment, the glass is now taken away from (carpometacarpal flexion and extension of the thumb). the mouth while also being turned over and emptied. Anatomically, flexion is the direction of segmental rota- This combined motion involves pronation of the fore- tion that brings ventral surfaces of adjacent segments arm-hand segment as an additional degree of freedom closer together, whereas extension is the direction of while the forearm-hand segment rotates (extends) segmental rotation that brings dorsal surfaces closer around a coronal axis at the elbow joint, and the seg- together. [Side-bar: Defining flexion and extension by ment again translates backward in space. The three- ventral and dorsal surfaces makes use of the true dimensional motion could be described by a single embryologic origin of the words ventral and dorsal, helical axis of rotation but is more commonly thought rather than using these terms as synonymous with ante- of as having sequential ICoRs. rior and posterior, respectively.] Example 1-3 Abduction and adduction of a segment occur around the same axis and in the same plane but in Continuing to use the forearm-hand segment and glass opposite directions. Abduction/adduction and lateral example, assume that the glass begins in the same posi- flexion generally occur in the frontal plane around an tion as in Figure 1-9. This time, however, the forearm- A-P axis, although carpometacarpal abduction and hand segment is moved exclusively by the biceps brachii; adduction of the thumb again serve as an exception. the humerus is fixed in space, thus eliminating the Anatomically, abduction is the direction of segmental translatory component of forearm-hand motion. The rotation that brings the segment away from the midline biceps brachii both flexes the forearm-hand around a of the body, whereas adduction brings the segment coronal axis and simultaneously supinates the forearm- toward the midline of the body. When the moving seg- hand segment around a longitudinal axis. The three- ment is part of the midline of the body (e.g., the trunk dimensional nature of the motion would be evident and the head), the rotatory movement is commonly because the glass would miss the mouth and, instead, termed lateral flexion (to the right or to the left). empty onto the shoulder. Panjabi and White3 used the term main (or primary) motion to refer to the motion Medial (or internal) rotation and lateral (or exter- of forearm-hand flexion and the term coupled (or sec- nal) rotation are opposite motions of a segment that ondary) motion to refer to the motion of forearm-hand generally occur around a vertical (or longitudinal) axis supination. in the transverse plane. Anatomically, medial rotation occurs as the segment moves parallel to the ground Direction of Displacement and toward the midline, whereas lateral rotation occurs opposite to that. When the segment is part of the mid- Even if displacement of a segment is confined to a sin- line (e.g., the head or trunk), rotation in the transverse gle axis, the rotatory or translatory motion of a segment plane is simply called rotation to the right or rota- around or along that axis can occur in two different tion to the left. The exceptions to the general rules for directions. For rotatory motions, the direction of move- naming motions must be learned on a joint-by-joint ment of a segment around an axis can be described as basis. occurring in a clockwise or counterclockwise direction. Clockwise and counterclockwise rotations are generally As is true for rotatory motions, translatory motions assigned negative and positive signs, respectively.5 of a segment can occur in one of two directions along However, these terms are dependent on the perspective any of the three axes. Again by convention, linear dis- of the viewer (viewed from the left side, flexing the placement of a segment along the x-axis is considered forearm is a clockwise movement; if the subject turns positive when displacement is to the right and negative around and faces the opposite direction, the same when to the left. Linear displacement of a segment up movement is now seen by the viewer as a counterclock- along the y-axis is considered positive, and such dis- wise movement). Anatomic terms describing human placement down along the y-axis is negative. Linear dis- movement are independent of viewer perspective and, placement of a segment forward (anterior) along the therefore, more useful clinically. Because there are two z-axis is positive, and such displacement backward (pos- directions of rotation (positive and negative) around terior) is negative.1 each of the three cardinal axes, we can describe three pairs of (or six different) anatomic rotations available Magnitude of Displacement to body segments. The magnitude of rotatory motion (or angular dis- Flexion and extension are motions of a segment placement) of a segment can be given either in degrees occurring around the same axis and in the same plane (United States [US] units) or in radians (International System of Units [SI units]). If an object rotates through a complete circle, it has moved through 360Њ, or 6.28 radians. A radian is literally the ratio of an arc to the radius of its circle (Fig. 1-10). One radian is equal to 57.3Њ; 1Њ is equal to 0.01745 radians. The magnitude of rotatory motion that a body segment moves through or
Copyright © 2005 by F. A. Davis. 10 ■ Section 1: Joint Structure and Function: Foundational Concepts Text/image rights not available. ▲ Figure 1-10 ■ An angle of 57.3Њ describes an arc of 1 radian. ▲ Figure 1-11 ■ When a joint’s range of motion is plotted on the y-axis (vertical axis) and time is plotted on the x-axis (horizontal can move through is known as its range of motion axis), the resulting time-series plot portrays the change in joint posi- (ROM). The most widely used standardized clinical tion over time. The slope of the plotted line reflects the velocity of the method of measuring available joint ROM is goniome- joint change. try, with units given in degrees. Consequently, we typi- cally will use degrees in this text to identify angular meters per second squared (m/sec2) and feet per sec- displacements (rotatory motions). ROM may be meas- ond squared (ft/sec2). Angular velocity (velocity of a ured and stored on computer for analysis by an elec- rotating segment) is expressed as degrees per second trogoniometer or a three-dimensional motion analysis (deg/sec), whereas angular acceleration is given as system, but these are available predominantly in re- degrees per second squared (deg/sec2). search environments. Although we will not be address- ing instruments, procedures, technologic capabilities, An electrogoniometer or a three-dimensional or limitations of these systems, data collected by these motion analysis system allows documentation of the sophisticated instrumentation systems are often the changes in displacement over time. The outputs of basis of research cited through the text. such systems are increasingly encountered when sum- maries of displacement information are presented. A Translatory motion or displacement of a segment is computer-generated time-series plot such as that in quantified by the linear distance through which the Figure 1-11 graphically portrays not only the angle object or segment is displaced. The units for describing between two bony segments (or the rotation of one translatory motions are the same as those for length. segment in space) at each point in time but also the The SI system’s unit is the meter (or millimeter or cen- direction of motion. The steepness of the slope of the timeter); the corresponding unit in the US system is the graphed line represents the angular velocity. Figure 1-12 foot (or inch). This text will use the SI system but shows a plot of the change in linear acceleration of a includes a US conversion when this appears to facilitate body segment (or a point on the body segment) over understanding. Linear displacements of the entire time without regard to changes in joint angle. body are often measured clinically. For example, the 6- minute walk6 (a test of functional status in individuals Introduction to Forces with cardiorespiratory problems) measures the distance (in feet or meters) someone walks in 6 minutes. Smaller Definition of Forces full body or segment displacements can also be meas- ured by three-dimensional motion analysis systems. Kinematic descriptions of human movement permit us to visualize motion but do not give us an understanding Rate of Displacement of why the motion is occurring. This requires a study of forces. Whether a body or body segment is in motion or Although the magnitude of displacement is important, at rest depends on the forces exerted on that body. A the rate of change in position of the segment (the dis- force, simplistically speaking, is a push or a pull exerted placement per unit time) is equally important. by one object or substance on another. Any time two Displacement per unit time regardless of direction is objects make contact, they will either push on each known as speed, whereas displacement per unit time in other or pull on each other with some magnitude of a given direction is known as velocity. If the velocity is force (although the magnitude may be small enough to changing over time, the change in velocity per unit be disregarded). The unit for a force (a push or a pull) time is acceleration. Linear velocity (velocity of a trans- in the SI system is the newton (N); the unit in the US lating segment) is expressed as meters per second system is the pound (lb). The concept of a force as a (m/sec) in SI units or feet per second (ft/sec) in US push or pull can readily be used to describe the forces units; the corresponding units for acceleration are encountered in evaluating human motion.
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 11 6000 4000 2000 0 -2000Acceleration ( mm/sec2) Text-40/00image rights not available. -6000 -8000 -10000 -12000 -14000 0 10 20 30 40 50 60 70 80 Time (100 frames = 1 sec) ▲ Figure 1-12 ■ Movement of a point on a segment can be displayed by plotting acceleration of the segment (y-axis) over time (x-axis). The slope and trend of the line represent increases or decreases in magnitude of acceleration as the movement continues. [Courtesy of Fetters, L: Boston University, 2003.] Continuing Exploration: A Force pull of gravity on the object’s mass with an acceleration of 9.8 m/sec2 (or 32.2 ft/sec2) in the absence of any Although a force is most simply described as a push resistance: or a pull, it is also described as a “theoretical con- cept” because only its effects (acceleration) can be weight ϭ (mass)(gravity) measured.4 Consequently, a force (F) is described by the acceleration (a) of the object to which the force or W ϭ (m)(g) is applied, with the acceleration being directly pro- portional to the mass (m) of that object; that is, Because weight is a force, the appropriate unit is the newton (or pound). However, it is not uncommon force ϭ (mass)(acceleration) to see weight given in kilograms (kg), although the kilogram is more correctly a unit of mass. In the US sys- or F ϭ (m)(a) tem, the pound is commonly used to designate mass when it is appropriately a force unit. The correct unit Because mass is measured in kilograms (kg) and for mass in the US system is the infrequently used slug acceleration in m/sec2, the unit for force is actually (1 slug ϭ 14.59 kg). kg-m/sec2 or, more simply, the newton; that is, a newton is the force required to accelerate 1 kg at 1 Continuing Exploration: Force and Mass m/sec2 (the pound is correspondingly the amount Unit Terminology of force required to accelerate a mass of 1 slug [to be described] at 1 ft/sec2). Force and mass units are often used incorrectly in the vernacular. The average person using the metric External forces are pushes or pulls on the body system expects a produce scale to show weight in that arise from sources outside the body. Gravity (g), kilograms, rather than in newtons. In the United the attraction of the Earth’s mass to another mass, is an States, the average person appropriately thinks of external force that under normal conditions constantly weight in pounds but also considers the pound to be affects all objects. The weight (W) of an object is the a unit of mass. Because people commonly tend to
Copyright © 2005 by F. A. Davis. 12 ■ Section 1: Joint Structure and Function: Foundational Concepts think of mass in terms of weight (the force of gravity Internal forces are forces that act on structures of acting on the mass of an object) and because the the body and arise from the body’s own structures (that slug is an unfamiliar unit to most people, the pound is, the contact of two structures within the body). A few is often used to represent the mass of an object in common examples are the forces produced by the mus- the US system. cles (pull of the biceps brachii on the radius), the liga- ments (pull of a ligament on bone), and the bones (the One attempt to maintain common usage but to push of one bone on another bone at a joint). Some clearly differentiate force units from mass units for forces, such as atmospheric pressure (the push of air scientific purposes is to designate lb and kg as mass pressure), work both inside and outside the body, but— units and to designate the corresponding force units in our definition—are considered external forces as lbf (pound-force) and kgf (kilogram-force).3,4 because the source is not a body structure. When the kilogram is used as a force unit: External forces can either facilitate or restrict 1 kgf ϭ 9.8 N movement. Internal forces are most readily recognized as essential for initiation of movement. However, it When the pound is used as a mass unit: should be apparent that internal forces also control or counteract movement produced by external forces, as 1 pound ϭ 0.031 slugs well as counteracting other internal forces. Much of the presentation and discussion in subsequent chapters of These conversions assume an unresisted accelera- this text relate to the interactive role of internal forces, tion of gravity of 9.8 m/sec2 or 32.2 ft/sec2, respec- not just in causing movement but also in maintaining tively. the integrity of joint structures against the effects of external forces and other internal forces. The distinction between a measure of mass and a measure of force is important because mass is a scalar Force Vectors quantity (without action line or direction), whereas the newton and pound are measures of force and have vec- All forces, despite the source or the object acted on, are tor characteristics. In this text, we will consistently use vector quantities. A force is represented by an arrow the terms “newton” and “pound” as force units and will that (1) has its base on the object being acted on (the use the terms “kilogram” and “slug” as the correspon- point of application), (2) has a shaft and arrowhead in ding mass units. the direction of the force being exerted and at an angle to the object acted on (direction/orientation), and (3) Because gravity is the most consistent of the forces has a length drawn to represent the amount of force encountered by the body, gravity should be the first being exerted (magnitude). As we begin to examine force to be considered when the potential forces acting force vectors (and at least throughout this chapter), the on a body segment are identified. However, gravity is point of application (base) of each vector in each figure only one of an infinite number of external forces that will be placed on the segment or object to which the can affect the body and its segments. Examples of other force is applied—which is generally also the object external forces that may exert a push or pull on the under discussion. human body or its segments are wind (push of air on the body), water (push of water on the body), other Figure 1-13 shows Sam Alexander’s leg-foot seg- people (the push or pull of an examiner on Sam ment. The weight boot is shaded in lightly for context Alexander’s leg), and other objects (the push of floor but is not really part of the space diagram. Because the on the feet, the pull of a weight boot on the leg). A crit- weight boot makes contact with the leg-foot segment, ical point is that the forces on the body or any one seg- the weight boot must exert a force (in this case, a pull) ment must come from something that is touching the on the segment. The force, called weightboot-on-leg- body or segment. The major exception to this rule is foot, is represented by vector WbLf. The point of appli- the force of gravity. However, if permitted the conceit cation is on the leg (closest to where the weight boot that gravity (the pull of the earth) “contacts” all objects exerts its pull); the action line and direction indicate on earth, we can circumvent this exception and make it the direction of the pull and the angle of pull in rela- a standing rule that all forces on a segment must come from tion to the leg; and the length is drawn to represent the something that is contacting that segment (including grav- magnitude of the pull. The force weightboot-on-legfoot ity). The obverse also holds true: that anything that con- is an external force because the weight boot is not part tacts a segment must create a force on that segment, although of the body, although it contacts the body. Figure 1-14 the magnitude may be small enough to disregard. shows the force of a muscle (e.g., the brachialis) pulling on the forearm-hand segment. The point of application CONCEPT CORNERSTONE 1-1: Primary Rule of Forces is at the attachment of the muscle, and the orientation and direction are toward the muscle (pulls are toward ■ All forces on a segment must come from something that is the source of the force) and at an angle to the segment. contacting that segment. The force is called muscle-on-forearmhand (represen- ted by the vector MFh). Although the designation of a ■ Anything that contacts a segment must create a force on force as “external” or “internal” may be useful in some that segment (although the magnitude may be small enough contexts, the rules for drawing (or visualizing) forces to disregard). ■ Gravity can be considered to be “touching” all objects.
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 13 MFh ▲ Figure 1-14 ■ Vector MFh represents the pull of a muscle on the forearm-hand segment. WbLf (40 N) foot segment, it must exert—in this case—a push on the segment. The force, footplate-on-legfoot, is repre- ▲ Figure 1-13 ■ Vector representation of the pull of the sented by vector FpLf with a point of application on the weight boot on the leg-foot segment (weightboot-on-legfoot [WbLf]), leg-foot segment and in a direction away from the with a magnitude proportional to the mass and equivalent to the source. The magnitude of FpLf will remain unspecified weight of the apparatus. until we have more information. However, the presence of the vector in the space diagram means that the force are the same for external forces such as the weight boot does, in fact, have some magnitude. Although the force and for internal forces such as the muscle. is applied at the point where the footplate makes con- tact with the foot, the point of application can also be The length of a vector is usually drawn propor- drawn anywhere along the action of the vector as long as tional to the magnitude of the force according to a the point of application (for purposes of visualization) given scale. For example, if the scale is specified as 5 remains on the object under consideration. Just as a mm ϭ 20 N of force, an arrow of 10 mm would repre- vector can be extended to any length, the point of sent 40 N of force. The length of a vector, however, does application can appear anywhere along the line of push not necessarily need to be drawn to scale (unless a or pull of the force (as long as it is on the same object) graphic solution is desired) as long as its magnitude is without changing the represented effect of the force labeled (as is done in Fig. 1-13). Graphically, the action (see Fig. 1-15B). In this text, the point of application line of any vector can be considered infinitely long; that will be placed as close to the actual point of contact as is, any vector can be extended in either direction (at possible but may be shifted slightly along the action line the base or at the arrowhead) if this is useful in deter- for clarity when several forces are drawn together. mining the relationship of the vector to other vectors or objects. The length of a vector should not be arbitrarily It is common to see in other physics and biome- drawn, however, if a scale has been specified. chanics texts a “push”’ force represented as shown in Figure 1-15C. However, this chapter will consistently use Continuing Exploration: Pounds and Newtons the convention that the base of the vector will be at the point of application, with the “push” being away from Although SI units are commonly used mostly in sci- that point of application (see Fig. 1-15A). This conven- entific writing, the SI unit of force—the newton— tion maintains the focus on the point of application on does not have much of a context for those of us the segment and will enhance visualization later when habituated to the US system. It is useful, therefore, we begin to resolve a vector into components. When to understand that 1 lb ϭ 4.448 N. Vector WbLf in the “push” of “footplate-on-legfoot” is drawn with its Figure 1-13 is labeled as 40 N. This converts to 8.99 base (point of application) on the object (see Fig. lb. To get a gross idea of the pound equivalent of any 1–15A), the representation is similar in all respects figure given in newtons, you can divide the num- (except name) to the force “strap-on-legfoot,” shown as ber of newtons by 5, understanding that you will be vector SLf in Figure 1-16. Vector SLf, however, is the underestimating the actual number of pound equiva- pull of the strap connected to either side of the leg- lents. foot segment. It is reasonable for vector FpLf in Figure 1-15A and vector SLf in Figure 1-16 to look the same Figure 1-15A shows Sam Alexander’s leg-foot seg- because the two forces “footplate-on-legfoot” and ment on the leg-press machine. The footplate is shaded “strap-on-legfoot” will have an identical effect as long in lightly for context but is not really part of the space on the rigid leg-foot segment as the point of application, diagram. Because the footplate is contacting the leg- direction/orientation, and magnitude are similar—as they are here. The magnitude and direction/orienta- tion of a force are what affect the object to which the force is applied, without consideration of whether the force is, in fact, a push or a pull.
Copyright © 2005 by F. A. Davis. 14 ■ Section 1: Joint Structure and Function: Foundational Concepts FpLf FpLf A B FpLf C ▲ Figure 1-15 ■ A. Vector representation of the force of the footplate of the leg-press machine on the leg-foot segment (footplate-on-leg- foot [FpLf]). B. The vector footplate-on-legfoot (FpLf) may be drawn with any length and with a point of application anywhere along the line of pull of the vector as long as the point of application remains on the leg-foot segment. C. The push of the footplate on the leg-foot segment is commonly shown elsewhere by placing the arrowhead of vector FpLf at the point of application. X SLf ▲ Figure 1-16 ■ The vector representing the pull of a strap ▲ Figure 1-17 ■ An unknown vector (X) can be named by connected to each side of the legfoot (strap-on-legfoot [SLf]) will identifying the segment to which it is applied and the source of the look the same as the push of the footplate on the leg-foot segment force (something that must be touching the segment). (Fig. 1-15A) because both have identical effects on the leg-foot seg- ment as long as the direction and magnitude are the same. CONCEPT CORNERSTONE 1-2: Force Vectors Are CONCEPT CORNERSTONE 1-3: Naming Forces Characterized By: We have already begun to establish the naming convention of ■ a point of application on the object acted upon. “something-on-something” to identify forces and label vectors. ■ an action line and direction/orientation indicating a pull The first part of the force name will always identify the source of the force; the second part of the force name will always identify the toward the source object or a push away from the source object or segment that is being acted on. object, at a given angle to the object acted upon. ■ length that represents and may be drawn proportional to its Figure 1-17 shows Sam Alexander’s leg-foot seg- magnitude (the quantity of push or pull). ment on the leg-press machine. A new vector is shown ■ a length that may be extended to assess the relation in this figure. Because vector X is applied to the leg-foot between two or more vectors or to assess the relation of the segment, the vector is named “blank-on-legfoot.” The vector to adjacent objects or points.
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 15 name of the vector is completed by identifying the is still the point from which the force of gravity appears source of the force. The leg-foot segment is being con- to act. The actual location of the CoM of any object can tacted by gravity, by the footplate, and by the femur. We be determined experimentally by a number of methods can eliminate gravity as the source because gravity is not within the scope of this text. However, the CoM of always in a downward direction. The footplate can only an object can be approximated by thinking of the CoM push on the leg-foot segment, and so the vector is in the as the balance point of the object (assuming you could wrong direction. The femur will also push on the leg- balance the object on one finger) as shown in Figure 1- foot segment because a bone cannot pull. Because vec- 18A to C. tor X is directed away from the femur, the femur appears to be the source of vector X in Figure 1-17. Although the direction and orientation of most Therefore, vector X is named femur-on-legfoot and can forces vary with the source of the force, the force of be labeled vector FLf. gravity acting on an object is always vertically downward toward the center of the earth. The gravitational vector Force of Gravity is commonly referred to as the line of gravity (LoG). The length of the LoG can be drawn to scale (as in a As already noted, gravity is one of the most consistent free body diagram, in which the length is determined and influential forces that the human body encounters by its magnitude) or it may be extended (like any vec- in posture and movement. For that reason, it is useful tor) when the relationship of the vector to other forces, to consider gravity first when examining the properties points, or objects is being explored. The LoG can best of forces. As a vector quantity, the force of gravity can be visualized as a string with a weight on the end (a be fully described by point of application, action plumb line), with the string tied or attached to the CoM line/direction/orientation, and magnitude. Unlike of the object. A plumb line applied to the CoM of an other forces that may act on point or limited area of object gives an accurate representation of the point of contact, gravity acts on each unit of mass that composes application, direction, and orientation of the force of an object. For simplicity, however, the force of gravity gravity on an object or segment, although not its mag- acting on an object or segment is considered to have its nitude. point of application at the CoM or center of gravity (CoG) of that object or segment—the hypothetical ■ Segmental Centers of Mass and Composition point at which all the mass of the object or segment of Gravitational Forces appear to be concentrated. Every object or segment can be considered to have a single CoM. Each segment in the body can be considered to have its own CoM and LoG. Figure 1-19A shows the gravita- In a symmetrical object, the CoM is located in the tional vectors (LoGs) acting at the CoMs of the arm, the geometric center of the object (Fig. 1-18A). In an asym- forearm, and the hand segments (vectors GA, GF, and metrical object, the CoM will be located toward the GH, respectively). The CoMs in Figure 1-19A approxi- heavier end because the mass must be evenly distrib- mate those identified in studies done on cadavers and uted around the CoM (see Fig. 1-18B). The crutch in on in vivo body segments that have yielded standard- Figure 1-18C demonstrates that the CoM is only a hypo- ized data on centers of mass and weights of individual thetical point; it need not lie within the object being and combined body segments.1 It is often useful, how- acted on. Even when the CoM lies outside the object, it ever, to consider two or more segments as if they were a single segment or object and to treat them as if they are AB GF GH GA GFh GA B C D GF GAfh A GA GAfh C ▲ Figure 1-19 ■ A. Gravity acting on the arm segment (GA), the forearm segment (GF), and the hand segment (GH). B. Gravity ▲ Figure 1-18 ■ A. Center of mass of a symmetrical object. B. acting on the arm and forearm-hand segments (GFh). C. Gravity act- Center of mass of an asymmetrical object. C. The center of mass may ing on the arm-forearm-hand segment (GAfh). D. The CoM of the lie outside the object. arm-forearm-hand segment shifts when segments are rearranged.
Copyright © 2005 by F. A. Davis. COM 16 ■ Section 1: Joint Structure and Function: Foundational Concepts COM going to move together as a single rigid segment (such as the leg-foot segment in the patient case). When two gravity vectors acting on the same (now larger) rigid object are composed into one gravitational vector, the new common point of application (the new CoM) is located between and in line with the original two seg- mental CoMs. When the linked segments are not equal in mass, the new CoM will lie closer to the heavier seg- ment. The new vector will have the same effect on the combined forearm-hand segment as the original two vectors and is known as the resultant force. The process of combining two or more forces into a single resultant force is known as composition of forces. Example 1-4 If we wish to treat two adjacent segments (e.g., the fore- ▲ Figure 1-20 ■ The CoM of the human body lies approxi- arm and the hand segments) as if these were one rigid mately at S2, anterior to the sacrum (inset). The extended LoG lies segment, the two gravitational vectors (GH and GF) act- within the BoS. ing on the new larger segment (forearm-hand) can be combined into a single gravitational vector (GFh) location of the CoM for a person in anatomic position applied at the new CoM. Figure 1-19B shows vector GA depends on the proportions (weight distribution) of on the arm and new vector GFh on the now combined that person. If a person really were a rigid object, the forearm-hand segment. Vector GFh is applied at the CoM would not change its position in the body, regard- new CoM for the combined forearm-hand segment (on less of whether the person was standing up, lying down, a line between the original CoMs), is directed vertically or leaning forward. Although the CoM does not change downward (as were both GF and GH), and has a mag- its location in the rigid body as the body moves in nitude equal to the sum of the magnitudes of GF and space, the LoG changes its relative position or alignment GH. Figure 1-19C shows the force of gravity (GAfh) act- within the body. In Figure 1-20, the LoG is between the ing on the rigid arm-forearm-hand segment. Vector person’s feet (base of support [BoS]) as the person GAfh is applied at the new CoM located between and in stands in anatomic position; the LoG is parallel to the line with the CoMs of vectors GA and GFh; the magni- trunk and limbs. If the person is lying down (still in tude of GAfh is equal to the sum of the magnitudes of anatomic position), the LoG projecting from the CoM GA and GFh; the direction of GAfh is vertically down- of the body lies perpendicular to the trunk and limbs, ward because it is still the pull of gravity and because rather than parallel as it does in the standing position. that is the direction of the original vectors. In reality, of course, a person is not rigid and does not remain in anatomic position. Rather, a person is con- The CoM for any one object or rigid series of seg- stantly rearranging segments in relation to each other ments will remain unchanged regardless of the position as the person moves. With each rearrangement of body of that object in space. However, when an object is segments, the location of the individual’s CoM will composed of two or more linked and movable seg- potentially change. The amount of change in the loca- ments, the location of the CoM of the combined unit tion of the CoM depends on how disproportionately will change if the segments are rearranged in relation the segments are rearranged. to each other. The magnitude of the force of gravity will not change because the mass of the combined seg- Example 1-5 ments is unchanged, but the point of application of the resultant force will be different. A more precise method If a person is considered to be composed of a rigid for mathematically composing two gravitational forces upper body (head-arms-trunk [HAT]) and a rigid lower into a single resultant force will be addressed later limb segment, the CoMs for each of these two segments when other attributes of the forces (the torque that will typically be located approximately as shown in each generates) are used to identify the exact position Figure 1-21A. The combined CoM for these two seg- of the new CoM between the original two CoMs. ments in anatomic position remains at S2 because the position of the body is the same as in Figure 1-20. When ■ Center of Mass of the Human Body the trunk is inclined forward, however, the point at which the mass of the body appears to be concentrated When all the segments of the body are combined and shifts forward. The new CoM is on a line between the considered as a single rigid object in anatomic position, original two CoMs and is located toward the heavier the CoM of the body lies approximately anterior to the second sacral vertebra (S2) (Fig. 1-20). The precise
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 17 AB ■ Center of Mass, Line of Gravity, and Stability ▲ Figure 1-21 ■ A. Location of the CoMs of the head-arms- In Figure 1-22, the LoG (GABC) falls outside the foot- trunk (HAT) segment and lower limb segment. B. Rearrangement of ball player’s left toes, which serve as his BoS. The LoG segments produces a new combined CoM and a new location for the has been extended (lengthened) to indicate its rela- LoG in relation to the base of support. tionship to the football player’s BoS. It must be noted that the extended vector is no longer proportional to upper body segment (Fig. 1-21B). This new CoM is the magnitude of the force. However, the point of physically located outside the body, with the LoG cor- application, action line, and direction remain accurate. respondingly shifted forward. Figure 1-22 shows a more By extending the football player’s LoG in Figure 1-22, disproportionate rearrangement of body segments. we can see that the LoG is anterior to his BoS; it would The CoMs of the two lower limb segments (segment A be impossible for the player to hold this pose. For an and segment B) and the CoM of the HAT segment (seg- object to be stable, the LoG must fall within the BoS. ment C) are composed into a new CoM located at point When the LoG is outside the BoS, the object will fall. As ABC, the point of application for the gravitational vec- the football player moved from a starting position of tor for the entire body (LoG ϭ GABC). standing on both feet with his arms at his sides, two fac- tors changed as he moved to the position in Figure 1- 22. He reduced his BoS from the area between and including his two feet to the much smaller area of the toes of one foot. His CoM, with his rearrangement of segments, also has moved from S2 to above S2. Each of these two factors, combined with a slight forward lean, influenced the shift in his LoG and contributed to his instability. When the BoS of an object is large, the LoG is less likely to be displaced outside the BoS, and the object, consequently, is more stable. When a person stands with his or her legs spread apart, the base is large side to side, and the trunk can move a good deal in that plane without displacing the LoG from the BoS and without falling over (Fig. 1-23). Whereas the CoM remains in approximately the same place as the trunk shifts to each side, the LoG moves within the wide BoS. Once again, it is useful here to think of the LoG as a plumb line. As long as the plumb line does not leave the BoS, the person should not fall over. Figure 1-24 shows the same football player in exactly the same position as previously shown (see Fig. 1-22), with vector GABC still in front of his toes. However, it now appears that the football player can ▲ Figure 1-22 ■ CoM of the football player’s left leg (A) and ▲ Figure 1-23 ■ A wide base of support permits a wide excur- the right leg (B) combine to form the CoM for the lower limbs (AB). sion of the line of gravity (LoG) without the LoG’s falling outside the The CoM (AB) combines with the upper trunk CoM (C) to produce base of support. the CoM for the entire body (ABC). The LoG from the combined CoM falls well outside the football player’s BoS. He is unstable and cannot maintain this position.
Copyright © 2005 by F. A. Davis. 18 ■ Section 1: Joint Structure and Function: Foundational Concepts ▲ Figure 1-25 ■ Given the very low CoM of the punching bag, the LoG remains within the base of support, regardless of the tipping of the bag from one position to another. ▲ Figure 1-24 ■ The football player’s segments are arranged than that of the man leaning from side to side in Figure identically to those in Figure 1-22, but by fixing his foot against the 1-23, the punching bag can “lean” farther without wall, he has expanded his BoS and is now stable. falling over because it is nearly impossible to get the very short LoG in the punching bag to displace outside maintain the pose. This is not a violation of the rule the BoS. With a very low CoM and a very short LoG, the that the LoG must fall within the BoS. Rather, the BoS punching bag is extremely stable. has been expanded. When a person grasps or leans on another object (or another person), that object (or per- CONCEPT CORNERSTONE 1-4: Stability of an Object or son) becomes part of the BoS. In Figure 1-24, the foot- the Human Body ball player’s BoS now includes not only his left toes but also all the space between his supporting foot and the ■ The larger the BoS of an object is, the greater is the stability wall he is leaning against. If the football player moves of that object. his head, arms, and trunk around, he will shift the loca- tion of the CoM as these segments are rearranged. ■ The closer the CoM of the object is to the BoS, the more However, he will remain stable as long as the LoG pro- stable is the object. jecting from the CoM remains somewhere within the extended BoS. ■ An object cannot be stable unless its LoG is located within its BoS. When the CoM of an object is close to the support- ing surface, movement of the object in space is less ■ Alterations in Mass of an Object or Segment likely to cause the LoG to fall outside the BoS. If you hold a plumb line that is 100 cm (~3 ft) long in your The location of the CoM of an object or the body hand with the weight at the end just above the ground, depends on the distribution of mass of the object. The the plumb line can be made to swing through a wide mass can be redistributed not only by rearranging arc over the floor with very little side-to-side movement linked segments in space but also by adding or taking of your hand. Conversely, a plumb line that is only 10 away mass. People certainly gain weight and may gain it cm (~4 inches) long and held just above the ground disproportionately in the body (thus shifting the CoM). will move through a much smaller arc with the same However, the most common way conceptually (as amount of side-to-side motion of your hand. If the BoS opposed to literally) to redistribute mass in the body is is the same size, an object with a higher CoM will be less to add external mass. Every time we add an object to stable than an object with a lower CoM because the the body by wearing it (a backpack), carrying it (a box), longer LoG (projecting from the higher the CoM) is or using it (a power drill), the new CoM for the com- more likely to be displaced outside the BoS. bined body and external mass will shift toward the addi- tional weight; the shift will be proportional to the Figure 1-25 shows a punching bag as it moves from weight added. side to side. The base of the punching bag is filled with sand; everything above the base is air. This distribution Example 1-6 of mass creates a CoM that nearly lies on the ground. Because the punching bag is not a segmented object, The man in Figure 1-26 has a cast applied to the right the position of the CoM within the punching bag is the lower limb. Assuming the cast is now part of his mass, same regardless of how tipped the bag might be. Even though the BoS of the punching bag is much smaller
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 19 ▲ Figure 1-26 ■ The addition of the weight of the cast has shifted the CoM. The addition of crutches enlarges the base of sup- port to the shaded area between the weight-bearing foot and crutches to improve stability. the new CoM is located down and to the right of the ▲ Figure 1-27 ■ The weight of the box added to the shoulder original CoM at S2. Because his CoM with the cast is girdle causes the CoM to shift up and to the right. The man laterally now lower, he is theoretically more stable. However, if leans to the left to bring the LoG back to the middle of his base of he could not bear weight on his right leg, his BoS would support. consist only of the left foot. The patient will be stable only if he can lean to the left to swing his LoG into his Introduction to Statics left foot. However, he remains relatively unstable and Dynamics because of the very small BoS (it would take very little inadvertent leaning to displace the LoG outside the The primary concern when looking at forces that act on foot, causing the man to fall). To improve his stability, the body or a particular segment is the effect that the crutches have been added. The crutches and the left forces will have on the body or segment. If all the forces foot combine to form a much larger BoS, adding to the acting on a segment are “balanced” (a state known patient’s stability and avoiding a large compensatory as equilibrium), the segment will remain at rest or in weight shift to the left. uniform motion. If the forces are not “balanced,” the segment will accelerate. Statics is the study of the con- Example 1-7 ditions under which objects remain at rest. Dynamics is the study of the conditions under which objects move. In Figure 1-27, the man is carrying a heavy box on his Isaac Newton’s first two laws will govern whether an right shoulder. The push of the weight of the box on object is static or dynamic. this shoulder girdle moves the new CoM up and to the right of S2. A LoG projecting vertically downward from Newton’s Law of Inertia the new CoM will move into the right foot (and poten- tially to the lateral aspect of the right foot if the box is Newton’s first law, the law of inertia, identifies the con- of sufficient weight). Because this is a relatively unstable ditions under which an object will be in equilibrium. position (even a small shift of the LoG to the right will Inertia is the property of an object that resists both the cause the LoG to be displaced outside the BoS), the initiation of motion and a change in motion and is man will lean to the left to “compensate.” The small directly proportional to its mass. The law of inertia rearrangement of segments caused by the left leaning states that an object will remain at rest or in uniform of the trunk does relatively little to relocate the CoM. (unchanging) motion unless acted on by an unbal- The goal of the body shift is not to relocate the CoM anced (net or resultant) force. An object that is acted but to swing the LoG back into the center of the BoS; upon by balanced forces and remains motionless is in with the LoG in the center of the BoS, any new shifts in static equilibrium. However, an object acted upon by the CoM or LoG from disturbances in position (pertur- bations) are less likely to displace the LoG to outside the BoS.
Copyright © 2005 by F. A. Davis. 20 ■ Section 1: Joint Structure and Function: Foundational Concepts balanced forces may also be in uniform motion, moving of an object, the greater the magnitude of net unbalanced force with a given speed and direction. Velocity is a vector needed either to get the object moving or to change its motion. A quantity that describes both speed and direction/ori- very large woman in a wheelchair has more inertia than does a entation. An object in equilibrium can have a velocity small woman in a wheelchair; an aide must exert a greater push of any magnitude (≥0), but that velocity remains con- on a wheelchair with a large woman in it to get the chair in motion stant. When velocity of an object is greater than 0, the than on the wheelchair with a small woman in it. object is in constant motion (dynamic equilibrium) that can be linear (as for translatory motion), angular (as Translatory Motion in Linear for rotatory motion), or a combination of both (as for and Concurrent Force Systems general motion). With regard to motion at joints of the body, dynamic equilibrium (constant velocity) of seg- The process of composition of forces is used to deter- ments of the body occurs infrequently. Therefore, mine whether a net unbalanced force (or forces) exists within the scope of this text, equilibrium will be simpli- on a segment, because this will determine whether the fied to mean an object at rest (in static equilibrium) segment is at rest or in motion. Furthermore, the direc- unless otherwise specified. tion/orientation and location of the net unbalanced force or forces determine the type and direction of Newton’s law of inertia (or law of equilibrium) can motion of the segment. The process of composition be restated thus: For an object to be in equilibrium, the of forces was oversimplified in Examples 1-4 and 1-5 sum of all the forces applied to that object must be zero. (see Figs. 1-19 and 1-21). The process of composition depends on the relationship of the forces to each other: ∑F ϭ 0 that is, whether the forces are in a linear, concurrent, or parallel force system. The equilibrium of an object is determined only by forces applied to (with points of application on) that object. Let us return to our case example of Sam There is no restriction on the number of forces that can Alexander and the weight boot. In Figure 1-13, we iden- be applied to an object in equilibrium as long as there tified the force of weightboot-on-legfoot (WbLf) on is more than one force. If one (and only one) force is Sam’s leg-foot segment. However, Figure 1-13 must be applied to an object, the sum of the forces cannot be 0. incomplete because WbLf cannot exist alone; other- Any time the sum of the forces acting on an object is wise, the leg-foot segment would accelerate downward. not zero (∑F ≠ 0), the object cannot be in equilibrium We also have not yet accounted for the force of gravity. and must be accelerating. Figure 1-28 is the same figure but with the addition of a new vector: gravity-on-legfoot (GLf). Vector GLf is app- Newton’s Law of Acceleration lied at the CoM of the leg-foot segment, is directed ver- tically downward, and has a magnitude proportional to The magnitude of acceleration of a moving object is the mass of the segment. The leg-foot segment typically defined by Newton’s second law, the law of accelera- has approximately 6.5% of the mass of the body.1 tion. Newton’s second law states that the acceleration (a) of an object is proportional to the net unbalanced GLf (48 N) (resultant) forces acting on it (Funbal) and is inversely proportional to the mass (m) of that object: a ϭ ᎏFumnᎏbal Because an object acted upon by a net unbalanced force must be accelerating, it is invariably in motion or in a dynamic state. The acceleration of an object will be in the direction of the net unbalanced force. A net unbalanced force can produce translatory, rotatory, or general motion. CONCEPT CORNERSTONE 1-5: Applying the Law of Acceleration (Inertia) To put the law of acceleration into simple words: A large unbal- WbLf (40 N) anced push or pull (Funbal) applied to an object of a given mass (m) will produce more acceleration (a) than an unbalanced small push ▲ Figure 1-28 ■ The forces of gravity-on-legfoot (GLf) and or pull. Similarly, a given magnitude of unbalanced push or pull on weightboot-on-legfoot (WbLf) are in the same linear force system an object of large mass will produce less acceleration than that when the leg-foot segment is at 90Њ of knee flexion. same push or pull on an object of smaller mass. From the law of acceleration, it can be seen that inertia (a body’s or object’s resist- ance to change in velocity) is resistance to acceleration and is pro- portional to the mass of the body or object. The greater the mass
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 21 Because Sam weighs 734 N (165 lb), his leg-foot seg- touching the leg-foot segment. However, the contact of ment will weigh about 48 N (10.8 lb). Because vectors the femur would be a push on the leg-foot segment and, WbLf and GLf are applied to the same segment, have in the position shown in Figure 1-28, would be away action lines that lie in the same plane, and act in the from the femur and in the same direction as WbLf and same line (co-linear and coplanar), these two vectors GLf. Also, the net downward force of WbLf and GLf are part of a linear force system. would tend to move the leg-foot segment away from the femur, minimizing or eliminating the contact of the Linear Force System femur with the leg-foot segment. A net force that moves a bony segment away from its adjacent bony segment is A linear force system exists whenever two or more known as a distraction force. A distraction force tends forces act on the same segment, in the same plane, and in to cause a separation between the bones that make up the same line (their action lines, if extended, overlap). a joint. In this case, however, we still need to account Forces in a linear force system are assigned positive or for a force of 88 N acting upward on the leg-foot seg- negative signs. We will use the same convention previ- ment to have equilibrium. ously described for translatory forces. Forces applied up (y-axis), forward or anterior (z-axis), or to the right (x- In the human body, the two bones of a synovial axis) will be assigned positive signs, whereas forces joint (e.g., the knee joint) are connected by a joint cap- applied down, back or posterior, or to the left will be sule and ligaments made of connective tissue. Until we assigned negative signs. The magnitudes of vectors in explore connective tissue behavior in detail in Chapter opposite directions should always be assigned opposite 2, capsuloligamentous structures are best visualized as signs. string or cords with some elasticity that can “pull” (not “push”) on the bones to which they attach. Figure 1- Determining Resultant Forces 29A shows a schematic representation of the capsu- in a Linear Force System loligamentous structures that join the femur and the tibia. [Side-bar: In reality, the capsule surrounds the adja- The net effect, or resultant, of all forces that are part cent bones, and the ligamentous connections are more of the same linear force system is determined by find- complex.] We will nickname the structures “Acapsule” ing the arithmetic sum of the magnitudes of each of the (anterior capsule) and “Pcapsule” (posterior capsule), forces in that force system (considering its positive or understanding that these two forces are representing negative value). All forces in the same linear force sys- the pull of both the capsule and the capsular ligaments tem can be composed into a single resultant vector. The at the knee. Because capsules and ligaments can only resultant vector has an action line in the same line as pull, the forces that are created by the contact of that of the original composing vectors, with a magni- Acapsule and Pcapsule in Figure 1-29A–C are directed tude and direction equivalent to the arithmetic sum of upward toward the capsuloligamentous structures (pos- the composing vectors. Because the vectors in a linear itive). Under the assumption that the pull of the cap- force system are all co-linear and coplanar, the point sule anteriorly and posteriorly in this example are likely of application of the resultant vector will lie along the to be symmetrical, the vectors are given the same length common action line of the composing vectors, and in Figure 1-29A. the resultant will have the same orientation in space as the composing vectors. The vectors for Acapsule-on-legfoot (AcLf) and Pcapsule-on-legfoot (PcLf) are drawn in Figure 1-29A We previously assigned vector WbLf a magnitude so that the point of application is at the point on the of 40 N (~8 lb). Vectors WbLf and GLf are in the same leg-foot segment where the fibers of the capsular seg- linear force system. The resultant of the two forces, ments converge (or in the center of the area where the therefore, can be found by adding their magnitudes. fibers converge). [Side-bar: Although the anterior and Because both WbLf and GLf are directed down, they posterior segments of the capsule also touch the femur, are assigned negative values of Ϫ40 N and Ϫ48 N, we are considering only the leg-foot segment at this respectively. The sum of these forces is Ϫ88 N. The two time.] The vector arrows for the pulls of AcLf and PcLf forces WbLf and GLf can be represented graphically as must follow the fibers at the point of application and con- a single resultant vector of Ϫ88 N. If Sam is not trying tinue in a straight line. A vector, for any given snapshot of to lift the weight boot yet, there should be no motion of time, is always a straight line. The vector for the pull of the leg-foot segment. If there is no motion (static equi- the capsule does not change direction even if the fibers librium), the sum of the forces acting on the leg-foot of the capsule change direction after the fibers emerge segment must total zero. Instead, there is (in Fig. 1-28) from their attachment to the bone. a net unbalance force of Ϫ88 N; the leg-foot segment appears to be accelerating downward. In order to “bal- In a linear force system, vectors must be co-linear ance” the net downward force, we must identify some- and coplanar. Vectors AcLf and PcLf are not co-linear thing touching the leg-foot segment that will be part of or coplanar with vectors WbLf and GLf. Therefore, they the same linear force system. cannot be part of the same linear force system. If vec- tors AcLf and PcLf are extended slightly at their bases, Figure 1-28 indicates that the femur is potentially the two vectors will converge (see Fig. 1-29B). When two or more vectors applied to the same object are not co- linear but converge (intersect), the vectors are part of a concurrent force system.
Copyright © 2005 by F. A. Davis. 22 ■ Section 1: Joint Structure and Function: Foundational Concepts CLf PcLf AcLf CLf (88 N) GLf (48 N) PcLf AcLf GLf (48 N) WbLf (40 N) WbLf (40 N) AB C ▲ Figure 1-29 ■ A. Schematic representation of the pull of the anterior capsule (AcLf) and posterior capsule (PcLf) on the leg-foot seg- ment. B. Determination of the direction and relative magnitude of the resultant (capsule-on-legfoot [CLf]) of concurrent forces AcLf and PcLf, through the process of composition by parallelogram. C. The resultant force CLf has been added to the leg-foot segment, with a magnitude equivalent to that of GLf ϩ WbLf. Concurrent Force System Example 1-8 It is quite common (and perhaps most common in the In Figure 1-29B, vectors AcLf and PcLf are composed human body) for forces applied to an object to have into a single resultant vector (CLf). Vectors AcLf and action lines that lie at angles to each other. A common PcLf are extended to identify the point of application point of application may mean that the forces are liter- of the new resultant vector that represents the com- ally applied to the same point on the object or that bined action of AcLf and PcLf. A parallelogram is con- forces applied to the same object have vectors that structed by starting at the arrowhead of one vector intersect when extended in length (even if the inter- (AcLf) and drawing a line of relatively arbitrary length section is outside the actual segment or object as we saw that is parallel to the adjacent vector (PcLf). The pro- with the CoM). The net effect, or resultant, of concur- cess is repeated by starting at the arrowhead of PcLf rent forces appears to occur at the common point of and drawing a line of relatively arbitrary length parallel application (or point of intersection). Any two forces in to AcLf. Both the lengths of the two new lines should be a concurrent force system can be composed into a sin- long enough that the two new lines intersect. Because gle resultant force with a graphic process known as the two new lines are drawn parallel to the original two composition by parallelogram. and intersect (thus closing the figure), a parallelogram is created (see Fig. 1-29B). The resultant of AcLf and ■ Determining Resultant Forces in PcLf is a new vector (“capsule-on-legfoot” [CLf]) that a Concurrent Force System has a shared point of application with the original two vectors and has a magnitude that is equal to the length In composition by parallelogram, two vectors are taken of the diagonal of the parallelogram. If the vectors at a time. The two vectors and their common point of were drawn to scale, the length of CLf would represent application or point of intersection form two sides of a ϩ88 N. parallelogram. The parallelogram is completed by draw- ing two additional lines at the arrowheads of the origi- Vector CLf in Figure 1-29C is the resultant of vec- nal two vectors (with each new line parallel to one of tors PcLf and AcLf in Figure 1-29B. Presuming nothing the original two). The resultant has the same point else is touching the leg-foot segment, vector CLf must of application as the original vectors and is the diagonal be equal in magnitude and opposite in direction to the of the parallelogram. If there are more than two vectors sum of GLf and WbLf because these three vectors are in a concurrent force system, a third vector is added to co-linear, coplanar, and applied to the same object. The the resultant of the original two through the same arithmetic sum of the three forces must be 0 because process. The sequential use of the resultant and one of (1) these vectors are part of the same linear force sys- the original vectors continues until all the vectors in the original concurrent force system are accounted for.
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 23 tem, (2) nothing else is touching the leg-foot segment, AcLf´ PcLf´ and (3) the leg-foot segment is not moving. α CLf β The magnitude of the resultant of two concurrent forces has a fixed proportional relationship to the orig- PcLf AcLf inal two vectors. The relationship between the two composing vectors and the resultant is dependent on α both the magnitudes of the composing vectors and the angle between (orientation of) the composing vectors. ▲ Figure 1-30 ■ The cosine law for triangles is used to com- In composition of forces by parallelogram, the relative pute the magnitude of CLf, given the magnitudes of AcLf and PcLf, lengths (the scale) of the concurrent forces being com- as well as the angle of application (␣) between them. The relevant posed must be appropriately represented to obtain the angle () is the complement of angle ␣ (180 Ϫ ␣). correct relative magnitude of the resultant force. Although the magnitude and direction of the resultant By substituting the variables given in the exam- force are related to both the magnitude and the angle ple, the magnitude of the resultant, CLf, can be between the composing forces, it is always true that the solved for using the following equation1: magnitude of the resultant will be less than the sum of the magnitudes of the composing forces. In Figure 1- CLf ϭ PcLf2 ϩ AcLf2 Ϫ 2(PcLf)(AcLf)(cos ) 29B, the sum of the lengths of PcLf and AcLf (if meas- ured) is greater than the length of CLf; that is, pulling If the value of 51 N is entered into the equation for directly up on the leg-foot segment (as seen with CLf) both PcLf and AcLf and an angle of 120Њ is used, vec- is more efficient than pulling up and anteriorly and tor CLf ϭ 88 N. As we shall see, the trigonometric pulling up and posteriorly, as AcLf and PcLf, respec- solution is simpler when the triangle has one 90Њ tively, do. angle (right triangle). Trigonometric functions can also be used to deter- When there are more than two forces in the con- mine the magnitude of the resultant of two concurrent current force system, the process is the same whether a forces. The trigonometric solution is presented below. graphic or trigonometric solution is used. The first two The trigonometric solution, however, requires knowl- vectors are composed into a resultant vector, the result- edge both of the actual magnitudes of the two compo- ant and a third vector are then composed to create a sing vectors and of the angle between them; that is, second resultant vector, and so on until all vectors are we would need to know the magnitudes of vectors accounted for. Regardless of the order in which the vec- AcLf and PcLf, as well as the angle between the vectors. tors are taken, the solution will be the same. Although These values are rarely known in a clinical situation. we could show this sequential process by using the four [Side-bar: Once sufficiently comfortable with the gra- vectors in Figure 1-29A, the procedure is generally use- phic composition of forces by parallelogram, the reader ful only for graphic solutions. It is unlikely that you will should be able to transfer this skill to visualize the be able to (or need to) compose multiple concurrent resultant of any two concurrent forces that can be vectors into a single resultant vector in a clinical situa- “seen” as acting on an object or body segment.] tion (other than as a very gross estimate). Continuing Exploration: Trigonometric Solution Returning to Sam Alexander’s weight boot, we have established that vectors GLf and WbLf have a net force Let us assume that PcLf and AcLf each have a mag- of Ϫ88 N and that CLf has a magnitude of ϩ88 N. nitude of 51 N and that the vectors are at a 60º angle Although a space diagram considers only one segment (␣) to each other. As done for the graphic solution, at a time (the leg-foot segment in this case), an occa- the parallelogram is completed by drawing AcLfЈ sional departure from that view is necessary to establish and PcLfЈ parallel to and the same lengths as AcLf clinical relevance. In Sam’s case, we must consider not and PcLf, respectively (Fig. 1-30). The cosine law for only the pull of the capsule on his leg-foot segment but also triangles can be used to find the length of the side the pull of his leg-foot segment on his capsule, because opposite a known angle once we identify the triangle Sam has injured his medial collateral ligament (part of of interest and the angle of interest that capsule). We can segue to consideration of this new “object” (the capsule) by examining the principle The reference triangle (shaded) is that formed by PcLf, AcLfЈ, and CLf (see Fig. 1-30). To apply the law of cosines, angle  must be known because vec- tor CLf (whose length we are solving for) is the “side opposite” that angle. The known angle (␣) in Figure 1-30 is 60Њ. If PcLf is extended (as shown by the dot- ted line in Fig. 1-30), angle ␣ is replicated because it is the angle between PcLf and AcLfЈ (given AcLfЈ is parallel to AcLf). Angle , then, is the complement of angle ␣, or:  ϭ 180Њ Ϫ 60Њ ϭ 120Њ
Copyright © 2005 by F. A. Davis. 24 ■ Section 1: Joint Structure and Function: Foundational Concepts ▲ Figure 1-31 ■ Newton’s third law (“for every action there is WbLf (40 N) LfWb (40 N) an equal and opposite reaction”) is commonly but incorrectly repre- sented by two vectors acting on the same object. WbLf (40 N) in Newton’s law of reaction. We will present a discus- ▲ Figure 1-32 ■ Weightboot-on-legfoot (WbLf) and legfoot- sion of the law of reaction before returning to its appli- on-weightboot (LfWb) are reaction forces or an interaction pair. cation to the joint capsule of Sam Alexander’s knee. Both forces exist by virtue of the contact between the two objects. Although separated for clarity, these two vectors will be in line with Newton’s Law of Reaction each other. Every force on an object comes from another object force. If we consider that gravity-on-legfoot might that touches or is contacting that object (acknowledg- more properly be named earth-on-legfoot (ELf), we ing again our conceit that gravity “touches” an object). can appreciate that the reaction force, legfoot-on- When two objects touch, both must touch each other and earth (LfE) actually represents that attraction that touch with the same magnitude. Isaac Newton noted this the mass of the leg-foot segment has for the earth compulsory phenomenon and concluded that all forces (that is, the earth and the leg-foot segment pull on come in pairs that are applied to contacting objects, are each other). Vector LfE is a force applied to the CoM equal in magnitude, and are opposite in direction. This of the earth, acting vertically upward toward the leg- is known as Newton’s third law, or the law of reaction. foot segment with a magnitude equivalent to the weight of the leg-foot segment. Newton’s third law is commonly stated as follows: For every action, there is an equal and opposite reac- Reaction forces are always in the same line and tion. This statement is misleading because it seems to applied to the different but contacting objects. The result in the incorrect interpretation shown in Figure 1- directions of reaction forces are always opposite to each 31. Newton’s third law can be more clearly restated as other because the two touching objects either pull on follows: When one object applies a force to the second each other or push on each other. Because the points object, the second object must simultaneously apply a of application of reaction forces are never on the same force equal in magnitude and opposite in direction to object, reaction forces are never part of the same force that of the first object. These two forces that are applied system and typically are not part of the same space dia- to the two contacting objects are an interaction pair gram. However, we will see that reaction forces can be and can also be called action-reaction (or simply reac- an important consideration in human function, tion) forces. because one segment never exists in isolation (as in a space diagram). Continuing Exploration: Reactions to Leg-Foot Segment Forces ■ Gravitational and Contact Forces Figure 1-29C showed the force vector of weightboot- A different scenario can be used to demonstrate the on-legfoot (WbLf). WbLf arises from the contact of sometimes subtle but potentially important distinction the weight boot with the leg-foot segment. If the between (and relevance of) a force applied to an object weight boot contacts the leg-foot segment, then the and its reaction. We generally assume when we get on a leg-foot segment must also contact the weight boot. scale that the scale shows our weight (Fig. 1-33). A per- Legfoot-on-weightboot (LfWb) is a reaction force son’s weight (gravity-on-person [GP]), however, is not that is equal in magnitude and opposite in direction applied to the scale and thus cannot act on the scale. to WbLf (Fig. 1-32). We did not examine LfWb ini- What is actually being recorded on the scale is the tially because it is not part of the space diagram contact (push) of the “person-on-scale” (PS) and not under consideration. It is presented here simply as “gravity-on-person.” The distinction between the forces an example of a reaction force. [Side-bar: In Figure 1- GP and PS and the relation between these two forces 32, the points of application and action lines of the can be established by using both Newton’s first and reaction forces are shifted slightly so that the two vec- third laws. tors can be seen as distinctly different and as applied to different but touching objects.] The force of grav- ity-on-legfoot in Figure 1-29A also has a reaction
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 25 creates an upward reaction of countertop-on-person (CP). The contact between the person and the coun- tertop create an additional contact force acting on the person, resulting in what appears to be a weight reduc- tion. It is not a decrease in GP, of course, but a decrease in PS. In the example of measuring someone’s weight, the reaction force (PS) to GP cannot be ignored be- cause it is the variable of interest. Situations are frequently encountered when the contact of an object with a supporting surface and its weight are used interchangeably. Care should be taken to assess the situation to determine whether the magni- tudes are, in fact, equivalent. The recognition of weight and contact as separate forces permits more flexibility in understanding how to modify these forces if neces- sary. ▲ Figure 1-33 ■ Although a scale is commonly thought to CONCEPT CORNERSTONE 1-6: Action-Reaction Forces measure the weight of the person (gravity-on-person [GP]), it is actu- ally recording the contact of the person-on-scale (PS). Vectors GP ■ Whenever two objects or segments touch, the two objects and PS are equal in magnitude as long as nothing else is touching the or segments exert a force on each other. Consequently, person. every force has a reaction or is part of an action-reaction pair. The person standing on the scale must be in equi- librium (∑F ϭ 0). If vector GP is acting down with a ■ The term contact force or contact forces is commonly used magnitude of Ϫ734 N (Sam Alexander’s weight), there to indicate one or both of a set of reaction forces in which must also be a force of equal magnitude acting up on the “touch” is a push rather than a pull. the person for the person to remain motionless. The only other object besides gravity that appears to be con- ■ Reaction forces are never part of the same force system and tacting the person in Figure 1-33 is the scale. The scale, cannot be composed (cannot either be additive or offset therefore, must be exerting an upward push on the each other) because the two forces are, by definition, person (scale-on-person [SP]) with magnitude equal to applied to different objects. that of GP (ϩ734 N). The force of scale-on-person, of course, has a reaction force of person-on-scale (PS) that ■ The static or dynamic state (equilibrium or motion) of an is equal in magnitude (734 N) and opposite in direc- object cannot be affected by another object that is not tion (down) but applied to the scale. Consequently, in touching it or by a force that is not applied to it. this instance, the magnitude of the person’s weight and the person’s contact with the scale are equivalent in ■ The reaction to a force should be acknowledged but may magnitude although applied to different objects. The vectors be ignored graphically and conceptually if the object to person-on-scale and scale-on-person occur as a result of which it is applied and the other forces on that object are not a push by the contacting objects. When reaction forces of interest. arise from the push of one object on another, they are often referred to as contact forces (FC). When con- Additional Linear Force tact forces are perpendicular to the surfaces that pro- Considerations duce them, the term normal force (FN) is also used.5,7 Contact forces, therefore, are a subset of reaction The equilibrium established in Sam Alexander’s leg- forces. foot segment as he sits with the dangling weight boot is dependent on the capsule (and ligaments) to pull Under usual conditions of weighing oneself, little upward on the leg-foot segment with the same magni- or no attention is paid to whether the scale is recording tude as that with which gravity and the weight boot pull the person’s weight or the person’s contact with the downward (see Fig. 1-29C). Because the capsuloliga- scale. The distinction between GP and the reaction mentous structures are injured in Sam’s case, we need force PS, however, can be very important if something to explore the forces applied to the capsule. If the capsule else is touching the person or the scale. If the person is pulls on the leg-foot segment with a magnitude of 88 N, holding something while on the scale, the person’s the law of reaction stipulates that the leg-foot segment weight (GP) does not change, but the contact forces must also be pulling on the capsule with an equivalent (PS and SP) will increase. Similarly, a gentle pressure force. If the capsule cannot withstand an 88-N pull, down on the bathroom countertop as a person stands then it cannot pull on the leg-foot segment with an 88- on the scale will result in an apparent weight reduction. N force; that is, the ability of the capsule to pull on the The pressure of the fingers down on the countertop leg-foot segment is dependent on the amount of ten- sion that the capsule can withstand. This requires an understanding of tensile forces and the forces that pro- duce them.
Copyright © 2005 by F. A. Davis. 26 ■ Section 1: Joint Structure and Function: Foundational Concepts HR (110 N) 34). If “hands-on-rope” (HR) has a magnitude of ϩ110 N (~25 lb), then “block-on-rope” (BR) must have a BR (110 N) magnitude of Ϫ110 N because the rope is in equilib- rium. If it is assumed that rope has a homogenous com- ▲ Figure 1-34 ■ The tensile forces of the pull of hand-on-rope position (unlike most biological tissues), the tension (HR) and the pull of the cement block on the rope (BR) produce two will be the same throughout the rope (as long as there forces of equal magnitude (110 N) that result in 110 N of tension is no friction on the rope), and the tension in the rope within and throughout the rope. will be equivalent to the magnitude of the two tensile forces acting on the rope.5 In Figure 1-34, both hands- Tensile Forces on-rope (ϩ110 N) and block-on-rope (Ϫ110 N) can be designated as tensile forces. Tension in the joint capsule, just like tension in any pas- sive structure (including relatively solid materials such Assume for the moment that the rope is slack as bone), is created by opposite pulls on the object. If before the man begins to pull on the rope. As the man there are not two opposite pulls on the object (each of initiates his pull, the man’s hands will accelerate away which is a tensile force), there cannot be tension in the from the block because the force pulling his hands object. Remembering that the connective tissue capsule toward his body (“muscles-on-hands” [MsH]) will be and ligaments are best analogized to slightly elasticized greater than the pull of the rope on the hands (“rope- cord, we first examine tension in a cord or rope. on-hands” [RH]). As the man’s hands get farther from the block, the rope will get tighter, and the force of If a man pulls on a rope that is not attached to any- rope-on-hands (RH) will increase. The acceleration of thing, no tension will develop in the rope, regardless of the hands will gradually slow down as the resultant of how hard or lightly he pulls, because there is no coun- MsH and RH diminishes. When MsH and RH are equal, terforce. The rope will simply accelerate in the direc- the man will be in equilibrium. Because RH and HR are tion of the man’s pull (with a magnitude equivalent to reaction forces (and always equal in magnitude), the the force of pull [Funbal] divided by the mass [m] of the tension in the rope (HR) will eventually be equivalent rope). If the rope is tied to an immovable block of to the magnitude of the man’s pull (MsH) (Fig. 1-35). cement, there will be two forces applied to the rope. The two forces are created by the only two things con- ■ Tensile Forces and Their Reaction Forces tacting the rope: the man’s hands and the block (Fig. 1- The interactive nature of reaction forces and net forces on an object can be seen as we continue our example. Hands-on-rope (HR) is a tensile vector and, therefore, must be equivalent in magnitude and opposite in direc- tion to the other tensile vector, block-on-rope (BR). Not only is the tensile vector block-on-rope (BR) equal to the other tensile vector (HR), but tensile vector BR is also equivalent in magnitude and opposite in direc- tion to its reaction force, rope-on-block (RB) (see Fig. 1- 35). Consequently, as long as the rope can structurally withstand the tension, the pull of hands-on-rope (HR) will be transmitted through the rope to an equivalent pull on the block (RB). The example of the man pulling on the rope and MsH (110 N) RH (110 N) HR (110 N) ᭣ Figure 1-35 ■ Equilibrium of the man BR (110 N) RB (110 N) will be achieved when the force of the rope-on- hand (RH) reaches the magnitude of muscles-on- hand (MsH). Rope-on-hand will not reach the 110 N magnitude needed to establish equilibrium until the tension in the initially slack rope reaches that magnitude as the man accelerates away from the block.
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 27 MsH (200 N) ᭣ Figure 1-36 ■ If the rope cannot withstand the tensile forces placed on it, it will break. Once the rope breaks, the force of muscles-on-hand (MsH) is unopposed, and the man will accelerate backward. cement block assumed so far that the rope could with- the assumption that there is 88 N of tension in the cap- stand whatever tension was required of it. If the rope is sule: (1) Does the magnitude of tension reach 88 N in damaged, it may be able to withstand no more than 110 the capsule immediately, and (2) can the injured cap- N of tension. If, however, the man in our example pulls sule (and ligaments) withstand 88 N of tension? Case on the rope (HR) with a magnitude of 200 N (45 lb), Application 1-1 applies the concepts from the example the rope will break. Once the rope breaks, there is no of tension in the rope to Sam’s joint capsule. longer tension in the rope. The man pulling on the rope (HR) with a magnitude of 200 N will have a net C a s e A p p l i c a t i o n 1 - 1 : Tension in the Knee Joint unbalanced force that will accelerate the hands (or the Capsule man) backwards (Fig. 1-36) until his muscles stop pulling (which will, it is hoped, happen before he The reaction forces of capsule-on-legfoot (CLf) and leg- punches himself in the stomach or falls over!). foot-on-capsule (LfC) have a magnitude of 88 N (see Fig. 1-37). [Side-bar: Vectors LfC and CLf should be co- Let us go back to Sam Alexander to determine how linear in the figure but are separated for clarity.] the tension example is applied to Sam’s use of the Legfoot-on-capsule is a tensile vector. Tension can weight boot. The equilibrium of Sam’s leg-foot segment occur in a passive structure only if there are two pulls was based on the ability of the capsule (CLf) to pull on on the object. Therefore, there must be a second tensile the leg-foot segment with a magnitude equivalent to vector (of ϩ88 N) applied to the capsule from some- GLf ϩ WbLf. If the capsule pulls on the leg-foot seg- thing touching the capsule at the other end. The second ment with a magnitude of ϩ88 N (as we established ear- tensile vector, therefore, must be femur-on-capsule (FC) lier), the leg-foot segment must pull on the capsule (Fig. 1-38), where the tensile vectors are effectively co- (legfoot-on-capsule [LfC]) with an equivalent force of linear but separated for clarity. The magnitudes of CLf, Ϫ88 N (Fig. 1-37). Two questions can be raised around CLf (88 N) CLf (88 N) FC (88 N) LfC (88 N) LfC (88 N) ▲ Figure 1-37 ■ The pull of the capsule-on-legfoot (CLf) must ▲ Figure 1-38 ■ The tensile forces of legfoot-on-capsule (LfC) have a concomitant reaction force of legfoot-on-capsule (LfC) that is and femur-on-capsule (FC) are shown with their interaction pairs, an 88-N tensile force on the joint capsule. capsule-on-legfoot (CLf) and capsule-on-femur (CF), respectively.
Copyright © 2005 by F. A. Davis. 28 ■ Section 1: Joint Structure and Function: Foundational Concepts LfC, and FC are equivalent because vector CLf is part of the same linear force system with WbLf and GLf (see Fig. 1-29C) and vectors LfC and FC are part of the same linear force system. The sum of the forces in both linear force systems is 0 because it is assumed that no move- ment is occurring. CONCEPT CORNERSTONE 1-7: Tension and Tensile GLf (48 N) Forces HLf (88 N) ■ Tensile forces (or the resultants of tensile forces) on an object are always equal in magnitude, opposite in direction, and WbLf (40 N) applied parallel to the long axis of the object. ▲ Figure 1-40 ■ As long as the 88 N force on the leg-foot seg- ■ Tensile forces are co-linear, coplanar, and applied to the ment from gravity (GLf) and the weight boot (WbLf) are supported same object; therefore, tensile vectors are part of the same by an equal upward force from the hand (HLf), the tension in the linear force system. capsule and ligaments will be zero (or negligible). ■ Tensile forces applied to a flexible or rigid structure of HLf is shown to one side of GLf and WbLf for clarity, homogenous composition create the same tension at all but assume that the supporting hand is directly below points along the long axis of the structure in the absence of the weight boot.] The magnitude of pull of the capsule friction; that is, tensile forces are transmitted along the (and ligaments) on the leg-foot segment would be neg- length (long axis) of the object. ligible as long as HLf had a magnitude equal and oppo- site to that of GLf and WbLf. As the upward support of Joint Distraction the hand is taken away, however, there would be a net unbalanced force down on the leg-foot segment that Joint capsule and ligaments are not necessarily in a con- would cause the leg-foot segment to accelerate away stant state of tension. In fact, if Sam started out with his from the femur. The pull or movement of one bony seg- leg-foot segment on the treatment table, there would ment away from another is known as joint distraction.8 effectively be no tension in his capsule or ligaments be- As the upward push of the hand decreases and the leg- cause the sum of the forces on the leg-foot segment foot segment moves away from the femur, the capsule from the “contacts” of gravity (gravity-on-legfoot) and will become increasingly tensed. The magnitude of the treatment table (table-on-legfoot) (Fig. 1-39) would acceleration of the leg-foot segment will be directly be sufficient for equilibrium (∑F ϭ 0). Although both proportional to the unbalanced force and indirectly the capsule and the weight boot are still attached to the proportional to the mass of the leg-foot segment and leg-foot segment, the magnitudes of pull would be neg- weight boot combined (a ϭ Funbal ÷ m). However, the ligible (too small to include in the space diagram). unbalanced force is difficult to quantify because it is constantly changing. Although the increase in capsular A situation similar to the leg-foot segment on the tension occurs concomitantly with the reduction in treatment table would exist if Sam’s foot were sup- hand support, the two forces are not equivalent in mag- ported by someone’s hand if his leg-foot segment and nitude because the leg-foot segment must move away weight boot were moved off the treatment table to the from the femur for the capsule to get tighter; that is, vertical position. In Figure 1-40, the hand is pushing up there must be a net unbalanced force on the leg-foot (ϩ88 N) on the leg-foot segment (hand-on-legfoot seg- segment to create the movement that causes the cap- ment [HLf]) with a magnitude equivalent to the pull of sule to get tighter. gravity and the weight boot (Ϫ88 N). [Side-bar: Vector The Continuing Exploration: Reactions to Leg- TLf (48 N) Foot Segment Forces presented calculation of accelera- tion of the leg-foot segment at one point in time (a GLf (48 N) static rather than dynamic analysis). However, the con- cepts are more important than the calculations, given ▲ Figure 1-39 ■ The forces of table-on-legfoot (TLf) and grav- that the weight of a limb, the support of the hand, and ity-on-legfoot (GLf) in this position are sufficient for equilibrium of the tension in the capsule in a true clinical situation are the leg-foot segment, with zero (or negligible) tension in the knee generally unknown. joint capsule.
Copyright © 2005 by F. A. Davis. Chapter 1: Biomechanical Applications to Joint Structure and Function ■ 29 Continuing Exploration: Acceleration in nitude of acceleration as the leg-foot segment, the joint Joint Distraction surfaces will not separate any farther than was required to initiate movement of the femur. Although we did not The leg-foot segment and weight boot together set the femur in equilibrium (did not stabilize the weigh 88 N. To calculate acceleration (a ϭ Funbal ÷ femur) in Case Application 1-1, there must be a force m), however, the mass (not just the weight) of the applied to the femur that is opposite in direction to leg-foot segment and weight boot must be known. capsule-on-femur for there to be effective joint distrac- Recalling that 1 N is the amount of force needed to tion. Joint distraction can occur only when the acceler- accelerate 1 kg at 1 m/sec2, weight (in newtons or ation of one segment is less than (or in a direction equivalently in kg-m/sec2) is mass (in kilograms) opposite to) the acceleration of the adjacent segment, multiplied by the acceleration of gravity, or: resulting in a separation of joint surfaces. W ϭ (m)(9.8 m/sec2) In the human body, the acceleration of one or both segments away from each other in joint distraction (the Solving for mass, a weight of 88 N is equivalent to 88 dynamic phase) is very brief unless the capsule and lig- kg-m/sec2 ÷ 9.8 m/sec2 ϭ 8.97 kg. Consequently, the aments (or muscles crossing the joint) fail. Sam’s leg- leg-foot segment and weight boot together have a foot segment will not accelerate away from the femur mass of approximately 9 kg. Assigning some arbitrary for very long before the distraction forces applied to values, assume that a downward force of Ϫ88 N is the adjacent joint segments (leg-foot and femur) are offset in this static example by an upward push of the balanced by the tensile forces in the capsule. Given that supporting hand of ϩ50 N and capsular tensile force Sam is still relaxed as the weight boot hangs on his leg- of ϩ10 N. The net unbalanced force on the leg-foot foot segment (we have not asked him to do anything segment (Ϫ88 ϩ 50 ϩ 10) is Ϫ28 N. Therefore: yet), the check to joint distraction (the pull of gravity and the weight boot) is the tension in the capsule (and Ϫ28 kg-m/sec2 ligaments). Sam presumably has a ligamentous injury a ϭ ᎏ9 kᎏg that is likely to cause pain with tension in these pain- sensitive connective tissues. If the distraction force a ϭ Ϫ3.11 m/sec2 remains, the capsule and ligaments may fail either microscopically or macroscopically (see Chapter 2). In When the hand in Figure 1-40 is no longer in con- the short term, we can prevent this problem by putting tact with the leg-foot segment and the tension in the the supporting hand back under the weight boot. If the capsule reaches 88 N, the leg-foot segment will stop upward push of the hand is sufficient, the tensile forces accelerating away from the femur and will reach equi- on the ligaments can be completely eliminated. librium. ■ Distraction Forces Continuing Exploration: Stabilization of the Femur The resultant pull of gravity and the weight boot on the leg-foot segment (composed into a single vector) can Because our primary interest is in Sam Alexander’s be referred to as a distraction force3 or joint distraction leg-foot segment and secondarily in the injured knee force. A distraction force is directed away from the joint joint capsule, the source of stabilization of the femur surface to which it is applied, is perpendicular to its (the other joint distraction force) was not a neces- joint surface, and leads to the separation of the joint sary component of our exploration. However, the surfaces. [Side-bar: It is important to note that the term principles established thus far will allow us to identify “distraction” here refers to separation of rigid nonde- that distraction force. formable bones. Distraction across or within a defor- mable body is more complex and will be considered in In Figure 1-41, WbLf and GLf are composed Chapter 2.] in-to a single resultant distraction force (GWbLf) of Ϫ88 N, with the leg-foot segment once again unsup- A joint distraction force cannot exist in isolation; ported. Vector GWbLf creates an 88 N tensile force joint surfaces will not separate unless there is a distrac- in the capsule that creates a pull of the capsule on tion force applied to the adjacent segment in the oppo- the femur (CF) with an equal magnitude of 88 N site direction. As the leg-foot segment is pulled away (see Fig. 1-41). A net distractive force of ϩ88 N from the femur, any tension in the capsule created by applied to the femur is necessary to stabilize the the pull of the leg-foot segment on the capsule results femur and create tension in the capsule. in a second tensile vector in the capsule (femur-on- capsule). If the femur pulls on the capsule, then the The femur is contacted both by gravity (GF) and capsule must concomitantly pull on the femur (see Fig. by the treatment table (TF) (see Fig. 1-41). To deter- 1-38). If there is no opposing force on the femur, the mine the net force acting on the femur, we must esti- net unbalanced downward force on the leg-foot seg- mate the mass or weight of that segment. Sam weighs ment will be transmitted through the capsule to the approximately 734 N, and his thigh constitutes femur; the femur will also accelerate downward as soon approximately 10.7% of his body weight.1 Conseque- as any appreciable tension is developed in the capsule. ntly, his thigh is estimated to weigh approximately 78 If the femur accelerates downward with the same mag- N. With the magnitudes of CF (Ϫ88 N) and GF (Ϫ78 N) known, it appears that the magnitude of TF
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