Understanding the scientific bases of human movemen

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Understanding The Scientific Bases of

Human Movement ALICE

L O'CONNELL,

ELIZABETH

B.

Ph.D.

GARDNER,

Ph.D.

Understanding

The

Scientific

Bases of

Human Movement

Understanding The Scientific Bases of

Human Movement ALICE

L O'CONNELL,

Ph.D.

F.A.C.S.M.: F.A.A.H.P.E.R. Associate Professor of Biomechanics

Boston University Sargent College of Allied Health Professions

ELIZABETH

B.

GARDNER,

Ph.D.

F.A.C.S.M.

and Physiology, Emerita Boston University Sargent College of Allied Health Professions Professor of Biology

The Williams

& Wilkins Co.

Baltimore 1972

^osSspJF

Copyright ©, 1972

The Williams & Wilkins Company 428 E. Preston Street Baltimore,

Md.

21202, U.S.A.

book is protected by copyright. No part of book may be reproduced in any form or by any means, including photocopying, or utilized by any information storage and retrieval system without written permission from the copyright owner. All rights reserved. This this

Made

in the

United States of America

Library of Congress Catalog Card SBN 683-06622-6

Composed and printed

at the

Waverly Press, Inc. Mt. Royal and Guilford Aves. Baltimore,

Md.

Reprinted 1973

21202, U.S.A.

Number

78-188932

is dedicated to Ruth B. Glassow who has stimulated and inspired many, including the authors, to pursue the study of human movement.

77ms book so

The authors wish

to record their gratitude to the Boston University Graduate School for a grant-in-aid which contributed to the preparation of this manuscript.

Foreword The term kinesiology, long used in referring to the science or study of human motion, has been giving way to the term biomechanics. Certainly this latter term is more self-explanatory and hence its rising popularity. In the past half century the change in the content of textbooks in this field reflects the changing concepts of kinesiology: the early texts were basically applications of the principles of anatomy and mechanics to human movement. Some physiology of muscle was included, and a little material on the central nervous system was added as that field advanced. The neuromuscular physiology which appeared in these early kinesiological texts it

was greatly

seems that way now

simplified, or perhaps

in the light of our present

knowl-

edge.

Early kinesiologists were concerned primarily with the anatomy of motion: with action occuring at the different joints with the identity of muscles that could produce these actions and. in many cases, with the use of this knowledge to develop those particular muscles to a high peak of efficiency. That this is still the case with a majority of the kinesiology instructors of undergraduate students today is illustrated indirectly by a survey made during the middle 1960's by Neuberger of Eastern Michigan University which was concerned with the visual aids used by these instructors. Of the replies reported in detail. 60% leaned very heavily to almost entirely on anatomical materials. Also, of the seven best known kinesiology texts on the market as this is written, all but two devote 50% or more of their content to musculoskeletal anatomy, and some labora-

manuals do likewise. However, we seem to be moving away from the applied anatomy concept; the trend in some instances has swung so far afield that there appeared a tendency to include the effects of emotions, physiological condition, progress in motor learning, etc. under a kinesiological umbrella. This text has no such ambitions: it is written for those kinesiologists who are concerned with the rapidly expanding ways and means of studying, and so arriving at a better understanding of. human motor control. tory

Today the modem kinesiologist asks many questions about human movement and eagerly seeks the answers. He not only wants to know what movement pattern(s) or sequence of movements is involved in a given skill,

but how those movements can be made more efficient. This in turn involves him in seeking answers to a number of other questions such as: What is the mass of the body, body part(s), or extremity? What is the angular or linear acceleration of a body, body segment, or extremity? What force does a body, or body part, exert as a re-

mass and acceleration? How, and in what direction, is this

sult of its

force applied to an external object, (ball, club, discus, etc.) or to the body itself (as in

jumping

or running)?

Where is the center of gravity of the body, not only when it is motionless but during performance of a motor skill?

Where

the center of gravity of any combination of an entire or partial extremity, the entire or upper trunk plus one or both upper limbs, the entire or lower trunk plus one or both of the lower limbs,

body

is

parts: e.g.,

etc.?

What is the angular velocity of a body segment at any given instant? What linear velocity becomes available at the distal end of a kinematic chain as a result of the angular velocity of a particular joint action, and which can be imparted to a ball or other throwing or striking implement? Finding these answers is made possible by our modern technology. The motion picture camera with speeds ranging up to thousands of frames per second provides the means for capturing a single motor performance for an indefinite period of study. Repeated viewing of the film with the aid of a time-motion study projector, a film editor-viewer or a microfilm reader gives ample opportunity for analysis. Finally, as has been illustrated by Plagenhoef* at the University of Michigan and then by Garrett et al.,t the computer can be elegantly used to *

Plagenhoef, S.

netics of Selected

C, 1962. An Analysis of the Kinematics and KiSymmetrical Body Actions. Doctoral Dissertation,

University of Michigan. t Garrett, R. E., Widule, C. J., and Garrett, G. E., 1968. Computer aided analysis. Kinesiol. Rev. pp. 1-4; Garrett, G. E., Widule, C. J., Reed, W. S., and Garrett, R. E., 1969. Human movement via

computer graphics. Paper presented

at convention of the

American

Association for Health, Physical Education and Recreation, Boston, 1969.

vii

VIM

FOREWORD

expedite matters. Many of the available methods for attaining these answers and for using various of .the investigative tools* available will be found in the text of this book.

Perhaps more difficult to answer are the questions asked by some of today's kinesiologists involving the initiation of movement and control of the body in performing motor skills. They are aware that true understanding of human movement requires knowledge of the means by which the central nervous system integrates proprioceptive input and coordinates the activity of the muscles so that each will contribute properly to the intended movement. However, knowledge of the functioning of the nervous system is now so extensive and is increasing so rapidly that it is difficult for the neurophysiologist, and impossible for the layman, to keep abreast of it. There is therefore a real need for material which will assist the kinesiologist in maintaining a general overview of advances in this area which may be of significance to his field. Up to the present, relatively few writers in the fields of physical education and physical therapy have been able to write in this field with assurance and authority. Hopefully, however, this situation is in the process of change. This text is one of *

Methodology of computer usage per

in this text.

se has not

been included

first to attempt to interpret and apply some of the expanding knowledge of neurophysiology to the field of motor performance. It includes four chapters on the neuromuscular bases of movement (Chapters 10 to 13), to supplement the too often scanty coverage of such information in undergraduate curricula. The inclusion of Section III on proprioceptive reflexes is unique with this text and is presented to provide a background for en-

the

larging the scope of kinesiological analysis. The final chapter deals with speculative postulations of reflex involvement in certain skills. It is offered in the hope that it may encourage the kinesiologist to include consideration of this aspect of human movement in his analysis and research, to recognize and investigate reflexes which may be assisting a performance, and to identify those which may be interfering and require voluntary inhibition. With such information available, he should be better equipped to understand the difficulties encountered by the beginner in learning a new skill, and why the use of one method or technique produces better results than another. He can then improve the best of the older techniques and design new and more effective methods based on his expanded knowl-

edge. A. L. O'C. E. B. G.

Contents SECTION TWO KINETICS

Foreword

chapter 6. The Laws of Motion and Energy Newton's Laws of Motion

Part I

BIOMECHANICS

Momentum Energy

SECTION ONE

KINEMATICS chapter

1

Skeletal

.

Joint Axes

Mechanisms

and Degrees

of

Freedom

3 3

Links and Chains

3

Movements

6

Joint

2. Mechanics of Muscle Action Muscle Classification Muscle Function Muscle Attachments and Their Effect on Function Application of Muscle Force to Skeletal

chapter

Levers

The Range

Classification of Levers

Torque

4. Motion Types of Motion

chapter

Human Motion Mechanics of Motion Linear Motion

Relation to

Rotational Movement Circular Motion 5.

chapter 9. Kinetic Analysis: Statics 113 Location of Line of Gravity by Segmental

Kinematics of

Human Movement

Analysis: The Scale Usefulness of Motion and Static Analysis Acceleration of Body Parts Analysis of Segmental Velocities

Value Determining Segmental Velocities

Equilib103 103 107

Method

113

Determinations of Estimates of Total Muscle Force 114 Compression Forces at the Acetabulum 119

39 39

Part II

NEUROMUSCULAR INTEGRATION

49 49 49 49 52

chapter

54

tal

56

Movement Analysis Sample of Movement

Gravity,

Locating the Center of Gravity Equilibrium

41 41

Moment Arms

chapter

32

8.

rium

36

Leverage

3.

29 29 29

Muscle Extensibility and Con-

tractility

cr\pter

Equilibrating Forces

chapter

35 of

77 78 81

85 85 86 88 88 98

chapter 7. Forces Mass. Weight, and Gravity Force Relations Typical Problems Met in Force Analysis Composition and Resolution of Forces

Man, and His

77

59 59 61 64 64 64 72

SECTION ONE

PHYSIOLOGY OF SKELETAL MUSCLE 10.

Structure and Chemistry of Skele-

Muscle Introduction Properties of Skeletal Muscle Structure of Skeletal Muscle The Nature of Contraction

127 127 127 127 134

Summary

144

chapter 11. Factors Which Affect the nitude of Contractile Tension The Magnitude of Contractile Tension

Mag147 147 ix

CONTENTS

Summary

159

Analysis

chapter

15.

flexes in

Motor

Involvement

of

Proprioceptive

Re223 223 226

Skills in

Motor

Skills

Neurokinesiological Analyses

SECTION

TWO

Types of Reflex Activity to be Identified Motor Skills Proprioceptors and Physical Education

NEUROPHYSIOLOGY chapter

12. Basic Neurophysiology The Neuron: Structure and Function Structure and Function of the Synapse

161 161 168

13. Basic Organization of the Neuromuscular System 179 The Sensory System: Structure and Function

chapter

of Receptors

179 184 188 191

The Motor System: Motor Units The Integrative System: Neural Circuits

Summary SECTION THREE

THE INTEGRATIVE ROLE OF THE PROPRIOCEPTIVE REFLEXES

appendix

in

231 231

A

Figure A.l. Link boundaries (at the joint centers) and percentage of distance of the cen233 ters of gravity from link boundaries appendix

B

Table B.l.

A, average segment characteristics of three cadavers dissected by Braune and Fischer (1889). B, average segment characteristics of two cadavers described bv Braune and Fischer (1892) 235 Table B.2. Average segment characteristics of 235 eight cadavers Table B.3. Location of centers of gravity of body

segments Table B.4. Specific gravity of body segments Table B.5. Segmental fractions of body weight according to somatotype Table B.6. Regression equations for calculating mass (in kg) of body segments Table B.7. Gravity and distance .

chapter 14. The Proprioceptors Associated Reflexes

and

Introduction Muscle Proprioceptors Joint and Cutaneous Receptors Labyrinthine and Neck Receptors Role of Reflexes in Skilled Movement

Their 193 193 194 209 212

219

APPENDIX

C

Table C.

1.

.

Tables of the trigonometric functions

.

236 237

238 238 239

241

Part I

BIOMECHANICS

Introduction There are two approaches to the study of the mechanics of human motion, the kinematic and the kinetic. The kinematic approach is purely descriptive, concerned with the geometry and temporal qualities of motion. While forces may be named or described, there is no attempt to quantify them, no concern for the size or direction of the forces involved. On the other hand, the kinetic approach to the study of movement is concerned with the forces that produce or change the state of rest or motion of a body. It determines the size and direction of the forces, their points of application, the ultimate results and resultant of the forces involved. The human body is a complicated apparatus, selfmaintaining, self-regulatory, and autonomous. It is also a system of levers whose cores are formed of long, short, or irregularly shaped bones linked together by capsules reinforced by ligaments to form joints. This system, although controlled and operated by the central nervous system's neural control of muscular tension, obeys all of the laws of mechanics involving both statics and dynamics. Statics deals with bodies at rest and forces in equilibrium, while dynamics is concerned with bodies in motion. A model who sits or poses for a photographer or artist is an example of statics as the body must be held motionless for a given period of time. The muscular effort required naturally depends on the assumed pose. The muscles must exert enough force to counteract the pull of gravity on the body parts so that equilibrium is established between gravitational force and the force exerted by the muscles. As long as the pose is held, this balance of forces is maintained and the situation remains static. As the force of gravity is constant, any increase or decrease in muscular force will result in movement, and the situation changes from static to dynamic.

Kinematic Analysis If

the kinesiologist limits himself to describing the

pose, i.e., the anatomical position of each joint and body segment, and names the muscles that he believes

are responsible for maintaining the pose, the analysis is anatomical as well as kinematic. If, under the changing conditions indicated above, he limits himself to stating whether there is (1) an increase in muscular force moving the body segment or segments against gravity or (2) a decrease in muscular tension which allows a segment or segments to move with gravity, and if he names the resulting movements and the segments moved, he is still making a kinematic analysis. Also, if when the pose is broken he determines the accelerations or velocities of joint and/or segment movement, or the direction and amount of change of the position of the center of gravity and its acceleration and/or velocity, he is continuing with the kinematic analysis of a now dynamic situation.

Kinetic Analysis

However,

the kinesiologist calculates the forces the muscles in supporting the pose and the compression forces being exerted on the major and/or weight-bearing joints as a result of the muscle tension and gravity, the analysis has become or force

if

moments exerted by

kinetic.

The first section of this part of the book is concerned with kinematics and presents the tools necessary for kinematic analyses of human movement. At the same time it presents many related concepts which should be of value to any student of human movement. Finally, all of these are applied to sample analyses of motor

skills in

Chapter

5.

The second

section deals with kinetics: the forces which act on the body, those which the body can exert, and the equilibrium which may or may not exist between them. As in Section I, the early chapters present the tools and concepts of analysis, while the use of these tools

Chapter

9.

in

a

sample analysis

is

presented in

SECTION ONE

CHAPTER

KINEMATICS

Skeletal

I

Mechanisms

The

skeletal system is the bony framework that supbody organs, protects many of them, and forms the hard core of all bodv segments. Its manv articula-

ports

tions provide mobility,

and

mobile articulations that

is

it is the function of these the concern of the kinesiolo-

gist.

JOINT AXES AND DEGREES OF FREEDOM Both the anatomist and the kinesiologist speak of

tween the

and as

a limited

joints as being uniaxial, biaxial, or multiaxial

having certain degrees of freedom (Steindler; Brunnstrom: Tern and Trotter). A joint with only one axis (uniaxial!: has one degree of freedom: that is to say, the articulating bones can move only in one plane. Examples in the human body include hinge and pivot joints. Hinge joints occur at the elbow, knee, interphalangeal. and ankle joints. The pivot joints are the -

atlantooccipital in the vertebral column and the radioulnar joints in the forearm. Joints that can move about two axes (biaxial) have two degrees of freedom and so

produce movement in two different planes. The wrist. the metacarpophalangeal and the metatarsophalangeal joints are biaxial. Joints that can permit movement in all three planes have three degrees of freedom but are called multiaxial by the anatomist rather than triaxial, as movement can occur in oblique planes as well as in the three cardinal planes. socket joints at the

Examples include the ball and hips and shoulders and the

numerous plane joints of the axial skeleton. In this instance the term "plane" is an adjective referring to the almost flat articular surfaces which can glide over one another, with movement being limited only by ligaments or by the joint capsule. Examples include those joints between the articular processes of the vertebrae and be-

LINKS

the degrees of freedom of the pianist's fingertips involved listing the joints occurring between the distal phalanges and the pelvis. These joints unite the various body segments which move upon each other in the manner of links in a chain. The concept of links and chains, first used by engineers, can be applied very elegantly by the kinesiologist to many phases of the study and analysis of movement.

Links in the Body first

degrees of freedom. For the kinesiologist there is a distinct advantage in using the term "degrees of freedom." While no one joint can have more than three degrees of freedom, the degrees at adjacent joints can be summed to express the total

amount

of freedom of motion of a distal

segment

proximal one. For instance, the distal phalanges of a pianist enjoy 17 degrees of freedom relative to his trunk: one degree at each of the distal and proximal phalangeal joints; two degrees at the metacarpophalangeal joints; two degrees at the wrist joint; one degree in the forearm at the radioulnar joints; one degree at the elbow; three degrees at the shoulder; relative

to a

three degrees at the acromioclavicular joint; three degrees at the sternoclavicular joint. Observation of many pianists might, however, lead us to add three more degrees of freedom arising from the motion in the vertebral column. This would express the freedom of the phalanges relative to the pelvis which is resting on the piano bench, rather than relative to his torso, making a total of 20 degrees of freedom available at the fingertips.

AND CHAINS

Summing up

Dempster was the

ribs and the vertebrae. These joints have such amount of movement at any one articulation that total movement of the torso occurs only because of the combined action of many or all of the joints and their

kinesiologist to adapt the link

concept to the problems involving kinetic and kinematic treatment of movements of the human body. Since engineering links involve overlapping articulating members held together by pins which act as axes of rotation. a link is considered to be a straight line of constant length running from axis to axis. A system of links is essentially a geometric entity for analysis of motion by geometric or kinematic methods. "...In engineering mechanisms the links move in relation to a framework, and this framework itself forms a link in the system. Thus, to transmit power, the links of machinery must

UNDERSTANDING THE

SCIENTIFIC

BASES OF HUMAN MOVEMENT

form a closed system in which the motion of one link has determinate relations to every other link in the system" (Reuleaux, quoted by Dempster). In the appendicular skeleton a body segment consists of a hard core made up of one or more bones enclosed in an irregular mass of soft tissue (muscle, connective

and skin). Ligaments and muscle tendons cross the joint between contiguous body segments, anchoring to the adjacent bone or bones and holding the segments together. The axis around which one segment moves on tissue,

another generally passes through one of the bones through an area near the joint; e.g., the axis for knee flexion and extension passes through the epicondyles of the femur (Fig. 1.13; see also Figs. 1.7 through 1.15 for locations of other joint axes). Thus during flexion and extension of the knee, the tibia moves around this axis which is constantly changing because of the shape of the femoral condyles, while at the same time it is gliding over these condyles. When the person is stand-

Kinematic Chains Reuleaux also introduced the term "kinematic chain" system of links. In engineering, the chain forms a closed system where, as quoted earlier, "the motion of one link has determinate relato refer to a mechanical

tions to every other link in the system,"

and "the closed

system assures that forces are transmitted in positive predetermined ways." Thus in engineering a kinematic chain is a closed system of links joined in such a manner that if any one is moved on a fixed link, all of the other links will move in a predictable pattern (Fig. 1.2). This

ing, the distal tibiofibular configuration rotates around an axis through the talus (and the two malleoli) when the leg moves over the talar head during forward or backward body sways (Fig. 1.14). Because the bones of the body rarely overlap at joints as at the ankle and, except for the atlantoaxial joint, have no pin-centered axes, and because at many joints movement can occur in different directions and planes, the engineering concept of links must be redefined to fit the need of the kinesiologist. Dempster has proposed the use of the term "link" in kinesiology as the distance between joint axes; e.g., the leg link becomes the linear distance between the joint axes passing through the distal end of the femur and the proximal end of the talus (or through the two malleoli), thus spanning both the knee and ankle joints (Fig. 1.1). Figure 9.2 in Chapter 9 on kinetics is a scale drawing made from actual

measurements of the in the same chapter.

links of the subject in Figure 9.1.

Leg Link

FIGURE FIGURE

1.1.

Leg

link.

1.2.

Types

of closed kinematic chains. All joints are

pin-centered and free to move.

Skeletal is

not true

m

the living body.

With

Mectunisms

tew exceptions the

system of skeletal links is not composed oi closed chains but of open ones, as the peripheral ends of the extrenu ties are tree (Pig. L.3). Forces may be transmitted in positive ways, predetermined by the central nervous system, but the central nervous system is notorious for never accomplishing the same act in exactly the same nay from one time to the next, even though the external results may appear similar. Thus when speaking of a living kinematic chain, we are usually speaking of a

FIGURE

1.4.

Example suggested by Steindler

of a

human

closed

kinematic chain.

in an open system, whose dimenby the linear distance from joint axis to joint axis, ignoring muscle mass, bone structure, and type of articulation between body segments. Although most living kinematic chains are open, Brunnstrom defines two closed kinematic chains in the body. The first is the pelvic girdle, which is made up of three bony segments united at the two sacroiliac joints and at the symphysis pubis. This can hardly be classified as a kinematic chain because normally no movement occurs at the joints mentioned. Dempster classes

series of links

arranged

sions are determined

the pelvis as a single triangular link (Fig. 1.3). The second closed kinematic chain in the body according to Brunnstrom is the thorax where the upper 10 ribs are jointed to the vertebral column and sternum. The rib cage, however, does constitute a system of closed kinematic chains because the upper 10 ribs of the left side cannot move without similar movement by the upper 10 ribs of the right side when they lift the sternum on inhalation. Steindler* considers a closed living chain (which he terms kinetic rather than kinematic) to exist in "all situations in which the peripheral joint of the chain meets with overwhelming resistance" (Fig. 1.4).

FIGURE

The human skeleton as a system of links. (From T.. 1955. Space Requirements for the Seated Operator Wright Air Development Center Technical Report 55159 Dempster.

1.3.

W

* While Brunnstrom and Steindler used the term "chain" modified by kinematic or kinetic respectively when referring to a series of body segments, they did not use the term "links" in their discussions.

UNDERSTANDING THE

SCIENTIFIC BASES OF

HUMAN MOVEMENT

JOINT MOVEMENTS The type and range of movement at any given joint depend upon the structure of the joint and the number of its axes, the restraint imposed by ligaments and muscles crossing the joint, and the bulk of adjacent tissue. A joint with three degrees of freedom may because of its structure have a very limited range of motion as was indicated earlier in regard to the intervertebral joints, while a joint with only one degree of freedom may have a large range of motion. For example, the forearm can move through an average range of 150° from the position in line with the arm to full flexion. The range may be increased by from 5 to 15° in the individual who has a smaller than average olecranon process or a deeper than average olecranon fossa which permits the forearm to hyperextend. Conversely, in an individual with overdeveloped biceps and brachialis muscles or with excessive adipose tissue, flexion may be limited by the very bulk of the soft tissue of the arm. Similar factors can also affect the range of mobility of

straightens the joint. While this definition is adequate it is not applicable to all joints. A more satisfactory one which can be applied to all except the

to a degree,

shoulder joints is based on the anatomical concept that flexion is the approximation of ventral or volar surfaces. This concept is based on the embryological development of the human fetus. Soon after the limb buds first appear in the embryo (Fig. 1.16. A), they project laterally with the thumbs and great toes uppermost (Fig. 1.16.B). As the limbs develop, they bend ventrad at the elbows and knees so that the apices of these joints are pointed outward and the palms of the hands and soles of the feet (the volar surfaces) face the torso (Fig. 1.16.C). Finally, both pairs of limbs rotate 90° but in opposite directions, the rotations taking place about the long axes of the limbs (Fig. 1.16.D). The upper

other joints. The following discussion presents a brief review of the motions occurring at the various joints, the axes around which these movements take place, and the plane in which the body part or parts are moved as a result of the joint action. As all movements are described as being initiated from the anatomical position (Fig. 1.5), each body part will generally* move in a plane parallel to one of the three primary planest of the body, i.e., the midsagittal, frontal (or coronal), or transverse plane. The midsagittal plane divides the body into right and left halves (Fig. 1.6. A), and any plane parallel to it is known as a sagittal plane. Similarly the primary frontal plane divides the body anteroposteriad into front and back halves (Fig. 1.6.B), and any plane parallel to it is known as a frontal plane. The third primary plane is transverse and divides the body into upper and lower halves (Fig. 1.6. C), and all horizontal planes parallel to it are also known as transverse planes. The typical motions are presented first, followed by special cases, e.g., movements of the forearm, foot, etc. (Figs. 1.7 to 1.15).

Flexion-Extension Flexion and extension are movements in which the moving segments travel in a sagittalf plane around a horizontal axis defined by anatomical frontal and transverse planest through the axis. Flexion is popularly considered to be a movement which decreases the angle between the moving part and the adjacent segment (as in

to

elbow or finger flexion), and extension is considered be a movement which increases this angle or

* It

is

recognized that

which are oblique

many movements

to the three

occur

at multiaxial joints

perpendicular planes, but these are

considered as combinations of the primary movements discussed here. t

As each primary plane divides the body into equal

halves, these

planes must pass through the center of gravity of the body.

FIGURE

1.5.

Anatomical position.

Skeletal Mrch.inisms

B FIGURE

1.6.

Medial planes of the body.

A. sagittal plane.

FIGURE

1.6.

B, frontal plane.

FIGURE

1.6.

C. transverse plane.

FIGURE L.

Major axes of the shoulder girdle as seen from above.

1.7.

retraction

of the

shoulder girdle;

T,

transverse axis for elevation

longitudinal axis for the limited rotational

axis at the acromial J. C.

ton

B..

end

and Smith. C

Company

movements

V. vertical axis for

of the clavicle.

VA

.

vertical axis,

of clavicle for scapular motion. (Joint axes in Figures

G..

1953.

In

Morris'

Human Anatomy,

protraction and

and depression of the shoulder

edited by

J. P.

17

to 1.15

and TA

drawn

New

Schaeffer

York:

.

girdle;

transverse

after Grant.

The

Blakis-

)

V

T-F

FIGURE

1.8.

Axes

for

movements

axis in the frontal plane for

at the shoulder joint. A. from the anatomical position

movements

of flexion

and extension; T-S. transverse axis

in

T-F, transverse

the sagittal plane for

movements of abduction and adduction; V. vertical axis running the length of the humerus for movements of inward and outward rotation of the arm. B, same axes but arm is abducted 90°. Note altered positions of T-F axes.

Skeletal

FIGURE

1.9.

ments are

flexion

Transverse Axis through the elbow

joint.

Mechanisms

Move-

and extension

B FIGURE

1.11.

transverse

extension

of the

metacarpals metacarpal

Axes

phalangeal for

axes

of the

axes

fingers:

ab-

not

and

V.

the

The

fingers;

plane

frontal

volar-dorsal

adduction

shown.

A.

fingers.

in

of

axis

the

B.

thumb.

for

flexion

through the

fingers

First

metatarsophalangeal

terphalangeal axes of the foot are similar.

T-F,

and distal

carpo-

and

in-

FIGURE 1.10. Axes of the forearm and wrist. L. long axis of the forearm for pronation and supination; T-F, compromise transverse axis in the frontal plane for wrist flexion

dorsal wrist.

axis

for

radial

and

ulnar

and extension; V. volarof the hand at the

deviation

10

UNDERSTANDING THE

FIGURE

1.12.

Axes

for

SCIENTIFIC BASES OF

movements

HUMAN MOVEMENT

at the hip joint. T-F. transverse axis

in

the frontal plane for flexion

and extension of the thigh. T-S, transverse axis in the sagittal plane for ab- and adduction of the thigh; V. vertical axis for inward and outward (lateral and medial) rotation of the thigh. Note location of axis through the lower limb

in B.

FIGURE in

the

1.14.

frontal

Axes plane

of the ankle

and

foot.

T-F. transverse axis

passing through both malleoli and the talar

head The only movements are dorsi- and plantar flexion. A and are compromise axes for inversion and eversion at the intertarsal through the talo-calcaneal joints. A through Chopart's joint and and talonavicular joints

FIGURE 1 axis m the axis

13.

Axes through the knee

frontal

plane

around which the

tibia

for

flexion

joint.

T-F.

and extension;

can rotate when the knee

is

transverse V.

vertical

flexed

FIGURE movement

1.15. of

Joint the

axes of the axial skeleton. A. axes on the vertebrae. 0-S. oblique axis

skull

the sagittal plane for lateral flexion of the head; axis in the frontal plane for dorsiV. vertical axis for

FIGURE

head

rotation, right

in

transverse

and ventriflexion of the head; and left.

1.15. B. axes for movement of one vertebra on the adjacent one below; 1. cervical. 2, thoracic. 3. lumbar vertebrae 0-S. oblique axis in the sagittal plane for movements of rotation combined with abduction; T-F. transverse axis in the frontal plane for movements of flexion and extension; T-S. transverse axis in the sagittal plane for movements of lateral flexion (abduction) right and left

11

T-F.

for

12

UNDERSTANDING THE

SCIENTIFIC BASES OF

HUMAN MOVEMENT ext

limb

auditory meatus

W

bud

pericardial liver bu

pericardial

swelling

FIGURE

1.16.

Langman, J, Williams

&

The human embryo. A. at 5 weeks: B. at 6 weeks; C. at 7 weeks: D. at 8 weeks. (From 1969. Medical Embryology Human Development. Normal and Abnormal Baltimore: The

Wilkins

Company

)

extremities rotate laterad so that the elbow points backward, the thumbs are outward, and the ventral and surfaces face forward. The lower extremity rotates mediad so that the knees point forward, the great toes are inward, and the ventral surfaces face backward, as do the soles of the feet (volar surface) when one is standing on the toes. The proximal surface of the limbs retains its embryological orientation in the regions of the axillae and groin. Because of this situation a large portion of the upper part of the thigh still presents some ventral surface on the anterior aspect, therefore a movement of the lower extremity forward and upward at the hip joint is an approximation of ventral surfaces and conforms to the definition of flexion (Fig. 1.17.B). Because the rotation is complete at the knee, flexion of this joint also meets the anatomical definition. Shoulder flexion and extension are not easily reconciled to either definition, so these movements are correlated with the direction of the movements at the hip: flexion of the arm at the shoulder is defined as movement forward and upward in the sagittal plane and extension as movement downward and backward in the same plane (Fig. 1.18). Flexion of the elbow, wrist, fingers, toes, and vertebral volar (palmar)

all conform to both concepts: i.e.. the anatomiconcept of approximating the ventral or volar surfaces and the popular concept of decreasing the angle between the body segments. Extension of these same joints is. of course, movement in the opposite direction. However, this agreement between anatomical and popular definitions breaks down when the concepts on which the definitions are based are applied to the movements occurring at the ankle joint. Study of Figure 1.19. A illustrates the divergence. Decrease of the angle between the foot and the leg, anatomical extension, is popularly called ankle flexion: "pointing the toes." anatomical flexion, is popularly known as ankle extension. If Figure 1.19.B is consulted one sees how the anatomical terminology is arrived at. Because of this paradoxical situation the term dorsiflexion has been adopted for anatomical extension/ popular flexion, and plantar flexion is the term applied for anatomical flexion/popular extension.

column cal

Abduction- Adduction This pair of movements takes place in the frontal plane and occurs at biaxial (metacarpophalangeal and

Skeletal

FIGURE

1.17.

A, hip extension.

FIGURE

1.17.

Mechanisms

B, hip flexion.

FIGURE

1.18.

A, shoulder extension.

FIGURE

FIGURE

1.19.

B,

lateral

1.18.

1.19. flexion.

Movements

the ankle joint. A, dorsi- and and Recording Joint Motion. of Orthopaedic Surgeons.)

(From

American Academy

1.

anterior surface of leg.

lateral

homo-

1' 2. Instep or dorsum of the foot, homologous to the back of the hand. 2'. 3, Sole of foot, homologous to palm of the hand. 3'. 4, Back of leg. homologous ventral surface of forearm. 4'. 5. Popliteal fossa homologous to the cubital fossa

logous to back of forearm.

FIGURE

shoulder flexion.

aspect of right lower extremity: C.

aspect of right upper extremity

plantar

B,

at

Measuring

of the forearm. 5'.

14

Skeletal

FIGURE

1.20.

Abduction-adduction. A, abduction

metatarsophalangeal) joints and at multiaxial (shoulder, and first carpometacarpal) joints. Abduction of the fingers and toes is movement away from the middle digit, while adduction is movement toward that digit. Abduction at a ball and socket joint (shoulder or hipl is movement of the limb upward and away from the midline (Fig. 1.20.A). At the glenohumeral joint the arm can be raised only 90° before the greater tuberosity of the humerus contacts the acromion process. Further abduction is accomplished by upward rotation of the glenoid fossa of the scapula (Fig. 1.21).* As a result the total range of abduction of the upper extremity can be as much as 180° In adduction of either the upper or lower extremity the limb may be drawn across the midline of the body (Fig. 1.20.B). hip.

.

Circumduction This *

may

occur at any biaxial or multiaxial joint and

is

See also discussion under "Movements of the Scapula," below.

at

both shoulders and at the right

Mechanisms

hip.

a combination of flexion-abduction-extension-adduction or the reverse, and it may involve rotation of the limb concerned. The extremity travels in a cone-shaped path with the apex at the fulcrum of the joint at which the movement originates (Fig. 1.22).

Horizontal Flexion-Extension Horizontal flexion is performed by the upper extremity from a position of abduction; the motion of the extremity is in a horizontal plane and the limb is carried forward across the front of the body. Horizontal extension is similar movement in the opposite direction (Fig. 1.23). There has been a tendency among therapists to use the term horizontal adduction for horizontal flexion because the arm is moved across the midline of the body. It should be noted, however, that those horizontal movements occur around the same axis through the head of the humerus as do flexion and extension of the

16

UNDERSTANDING THE

SCIENTIFIC BASES OF

HUMAN MOVEMENT

180

180

RANGE OF TRUE GLENOHUMERAL MOTION

180*

COMBINED GLENOHUMERAL AND SCAPUL0TH0RACIC MOTION

FIGURE of the

1.21.

Recording

FIGURE

1.20.

B.

adduction at both shoulders and at the right hip

Glenohumeral motion. Note the upward

Surgeons.)

Joint

Motion.

rotation

completed (From Measuring and American Academy of Orthopaedic

scapula as abduction

is

Skeletal Mcch.
•»

a

W mutw If'

w * »» S**tL,caEH

IP

at

"

.

!'"'

^^

K^afA! .

53

Pill _

IiKa" jrJ*

_J«i."H_

r^ f iirr

tcSM «»

g

-l

*LrkS jfciV.i. a^M? -'AIM

Bar

Vav

90

E>

'

!7JBP 1 IN Jlpr' 1

\vf

»*.* ai-a

"IE

J*» *Li

iSMI

:crajj

.»•»

BEt*' !

FIGURE

a

«Lii **&

iirr

:'i>
'v

FIGURE

4.5. Relation

between time,

= 128 8

ft/sec

velocities, length of drop,

and the acceleration due to

gravity.

53

UNDERSTANDING THE

54

SCIENTIFIC BASES OF

HUMAN MOVEMENT

ROTATIONAL MOVEMENT Rotation of a body about a fixed point or axis causes any portion of that body to travel in a circular path as it undergoes angular displacement. As this occurs when any body segment moves on another (rotatory motion), it behooves the kinesiologist to have some

mass about the axis of rotation. The average of the sum of the perpendicular distances (^ r) is also called the radius of gyration k, so that:

understanding of the laws governing such action.

The

Torque or

Moment

of Force

it is the size of the moment that increases or decreases the angular velocity of a body and so produces acceleration or deceleration of the

in a straight line:

its

moment

Eq. 4.9

is

defined as the distance

of inertia.

The moment

of inertia

is

the measure of the resist-

ance of a body at rest to rotatory motion or, if rotating, to change its state of rotation. As torque exerts a turning force on a body, it is analogous to the force in the equation F = ma. Therefore it can be stated that:

movement.

= Io

Angular Velocity and Acceleration

As a body rotates about a fixed axis, each and every particle of that body travels in an arc and moves through the same angular displacement 6 (the Greek letter theta) in the same amount of time. The body may rotate with a fixed speed or constant velocity, or the rotatory speed may increase or decrease. The same principles as those used in determining linear velocities

radius of gyration itself

2

from the axis of rotation of a point at which the total mass of a body might be concentrated without changing

In dealing with rotational movements or motion, the moment or torque (see Chapter 3 under "Torque") fulfills the same function that force performs for motion

rotational

mk

/ =

and accelerations

are applied in calculating the

Eq. 4.10

Torque is equal to the product of the moment of inertia and the angular acceleration and so is analogous to force F as / is analogous to mass m and a to acceleration a in linear motion.

Angular Motion Linear

momentum P equals the momentum A

velocity, so angular

moment

of inertia

product of mass and the product of the

is

and the angular velocity:

angular counterparts:

A

= Iw

Eq. 4.11

Angular

Linear s

Eq. 4.6 I

IF THE SYSTEM IS CLOSED AND THERE IS NO EXTERNAL TORQUE ACTING ON THE SYSTEM, THE TOTAL ANGULAR MOMENTUM REMAINS UNCHANGED, EVEN THOUGH THE MOMENT OF INERTIA MAY BE ALTERED This is the law of conservation of angular momentum. Note that while the quantity of mass remains unchanged, r or k (and so r 2 or .

h

Angular velocity

uj

-

to

/i

(the

analog of linear velocity

a

(the

Greek

-

Eq. 4.7 to

Greek letter omega) is the u, and angular acceleration

letter alpha)

is

the analog of linear ac-

celeration a.

Moment

of Inertia

discussing Newton's First Law (see Chapter 6) it is mentioned that the terms mass and inertia could be used interchangeably. In rotational movement the moment of inertia / is the analog of mass in linear motion. If a body is divided into a large number of very small parts, and a typical particle has a mass which is at a perpendicular distance r from the axis of rotation, then its contribution to the moment of inertia / is mr 2 Under these conditions the moment of inertia will be equal to the sum of all such contributions from all portions of the body, and: In

m

.

k 2 ) can be changed. A man stands on a frictionless turntable holding a 15-pound barbell in each hand; his arms, with extended elbows, are abducted 90° as in Figure 4.6. He is started spinning at a rate of one rps (revolution per second, 360°/sec). While he is spinning at this rate he pulls the barbells into his shoulders so that they are only 6 inches from his axis of rotation. At the start of the experiment the barbells were held at 3 feet from the rotational axis. Under the circumstances any change in the moment of inertia of his body will be so extremely small it can be ignored. Therefore as / =

at

1

mr 2

(from Eq.

4.8)

rps /

=

15 1b

X

3

ft

2

9 I

The distance

=

X mr2

Eq. 4.8

135 lb/ft

any particular portion is constant only for one given axis. As a body or body segment can rotate about different axes, r will change with each change of

A =

Thus the moment of inertia is not a fixed constant body but is dependent on the distribution of the

A =

axis. for a

r for

(from Eq. 4.11)

Im 135 lb/ft

X

1 rpsi

Motion

FIGURE

When

4.6.

Subject on frictionless turntable.

the hands draw the weights close to the body, is unchanged but r has shrunk to 0.5 foot.

TABLE

Analogies Between Linear and Rotational Motion

4.1.

mass

their

6i>

Linear motion

Angular motion

Then Distance

A =

15 1b

X

0.5 ft 2

X

Acceleration a

9

as

A

is

unchanged 135 lb/ft

rps,

=

=

3.75 lb/ft

135 lb/ft

X

The

greater the

mass

=

X

mass

drawn

Moment

of inertia /

Force

F

Moment

or torque of a couple

mentum

at a greater distance

man

will rotate.

closer to the axis, r or k

is

2

When

the

with time

mv

from the axis

,

—

in a

closed system in-

'

Work done

tB

Torque

rate

=

angular

mv

- t of conservation of momen-

momentum A

Kinetic energy

*

Law

r

lai

2

= Iu

2

of change

momentum ijOJi

t,

Law tum

= la

t

Angular

Fs Force = rate of change of mo-

decreased so that the moment of inertia becomes smaller and, as the angular momentum remains unchanged, the increase in the angular velocity must balance the equation. is

m

Work done

36

of rotation, the slower the

Mass

F = ma Momentum P = mv Kinetic energy Vi mv

rp< :

3.75 lb/ft rps,

Angle 6 Angular velocity w Angular acceleration a

s

Velocity v

rps,

—

I

—

*

of

with time

t*J

1

of conservation of angular

momentum

in

a closed sys-

volving no resultant or out-

tem with no external torque

side forces

acting on the system

UNDERSTANDING THE

56

SCIENTIFIC BASES OF

of these principles forms the basis of control in from figure skating to tumbling to rebound tumbling to more elaborate competitive dives. In the air the diver's (or tumbler's) body rotates about its cen-

Use

many

skills,

and the performer can regulate the speed of his rotation by the posture or postures he assumes. If his position is a tuck, the radius of gyration mentioned earlier will be quite short, making the moment of inertia comparatively small, and he will spin rapidly about a transverse axis in the frontal plane. Thus a good diver can complete one and a half somersaults from the low (1 meter/3-foot) diving board. If the performer feels that ter of gravity

spinning too fast, he can decrease his angular velocity by opening his tuck slightly, increasing the radius and thus his moment of inertia. On the other hand, if he feels that, at his present speed of rotation he will not complete the number of somersaults that he planned,

he

is

HUMAN MOVEMENT he can increase his rate of spin by tightening up his tuck (or pike). In the twisting dives or leaps or the skating spins, the about an axis running lengthwise of

performer rotates his body, and he hence his angular the position of his

controls his

moment

of inertia,

and by

velocity, during the twist or spin

just as the man did on the fricthese techniques, many a diver has "saved" a dive that started to go wrong.

arms

tionless turntable.

By

Analogies between Linear and Rotational Motion

Throughout the preceding section the analogies occurring between rotational and linear motion have been indicated. Table 4.1 summarizes these analogies and serves as a reminder to the reader of the many important concepts and laws that have been presented.

CIRCULAR MOTION This discussion is concerned with a body moving in a circular path, whether it is a ball on a string or a space capsule in orbit. It can also be applied to the movement of a distal end of a body kinematic chain moving in an arc about an axis through a proximal joint. Velocity

it

the body

is

= rad/sec x

1

rad

= =

moving at uniform speed,

r

Eq. 4.15

As 27rrads = 360°,

and Distance

in a given time an arc a which subtends an angle 6 whose sides are formed by the radii rand r' (Fig. 4.7). The linear distance s that the body (or distal end of a body segment) travels during the time t is determined by equating two ratios: (1) that of d to 360° (the number If

t,

LV

360°

57.2957°

Then

travels through

of degrees in a circle)

and

(2)

that of the arc s to the

circumference of the circle of which

it is

an

arc:

s 27T7-

360°

360°

and the

linear velocity of the

LV

3(i0°

X

2wr

body

X

Eq.4.12

will be:

2nr

LV

degrees/sec

Xr

Angular Acceleration Newton's First Law of inertia states that "a body will continue motion at a constant velocity in a straight line unless acted on by an outside force." As the body under discussion is moving in a circle at a constant velocity, there must be a. force of some sort that keeps it in the circular path. But a force F acting on a body implies an acceleration: F = ma. Velocity is a vector quantity and so has magnitude and direction in a straight line. The velocity at any one point on the circular path is in a straight line tangent to that path and perpendicular to the radius at that point (Fig. 4.8). As the body continues to travel in a

Eq. 4-13

The

solution of such problems can be simplified by converting 6 to degrees per second (0/t) and dividing the result by 360° to arrive at revolutions per second (rps). Using LV for linear velocity, the equation then first

becomes LV

= rps x

2tt r

Eq. 4.14

On the other hand, even more arithmetic can be avoided by converting degrees per second to radians per second (rad/sec) and then multiplying by the radius of the arc so that

Eq. 4.16

57.2957

FIGURE

4.7.

Movement

in a circle.

57

Motion

FIGURE

4.8.

acceleration

circle,

ing

A

to C. graphic

See

method

for deriving

the direction of the velocity

and each new vector

is

constantly chang-

also perpendicular to the

is

radius at each new point (Fig. 4.8. A). If velocity vectors drawn in the same directions as i n Figure 4.8.A with a common origin X. with vector as v and vector as (.•'. vector YZ is the change in velocity. Ac. during are

XY

XT.

(Fig. 4.S.C).

This demonstrates that the change in

velocity, or the acceleration that keeps the

body

in the

toward the center of Figure 4.8.D. This radial or

circular path, travels along a radius

the circle as illustrated in centripetal acceleration a R

the factor that keeps the

is

body moving in its circular path. When a body is moving in a circular path at a constant speed v and with a constant angular velocity ta, it has an acceleration which always points toward the center of the circle and whose magnitude is: aR =

where angular velocity

where velocity

is

a;

2

physics text

Eq. 4.17

r

in radians per second, or

*

=

2™ V

is

any college

interested.)

When motion vectors

velocity

in a circular will

have

path

is

not constant, the

different

lengths,

as

in

Figure 4.9.A. If the vectors of u, and c 2 are drawn as before (Fig. 4.9.B) with the same angle 6 between them, as in Figure 4.8. the vector Ac represents the resultant change in velocity. In Figure 4.9. C this change of velocity has been resolved into two components, the change in radial velocity au r which results from the change in direction, and the change in tangential velocity Ac T which results from the change in the magnitude of the velocity. The tangential acceleration aT

is:

Qt =

Sir Eq. 4.20 At

where tangential acceleration is equal to the change in tangential velocity divided by the change in time, and it will be in the direction of the changing velocity (Fig.

As a R and a T form two sides of a right triangle, the actual acceleration is:

Eq 418

U^r

-

in degrees per second,

aR

and

=-

man Eq. 4.19

Tangential acceleration as centrifugal

layman terminology forced to

is

the student

Tangential Acceleration*

*

where v*

if

4.9.D). is

/ a

D. direction of the

(Sv).

tions of these equations are available in

XZ

the time that point P has moved to P' (Fig. 4.8. B). If we extend the tangent line for c' backward until it intersects the^ tangent for c at point X'. and then d raw vecXZ'. and YZ' similar to XY, XZ. and YZ. tors the direction of Y Z' or Ac will be parallel to the line

OX'

instantaneous acceleration

text for discussion

linear velocity.) (The

mathematical deriva-

move

ton's First

may be more

acceleration. results

6.

recognizable to the lay-

the "centrifugal force" of

from the inertia of the body which

in a circular path.

Law. Chapter

N.B.:

See the definition of

inertia.

is

New-

UNDERSTANDING THE

58

SCIENTIFIC BASES OF

HUMAN MOVEMENT

B

A

FIGURE C and D

4.9.

Graphic method for deriving instantaneous acceleration. a« and a r

illustrate the directions of

a = y/a T 2

A and

B. tangential acceleration;

.

also

+ aH 2

when

aR

=

Eq. 4.23

M

Avt

Aar

Eq.4.21 At

and

Thus when a a« =

Eq. 4.22

situation is one of varying speed, a R will change from one point to another, and a T will also vary from one instance to another.

BIBLIOGRAPHY Atkins, K. R., 1966. Physics.

M.

Broer,

W.

R.,

New

York: John Wiley

1966. Efficiency of

Saunders Company. Cooper, J. M., and Glassow, R.

&

Resnick, R., and Halliday, D., 1960. Physics— for Students of Science

Sons, Inc.

Human Movement.

Philadelphia:

B.

C. V.

B., 1968. Kinesiology. St. Louis:

The

Mosby Company.

Dyson, G. H.

London

G., 1964.

Mechanics of Athletics. London: University of

Northwestern University. Amer.

J.

Phys. Med. 46:334.

H., and Frederick, D., 1964. Engineering Mechanics, and Dynamics. New York: The Ronald Press. Rasch, P. J., and Burke, R. K., 1967. Kinesiology and Applied Ana-

D.

Statics

tomy. Philadelphia: Lea

I.

New York: John Wiley & Sons, Inc. Human Motion. New York: Appleton-

G., 1963. Analysis of

Century-Crofts.

C Thomas,

Charles

Timoshenko

Press, Ltd.

ceedings of Exploratory and Analytical Survey of Therapeutic Ex-

Pletta,

M.

Steindler, A. 1964. Kinesiology of the

O'Connell, A. L. 1966. Ingredients of coordinate movement. In Proercise at

and Engineering, Part Scott,

&

Febiger.

Ed.

New

4.

S.,

Wells,

and Young, D.

and

Tricker, B.

New York: American

K.

Body. Springfield,

111.:

H..

1956.

Engineering Mechanics,

York: McGraw-Hill Book Company.

Tricker, R. A. R.,

ments.

Human

Publisher.

F.,

1966.

J. K.,

1967.

The Science of MoveCompany, Inc.

Elsevier Publishing

Kinesiology.

Philadelphia:

W.

B.

Saunders

Company. Williams, M., and Lissner, H. R., 1962. Biomechanics of

Motion. Philadelphia: W. B. Saunders Company.

Human

SECTION ONE

CHAPTER

KINEMATICS

Kinematics of

Human Movement movements that were executed by the performer, but such generic terminology is insufficient for the kinesiologist, who should use those words or phrases most widely understood and applicable. Kinematics, as stated above, is a descriptive art.

Kinematics is concerned with the geometry and temporal qualities of motion. It is descriptive and is thus concerned with displacements, velocities, and accelerations of a body or body parts. There is no concern for the forces involved with the production of any of these phenomena other than that of their identi-

Adjectives are needed in order to describe anything, and the adjectives and verbs used in kinematics of

fication.

The

movement analysis movements and/or

ex-football player or football fan will follow the

sequence of events after the ball has been snapped and. as the action unfolds, he will know almost immediately what play is being made long before it is completed. If asked, he could diagram the play, indicating the line shifts and moves of each player in technical football-ese such that any coach could present the play to his squad and they could reproduce it. But this onlooker cannot do the same for a single skill, such as a punt or a tackle, in technical anatomical terms such that the kinesiologist ignorant of football could reconstruct the exact movements occurring at each of the performer's joints and the forces that produced them. Being able to describe a skilled performance, even an unskilled one. requires a precise vocabulary as well as a complete understanding of the skill. To say that the ballet dancer performed an entrechat, a fencer made a riposte in tierce, or that a gymnast performed a dislocate on the rings would have instant meaning only

to

the

initiate

of the

5

discipline.

would immediatelv visualize the entire

The

are terms used to describe joint resulting

body

positions. Probably

the most widely accepted terminology is that used in anatomy and medicine. The American Academy of Orthopaedic Surgeons has published a booklet which they hope will lead to a standardized terminology in the area (see Chapter 1 under "Joint Movements"). Kinesiologists and other students of human movement are often dissatisfied with these standard anatomical terms inasmuch as the latter lack precision. This is particularly true of the engineers designing space suits who have built their own vocabulary of descriptive terms (Fig. 5.1). This latter terminology, combined with suitable angular measures, is as precise as the angles and directions measured.

Anatomists and kinesiologists may in time come to adopt this or a similar system of notation. However, in the

meantime

it

is

still

possible, using the

more

familiar anatomical terminology (see Chapter 1 under "Joint Movements"), to give an adequate description

initiate

of

any sequence of movements.

series of bodilv

MOVEMENT ANALYSIS Any kinematic

analysis

scription of the bodily

should

with a

de-

5.

movements which take place

6.

during the performance of the the following data:

skill

start

if

any: gravity or eccentric

muscular contraction (muscle being lengthened while exerting tension).

and should include

Where

7.

this force

is

applied.*

Many

students have been taught to think of the skeleton as a system of levers acted on by muscles,

1. The name of the movement and the time or frame number at which the movement starts and finishes. 2. The joint at which the movement occurs. 3. The lever, i.e.. the segment or segments making up the kinematic chain being moved as a result of the joint action. 4. The force producing the movement, muscular shortening (i.e.,

isotonic or concentric contraction), gravity, or

Where this force is applied.* The force resisting the movement,

*

Muscular tension, either while

a

muscle

is

shortening (isotonic

or concentric contraction) or being lengthened by an outside force (eccentric

or

lengthening

contraction),

is

being applied to the moving part or segment,

some other imposed

is

force.

considered

as

to the lever. Gravity

always considered as being applied at the center of gravity of the

lever.

59

always i.e.,

60

UNDERSTANDING THE

SCIENTIFIC BASES OF

HUMAN MOVEMENT

UPWARD

Si\

V^

(-1)

s=o° F=0°

LEFT

SIDEWARD

(-Y)

S= 1 F=

270°

T= 270°

-PBa BBBB-4 BBB

BBBI IBBBB' BBBBI IBB BBI IBBBI IBBBI IBB iJBBI IBBBI iBBBBi

BBBBbTI

BBBr JBBB vmw * aaaa |£pbb iai

r

BBaaapaaa BBP**' BBP

rill jbbB .-a*

r'^ IBBBB ^aaa in

! MM

iiiibi

•'< IHI^-.

*bb| «uiH Bar* BBBf BBBBH .

IBBr^iflB

MM

•"--

m^n-m .BBBB

--BB

r.4 JBfl

-* IBB

BBBBI BBBr

'

nn'.burc bod) rotation in film enalysis

American Association and Recreation, undated

at research section of the ication,

bouse,

I

E.,

and Cooper, J

M.

tor

Paper presented

of

Health, Physical

I

I960. Kinesiology

Selected

Iniversitj of

Reuschlein,

C

V

Moabj

A.

1958

1.

Electromyographic

movements

muscles during

of the tree

ti>ot

stud)

of

certain

IV

I...

L962,

ai

Body

An

the forward SOmerSOUlt

prepared 0*Connell,

Symmetrical Michigan on

the University

Human Movement Actions.

1969. Ixibomton.

Manual

ami during standing.

for Kinesiology.

Boston

Roebuck, -enied

).

ai

A..

Jr.,

L966.

"i

Wisconsin.

Kinesiology

in

engineering.

the Kinesiology Council, Convention

nhoef, S

C

.

1962.

An

Analysis of the Kinematics and Kinetics

Paper pre

American and Recreation, March of

the

1966

Roebuck.

I'niversiiy Bookstore. ;

>!

leg

Association tor Health. Physical Education, 1...

Doctoral

analysis of die speed of rotation for (he trampoline. Unpublished paper

I

onell, A.

75

-I.

A..

Jr..

bility evaluations.

19(iS.

Hum.

A system Factors

10.

l

nutation for space suit

mo

SECTION TWO KINETICS

The Laws

CHAPTKK

of

6

Motion and Energy NEWTON'S LAWS OF MOTION

The bases for the modern study of motion were laid by Sir Isaac Newton in the 17th century when he formulated his three laws of motion. The early translation from the original Latin is in language which is, to us,

AT REST, OR IN A STATE OF UNIFORM MOTION IN A STRAIGHT LINE UNLESS ACTED ON BY AN APPLIED FORCE Today this has a far more explicit meaning to the average individual than it did a few years ago before the space program. Apollo moon rockets remained at rest on their launching pads at Cape Kennedy until ignition of their rocket fuel. They continued to accelerate as long as the "burn" lasted arid, when the rocket motors cut off, they had reached escape velocity. The command capsule continued in the same straight line until (1) it was acted on by a short burst of rocket fire for a course cor.

archaic:

rather

Law. "Every body persists in its state of rest uniform motion in a straight line unless it is compelled to change that state by forces impressed on it." Second Law. "The change of motion is proportional to the motive power impressed, and is made in the direction of the right (straight) line in which the force is impressed." Third Law. "To even action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts" Resnick and Halliday). These original statements read by modern eyes seem First

or of

rection,

-

esoteric

and the meaning

difficult to grasp.

During the years since Newton's lifetime, these statements have been worked over and reworded to make their meaning clear to each generation of students. The third law above is very simply expressed today as follows: TO ENTRY ACTION THERE IS AN EQUAL AND OPPOSITE reaction Thus forces work in pairs. When a man's foot presses on the ground as he walks, the ground pushes back with an equal but opposite force. The ground is acted upon as the foot strikes it, it reacts with an equal force in the opposite direction, and the

change in its state of movement or of rest, i.e., inertia, as it had on earth. It will take the same amount of force to put it in motion (accelerate it), to stop it moving, or to change its direction as it did on earth. The weightlessness of space does not change this, nor does the one-sixth gravity of the moon. The only difference is the absence of or the decrease in the force pulling an object toward the center of a planet or

.

individual

moves

in that

direction (Fig. 6.1).

into the pull of the gravitational

This law is also described as the law of inertia because it describes the quality of needing a force to change the state of rest or of motion of a body, and it implies as a consequence a resistance to such a change. The concept of inertia is sometimes applied interchangeably with that of mass. A body in space or on the moon will have the same mass or resistance to

I

somewhat

and (2) it came moon.

force of the

As the

satellite, gravity.

running long jumper takes off. his take-off foot thrusts against the board, the board pushes back at him, and he is propelled through the air by a force equal to the thrust of his take-off foot.

One of the problems met with in the space program was this same law of action and reaction. When there is no large mass such as the earth or moon to react against the thrust of a foot or a hand, the body itself responds by turning or moving in the opposite direction. This can be illustrated in the laboratory by someone standing on a freely movable turntable with one arm abducted 90°. Regardless of whether he swings the arm to the left or right in horizontal flexion or extension, the

table reacts

The

first

by turning him law

is

FIGURE

in the opposite direction.

cited today as follows: a

6.1.

Action and reaction. The foot lands and exerts pres-

sure on the ground: the ground reacts with an equal but opposite

body remains

force

77

UNDERSTANDING THE

78

SCIENTIFIC BASES OF

HUMAN MOVEMENT

The first law leads into the second, which is concerned with force, mass, and acceleration. Actually, this law as Newton stated it uses the term motion as we use momentum today: as the product of mass and velocity, mv (Resnik and Halliday; Tricker and Tricker) and refers to the change in momentum. Taking this law as Newton stated it, the equation is: mi', -

—

0'i

F =

h vi

— -

—

more familiar one:

t'o)

Eq. 6.2 to

vo

Eq. 6.3

h-k But

- v

(u,

)/(t

,

-

t

celeration (Eq. 4.2).

mv.

)

is the formula Hence:

for

determining ac-

Eq. 6.1

M

F

m

is the velocity at time zero, u, is the mass, l» the velocity at time!, At is the elapsed time between t and ti, and F is the (vector) sum of all of the forces acting on the body. This equation can be treated alge-

where

braically as follows to achieve the

is

and thereby

=

ma

Eq. 6.4

achieved the more modern statement of a body of mass, m. has an acceleration. a, THE FORCE ACTING ON IT IS DEFINED AS THE PRODUCT OF ITS MASS AND ACCELERATION. the law that:

is

if

MOMENTUM Momentum

the product of the mass of a body and as such it is a vector quantity, a quantity of motion possessed by a body: P = mv. If mass is in pounds and velocity in feet per second, momentum P is expressed as pound feet per second; similarly if mass is in kilograms and velocity in meters per second, P is expressed as kilogram meters per secits

velocity,

is

and

ond.

Application of Force and Changes in

an interval as possible. The resulting tangential

and

radial accelerations* are responsible for the strained ligaments, damaged bursae. and possibly torn muscles that pitchers occasionally suffer. It should also be remembered that a small force applied for a long enough period of time can achieve results comparable to a far larger force applied for a brief interval only. As a consequence some authors write the equation:

F

Momentum

The equation of Newton's original statement of his second law: mv

brief

,

Eq

6 5

and speak of F x in

Eq. 6.6

as the impulse that causes the change

momentum.

Receiving a Force

At

involving the time rate of change of momentum can be used for determining the force involved in striking a ball if the necessary data are available: the duration of the contact of the implement with the ball, the velocities of the ball before and after it was struck, and the mass of the ball. It is only in recent years, since ultra-high speed photography has made the acquisition of such data possible, that the use of this formula has become practicable in sports analysis. Unfortunately, as the speed of the struck ball also includes that contributed by the elastic qualities of the ball and possibly those of the striking implement as well, the estimate of the force involved will not be entirely accurate. However, in looking at the above equation, it becomes obvious that the smaller the amount of time through which the force is applied, the greater will be the change in momentum. Conversely, if the same force is applied over a longer period of time, there must be a decrease in the change of momentum per unit of time. In projectile skills, then, when force may be applied over a relatively long period of time as in throwing, greater force is needed to achieve a high velocity. This explains why baseball pitchers may hurt their arms when they attempt to achieve maximal velocity of the ball by applying a maximal amount of force over as

t

x At

When

considering Equations 6.5 and 6.6 from the op-

posite viewpoint, that of catching a fast ball or landing

from a high jump, the greater the increase of time

consumed

in the

in catching the ball or for

amount

completing

the landing, the less will be the force felt by the catcher jumper at any one instant while his body is receiving the force and the less will be the damage to body tissues. In catching a fast ball, even with a catching mitt, the hands are stretched forward to meet the ball and are drawn toward the body as the ball contacts the glove. All of this increases the time during which the momentum of the ball is absorbed by the catcher. Landing pits for long and high jumps were originally filled with sand, then with sawdust, wood shavings, or tanbark. Today most of them are filled with chunks or sheets of foam rubber. The jumper, if he lands on his feet (this is chiefly in the case of the long jump), dorsiflexes at the ankles, flexes the knees, hips, and spine, and may go into a roll. The high jumper or pole vaulter may land on his shoulders and back or with a roll or both. In any case the force generated by his momentum is spread over as large an area of his body as possible so that any one area is impacted by a comparatively small amount of force. This type of landing also increases the duraor

*

See section on "Circular Motion." Chapter

4.

I

which in turn decreases the per unit of time so thai there is less force for the body to absorb. The landing medium absorbs some of the momentum, as the particles sand, shavings, tanbark, or foam rubber fly in all actions when some of the jumper's momentum is transferred to the landing medium. The ultimate retion of the landing process,

change of

momentum

by the jumper at any one instant any one area of bis body is deereased to the point where there is little it' any damage to the tissues. The

sult is that the force felt

of

'20

per second.

feel

79

Motion and Entrg)

is

The quarterback

sidestep

at

as a single unit after the tackle

react

resulting

momentum

the

of

is

made

so

tackle quarterback

I

he

sys

determined as follows. 1. Determine and X and Y momenta com ponents (Xmv and Ymv) of the two bodies: tern

is

Problem

^Xmu

m .r,

cos 0°

200 lb

x

i

m,D, cos 60°

20 ft/sec

x

1

]£

Ymv

=

6400

=

m,i',sin0 o +

as sin 0° =

160

i

1000 lb ft/sec + 2400 lb

".

The above

made

of

an angle of 60°. The quarter back is still moving at 30 feel per second and weighs 160 pounds as against the 200 pounds of the tackle (Kig. 6.2). As the tackle grabs the quarterback they that the tackle

boxer rolls with the punch tor the same purpose: to decrease the force with which he is hit by increasing the time of contact, which in turn decreases the change in momentum per unit of time. It is the abrupt change in momentum from a large quantity to little or none that is harmful to the body (e.g., an automobile collision). All of this information is implicit in Newton's First Law, and in the equation F \ f = mr, - mv n-ation of

jws

II)

30 ft/sec

•

0.5

ft /sec

lb ft/sec

m

2

u 2 sin60°

0,

Momentum

illustrations all

have spoken of

X Ymv

momentum

time as is conveniently possible, or that some of the momentum has been transferred under each of the circumstances. This slow deceleration, as has been pointed out, is accomplished so that the amount of force applied to the body at any given instant is as small as it can be made while the momentum is being altered. These facts imply that momentum is not lost or destroyed,

=

160 lb x 30 ft/sec x 0.8660

= 4156.8 lb ft/sec

as being "absorbed"' over as long a period of

and

it is

.

where ing

RMV

Hi,l/,

- iR|0| -

^ RMV

momenta

is

m

3

u3

.

.

.

(before) =

^ RMV

the magnitude of the

sum

after the interaction. This

2

Y.Xmv

2

+

X Ymu

!

6400 2 + 4156.8 2

RMV

not.

If one observes the interaction of two or more bodies with m.i',. m 2 v 2 m 3 v 3 ...etc in a closed system in which there are no resultant or outside forces involved and records the sum of the various momenta both before and after the interaction, he will find that:

£

Problem 2. The resulting momentum (RMV) can be obtained by use of trigonometric functions or the Pythagorean theorem.

The

resulting or final velocity, v f

,

of the system can

by dividing the resultant by the resultant mass RM.

easily be obtained

RMV

7631.4 lb ft/sec

momentum

(after)

of the result-

is

the

law of

conservation of momentum. At the same time it must not be forgotten that the law of conservation of mass also applies to the above interaction so that:

z

.

(before) =

X RM




Whatever happens, the momentum lost by one obbody is gained by another. The chunks of rubber in the landing pit fly around as the jumper lands, and

ject or

momentum is not transferred to the rubber eventually absorbed by the earth, whose mass is so enormous that we cannot measure any change that is caused. Similarly, when the outfielder catches a line drive, the momentum of the ball (whose mass is extremely small) is transferred to his body and then, as with the jumper, to the earth. However, consider the problem of the football quarterback earning the ball being tackled by a lineman. The tackle is approaching the quarterback with a velocity whatever is

FIGURE

Schematic of a collision on the football field. T. 200 pounds and traveling at 20 feet per second: quarterback carrying the ball weighs 160 pounds and travels 30 feet per second See text. 6.2.

tackle weighing Q. at

RM

+

=

m.}

=

200 lb

HUMAN MOVEMENT

SCIENTIFIC BASES OF

UNDERSTANDING THE

80

By

im

+

substitution

160 lb t

=

0.167 ft 10.6 ft/sec

= Vf

360

=

lb.

=

RMV

F =

0.0157 or 0.016 sec

—

mv\

mva

t

_ 360

360

=

_ -7632.0

deter-

360 lb

X

21.2 ft/sec

53

arctangent

y"l

ft/sec

lb.

0.016 sec

^ Ymv

=

-

ft

21.2 ft/sec

= -47.7

side o of a right triangle.*

6

X

0.016 sec

The direction in which the system moves is mined trigonometrically as J^ Xmv ~ side a and

~

lb

X

104 lb

ft

Assuming that the hockey players hit the boards at an angle of 60° with their slide, the impact force would

Ymv Xmv

be halved. For example, taking the line of slide as the x axis the Y component would be and

4156.8

arctangent 6400.0

Fx

=

arctangent 0.64937

=

33°

=

F

cos 60°

= 47.7 x 10' x 0.5 0'

= 23.85 x 10 4 lb

and quarterback would move onward with a velocity of 21.2 feet per second and at an angle of 33° with the direction in which the tackle was travel-

The

ft

tackle

were no other forces acting on them. However, gravity and friction with the ground acting on the tackle's body combine to pull the quarterback down and they both come to a halt very shortly. If it is assumed that such a collision occurred on ice and between hockey players, and if the two men slid as one mass almost without friction, they would travel at 21.2 feet per second until they slammed into the side wall, where their combined velocity would become zero feet per second. If as they hit their bodies were compressed approximately 2 inches (0.167 foot) by the force of impact, we can determine the force with which they hit the boards if the angle at which they hit is known. If it is assumed that this angle is 90°, the resulting force will be maximal for the circumstances. In order to determine the amount of this force, the impulse equation (Equation 6.6) is used: Ft = mv - mv The length of time of the impact is also unknown but can be determined from the available data. The amount of compression (0.167 foot) is the distance that the bodies travel during the impact. As their velocity changes from 21.2 to feet per second during this time, the average velocity v a is used in the final determination by substitution in the formula v = s/t or t = s/v. ing, if there

,

Vf

—

.

of the slide with the sides of the

The smaller the angle

rink, the less will be the force with

which the players

hit.

The effect of increasing the time of impact can be graphically illustrated if a comparison is made of the force generated when a 100-pound boy drops 3 feet from a high bar and lands (1) "hard" with little or no give in his joints so that the total give of his body (shoes, approximately 1 inch; and (2) when he lands softly on the balls of his feet and his joints all give so that the total drop of his center of gravity after his feet touch the floor is approximately 10 inches. Both of these problems can be attacked in the same manner as the previous ones. The time for a 3-foot drop is 0.43 second (see Table B.l, Appendix B) so that his velocity at the end of the drop is 13.8 feet per second (v = u + gt). The average velocity during impact from feet per second is 6.9 feet per second. The 13.8 to distance during impact is 1 inch or 0.083 foot. Then

feet, spine, etc.) is

t

=

0.083

ft

6.9 ft/sec 0.

012 sec

and as mv\

—

mva

Vi t

0-21.2

ft

/sec

Eq.6.7

=

100 lb

X

-

100 lb

X

13.8 ft/sec

0.012 sec

=

11.5

X

10 4 lb

ft

10.6 ft/sect *

The symbol ~

t

This velocity

than acceleration.

is

is

to be read "equivalent to."

negative as

it is

the velocity of deceleration rather

In the second instance, the soft landing, the time of drop, and the velocities are the same, but the distance has increased 10 times, so

Laws -

I

0.

ft

of

Motion and Energy

81

the time lapse from toe touch to deepest bend W8I

second; then

/sec

1

13801b sec

and 1623.5 lb

y

This

0.12 sec 11.5 \

On

the other hand,

10* lb

the hoy uses a greater

it'

amount

eccentric, lengthening contraction sufficient to slow

time would be increased on his feet would be desed. Suppose that the boy was photographed and

his joint actions further, the

beyond

this point

and the

force

is considerably less force than in either of the two previous examples; i.e., 1.4% of that of the hard landing and only 14% of that of the first soft landing. These exercises serve to illustrate some of the typical forces to which the human body can be subjected without sustaining serious injury, and they provide objective evidence for using controlled "give" when one is landing

from a height.

ENERGY mentioned earlier, and is often described as the capacity to do work. The breakdown of gasoline provides the energy to drive the internal combustion engine: the splitting of the high energy phosphate bond (~ (?)) of adenosine Energy per

triphosphate

se

is

iAT-r

a scalar quantity, as

~

~ ®)

(f

energy of any given mass depends solely on its velocity, while the potential energy of that mass depends solely

upon

muscular contraction. These are samples of chemical energy. Mechanical energy, which is our concern at the moment, involves both potential and kinetic energy.

position.

Conservation of Energy

provides the energy

for

its

The law that,

of conservation of mechanical energy states

in a closed

forces present, the

potential energy

equal to a constant for that system:

PE + KE

Potential Energy Potential energy (PE) is also spoken of as gravitational energy or energy of position. A 20-pound boulder balanced on the edge of a 70-foot cliff has a potential energy of 1400 foot pounds. It could do considerable

highway at the foot of the cliff. A gymnast hanging from the stationary rings (Fig. 6. 3. A) has a certain amount of potential energy. If he weighs 140 pounds and his center of gravity has been raised 0.5 feet when

damage

is

system where there are no outside sum of the kinetic energy and the

to a

he leaped upward to grasp the rings, he increased his potential energy to 210 foot pounds in relation to the floor.

= a constant

no change in the total amount of energy Thus there is no change in the sum of the potential and kinetic energies for that system at

so that there

is

in the system.

any one instant in any given situation. Suppose the gymnast in Figure 6.3 is on the flying rings and swings through an arc of 63° as in Figure 6.4. At each high point of his swing he has only potential energy, and at the lowest point he has only kinetic energy. As PE + KE at the high point will equal a constant and as PE + KE at the low point will equal the

same constant: PE

= weight

=

height

(or.

as wt. =

mass x

PE + KE

ace. of g:)

mgh

=

PE + KE

Eq. 6.8 (at the

top of the swing)

mgh

the gymnast above changes his position to the front uprise (Fig. 6.3.B) his center of gravity will rise approxIf

imately another 4 feet and his potential energy will increase further to some 650 foot pounds (4 x 140 lb) in relation to the floor.

(at

the low point of the swing)

+0

=

0+

mgh

=

4 mv

top of swing

';

mt

!

2

bottom

of swing

the gymnast's center of gravity rises 5 feet during the swing and he weighs 140 pounds, the constant is most easily calculated at the top of the swing: If

Kinetic Energy

Kinetic energy (KE) as of motion.

body

is

The amount

its

name

implies,

is

the energy

of kinetic energy in a

KE

(mg =

moving

determined by: Eq. 6.9

(Kinetic energy equals one-half the mass times the square of the velocity.) Thus the quantity of kinetic

PE

= 140 lb x 5

ft

= 700 lb

ft

KE is at the top of the swing, 700 pound feet is the constant energy for the system. As PE + KE will always equal 700 pound feet, it is a very simple matter to determine the kinetic energy for a drop of any given disAs

=

weight)

tance.

82

UNDERSTANDING THE

SCIENTIFIC BASES OF

HUMAN MOVEMENT

B FIGURE

6.3.

Gymnast on the

\ \ \ \ \ \ \ \ \

still

rings. A. hanging; B. front uprise. X. location ot center of gravity

If

\

will

there has been a drop of be found as follows:

140 lb x

1

foot the kinetic energy

PE + KE

= 700 lb

ft

KE

= 700 lb

ft

KE

= 700 lb

ft

= 560 lb

ft

1 ft

+

- 140 lb

ft

And

if one then wishes to calculate the velocity of the swing at that point:

U

1401b

560 lb

ft

560 lb

ft

560 lb

ft

=

560 lb

ft

=

8

w

2 \32.2 ft/sec 2 /

70 1b 32.2 ft/sec 2

r

2

X

32.2 lb

ft

/sec 2

70 1b

FIGURE

6.4.

Gymnast on

flying rings.

See

text

ft

X

32.2

ft

/sec 2

/

6

\

ft

stv

257l$Ti

:

MKS'

the v

•

s\stcm

ws of

Motion and Energy

a force of

V new

tons

bj.

moves

meters, or in the English system, a force of

/'

ma\ move a bodj feet. However, in the MKS amount ot work accomplished is expressed as \

uvo r

the

not.

Thus

in

any system or

flight

path where friction

is

negligible, the potential energj

at the highest poinl oi' the path will provide the constant tor any other point in

and

the path,

this potential

energy will be equal to the

mal kinetic energy of the system.

as

the

exercise

physiologist

blithely

a

bod>

pound system joules,

states,

in

"kilogram- meters."

MKS:

meter does joule newton moving a body work; similarly 1 dyne moving a body 1 centimeter does 1 erg of work, while in the English system a force foot does 1 foot pound of of 1 pound moving a body work. Power is defined as the time rate of performing work: e.g., so many joules per second or so many foot pounds per second. One horsepower is 550 foot pounds of work per second. 1

I

1

oi

1

IV rk

and Pou

When

er

F

applied so that a body is moved a work has been accomplished: work Fx. Work makes use of the same units as energy; in I

d

a force

distance

is

.v.

BIBLIOGRAPHY Atkins. K. R.. 1966. Physics.

Dyson, G. of

H

London

G..

1964.

New

York: John Wiley

Mechanics

of Athletics.

&

Sons. Inc.

London: University

Press. Ltd.

and Frederick. D.. 1964. Engineering Mechanics. s and Dynamics. New York: The Ronald Press. Resnick. R.. and Halliday. D.. 1960. Physics— for Students of :nd Engineering. Part I. New York: John Wiley & Sons. 5 Plena.

Inc.

D.

H..

Rogers.

E.

M..

1960.

Physics for the Inquiring Mind.

Princeton:

Princeton University Press.

Timoshenko, S., and Young, D. H.. 1956. Engineering Mechanics; Ed. 4. New York: McGraw-Hill Book Company. Tncker. R. A. R.. and Tricker, B. J. K., 1967. The Science of Movement. New York: American Elsevier Publishing Company. Inc. *

Meter, kilogram, second system of notation.

SECTION TWO KINETICS

CHAP IKK

7

Forces number only and is not to be confused with g, which is the downward acceleration arising from the gravita-

The one universal force, that of gravitation, was first defined by Sir Thomas Newton in the 17th century. This Law of Universal Gravitation states that: \NY rWO BODIES IN THE UNIVERSE HAVE A GRAVITATIONAL ATTRACTION FOR ONE ANOTHER IF THEIR MASSES ARE W. AND m, AND THEIR DISTANCE APART, T, IS LARGE COMPARED WITH THE SIZE OF EITHER. THEN THE FORCE ON EITHER BODY POINTS DIRECTLY TOWARD THE OTHER BODY

moon. As the mass of humanity comparison to that of a planet, it can be omitted when calculating the gravitational attraction of a planet to any body or object on or tional pull of a planet or

and

.

,

its artifacts is

near

its

so small in

surface: g =

AND HAS THE MAGNITUDE

where

F

=

G

in,

mjr

G

G mjr*

m

the gravitational constant, p is the planeand r is the distance to the center of the planet at that point. As this gravitational force is acting between the center of the planetary mass and the object or objects that it is attracting on or near its surface, the direction of this force is always vertically downward and produces a constant acceleration on these objects.

2

is

tary mass,

where F is the gravitational force, m and m 2 the respective masses of the two bodies, r the distance between them, and G is a gravitational constant having the same value for all bodies. It should be noted here that G is a ,

MASS, WEIGHT, AND GRAVITY Mass per

matter and as measure of the quantity of matter to which inertia is ascribed. Mathematically mass is defined by the following formulae: m = F/a (from Newton's Second Law (Chapter 6)) and m = Vp, where m is mass, F is force, a is acceleration. V is volume; and p (the lower case Greek letter rho) is the density of the mass. Weight, on the other hand, is defined as a force with which a body is attracted toward the earth (or planese is defined as a quantity of

If

tary or lunar type of mass). Therefore weight

The

If

mass

dynes,

W

in

is

grams and g

If

mass

weight

,

10

is

grams x 980.6 cm/sec 2 = 98.06 dynes

in kilograms

is

and g

in

meters per second 2 weight ,

is

mass

is

in

in

10 kg / 9.806 m/sec 2 = 98.06 newtons

pounds and g

in feet per

second 2 weight ,

is in

is

in

pounds,

last

= 10 slugs x 32.17 ft/sec 2 = 321.7 pounds

two items above illustrate the confusion American and English systems of nomen-

poundals,

e.g.

W=

,

As the amount of gravitational attraction is inversely proportional to the square of the planet's radius (see above), the shape of the planet affects the amount of force exerted at any given point on its surface. For example, on earth this gravitational force causes an object to be accelerated 980.6 cm/sec 2 (CGS* system) or 32.17 ft/sec 2 (English system) when it is falling at 45° latitude, i.e., halfway between the equator and either pole. However, because the earth is flattened at its poles, making the distance to its center slightly less at these two

e.g.:

W= If

centimeters per second

in

second 2 weight

wise.

e.g.:

W= newtons.

is

in feet per

This confusion extends somewhat to the metric system also, as common practice habitually uses the terms gram and kilogram as well as pound to define both mass and weight. However, although the difference between mass and weight has been noted above, this text follows the common practice and uses the terms grams, kilograms, and pounds as units of both mass and weight until such time as it is deemed advisable to do other-

physicist:

in

and g

clature.

which draws people or objects on or near the surface of a planet toward its center. Mathematically the formula defining this = mg, where force is is weight, m is mass, and g is the acceleration of gravity. According to the

2

in slugs

existing in the

of the gravitational attraction

W

is

W

a result

is

mass

e.g.:

a

10 lb (mass) / 32.17 ft/sec 2 = 321.7 poundals

*

85

Centimeter, gram, second system of notation.

UNDERSTANDING THE SCIENTIFIC BASES OF HUMAN MOVEMENT

86

TABLE

7.1.

Variation of

Altitude

ft

a'

TABLE

with altitude at 45° latitude

#

Altitude

ft/sec'

m

g m/sec*

Equator

9.806

32.174

Variation of

7.2.

A'

with latitude at sea level

Feet /sec*

Latitud(

Meters/sec"

0°

32.0878

9.78039

10°

32.0929

9.78195

1,000

32.170

1,000

9.803

20°

32.1076

9.78641

4,000

32.161

4,000

9.794

30°

32.1302

16,000

32.124

8,000

9.782

40°

32.1578

9.79329 9.80171

60,000

31.988

16,000

9.757

50°

32.1873

9.81071

100,000

31.865

32,000

60°

32.2151

500,000

30.631

100,000

9.708 9.598

70°

32.2377

9.81918 9.82608

80°

32.2525

9.83059

90°

32.2577

9.83217

Doints. a falling obiect

Pole

wol ild be accelerated 983.2 cm/

2 in the polar regions. When falling sec 2 or 32.257 ft/sec at the equator, where the distance to the center is greatest, the same object falling at sea level would undergo 2 2 In less acceleration; i.e., 978 cm/sec or 32.08 ft/sec other words, the object would be heaviest when weighed ,

,

.

at either pole, lighter

when weighed at 45° latitude, and lightest of all on a

lighter still at equator sea level,

mountain peak along the equator,

i.e.,

in Bolivia,

Kenya,

Sumatra; see Tables

or

day).

As

7.1

and

7.2

this gravitational force

is

(Resnick and Halliacting between the

center of the planetary mass and the objects which it is attracting on or near its surface, this force is always

acting vertically

downward and produces

a constant

acceleration, g. Thus g may or may not be one of more forces which interact in or on a body.

two or

FORCE RELATIONS parallel forces: the weight of the

Linear Forces on the basis of the relationship between their action lines. This is the All interacting forces are defined

the same straight line (Fig. 7.1). As an example, in a given situation forces of 10, 3, and 5 pounds are pushing a body to the right, while at the same time forces of 2 and 4 pounds are pushing the same body to the left. These forces have direction and magnitude and so are vector forces which can be graphically illustrated by arrows whose length is scaled to represent the amount of force (Fig. 7.2). Vectors pointing to the right or upward are by convention considered as positive, while those pointing in the opposite directions are considered negative. Adding graphically in Figure 7.2.B, we end up with a positive vector 12 units long which is equivalent to 12 pounds.

simplest combination, as

all forces

must

lie in

Mathematically:

R

where

R

is

=ZF

the resultant and

F

Eq. 7.1 is

the force or forces in-

volved:

R

=

+10-4

+ 5 +

3-2= 18-6 =

+12

lb

Parallel Forces

10-pound shot held the weight of the forearm and hand acting at the center of gravity of the two segments, and the unknown upward pull of the biceps, th supporting the forearm hand and weight. b ,* These forces acting at various measurable distances from the axis form a third class lever system with the axis A in the elbow joint, the unknown force of the biceps as the supporting effort, and the combined weights as the resistance to be balanced. (For the soluin the

W

hand,

W

s

,

,

TMF

tion of this

problem see below under Problem

Force Couples

A special case of parallel forces occurs when there are two forces of equal magnitude acting at a distance from each other and in opposite directions. Under these circumstances they produce a turning action as in Figure 7.4, in which two boys cooperate to turn the boat end for end. As long as the force exerted by boy A is equal and opposite to that exerted by boy B, the boat will not go anywhere; there will be no linear displacement or acceleration, and the resultant of the two forces will be zero. But the boat is turned. The forces acting in this situation are known as a couple and the moment or torque is the product of one of the equal and opposite forces multiplied by the distance be-

tween them:

In this case the action lines of the forces under consideration are parallel to one another as in the examples of the lever classes presented earlier (see Chapter 3 under "Classification of Levers"). As these forces are applied at some distance from each other, a slightly different approach is necessary to determine the mag-

nitude of one unknown (force or moment arm) when the remainder of the forces and moment arms are known. Given the system in Figure 7.3, we have three

9.

Fd

Eq.

where the subscript c refers to the couple. In the above illustration each boy is exerting a 30-

pound force and they are 7 feet apart. Boy A is exerting a clockwise force, thus the resultant of the two forces *

The abbreviation

of

TMF

for total

out this text to refer to the total exerting at the instant under study.

muscle force

amount of

is

used through-

force that a muscle

is

87

Forces

FIGURE

7.1. Linear force

system; two pairs of forces. A. force exerted by the arms; F the force exerted by the floor

the rope; X. the force exerted by the feet;

FIGURE

7.2.

representing

R. the force exerted by

,

Vector diagram of forces described in text. A. vectors each of the different forces (see text); B. vectors

added graphically.

will

be zero:

FIGURE in

total

R but the

moment

=

- 30 - 30

Parallel force system.

W

fh

,

W

s

,

the weight of the shot held

the weight of the forearm and hand;

TMF b

,

the

muscle force exerted by the biceps to support the system.

=

Concurrent Forces

or torque: t c = 30 lb x 7

= 210 lb

7.3.

the hand:

ft

ft

Forces whose action lines meet at a point are concurSuch forces may be applied to a body from two or more different angles so that projections of their rent.

UNDERSTANDING THE

88

HUMAN MOVEMENT

SCIENTIFIC BASES OF

FIGURE

7.5.

Concurrent forces.

A, lines of force intersect inside

the body; B. lines of force intersect outside the body

that gravity has a clockwise action on the body and only the counterclockwise tension in the calf muscle maintains erect posture (Fig. 7.6). With all of these concepts as a basis, it is possible to proceed to solving problems that arise in the study of human movement, in particular those involving kinetics.

FIGURE 7.4. Example of a force couple in which equal and opposite forces are applied to each end of the boat. The torque or moment of this couple r b is the product of one force and the (

distance between

)

them

action lines will cross. This intersection need not be inside the body. Figure 7.5A presents a simple situation where the action lines intersect within the body, while B illustrates an intersection outside the body. Another example occurs in normal quiet standing where the line of gravity falls slightly anterior to the ankle joint so

Kinetics itself is concerned with the forces that either produce or change the state of rest or motion of a mass, living or inert. When studying the kinetics of the human body we must be aware of two kinds of forces which act on it, internal and external. Internal force is exerted by muscular tension, either while shortening or being lengthened by an outside force. The major external force

TYPICAL PROBLEMS MET In the study of various aspects of movement and posture, two general types of situations arise in a force analysis. of Finding the Resultant of Two or Forces. This involves what is known as the composition of forces and is the process of reducing any number of forces acting on a body to a single force, known as the resultant, whose action will be the same as that of the combined original forces. This entails deter1.

The Problem

More

mining both the magnitude and the direction of the

re-

sultant. 2.

The Problem

Necessary

of Finding the Equilibrium Force (or

Moment Arm)

in

is,

of course, gravity.

As

gravitational force pulls

bodies toward the center of the earth, its action line is always vertically downward. Other external forces include those of impact as in catching or striking, falling or contact in sports, water resistance when swimming, the force exerted by a fiberglass pole in vaulting, etc. The action line of these external forces other than gravity depends upon the situation being analyzed. As all of these forces, both internal and external, have direction and magnitude they are vector quantities and subject to vector analysis. all

IN

FORCE ANALYSIS

Simple problems of this type have been presented Chapter 3. A more complex problem is illustrated

in in

Starting with known factors such as the magnitude of the two weights involved (forearm and hand plus the weight held) and their moment arms, as well as the point of attachment and the action line of the biceps muscle, the total muscle force that the biceps is exerting (TMF) can be calculated. In other words, when a body is in equilibrium and certain factors are known,

Figure

the the

7.3.

unknown factors can be determined because both sum of the moments and/or the sum of the forces

must be equal

to zero.

an Equilibrium Situation.

COMPOSITION AND RESOLUTION OF FORCES Many

times when a number of forces are acting on

as the composition of forces. This problem of finding a

desirable to find a single force or resultant that will have the same effect on the body as that of the

resultant of two or more forces has as its concomitant the problem of resolving a single force into two or more cor ponents such that the combined action of the two

a

body

it is

combined

forces that

it

replaces. This process

is

known

Forces

new

forces will

be equivalent

to that

The most common situation human movement will encounter force.

resolve a single force into

of the original

that the Btudent of is

the necessity

to

two components.

more combe the same as that

single force can he replaced by two or

ponents whose combined action

will

The simplest situation of resolving two components involves the construction

of the original force. rce into

of a parallelogram of forces

forms the diagonal

such that the original force each of the above cases

(Fig. 7.7). In

components of the force F are indiand Q. P and Q and P' and Q\ These

the pairs of force

ed by

P

unison on a body, will produce the same

as the single force F. However, it is more dethe direction that each of the two

sirable to specify

replacements must take, and the most frequent procedure is to resolve the force F into two components at right angles to each other. These directions are normally horizontal and vertical, forming an X component on the horizontal or x axis, and a Y component along the vertical or y axis (Fig. 7.8). Thejbrce F is at 45° with the horizontal, and the vector AB represents the force. The x and y axes are drawn through point A. the origin of the force. Perpendiculars from point B to the two axes define the two components, vectors

AC

is

a

(TMF)

forces to dislocating decnm pressing forces as the angle of pull of a muscle increases beyond 90°. The rotatory component, on the other hand, has constant direction in that it is always perpendicular to the long axis of the bone or segment that it moves, while the secondary component always follows the long axis of that bone either toward or away from

the

moving

muscle

joint,

depending on the angle

at

which the

pulling.

is

.

forces, acting in effect

technique used in resoh in^ the total muse Ito\ a given muscle into its rotatory and secondary components. Secondary components arc frequently much larger than rotators components but are so called because they may change in direction from

This

force

stabilizing compressing

^hical Resolution of Forces

Any

8'.

(the

X

component) and

AD

(the

Y compo-

nent).

Figure 7.9 illustrates the resolution of the biceps into these two components. The action line of the rotatory component originates at point A, the intersection of the biceps action line with the long axis of the forearm, x-x, and is perpendicular to this axis. The action line of the secondary component is coincident with the long axis, and its direction can easily be determined by constructing the parallelogram of forces around a given of the biceps. To illustrate this, let us assume a of 100 pounds. Vector AB is drawn along the biceps action line and is 100 units long. In A of Figure 7.9 the forearm is extended and the action line of the biceps makes an angle of approximately 8° with the long axis of the forearm. With a of 100 pounds the rotatory component, vector

TMF TMF

TMF

FIGURE 7.7. Resolution of force F P and Q' and P" and Q" by 1

.

into

two components: P and

constructing

Q.

parallelograms of

force around force F.

FIGURE

7.6.

muscle force gravity

The concurrent force system f

.

of gravity g and soleus maintaining the upright posture. CG. center of

FIGURE

7.8.

Resolution of force F into vertical and horizontal

components. See

text.

90

UNDERSTANDING THE

SCIENTIFIC BASES OF

HUMAN MOVEMENT

est*

FIGURE 7.9. Resolution of total muscle AB '^TMF b is maintained at 100 pounds

force of biceps

component Note the change in direction beyond 90° as in D and E See text

of the

(TMFb

)

into rotatory

and secondary components. component: AD. secondary

x-x. long axis of forearm; AC. rotatory

secondary component when the angle

of pull

6 increases

AC,

bow

net

secondary component increases. But its direction is now away from the joint; the secondary force has become one of dislocation-decompression. In D the muscle is pulling the for earm away from the joint with a 92-pound force (vector AD), and the rotatory force is 41 pounds (vector AC). Note that the sum of the two components is always

is only 15 pounds, while^ the stabilizing comporepresented by vector AD is 98 pounds. As the elbow is flexed the angle of pull of the biceps action line increases, as does the rotatory component. Maximal rotatory force is achieved at C when the action line of the biceps is at 90° with the forearm axis and all of the 100 pounds of contractile force are available for supporting or moving the forearm. With continuing el-

flexion the rotatory

component decreases

as the

91

Forces greater than the IMF of 100 pounds. The two com ponent force vectors form the two legs of a right angle triangle, with the biceps TMF as the hypotenuse, bo the Pythagorean theorem is applicable: "The square oi the hypotenuse is equal to the sum of the squares of the two sides.'" Consequently,

AH

>AC«

AD'

Frequently in calculating muscle forces the problem is simplified by resolving the TMF into horizontal and vertical components rather than into stabilizing or dislocating forces. An example of this treatment is given under Problem 10 below.

Graphical Composition of Forces

A number of forces acting on a single body can be reduced to a single force whose action will be the equivalent of the

When

is

assumed

combined action of all of the

the reverse of resolving a force into 7.7 and 7.8. The pair of concurrent forces becomes the sides of a parallelogram, and the single resultant force R is the diagonal of the parallelogram which originates from the junction of the is

two components as in Figures

forces.

two forces. P of 10 pounds, and pounds, are acting on a single body.

In Figure 7.10 7.5

Q

of

Problem 1. Find the magnitude and the direction of the resultant (the single force that can replace P and

Q) Solution.

by drawing

Construction

a

paralle logr am

line BD__p>arallel to vector

parallel to vector

AB.

D

is

A

of

AC, and

forces

line

CD"

the point of intersection of

Tensor

the point of

the paralleloint

On

fascia lata

1

Cluteus minimus

2

Gluteus medius

4

and knowing the directions of the action the proportional resultant force can be determined. Figure 7.13 indicates the action lines from the trochanter A. That of the tensor fascia lata, AB, is shown as attaching where it does as this point is so close to its junction with the iliotibial band. The action of the gluteus medius is represented by^ AC and of the gluteus minimus by line AD. a, b, and c are the respective vectors. The resultant of these three muscles may be determined graphically in two different ways: (a) by the parallelogram methods illustrated above, and (b) by constructing a polygon of forces. Problem 2. Find the resultant of the combined force of the three abductors. a. Solution by parallelograms. With three forces it is necessary^ to^ construct two parallelograms. The three vectors a, b, and c are drawn from a common this basis

lines,

POUNDS

FIGURE

is

original forces.

can be treated by both methods. A graphical solution of a linear force problem is presented above under "Linear Forces." Finding the resultant of a pair of con-

two

equilateral.

If we consider the three abductors, gluteus medius, gluteus minimus, and tensor fascia lata, we face a different problem. Each of these three muscles has a different shape and mass and a slightly different angle of pull, although they all attach on or near the greater trochanter of the femur. Inman determined their action line on the pelvis and also postulated that their contribution to abduction was proportional to their mass. He found the following proportions:

is

current forces

Under such circumstances

erse ction of the action line of each muscle part. AB and AC are the vectors of the two parts and AD is the resultant force of the muscle.

desirable to manipulate forces in this manner there are two general approaches, graphical and mathematical. Parallel forces can best be composed mathematically, while linear and concurrent forces it

that each part contributes equally to the re

sultant force.

grams are

AH

AC

AB«

BD' and CD". The vector AD is the resultant of forces /'and (I The graphical solutions of muscular forces are illustrated in Figures 7.11 and 7.12. For these muscles it

7.10. Graphical composition of forces

P and

Q. See text

UNDERSTANDING THE

92

FIGURE

7.

two heads

origin

11.

Graphical composition of the forces exerted by the gastrocnemius.

of the

A

(Fig.

In this case, for reasons of clarity th£

first

and

7.13.B).

SCIENTIFIC BASES OF

at the

same angles

as in Figure

7. 13.

HUMAN MOVEMENT

FIGURE 7.12. Graphical composition heads of the pectoralis major.

of the sternal

and clavicular

parallelogram is drawn between vectors a and c with a resultant vector R'._JThe second parallelogram is drawn between vectors R' and b to find the final resultant vector R (Fig. 7. 13.B). b. Solution by a polygon of forces. The first vector, a, is drawn at the same angle with the horizontal as in the drawing of Figure 7. 13. A. The second vector, b, ^s added to the open end of the first, and the third, c, similarly added to the second. The vector that closes the polvgon is the resultant R of the three forces (Fig. 7.13.C).

Mathematical Composition of Parallel Forces

The composition of parallel forces requires the use moments: THE MOMENT OF THE RESULTANT (R X MA R OF A PARALLEL FORCE SYSTEM ABOUT ANY GIVEN POINT MUST BE EQUAL TO THE SUM OF THE MOMENTS OF THE INDIVIDUAL

of the principle of

)

forces about the same point This .

R

MA H

is

expressed by:

=Y. M

is

^M

Abductor forces acting at the hip joint. A. pelvis with abductor action lines (redrawn after Inman): B.

7.13. joint

graphical solution of Problem 2. a by parallelograms: solution of Problem

2b

MA

C.

graphical

by a polygon of forces.

downward

as in Figure 7.14. Reading 1. 2. and 4 pounds. There is a distance of 2 feet between the 3- and the 1-pound weight and between the 2- and the 4-pound weight, with

acting vertically Eq. 7.3

R is the resultant (the sum of the forces), R the moment arm of the resultant, and is the algebraic sum of the moments of the parallel forces. As an illustration, assume that four parallel forces are where

FIGURE and hip

from

1

left

foot

to right they are

between the

1-

3,

and the 2-pound weights

(Fig.

7.14A).

Problem

3.

Find the resultant of the four forces (the

Forces B

A

C

93 D

4

FIGURE

7.14.

Problem

diagram of

A.

3.

parallel force

system;

B.

diagram of Solution

diagram of

C,

a;

Solution b

whose direction and magnitude would have

single force

the

same

effect as the four original forces).

^

MA

Solution A. R \ M. Let the origin of the R = A. be the given point around which the moments of the various forces will be determined. The moment for force A will then be zero. Forces B, C, and D will tend to rotate the body in a clockwise direction and so will be positive: first force.

R = 31b +

llb

+ 21b +

41b

= 10 1b

£M

MA B

(B x

x

= (lib

= 2ftlb

10 lb \

)

2ft)

+ (C x

ft

lb

MA K

= 28

ft

lb

.\/.4 R

= 2.8

)

+ (2 lb x 3ft) +

+ 6ftlb +

= 28

MA C +(Dx MA D 20

A (Fig.

x5ft)

lb

ft

The 10-pound downward resultant of

ft

(4 lb

)

is

2.8 feet to the right

R

>

be considered positive; those to the

MA R MA H

=

£ Af

= (41b = 8

ft

= - 2

MA

B

is

the

(+

D

2ft)

-

=

x

x

ft

MA D

(3 lb

lb - 9 ft lb

= - 0.2

The 10-pound C. This

£M

=

left

would

-

I

- (A x

x 3ft) 1 ft

MA A

(1 lb

x

)

- (B x

MA B

W

XM

=

W, x MA, +

= 2.8 lb x 4.3 = 12.04 in. lb

= 24.28

in. lb

3.7 lb x

MA R

= 24.28

in. lb.

MA R

= -

lb

24.28

lb

W

in.

+

f

equal forearm, and

MA h

h

x

+

0.9 lb x 13.5 in.

12.24 in. lb

in. lb

3.7 1b

ft

6.6 in.

resultant

same

0.9 lb

equal weight, the subscript the subscript h equal hand; then:

Let

)

1 ft)

+

= 2.8 lb

= 3.7 lb

venience, positive forces precede the negative ones in the equation:

10 lb

MA R R

tend to rotate the body in a counterclockwise direction and so are considered negative. For the sake of con-

R

x

7.14.B).

Solution B. Let the origin of the third force, C, be the given point; then the moment of C will be zero. R will equal 10 pounds as in A. Forces to the right of C will tend to rotate the body in a clockwise direction and so will

7.14.C). The student is invited to try solutions using any other point to further illustrate this principle. The use of the principle of moments in the study of human kinetics is frequently applied to locating the center, or just the line, of gravity of two or more body segments as illustrated in the following problem. Problem 4. The hand of a 150-pound man weighs 0.9 pound and his forearm weighs 2.8 pounds. The center of gravity of the hand is 13.5 inches from the elbow axis (50.6% of the length of the hand from the wrist plus the length of the forearm, Table B.3, Appendix B), the center of gravity of the forearm is 4.3 inches from the same axis (43% of the length of the forearm from the elbow axis), Figure 7.15. Locate the center of gravity of the combined forearm and hand. Solution. Since this is a problem concerned with determining the location of a common center of gravity for two adjacent segments, the length of the horizontal moment arm of this common center will solve the problem, and the principle of moments is used: (Fig.

is 0.2 of a foot to the left of force location as found in solution A above

The center lies 6.6

of gravity of the combined forearm and hand inches distal to the axis through the elbow.

UNDERSTANDING THE

94

SCIENTIFIC BASES OF

HUMAN MOVEMENT (Values for these functions for any angle up to 90° are

found

in

Appendix

C).

The

relationship of the two sides of the triangle to the hypotenuse is illustrated in Figure 7.17. In these triangles the length of the hypotenuse is always the

radius of the circle and, as the angles change, it is easy to see the changes in the length of the side adjacent and the side opposite.

0-9 2*8

FIGURE

Drawing

7.15.

lbs

lbs of situation.

Problem

4.

Trigonometric Functions Needed for Solution of Force Problems* In the mathematical composition of concurrent forces, rectangular components as described above under "Graphical Resolution of Forces" are always used. In order to determine the magnitude and direction of the resultant of concurrent forces, it is necessary to make use of cetain trigonometric functions known as sines and cosines and tangents and cotangents. These are terms identifying certain constant relationships between the sides and the hypotenuse of a right angle triangle as follows.

Given the

SIDE

ABC

right triangle

FIGURE

7.16.

ft

ft

Let:

BAC be known as (the Greek letter theta) side AC be known as side a or the side adjacent to 6 side BC be known as side o or the side opposite side AB be known as h or the hypotenuse of the right angle

A

Right triangle, ABC. angle BAC; AC. the side adjacent (side a) to BC. the side opposite (side o) to 8 AB. the hypotenuse (side h) of the triangle

(Fig. 7.16):

.

:

:

ft

triangle (Fig.

7.16)

There are certain constant relationships

for these three

sides, regardless of the size of the triangle,

dependent solely on the

size of the angle

6.

which are These re-

lationships are expressed as: side opposite

or

sine 6

sin 8

o = -

hypotenuse

h

side adjacent

cosine 6

or

cos

a = —

hypotenuse

h

side opposite

tangent

or tan

8

=

side adjacent

or

8

-

a

side adjacent

cotangent

o

cot

=

a -

side opposite

FIGURE *

Students who are familiar with trigonometry and the use of

trigonometric functions

may

skip this section.

7.17.

Hypotenuse

of the right triangles

changed. Note changes in the relationships and side opposite as y changes.

of the

remains unside adjacent

Forces

\ematical Composition and Resolution Concurrent Forces

When

of

Referring back to Figure 7.9B, the action line of the biceps brachii tonus an angle of 50° with the long axis of the forearm. It it is assumed, as in the graphic solution, that the biceps is exerting 100 pounds of tension, the magnitude of the rotatory and compression forces can he determined by use of sines and eosines. Problem 5. To determine the magnitude of the rota tory and compressive components o( the hiceps brachii, given a contraction force of 100 pounds and an action line at 50° with the long axis of the forearm (Fig. 7.18). Solution. The long axis oi the forearm forms the side adjacent to the angle 6 of the right triangle ABD so side a is equivalent to the compression force. If this long axis is considered the x axis of a pair of x and y coordinates, then the compression force becomes the component of the force F. The rotatory component, then, lies on the y axis and is the Y component of the force F. Side BD forms side o of the right triangle ABD and is

X

equal to the

Y component -

the bicep8

is

exerting LOO pounds of tension

9.o pounds and the compression force is 64.3 pounds (compare with the graphical solution accompanying Fig. 7.9, above).

However, unless one is working with similar problems involving rotatory and stabilizing forces acting on long bones, it is more satisfactory to resolve the forces being studied into horizontal or X components and vertical or Y components. Unless the angles that each force makes with the horizontal are known, a force diagram is drawn, reproducing the action lines of the forces in question, and a horizontal x axis is added, intersecting the action lines. (The y axis is not always needed but can be drawn in if desired.) Figure 7.19 is a force diagram of Problem 1, above. Both the x and y axes have been added at the origin of the two forces. The action line of force P lies in the first quadrant of the x-y system, so both the and Y components are positive. On the

X

of the force F. Hence:

therefore

X

=

F

therefore

Y

=

F

Eq. 7.4

cos 6

F

and sin 8 =

—

Eq. 7.5

;

F

X

Y

= 100 cos 50°

= 100 sin 50°

= 100 \ 0.64279

= 100 x 0.76604

= 64.3

= 76.6

FIGURE

7.18.

FIGURE

Problem

5. A,

sion force exerted by biceps;

between

TMFb

and Cfb

vertical y

drawing;

TMF b

.

1

with horizontal x and

diagram. Rfb rotatory force exerted by biceps; Cfb compresmuscle force exerted by biceps [TMF b = 100 lb); e. the angle

B. force

total

Force diagram from Problem axes added. See text.

7.19.

.

.

SCIENTIFIC BASES OF

UNDERSTANDING THE

96

other hand, the action line of force Q lies in the fourth component is positive, the quadrant and, while the Y component is downward and therefore negative. By measuring the angles on the diagram, force P is at 35°

X

with the x axis and force Q forms a downward angle of 25° with the same axis. The X component of the resultant force will equal the sum of the X components of the forces involved, and similarly the Y component of the resultant will equal the sum of the Y components of the forces involved:

Z *\x

R„

=

Rx

-E*.

F 2X

-

Problem

The resultant equals 15.2 pounds, but the direction has yet to be located: i.e., the angle that the resultant makes with the

horizontal:

y~o

£F.Y- F Y

Rv =

£*\

-

8 = arcsin 2.5/15.2

= arcsin 0.16453

Solution B. The magnitude and direction can also be determined as follows:

FY 3

2

Tangent =

o/a; o

~ Y component, and

-35°

-25°

Sine

0.57358

0.42262

Cosine

0.81915

0.90631

= 10

Q

lb

= arctan

= 7.5 lb

= 9°

Q

cos 25°

= (10 lb x 0.81915)

+

+ 6.7973

or a/cos

horizontal

28'

6.

(7.5 lb x 0.90631)

h -

R

= o/sin 6

lb

= 2.5/0.16447

6.8 lb

= 15.2 lb

= 15.0 lb

The

Y/X

This gives the direction of the resultant but not its magnitude; which is equivalent to the hypotenuse of the right triangle. Cos 6 = a/h and sin 6 = o/h: so either can be used to find h, the hypotenuse, as h = o/sin

Solution.

+

com-

= arctan 1.66666

Angle with horizontal

= 8.2 lb

X

= arctan 2.5/15.0

P

= 8.195 lb

~

7.19).

Table of known data

cos 35° +

a

ponent; then:

P and Q (Fig.

P

divided by R.

= 9° 28'

Ry =

=

Y

is

Y/R

6 = arcsin

3

Force

Rx

sine = 0/h*

This angle is the angle whose sine This is expressed as:

Find the resultant (magnitude and direc-

6.

r?~hand

and

FX

and:

tion) of forces

HUMAN MOVEMENT

is 15.2 pounds upward at an angle of 9° found by the Pythagorean theorem.) The problem presented in Chapter 4 under "Time Related to Motion Velocity" can be solved in the same manner as the above problem, but there is a negative X displacement of 5 miles northwest and a zero Y displacement of 2 miles due east;

The

component

of forces

P

and Q

is

15

pounds.

resultant

28' (cf. results

—

R^

=

Psin 35° - Q

sin 25°

= (101b x 0.57358) - (7.51b x 42262)

= 5.7358

1b -

3.16965 lb

Displacement Angle* Sine Cosine

= 5.7 lb - 3.2 lb = 2.5 lb

The vertical component of forces P and Q is 2.5 pounds. The resultant being sought is the hypotenuse of the whose two sides are represented respectively by vectors 15.0 and 2.5 units long. Solution A. By using the Pythagorean theorem the magnitude of the resultant can be determined:

*

5 miles

NW

135°

2 miles 0°

+0.70711 -0.70711

+1

E

NE

3.3 miles 45°

+0.70711 +0.70711

Angles are measured with east as 0° on the

X

axis.

right triangle

R2

= 15 2

Problem 7. To find the resultant of the distance direction traveled.

= 225 + 6.25

= 231.25

R

=

/? x

=

D, cos 135° +

D

2

cos 0°

+

D

/?v

=

D

D

2

sin 0°

+

D

Rx

= 5 cos 135°

2.5-

4

t

sin 135°

+

+

2 cos 0°

= (5 x -0.70711)

+

(2

x

*

The symbol ~

is

3

cos 45° sin 45°

+

3.3 cos 45°

1)

+

(3.3

V231.25

= 15.2 lb

3

to be read "equivalent

to."

x 0.70711)

and

97

Forces

•

0.79 mile


>i

striated

Peachej

i"

"i

Peachey,

H

\.

.

K. and Hanson,

J., I960 The molecular basis of con tree tnctkm of Muscle Vol. I. edited b) G. II New York: Academic Pros. p. 183. EL. 1965 rhe mechanism of muscular contraction. Sci.

In v

y.

H

D.,

H

ite,

in

EL,

1969

Hie mechanism

A. A.. Klaupiks. D.,

of

and Davies, R

muscular contraction, EL,

1964

ATP

muscle doing negative work. Science 144: 1577. kuhl, D., 1966. Local factors in muscle performance. ITier

s

changes

Amer.

J.

Us. 46: 473.

1968. Effect of direct tetanixation

on twitch tension

and

C. and

Porter. K.

intrafibrillar

R...

restoring

1966. force

Muscle relaxation: evidence in

vertebrate striated

for

muscle.

7 14.

Perry. S. V.. 1960. Introduction to the contractile processes in striated

muscle. In 77ic Contractile Process: Proceedings of a

sponsored by the

New York

Brown and Company,

p. 63.

Heart

Assocation.

Porter,

cells

/*//

R,

K,

Science

reticulum and 26

ti

n

!09

1959. Intracellular

impulse COD

129: 721.

and Fran/mi Armstrong, Amer. 212(3): 72.

.

14

('.,

1965.

The sarcoplasmic

Sci

M

1).

New

19(17.

.

ihe Neurophysiology of Postural Mecha

York: Plenum Press.

.

Muscle as

1962,

a Tissue.

New

York:

Sechenov, I.. 1935, Selected Works. Moscow and Leningrad: Stale Publishing House 1863 quote). (

of forearm flexors

symposium

Boston:

and extensors.

M.. and Padykula, H.

J.

Appl. Physiol.

A., 1962.

of individual muscle fibers of the rat.

Szent Gyorgyi,

Muscle.

New

A.,

1953.

21:

1435.

Histochemical classification Amer. J. Anat. 110(2): 103.

Chemical Physiology of Body and Heart

York: Academic Press.

Szent Gyorgyi, A. G., 1960. Proteins of the myofibril. In Structure in

developing cat leg muscles. Acfa Physiol. Scand. 74: 319. Parsons.

and

k and Horvath, S. M„ McGraw-Hill Book Company.

.Stein, J.

M

B..

rhe urcoplasmic

Muscle

Singh, M., and Karpovich, P. V., 1966. Isotonic and isometric forces

ind, L.. and Molnar, J., 1962. Biochemical control of relaxation in muscle systems. In Muscle as a Tissue, edited by K. Rodahl and Horvath. New York: McGraw-Hill Book Company, p. 97. S

Nystrom,

R

of Skeletal

sartorius, J. Cell

muscle

K.

rusms.

-

m

reticulum Roberts, T.

Rodahl.

213(6): 18.

Huxley,

Porter.

L965

frog

L

duction

tion.

D

I

tubules

126

rhe contraction of muscle. Sci. Amir,

1958

EL,

K

Physiol. (London) 111:

J.

and Chemistry

Little,

and Function of Muscle. Vol. II, edited by G. H. Bourne. New York: Academic Press, p. 1. Walker. S. M.. and Schrodt, G. R., 1967. Contraction of skeletal muscle. Amer. J. Phys. Med. 46: 151. Walls. E. W., 1960. The microanatomy of muscle. In Structure and Function of Muscle. Vol.

Academic Wells,

J.

B..

slow and

I,

edited by G. H. Bourne.

New

York:

Press, p. 21. 1965.

fast

Comparison of mechanical properties betweeen

mammalian

muscles. J. Physiol. (London) 178: 252.

SECTION ONE

CHAPTKK

PHYSIOLOGY OF SKELETAL MUSCLE

Factors

Which

Affect the

Magnitude

11

of

Contractile Tension THE MAGNITUDE OF CONTRACTILE TENSION When a single adequate pulse is applied to a whole muscle, the muscle will respond with a quick contraction, followed immediately by relaxation. Such a response is called a twitch. Its magnitude will vary with the number of muscle fibers which respond to the stimulus and this will vary directly with the intensity of the pulse up to a finite maximal intensity. The twitch is an indication of force development by the muscle. After a short latent period tension becomes evident and rises in a hyperbolic manner to a peak (the contraction period). It then declines over a slightly longer time course to zero (the relaxation period) (Fig.

The tension developed by a contracting muscle is influenced by a number of factors such as the characteristics of the stimulus, the length of the muscle both at the time of stimulation and during the contraction, and the speed at which the muscle is required to contract.

The Stimulus Most oi what has been learned about muscle has been derived from studies using stimulation by electrical pulses. Although it is an artificial stimulus, electricity has distinct advantages for experimental purposes because it can be precisely controlled. The intensity, form (time course of rise to and duration of peak intensity), and frequency of pulses can be arbitrarily selected and varied as desired. Measurable responses of the muscle can be correlated with the quantitated stimulus char-

11. LA)

The time

course of the development of overt tension is influenced by the interaction of the contractile components of the fibrils with the elastic components of the muscle. Figure ll.l.B illustrates the sequence of events and their influence on the shape of in the twitch

acteristics.

A

curarized muscle

may be

stimulated directly by

pulses applied to the muscle tissue or indirectly

the twitch curve. 1. The active state is evident even before tension appears. It reaches full intensity abruptly, is maintained for about half of the contraction period, and then progressively declines during the rest of the contraction

by

pulses applied to its motor nerve fibers. The response of the whole muscle, of a single motor unit, or of one

muscle fiber

may be

studied under controlled con-

ditions.

period. 2. The contractile components begin to undergo activation during the latter half of the latent period. As they shorten, the elastic components of the muscle are

The Single Pulse

Response of Muscle to a Single Pulse. If a single pulse of adequate intensity is applied directly to a muscle fiber, the fiber will respond in an all-or-none fashion. Increasing the intensity of the pulse will not increase the magnitude of the fiber's response. It is important to mention here that the all-or-none response of the muscle fiber is determined by the all-or-none

stretched and begin to exert passive elastic tension. Elastic tension is low at first. During this time the contractile elements are able to shorten rapidly. 3. When the active state is at full intensity, about halfway through the contractile period, the elastic tension is rising rapidly. 4. As the active state begins to decline in the latter half of the contractile period, its intensity is still sufficient to continue to stretch the elastic components, and tension continues to mount but at a decreasing rate. The twitch curve begins to round off. 5. At the peak of the twitch curve, tension in the contractile and elastic elements is in equilibrium. 6. Beyond the peak, as the active state continues its decay, developed tension falls below elastic tension and

character of its excitation and not by any all-or-none limitations inherent in the contractile mechanism itself. The production of local graded contractions by applying small electric currents through microelectrodes to specific selected areas of the sarcomere was mentioned earlier. These contractions are not propagated along the fiber but are restricted to the region stimulated. They are not all-or-none but vary directly with the intensitv of the stimulus.

147

SCIENTIFIC BASES OF

UNDERSTANDING THE

148

HUMAN MOVEMENT fiber is stimulated to contract isometrically until its full active state has been developed. Then it is suddenly released to a slightly shorter length. Tension falls immediately but is quickly re-developed, at a rate exceeding that in a normal twitch. The peak level, however, is lower. By varying the time of release and plotting

A FIGURE stimulus;

B

11.1.

A, Tension development

latent

/.

period;

2.

in

contraction

a

muscle twitch.

period;

3.

S.

relaxation

and tension development. superimposed on the twitch tension

period. B. Relationship of active state

Active state (broken

line)

curve (solid

peak of active

line).

7.

is

state;

2.

elastic

components

begin to exert tension; 3. tension rises rapidly; 4. as active state

begins to decline, tension continues to 5. at the

are

in

rise

but at a decreasing rate;

peak of the twitch curve, contractile and

elastic tensions

equilibrium; 6, as the active state continues to decay, recoil

of the elastic

and tension

components stretches out the

falls;

7.

active state

has returned to zero See text

decay

is

contractile elements complete before tension

for further discussion.

the elastic components recoil, stretching out the contractile components. Overall tension falls. 7. Decay of the active state is completed before tension returns to zero. The fact that tension outlasts the active state is partially explained on the assumption that the breaking of cross bridges requires more time than their formation. Therefore the recoil of the elastic components lengthens the contractile material less rapidly than the rate of decay of the active state (Hanson and Lowy: Walker and Schrodt*). The time course and intensity of the active state are studied by the techniques of quick stretch and quick release. Because of the elastic components and the viscosity of muscle tissue, the externally measured force exerted in a twitch is less than the full capability of the contractile material: that is, less than the intensity of the active state. The viscoelastic effect may be counteracted and the full tension characteristics of the contractile elements registered by employing quick stretch or quick release. If, coincident with stimulation, the muscle is given a short, quick stretch which pulls out the elastic elements just slightly beyond what their effective excursion would be, the muscle is relieved of the necessity of stretching out the elastic components and its full tension is revealed. By this means the onset, rise time and duration of the peak intensity of the active state can be determined.

The time course of the decay of the active state is studied by the method of quick release, in which the

of

* For references appearing Chapter 10.

re-developed tension against time, a curve reflecting the decline of the active state is obtained (Fig. ll.l.B). Characteristics of the Single Pulse and Their Influence on the Muscle Twitch. An adequate stimulus may be defined as any environmental change, external or internal, which arouses in the contractile material an active state of sufficient magnitude to produce measurable tension. Whether natural or artificial the environmental change must meet certain minimal requirements in regard to its basic characteristics: the magnitude or intensity of the change, its abruptness or rate of rise, and the duration of its application. Within physiological limits, increase above minimum in any of these will induce an increased response in the muscle. f

in this chapter, see

Bibliography

at

end

A

must have a certain minimal minimum is an inverse measure of the irritability of the tissue: the smaller the minimal intensity, the greater the irritability. The minimal effective intensity is designated the threshold or liminal stimulus. These terms refer to the weakest stimulus which will evoke a barely perceptible response. Subthreshold and subliminal refer to a stimulus of inadequate intensity. As the intensity of the single pulse is increased above 1.

Intensity.

single electrical pulse

intensity to be effective.

The

minimal, contractile tension

level of the

in the

muscle increases progressively as

more and more muscle fibers. Finally an intensity is reached which evokes the maximal response of which the muscle is capable. Presumably all fibers are then active. Further increase in intensity will not be accompanied by further increase in contraction. The weakest stimulus intensity which will evoke maximal

a result of the activation of

contraction of a muscle

is

called the

maximal stimulus.

Abruptness or Rate of Rise. A weak but adequate pulse with a rapid rate of rise from zero to its pre-set intensity will evoke a stronger contraction than will a pulse of the same intensity with a slower rise. A minimal rate is required even for an intense stimulus. If intensity rises too gradually, there will be no response at all; the stimulus is then ineffectual. For any stimulus of adequate intensity, the more abruptly it is applied the greater will be the response it evokes, within 2.

the limits of the muscle's capacity. rapidly

it

need

A common

rise to

The

produce a given

greater the intensity the less

level of response.

experience illustrates the principle.

If

the hand

is

plunged abruptly into hot water of about 110°F. the response (sensation of heatl resulting from the abruptness of the change in skin temperature from about 93° to near 110°F will be greater than if the change is made gradually by first immersing the hand in water at skin

temperature and then slowly raising the temperature to 110°. If the rate of temperature change is too slow, the change will be imperceptible. 3.

Duration. For a stimulus of adequate intensity and

the duration of

Within

its

peak intensity

will

influence

its

limits, the longer its duration the greater will

rise rate,

effectiveness.

be the muscle's

response. Exclusive limits are found at both extremes: the duration

can be so short that no response will occur in spite of the fact that the same intensity and abruptness would be sufficient with longer duation, or the duration can be so long that the response decreases

t In a single fiber, only

produce an increase in

its

an increase

in the

frequency of stimuli will

all-or-none response.

Magnitude together.

until

laboratory

uls

when

the latter

direct current

is

is

.1

common

experience

in

the

used to stimulate tissue The muscle

\

the closing of the circuit but ceases to respond .b current

st

!*

of Contractile Tension

flow continues at the constant (peak) level.

The duration

of the peak

intensity has exceeded the response capabilities of the tissue

The

and duration of direct production o( a barely perceptible contraction is presented in the intensity-duration curve shown in Figure 11.2. Note that both the upper and lower ends oi the curve become straight lines, one vertical and the other horizontal, neither meeting the coordinate. The upper end indicates that even a very strong stimulus must be applied tor at least a minimal duration to be effective. The lower end shows that below a certain minimal intensity a stimulus will not induce a response regardless of its duration. Between these limits, the greater the intensity, the less duration is required to produce a response. For a stimulus of constant intensity and rate of rise, the longer its duration the greater will be the response up to a finite limit, beyond which effectiveness diminishes progressively to zero (Fig. 11. 3. A). The duration required for a given stimulus to evoke a perceptible response is its excitation time and is. within the limits discussed above, inversely related to the intensity. If a stimulus of constant intensity and duration is applied at various rates of rise, effectiveness will be directly related to the rate. The more abruptly the stimulus is applied the greater will be the muscle's response. As the rate decreases, the response will diminish until ultimately, regardless of intensity, the stimulus becomes ineffectual (Fig. 11.3.B). The decreased effectiveness of a constant stimulus intensity at long duration and/or low rate of rise is designated adaptation or accommodation. Many relationship of intensity

current

in

the

A

FIGURE rise

and

shown

in

11.3.

B

Comparison

duration

peak

of

of single pulses in regard to rate of intensity

Hypothetical

responses are

(Note: the time scale of the pulses

circle insets

greatly

is

exaggerated as compared with that of the responses A, duration of pulse. Four pulses of identical intensity and rate of rise but with )

shown: /, duration too moderate duration: probably the most

effectiveness

different durations are

long:

low.

effective;

2,

duration: of rise. different

less effective; 4,

duration too short:

ineffectual

3.

short

B, rate

Three pulses of identical intensity and duration but with rise

times are shown;

effective stimulus; 2,

/.

the most rapid

less rapid rise:

least effective (response

much reduced

rise:

less effective; 3,

the

least

most rapid:

or absent).

adapt to a gradual or persistent stimulus. The physiological changes which are induced by the stimulus are apparently reversed at a rate which is faster than their development under the existing conditions. In the case of muscle tissue, excitatory processes may be inadequate to activate the tissue or, if activated, the magnitude or persistence of the active state may be insufficient to stretch out the elastic components enough to produce overt tension. The rate of rise and duration of electric pulses may be varied as required for the principle under study. For most studies of concern to us, a pulse of rapid rise and short duration is used, with variations in its intensity appropriate to experimental objectives. tissues besides muscle

Repetitive Stimulation

DURATION FIGURE

11.2.

(msec)

Intensity-duration curve. The upper limb of the

curve indicates that even a very strong stimulus must be applied for at least a

shows

minimal duration

that

below

a

in

certain

order to be effective The lower limb minimal intensity a stimulus will not

induce a response regardless of intensity

its

duration

and duration are inversely related

Between these

limits

Response to Repetitive Pulses. If an adequate stimulus is applied to a muscle fiber repeatedly at a rate rapid enough so that each succeeding stimulus reactivates the contractile elements before the previous tension has completely subsided, successive responses summate, each building upon the previous until a maximal level is achieved. If stimulation is continued, the contraction peak is maintained at this level. Such a response is known as tetanus or tetanic contraction. Ultimately, fatigue will cause the peak level to decline progressively. When stimulation ceases, contraction terminates and the fiber relaxes, tension subsiding quickly to zero. If, however, the repetitive stimulation

UNDERSTANDING THE

150

SCIENTIFIC BASES OF

too prolonged, contracture will result and relaxation be very much slowed as compared with normal. Unlike rigor, contracture is reversible. is

will

Effect of Frequency of Pulses upon Response. The frequency of stimulation, usually expressed as cycles per second (some investigators employ the recently adopted physical unit, hertz), determines both the shape and the magnitude of a tetanic contraction traced on a myograph by an excised muscle. When pulses are delivered with a period which places successive stimuli during the relaxation phase of the preceding response, the contraction approaches a tremor and a scalloped tracing results. This is incomplete tetanus. With a period which is short enough to restimulate during the contraction phase, the tracing is smooth. This is complete tetanus (Fig. 11.4). Within physiological limits, the shorter the period (i.e., the greater the frequency) the smoother the curve and the greater the tension development will be. If, however, the period is shortened beyond a certain point, the refractory period will be encountered. The absolute refractory period is a short space of time immediately following stimulation during which the muscle cannot be reexcited regardless of stimulus intensity. This is followed by a longer period, the relative refractory period, during which irritability is gradually regained and the tissue will respond to a stimulus which is appropriately greater than threshold. The earlier the pulse falls in the relative refractory period the greater its intensity must be to be effective. Although both portions of the refractory period last for a finite time, in muscle both have been completed before tension begins. Stimulation of skeletal muscle in the living animal is normally accomplished by a train of impulses transmitted to the muscle fibers over their motor neurons. The magnitude and duration of the impulses in any neuron are essentially constant but frequencies vary, sometimes over a wide range. Therefore the latter is the characteristic which influences the response of muscle fibers and is of concern to us in studying human

movement.

The frequency of impulses in human motor neurons generally ranges from 20 to 40 per second. At such stimulation rates, the normal response of the muscle an incomplete tetanus. Muscle contractions, appear smooth because excitation by the various motor neurons is not synchronous and hence

fiber

is

however,

HUMAN MOVEMENT motor units respond out of phase. The relaxation in one motor unit is offset by contraction in another. Any oscillations of tension which might occur are further smoothed out by the transmission of tension to the lever through a common tendon. The tension developed by the muscle depends upon the number of motor units activated and the frequency of activiation of the muscle fibers composing each unit. During maximal exertion the frequency of motor impulses may be great enough to produce complete tetanic contractions. If so, synchrony of responses in motor units develops and tremor results, a common experience in all-out effort. Tetanus-Twitch Ratio. The tension developed in response to repetitive pulses is greater than that evoked by a single pulse of the same magnitude. The tetanustwitch ratio varies with different muscles and may be as great as 5 (rat gastrocnemius). To explain the greater tension developed in a tetanic contraction, it has been postulated that in a twitch the short duration of the active state allows too few bridge movements to permit

the contractile material to shorten enough to fully stretch out the elastic components before the active state begins to subside. Hence the full capacity for tension production cannot be realized. Repetitive stimulation, however, by maintaining the active state, permits continuation of bridge activity. The stretching of the elastic components is developed.

is

completed and

full

tension

Post-tetanic Potentiation in Muscle. In many muscles, especially when curarized, if twitch responses to single pulses are recorded before and immediately after a period of tetanic stimulation, the post-tetanic twitch shows an increase in magnitude and a steeper rise of tension than the pre-tetanic control. This phenomenon is known as post-tetanic potentiation (PTP). The effect occurs whether the muscle is stimulated directly or indirectly by its motor nerve. Potentiation is maximal shortly after the repetitive stimulation and then decays exponentially at a rate which is dependent on both the frequency of pulses and the number delivered in the train. Short trains produce potentiation without any alteration of the twitch duration, but longer trains result in lengthening of the contraction time and of the half-relaxation time (the time required for tension to drop to 50% of its peak value).

has been suggested (Close and Hoh), the of PTP is located within the muscle fibers, it may involve prolongation of the active state with a resulting increase in the number of fully activated myofibrils in the fiber. Or it may be due to an increased + liberation of some activator substance, perhaps Ca ", which induces an increase in the number of bridges formed between actin and myosin and in the rate of If,

as

mechanism

their cycling.

FIGURE

11.4.

Response

to

repetitive

stimulation.

single twitch in response to a single stimulus: curve 2.

response to low frequency repetition of the stimulus; complete tetanus in response to higher frequency repetition the stimulus Same stimulus used in all three

tetanus curve of

Curve 1. incomplete

3.

Conclusion

in

To be adequate, stimulation must consist of an appropriate combination of intensity, rate of rise.

Magnitude duration and frequency to excite muscle fibers and to activate their contractile material sufficiently to pro duce measurable tension. Within the limits discussed, the adequacy of the stimulus is determined by the interaction of these mutually interdependent charec teristies. In the living body an adequate environmental change results in the generation and conduction of a

shorten.

active

of Contractile Tension

Their shortening,

as

161

discussed above under

by

stretching of the elastic components. By current usage, an isometric contraction is one in which the external length of the muscle remains unchanged. (Isometric contraction is sometimes called static contraction.) state,

offset

is

Isotonic Contraction. If the internal force produced the muscle exceeds the external force of the resistance and the muscle shortens, producing movement, the contraction is isotonic. (This is sometimes called a concentric or a shortening contraction.) Energy utilization is greater than that required to produce tension which will balance the load, and the extra energy is used to shorten the muscle. During isotonic contraction work is done on the load by the muscle. This is

action potentials in motor neurons which becomes responsible tor excitation of the muscle fibers. The frequency of impulses reflects the effectiveness of the stimulation and determines the magnitude of the muscle tension developed. Because the magnitude and form of the nerve impulses are constant for any given set of body conditions, the frequency of these impulses is the most significant characteristic in determining the

by

muscle's response.

positive work. Work is the product of the load and the distance that the load is lifted.

train

of

Muscle Length

W

The most obvious property

of muscle

to develop tension against resistance.

capacity length of the

is its

The

the time of activation markedly affects its ability to develop tension and to perform external work. Muscle tension may be measured in terms of the greatest load which can just be lifted or as the maximal tension read-out on a strain gauge or tensiometer. When a muscle contracts, the contractile material itself shortens, but whether the whole muscle shortens or not depends on the relation of the internal force developed by the muscle to the external force exerted by the resistance or load. The terms "force" and "tension" are often used erroneously as synonyms. Tension is a scalar quantity having magnitude only, while force is a vector quantity having both magnitude

muscle

and

at

direction.

The term tension

is

used

in

this

discussion to refer to the magnitude of the pull of the muscle as it would be registered on a strain gauge arranged in line with the muscle axis. Internal force is used to refer to the moment of the tension magnitude acting in the direction of the action line of the muscle under given conditions, and external force refers to the moment of the resistance opposing the muscle.

Types of Muscle Contraction 1945 Fenn identified three types of muscle contraction according to the length change, if any, induced by the relationship of internal and external forces. The three types are here designated isometric, isotonic, and eccentric contraction. Isometric Contraction. If the internal force generated by the contractile components does not exceed the external force of the resistance and if no change of muscle length occurs during the contraction, the contraction is isometric. The available energy* expended by the muscle and the tension produced against the resistance may be considered to be in equilibrium. No contraction in the body is purely isometric because at the fibril level the contractile components do In

*

Available energy

is

that portion of the released energy which

available to the muscle for production of tension and/or work.

is

where

W

is

work

pounds, and d

=

F

x d

Eq. 11.1

pounds, F is force of the load in the distance in feet that the load is

in foot

is

lifted.

A

muscle can develop greater tension in isometric in isotonic contraction because none of the available energy is expended in shortening. In isotonic contraction the greatest load that the muscle can lift is about 80% of its maximal isometric tension. Eccentric Contraction. If to an already shortened muscle an external force greater than the internal force is added and the muscle is allowed to lengthen while continuing to maintain tension, the contraction is called eccentric. (The term lengthening contraction is sometimes used.) The energy expended by the muscle is less than the tension exerted on the load, but the muscle acts as a brake controlling the movement of the load. In eccentric contraction a muscle can sustain greater tension than it can develop in isometric contraction at any given equivalent static length. During an eccentric contraction negative work is done by the load on the muscle. Negative work is meas= F x d, the ured in the same units as positive work, only difference being that d is the distance that the load is lowered. The amount of negative work performed is than

W

the

same

lifting the

as the amount of positive work involved in same load the same distance. While chemical

energy is expended by the muscle in both instances, the energy cost of the negative work is considerably less than for the positive work. The difference is indicated by a lower oxygen uptake during the negative work, being about one-tenth as much in human subjects. Other estimates have placed the cost at one-third to onethirteenth of that required for the equivalent amount of positive work. Eccentric contractions are very common. Every

movement

is controlled by an contraction. Examples include sitting, squatting or lying down, bending forward or sideward, going down stairs, stooping, placing any object down onto a surface, etc. In eccentric contractions the active muscles are those which are the antagonists of the same

eccentric

in the direction of gravity

152

UNDERSTANDING THE

movement when

it

is

made

SCIENTIFIC BASES OF

HUMAN MOVEMENT eccentric force was lowest of the three at the start (shortened position, elbow at 140°), it had exceeded isotonic force when the angle reached 120°, and by 100° it had surpassed isometric force as well (Fig. 11.5.B). Daily activities involve a continual shifting from one to another type of contraction and of combinations of the three types, as required. During movements the changing lengths of lever arms and of angles of pull, for both muscle and load, introduce complexities which require complicated processes of neuromuscular integration to properly adjust the number and actitivy of motor units to the task.

against gravity. Sitting or

controlled by leg extensors, not flexors; lying down, by hip flexors, not extensors; lowering a load, by shoulder flexors, not extensors, etc. Electromyograms show not only that anatomically antagonistic

squatting

is

muscles are actively controlling the eccentric movement but also that the electrical activity in these muscles is less than when the same muscles are contracting isotonically to do the same amount of positive work with the same load over the same distance and at the same speed.

Some textbooks still state that a muscle develops its greatest tension during isometric contraction. The work of Singh and Karpovich demonstrates that the force

Relationship of Muscle Tension to Length

developed by elbow flexor and extensor muscles in eccentric contraction exceeds that in both isotonic and isometric contraction at most muscle lengths. Using an instrument designed by Singh, they measured muscle force through the entire range of motion at the elbow joint, simultaneously recording the angle through which the forearm was moving. With the elbow flexors,

The initial length of a muscle, i.e., its length at the time of stimulation, influences the magnitude of its

A stretched contracts more forcefully than when it is unstretched at the time of activation. This is true whether the contraction is isometric, isotonic, or eccentric. Within physiological limits, the greater the initial length, the greater will be the muscle's tension capability. Parallel-fibered muscles exert maximal total contractile response to a given stimulus.

muscle

eccentric force was consistently the greatest of the three over the entire range of motion and isotonic was least (Fig. 11. 5. A). With the elbow extensors, although

60r

50

40

30

Eccentric Force

20

Isometric Force

.

-Concentric Force * Starting

10

50°

60°

_L

_L

_L

70°

80°

Elbow Angle

FIGURE

Angle

90° in

100°

110°

120°

130° 140°

Degrees

Concentric, isometric, and eccentric tension curves. A. curves of maximal concentric, isometric, and eccentric tension of forearm flexor muscles. B. curves of maximal concentric, isometric, and eccentric tension of forearm extensor muscles. (From Singh. M.. and Karpovich. P. V. 1966 Isotonic and 11.5.

isometric forces of forearm flexors and extensors. J.

Appl

Physiol. 21

:

1435

)

1o

Magnitude of Contractile Tension

50

40

30

:c

to

-

-

Eccentric Force Isometric Force

^•—•-Concentric

J

50°

60°

70° Elbow

*

flex.

The stretch response should not be confused with the stretch reThe latter is a response mediated by the nervous system, while

the former

is

a property of

muscle tissue independent of nerve.

I

90°

Angle

FIGURE

tension at lengths only slightly greater than rest length. Muscles with other fiber arrangements have maxima at somewhat greater relative stretch. In general, optimal length is close to the muscle's maximal body length, i.e.. the greatest length that the muscle can attain in the normal living body. This is about 1.2 to 1.3 times the muscle's rest length. Tension capability is less at shorter and longer lengths. Therefore a muscle can exert the greatest tension or sustain the heaviest load when the body position is such as to bring it to its optimal length. In isotonic contractions the increased tension and longer length permit greater shortening, hence more work can be done or. alternatively, the same work can be done at lower energy cost. The diminished energy cost of eccentric contraction is in part due to this stretch response," but other factors are also involved, as evidenced by the capacity to produce greater tension than with either isometric or isotonic contractions at most equivalent lengths. The relationship of tension to muscle length may be presented graphically in the form of a tension-length curve in which tensions in an isolated muscle are plotted against a series of muscle lengths from less than to greater than the resting length (Fig. 11.6). Both the passive elastic tension (curve 1) exerted by the elastic components in the passively stretched muscle and the total tension (curve 2) exerted bv the activelv con-

I

80°

in

Force

Angle

* Starting

I

I

I

100° 110° 120° 130° 140°

Degrees

11.5. B.

muscle are plotted. Since total tension represents the sum of elastic tension plus the developed tension of the contractile elements, the latter may be found by subtraction and is represented by the area between the curves for total and elastic tensions. Values for developed tension are shown as curve 3. Note the following facts regarding developed tension. tracting

a.

At less than SO^r of

rest

length the muscle cannot develop

contractile tension. b.

At normal

(intact) rest length the

muscle

is

already in slight

passive elastic tension. At this length the muscle produces

developed tension c.

When

(total tension

contraction

although total tension

is

is

minus passive

its

greatest

elastic tension).

initiated at a length longer than rest length,

greater than at rest length, developed tension

has already diminished and declines progressively at

all

greater

lengths. d. At extreme lengths (far right end of the curves) total tension would ultimately become equal to elastic tension, developed tension

being zero.

Maximal

tension is assumed to be lengths are such that maximal single overlap of actin and myosin filaments exists. At greater lengths the number of cross links diminishes as overlap decreases, and at shorter lengths double overlap results in reduced tension as a result of the antagonistic action of bridges. Gordon and colleagues investigated the tension-length relationship in frog skeletal muscle fibers at various sarcomere lengths. Their results are plotted in Figure 11.7. In these fibers the mean sarcomere length was 2.5 n and the mean

developed

contractile

when

sarcomere

UNDERSTANDING THE

154

HUMAN MOVEMENT

SCIENTIFIC BASES OF

filament lengths were 1.5 n for myosin and 1.0 ^ for Maximal tension was developed at sarcomere lengths of 2.0 to 2.25 m- At greater lengths tension decreased linearly, becoming zero at about 3.65 n. At shorter lengths tension declined gradually with decreasing length until about 1.7 m and then dropped abruptly

process most probably associated with significant lengths on the tension-length curve presented above. Figure 11.8, A through D, presents four of these stages, as follows.

actin.

to zero at about 1.27

A. Sarcomere length 2.5

\i.

the myosin filaments of the

pi.

A

Actin filaments only partially overlap bands. Therefore not

attach. Tension capability at this length

Drawing upon these data and the electron microscope evidence of filament relationships in contracted muscle, we may postulate the stages of the sliding filament

B. Sarcomere length 2.25

drawn

into the

potentially

all

A band

fi.

Ends

is

all

about 85% of

bridges can

maximum.

of the actin filaments have been

to the edge of the

pseudo-H zone. Here,

bridges can attach and maximal single overlap occurs.

Tension capability at this length is maximal. C. Sarcomere length 2.0 n. Ends of the actin filaments have reached the center of the sarcomere. Maximal overlap still exists and tension capability is still maximal. D. Sarcomere length 1.5 p. Z discs have collided with the ends of the myosin filaments. Ends of the actin filaments have passed beyond the pseudo-H zone limits and entered the antagonistic bridge areas on the opposite sides of the sarcomere. Tension capability is reduced to about 45% of maximal.

The work

of Gordon and colleagues thus provides further evidence favoring the sliding filament theory and

supporting the concept of a quantitative relationship between tension and the number of bridges linking actin

and myosin filaments.

Speed of Contraction 100

% FIGURE 1.

Most isolated nonloaded muscles normally shorten by about 50% or less of their rest length. The absolute amount by which any muscle can shorten depends upon the length and arrangement of its fibers, the greatest shortening occurring in the long parallel-fibered muscles such as the biceps and sartorius. In intact muscle, shortening is further limited by the structure of joints,

of rest length

Tension-length curves for isolated muscle. Curve tension in a muscle passively stretched to lengths, curve 2, total tension exerted by muscle

11.6.

passive

increasing

elastic

contracting actively from increasingly greater

initial

lengths; curve 3.

developed tension calculated by subtracting elastic tension values

on curve

1

from

total tension

values at equivalent lengths on curve 2.

SARCOMERE LENGTH

FIGURE

11.7. Tension-length curve for frog

muscle

(^1

sacomere lengths. The

letters on the Note that tension is zero both at the shortened length of 1.27^ and at the extended length of about 3.7// is maximal over lengths 2 00^ and 2 25 n and declines rapidly below 1.67/j and above 2.25 M (After Gordon. A. M. Huxley. A. F., and Julian. F. J 1966 Variation in isometric tension with sarcomere length in vertebrate muscle fibers.

at various

tension curve and the broken vertical lines relate tension to significant sarcomere lengths

.

.

.

J.

Physiol

fibrils at

(London) 184: Fig

some

of the lengths.

12. p

185.) See Fig

1

1.8 for

diagrams of probable filament relations

in

myo-

Magnitude

of Contractile Tension z

z

B

"

C

_

,,

,

J

FIGURE 11.8. Schematic drawings of filament relationships at various stages of the sliding filament process associated with significant lengths on the tension-length curve presented in Fig 117 M. myosin. A. actin: Z. Z disc A, sarcomere length 2.5 m Actin filaments are partially overlapping the myosin filaments in the A band B. sarcomere length 2.25 ft. Actin filaments are in maximal single overlap with the bridgecontaining regions of the myosin filaments C, sarcomere length 2.0 m Actin filaments have reached the center of the A band D. sarcomere length 1.5 m Z discs have collided with the ends of the myosin filaments Ends of the actin filaments have passed into the bridge area of the opposite half of the sarcomere .

the

resistance

of

antagonists,

and any load which

opposes the muscle. Intrinsic

The

Speed of Shortening

speed of a muscle reflects the rate of shortening at the sarcomere level. It is limited by the rate at which bridges can attach, move, and detach and by the rates of the chemical reactions involved. With muscle attachments severed, shortening speed of the contractile material is maximal but no tension is developed. A muscle can produce tension only when shortening against resistance, and the amount of tension developed is equal to the load. When shortening against resistance, speed varies inversely with the load. Therefore in isotonic contraction the less the resistance the more nearly maximal is the rate of shortening. This may be explained as follows: the active state arises abruptly upon stimulation and persists for a relatively fixed period of time; the less resistance which is met by the contractile material the more readily the bridges function and the greater the distance of shortening accomplished during the persistence of the active state. When a muscle is required to shorten more rapidly against the same load, less tension is produced than when shortening more slowly. This may be due to the fact that fewer links are formed between actin and myosin in the shorter time available and that the bridges which do form are detached more quickly. Consequently at higher speeds fewer bridges will be intrinsic shortening

attached at any given moment and less tension is produced. Hill has pointed out that the load determines the rate of the chemical reactions associated with contraction and that the magnitude of the velocity depends on the difference between the actual load being lifted and the maximal magnitude of force of which the muscle is capable. Barany's work supports the correlation of ATPase activity and speed of contraction in 14 different muscles of mammals, lower vertebrates, and invertebrates.

In isometric contraction, different levels of tension are

achieved at the same rate. Since no shortening is involved beyond that needed to stretch out the elastic components, the rate of tension development is constant, determined by the active state. Hence the time to reach any given tension will be proportional to the tension: lower tensions will be achieved sooner than higher tensions.

Force-Velocity Relation a scalar quantity, lacking the component of while velocity is a vector quantity having both magnitude and direction. Therefore the term speed has been used in the discussion of the rate of intrinsic shortening of the contractile material and the rate of tension development within the muscle, for direction was not significant. The term velocity is used to discuss the rate of muscle shortening against external

Speed

is

direction,

UNDERSTANDING THE

156

SCIENTIFIC BASES OF

resistance, i.e., the rate of movement, for in such considerations direction is an influential factor. The velocity at which a muscle shortens is influenced by the force that it must produce to move the load. In isotonic contraction the relationship is evidenced by the decrease in velocity as the load is increased (Fig. 11.9, solid line curve). Shortening velocity is maximal with zero load and reflects the intrinsic shortening speed of the contractile material. Velocity reaches zero with a load just too great for the muscle to lift; contraction is then isometric and maximal force can be produced.* When more muscle fibers are activated than are needed to overcome the load, the excess force is converted into increasing velocity and therefore greater distance of movement. A commonly experienced example is the exaggerated movement which occurs when one lifts a light object anticipated to be much heavier. In eccentric contraction, values for shortening velocity become negative and the muscle's ability to sustain tension increases with increased speed of lengthening, but not to the extent which might be expected from extrapolation of the shortening curve (see broken line in Fig. 11.9 extending the curve from the hyperbola of the force-velocity curve below the abscissa into the area of lengthening velocity). In isotonic contractions the length and tension of the elastic components do not change once they are sufficiently stretched to permit the load to be raised. Therefore isotonic twitch myograms reflect the velocity as well as the extent of shortening. When an excised frog gastrocnemius muscle records twitch responses with different loads, the lighter the load the higher is the twitch curve and the steeper its rising slope (Fig. 11.10). In other words, the lighter the load the greater the amount of shortening per unit of time. With a light load (Fig. 11. 10. A) the muscle's maximal velocity as measured over the steepest part of the contraction is 17 in 0.01 second or 1.7 meters per second. With a moderate load (Fig. 11.10.B) velocity is 12 in 0.01 second or 1.2 meters per second, and with a heavy load (Fig. 11.10.C) 5 in 0.01 second or 0.5 meter per second.

HUMAN MOVEMENT optimum are uneconomical because must be maintained over a longer time, hence more energy is expended to achieve the same amount of shortening. Velocities above optimum waste energy because of the need to employ a greater number of muscle fibers to achieve the same force. A plausible Velocities below

force

explanation is as follows: (1) the tension developed will be proportional to the number of bridges attached at any moment; (2) chemical reaction rates dictate that a finite and constant time is required for a bridge to attach; (3) the greater the speed at which the active sites on the actin filament move past the myosin bridges, the fewer the bridges that can attach and the less will be the tension; (4) as a result, more muscle

LOAD

FIGURE

Relationship of velocity of shortening to tension

11.9.

isotonic contraction (solid

in

(g)

line curve).

As the

load

increased,

Is

velocity of shortening decreases, reaching zero with a load just too

great

for

velocities

the

muscle to

eccentric

In

lift

become negative and

speed of lengthening (broken

contraction,

shortening

increases with increased

tension

line).

mm

mm

mm

Optimal Velocity

The force-velocity relationship may also be stated conversely in terms of the influence of velocity upon force: a rapidly contracting muscle generates less force than does one contracting more slowly. From the standpoint of efficiency, however, energy expenditure is least

when work

is

done

at

moderate

velocity.

With any

given load, if the velocity of shortening is gradually increased, the work output of the muscle rises at first, reaches a peak, and then declines. Therefore, for any load the optimal velocity lies somewhere intermediate to the slowest and fastest shortening rates. Furthermore, the greater the load the lower is the optimal velocity with which it can be moved.

FIGURE 11.10. Shortening velocity of excised frog gastrocnemius with three different loads. Stimulus (s): a. 001 second taken at steepest part of contraction; b. amount shortening

of

light

load,

steepest *

As previously

stated, greater forces can be sustained in eccentric

1.7

m

part

per

contraction than can be produced in either isometric or isotonic con-

per 0.01

traction.

is

5

mm

in

of

sec.

sec or

during

millimeters

maximal the B.

1.2

of

rate

contraction)

with

m

per 0.01 sec or

a

17

is

moderate

per sec. 5

that

m

time

shortening

C.

(as

mm

load,

unit

A.

with

over

taken per

01

velocity

is

sec

12

a

the or

mm

with a heavy load, velocity

per sec.

Magnitude l

\IU F

il.

l.

Compajrisoa of

fust

ami >lo« muscle

TiIhts: Histological, physiological,

of Contractile Tension

and biochemical

K.1M

Slow

Histological differences

and color

S

Fibers smaller ami redder because

Fibers large ami pale

"i

greater

myo

globin content

plasm

Granular sarcoplasm

Agranular sarcoplasm

Many

Fibrils

Fewer

fibrils

mitochondria but few Narrow / discs l-ir^e

Z discs

T

d

system

End

plates

Blood supply

)

A-I junctions

SR

sparse and rudimentary;

is

Single innervation bj Large somatic motor neurons with fast

fibrils

Numerous small mitochondria Wider / discs (about 2

number

Sarcoplasmic reticulum abundant and well developed;

T tubules (and triads) at Innervation

m

found

at

system, when present,

'1'

the / lines

Innervation by small, slow -conducting somatic motor neurons; multiple innervation

conduction rates

Discrete (en plaque) end plates with

many

sarcolem-

mal folds Few capillaries except those shared with adjacent

in

some

species.

Some

autonomic neurons Diffuse (en grappe) end plates with few or no junctional folds

slow-

fibers

Dense capillary supply, located angles between fibers

at

the interstitial

Physiological differences

Contraction cycle

Slower cycle (2-3 x); graded contraction

Rapid contraction-relaxation cycle

in

some

muscles Vetanus Potentials

Rapid onset of tetanic fusion but only at high frequencies and short-lasting Higher resting potential Larger end plate and action potentials

Slower onset of fusion but at lower frequencies and of longer duration

Lower

resting potential

Smaller end plate and action potentials, latter often

graded Active state

Endurance Tension capacity Elasticity

Rapid initial decay of active state Rapid fatigue Higher tension which develops rapidly Lower coefficient of elasticity

Slower

initial

decay

Greater endurance

Lower tension and slower development Greater elasticity

Biochemical differences

Metabolism Myoglobin content Glycogen Na and K"

Metabolism primarily

ATPase Low

glycolytic (as indicated

by high

High

Large glycogen storage

Variable glycogen storage

Less Na" and more K~; rapid loss of

K

+

during stim-

acids

More Na" and

Differences in concentration of various

fibers must be recruited to achieve the necessary force. Optimal speed probably reflects the greatest speed which will still allow a sufficient number of bridges to

attach to provide the required tension. Most individuals will unconsciously perform at optimal velocity if allowed to do so. Optima vary for the same individual with different loads and in different types of activity and among individuals for the same load or activity. In athletic performance efficiency is often sacrificed for other objectives, and rates above optimum are deliberately adopted. There is evidence that strength increase in a trained muscle is due to an increase in the number of fibrils per fiber. This would mean that there would be more bridges within the fiber which could attach per unit of time, and the force capability at any given velocity would be correspondingly greater. A trained muscle should be able to develop a given force more rapidly. In other words training should be expected to improve both force and optimal velocity of contraction. In isolated vertebrate muscle the Q of velocity* is 10 refers to the extent to

KT

;

rate of

K+

depletion dimin-

amino acids

Although studies have not been made directly in man, empiric evidence suggests that the velocity of muscle shortening is improved by "warming up." Experiments on the velocity effects of local warming or increased core temperature might indicate that artificial prewarming of athletes to a safe degree would improve their performance records. 2.5.

Slow and Fast Muscle Fibers Although

the

previous

which the rate of a chemical reaction

discussion

striated muscle in general, there

has

considered

abundant evidence

is

there are two types of skeletal muscle, distinguishable by speed of contraction and endurance. Almost 100 years ago Ranvier observed that some muscles of the rabbit were redder in color and that those muscles contracted in a slower and more sustained manner than did the paler muscles of the same animal. Since then the designations of red and that

is

increased by a 10°C. rise in temperature.

within the range 15 to 35°C.

Q

less

ishes with continued stimulation

ulation

Amino

Oxidative metabolism (as indicated by high succinic

dehydrogenase activity)

activity)

A Q 10

It

is

usually measured

of 2 indicates that the rate

doubled. For most biochemical reactions Q,„

lies

between

2.5

and

is

3.0.

UNDERSTANDING THE

158

SCIENTIFIC BASES OF

white muscles have become synonymous with slow and fast contraction. In addition to a slower contractionrelaxation cycle, red muscles have lower thresholds, tetanize at lower frequencies, fatigue less rapidly, and are more sensitive to stretch than the faster white

muscles. expected, individual muscle fibers behavior. contractile in differences these Investigations by a number of workers both here and abroad have revealed histological and biochemical differences which distinguish the two types of muscle fibers and which correlate with the physiological differences between fast (white) and slow (red) muscles.

As

might be

reflect

These are listed in Table 11.1. Examination of the table suggests obvious relationships between the several categories which are consistent with differences in speed and endurance. The

FIGURE activity

muscle

11.11.

HUMAN MOVEMENT larger size, greater density of fibrils, and lower viscosity of the white fibers should contribute to greater

speed. Their abundant sarcoplasmic reticulum and T tubules located at the A-I junctions should, by providing for the transport of glycolytic enzymes and for large scale release of Ca ++ ions in the vicinity of the cross bridges, also favor fast response. Conversely, in the red fibers reduction in the sarcoplasmic reticulum and sparsity of T tubules are appropriate to slower response. The discrete end plates and multiple folds in the subneural sarcolemma of the fast fibers may be expected to provide for increased activation. Where multiple innervation exists in slow fibers, it is consistent with the small graded action potentials which distinguish them from the fast fibers with their rapidly rising, larger potentials.

The

large glycogen stores

and high ATPase

Fast and slow muscle fibers. Rat muscle fibers stained to show succinic dehydrogenase (x230) of the medial head of the gastrocnemius muscle showing three types of

A. cross section fibers:

A, fast fibers

(light);

B.

intermediate fibers (slow); C. slowest fibers (dark)

(x125) from the soleus muscle shows only fibers are

indicated by arrows. (From Stein.

of individual

muscle

fibers of the rat.

fiber J.

types B and C M..

Amer.J.Anat

1

Two

and Padykula, 10(2); Fig 5. p

H.

B.

cross section

type C (darker) and one type B (lighter) A.,

1962. Histochemical classification

121 and

Fig. 12. p.

123.)

activity

Magnitude in the white fibers will favor speed of response but quicker fatigue is to be expected because of their lesser blood supply ami predominantly glycolytic metabolism. rhe increased endurance of the red fibers is consistent with their rich blood supply and abundance of mitochondria which support an essentially oxidative their small Furthermore, diameters metabolism.

provide a greater surface for exchange of gases, ions, provided by an equivalent is Rapid K depletion fibers. progressively alters the ionic gradients and ultimately limits the ability of the last fibers to perform work. In the slow fibers presumably a steady state is reached in which K gain resulting from recovery processes is in equilibrium with the loss incurred during contraction. As a result, endurance is enhanced. The greater elasticity and slower initial decay of the active state which is characteristic of the red fibers can account for their mechanical fusion at lower frequencies and for their prolonged twitch times. Most of man's striated muscles contain both types of fibers but in differing proportions which determine the color of each muscle. Some show a characteristic arrangement or zonation of the fiber types within the in others the two types are randomly muscle; distributed. In such muscles as the gastrocnemius,

and metabolites than mass of larger white

of Contractus Tension

161

and flexor digitorum longus, last fiber predominate, although slow fibers may also be present In many mammals the soleus muscle appears to consist entirely of slow libers. The preponderantly slow fibercd muscles arc the antigravity muscles, adapted for continuous body support. Their sensitivity to stretch results in a continuous mild (tonic) activity even at rest. The predominantly last-fibered muscles are phasic muscles which produce quick postural changes and fine skilled movements. At rest they are electrically tibialis anterior,

silent.

Recent studies indicate that in mammals the slow red should be further subdivided on the basis of differences in enzymatic activity, especially succinic dehydrogenase and ATPase activity, into two subcategories. As a result, muscle fibers may be classified into three types, which have been designated as A, B fibers

and

C.

Type A

fibers represent the classic fast, pale

B and C represent two types of slow red fibers. Stein and Padykula found all three types present in the rat gastrocnemius, with type A predominant (Fig. 11. 11. A). In the soleus, A fibers were absent but the muscle contained both types B and C whereas types

fibers,

(Fig.

ll.ll.B).

The muscle

significance of slow and fast characteristics of fibers is discussed further in Chapter 13.

SUMMARY Important aspects of Chapters 10 and 11

summarized as follows. The study of muscle from many by

many

may be

different animals

and

different investigators has revealed that the

following contractile features are

common to all (Hanson

and Lowyi. 1. Muscles cardiac

of every type — striated, smooth, and — contain actomyosin. and actomyosins prepared

from different muscles react with ATPase in the same manner. Furthermore, no differences have been found in the ATPase from different muscles or species. 2. In all types of muscle the molecules of the contractile proteins are grouped into filaments which are thick enough to be seen in electron micrographs. 3. For striated muscle there is as yet no convincing evidence that the filaments themselves shorten, but there is abundant evidence which is compatible with the sliding filament theory. 4.

Tropomyosin

is

a constituent of the contractile

systems of all muscles. Of the two kinds of tropomyosin which have been recognized, one, tropomyosin B, is

common

to all muscles. (Tropomyosin A, or paramyosin. is found in molluscan muscle and appears to be associated with its "catch" mechanism by which tension is maintained to hold bivalve shells closed for long periods of time without evidence of fatigue and with very small expenditure of energy.) 5. The tension exerted by active muscle is a function of its length and is maximal at about the greatest length

that the muscle can assume in the living animal. Tension decreases nearly linearly above and below this length. The shape of the tension-length curve for isometric contraction is similar for all muscles tested. 6. A muscle which is being lengthened while it is contracting can maintain greater tension than it can develop at any given equivalent static length. Therefore, tensions greater than the isometric maximal can be recorded in eccentric contraction. 7. Velocity of shortening of the contractile material decreases with increasing load in a hyperbolic manner. When velocity is expressed as a percentage of maximal velocity at zero force and force is expressed as a percentage of maximal force at zero velocity, the forcevelocity curve is essentially the same for all muscles. 8. Quick release produces the same effects in all muscles: tension falls immediately and then re-develops to a value characteristic of the shorter length. 9. The rate of the development of tension depends on the intrinsic shortening velocity of the contractile material, compliance of the elastic components, and the rate of decay of the active state. The existence of such extensive similarities among muscles provides a measure of confidence in assuming that information on the properties of muscle derived

from animal studies may also apply to man. However, reasonable caution and good judgment must be exercised

when

direct confirmation

is

lacking.

SECTION TWO

THAPTKK

NEUROPHYSIOLOGY

12

Basic Neurophysiology when, how

Skeletal muscles are under the control of the nervous system which determines which muscles shall contract.

in force

fast, to

and

what extent, and with what changes

velocity from

moment

to

moment.

THE NEURON: STRUCTURE AND FUNCTION Morphology of the Neuron

sisted

from the days before techniques

for identification

of myelin were as refined as they are today. Larger

The nervous system is composed of two types of cells: neurons and neuroglia. We are concerned only with the first oi these. Neurons are usually greatly elongated cells with diameters ranging from 0.5 n in small unmyelinated fibers to 22 n in the largest myelinated fibers. Some exceed 1 meter in length. They are specialized to receive, conduct, and transmit excitation. A generalized neural cell or neuron consists of four morphologically and physiologically distinct portions: a receiving pole, a terminal transmitting pole, an intervening conducting segment, and a cell body or soma. Each is specialized for its particular role in the cell's function. Most neurons possess two types of protoplasmic processes extending outward from the nucleated soma: dendrites and axons. The processes vary in length and in the amount and extent of their branching. Dendrites are usually multiple, short, and highly branched. The space occupied by their three dimen-

axons are enclosed in increasingly more numerous sheathing layers formed by more and more windings of sheath cell processes. As the folds become tightly packed together, most of the cytoplasm is squeezed out so that the sheath is ultimately composed of concentric

layers

of

lipid-rich

cellular

membrane. The

myelin sheath of the larger axons is segmented rather than continuous, and each segment is contributed by a single sheath cell. The length of the segments and the thickness of the myelin are fairly constant for neurons of a given caliber, larger axons having longer segments and thicker sheaths. The segments are separated by short, unmyelinated gaps, the nodes of Ranvier. Collateral branches, when present, arise at nodal gaps and leave the parent axon at approximately right angles (Fig. 12.1).

The

area of axon outgrowth from the nucleated poris known as the axon hillock. Nerve impulses are generated in the initial segment of the axon which, even in myelinated fibers, is unmyelinated. Axon ends usually divide distally into a spray of terminals, the telodendria, which lose the myelin sheath and end tion of the cell

is often extensive. They constitute the receiving pole of the cell. Axons are usually single, long and, although one or more collateral branches may occur,

sional spread

they are relatively unbranched except at their ends. The axon is responsible for both conduction of excitation and its transmission to other cells.

in

naked

The

tips.

body or soma of the neuron is the metabolic center of the cell where, under control of its single nucleus, proteins and other metabolically important substances (enzymes, transmitter substances, neurohormones, etc.) are elaborated. The fact that materials are moved from the cell body into and along the neuronal processes has been well established in the last decade. The channel for transport is probably provided by the endoplasmic reticular system of the cell body which connects with microtubules in the cytoplasm of the neural processes. When severed from the nucleated portion of the cell, a nerve process will soon degenerate because it is no longer supplied with essential materials, p.nd a new process will grow out from the cut stump which is still attached to the cell body. Location of the soma is the major difference among

An axon

generates action potentials (nerve impulses) and conducts them from the receiving portion of the cell to the transmitting region. It is a delicate cylinder of neural cytoplasm with a limiting membrane, the axolemma. It varies in length and in diameter in different types of neurons. Axons are enclosed in a cellular sheath of lipid material, the myelin sheath, which serves to separate individual axons from one another and from adjacent neural components. The myelin sheath is formed by concentric wrappings of membranous processes from sheath cells, called oligodendrocytes within the central nervous system and Schwann cells in the peripheral nervous system. Small axons which are invested by only a single layer of sheath cell process are called "unmyelinated" fibers, a term which has per-

161

cell

162

UNDERSTANDING THE

SCIENTIFIC BASES OF

nerve cells. In vertebrates the nucleated portion of most neurons, including motor neurons and interneurons, is a part of the receiving region of the cell. The somata of sensory neurons from the skin, however, have been displaced centripetally along the course of the axon where they are better protected from injury than if situated peripherally with the receiving structures (Fig. 12.2). In these cells the dendrites communicate directly with the axon, and the cell body does not participate in the reception of excitation. In other words, the part of the fiber, often myelinated, which conducts toward the cell body and the portion leading from the cell body are both parts of the axon. Only the peripheral terminals are dendrites, while the central terminals are telodendria.

Mitochondria are present in axons, especially at the nodal areas, and are numerous in the cell body and in both the receiving and transmitting portions of the neuron, being abundant in the latter. Ribosomes are mostly restricted to the cell body. Minute and unique neurofilaments, whose function is as yet unknown, are distributed throughout the cytoplasm. Normally, excitation is conducted only from the re-

HUMAN MOVEMENT ceiving to the transmitting pole of the cell. This polarity derives from the fact that nerve cells in the body are stimulated only at the receiving end, because it can be shown experimentally that an axon which is stimulated at a point along its length is capable of conduction in

both directions.

Neurons may be classified as either receptor neurons or synaptic neurons on the basis of the type of input which they receive. Receptor neurons are those which

and transduce environmental energy such as sound, heat, or chemical or electrical energy. They are specialized to be excited by specific types of stimuli, and their dendritic portions are appropriately modified in structure (Fig. 12. 2. B). Synaptic neurons (Fig. 12. 2. C) receive information from other neurons by means of synaptic transmission. Their dendritic geometry may be extensive and complex, providing a wide field for reception of a great number and variety of inputs, all of which are already encoded in the manner characteristic of nervous system communication. receive

light,

Physiology of the Neuron: Excitation and Conduction

Membrane Theory

of

Irritability is a characteristic property of all protoplasm, but types of cells differ widely in the extent to which they display the property. It is most highly developed in nerve and skeletal muscle cells. Excitation is induced in a cell by appropriate stimulation and is associated with chemical and electrical changes which spread over the cell membrane. In many types of cells the change is graded and spreads decrementally from the point of stimulation. In nerve axons and muscle cells, however, if the stimulus is adequate, the change is conducted without decrement as an all-or-none action potential. Adequate spread of excitation evokes the characteristic response of the cell. In a muscle cell the response is contraction; in a gland cell, secretion. The essential function of a nerve cell is to transmit excitation to other cells, and it responds by releasing a chemical transmitter substance at its synaptic terminals. Although neurons may be artificially excited by a number of different kinds of stimuli, with a few exceptions their normal stimulus is the action upon their membranes of the chemical transmitters released by other neurons. Exceptions include the stimulation of receptor neurons by pressure, vibration, heat, etc., and in a few instances, rare among the vertebrates, direct

electrotonic excitation of one neuron bv another.

Resting

FIGURE

12.1. Multipolar neuron. A. dendrites of the receiving axon or conducting segment C. telodendria and terminal arborizations of the transmitting pole 7. nucleus: 2, cell body or soma (/ and 2 compose the metabolic center of the cell): 3. axon hillock: 4. initial segment of axon; 5. myelin sheath segment: 6. node of Ranvier (naked membrane exposed); 7. collateral branch of the axon pole.

B.

Membrane

Potential

Action potential generation and conduction in the neuron axolemma and the muscle sarcolemma are essentially identical. The present discussion describes these events as they occur in the neuron. The axon membrane in the unexcited or resting state is polarized as a result of a differential distribution of ions on the two sides of the membrane. The cations potassium (K + ) and sodium (Na + ), the anion chloride

Basic Neurophysiology

163

DENDK

AV\ (

ORIGIN

impulse tniti.»'

AX.N tail

ZONE

C

or

none

cono.

TELOOENDRIA tchemical trans'"

-

OUTPUT

A FIGURE

12.2.

C

B Types

of neurons. A,

diagrammatic representation of the three functional portions of the

neuron, showing the dendritic or receiving pole where excitation causes graded electrogenesis; the axonic or conducting

segment which conducts the all-or-none impulses originating in the initial segment; and the where excitation is transmitted by chemical means from the tips of the telo-

synaptic or transmitting pole

B and C: representative types of neurons B, receptor neurons. /, special sensory neuron; 2. cutaneous neuron Note that the nucleated portion is situated along the course of the axon These cell bodies

dendria

are located

in

motor neuron; cleated portion

the dorsal root ganglia or 2. is

interneuron located

Cold Spring Harbor

in

in

homologue

a

These are the most

types

the receptor pole of the neuron

Symp Quant

Biol.

30:

393

in

C,

synaptic neurons.

1

the nervous system

(After Dowling, J

E

.

Note that the nuand Boycott. B B 1965 .,

)

I. and certain organic anions are the ions most importantly concerned. The differences in ion concentrations reflect the selective permeability of the membrane. While K~ and CI" pass readily through the membrane, Na~ passes only with difficulty and then is promptly ejected by an active transport mechanism which, although little understood, is called the sodium pump. Sodium accumulates in the intercellular fluid outside the cell in a concentration which is about 10 times greater than that inside the cell. Potassium is about 30 times more concentrated inside the cell than outside, and the chloride concentration is about 14 times greater outside than inside. When there is an unequal distribution of an ion on two sides of a permeable membrane, two types of forces act upon that ion. First, the chemical gradient results in a diffusion force whereby ions may be drawn through the membrane away from the area of their own greater concentration into the area of their own lesser concentration. Consequently, in the resting nerve cell the chemical gradients tend to drive sodium into and potassium out of the cell. The second force is electrostatic attraction whereby an electrically charged area attracts ions of opposite charge and oppositely charged ions attract each other. The driving voltage for a particular ion is the difference between the value of the membrane

'CI

of a dorsal root ganglion

common

potential

and the equilibrium potential*

for that ion.

In the resting condition the ionic currents balance each other exactly and the membrane potential remains

constant. In the resting cell

Na +

kept out by the action of the and holds a considerable amount of Cl~. Although CI", being in higher concentration in the intercellular fluid, tends to diffuse into the cell, its inward diffusion force is balanced by the electrical attraction of the positively charged Na + and the chloride ions remain in equilibrium. Inside the cell, large, negatively charged protein molecules, which were formed within the cell and are too large to pass through the membrane, exert an electrical attraction on cations. Therefore, since Na + cannot remain within the cell, K + is drawn into the cell and to a significant extent held there, accounting for the high concentration of potassium on the inside. The inward attraction ex-

sodium

pump,

,

attracts

,

*

The equilibrium potential is the electrical difference which must membrane to maintain the ionic concentration gradi-

exist across the

ent. Its

magnitude

for a particular ion

is

concentrations of the ion. equation.

and efflux and external value may be calculated from the Nernst

just sufficient to equalize the influx

and depends on the Its

The equilibrium

potential for Cl~

perimentally measured resting cle cells.

ratio of internal

membrane

is

identical with the ex-

potential of nerve

and mus-

164

UNDERSTANDING THE N

SCIENTIFIC BASES OF

HUMAN MOVEMENT

SIDE

+ + ,

\v''-''.

, .

.'.'-''. ; ', ».-iTJ

+ + + + +

!^r4.

+

+

-t-

-4-

+

•+-

-*-

+

+•

-f

-I-

-t-

+ +

+ -f

OUT SIDE Resting membrane. A Because of permeability properties of the membrane. K^and CI pass + passes with difficulty and is promptly ejected by the sodium pump readily through the membrane. Na + mechanism As a result. K accmumulates inside the cell and Na+ outside Organic anions, each indicated

FIGURE

12.3.

t>VVv are t0 ° of the

'

ar 9 e t° leave the cell

membrane

is

See

text. B.

As

K + by the organic anions is opposed by the chemical gradient tending to drive it out. Because the two antagonistic forces are not in equilibrium, the outward-directed chemical force being stronger than the inward-directed electrical attraction, enough K + remains outside to produce a balance between the two erted on

12. 3. A). As a final result, there is a deficiency of positive ions inside the cell and the net charge on the inside of the membrane is negative, while the charge on the outside, where there is an excess of

forces (Fig.

positive ions,

The

is

positive (Fig. 12. 3. B).

is often called a potassium podue to the excess K + in the intercellular fluid. However, it is the active removal of Na + at a rate equal to the net rates of entry of K + and CI that is the means by which the cell maintains its normal concentration differences. Other ions distribute themselves in a Donnan equilibrium as indicated by their

resting potential

tential because

it

is

concentration ratios.

The electrical difference between the inside and the outside of the membrane of an unexcited cell is the membrane or resting potential* and ranges from 40 to 100 millivolts (mv) in nerve and muscle cells. Excitation: Generation of Action Potential

The

forces acting on

a result of the differential distribution of ions, the interior

negatively charged while the exterior

sodium and potassium across

membrane are importantly involved in cell excitation. Any change in conditions producing an alteration in membrane permeability will result in movement the resting

of these ions in response to their driving forces,

and the

resting equilibrium will be upset. The time course of change in permeability of the cell membrane to a particular ion is expressed as conductance for that ion. Any change in conductance for one ion will appreciably

is

positively charged.

and hence alter the membrane any change in membrane poten-

affect that of other ions

potential. Conversely, tial will

influence conductances. Stimulation of a nerve

axon appears to increase membrane permeability, and hence conductance, to Na + at the point of stimulation. Sodium, driven by both its chemical and electrical gradients, passes into the cell. Because sodium is a positively charged ion, its entry decreases the negativity

and at the same time and to a similar extent decreases the positivity outside. In other words the value of the resting potential is lowered toward zero; the membrane is being depolarized. A change of resting potential in the depolarizing direction constitutes a inside

local excitatory state of l.e.s. (Fig. 12.4,

All biological "potentials" are actually potential differences.

B).

net influx of sodium is slow at first, but the depolarization caused by its entry produces a self-regenerative increase in permeability so that the rate of Na + influx and the rate of depolarization increase exponentially. This is an example of positive feedback. If the resting potential drops to a sufficient extent, it will reach a critical level which is characteristic and constant for each cell. In mammalian nerve and muscle cells critical depolarization levels range between 10 and 20 mv below the resting potential. At the critical level something happens suddenly which seems to throw the sodium gates wide open. Sodium rushes in to such an extent that the membrane potential in the stimulated area passes beyond zero and the polarization is reversed: the membrane becomes negative outside and positive inside at the point of stimulation. This almost instantaneous change in potential, which appears on the oscilloscope as a spike (Fig. 12. 4. C), is known as the action potential. t The more intense the stimulus the t

The

action potential

is

a

sodium potential and leads

nation of the hypothesis as the *

A and

The

tion potential.

sodium theory

to the desig-

of the nature of the ac-

Basic Neurophysiology

sooner depolarization reaches the critical level ami the earlier the spike appears The rising phase of the spike (depolarization) is acSodium conductance counted for by the influx of Na rises rapidly, in 0.1 to 0.2 millisecond tins), and then tails rapidly to a low level. Potassium conductance does not change appreciably at first hut then becomes

A

B

.

is inactivated. An efresponsible for the falling phase of the spike (repolarization). The separation in time of these two conductance changes accounts for the change in the membrane potential which constitutes the action

marked

as

K

flux oi

sodium conductance is

potential.

The magnitude

of the action potential

is

the alge-

between the resting and the active (reversed) potential values as recorded from the inside difference

braic

of the

memhrane.

tential is

which

In Figure 12. 4. C the

membrane

po-

mv (inside) in the resting state mv upon adequate excitation. The

70

is

reversed to - 30 potential is

action

70

mv

therefore: -

(+30 mv)

100

mv

Eq. 12.1

No action potential occurs unless depolarization reaches the critical level. If the stimulus is inadequate and does not result in the entry of enough sodium to depolarize the membrane sufficiently, the local excitatory

state

soon

tive reactions,

dies

away

mainly

K-

(Fig. efflux,

12. 4. B).

Regenera-

restore the resting

The sodium pump gradually ejects the sodium, and eventually the resting ion distribution is regained. If. however, subsequent subthreshold stimuli are applied in such rapid succession that each succeeding stimulus evokes its l.e.s. before that of the preceding stimulus has dwindled away, the local excitatory states polarization.

summate. When the critical level action potential will be generated. This

will

designated

summation

is

reached, an

phenomenon

is

of inadequate stimuli (Fig.

12.4TJ).

TTT 1

FIGURE the

12.4.

membrane

2 3

1

Membrane potential

in

cate application of stimulus is

—70 mv

reach the

(inside)

2 3

changes with time. Ordinate: millivolts. Abscissa: time. Arrows indi-

potential

A, resting state:

membrane

Depolarization of at least 15

critical level of

-55 mv

mv

is

potential

required to

B, local excitatory state (l.e.s.)

Application of an inadequate stimulus (arrow) partially depolarizes

Recovery of Resting State

The action potential occupies a definite distance of the nerve fiber (5 to 6 cm in the largest mammalian neurons), lasts for a definite duration (about 0.4 ms), and then the membrane potential is rapidly returned to the resting state (Fig. 12.5). Because the action of the sodium pump is relatively slow, recovery of the resting polarization requires a faster mechanism. The outflow of K~, as its conductance increases, brings about the almost immediate recovery of positivity outside and negativity inside. Potassium is driven outward by its chemical gradient and also by the reversal of the electrical force as the cell interior becomes positively charged. Ion distributions will be eventually restored by the slower action of the sodium pump. It has been suggested that the same carrier which actively transports Na + out of the cell brings K~ back, exchanging the ions one for one. Although no metabolic energy is required for the electrical changes during activity, energy is required for the action of the sodium pump by means of which ion con-

the

membrane, producing

a

l.e.s.

Since

critical level is

not attained

dwindles away and the resting state is restored (2). C, action potential. An adequate stimulus (arrow) induces a l.e.s. (1) which quickly reaches the critical level and the membrane potential suddenly reverses to +30 mv (inside), producing an action potential of 100 mv (2) Restoration processes act promptly and start to return the membrane to its resting po(1),

no spike

is

generated. The

l.e.s.

The process slows down, producing the short negative afterwhich the membrane is still slightly depolarized and hence abnormally irritable. This is followed by a longer lasting positive afterpotential (4) which is an overshoot resulting in slight hyperpolarization The resting poand subnormal irritability tential (5) is gradually regained. D, summation of inadequate stimuli. The l.e.s. of three inadequate stimuli summate (1) to reach the critical level (2), and an action potential spike is generated, followed as in C by negative (3) and positive (4) afterpotentials. and ultimate recovery (5) tential.

potential (3), during

centration gradients are restored and maintained. Ion pumping goes on continuously, even during the generation of the action potential. A small increase in energy expenditure occurs during recovery and is related to

UNDERSTANDING THE

166

+

++ +

-H

+

SCIENTIFIC BASES OF

+++++-+++

HUMAN MOVEMENT

+ 4.

4.

^ #y. -s+

4.

4. 4:

+,

+

+'+

4.

^^

+/-y^

2

1

H-4--F-r--t-4--»-4-4--«-4-i-

^

FIGURE

12.6. Saltatory conduction in a myelinated fiber. 1. node of Ranvier; 2. inactive node of Ranvier; ms. myelin sheath segment Arrows indicate flow of action current. Direction active

-f 4- 4-

12

of impulse

4

3

from left to right In myelinated nerve from node to node without depolarization of internodal portions of the axon Only a section of the surconduction

is

fibers the action current skips

5

face

membrane

is

shown

is

profile,

with the outside of the fiber

above and the inside below

•

+•

— — — + + + + +

-»-

-f-

+

•f-

1-

+ + +

axon terminals. Conduction

12

4

3

+-(-+--(-+

-

5

+-

+ 4

4-4-4

12 ++ + +

+

FIGURE

12.5.

3

Electrical

conduction, and recovery

4

5

+ +

F

4-

4

changes associated with excitation, in an unmyelinated cell membrane,

Resting membrane is positively charged on and negatively charged on the inside (Charges are shown on only one surface of the profile Numbers indicate sequential areas of the membrane b, excitation. An adequate stimulus

a,

resting

the

membrane.

outside

)

(arrow)

results

reversal

in

stimulation (area

1)

An

of polarity

(negativity)

at

the point of

action current then flows from positively

charged (outside) inactive area 2 to negatively charged (outside)

and along the inside of the membrane, emergc to e conduction, c: Emergence of action current stimulates the membrane of inactive area 2. which then becomes depolarized The action current now flows from presently inactive area 3 to newly active area 2 d and e: the action potential is self-propagated along the fiber to areas 3 and 4 Recovery: in c. d, and e. the area behind the action potential gradactive area

1.

into

ing through the inactive area

ually

recovers

its

original

polarization

(positively

charged on the

outside).

is accomplished by selfpropagation of the disturbance away from the site of origin. Adjacent inactive areas on the outside of the membrane are still positively charged at the resting level. Since the extracellular fluid is electrolytic, a small current, the action current, flows from the positively charged inactive region to the negatively charged active area where it passes in through the membrane, through the cell fluids, and out again through the inactive region. Although small, the current is sufficiently strong memto constitute a stimulus capable of increasing brane permeability as it emerges. The same sequence of excitatory events is repeated here: sodium moves in, polarity is reversed at this point, and the action potential has progressed along the fiber. This is repeated again and again until the action potential reaches the end of the axon terminal (Fig. 12.5). The generation of an action potential involves relatively few ions, so few as to be chemically undetectable although the electrical changes are readily measurable. Therefore a neuron can continue to conduct for hours without any cessation of activity to provide for restoration of the original ionic

distributions.

In unmyelinated fibers such as the autonomic postganglionic neurons and the afferent fibers for dull pain, conduction is accomplished by progression of the action potential down the fiber as described above. In myelinated fibers, however, the action current skips from node to node without depolarizing the internodal portions of the axon (Fig. 12.6). This is saltatory conduction, and the velocity of conduction is considerably greater than in unmyelinated fibers. Because the same exchange of ions occurs less times for a given length of fiber, saltatory conduction involves fewer ions and hence requires less energy for recovery. This is an

advantage which permits myelinated fibers to continue transmitting for some time even in the absence of the redistribution of ions by increased activity of the

oxygen.

pumping mechanism.

Some Conduction of Action Potential

The action potential which is generated at the site of stimulation must be conducted along the fiber to the

Characteristics of

Nerve Conduction

Refractory Periods. As the action potential travels along the fiber surface, it consists of a wave of negativity followed by an area of gradually recovering positivity

16.

Basic Neurophysiology

rABLE

IS,

Comparison of classification systems and sensory nerve fibers

1.

for

motor

Stmoq Group

lVnmnittumt

Vrloolv

etrr

llr.'up

Ih.llml.'l

U

-

\ ,l.i,

•',

;

n\

Origin in',,

i

it.

\

19

GO 100

20 -

15 ;U>

1

i

B C

Muscle

90 TO

6 12

12

30

3

15

0.5-2

0.5-1 ling to

libers

II

Classified according to Lloyd.

-

IF. intrafusal fibers;

J

GTO,

Fig.

Blood vessels?)

ANS ANS

Spindle,

30 to

Spindle, skin,

12-30

Pain

joint joint

III

2-6

IV

0.5-1

preganglionic

postganglionic

0.5-2

Pain

ANS. autonomic nervous system.

While an area

is in

period.

Afterpotentials. During the relative refractory peamplitude and velocity of the spike are altered, reflecting changed conditions in the fiber. In some neurons the latter portion of the downward course of the spike is considerably less rapid than its rise, showing a marked concavity before reaching its initial level. This is the negative afterpotential because it indicates a delay in return to the resting potential and a prolongation of some slight depolarization (hence negativel. During this period of 12 to 80 ms, the membrane is hyperexcitable or supernormal and hence more easily restimulated. The recovery may continue into a hyperpolarized state, the positive afterpotential, which persists for a much longer time, up to 1 full second, during which the membrane is subnormal in excitability riod both the

(Fig.

GTO,

70 120

Golgi tendon organ.

L2.5). it

12

Erlanger and Gasser.

-

state,

(i

Spindle IKs i

is in its reversed (active) absolute refraction and cannot be restimulated. During recovery, the membrane is relatively retractor., a state which lasts many times longer than the absolute refractory period. Intense or sustained stimuli may restimulate the original site during repolarization. The refractory periods are almost entirely due to the conductance changes. Inactivation of sodium conductance decreases excitability because a greater depolarization would be required to produce further increase in Na~ conductance to a point where its net influx would exceed the efflux of K~. Elevation of K> conductance gradually restores resting polarization and hence excitability. Thus the inactivation of Na + conductance and the elevation of K^ conductance account for both the absolute refractory period and the relative refractory

i

12 22

1

\\.>ns to Bpindle IFs

12.4.

C and

D).

The

positive afterpotential

is

probably due to a delay in the restoration of K + conductance to its normal level. Frequency of Impulses. In general, natural stimuli are of sufficient duration to reactivate the membrane after the absolute refractory period. For this reason neurons normally carry trains of impulses. A single electric shock may produce a single action potential but only because its duration does not outlast the refractory period of the fiber. The stronger the stimulus the earlier

it will re-excite, the shorter will be the time span between impulses and the greater the frequency. Because each action potential is followed by an absolute refractory period, action potentials cannot summate* but remain separate and discrete and, because of

the characteristics of the relative refractory period, neurons do not conduct impulse frequencies as high as the absolute refractory period would suggest. A fiber with a spike duration of 0.4 ms might be expected to conduct impulses at a frequency of 2500 per second but its upper limit will actually be closer to 1000 per second. For reasons not well understood, frequencies rarely approximate their possible maxima. Motor neurons usually conduct at frequencies of 20 to 40, rarely as high as 50, impulses per second, and upper limit frequencies for sensory neurons normally lie between 100 and 200 impulses per second instead of between 800 and 1000. Velocity of Conduction. Velocity of conduction depends not only on myelination but, more importantly, on the diameter of the fiber. It can be fairly accurately predicted from the following equation. Velocity in m/sec = 6 x diameter in ^

Eq. 12.2

Hence the largest motor and sensory nerve fibers, with diameters near 20/i, have conduction velocities up to 120 meters per second. f In small unmyelinated fibers, velocities range from 0.7 to 2 meters per second. Large fibers not only conduct more rapidly than small fibers but characteristically have lower stimulus thresholds and larger spikes with shorter durations. Classification of Nerve Fibers. As a result of the classic experiments of Erlanger and Gasser in 1937, nerve fibers are classified into three major groups, A, B, and C, on the basis of conduction velocities. Group C contains the unmyelinated postganglionic fibers and group B the small myelinated preganglionic fibers of the autonomic nervous system. Group A includes the *

The

l.e.s.

displays no refractoriness and hence

summation

is

possible at subthreshold levels. t In 5

mm

mammalian per second.

skeletal

muscle

fibers,

conduction velocity

is

about

UNDERSTANDING THE

168

SCIENTIFIC BASES OF

large, rapidly conducting myelinated somatic fibers. It has been further divided into four subgroups: alpha (a), beta (0), gamma (7) and delta (5) on the basis of velocity and diameter. The fastest fibers are those with the largest diameters. Sensory nerve fibers have more recently been separately classified by Lloyd according to diameter into groups I, II, III, and IV, with corresponding velocities. These do not correspond exactly in size and velocity to the subgroupings of the Erlanger-Gasser class, but among afferents from the skin and muscles

HUMAN MOVEMENT group

and

approximates A„ and group

I

Ay

II

approximates

Aa

In order to avoid confusion, use of the alphabetical designations is restricted to efferent fibers and .

the numerical designations to afferent fibers. Table 12.1 provides a comparison of the two classifications as related to both motor and sensory fibers. As indicated by the table's columns for termination of motor fibers and origin of sensory fibers, it is obvious that specific structures tend to be innervated by fibers of quite specific

size

and conduction

characteristics.

STRUCTURE AND FUNCTION OF THE SYNAPSE The

functional unit of the nervous system is not the sinneuron but consists of a chain of at least two and usually three or more neurons which connects receptor with effector. Each neuron in the chain remains a separate and discrete entity. Its axon terminal ends in close proximity to the receiving structures of other neurons but there is no protoplasmic continuity between neurons. Dendritic branches and axon telodendria interweave to form a complex feltwork known as the neuropil. The region of functional contact between neurons is the synapse, across which excitation must be transmitted. The synapse is probably the most important aspect of neural organization; in fact, its importance cannot be overemphasized. It is responsible for the physiological continuity of conduction through the neural chains and it is the site in the nervous system where occurs the modification of communication without which integrated response would be impossible. In the neuron itself, nerve impulses are transmitted in an all-or-none fashion in both magnitude and velocity, and these properties vary only with changes in the condition of the fiber. At the synapse, however, transmission is not all-or-none and may be amplified, reduced, or even completely blocked. As a result, the signal transmitted by a subsequent neuron in the chain may be quite different from the original input. Furthermore, blocking of some synapses and concurrent facilitation of transmission at others serve to determine the distribution of communication by directing it into spegle

cific

channels.

Morphology

of the

Synapse

arborizations which form nests, brushes, or baskets, and sometimes they are simply naked terminals which climb along a dendrite for some distance or cross it at right angles. Presynaptic terminals contain mitochondria, neurofilaments, and numerous minute vesicles. 200 to 1000 A in diameter, which are often clustered against the presynaptic membrane. Vesicles occur in a variety of

shapes and

sizes.

The synaptic

continuous with the intercellular ranges from 100 to 200 A. It is usually occupied by vague dense material which forms a thin dark plate between the apposed membranes and is sometimes thicker on the postsynaptic side. The contact area on the postsynaptic neuron may be called the subsynaptic membrane or the receptive site. While morphological specialization has not been clearly

space. In width

cleft is it

some electron microscope studies have shown what appear to be delicate hooklike fibrillar extensions which make contact with presynaptic fibrils in the synaptic cleft (Fernandez-Moran). Biochemical and physirevealed,

studies suggest the presence of ion-specific channels or pores and specific reactive chemical groups. Synaptic contacts occur on dendrites and, in neurons whose nucleated portion is located within the receiving pole of the cell, on the soma. Synapses formed by contact of axon terminals with postsynaptic dendrites are axodendritic synapses; those formed by contact with

ological

the cell body are axosomatic synapses. There are also synapses in which an axon terminal makes contact with another axon terminal or even with the initial segment of another neuron; these are axoaxonic svnapses (Fig. 12.8).

A

synapse

axon terminal of the transmitting or presynaptic neuron, separated by a fluid-filled space, the synaptic cleft, from the receiving membrane of the postsynaptic neuron. The axon ending and the postsynaptic membrane are closely contiguous. The two apposing membranes are stongly adherent and there is evidence that they may be held together by a special synaptic cement. Each telodendron terminates in a specialized unmyelinated ending which may take one of several forms. (Fig. 12.7) consists of the specialized

Frequently endings are bulbous swellings known as or boutons, but sometimes they are diffuse

knobs

Each presynaptic neuron makes synaptic contact

many different postsynaptic neurons, often sending several telodendria to each. Each postsynaptic neuron receives multiple axon terminals from many different with

presynaptic neurons: a single motor neuron in the spinal cord may have more than 1000 synapses occupying 40^ of its receiving surfaces. The scope of a neuron's influence may be further extended by collateral branches of its axon which may have destinations quite different from that of the parent axon. In some cases a collateral may even turn back into the dendritic field of its own neuron as a recurrent collateral.

169

Basic Neurophysiology

and others have shown by tunc lapse photographJ

Physiology of the Synapse iptic

cinematography that nerve libers are living, squirmi moving streams through which a peristaltic How of chemical supplies is driven from (he cell body at rate oi' from to a few millimeters per day. These materials

Transmission

an active neuron, nerve impulses travel out into all many tiny terminal branches ami into as manj synapses. Abundant evidence indicates that synaptic transmission is accomplished in most instances by a chemical process. (Electrical transmission, which is known to occur in many invertebrates such as the :ish. squid, and annelid worm, has recently been In

of

i

its

l

may

include the neurosecretions, as well as materials nourish and replenish the neural processes and, in motor nerves, substances which significantly influence the fast-slow response characteristics of muscle fibers. The depolarization produced in the membrane of the presynaptic terminals by the arrival of an impulse is assumed to trigger an excitation-secretion coupling mechanism which causes the rupture of the synaptic vesicles. A quantal amount of transmitter substance is ejected into the synaptic cleft by the bursting of each vesicle. Impulses probably do not initiate transmitter release but simply accelerate a secretory process which goes on continually at a low rate. The amount released is proportional to the magnitude of the impulse, and it has been calculated that, for each 30 mv of action potential, transmitter release is increased 100-fold. An impulse probably causes all of the vesicles in immediate juxtaposition to the membrane to rupture and also mobilizes other vesicles for subsequent release by causing them to move into the strategic area. The transmitter substance, which diffuses across the intervening space in a few microseconds, reacts with the specific chemical groups at the receptor site of the postsynaptic membrane. It has been suggested (Eccles, 1964, 1965) that these sites are associated with fine channels or pores which are somehow opened by the chemical reaction to permit ions to flow through the to

.

some vertebrates but is as yet unknown mammals.! The nerve impulse itself does not cross the interneuronal gap but rather, upon its arrival at identified in

in

ends, it causes the secretion of a chemical transmitter substance. The minute vesicles revealed by electron microscope photographs of presynaptic terminals are presumed to contain storage units of the transmitter, which may have been manufactured in the vesicles or. more likelv. in the nerve cell bodv. Weiss

membrane

FIGURE

12.7.

Diagrammatic

nents of the synapse.

1.

representation

of

the

compo-

presynaptic telodendrion; 2. bouton; 3,

vesicles: 4. synaptic cleft: 5. postsynaptic neuron: 6. receptor site or subsynaptic

membrane

FIGURE

12.8.

A. B. and C.

neuron

/

are axodendritic

forms an axoaxonic synapse

(c)

on

a

at

result, a

and 2 make synaptic connections with interneurons (b) A collateral from the axon of terminal of neuron 2 just prior to its axodendritic synapse on C

Types of synapses. Afferent neurons

Some synapses

many

times their normal rates (Fig. 12.9). small change occurs in the resting potential of the postsynaptic membrane at the subsynaptic site. The potential difference between this and adjacent unstimulated areas of the membrane is the postsynaptic potential (PSP).

As a

(a),

Horizontal arrow indicates direction of conduction.

1

others are axosomatic

170

UNDERSTANDING THE

SCIENTIFIC BASES OF

VESICLE

HUMAN MOVEMENT

a*

PRESYNAPTIC

MEMBRANE

SYNAPTIC CLEFT

POSTSYNAPTIC

MEMBRANE

CELL INTERIOR

Hypothetical explanation of synaptic transmission. A synaptic vesicle is releasing exsubstance (stippled) which diffuses across the synaptic cleft 7. a specific group of atoms on the postsynaptic membrane (rectangle) is so oriented that it occludes the pore which passes through the postsynaptic membrane 2. the excitatory transmitter has already interacted with chemical groups at the receptor site, producing a change in molecular configuration which has "opened" the pore. This enables

FIGURE

12.9.

citatory transmitter

sodium to enter and.

later,

potassium to leave the cell. 3, a narrower channel is shown which requires a difpresumably an inhibitory one. (From Gardner, E. B.. 1967. The neurophysio-

ferent transmitter substance, logical basis of

motor learning

—

a

review

J.

Amer. Phys. Ther Ass 47: 1115)

The action of the transmitter upon the subsynaptic membrane does not directly induce an action potential. The membranes of dendrites and soma (with some exceptions among dendrites of certain brain cells) are and incapable of generating acpotential spikes. Therefore the PSP is a local, graded, nonpropagated change in the resting potential which spreads electrotonically from the point of origin. The potential change gradually diminishes (decrements) as it spreads. The initial segment of the electrically inexcitable

tion

axon, however, is electrically excitable and has the lowest threshold of any part of the cell membrane. If the PSP is an excitatory change (depolarization) and if its magnitude reaches the critical level of the axon membrane, an action potential will be generated in the initial segment and conducted nondecrementally over the axon. Except for the interposition of the electrically inexcitable receptor portion of the cell, the sequence of events in synaptic excitation appears to be similar to that described for direct stimulation of the axon. The chemical transmitter substances do not remain long in the synaptic cleft but are soon destroyed, each by a specific enzyme. Almost immediate destruction of transmitter is essential to neural regulation of activity because its persistence and accumulation would result in exaggerated and uncontrolled responses.

There are two types of transmitter substances, those which are excitatory and those which are inhibitory. A transmitter is excitatory if it exerts a depolarizing effect upon the postsynaptic membrane, thus bringing its resting potential toward the firing level. It is inhibitory if it decreases the possibility of firing either by hyperpolarizing the resting membrane or by stabilizing it, possibly by combining with the chemical groups of the receptor site in a way which prevents activation. A postsynaptic neuron has many synapses on its surface, some of which are excitatory and some inhibitory, and both types are often active at the same time. The constant interplay of excitatory and inhibitory activity results in a fluctuating membrane potential in the initial segment which, at any moment, is the algebraic sum of these depolarizing and hyperpolarizing influences. Unsuccessful attempts have been made to correlate morphological differences among synapses with excitatory and inhibitory action. Some evidence indicates that excitatory synapses may have wide clefts and broad, continuous postsynaptic plates and may be located on more distal portions of dendrites, while inhibitory synapses may have narrower clefts and thinner, discontinuous plates and may be located upon dendritic trunks and soma surfaces. The situation, however, is not a simple one. Many intermediate and exceptional

Basic Neurophysiology

tonus are found. In

fact

some

oi the larger terminals

show both synaptic types on the same postsynaptic dendrite. Attempts have also been made to correlate differences in the design of presynaptic endings (knobs, baskets,

trails, etc.) with excitation and inAt the present tune no hard and last eon

brushes,

habitation.

drawn linking tine structure and synap However, the spatial distribution of active terminals in relation to each other and to the axon hillock may be important. Because of the deeremental nature of conduction in the receptive membranes, synapses far out on dendrites should be expected to exert less influence than those closer to the cell body, and synapses on the soma near the axon hillock should be the most effective. The possibility exists, however, that large dendrites may have electrically excitable sections which could act as booster stations for their otherwise deeremental conduction. Another interesting thought elusions can be tie

is

function.

that strategically placed inhibitory terminals could

markedly alter the effectiveness of excitatory endings. The chemical identity oi the transmitters which act at neural junction outside the central nervous system is well known. At the neuromuscular junction release of acetylcholine (ACh) by motor neuron terminals excites the end plate membrane of the muscle fiber. Acetylcholine is released at all autonomic ganglia by the preganglionic neurons and is the transmitter at all parasympathetic and some sympathetic neuroeffector junctions. For the majority of sympathetic junctions, the transmitter is norepinephrine (nor-E). The transmitters which operate at synapses within the central nervous system have been identified in only one instance: ACh is known to be liberated by terminals of recurrent collaterals of motor neurons at their syn-

apses with certain cells

(Renshaw

cells) in the spinal

ACh and

nor-E may be widely involved in central nervous system transmission, and it seems equally likely that others are also concerned, especially in the brain. Candidates include gamma-aminobutyric acid (GABA). histamine, 5-hydroxytryptamine (serotonin), and dihydroxyphenylalanine (dopamine), all of which are present in significant amounts. There is growing evidence in the literature on synaptic transmission in the invertebrates of a direct excitatory role for L-glutamic acid and for cord.

It

GABA

seems certain that both

as an inhibitory transmitter. Variations in the shape and size of synaptic vesicles may be related to the transmitter contained. Clear, nongranular vesicles, 200 to 400 A in diameter, probably contain ACh. and dense, granulated vesicles, 800 to 900 A in diamter. each hold a dense spherical droplet which may be nor-E. Other vesicles differing from these may contain other transmitters. The action of ACh is fairly well understood. Its reaction with the postsynaptic membrane produces a permeability increase which results in a rapid, localized depolarization of short duration. It is then quickly destroyed by the enzyme acetylcholinesterase (ACh-ase), which hydrolyzes it to choline and acetic acid. Destruction of the transmitter is necessary to avoid per-

171

sistent and convulsive responses. Several chemical substances (e.g., eserme and neostigmine) inhibit ACh ase. preventing destruction of ACh, and much has been learned about this neural transmitter through the use oi these agents. They have also proven useful in the management of myasthenia gravis, a disease character ized by weakness and extreme muscular fatigue resulting from subnormal release of ACh by motor nerve terminals.

The classic concept of synaptic function is that each neuron releases the same kind of transmitter at all of its terminals (Dale-Feldberg law) and that the transmitter has either an excitatory or inhibitory effect on all of the postsynaptic neurons upon which it acts. The unitary nature of neuron secretion is universally accepted. There is, however, considerable evidence that the sign or -) of a transmitter's action may be determined ( + by properties of the postsynaptic cell. In the autonomic nervous system ACh is excitatory for some effectors (for example, smooth muscle of gut and bladder) and inhibitory for others (cardiac muscle). Norepinephrine exerts both effects but oppositely in the various tissues. Furthermore, instances are known in which the effect may be reversed by hormonal influences acting on the innervated tissue. For example, the smooth muscle of the pregnant uterus is excited while that of the nongravid organ is inhibited by ACh. Also, in some simple vertebrate nervous systems clear cut instances have been found in which the same presynaptic neuron excites some postsynaptic neurons and inhibits others, presumably by the same transmitter (Kandel and Wechtel).

Synapses control the normal impulse traffic through the nervous system, determining the amount and pattern of information input and the consequent behavior of each neuron and group of neurons. Synaptic integrative action is based upon the interplay of antagonistic influences: facilitation and inhibition. Synaptic Facilitation Excitation in a presynaptic neuron does not necessarin transmission at every synapse which its terminals encounter. A certain amount of resistance is inherent in each junction and reflects the critical level of depolarization which is required to fire the postsynaptic neuron. Synaptic resistance varies from synapse to synapse and at each synapse is subject to temporary or persistent modification. If the transmitter is excitatory, the PSP will be a reduction in membrane potential; i.e., a partial depolarization. Such a decrease in the electric charge across the postsynaptic membrane is the excitatory post synaptic potential or EPSP and represents a reduction of synaptic resistance toward the firing level (Fig. 12. 10. A). This is known as facilitation. The action of the excitatory transmitter upon the postsynaptic membrane is thought to result in a general increase in membrane permeability, an opening of all ionic pores. The most notable ion movement, however, is that of Na + because of its greater electrochemical driving force. ily result

UNDERSTANDING THE

172

SCIENTIFIC BASES OF

HUMAN MOVEMENT graphical distribution within a single modality; proprioceptive feedback of information concerning body position or movement; and supraspinal influences from brainstem, cerebellum, and cortex.

t

Synaptic Inhibition

EPSP 70

mV

75

mV

1

B

A

FIGURE

12.10.

Facilitation

and

resting potential of the postsynaptic critical

EPSP

level

is

of about

ther excitation facilitated

of about 5

in

membrane

is

neuron.

B,

inhibitory

mv (membrane

Excitatory transmitter equivalent to this inhibited

inhibition

synapses. The -70 mv and its

-55 mv A. excitatory transmitter has evoked an 7 mv (membrane potential is now -63 mv). Furequivalent to 8 mv will be required to fire this

postsynaptic

duced IPSP

Postsynaptic Inhibition. Although involving different transmitters, both excitatory and inhibitory synapses are presumed to have the same general manner of function. In both, quantal packets of transmitter are re-

transmitter

potential

20 mv

will

is

has

now -75

be required to

in-

mv). fire

postsynaptic neuron

When a number of excitatory volleys arrive simultaneously or in close succession at several synapses of a cell, each contributes its small amount to the postsynaptic depolarization. If summation of EPSPs reaches the neuron's critical level, a spike potential is generated in the initial segment and conducted along the fiber. The rise of the action potential wipes out the EPSP, probably by antidromic invasion of the soma. However, if the total excitatory effect is in excess of the threshold level or if the presynaptic bombardment is sustained, the initial segment will be repeatedly restimulated and the postsynaptic impulse frequency will rise accordingly. The frequency of impulses in the postsynaptic axon will therefore depend upon the amount of facilitatory transmitter substance released. The greater the amount the earlier in the relative refractory period will another spike be generated. Summation of excitatory effects occurring at a number of synapses on the same postsynaptic neuron and involving terminals of presynaptic neurons from a variety of sources is known as spatial summation. The partial depolarization of the postsynaptic membrane by concurrent subliminal inputs makes the neuron more ready to respond. As a result it may be fired by a subsequent input which alone would have been inadequate. Facilitation may also be accomplished by a high frequency of impulses arriving over a single presynaptic terminal. Such temporal summation is probably less effective, except perhaps at the synapses of receptor neurons. In both spatial and temporal summation, each quantum of transmitter contributes toward the possibility of ultimate firing. If a response is already ongoing, facilitatory inputs will cause amplification of the response by increasing the frequency of the postsynaptic impulse train.

Sources of synaptic facilitation include the following: multiple sensory inputs which provide summation as a result of differences in modalities and/or in topo-

leased and react at receptor sites on the subsynaptic membrane, producing a momentary increase in permeability. Eccles (1964, 1965) conjectured that the action of the inhibitory transmitter differs from that of excitatory transmitter in that it produces a selective rather than a general permeability increase by opening pores for penetration restricted to small ions. The flow of current through the membrane of an inhibitory synapse + or the inis probably due to either the outflow of flow of CI" or both, with a concomitant increase in internal negativity (Fig. 12.10.B). The resulting hyper-

K

is the inhibitory postsynaptic potential (IPSP), and it opposes the EPSP. Consequently a greater summation of excitatory transmitter is required to lower the resting polarization to firing level. This type of inhibition is known as postsynaptic inhibition. It is the basis of reciprocal inhibition of antagonistic muscles,

polarization

in coordinated motor activity. Excitatory input from afferent neurons is transformed into inhibition at appropriate points in the neural network by the interposition of inhibitory interneurons (Fig. 12.11). These are special short-axon neurons which release an inhibitory transmitter at their synapses, thus making it harder to fire the postsynaptic neuron. Therefore, all inhibitory pathways must contain at least three neurons and all pathways involving only an afferent and an efferent neuron (monosynaptic chains) must be excitatory. Conductive delays substantiate the inclusion of at least two synapses in even the most direct in-

an essential factor

hibitory pathways in

mammals.

There are two forms of postsynaptic inhibition which merit special mention: recurrent or "surround" inhibition and disinhibition. Recurrent or Surround Inhibition. A particular type of

B 12.11. A simple inhibitory circuit. A neuron (7) synapses with an efferent neuron (A) and on a short inhibitory interneuron (2). Both of these synapses are excitatory and both neurons are activated. However, since the interneuron (2) secretes inhibitory

FIGURE

[B) will be inhibited and will fire synapses (not shown) induce sufficient

transmitter, the efferent neuron

only

EPSP

if

other

excitatory

to reach the critical level.

Basic Neurophysiology

postsynaptic inhibition, in which active colls in sensory motor projection systems inhibit adjacenl neurons, has received considerable attention among neurophysiotogists. rhe pathway foi tins inhibition involves re current collaterals which leave motor axons before they •merge from the gray matter of the cord. They pass back or

into the cord and excite special short inhibitory inter neurons called Renshaw cells. A Renshaw cell responds to a single stimulus with a high frequency burst of impulses and the release of inhibitory transmitter, with a consequent reduction of excitability in the inciting and adjacent neurons upon which its terminals impinge (Fig. 12.12). More strongly stimulated cells exert a stronger inhibition on their neighbors than that which they receive and hence the excitatory difference between them is exaggerated. The exact function oi this recurrent or surround inhibition is not yet clear. In motor neurons it presumably plays a role in localizing activity within a muscle and so may be oi value in distributing motor unit activity for fine

movements (Wilson). A similar mechanism in senmay serve to sharpen contrast (Brooks).

sory pathways

173

Disinhibition. Not only d Renshaw cells inhibit ad motor neurons but they may also inhibit an al read] existing inhibition and thereby facilitate neurons jacenl

of the

motor

pool.

Motor neurons are

subject to a tonic

inhibitory influence by some as yet unidentified inter neurons, probably reticulospinal fibers. Through in hibitory synapses on these cells, the

Renshaw

cells

de

press their inhibitory action and thus release the motor

neurons from the inhibition. This then is a facilitation by disinhibition (Fig. 12.13). The fact that this is not a usual type of facilitation is supported by both electrophysiological and pharmacological evidence: membrane potential changes are /Vyperpolarizations, and the effect is blocked by strychnine and tetanus toxin, drugs which block postsynaptic inhibitory synapses but do not affect excitatory junctions.

Normally, both facilitation and inhibition are occurring simultaneously but to different extents at the multitude of synapses of a postsynaptic neuron. The postsynaptic cell will fire whenever the algebraic sum of the two antagonistic influences is sufficient to depolarize it to its critical level, and the greater the sum the greater will be the frequency of impulses generated.

Presynaptic Inhibition. Within the last decade physiological experimentation has established the existence of presynaptic inhibition. As the name im-

FIGURE

12.12.

collateral

from motor neuron

Recurrent or surround inhibition.

A

A

recurrent

re-enters the ventral gray matter

and synapses with a short inhibitory neuron, the Renshaw cell. C (cell body crosshatched) The Renshaw cell sends terminals to surrounding motor neurons, where its inhibitory transmitter diminishes their

irritability.

FIGURE by

way

inhibits in

is under inhibition from an unknown source from the motor axon activates a Renshaw cell (2) which the inhibitory neuron The more strongly the motor neuron is stimulated the greater is the reduction

12.13. Facilitation by disinhibition. The motor neuron

of an inhibitory interneuron

the incident inhibitory influence

sponse

plies, the effect is exerted upon the presynaptic neuron rather than upon the membrane of the postsynaptic cell. The pathway for presynaptic inhibition appears to involve neural circuits in which the inhibiting neuron synapses upon the axon of the presynaptic neuron close to its own termination (Fig. 12.14). The electron microscope has revealed the existence of small boutons making synaptic contact with telodendria near their large end knobs. These axoaxonic synapses are believed to be the morphological basis of presynaptic inhibition. Pharmacological evidence indicates that the transmitter substance is quite different from that of postsynaptic inhibition. First, the presynaptic inhibitory effect is not blocked by strychnine or tetanus toxin, both of which are powerful antagonists of postsynaptic inhibition, and second, it is sensitive to picrotoxin, a con-

(/).

A

collateral

Disinhibition

would thus contribute to enhancement

of the muscle's re-

UNDERSTANDING THE

174

SCIENTIFIC BASES OF

HUMAN MOVEMENT the central nervous system can control

12.14. Presynaptic inhibition. A hypothetical circuit mediating presynaptic inhibition. An afferent neuron from a muscle spindle (7) is shown making an excitatory connection with a motor

FIGURE

neuron

(2) to its

own

extensor muscle.

A

collateral

branch of the

neuron activates a short interneuron (3), whose terminals synapse with the axon of an afferent neuron {4) which is making an excitatory connection with an efferent neuron (5) to

afferent

the

antagonistic flexor

extensor afferent

(7) will

muscle.

As

a

result,

excitation

over the

diminish the excitatory influence upon the

antagonistic flexor motor neuron by presynaptic inhibition.

vulsant drug which has no action upon postsynaptic inhibition.

The

distinctive characteristic of presynaptic inhibi-

that EPSPs of the postsynaptic neuron are depressed without any measurable hyperpolarization of its membrane. There is good evidence that the depression is due to a partial depolarization of the presynaptic axon which reduces the magnitude of the action potential invading its terminals. For example, if a depolarization of 10 mv has been induced at the axoaxonic synapses, the spike of the presynaptic neuron will be reduced by 10 mv from its usual level. As mentioned earlier, transmitter release is proportional to the magnitude of the action potential. Consequently, when these smaller potentials reach the end knobs, less excitatory transmitter is released and the EPSP is proportionately lessened. The reduction in transmitter probably reflects a decrease in the number of ejecting vesicles, because there is no evidence that the size of individual quanta is affected. A lesser amount of transmitter results in a lower impulse frequency in the postsynaptic neuron and therefore a decreased response. When the giant axons of the squid were presynaptically depolarized by electric current, a 5% reduction in the magnitude of the presynaptic spike caused a 50% reduction in the postsynaptic response. The neurons which produce presynaptic inhibition often fire repetitively and the presynaptic spike depression may last as long as 100 ms. It is also possible that antidromic impulses traveling centrifugally in the dorsal roots may collide with the orthodromic incoming impulses and reduce their magnitude in that way. Presynaptic inhibition provides a mechanism whereby tion

is

its input by completely suppressing weak or extraneous sensory inflow and can adjust the effectiveness of signals from one part of the body in relation to conditions prevailing in another part. Most important, it can modulate or eliminate undesirable input from one specific source without altering the postsynaptic neuron's sensitivity to input from other sources. This is in sharp contrast to postsynaptic inhibition, in which the excitability of the postsynaptic neuron is depressed. In the central nervous system of vertebrates, presynaptic inhibition is widespread at all spinal cord levels, occurs commonly in the brain, and has been found in interactions between cord and brain. There is increasing evidence that all afferents entering the cord from the skin and other peripheral receptors may exert presynaptic influence upon adjacent neurons and upon themselves. Pyramidal tract cells are thought to reduce stretch reflex activity by imposing presynaptic inhibi-

upon spindle afferents (Fig. The existence of presynaptic

tion

12.14).

facilitation

through an

increase of transmitter release by the presynaptic neuis suspected though as yet unproven (Ganong). Both recurrent (Renshaw) postsynaptic inhibition and presynaptic inhibition are feed back inhibitions: an active neuron sends collaterals back to produce inhibition at an earlier point in the transmission pathway. In the cerebellum a feed forward inhibition has been demonstrated. Basket cells and Purkinje cells are both excited by the same input but the basket cells send terminals forward to inhibit the Purkinjes. Presumably the mechanism limits the duration of excitation produced by any

ron

given afferent volley. The total subsynaptic area, dendritic plus somatic. of a postsynaptic neuron is relatively enormous as compared with a single synapse and the number of presynaptic terminals impinging on a postsynaptic cell may be very large. Since both facilitatory and inhibitory synapses are represented, both effects may be exerted upon the cell simultaneously. The magnitude of the depolarizing current through the postsynaptic neuron's initial segment will be determined by the number of active synapses and the algebraic sum of the two antagonistic influences. As long as the excitatory influence exceeds the inhibitory influence by at least the critical

amount, the neuron

By

will fire.

selective facilitation of

some synapses and

inhibi-

may be

directed into proper outflow channels. Muscles which should contract do so. and those which would interfere with the movement are caused to relax by cessation of outflow to them. tion of others, excitation

Other Properties of Synapses

The

properties characteristic of synaptic transmission

compatible with the accepted chemical theory. They differ in several respects, however, from the electrochemical conduction of action potentials in the nerve fiber. Synaptic Delay. Transmission across the width are

j

Basic Neurophysiology

1

100 to 200 .V of the synaptic cleft requires

man and up

ms

0.

l

to 0.3

ms

to

l

in

teraction with the postsynaptic

membrane

are




other animals. As compared to conduction velocites of over 100 meters per second in e nerve fibers, synaptic transmission is nearly 2 mil hon times slower. Most of the delay is consumed by transmitter release, as both diffusion and chemical inin

& &

the cat. J. Neurophysiol. 28: 71.

and Boycott. B. B.. 1965. Neural connections of the retina. Cold Spring Harbor Symp. Quant. Biol. 30: 393. Doyle. A. N\. and Mayer. R. F.. 1969. Studies of the motor units in the cat. Bull. Sen. Med. Maryland 54: 11. Eccles. J. 1960. Neuron physiology, introduction. In Handbook of Physiology: Seurophysiology. Vol. I. edited by J. Field and H. W. Magoun. Baltimore: Waverly Press. Eccles. J. C. Magni. F.. and Willis. W. D.. 1962. Depolarisation of central terminals of Group I afferent fibers from muscle. J Physiol. Eccles.

Synaptic and ephaptic transmission, In Hand Neurophysiology, Vol. I. edited by J Field

significance of cell size in spinal motoneurons,

McPhedran. A.

Press.

principles of sensory receptor action. Physiol.

Rev. 41:

Dowling.

1969

..

II \\ Magoun, Baltimore: Waverly Press Henneman, E., Somjen, ') serving the effector {K). As a result, impulses arising in a number of receptor organs

tion. This is the pathway of the passive stretch reflex and includes only the afferent neuron from the muscle spindle stretch receptor and the efferenl (motor) neuron to the muscle fibers, the two neurons synapsing in the spinal cord without an intervening interneuron (set' Rg. 14.8, Chapter 14). Three or More Neuron Circuits. Except for the stretch reflex circuit, pathways in the spinal cord and

converge upon a single effector, signal to the muscle. Again the

brain contain

all

three types of neurons. Circuits of only

three neurons, one of each type, are the exception rather than the rule, however. Generally the central portions of

the

circuit,

the

intemeurons, are multiple,

forming

and networks. As a result, basically simple circuits are converted into complex pathways. There are essentially three basic circuit types: (a) divergent circuits, by which a single receptor may influence many effectors; ib) convergent circuits, by which a chains

number

of different receptors

effector; (c) repeating circuits, is

multiplied a

number

may

influence the

same

by which a single input

of times.

plified but serves to present the basic principle.

Repeating Circuits

Two basic types of repeating circuits are known, reverberating and parallel circuits, in both of which a single input results in repetitive firing of efferent neurons. Figure 13. 10. A illustrates a reverberating circuit. The afferent neuron (7) synapses with a chain of interneurons (2, 3, 4), with the fourth neuron transmitting the signal to the efferent neuron (5). Impulses traveling this circuit, however, reverberate through an axon collateral from one of the chain (3) to restimulate a neuron (2) situated earlier in the chain. Repetitive firing will continue to activate the effector (F) until terminated by fatigue or by inhibition imposed through another circuit.

The parallel linear chain of

Divergent Circuits

afferent

A shows

diagrammatically a simple divergent circuit. The axon of the afferent neuron (1) branches to synapse with two intemeurons (2) which also send collaterals, each synapsing with a different efferent neuron (3). As a result, impulses originating in the single afferent reach four different effectors. A, B, C. and D. The circuit pictured is overly simplified and geometric in form. The central components, namely the intemeurons. characteristically occur not singly but in chains of different lengths and with a variety of branching patterns so that a stimulus impinging upon a single receptor organ, e.g.. the eye. can evoke responses involving many parts of the body. Further illustrations include a noxious skin stimulus which evokes a mass withdrawal response, or a sudden loud sound which may produce a total body response. Figure

13.9.

Convergent Circuits Figure 13. 9. B presents a simple convergent pattern. Axons of several afferent neurons (1) synapse with two intemeurons (2) whose axons in turn synapse with a

thus amplifying the is over sim

diagram

(1)

circuit

shown

is

intemeurons

(2, 3,

in Figure 13.10.B.

and

with the efferent neuron

from these intemeurons

(2

and

4)

A

connects the

(5).

Collaterals

3) also project to

syn-

napses of the efferent neuron, however, so that in the case illustrated a single input will stimulate the efferent not once but successively (three times), with synaptic delay determining the order of their arrival. Complexity is readily introduced into the repeating circuits by collaterals supplying additional efferent pathways or interconnecting with other circuits of the same or different type. Interposition of inhibitory neurons assures suitable direction of excitatory flow.

Combination of Circuits

The

possibilities of combination are obviously limitConsider the relatively simple combinations suggested by the diagrams in Figure 13.11 and determine the response which will be induced in each effector. With further modification by appropriate facilitation and inhibition, these combinations of simple circuits provide a basis for the observed complexity of response less.

patterns.

SUMMARY The neurons of the nervous system are arranged in a complex but orderly manner. Afferent and efferent neurons are combined to form the peripheral nerves which connect receptors and effectors with the central nervous system. The interneural chains, which complete the circuits from receptor to effector, compose the spinal cord and brain. Cell bodies serving specific functions are aggregated into clusters (nuclei or centers)

from which axons travel in groups (tracts) to distant destinations. Collateral branches leave the longer tracts to provide interconnections with centers at various levels of brain

and

cord. Interconnections

modified influences, provide properly

movement

patterns.

by

facilitatory

the

framework

between circuits, and inhibitory for

coordinated

SECTION THREE

THE INTEGRATIVE ROLE OF THE PROPRIOCEPTIVE REFLEXES

CHAPTER

14

The Proprioceptors and Their Associated Reflexes* INTRODUCTION man. are born with genetically which are preprogrammed by modification of synaptic transmission to produce stereotyped response patterns useful to the species. These are not learned. They are present at birth or appear as the developing nervous system progresses to com-

and stabilizers. All of these must be precisely regulated in regard to their intensity, speed,

All animals, including

built-in neural

of synergists

circuits

duration, and sequential changes in activity from the beginning to the end of the movement. This requires a considerable amount of integrative function which is largely automatic and unconscious. Muscle-to-muscle integration is accomplished by

pletion. They represent the heritage of the species. Genetic material prescribes and ontogeny builds the components for the types of movement characteristic of human behavior. These components include not only the bones, joints, muscle attachments, and nerve supply, both afferent and efferent, but also appropriate interconnections and patterns of facilitation and inhibition. A child is born with a repertory of a few hundred movements which compose the raw material of motor learning. The modification and recombination of these in all possible ways results in the acquisition of additional motor patterns, some of which are very different from inherited patterns. These are learned motor

basic reflex reactions which are initiated by receptors strategically located to feed back information to the central nervous system. Information must be received

continuously regarding body position, muscle length tension, speed, range and angle of movement, acceleration of the body or its parts, and balance and equilibrium. This information must then be integrated by cord and lower brain centers and converted into a suitable modification of the impulse outflow to produce immediate adjustment of each muscle concerned. As the state of a muscle changes, the information input will also change, evoking re-modifications in neverending succession. Much of the information also becomes available to centers in the conscious areas of the brain where it may be sorted, analyzed, interpreted, and converted into an outflow of signals to appropriately modify voluntary body movements as occasion

and

skills.

Stereotyped responses in the form of simple human are well known, such as the stretch reflex, withdrawal (flexion) reflex; extensor thrust and the positive supporting (extension) reflexes; crossed extensor reflex; righting reflexes; placing reactions and others. Some of these appear to be very simple; others must be amazingly complex. Some are fully formed at birth, while others develop as natural expansions of these during the maturation of the neuromuscular system. Each of these patterns consists of a coordinated combination of several to many joint movements, and each joint movement further consists of a coordinated combination of muscle actions: contraction of prime movers, relaxation of antagonists, and supporting contractions reflexes

demands. Voluntary movement requires a foundation of automatic responses which assure a proper combination of mobility and stability of body parts. Since activity occurs in many muscles simultaneously or sequentially, precise regulation

*

Some

of the material

and

in Quest,

1969, pp. 1-25,

essential. Fortunately, neural con-

muscles,

and termination

of the

and direcmovement. Volition does

not normally include control of individual muscles, although the human capability of doing so and even of controlling single motor units has been amply demon-

several figures in this chapter have

Monograph XIL

of

initiation, regulation of speed, force, range, tion,

appeared previously

is

whether activity is unconscious or deliberate, is mostly involuntary: muscles are smoothly regulated by reflex mechanisms. The voluntary contribution to movement is almost entirely limited to

trol

and

are reproduced here with permission of the publishers.

193

194 strated.

UNDERSTANDING THE For example, reaching

SCIENTIFIC BASES OF

for

an object

tarily prescribed as to direction, speed,

is

volun-

and the object

sought; but the functional features of shoulder girdle fixation, elbow extension, wrist stabilization, and finger movement are regulated by subcortical mechanisms.

Two

contrasting neurophysiological hypotheses exist regarding the subconscious regulation of the many muscles concerned in the behavior subserved by neural circuits; the central control hypothesis and the periph-

The central control hypothan unspecific input, or even the central nervous system itself, triggers activity in a nerve net which has been genetically structured (in terms of mutual excitatory and inhibitory influences) eral control hypothesis.

esis postulates that

so that, once activated, the total pattern proceeds automatically. In invertebrates numerous instances are known in which specifically identifiable interneurons drive "follower" cells to produce relatively complex coordinated movements. Because in these cases the interneurons must be continually active for the response to proceed, it is obvious that they must be under the control of a preceding triggering stimulus.

The stimulus may be

a single brief event. Once initithe movement sequence proceeds without requiring further regulatory input. In vertebrates some instances are known in which stimulation of selected hypothalamic or cortical regions elicits complex motor acts or sensory experience, and the response often greatly outlasts the stimulus. However, nothing is known regarding the neurons or pathways involved (Willows and Hoyle).

ated,

The peripheral control hypothesis,

in contrast, at-

sequential activation of various component circuits of the nerve net to specific inputs from protributes

prioceptors, each of which triggers either excitatory or inhibitory output to appropriate muscle pathways. Most neurophysiologists have favored the latter hypothesis because proprioceptive loops have been so convincingly demonstrated. It seems highly probable that in man there is no movement pattern controlled purely by either one of these methods but rather that features of both are involved. While man's basic motor patterns are probably genetically coded in the species and laid down during development as specifically structured interconnected nerve circuits, activity within the nerve nets appears to be adjusted to the momentby-moment changing conditions of environment and/or body orientation (changes which could not be anticipated genetically) by proprioceptive feedback. Although little is known regarding the exact struc-

HUMAN MOVEMENT ture of vertebrate nerve nets, a large

amount

of in-

formation has accumulated regarding proprioceptors

and fects

their structure,

upon the

mode

of function,

activity of muscles.

Some

and

reflex ef-

of this knowl-

edge has been derived from clinical and laboratory studies on humans, but the greater portion and the most detailed information has, of course, come from the study of other mammals, especially monkeys and cats. It is not wise to transpose indiscriminately from one species to another but, where interspecies similarities and instances of parallel evidence exist, speculation regarding the operation of the same mechain man is justified, especially as the basis for the formulation of hypotheses and for the design of experiments. According to Sherrington, proprioceptors are those end organs which are stimulated by actions of the body itself. They are somatic sensory organs located so as to secure inside information and to effectively bring about cooperation and coordination among muscles. The nervous system uses these sensory receptors to modify and adjust muscle function so that peripheral automatic (subconscious) regulation will dominate in most of our so-called voluntary or volitional movements. When proprioceptors are stimulated by movement or position, impulses traverse neural chains to act upon muscles in diverse and interrelated ways. By exciting various proprioceptors, contraction of any muscle tends to organize other muscles to cooperate with it. In the parlance of the electronic engineer, these reflexes operate as negative feedback loops by means of which

nisms

motor activity becomes

in

large

measure

self-regu-

words, aspects of the movement process muscle tension, absolute muscle length,

lating. In other

such

as velocity of change in muscle length, joint angle, joint movement, head position, and contact with surfaces

act as stimuli to initiate signals in nerve fibers which are fed back into the central nervous system. In some way as yet unknown, this information is compared with the desired pattern which nature or conditioning has established. If the afferent signal indicates any divergence from this pattern, centers in the nervous system modify efferent signals so that the activity of the proper muscles is appropriately increased or decreased to correct the difference. Proprioceptors may be conveniently classified into three groups: the muscle proprioceptors, the proprioceptors of the joints and skin, and the labyrinthine and neck proprioceptors.

MUSCLE PROPRIOCEPTORS The muscle proprioceptors are the neuromuscular spindles and the Golgi tendon organs, both of which are incorporated into the gross structure of the muscle itself.

Neuromuscular Spindles Neuromuscular spindles are highly specialized sense organs which are distributed among the bundles of contractile fibers in the muscles. They are found through-

Proprioceptors and Associated Reflexes

mass of the muscle but tend to be more con There are more centrated in the central portion. spindles in man's phasic muscles than in his tonic (postural) muscles, as would be expected since the former more precise control. The neuromuscular require spindle is probably the most important and surely the

the spindle reflect both the rale of change in length (phasic response) ami the ultimate length finally achieved and maintained (tonic response) Both as peels of muscle length are signaled by variations in the firing frequency of the afferent neurons serving the re-

out the

most

complex

of

proprioceptive

the

ceptor.

One

receptors.

expects, and finds, structural complexity associated with functional complexity. In the case of the muscle spindle, its structure presents an outstanding duality which is also reflected in its function. usually

The muscle spindle

is

sensitive to length and.

Structure of the Spindle details vary slightly from spindle to depending upon the particular function of the muscle in which the spindle lies. In general, each consists of a fluid-filled capsule 2 to 20 long and enclosing 5 to 12 small specialized muscle fibers (Fig. 14.1). These are known as intrafusal fibers, to distinguish

Structural

spindle,

when

mm

stretched, responds to both constant length, as in maintained position or posture, and changing length, as

during movement.

The

firing of the sensorv

neurons of

rxGWm

Figure 14.1.

The neuromuscular

spindle. A, a

muscle spindle

in situ, lying in parallel

with the extrafusal

muscle fibers. A spindle consists of a fluid-filled capsule containing small intrafusal muscle fibers of which there are two types The two outer intrafusals are nuclear bag fibers which are percapsular The three lying centrally are nuclear chain intrafusal fibers and are intracapsular. Innervation is omitted (From Gardner. E B. 1969. Proprioceptive reflexes and their participation in motor skills Quest XII: Fig. 1-B. p 5 B, photomicrograph of a spindle in cross section, demonstrating its multi-layered capsule and the diameters of the nuclear bag UFb) and nuclear chain (IFc) intrafusal muscle fibers Gamma efferent axons (Ge) and extrafusal muscle fibers (EF) are identified (From Truex, R. C. and Carpenter, M B 1969 Human Neuroor contractile

)

.

anatomy. Ed

6. Fig

9-

1

2.

p

1

84

195

Baltimore: The Williams

&

Wilkins Company.)

196

SCIENTIFIC BASES OF

UNDERSTANDING THE

them from the contractile or extrafusal fibers of the muscle. The latter, when stimulated by their large alpha motor neurons, contract to produce the muscle's tension.

Intrafusal fibers differ from contractile muscle fibers

HUMAN MOVEMENT Most muscle spindles receive two types of afferent innervation, designated primary and secondary. The two types of afferent neuron are distinguished by differences in sensory ending and by differences in axon The primary afferent neurons terminate in an-

size.

in several ways. Their diameters range from one-tenth to one-fourth the diameter of the contractile fiber:

nulospiral

their nuclei are concentrated in the central or equato-

afferents

rial region rather than being distributed throughout the fiber; and the contractile material is restricted to

the polar ends. There are two distinct sizes of intrafusal fibers (Fig. 14.1). Each spindle contains one to three large intrafusal fibers ranging from 12 to 26 \i in diameter and one to eight smaller intrafusal fibers with diameters ranging from 4 to 12 n- These two types of receptor cells differ not only in diameter but in length. The smaller fibers are contained entirely within the spindle capsule (intracapsular fibers), while the larger fibers pass well beyond the capsule (percapsular fibers). In the large fibers the centrally located nuclei are aggregated into a swollen baglike region in the equatorial portion and hence these are called nuclear bag fibers. There may also be a single-file projection of

known as the myotube, extending outward from the bag region on either side. The smaller intrafusal fibers contain only a single column of nuclei through their equatorial region. These are known as nuclear chain fibers. The large bag fibers, extending well beyond the capsule, attach to the connective tissues and endomysia of the contractile muscle fibers. In some instances, a single nuclear bag fiber may pass several nuclei

through several capsules each having a nuclear bag. In such cases the structure is known as a tandem spindle. Although the significance of tandem spindles is not yet clear, they are probably concerned in the intricate role of the spindle in muscle regulation. The chain intrafusals, fully contained within the capsule, attach to the inner surface of the capsular connective tissue at either end. The two types of intrafusal fibers also vary in viscosity, the bag fibers being more viscous than the chain fibers. Viscosity may determine the type of contraction of the intrafusal fibers, a characteristic which has an important influence upon its function. The innervation of the two types of intrafusal fibers also differs.

Spindle Innervation

Afferent Innervation. Afferent (sensory) neuron endings are intimately associated with the intrafusal fibers and are stimulated mechanically when the fibers are stretched. Impulses then pass over the axons and enter the spinal cord by the dorsal roots. Within the spinal gray matter they distribute to a number of pathways. Most prominent, however, is their influence upon their own muscle group. In general, spindle afferents exert

an excitatory

upon the muscle in which they lie, a facilitatory effect upon synergistic muscles, and an inhibitory effect upon antagonistic muscles.

An important

effect

exception

is

discussed later.

regions

endings

of the

which

intrafusal

coil

fibers

around the nuclear while the secondary

terminate juxta-equatorially. i.e.. farther toward the striated polar regions, either in smaller coils or in flower-spray endings. Axons of primary afferents are large group I fibers (known as la), while the secondaries fall into group II of the Lloyd classification. The two types of afferent neurons distribute differently to the two types of intrafusals (Fig. 14.21. Each spindle has only one primary afferent. It enters the spindle and branches to supply an annulospiral ending to each of the intrafusal fibers of the spindle, both bag and chain. Each spindle receives one to five of the smaller secondary afferents. Their endings are restricted almost entirely to the chain fibers.* and their axons rarely branch, the axon to ending ratio being essentially 1:1. The two types of afferents differ in sensitivity, the primary afferents having much lower thresholds to stretch than do the secondaries. Only a few millimeters of stretch per second are sufficient to activate them. Furthermore, the primary afferents signal both phasic and tonic stretch, while the secondaries signal tonic length only.

The

primary-

afferent neuron signals the phasic length state of the its impulse frequency during phasic response. The frequency reflects, not length as such, but rare of change in length, i.e.. velocity of the stretch. When stretching is com-

muscle by changes

in

stretch. This is the

pleted, the frequency of discharge drops to a constant level appropriate to the new tonic length. This is the tonic response (Fig. 14.3). When a small stretch is rapidly imposed, the phasic response frequency rises sharply and then drops markedly when stretching ceases. The difference between the maximal frequency attained during the phasic portion of the stretch and the level to which frequency settles in the tonic response is called the dynamic index. The tonic value is taken at 0.5 second after the final position has been reached. The contrast between the responses of primary and secondary spindle afferents during a slow stretch is illustrated in Figure 14.4. Efferent Innervation. Intrafusal fibers are supplied with motor innervation in the form of small gamma-

sized neurons known as gamma motor neurons or fusiform neurons, whose cell bodies lie in the anterior horn of the gray matter of the spinal cord. Axons of these neurons leave the ventral root, travel to the mus-

and terminate in motor endings on the contractile polar end regions of the intrafusal fibers. Each spindle receives 7 to 25 cle in the appropriate spinal nerve,

(average 10 to 15) efferent neurons. Impulses traveling * There is some disagreement among authorities regarding the completeness of the limitation of secondary afferents to the chain intrafusal muscle fibers.

Proprioceptors and Associated Reflexes

197

2

3T^ FIGURE

14.2.

Intrafusal fibers

with equatorial region

A

picture

filled

and

their innervation.

A

large nuclear

bag

intrafusal fiber

is

shown above

with nuclei and with contractile polar ends extending beyond the limits of the

nuclear chain fiber

is

shown below,

smaller

in

diameter and. shorter, with the characteristic

is pictured on the two types of intrafusal ends in coiled terminals (annulospiral endings) on the nuclear region of each intrafusal, while the smaller secondary afferents (2) have branched endings (flower sprays) located on the outer parts of the nuclear region, and appear only on the chain intrafusals Efferent innervation is also shown The gamma (fusimotor) neuron (3) ends in gamma plates located quite far out on the polar regions of the nuclear bag fiber The nuclear chain fiber receives another type of gamma neuron (4) which terminates in gamma trail endings situated closer to the equatorial region (From Gardner. E. B. 1969. Proprioceptive reflexes and their participation in motor skills. Quest XII: Fig. 1-B. p. 5)

row The

single fibers

of nuclei

in

its

central

region

Afferent innervation

neuron

single large primary afferent

(/)

to the connective tissues of the extrafusal muscle fibers,

1 2

FIGURE 14.3. Phasic and tonic responses of the spindle primary afferent neuron to interrupted stepwise increases in muscle length. Each stretch, denoted by the solid bars on the time scale, involved the same amount of stretch but was imposed at

a

different

rate

as indicated by the time. After each stretch

new length was maintained, a. stretch was applied: b, the beginning

the

at the

was

completion of the stretch:

attained:b'.

stretches

Note

c'.

initial

frequency before

of the stretch:

frequency after the

d.

c,

frequency

new

length

and b". c". d". responses to subsequent

d'

that

during each stretch

the

frequency (b

new constant

to

c.

the

impulse discharge increased phasic response) but dropped

of

each new length atThe small double-ended vertical arrows represent the dynamic index for each stretch Hypothetical, based on Matthews. P. B 1968 Central regulation of the activity of skeletal muscle In The Role of the Gamma System in Movement and Posture. Revised edition. New York: Association back to

tained

a

(d.

the

tonic

level consistent with

response).

C

for

Aid of Crippled Children.)

over

these fusimotor neurons evoke contraction of the polar ends of the intrafusal fibers just as impulses in the large alpha motor neurons evoke contraction in the large contractile fibers, but contraction of the exerts no detectable influence on muscle an intrafusal fiber is connected at both ends either to the interior wall of the capsule or intrafusals

tension. However, since

the shortening of its polar ends imposes a stretch upon the nuclear region (Fig. 14.5). Afferent endings are activated just as they are during passive stretch of the whole muscle. Stretch produced by gamma innervation may be referred to as internal stretch, while stretch imposed by gravity, an outside force, or shortening of an antagonistic muscle is designated as external stretch. Impulses generated in afferent neurons, whether by internal or external stretch, traverse neural circuits to the muscles. The structural duality of the muscle spindle is also reflected in its motor innervation: there are two types of gamma neurons (Fig. 14. 2. A). Existence of the two types is supported by anatomical, physiological, and pharmacological evidence. Anatomically there are two types of endings differentially located on the intrafusal fibers. There are endings found on the polar regions of nuclear bag fibers which resemble smaller versions of motor endplates, and these endings are known as gamma plates. On the nuclear chain fibers, more diffuse endings occur known

as

gamma

trails.

Gamma

trails are situated

more cen-

on the polar regions, i.e., closer to the equatorial region, sometimes overlapping the secondary sensory endings, while the plate endings occur more distally. Furthermore, in the muscle nerve there occur two distinct types of gamma-sized axons: one type is thickly myelinated, the other thinly myelinated. The two types show no separable differences in their axon diameters, overlapping over the whole range. Physiologically, some gamma neurons conduct rapidly and may be called fast fusimotor neurons, while others conduct more slowly (slow fusimotor neurons). There is little overlapping in conduction velocities between the two groups. Indirect evidence suggests that the thickly myelinated axons are the fast-conducting ones, although this needs further substantiation.

trally

UNDERSTANDING THE

198

BASES OF HUMAN MOVEMENT

SCIENTIFIC

Furthermore, the gamma neurons display two different types of influence on the primary afferents. Some gamma neurons, when stimulated concurrently with external stretch of the muscle, appear to produce an in-

;

crease in the dynamic index. In other words their action exaggerates the phasic response of the primary endings. Such neurons are called dynamic fusimotor neurons. Other gamma neurons markedly decrease the

dynamic index. These are known as static fusimotor neurons. They may also increase the tonic response this effect is less noticeable than is the reduction of the phasic response (Fig. 14.6). Pharmacologically, evidence for two separate gamma systems is found in Rushworth's investigation of drug effects on phasic responses as compared with tonic responses. He found that barbiturates depressed the tonic response but left the phasic responses unaffected, and that procaine completely suppressed the tonic response and reduced the phasic response. Moreover, Rushworth found that the phasic response was exaggerated in cerebellar disease, while the tonic response was absent. Other studies have shown that the anterior

somewhat, but

B FIGURE

fibers

>

Dynamic index

of

FIGURE

./,Secondary

/

Responses of primary and secondary spindle Upper curve the phasic frequency response of a priafferent neuron during

stretch of the muscle. afferent

under

Lower

same

new

length.

The

and

directly following a

Broken

vertical

was completed, and last

slow

curve, response of a secondary spindle

conditions.

point at which stretch at this

(mm)

14.4.

mary spindle

indicates

line

the muscle

point on each curve

is

was

held

the frequency

new length had been maintained for 0.5 second. Note the marked drop (dynamic index) in the response of the

recorded after the primary

afferent,

stretch,

while the secondary afferent has responded only to ab-

indicating

that

it

is

responding to velocity of

solute length. (Adapted from Harvey. R

J., and Matthews. P. B. C. 1961. The response of de-efferented muscle spindle endings in the cat's soleus to slow extension of the muscle. J. Physiol. (London) 157: 370. Cf. Matthews. P. B. C. 1968. Central regulation

of the activity of skeletal muscle. In The Role of the

Movement and

tem

in

New

York: Association for Aid of Crippled Children.)

Posture,

the

cerebellum

normally

inhibits

dynamic

activity, suggesting that in disease of the cere-

reduced or absent. Such differon the two types of response support their

this inhibition is

correlation with separate structural properties.

to relate the dynamic gammas to the plate endings, and therefore the nuclear bag fibers, and the static gammas to the trail endings and the chain fibers. For

-l_i-

afferents.

The intra-

It is

/

Stretch

fiber.

to

/

/

intrafusal

not known for sure which gamma axon is related which type of motor ending and, coincidentally, which type of intrafusal fiber. Indirect evidence seems

X

/*

an

are

ential effects

/

of

state with polar

bellum

Primary

stretch

attached. A, a nuclear bag fiber in the neutral ends uncontracted B. the same fiber under gamma (fusimotor) stimulation. Polar ends have contracted, putting the nuclear region under stretch.

fusal

gamma

/:\

Internal

bars on either side represent tendons to which the

vertical

lobe

120

14.5.

revised

edition.

Gamma

Fig.

1

7.

p.

Sys32.

example, the primary afferent neurons have their endings on both types of intrafusal fibers and are influenced by both gammas; however, the secondary afferents, whose endings are restricted almost entirely to chain fibers, are activated only by static gammas. Examined from another viewpoint, if dynamic gammas end in the plate endings on the nuclear bags as suggested, they should influence the primary afferents but not the secondaries, which is the actual experimental finding. And if static gammas end on chain intrafusals, they should affect both types of afferent ending, and, in fact, the static fusimotor neurons can drive* both primary and secondary afferents. So it seems likely that the dynamic fusimotor neurons distribute to the nuclear bag intrafusal fibers and the static fusimotors to the chain intrafusals. Evidence is accumulating that the type of contraction of the two intrafusal fibers differs. This is significant because the nature of intrafusal contraction may determine the afferent response, while the particular type of gamma neuron simply activates a particular type of intrafusal fiber. Smith has shown that bag fibers contract slowly on a local, graded manner. This would *

A

neuron

is

considered to drive another neuron

that other neuron to respond one for one to

limited

number

of cvcles.

its

when

it

causes

frequency over a

Proprioceptors and Associated Reflexes

be consistent with the phasic response in which fire quency increases with the rate of stretch. The smaller chain fibers, however, contract in a faster, twitchlike manner, and complete tetanus can be evoked in them bj about 15 impulses per second. This would be com patible with the tonic response toa maintained stretch. the extent of tetanus in the intrafusal fiber determining the frequency in the afferent neuron. Since the primary .rents ser\e both types of intrafusal fiber, they would be expected to signal both phasic and tonic stretch, while the secondaries, associated mainly with the chain fibers, would signal only tonic length; this is in fact the case. Further, if it is the dynamic type of

gamma

which serves the bag

and evokes

fibers

appropriate

to

14.7).

After

gamma

chain

denervation,

fibers

axons has been demonstrated histologically Ada! and Barker (1965) and electrophysiological^ by Bessou, Emonel Denand, and Laporte U!>

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Cot

Cos

B.1443 .1248 .1054 .0860 .0667 8.0476 .0285 8.0095 7.9906 .9718 7.9530 .9344 .9158 .8973 .8789 7.8606 .8424 .8243 .8062 .7882 7.7704 .7525 .7348 .7171 .6996 7.6821 .6647 .6473

.99255

787 817 846 876

.99237

233 230 226 222

.99200 197 193 189

215 211 208 204

Cot

3 4 5

6 7

8 9

10 11 12

13 14

15

186

41

.99182 178 175 171 167 .99163 160 156 .6301 152 .6129 148

40

20

39 38 37 36 35 34 33 32

21

.99144 141 137 133 129 .99125 122

30 29 28 27 26

30

26 24 23 22

35 36

.2531 .2375 .2220 .2066

118 114 110 .99106 102

Tan 82°

31

22 23 24 25 26 27 28 29 31

32 33 34

.13917 .14054 046 084 .13975 113 .14004 143 033 173 .14061 .14202

090 119 148 177

232 2G2 291 321

.14205 .14351 234 3M

263 292

320

410 440 470

.14349 .14499

378 407 436 464

529 559 588 618

.14493 .14648

522 551 580 608

Cot 7.1

.1004 .0855 .0706 .0558

.9827 990 6.9682 .98986 .9538 982 .9395 978 .9252 973 .9110 969 6.8969 .98965 .8*28 961 .8087 957 .8548 953 .8408 948 6.8269 .98944

678 707 737 767

.8131 .7994 .7856 .7720

.14781 .14945 810 .14975 838 .15005

867 896

034 064

.14925 .15094 124 954 153 .149i>2 .15011 183

6.6912 .6779 .6646 .6514 .6383 6.6252 .6122 .5992 .5863 .5734 6.5606 .5478 .5350 .5223 .5097 6.4971 .4846

.15069 .15243

41

272 126 302 332 155 362 184 .15212 .15391 241 421 .4721 270 431 481 .4596 299 .4472 327 511 .15356 .15540 6.4348 570 .4225 385 .4103 414 600 .3980 442 630 .3859 471 660 .15500 .15689 6.3737 529 719 .3617 557 749 .3496 779 .3376 586 .3257 615 809 .15643 15838 6.3138

16

083 079 075 071

14 13 12

.99067

10

063 059 055 051

9

51

8

52 53 54 55 56 57 58 59 60

11

7

6 6

4 3 2 1

940 936 931 927

.14637 .14796 6.7584 .98923 .7448 826 919 666 .7313 914 695 856 .7179 910 723 886 .7045 906 752 915

40

091

028 019 015

011 7.0410 .99006 .0264 .99002 7.0117 .98998 6.9972 994

21

.99087

Cos

154 .99027

20

42 43 44 45 46 47 48 49 50

Sin

Tan

37 38 39

19 18 17 16

098 094

.13773 .13906 7.1912 .99047 802 935 .1759 043 831 .1607 039 965 .1455 860 .13995 035 889 .14024 .1304 031 .13917 .14054 7.1154 .99027

Cos

51

60 49 48 47 46 46 44 43 42

.99219

1

2

16 17 18 19

.12995 .13024 .13053 .13165 7.5958 081 .5787 195 110 224 .5618 139 .5449 254 168 284 .5281 .13197 .13313 7.5113 226 343 .4947 254 372 .4781 283 402 .4615 312 .4451 432 .13341 .13461 7.4287 370 .4124 491 399 .3962 521 427 550 .3800 456 580 .3639 .13485 .13609 7.3479 514 .3319 639 543 .3160 669 572 .3002 698 600 .2844 728 .13629 .13758 7.2687

658 687 716 744

251 248 244 240

1

60 59 58 57 56 66 54 53 52

Sin

040

213

097

Cos

Tan

Cot

81

c

.98902

897 893 889 884 .98880

876 871 867 863

60 59 58 57 50 65 54 53 52 51

60 49 48 47 40 46 44 43 42 41

40 39 38 37 36 35 34 33 32 31

30 29 28 27 26 25 24 23 22

854 849 845 841

21 20 19 18 17 16

.98836

15

832 827 823 818

14 13 12 11

.98814

10

.98858

809 805 800 796 .98791

787 782 778 .773 .98769

Sin

9 8

7 6 6 4 3 2 1

244

Appendix C

10° Sin

Cos Cot Tan 15838 6.3138 98769 764 .3019 868 760 .2901 898 755 .2783 928 751 .2666 958 15988 6.2549 .98746 741 .2432 16017 737 .2316 047 732 .2200 077 728 .2085 107

Sin

Tan

Il Cot

Cos

Sin

Tan

e

Cot

Cos

|

o 1

2 3

4 5 6 7

g 9 10 11

12 13 14

16 16 17 18 19

20 21 22 23 24

15643 672 701 730 758 15787 816 845 873 902 15931 959 15988 16017 046 .16074 103 132 160

16137 167 196

226 256 16286 316 346 376 405

189 .16218 .16435

465 495 525 555

246 275 304 333

25 .16361 .16585 615 390 26 645 419 27 674 447 28 704 476 29 30 .16505 .16734 764 533 31 794 562 32 824 591 33 854 620 34 35 .16648 .16884 914 677 36 944 706 37 .16974 734 38 763 .17004 39 40 .16792 .17033 063 820 41 093 849 42 878 L23 43 153 906 44 45 .16935 .17183 213 964 46 243 47 .16992 273 48 .17021 303 050 49 50 .17078 .17333 107 303 51 393 136 52 423 164 53 453 193 54 55 .17222 .17483 513 250 56 279 543 57 573 308 58 603 336 59 60 .17365 .17633

Cos

1

Cot

6.1970 .98723 718 .1856 714 .1742 709 .1628 704 .1515 6.1402 .98700 695 .1290 690 .1178 686 .1066 681 .0955 6.0844 .98676 671 .0734 667 .0624 662 .0514 657 .0405 6.0296 .98652 648 .0188 643 6.0080 638 5.9972 633 .9865 5.9758 .98629 624 .9651 619 .9545 614 .9439 609 .9333 5.9228 .98604 600 .9124 595 .9019 590 .8915 585 .8811 5.8708 .98580 575 .8005 570 .8502 565 .8400 561 .8298 5.8197 .98556 551 .8095 546 .7994 541 .7894 536 .7794 5.7694 .98531 526 .7594 521 .7495 516 .7390 511 .7297 5.7199 .98506 501 .7101 496 .7004 491 .6900 .6809 486 5.6713 .98481

Tan 80°

Sin

60 59 58 57 56 55 54 53 52

1

2 3

4 6 6 7

8

51

9

60 49 48 47 46 46 44 43 42

10 11 12 13 14

16

41

16 17 18 19

40

20

39

21

38

22 23 24 26 26 27 28 29

37 36 35 34 33 32 31

30 29 28 27 26 25 24 23 22 21

20

30 31 32 33

34 35 36 37 38 39 40

19 18 17 16

41

16

45

14 13 12 11

46 47 48 49 50

10 9 8 7

6 6

4 3

2 1

'

42 43 44

51 52 53

54 65 56 57 58 59 60

17365 .17633 5.6713 98481 393 .6617 663 476 422 693 .6521 471 451 723 .6425 466 479 753 .6329 461 17508 .17783 5.6234 .98455 537 813 .6140 450 565 843 .6045 445 594 873 .5951 440 623 903 .5857 435 .17651 .17933 5.5764 .98430 680 963 .5671 425 708 .17993 .5578 420 737 .18023 .5485 414 766 053 .5393 409 .17794 .18083 5.5301 823 113 .5209 852 143 .5118 880 173 .5026 909 203 .4936 .17937 .18233 5.4845 966 263 .4755 .17995 293 .4665 .18023 323 .4575 052 353 .4486 .18081 .18384 5.4397 109 414 .4308 138 444 .4219 166 474 .4131 195 504 .4043 .18224 .18534 5.3955 252 564 .3868 281 594 .3781 309 624 .3694 338 654 .3607 .18367 .18684 5.3521 395 714 .3435 424 745 .3349 452 775 .3263 481 805 .3178 .18509 .18835 5.3093 538 865 .3008 567 895 .2924 595 925 .2839 624 955 .2755 .18652 .18986 5.2672 681 .19016 .2588 710 046 .2505 738 076 .2422 767 .2339 106 .18795 .19136 5.2257 824 .2174 166 852 197 .2092 881 227 .2011 257 .1929 910

.98404

399 394 389 383

Cot

Tan 79°

.19081

59 58 57

1

2 3

56 55 54 53 52 51 50 49

48 47 46 45 44 43 42

4 6

109 138 167 195 .19224

6 7 8 9

252

10

.19366

11

12 13 14

395 423 452 481

15

.19509

16 17 18

538 566 595 623

281

309 338

19438 5.1446 468 .1366 498 .1286 529 .1207 559 .1128 .19589 5.1049 619 .0970 649 .0892 .0814 680 710 .0736 .19740 5.0658 770 .0581 .0504 801 831 .0427 861 .0350 .19891 5.0273 921 .0197 952 .0121 .19982 5.0045 .20012 4.9969 .20042 4.9894 073 .9819

41

19

.98378

40

373 368 362 357

39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19

20 .19652 21 680 22 103 .9744 709 .9669 23 737 133 164 .9594 24 766 25 .19794 .20194 4.9520 224 .9446 26 823 254 851 .9372 27 880 285 .9298 28 315 .9225 29 908 30 .19937 .20345 4.9152 376 .9078 31 965 406 .9006 32 .19994 33 .20022 436 .8933 051 466 .8860 34 36 .20079 .20497 4.8788 527 .8716 36 108

.98352

347 341 336 331 .98325

320 315 310 304 .98299

294 288 283 277 .98272

267 261 256 250

37 38 39

40 41

18 17 16

42 43 44

.98245

16

240 234 229 223

14 13 12

45 46

.98218

10

212 207 201 196

9 8

.18938 .19287 5.1848 .98190 967 317 .1767 185 .18995 347 .1686 179 .19024 378 .1606 174 052 408 .1526 168 .19081 .19438 5.1446 .98163

Cos

60

Sin

11

7 6

6

4 3 2 1

/

47 48 49 60 51

.98163 157 152 146 140 .98135 129 124

118 112 .98107 101

096 090 084 .98079

073 067 061 056

78°

50 49 48 47 46 45 44 43 42 41

40 39 38 37 36 36 34 33 32

.98021

016 010 .98004 .97998 .97992

987 981 975 969 .97963

958

52 53 54 65 .20649 .21104 4.7385 .97845 677 134 .7317 839 56 164 .7249 57 706 833 734 195 827 .7181 58 225 59 763 .7114 821 60 .20791 .21256 4.7046 .97815

Tan

51

044 039 033 027

770 .8147 910 336 .20364 .20800 4.8077 .97905 .8007 899 393 830 .7937 893 421 861 .7867 887 450 891 921 .7798 881 478 .20507 .20952 4.7729 .97875 .7659 869 535 .20982 563 .21013 .7591 863 857 592 043 .7522 073 .7453 851 620

Cot

59 58 57 56 55 54 53 52

.98050

557 .8644 952 136 .8573 946 165 588 618 .8501 940 193 .20222 .20648 4.8430 .97934 679 928 250 .8359 922 279 709 .8288 739 307 .8218 916

Cos

60

Sin

31

30 29 28 27

26 25 24 23 22 21

20 19 18 17

16

15 14 13 12 11

10 9 8 7 6

6 4 3 2 1

'

245

Appendix C

13°

12°

T

S.n .207 iM

1

BM

2

84S

o

877

4

905

5

.20933

6

888

7

.209911

S

.21019

9 10 11

12 13 14

15 16 17 IS 19

20

047 .21076 104 138 161 188 .21218 246

275 303 331 .21360

21

22 23 24 25 26 27 2S 29 30

417 445 474 .21502 530 559 587 616

31 32

672

33 34 35 36 37 38 39

.21644

701 729

758 .21786

814 843 871 899

40

.21928

41

956

42 43 44 45 46 47 48 49 50

.21985 .22013

51 52 53

54

240 268 297 325

55

.22353

56 57 58 59 60

382 410 438 467

041 22070 098 126 155 183 .22212

.22495

Cos

Tan

Cot

Cos

4.7046 .97815 .6979 BOO 988 .6912 316 803 797 347 .6845 .6779 791 877 4.0712 .97784 .81408 .6646 778 488 77: 469 .6580 499 .6514 766 .644J. 760 529 .21560 4.6382 .97754 .6317 748 580 742 .6252 621 735 651 .6187 .6122 729 688 .21712 4.6057 .97723 .5993 717 743 773 711 .5928 .5864 705 S04 .5800 834 698 21S64 4.5736 .97692 S95 .5073 686 .5609 925 6S0 .5546 673 956 667 21986 .54S3 .22017 4.5420 .97661 047 .5357 655 .5294 648 07S .5232 642 10S .5169 139 636 .22169 4.5107 .97630 .5045 623 200 .4983 617 231 .4922 261 611 292 .4860 604 .22322 4.4799 .97598 .4737 353 592 .4676 585 383 .4615 579 414 .4555 573 444 .22475 4.4494 .97566 .4434 560 505 .4373 536 553 .4313 547 567 4253 597 541 .22628 4.4194 .97534 .4134 658 528 4075 689 521 719 .4015 515 .3956 508 750 .22781 4.3897 .97502 .3838 496 811 .3779 489 842 .3721 872 483 .3662 476 903 .22934 4.3604 .97470 .3546 964 463 .22995 .3488 457 .3430 .23026 450 .3372 056 444 .23087 4.3315 .97437 .21256

Cot

I

77°

Tan

Sin

60 59

58 57 56 55 54 53 52 51 50 49 4S 47 46 45 44 43 42 41

40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24

Cos Sin Tan Cot 6 .22495 .23087 4.3315 .97437 .3257 430 523 117 i .3200 552 148 424 2 .3143 417 179 580 3 209 .3086 411 608 4 5 .22637 .23240 4.3029 .97404 271 .2972 665 398 6 7 8 9 10 11

12 13 14

15 16 17 18 19

20 21

23 22 21

22 23 24 25 26 27 28 29 30 31 32 33 34 36 36 37 38 39

20

40

19 18 17 16

41

15 14 13 12 11

10 9 8 7 6 5 4 3

2 1

'

14°

42 43 44 45 46 47 48 49 50 51 52 53

54 55 56 57 58 59 60

693 722 750

301 .2916 332 .2859 363 .2803 .22778 .23393 4.2747 807 424 .2691 835 455 .2635 485 .2580 863 .2524 892 516 .22920 .23547 4.2468 578 948 .2413 .22977 608 .2358 .23005 639 .2303 670 .2248 033 .23062 .23700 4.2193 731 .2139 090 762 .2084 118 793 .2030 146 823 .1976 175 .23203 .23854 4.1922 231 885 .1868 260 916 .1814 946 .1760 288 .1706 316 .23977 .23345 .24008 4.1653 373 .1600 039 .1547 401 069 .1493 429 100 .1441 458 131 .23486 .24162 4.1388 .1335 514 193 .1282 542 223 .1230 571 254 599 285 .1178 .23627 .24316 4.1126 .1074 656 347 684 377 .1022 .0970 712 408 740 439 .0918 .23769 .24470 4.0867 .0815 797 501 .0764 825 532 .0713 853 562 .0662 882 593 .23910 .24624 4.0611 .0560 938 655 966 .0509 686 .23995 .24023

717 747

.0459 .0408

391

384 378

59 58 57 56 56 54 53 52 51

325 318 311 .97304

365 358 351

345 .97338 331

298 291 284 278 .97271

264 257 251 244 .97237

230 223 217 210

|

76

Tan c

Sin

6 .24192 220 i 249 2 277 3 305 4 5 .24333 362 6 7 390 8 418 9 446 10

.24474

11 12 13

503 531 559 587

14 15

.24615

41

18 19

644 672 700 728

40

20

.24756

39 38 37 36 35 34 33 32 31 30 29 28 27 26

21

784 813 841 869

16 17

22 23 24 25 26 27 28 29

.24897

925 954

30

.24982 .25010 .25038

31 32 33

066 094 122 151 .25179

.97203 196 189 182 176 .97169 162 155 148 141

25 24

34 35 36

23 22 21

37 38 39

207 235 263 291

20

40

.25320

19 18 17 16

41 42 43

348 376 404 432

.97134 127 120 113 106 .97100

15

093 086 079 072

9

14 13 12 11

10 8 7 6 5

4 3

2 1

4.0108 .97030

.24192 .24933

Cot

60

50 49 48 47 46 45 44 43 42

.97371

.24051 .24778 4.0358 .97065 079 .0308 809 058 840 .0257 108 051 .0207 136 871 044 902 .0158 037 164

Cos

1

Sin

'

Tan

Cot

Cos

.24933

4.0108 .0058 4.0009 3.9959 .9910 3.9861 .9812 .9763 .9714 .9665 3.9617 .9568 .9520

.97030

964 .24995 .25026

056 .25087 118

023 015 008 .97001

.96994

987 149 980 180 973 211 966 .25242 .96959 273 952 304 945 .9471 335 937 .9423 366 930 .25397 3.9375 .96923 .9327 916 428 .9279 459 909 .9232 490 902 .9184 894 521 .25552 3.9136 .96887 .9089 880 583 .9042 873 614 .8995 645 866 .8947 858 676 .25707 3.8900 .96851 .8854 844 738 .8807 837 769 .8760 829 800 .8714 822 831 .25862 3.8667 .96815 .8621 807 893 .8575 924 800 955 .8528 793 .8482 786 .25986 .26017 3.8436 .96778 .8391 771 048 .8345 764 079 .8299 756 110 .8254 749 141 .26172 3.8208 .96742 .8163 203 734 235 .8118 727 266 .8073 719 .8028 712 297 .26328 3.7983 .96705 .7938 697 359 .7893 390 690 .7848 421 682 .7804 675 452 .26483 3.7760 .96667 .7715 515 660 .7671 546 653 577 .7627 645 .7583 608 638 .26639 3.7539 .96630 .7495 670 623 .7451 701 615 .7408 733 608 .7364 764 600

44 45 .25460 488 46 47 516 545 48 49 573 60 .25601 629 51 657 52 685 53 54 713 55 .25741 769 56 57 798 826 58 59 854 60 .25882 .26795 3.7321 .96593

Cos

Tan

Cot

75°

Sin

60 59 58 57 56 56 54 53 52 51

60 49 48 47 46 45 44 43 42 41

40 39 38 37 36

35 34 33 32 31

30 29 28 27 26 25 24 23 22 21

20 19

18 17 16

15 14 13 12 11

10 9 8 7

6 5

4 3

2 1

/

246

Appendix C

15 '

Sin

Tan

.25S&2 .26795

3

910 938 966

4

.25994

6

.26022 .26951 050 .26982 079 .27013

1

2

6

7 8 9 10 11 12 13

14

15 16 17 18 19

20 21

22 23

107

359 387 415

500 528 556

.26976

40

27004 032 060 088

47 48 49 60 51 52 53 54 66 56 57 58 59

60

201 232

326 357 388

.26443 .27419 471 451

39

44 46 46

044

.26303 .27263 294 331

24 25 .26584 26 612 27 640 28 668 29 696 30 .26724 752 31 32 780 33 808 34 836 36 .26864 892 36 920 37 948 38

41 42 43

826 857 888 920

076 135 .26163 .27107 191 138 219 169 247 275

116

27144 172

200 228 256 .27284

312 340 368 396 27424 452 480 508 536 27564

Cos

c

482 513 545 .27576

607 638 670 701

16° Cot

Sin

Tan

17° Cot

Cos

Tan

Cot Cos 29237 30573 3.2709 95630 265 605 .2675 622 293 637 .2641 613 321 669 .2607 605 348 700 .2573 596 Sin

|

3.7321 .96593 585 .7277 578 .7234 570 .7191 562 .7148

3.7105 .7062 .7019 .6976 .6933 3.6891 .6848 .6806 .6764 .6722 3.6680 .6638 .6596 .6554 .6512 3.6470 .6429 .6387 .6346 .6305 3.6264 .6222 .6181 .6140 .6100

.27732 3.6059 764 .6018 795 .5978 826 .5937 858 .5897 .27889 3.5856 921 .5816 952 .5776 .27983 .5736 .28015 .5696 .28046 3.5656 077 .5616 109 .5576 140 .5536 172 .5497 .28203 3.5457 234 .5418 266 .5379 297 .5339 329 .5300 .28360 3.5261 391 .5222 423 .5183 454 .5144 486 .5105 .28517 3.5067 549 .5028 580 .4989 612 .4951 643 .4912 .2*675 3.4874

Cot

Cos

Tan 74°

.96555

547 540 532 524 .96517 509 502

494 486 .96479

471 463 456

448 .96440

433 425 417 410 .96402

394 386 379 371 .96363

355 347 340 332 .96324

316 308

60 59

1

58

2 3

57

56 66 54 53

4

52 51

8 9 10

60 49 48 47 46 46 44 43 42 41 40 39 38 37 36 36 34 33 32 31 30 29 28 27 26 26

301

24 23 22

293

21

.96285

20

277 269

19 18 17 16

261 253 .96246 238

230 222 214 .96206 198 190 182 174

.96166 158 150 142 134 .96126

Sin

16 14 13 12 11

10 9 8 7

6 5 4 3

2 1

6 6 7

11 12 13 14

15 16 17 18 19

20 21

22 23 24 26 26 27

28 29 30 31

32 33 34 36 36 37 38 39 40 41

42 43 44 46 46 47 48 49

50 51

52 53 54 65 56 57 58 59

60

27564 28675 3.4874 592 706 .4836 .4798 620 738 648 769 .4760 676 801 .4722 27704 2S832 3.4684 731 864 .4646 759 895 .4608 927 .4570 787 815 958 .4533 27843 28990 3.4495 .4458 871 29021 .4420 899 053 927 0S4 .4383 955 116 .4346 27983 .29147 3.4308 .4271 28011 179 .4234 039 210 242 .4197 067 274 .4160 095

.96126 118 110 102

094 .96086

078 070 062 054 .96046

037 029 021 013 .96005 .95997

989 981 972

60 59 58 57 56 56 54 53 52 51 50 49 48 47 46

45 44 43 42 41

.28123 .29305 3.4124 .95964 40 956 39 337 .4087 150 948 38 368 .4050 178 940 37 400 .4014 206 432 .3977 931 36 234 .28262 .29463 3.3941 .95923 35 915 34 495 .3904 290 907 33 318 526 .3868 .3832 898 32 558 346 890 31 590 .3796 374 .28402 .29621 3.3759 .95882 30 874 29 653 .3723 429 865 28 685 .3687 457 857 27 .3652 485 716 849 26 748 .3616 513 .28541 .29780 3.3580 .95841 25 811 .3544 832 24 569 843 .3509 824 23 597 .3473 816 22 875 625 807 21 906 .3438 652 .28680 .29938 3.3402 .95799 20 .3367 791 19 708 .29970 .3332 782 18 736 .30001 .3297 774 17 033 764 766 16 065 .3261 792 .28820 .30097 3.3226 .95757 16 .3191 749 14 128 847 .3156 740 13 160 875 .3122 732 12 192 903 724 11 .3087 224 931 3.3052 .95715 10 .30255 .28959 707 9 287 .3017 .28987 698 319 .2983 8 .29015 690 351 .2948 7 042 .2914 681 6 382 070 6 .29098 .30414 3.2879 .95673 664 .2845 4 446 126 .2811 656 3 478 154 .2777 647 2 509 182 639 1 .2743 541 209 .29237 .30573

Cos

Tan 73°

Sin

2

3

4 5 .29376 .30732 3.2539 .95588 404 6 764 .2506 579 432 7 796 .2472 571 460 8 828 .2438 562 9 487 860 .2405 554 10 .29515 .30891 3.2371 .95545 543 11 923 .2338 536 571 12 955 .2305 528 599 .30987 13 .2272 519 626 .31019 14 .223s> 511 15 .29654 .31051 3.2205 .95502 682 16 083 .2172 493 710 17 115 .2139 485 737 147 .2106 476 18 765 467 19 178 .2073 20 .29793 .31210 3.2041 .95459 821 21 242 .2008 450 849 22 274 .1975 441 876 23 306 .1943 433 904 338 .1910 424 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

40 41

42 43 44 45 46 47 48 49 50 51 52 53 54 65 56 57 58 59

60

3.2709 .95630

Cot

1

'

.29932 .31370

960 .29987 .30015

043

402 434 466 498

3.1878 .95415 .1845 407 .1813 .1780 .1748

398 389 380

.30071 .31530 3.1716 .95372 098 562 .1684 363 126 594 .1652 354 154 626 .1620 345 182 337 658 .1588

3.1556 .95328 .1524 319 .1492 310 .1460 301 .1429 293 .30348 .31850 3.1397 .95284 376 882 .1366 275 914 403 .1334 266 431 946 .1303 257 459 .31978 .1271 248 .30486 .32010 3.1240 .95240 042 .1209 514 231 542 074 .1178 222 670 106 .1146 213 597 139 .1115 204 .30625 .32171 3.1084 .95195 653 203 .1053 186 .1022 680 235 177 267 708 .0991 168 299 .0961 159 736 .30763 .32331 3.0930 .95150 791 363 .0899 142 819 396 .0868 133 846 428 .0838 124 874 460 .0807 115 .30902 .32492 3.0777 .95106

.30209 .31690

237 265 292 320

Cos

722 754 786 818

Cot

Tan 72°

Sin

60 59 58 57 56 65 54 53 52 51

60 49 48 47 46 46 44 43 42 41

40 39 38 37 36 35 34 33 32 31

30 29 28 27 26 25 24 23 22 21

20 19 18 17 16

15 14 13 12 11

10 9 8 7 6

6

4 3 2 1

247

Appendix C

20°

19°

18°

'

T

Sin

Tan

Cot

32492 3.0777 524 .0746 i 957 556 .0716 2 .0686 a .30985 688 4 .31012 621 .0655 5 .31040 32653 3.0625 .0595 6 068 885 095 717 .0565 7 123 .0535 749 8 151 .0505 188 10 .3117$ 32S14 3.0475 .0445 11 206 S46 ^7^ 233 .0415 13 261 911 .0385 14 .0356 943 us 15 31316 32975 3.0326 344 33007 .0296 16 372 .0267 17 040 .0237 399 072 18 427 .020S 104 20 .31454 33136 3.0178 21 4S2 .0149 169 22 510 .0120 201 537 .0090 233 24 565 .0061 266

30902 929

u

U

a

25 26

27 as 29

30 31 32

33 34 35 36 37

3S 39

40 41

42 43 44 45 46 47 45 49 50 51 58

53

M 55

56 57 5s 59

60

Cos 95106 097 OSS 079

070 95061 052 043 033 024 95015 95006 94997 9SS 979 94970 961 952 943 933 94924 915 906 897 888

.31593 .3329S 3.0032 .94878 869 620 330 3.0003 64 B 860 363 2.9974 675 395 .9945 851 703 427 .9916 842 .31730 .33460 2.9887 .94832 758 492 .9858 823 786 524 .9829 814 813 .9800 805 557 841 589 .9772 795 .31868 .33621 2.9743 .94786 896 777 654 .9714 768 923 686 .9686 951 718 .9657 758 .31979 .9629 749 751

.32006 .33783 2.9600 034 .9572 816 061 848 .9544 089 .9515 881 116 913 .9487 .32144 .33945 2.9459 171 .33978 .9431 199 .34010 .9403 227 043 .9375 254 075 .9347 .32282 34108 2.9319 309 140 .9291 337 173 .9263 364 205 .9235 392 238 .9208 .32419 34270 2.9180 447 303 .9152 474 335 .9125 502 368 .9097 529 400 .9070 .32557 34433 2.9042

Cob

Cot

Tan 71°

.94740

730 721 712 702 .94693 684

674 665 656 94646 637 627 618 609 94599 590 580 571 561 .94552

Sin

Sin

Tan

Cot

Cos

32557 34433 2.9042 94552 542 584 465 .9015 498 .8987 533 612 523 639 530 .8960 563 .8933 514 667 2.8905 94504 .32694 34596 628 .8878 495 722 661 485 749 .8851 .8824 476 777 693 .8797 726 466 804 .32832 34758 2.8770 94457 791 .8743 447 859 824 .8716 887 438 .8689 914 856 428 889 .8662 418 942 .32969 .34922 2.8636 .94409 954 .8609 399 .32997 .8582 390 .33024 .34987 .8556 051 .35020 380 052 .8529 370 079

33106 .35085 2.8502 .94361 118 .8476 351 134 150 .8449 342 161 183 .8423 189 332 216 .8397 322 216 .33244 .35248 2.8370 .94313 281 .8344 303 271 314 .8318 293 29S

346 .8291 326 284 379 .8265 274 353 33381 .35412 2.8239 .94264 445 .8213 254 408 477 .8187 245 436 510 .8161 463 235 543 .8135 490 225 .33518 .35576 2.8109 .94215 608 .8083 545 206 641 .8057 196 573 674 .8032 186 600 707 .8006 627 176 .35740 2.7980 .94167 .33655 772 .7955 157 682 805 .7929 710 147 838 .7903 737 137 .7878 871 127 764 .33792 .35904 2.7852 .94118 937 .7827 108 819 .7801 098 846 35969 .7776 088 874 36002 035 .7751 901 078 33929 36068 2.7725 .94068 101 .7700 956 058

.33983 .34011

038 .34065

093

134 167 199

.7675 .7650 .7625

049 039

029

36232 2.7600 .94019 265 .7575 .94009 298 .7550 .93999 331 .7525 989 364 .7500 979

120 147 175 .34202 .36397 2.7475 .93969

Cos

Cot

Tan

70°

Sin

Tan

Cot Cos 6 .34202 .30397 2.7475 93969 95'.) i 229 .7450 430 919 2 257 .7425 463 3 284 .7400 939 496 4 5 6 7

8 9

10 11

12 13 14

Sin

311

529

.7376

.34339 .36562 595 366

2.7351 .7326 .7302 .7277 .7253

393 421 448

628 001

694

.34475 .36727

503 530 557 584

760 793 826 859

16

.34612 .36892

16 17

639 925 666 958 694 .36991 721 .37024

18 19

20 .34748 21 775 22 803 23 830 24 857 26 .34884 26 912 27 939 28 966 29 .34993 80 .35021 31 048 32 075 33 102 34 130 36 .35157 36 184 37 211 38 239 39 266 40 .35293 41 320 42 347 43 375 44 402 46 .35429 456 46 484 47 48 511 49 638 60 .35565 592 61 619 52 647 53 674 54 66 .35701 728 66 755 57 782 58 69 810 60 .35837

Cos

.37057

090 123 157

190 .37223

256 289 322 355

929 93919 909 899 889 879

2.7228 .93869 .7204 859 .7179 849 839 .7155 .7130 829 2.7106 .93819 809 .7082 799 .7058 .7034 789 779 .7009 2.6985 .93769 .6961 759 748 .6937 738 .6913 728 .6889 2.6865 .93718 .6841 .6818 .6794 .6770

708 698 688 677

60 59 58 57 50 55 54 53

52 51

50 49 48 47 46 46 44 43 42 41

40 39 38 37 36 35 34 33 32 31

.37388 2.6746 .93667 657 422 .6723 647 455 .6699 637 488 .6675 626 521 .6652 .37554 2.6628 .93616 606 588 .6605 596 621 .6581 654 585 .6558 575 687 .6534 .37720 2.6511 .93565 555 754 .6488 544 .6464 787 534 .6441 820 524 853 .6418 .37887 2.6395 .93514 503 920 .6371 493 .6348 953 .6325 483 .37986 .6302 472 .38020

30

2.6279 .93462 452 .6256 441 .6233 431 .6210 .6187 420 2.6165 .93410 400 .6142 .6119 389 379 .6096 .6074 368 2.6051 .93358

10 9 8

.38053

086 120 153 186 .38220

253 286 320 353 .38386 |

Cot

Tan 69°

Sin

29 28 27

26 25 24 23 22 21

20 19 18 17 16

16 14 13 12 11

7

6

6 4 3

2 1

/

248

Appendix C

21 /

1

2 3

4 6 6 7

8 9

10 11

12 13 14

15 16 17 18

Sin

217 .36244 271

19

.36379

21

406 434 461 488

32 33 34 35 36 37 38 39 40 41 42 43

Cot

2.5826 .93253 .5804 243 .5782 232 .5759 222 .5737 211 854 .38888 2.5715 .93201 .5693 190 921 .5671 180 955 .5649 169 .38988 .5627 159 .39022 .39055 2.5605 .93148 .5583 137 089 .5561 127 122 .5539 116 156 .5517 106 190 .39223 2.5495 .93095 .5473 084 257

.36108 .38721 135 754 162 787 190 821

20

31

22°

.36515

542 569 596 623

290 324 357

.5452 .5430 .5408

074 063 052

.36650 .39391

2.5386 .5365 .5343 .5322 .5300 2.5279 .5257 .5236 .5214 .5193 2.5172 .5150 .5129 .5108 .5086 2.5065 .5044 .5023 .5002

.93042

677 704 731 758

425 458 492

626

.36785 .39559

812 839 867 894

593 626 660 694

.36921 .39727 948 761 .36975 795 .37002 829

031 020 .93010 .92999 .92988

978 967 956 945

Cot

Tan 68°

80

10

40

20

39 38 37 36 35 34 33 32 31 30 29 28 27 26 26 24 23 22

21

21

20

924 913 902 892

19 18 17 16

.92881

16

870 859 849 838

14 13 12

.92827

10

816 805 794 784

9

.92773

6 4 3 2

762 751 740 729 .92718 |

Sin

11

8 7

6

1

'

Sin

23°

Tan

Cot

5 6 7

8 9 11

12 13 14

16 16 17 18 19

22 23 24 26 26 27 28 29

.37595 .40572 2.4648 .4627 622 606 649 640 .4606 676 674 .4586 .4566 703 707 .37730 .40741 2.4545 .4525 757 775 784 809 .4504 811 843 .4484 .4464 838 877 .37865 .40911 2.4443 .4423 892 945 .4403 919 .40979 946 .41013 .4383 047 .4362 973 .37999 .41081 2.4342 .38026 .4322 115 053 149 .4302 .4282 080 183 .4262 107 217

.38134 .41251 2.4242 .4222 161 285 .4202 188 319 .4182 215 353 .4162 241 387 80 .38268 .41421 2.4142 .4122 455 295 31 .4102 322 490 32 524 .4083 349 33 .4063 558 34 376 36 .38403 .41592 2.4043 .4023 430 626 36 .4004 37 456 660 .3984 38 483 694 .3964 39 728 510 40 .38537 .41763 2.3945 797 .3925 41 664 .3906 42 691 831 .3886 43 617 865 .3867 44 644 899 46 .38671 .41933 2.3847 .3828 46 698 .41968 .3808 47 725 .42002 .3789 48 752 036 .3770 49 070 778 60 .38805 .42105 2.3750 .3731 51 832 139 .3712 52 859 173 .3693 53 207 886 .3673 54 242 912 66 .38939 .42276 2.3654 310 .3635 56 966 345 .3616 67 .38993 379 .3597 58 .39020 .3578 413 59 046 60 .39073 .42447 2.3559

Cot

Tan

Cot

67

c

/

Cos

6 .37461 .40403 2.4751 .92718 .4730 707 488 436 i 697 516 470 .4709 2 .4689 642 504 686 3 538 .4668 675 569 4

59 58 57 56 66 54 53 52 51 60 49 48 47 46 45 44 43 42 41

.92935

029 44 862 45 .37056 .39896 083 46 930 110 47 963 137 .39997 48 164 .40031 49 .4981 60 .37191 .40065 2.4960 218 51 098 .4939 52 245 132 .4918 272 53 166 .4897 54 299 200 .4876 56 .37326 .40234 2.4855 56 353 267 .4834 57 380 301 .4813 58 407 335 .4792 59 434 369 .4772 60 .37461 .40403 2.4751 Cos

/

Cos

35837 .38386 2.6051 .93353 420 .6028 318 864 337 891 453 .6006 487 327 .5983 918 520 .5961 316 945 35973 .38553 2.5938 .93306 587 295 36000 .5916 620 .5893 285 027 654 .5871 274 054 081 687 .5848 264

298 325 352

22 23 24 26 26 27 28 29 30

Tan

c

.92664

653 642 631 620 .92609

598 587 576 565 .92554 643

532 621 510 .92499

488 477 466 455

60 59 58 57 56 66 54 53 52

1

2 3

4 6 6 7

8

51 50

9 10

49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32

11 12 13 14

15 16 17 18 19

20

31

21 22 23 24 26 26 27 28 29

.92388

30

30

377 366 355 343

29 28 27 26 26 24 23 22

31 32 33 34

.92444

432 421 410 399

.92332 321

310 299 287

21

.92276

20

265 254 243 231

19 18 17 16

.92220

16

209 198 186 175

14 13 12 11

.92164 152 141 130 119 .92107

10

096 085 073 062

4

9

8 7

6 6 3

2 1

.92050

Sin

/

35 36 37 38 39 40 41 42 43

44 46 46 47 48

Sin

Tan

Cot

.39073 .42447 100 482 127 516 153 551

Cos

2.3559 .92050

60

.3539 039 .3520 028 .3501 016 .3483 .92005

59 58 67 56

.91994

66 54

180 585 .39207 .42619 2.3464 234 654 .3445 260 688 .3426 287 722 .3407 314 757 .3388 .39341 .42791 2.3369 367 826 .3351 394 860 .3332 421 894 .3313 448 929 .3294 .39474 .42963 2.3276 601 .42998 .3257 528 .43032 .3238 655 067 .3220 681 101 .3201 .39608 .43136 2.3183 635 170 .3164 661 205 .3146 688 239 .3127 715 274 .3109 .39741 .43308 2.3090 768 343 .3072 795 378 .3053 822 412 .3036 848 447 .3017 .39875 .43481 2.2098 902 616 .2980 928 650 .2962 955 585 .2944 .39982 620 .2925 .40008 .43654 2.2907 035 689 .2889 062 724 .2871 768 .2853 088 115 793 .2835 .40141 .43828 2.2817 168 862 .2799 195 897 .2781 .2763 221 932 248 .43966 .2745 .40275 .44001 2.2727 .2709 301 036 .2691 328 071 .2673 355 105 .2655 381 140

982 971 959 948 .91936

925 914 902 891

Cot

Tan 66°

51

60 49

48 47

.91879

46 46

868 856 845 833

44 43 42 41

.91822

40

810 799

39 38 37 36

787 775 .91764 762 741 729 718 .91706

694 683 671

660 .91648

636 625 613 601 .91690

36 34 33 32 31

30 29 28 27 26 25 24 23 22 21

20

678 666 655 543

19 18 17 16

.91531

15

619 508 496 484

14 13 12

49 60 .40408 .44175 2.2637 .91472 210 .2620 51 434 461 244 .2602 52 449 461 279 .2584 53 488 437 314 .2666 425 54 614 66 .40541 .44349 2.2549 .91414 567 384 .2531 402 66 594 418 .2513 390 57 .2496 621 453 378 58 59 647 488 .2478 366 .44523 2.2460 60 .40674 .91355

Cos

53 52

Sin

11

10 9 8 7

6 5 4 3 2 1

i

249

Appendix C

|

Tan

Cot

1

2

700

m

59.!

ni

662

.;

4

6 6 7

S

\>o BOO

m

1

913

.40939

11 12 13

966

15 16 17 18 19

20 21

22 23 24 35 26 27 28 29

30 31

32 33 34

36 36 37 38

39 40 41

42 43 44 45 46 47 48 49

60 61

52 53 54

2408

627

319 307

.3390 2.2373 .91295 2355 383 rsa 272 707 .2338 .2320 802 380 248 SJ7 .2303 .91236 .44873 2.2286

.40808 .44697

10

14

668

2268

907 942

.2251 .2234 .2216

.40992 .41019 .44977 045 .45012

224 212 200 188 .91176 164 152 140

.41072 .45047 2.2199 .2182 0S2 09S .2165 125 117 152 .214S 151 .2130 128 17S 1S7 .41204 .45222 2.2113 .91116 104 231 257 .2096 092 257 292 .2079 080 327 .2062 2S4 .2045 068 362 310 .41337 .45397 2.202S .91056

363 390 416 443

432 467 502 638

.41469 .45573

496 522 549 675

608 643 678 713

.41602 .45748

628 655 681 707

784 819 854 889

.41734 .45924

760 960 787 .45995 813 .46030 840 065 .41866 .46101 892 136 919 171

945 972

206 242

.41998 .46277 .42024 312 051 348

077 104

383 418 66 .42130 .46454 56 156 489 57 183 525 68 209 660 69 235 595 60 .42262 .46631

Cm

Cot

|

.2011 .1994 .1977

044 032 020

.1960 2.1943 .1926 .1909 .1892 .1876 2.1859 .1842 .1825 .1808 .1792 2.1775 .1758 .1742 .1725 .1708 2.1692 .1675 .1659 .1642 .1625 2.1609 .1592 .1576 .1560 .1643 2.1527 .1510 .1494 .1478 .1461 2.1445

.91008 .90996

Tan

65°

/

Cos

2.2460 .01355 .2443 343 331 2426

.40074 .44523

26*

25°

24° Sin

984 972 960 948 .90936

924 911 899 887

60 59 5S 57 56 55 54 53 52 51

50 49 4S 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27

26 25

.90875

24 23 22 21 20

863 851 839 826

19 18 17 16

.90814

15

802 790 778 766

14 13 12 11

.90753 741

10

729 717 704

8 7 6 5

.90692

680 668 655 643

9

4

3 2 1

.90631

Sin

'

Sin

Tan

Cot

Cos

6 .42202 .46631 2.1445 .90631 .1429 618 666 i 388 Q 702 .1413 606 315 594 341 737 .1396 3 .1380 682 367 772 4 6

6 7

8 9 10 11 12 13 14

15 16 17

18 19

20 21 22

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

40 41 42

43 44 45 46 47 48 49 60 51 52

63 54 66 56 57 58

69 60

.42394 .4680S 2.1364 .90569 557 .1348 420 843 .1332 545 446 879 532 914 .1315 473 .1299 520 499 950 .42525 .46985 2.1283 .90507 .1267 495 552 .47021 483 578 056 .1251 .1235 470 604 092 .1219 458 631 128 .42657 .47163 2.1203 .90446 433 199 .1187 683 234 .1171 421 709 270 .1155 408 736 396 762 305 .1139 .42788 .47341 2.1123 .90383 377 .1107 371 815 358 841 412 .1092 346 867 448 .1076 .1060 334 894 483 .42920 .47519 2.1044 .90321 946 555 .1028 309 .1013 296 972 590 626 .0997 284 .42999 .43025 662 .0981 271 .43051 .47698 2.0965 .90259 077 733 .0950 246 769 .0934 233 104 805 .0918 221 130 .0903 208 156 840

2.0887 .90196 .0872 209 912 183 948 .0856 171 235 .0840 158 261 .47984 .0825 146 287 .48019 .43313 .48055 2.0809 .90133 091 .0794 120 340 127 .0778 108 366 163 .0763 095 392 082 198 .0748 418 .43445 .48234 2.0732 .90070 270 .0717 057 471 306 497 .0701 045 342 623 .0686 032 378 .0671 019 649 .43575 .48414 2.0655 .90007 .0640 .89994 602 450 .0625 628 486 981 521 .0609 968 654 .0594 680 657 956 .43706 .48593 2.0579 .89943 629 .0564 733 930 .0549 759 665 918 785 701 .0533 905 .0518 892 811 737 .43837 .48773 2.0503 .89879

.43182 .47876

Cos

Cot

Tan 64°

Sin

Tan

Sin

60 59 68 67 56 66 54 53 52

1

2

889

845

3

Jit.

881

S 12

917

4

6 6 7

8

51

9

60 49 48 47 46 46 44 43 42

10 11

12 13 14

15

41

16 17 18 19

40

20

39 38 37 36 36 34 33 32

21 22 23

31

30 29 28

24 25 26 27 28 29 30 31

24 23 22 21

32 33 34 35 36 37 38 39

20

40

19 18 17 16

41 42 43

27

26 26

15 14 13 12 11

10 9 8 7 6 5 4 3 2 1

'

.43837 .48773 883 809

44 46 46 47 48 49 60 51

52 53 54 66 56 57 58 59 60

.43968 .43094 18989 .44020 .19020 .

046 072

002 098

.44098 .49134 124 170 151 200 177 242

203

278

.44229 .49315 255 351 281 387

307 333

Cot

Cos

2.0503 .diss .0473 .0458 .0443 2.0428 .0413 .0398 .0383 .0368 2.0353 .0338 .0323 .0308 .0293 2.0278

.89879 S07

.0263 .0248 .0233 .0219

423 459

854 341

828 .89816 803 790

777 764 .89752

739 726 713 700 .89687

674 662 649 636

.44359 .49495 2.0204 .89623 .0189 385 532 610 .0174 411 568 597 .0160 437 584 604 .0145 464 640 571 .44490 .49677

516 542 568 594

2.0130 .89558

713 749 786 822

.0115 .0101 .0086 .0072

545 532 519 506

60 50 58 57

56 55 54

53 52 51

50 49 48 47

4G 45 44 43 42 41

40 39 38 37 36 35 34 33 32 31

.44620 .49858 2.0057 .89493 .0042 480 646 894 .0028 467 672 931 454 698 .49967 2.0013 1.9999 441 724 .50004 .44750 .50040 1.9984 .89428 415 .9970 776 076 .9955 402 802 113 .9941 389 149 828 .9926 376 854 185 .44880 .50222 1.9912 .89363 .9897 350 906 258 337 932 295 .9883 324 958 .9868 331 311 .44984 .9854 368

30

1.9840 .89298 285 .9825 272 .9811 259 .9797 245 .9782 1.9768 .89232 .9754 219 .9740 206 .9725 193 .9711 180 1.9697 .89167 .9683 153 .9669 140 .9654 127 .9640 114

16

.45010 .50404 036 441

062 088 114

477 514 550

.45140 .50587 166 623 192 660

218 243

696 733

.45269 .50769

295 806 321 843 347 879 373 916 .45399 .50953 1.9626 .89101

Cos

Tan

Cot

63

a

Sin

29 28 27 26 25 24 23 22 21 20 19 18 17 16 14 13 12 11

10 9 8 7 6 6 4 3 2 1

/

250

Appendix C

28°

27° /

1

2 3 4

Sin

477 503 554 580 606 632 .45658

12 13 14

684 710 736 762

15

.45787

16 17 18 19

813 839 865 891

20

.45917

21 22 23

942 968

11

24 25 26 27 28 29 30 31 32 33 34

35 36 37 38 39 40 41 42

43 44 45 46 47 48 49 60 51

52 53 54 65 56 57 58 59 60

Cot

Cos

1.9626 .89101 087 .9612 074 .9598 061 .9584 063 048 .9570 099 .51136 1.9556 .89035 021 .9542 173 .9528 .89008 209 .9514 .88995 246 981 .9500 283 .51319 1.9486 .88968 955 .9472 356 942 .9458 393 928 .9444 430 915 .9430 467 .51503 1.9416 .88902 888 .9402 540 875 .9388 577 862 .9375 614 848 .9361 651 .51688 1.9347 .88835 .9333 822 724 .9319 808 761 .9306 795 798 782 .9292 835

.45399 .50953 425 .50989 451 .51026

5 .45529 6 7 8 9 10

Tan

.45994 .46020 .46046 .51872

072 097

909 946

123 .51983 149 .52020 .46175 .52057 201 094 226 131 252 168

278

205

.46304 .52242

330 355 381 407

279 316 353 390

.46433 .52427

458 484 510 536

464 501 538

575

.46561 .52613

587 613 639 664

650 687 724 761

.46690 .52798

716 742 767 793

836 873 910 947

.46819 .52985

844 870 896 921

53022 059 096 134

.46947 .53171

Cos

Cot

1.9278 .88768 .9265 755 .9251 741 .9237 728 .9223 715

1.9210 .9196 .9183 .9169 .9155 1.9142 .9128 .9115 .9101 .9088 1.9074 .9061 .9047 .9034 .9020 1.9007 .8993 .8980 .8967 .8953 1.8940 .8927 .8913 .8900 .8887 1.8873 .8860 .8847 .8834 .8820

'

.88701

688 674 661 647 .88634 620 607 593 580 .88566 553 539

59 58 57 56 66 54 53 52 51 50

49 48 47 46 46 44 43 42 41

40 39 38 37 36 36 34 33 32 31

30 29 28 27

26 25 24 23 22 21 20 19

.88499

16

485 472 458 445

14 13 12

.88431

10

417 404 390 377

9 8 7 6

11

6 4 3 2 1

Tan

Sin

1

973

2 3 4

.46999 .47024

6 6

.47076 .53358 101 395 127 432 153 470

7

8 9

10 11

12 13 14

16 16 17 IS 19

20

Cos

/

050

178

229 255 281 306

37

38 39 40 41 42 43 44 46

46 47 48 49 60 51

52 53 54 65 56 57 58 59

1.8676 .88158 .8663 144 .8650 130 .8637 117 .8624 103

582 620 657 694

.47332 .53732

358 383 409 434

1.8611 .88089 .8598 075 .8585 062 .8572 048 .8559 034

769 807 844 882

.47460 .53920

537 .54032 662 070

36 36

.8755

957 511 .53995

.47588 .54107 614 145 639 183

665 690

254 240

1.8741 .88226 .8728 213 .8715 199 .8702 185 .8689 172

507

486

31 32 33 34

.8768

.47204 .53545

22 23 24

26 26 27 28 29 30

1.8807 .88295 .8794 281 .8781 267

208 246 283 320

21

60

1.8807 .88295

62°

Cot

c

Tan

Sin

Cot

Cos

1.8040 .8028 .8016 .8003 .7991 1.7979 .7966 .7954 .7942 .7930 1.7917 .7905 .7893 .7881 .7868 1.7856 .7844 .7832 .7820 .7808 1.7796 .7783 .7771 .7759 .7747 1.7735 .7723 .7711 .7699 .7687

.87462

60

448 434 420 406

59 58 57 56 56 54 53 52 51 60 49 48 47

1.7675 .7663 .7651 .7639 .7627 1.7615 .7603 .7591 .7579 .7567

.87036 021 .87007 .86993

|

18 17 16

349 336 322 308

Tan

.46947 .53171

60

526 512

.88363

Sin

29 /

220 258

.47716 .54296 741 333 767 371

328

203

.48354 .55241

379 405 430 456

279 317 355 393

.48481 .55431

Cos

Cot

7

60 49 48 47 46 46 44 43 42

10

8 9

.48862 .56003

22 23 24 26 26 27 28 29 30 31 32 33

21

1.8291 .87743 729 .8278 715 .8265 .8253 701 .8240 687

20

40

19 18 17 16

41

1.8228 .87673 .8215 659 .8202 645 .8190 631 .8177 617

16

1.8165 .87603 .8152 589 .8140 575 .8127 561 .8115 546 1.8103 .87532 .8090 518 .8078 504 .8065 490 .8053 476 1.S040 .S7462

10

42 43 44 45 46 47 48 49 50

9

51

8 7 6

52 53 54 55 56 57 58 59

Tan

1

61

c

Sin

14 13 12 11

5

4 3 2 1

60 /

697 736 774

16 16

34 35 36 37 38 39

25 24 23 22

659 684 710

786 888 811 926 837 .55964

35 34

26

.48608 .55621 634 659

.48735 .55812 761 850

1.8482 .87951 .8469 937 .8456 923 .8443 909 .8430 896

798 784 770 756

469 507 545 583

12 13 14

21

33 32 31 30 29 28 27

506 532 557 583

11

39 38 37 36

.8341 .8329 .8316 .8303

.48099 .54862 124 900 150 938 175 .54975 201 .55013 .48226 .55051 252 089 277 127 303 165

53 52 51

20

522 560 597 635

786 824

4 6 6

40

1.8354 .87812

048 073

3

41

.47844 .54484

.47971 .54673 .47997 711 .48022 748

1

2

1.8546 .88020 .8533 .88006 .8520 .87993 .8507 979 .8495 965

409 446

869 895 920 946

.48481 .55431

17 18 19

1.8418 .87882 868 .8405 854 .8392 840 .8379 .8367 826

793 818

|

60 59 68 57 56 65 54

888 913 938 964

041 079 117 156

.48989 .56194 .49014 232

040 065 090

270 309 347

.49116 .56385 424 141 166 462 192 501

217

539

.49242 .56577

268 293 318 344

616 654 693 731

.49369 .56769

394 419 445 470

808 846 885 923

.87391

377 363 349 335 .87321

306 292 278 264 .87250

46 45

235 221 207

42

44 43

193

41

.87178 164 150 136 121 .87107

40

093 079 064 050

978 .86964

949 935 921 906

39 38 37 36 35 34 33 32 31

30 29 28 27 26 26 24 23 22 21

1.7556 .86892 .7544 878 863 .7532 849 .7520 834 .7508 1.7496 .86820 805 .7485 791 .7473 .7461 777 762 .7449 1.7437 .86748 .7426 733 719 .7414 704 .7402 690 .7391

20

580 619 657 696

1.7379 .86675 .7367 661 .7355 646 632 .7344 .7332 617

5 4

.50000 .57735

1.7321 .86603

.49495 .56962 521 .57000

546 571 596

039 078 116

.49622 .57155 647 193

672 232 697 271 723 309 .49748 .57348 773 386 798 425 824 464 849 503 .49874 .57541

899 924 950 .49975

Cos

|

Tan

Cot

60°

Sin

19

18 17 16

16 14 13 12 11

10 9 8 7

6

3

2 1

Appendix C

30° '

Sin

6 1

2

3 4 5 6 7

8 9

10 11

12 13 14

15 IS 17 IS 19

20 21

22 23 24 36 26 27 28 39

30 31 32 33 34 36

36 37 38 39 40 41 42 43 44 46 46 47 48 49 60 51

62 53 54 66 56 57 58 59 60

1

Tan

Cot

.60000 .57735 774 025

050 076

813 S51

101

890

.50126 .57929 151 .57968 176 .5S007 201 046 •227

085

.50252 .58124 277 162

301 327 352

201

240 279

.50377 .58318

403 428 453 47S

357 396 435 474

.50503 .58513

31 Cob

1.7321 .86803 .7309 588 .7297 573 .7286 559 .7274 544

1.7262 .86530 .7251 515 .7239 501 .7228 486 .7216 471

1.7205 .86457 .7193 442 .7182 427 .7170 413 .7159 39S 1.7147 .86384 .7136 369 .7124 354 .7113 340 .7102 325 1.7090 .86310 .7079 295 .7067 281 .7056 266 .7045 251

552 638 553 591 578 631 603 670 .50628 .5S709 1.7033 .86237 748 654 .7022 222 679 7S7 .7011 207 704 826 .6999 192 729 865 .6988 178 .50754 .58905 1.6977 .86163 779 944 .6965 148 804 .58983 .6954 133 829 .59022 .6943 119 854 061 .6932 104 .50879 .59101 1.6920 .86089 140 904 .6909 074 179 929 .689S 059 954 218 .6887 045 .50979 258 .6875 030 .51004 .59297 1.6864 .86015 029 336 .6853 .86000 054 376 .6842 .85985 079 415 .6831 970 454 104 .6820 956 .51129 .59494 1.6808 .85941 154 533 .6797 926 179 573 .6786 911 204 612 !6775 896 229 651 .6764 881 .51254 .59691 1.6753 .85866 279 730 .6742 851 304 770 .6731 836 329 809 .6720 821 354 849 .6709 806 .51379 .59888 1.6698 .85792 404 928 .6687 777 429 .59967 .6676 762 454 .60007 .6665 747 479 046 .6654 732 .51504 .60086 1.6643 .85717

Cos

Cot

Tan 59°

1

Sin

Sin

60

6

59 58 57 56 66 54 53 52

i

51

60 49 48 47 46 46 44 43 42

2 3

4

15

41

40

20 21

22 23

24 26 26 27 28

31

29

30 29 28 27 26 26 24

30 31

32 33 34

35

30 29 28

275 299 324 349

.52621

7 6

5

4 3

2 1

53

54 56 56 57 58 69

320 360 400 440

.52374 .61480

45 46 47 48

8

200 240

.52250 .61280

522 547 572 597

49 50 51 52

50 49 48 47 46 46 44 43 42 41

1.6319 .85264 .6308 249 .6297 234 .6287 218 .6276 203 1.6265 .85188 .6255 173 .6244 157 .6234 142 .6223 127 1.6212 .85112 .6202 096 .6191 081 .6181 066 .6170 051

200 225

.52498

9

066 091

33 32 31

41 42 43 44

11

041

57 56 56 54 53 52 51

35 34

40

10

68

02487 527 568 608 649 62689 730 770 811 852 62892 933 62973 63014 055 63095 136 177 217 258 63299 340 380 421 462 63503 544 584 625 666 63707 748 789 830 871

310 294 279

19

14 13 12

59

62092 53017

60

1.6372 .85340 .6361 325

20

15

1.6643 .85717 .6632 702 .6621 687 .6610 672 .6599 657

.52126 .61080 151 120 175 160

21

18 17 16

Tan

1.6479 .85491 .6469 476 .6458 461 .6447 446 .6436 431 1.6426 .85416 .6415 401 .6404 385 370 .6393 .6383 355

399 423 448 473

23

Sin

Coa

.51877 .60681 902 721 927 761 952 801 .51977 841 .52002 .60881 026 921 051 .60960 076 .61000 101 040

36 37 38 39

22

205 245

Cot

6 .51628 .60284 1.6588 .85642 .6577 627 653 324 6 678 364 .6566 612 7 703 403 .6655 597 8 728 .6545 582 443 9 10 .51753 .60483 1.6534 .85567 778 522 .6523 551 1) 803 562 .6512 536 12 828 .6501 521 602 13 852 642 .6490 506 14 16 17 18 19

39 38 37 36 36 34 33 32

32°

Tan

.51504 .600S6 529 126 554 165

579 604

o

520 561 601 641 .61681 721 761 801 842 .61882

646 922 671 .61962 696 .62003 720 043 .52745 .62083

770 794 819 844

124 164 204

245

.52869 .62285

893 918 943 967

325 366 406 446

.6351 .6340 .6329

1.6160 .6149 .6139 .6128 .6118 1.6107 .6097 .6087 .6076 .6066

40 39

38 37 36

27

26 25 24 23 22 21

20 19 18 17

020

16 15 14

.85005 .84989

13 12

.85035

974

H

.84959

to

943 928 913 897

9

1.6055 .84882 .6045 866 .6034 851 .6024 836 .6014 820

5

8 7 6 4 3

2 1

60 .52992 .62487 1.6003 .84805 r

Cos

2f>,

Tan

Cot

58

c

Sin

/

.53115 140 164 189 214

53238 263 288 312 337 53361 386 411 435 460 53484 609 634 558 583 .53607

632 656 681 705 53730 754 779 804 828

Cot

Cos

1.6003 .84805 789 .5993 .5983 774 .5972 759 .5962 743

1.5952 .5941 .5931 .5921 .5911

84728 712 697

60" 69 r>s

57 66 65 64

681

63 62

666

51

1.5900 .84650 .5890 635 .5880 619 .5869 604 .5859 588

50 49 48

1.5849 84573 .5839 557 .5829 542 526 .5818 .5808 511 1.5798 .84495 .5788 480 .5778 464 .5768 448 .5757 433 1.5747 .84417 402 .5737 .5727 386 .5717 370 :5707 355

45 44 43 42

84339 324 308 292 277

30

1.5697 .5687 .5677 .5667 .5657

47 40

41

40 39

38 37 36 35 34 33 32 31

29 28

27 26

1.5647 .84261 .5637 245 230 .5627 214 .5617 .5607 198

25 24 23 22

1.5597 .84182 167 .5587 151 .5577 .5567 135 .5557 120

20

.54097 .64322 1.5547 .84104 .5537 088 122 363 404 .5527 072 146 446 .5517 057 171 041 487 .5507 195 .54220 .64528 1.5497 84025 569 .5487 84009 244 610 .5477 .83994 269 978 652 .5468 293 693 962 .5458 317 54342 .64734 1.5448 .83946 930 775 .5438 366 817 .5428 915 391 899 415 858 .6418 899 .5408 883 440 .83867 54464 .64941 1.5399

15

.53853 .63912

953 877 902 .63994 926 .64035 076 951 .53975 .64117 158 .54000 199 024

049 073

Cos

240 281

Cot

Tan 57°

Sin

21 19 18 17

16 14 13 12

11

10 9

8 7 6

5 4 3 2 1

Appendix C

252

34°

33° I

6 i

2 3

4 5

6 7

8 9 10 11

12 13

14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32

33 34 35 36 37 38

39 40 41

42 43 44 46 46 47 48 49 60 51 52 53 54 55

56 57 58 59 60

Sin

Tan

.54464 .64941 488 .64982 513 .65024

537 561

065 106

.54586 .65148 610 189 635 231

659 683

272 314

.54708 .65355

732 756 781

805

397 438 480 521

.54829 .65563

854 878 902 927

604 646 688 729

.54951 .65771

975 .54999 .55024

048

813 854 896 938

.55072 .65980

097 121 145 169 .55194

218 242 266 291 .55315

339 363 388 412 .55436

460 484 509 533

66021 063 105 147 66189 230 272 314 356 66398 440 482 524 566 66608 650 692 734 776

.55557 .66818 581 860

605 902 630 944 654 .66986 .55678 .67028 702 071 726 113 750 155 775 197 .55799 .67239

Cot

Cos

1.5399 .83867 851 .5389 835 .5379 819 .5369 .5359 804

1.5350 .83788 .5340 772 .5330 756 .5320 740 .5311 724 1.5301 .83708 .5291 .5282 .5272 .5262

692 676 660 645

1.5253 .83629 .5243 613 .5233 597 .5224 581 .5214 565

1.5204 .5195 .5185 .5175 .5166 1.5156 .5147 .5137 .5127 .5118 1.5108 .6099 .5089 .5080 .5070 1.5061 .5051 .5042 .5032 .5023 1.5013 .5004 .4994 .4985 .4975 1.4966 .4957 .4947 .4938 .4928 1.4919 .4910 .4900

60 59

i

2 3

51

60 49

48 47

46 45 44 43 42 41

40

533 517 501 485

39 38 37 36 36 34 33 32 31

453 437 421

405 .83389 373

356 340 324 .83308

292 276 260 244

30 29 28 27 26 26 24 23 22 21

.83228

20

212 195

19 18

179 163 .83147

17

131

115

098 082

16

15 14 13 12 11

.83066

10

050 034 017

8

9

.4891 .4882 .83001

7 6

282 324 366 409

1.4872 .82985 .4863 969 .4854 953 .4844 936 .4835 920

6 4

.55919 .67451

1.4826 .82904

823 847 871 895

Cos

Cot

Tan 56°

Sin

Cos Sin Tan Cot 6 .55919 .67451 1.4826 .82904

58 67 56 55 54 53 52

.83549

.83469

35° /

f

3

2 1

t

4 6 6

7

943 968 .55992 .56016

493 536 578 620

.56040 .67663

064 088

705 748 790 832

.4816 .4807 .4798 .4788

887 871 855 839

1.4779 .82822 .4770 806 .4761 790

.4751 773 112 .4742 757 136 10 .56160 .67875 1.4733 .82741 917 .4724 724 184 11 .4715 708 12 208 .67960 .4705 692 232 .68002 13 045 .4696 675 256 14 16 .56280 .68088 1.4687 .82659 .4678 130 643 16 305 173 .4669 626 17 329 215 .4659 610 18 353 258 .4650 593 19 377 20 .56401 .68301 1.4641 .82577 343 .4632 561 21 425 386 .4623 544 22 449 429 .4614 528 23 473 .4605 471 511 24 497 26 .56521 .68514 1.4596 .82495 .4586 557 478 26 545 .4577 600 462 27 569 .4568 642 446 28 593 .4559 429 685 29 617 30 .56641 .68728 1.4550 .82413 .4541 771 396 31 665 .4532 814 380 32 689 .4523 857 363 33 713 .4514 347 900 34 736 36 .56760 .68942 1.4505 .82330 .68985 .4496 314 36 784 .4487 297 37 808 .69028 .4478 281 071 38 832 .4469 114 264 39 856 40 .56880 .69157 1.4460 .82248 .4451 231 200 41 904 .4442 214 243 42 928 198 286 .4433 43 952 329 .4424 181 44 .56976 46 .57000 .69372 1.4415 .82165 416 .4406 148 024 46 .4397 132 459 47 047 502 .4388 115 48 071 545 .4379 098 49 095 50 .57119 .69588 1.4370 .82082 631 .4361 065 51 143 048 675 .4352 52 167 718 .4344 032 53 191 .82015 761 .4335 54 215 66 .57238 .69804 1.4326 .81999 .4317 982 847 56 262 965 891 .4308 57 286 .4299 949 934 58 310 .4290 932 59 334 .69977 60 .57358 .70021 1.4281 .81915

8 9

Cos

Tan

Cot

55

a

Sin

60 59 58 57

56 65 54 53 52 51 50 49 48 47

46 46 44 43 42

1

2 3 4

6 6 7

8 9

10 11 12 13 14

15

Sin

Tan

Cot

.57358 .70021 381 064 405 107 429 151 453 194 .57477 .70238 501 281

524 548 572

325 368 412

.57596 .70455

619 643 667 691

499 542 586 629

.57715 .70673

738 762 786 810

717 760 804 848

41

16 17 18 19

40

20

.57833 .70891

39 38 37 36 36 34 33 32

21

857 935 881 .70979 904 .71023 928 066

31

30 29 28 27 26 25 24 23 22

22 23

24 25 26 27 28 29

047 285 30 .58070 .71329 094 31 373 32 118 417 33 34

35 36

21

37 38 39

20

40

19 18 17 16

41

15 14 13 12 11

10 9 8

7 6 5

4 3 2 1

/

.57952 .71110 976 154 .57999 198 .58023 242

42 43 44 46 46 47 48 49 50 51

52 53 54 65 56 57 58 59 60

141 461 505 165 .58189 .71549

212 236 260 283

593 637 681 725

.58307 .71769

330 354 378

813 857 901 946

401 .58425 .71990 449 .72034

472 496 519

078

567 590 614 637

255 299 344 388

122 167 .58543 .72211

.58661 .72432

684 708 731 755

477 521 565

610

.58779 .72654

Cos

Cot

Cos

1.4281 .81915 .4273 899 .4264 882 .4255 865 .4246 848

1.4237 .81832 .4229 815 798 .4220 .4211 .4202

782 765

1.4193 .81748 .4185 731 714 .4176 .4167 698 .4158 681 1.4150 .81664 .4141 647 .4132 631 .4124 614 597 .4115 1.4106 .81580 .4097 563 .4089 546 .4080 530 .4071 513 1.4063 .81496 479 .4054 462 .4045 445 .4037 .4028 428 1.4019 .81412 395 .4011 378 .4002 .3994 361 .3985 344 1.3976 .81327 .3968 310 .3959 293 276 .3951 259 .3942 1.3934 .81242 225 .3925 .3916 208 .3908 191 .3899 174 1.3891 .81157 140 .3882 123 .3874 106 .3865 089 .3857 1.3848 .81072 055 .3840 038 .3831 021 .3823 .3814 .81004 1.3806 .80987 .3798 970 .3789 953 .3781 936 .3772 919 1.3764 .80902

Tan

54°

Sin

60 59 58 57 56 66 54 53 52 51

60 49 48 47 46 45 44 43 42 41

40 39 38 37 36 35 34 33 32 31

30 29 28 27 26 25 24 23 22 21

20 19 18 17 16

15 14 13 12 11

10 9

8 7 6

6 4 3

2 1

/

Appends

37°

36'

6 i *

3 4

5 6

7 s 9 10 11

12 13 14

15 16 17 18 19

20 21

22 33 34 26 26 27 2S 29 90 31

32 33 34 35

36 37 38 39 40 41

42 43 44 46 46 47 48 49 60 51

52 53 54 66 56 57 58 59 60

Sin

Tan

.58779

79654 699 743 788 B39 72877

bu 8M 849 879 5S896 aao 943

Cot

921

72906

987 .73010 .58990 055 .59014 .73100 037 144 061 189 234 084 108 278 .59131 .73323 154 368 178 413 201 457 225 502 .5924S .73547 272 592 295 637 318 681

342

726

.59365 .73771

389 412 436 459

816 861

906 951

.594S2 .73996 506 .74041

529 552 576

086

131 176 .59599 .74221

622 646 669 693

267 312 357 402

.59716 .74447

739 763 786 809

492 538 583 628

.59832 .74674

856 879 902 926

719 764 810 855

.59949 .74900

972

946

.59995 .74991 .60019 .75037

042

082

.60065 .75128 089 173 112 219 135 264 158 310 .60182 .75355

Cos

Cot

Cos

1.3764 .80902 .375,''

.3747 .3739 .3730

1.3722 .3713 .3705 .3697 .36S8

1.3680 .3072 .3663 .3655 .3647

'

888 867 850 833 .80816 799 782 765 748 .80730 713 696 679 662

1.363S .80644 .3630 627 .3622 610 .3613 593 .3605 576

60 59 58 57 56 56 54 53 52 51

60 49 48 47 46 46 44 43 42

6 i

2 3 4

6 6 7

8 9

10

1.3514 .80386 .3506 368 .3498 351 .3490 334 .3481 316

30 29 28 27 26 26 24 23 22

21

22 23

24 26 26 27 28 29

30 31

21

32 33 34 36 36 37 38 39

1.3432 .80212 .3424 195 .3416 178 .3408 160 .3400 143 1.3392 .80125 .3384 108 .3375 091 .3367 073 .3359 056

20

40

19 18 17 16

41

1.3351 .80038 .3343 021 .3335 .80003 .3327 .79986 .3319 968

10 9 8 7 6 5 4 3

1.3311 .79951 .3303 934 .3295 916 .3287 899 .3278 881

13 12 11

2 1

1.3270 .79864

Tan 53°

Sin

/

251 274 .60298 .75584 321 629 344 675 367 721 767 390 .60414 .75812

553 088 576 134 699 180 622 226 .60645 .76272 668 318 691 364 714 410

20

14

447 492 538

.60529 .76042

41

16

228

14

40

1.3473 .80299 .3465 282 .3457 264 .3449 247 .3440 230

Cot

15

12 13

1.3597 .80558 .3588 541 .3580 524 .3572 507 .3564 489 1.3555 .80472 .3547 455 .3539 438 .3531 420 .3522 403

31

Tan

437 858 460 904 483 950 506 .75996

11

16 17 18 19

39 38 37 36 35 34 33 32

Sin

.60182 .75355 205 401

42 43 44 46 46 47 48 49 60 51 52 53 54

56 56 57 58 59 60

738

456

.60761 .76502

784 807 830 853

548 594 640 686

.60876 .76733

899 922 945 968

779 825 871 918

.60991 .76964 .61015 .77010

038 061 084

057

103 149 .61107 .77196 130 242 153 289 176 335 199 382 .61222 .77428

245 268

475

291

568 615

314

521

.61337 .77661

360 383 406 429

708 754 801

848

.61451 .77895

474 941 497 .77988 520 .78035 543 082 .61566 .78129

Cos

38° '

Cos

1.3270 .79864 .3262 846 .3254 829 .3246 811 .3238 793

1.3230 .79776 .3222 758 .3214 741 .3206 723 .3198 706 1.3190 .79688 .3182 671 .3175 653 .3167 635 .3159 618 1.3151 .79600 .3143 583 .3135 565 .3127 547 .3119 530 1.3111 .79512 .3103 494 .3095 477 .3087 459 .3079 441 1.3072 .79424 .3064 406 .3056 388 .3048 371 .3040 353

60 59 58 57 56 66 54 53 52

Sin Tan Cot Cos 6 .61566 .78129 1.2799 .78801 i

2

3 4

704 410 .2753 726 457 .2740 749 504 .2738 772 551 .2731 .61795 .78598 1.2723 818 645 .2715 841 692 .2708 864 739 .2700 887 786 .2693 .61909 .78834 1.2685 932 881 .2677 955 928 .2670 .61978 .78975 .2662 .62001 .79022 .2655 .62024 .79070 1.2647 046 117 .2640 069 164 .2632 092 212 .2624 115 259 .2617 .62138 .79306 1.2609 160 354 .2602 183 401 .2594 206 449 .2587 229 496 .2579

8 9

10

34 33 32 31

11 12 13

14

16 16 17 18 19

20 21

22 23 24 26 26 27 28 29 30

30

1.2993 .79247 .2985 229 .2977 211 .2970 193 .2962 176

26 24 23 22 21

35 36

1.2954 .79158 .2946 140 .2938 122 .2931 105 .2923 087

20

40

19 18 17 16

41

1.2915 .2907 .2900 .2892 .2884 1.2876 .2869 .2861 .2853 .2846 1.2838 .2830 .2822 .2815 .2807 1.2799

.79069 051

16

Tan

Sin

52°

29 28 27

26

31

32 33 34

37 38 39

.62251 .79544

274 297 320 342

591 639

686 734

.62365 .79781

388 411 433

829 877 924

456 .79972 .62479 .80020

502 524 547 570

067 115

615 638 660 683

306 354 402 450

.78980

10

42 43 44 45 46 47 48 49 60

962 944 926 908

9

51

8

52 53 54 55 56 57 58 59

.62819 .80738

60

.62932 .80978

033 .79016 .78998

.78891

873 855 837 819

14 13 12 11

7

6 6

4 3 2 1

.78801 '

.2792 .2784 .2770 .2709

.61681 .78363

7

60 49 48 47 46 46 44 43 42

40 39 38 37 36 35

175

222 209 316

5

51

41

589 612 635 658

6

1.3032 .79335 .3024 318 .3017 300 .3009 282 .3001 264

Cot

2b/

(

163 211 .62592 .80258

.62706 .80498

728 751 774 796 842 864 887 909

Cos

546 594 642 690 786 834 882 930 Cot

783 765 747 729

1.2761 .78711

694 676 658 640

60 59 58 57 56 56 54 53 52 51

604 586 568 550

60 49 48 47 46

.78532

46

514 496 478 460

44 43 42

.78442

40

424 405 387 369

39 38 37 36

.78351

36

333 315 297 279

34 33 32

1.2572 .78261 243 .2564 225 .2557 206 .2549 188 .2542 1.2534 .78170 152 .2527 134 .2519 116 .2512 098 .2504

30

1.2497 .78079 061 .2489 043 .2482 025 .2475 .2467 .78007

20

1.2460 .77988 970 .2452 952 .2445 934 .2437 .2430 916 1.2423 .77897 .2415 879 861 .2408

15

843 824

7 6

.2401 .2393

.78622

1.2386 .77806 788 .2378 .2371 769 751 .2364 .2356 733 1.2349 .77715

Tan 51°

Sin

41

31

29 28 27 26 25 24 23 22 21 19 18 17 16 14 13 12 11

10 9 8

5

4 3 2 1

/

254

Appendix C

40°

39° Sin

o i

2 3

4 5 6

7

8 9

10

n

12 13

14 15 16 17 18 19

20 21

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

40 41

42 43 44 45 46 47 48 49 50 51

52 53 54 56 56 57 58 59 60

Tan

Cot

Cos

Sin

Tan

.62932 .80378 955 .81027 .62977 075 123 .63000 171 022

1.2349 .77715 696 .2342 678 .2334 .2327 660 .2320 641

.64279 .83910 301 .83960 323 .84009

.63045 .81220

1.2312 .77623 .2305 605 .2298 586 .2290 568 .2283 550

.64390 .84158

068 090

268 316 364 413

113 135 .63158 .81461 1.2276 .77531 510 .2268 513 180 558 203 .2261 494 606 225 .2254 476 .2247 248 655 458 .63271 .81703 1.2239 .77439 293 752 .2232 421 316 800 .2225 402 338 849 .2218 384 898 .2210 361 366 .63383 .81946 1.2203 .77347 406 .81995 .2196 329 428 .82044 .2189 310 092 451 .2181 292 141 473 .2174 273 63496 .82190 1.2167 .77255

238 .2160 236 287 .2153 218 563 336 .2145 199 385 585 .2138 181 .63608 82434 1.2131 .77162 483 630 .2124 144 518 540

531 653 580 675 629 698 63720 82678 727 742 776 765 825 787 874 810 .63832 82923 854 82972 877 83022 899 071 120 922 63944 83169 966 218 .63989 268 .64011 317 033 366 .64056 83415 078 465 514 too 123 564 145 613 .64167 83662 712 190 212 761 234 811 256 860 64279 83910

Cos

Cot

.2117 .2109 .2102

1.2095 .2088 .2081 .2074 .2066

1.2059 .2052 .2045 .2038 .2031

1.2024 .2017 .2009 .2002 .1995 1.1988 .1981 .1974 .1967 .1960

1.1953 .1946 .1939 .1932 .1925

125 107

088 77070 051 033 77014 76996 76977 959 940 921 903 76884 866 847 828 810 76791 772 754 735 717 76698 679

1.1918

661 642 623 76604

Tan

Sin

50°

346 368

059 108

41 Cot

Cos

1.1918 .76604 .1910 586 .1903 567 548 .1896 .1889 530 1.1882 .76511 492 .1875 .1868 473

412 208 258 435 307 .1861 455 457 .1854 436 479 357 64501 .84407 1.1847 .76417 457 398 524 .1840 546 507 .1833 380 556 .1826 361 568 342 590 606 .1819 64612 .84656 1.1812 .76323 635 304 706 .1806 657 756 .1799 286 679 806 .1792 267 .1785 248 701 856 .64723 .84906 746 .84956 768 .85006

790 812

057 107

.64834 .85157

856 878

207 257 308 358

901 923 .64945 .85408

1

2 3

4 5 6 7

8 9 10 11

12 13 14

192 173 154 1.1743 .76135 .1736 116

22 23

1.1708 .76041

.1702 458 022 .64989 509 .1695 .76003 .65011 559 .1688 .75984 033 609 .1681 965 .65055 85660 1.1674 .75946 .1667 077 710 927 100 761 .1660 908 122 811 .1653 889 144 862 .1647 870 .65166 85912 1.1640 .75851 188 85963 .1633 832 .1626 210 86014 813 .1619 232 064 794 254 .1612 775 115 .65276 86166 1.1606 .75756 298 216 .1599 738 320 267 .1592 719 .1585 700 342 318 364 368 .1578 680 .65386 86419 1.1571 .75661 .1565 642 408 470 .1558 623 430 521 .1551 452 572 604 474 623 .1544 585 .65496 86674 1.1538 .75566 547 725 .1531 518 776 .1524 528 540 827 .1517 509 562 878 .1510 490 584 65606 86929 1.1504 75471

Cot

49°

Tan

Sin

492 543 595 646

.65978 .66000

21

097 078 059

.65825 .87441

16 17 18 19

20

.1729 .1722 .1715

672 082 694 133 .65716 .87184 738 236 759 287 781 338 803 389

847 869 891 913

24

25 26 27 28 29 30 31

32 33 34 35 36 37 38 39 40 41

42 43 44 45 46 47

48 49 60 51

956

022

749 801 852 904

.66044 .87955 066 .88007

088 109

059 110 162

131 .66153 .88214 175 265 197 317

218 240

369 421

.66262 .88473

284 306 327 349

524 576 628 680

.66371 .88732

393 414 436 458

784 836 888 940

.66480 .88992 501 .89045 523 097 545 149 566 201 .66588 .89253

610 632 653 675

306 358 410 463

.66697 .89515

718 740 762 783

52 53 54 55 56 57 58 69

.66805

60

.66913

827 848 870 891

Cos

c

Tan

.65935 .87698

210

.1771 .1764 .1757 .1750

Sin

.65606 .86929 628 .86980 650 .87031

15

1.1778 .76229

967

Cos

/

Cot

Cos

1.1504 .1497 .1490 .1483 .1477 1.1470 .1463 .1456 .1450 .1443 1.1436 .1430 .1423 .1416 .1410 1.1403 .1396 .1389 .1383 .1376 1.1369 .1363 .1356 .1349 .1343 1.1336 .1329 .1323 .1316 .1310

.75471

452 433 414 395 .75375

356 337 318 299 .75280 261 241

222 203 .75184 165 146 126 107

44 43 42 41

39 38 37 36 35 34 33 32

.75011

.74992

973 953 934 915

1

60 49 48 47 46 45

40

1.1270 .74799 .1263 780 760 .1257 .1250 741 .1243 722 1.1237 .74703 .1230 683 .1224 664 .1217 644 625 .1211 1.1204 .74606 .1197 586 567 .1191 .1184 548 528 .1178 1.1171 .74509 .1165 489 .1158 470 .1152 451 431 .1145

Tan

51

069 050 030

1.1303 .74896 .1296 876 .1290 857 .1283 838 .1276 818

48°

59

58 57 56 65 54 53 52

.75088

567 620 672 725 89777 1.1139 .74412 .1132 392 830 .1126 373 883 935 .1119 353 89988 .1113 334 90040 1.1106 .74314 Cot

60

Sin

31

30 29 28 27 26 25 24 23 22 21 20 19 18 17 16

15 14 13 12 11

10 9 8 7

6 5

4 3 2 1

'

Appendix

42°

6 i

Sin

Tan

.66913 085

90040 093

Cot

956

3 4

5

6 7 s

9 10 11

12 13

M

15 16 17 IS 19

20 81 22

23 24 25 26 27 28 29

30 31

32 33

34 35 36 37 38

39 40 41

42 43 44 45 46 47 48 49 60 51

52 53 54 55 56 57 58 59 60

146 199 .66999 SSI .67021 .90304 043 357

064 0S6

410

463 107 516 .67129 .90569 151 SSI 172 674 727 194 215 781 .67337 .90834 258 B87 940 380 301 .90993 323 .91046 .67344 .91099 153 366 206 387

409 430

259 313

.67452 .91366

473

495 516 538

419 473 526 580

.67559 .91633 6>7 580 602 740 794 623

645

847

.67666 .91901 688 .91955 709 .92008

730

062

752 116 .67773 .92170

795 816 837 859

224 277 331 385

.67SS0 .92439 901 493

923 944 S65

547 601 655

.67987 .92709 .68008 763

029 051 072

817 872 926

Cos

1.1106 .74S14 .1100 398 276 .1093 .1087 356 .1080 337 1.1074 .74217 19S .1067 17S .1061 159 .1054 139 .1048 1.1041 .74120 100 .1035 .1028 OSO .1022 061 041 .1016

1.1009 74022 .1003 .74002 .0990 .73983 .0990 963 .0983 944 1.0977 .73924 .0971 .0964 .0958 .0951

904 885 865 846

1.0945 .0939 .0932 .0926 .0919 1.0913 .0907 .0900 .0S94 .0888 1.0881 .0S75 .0869 .0862 .0856 1.0850 .0843 .0837 .0831 .0824 1.0818 .0812 .0805 .0799 .0793

.73826

806 7>7

767 747 .73728

708 688 669 649 .73629

610 590 570 551

.68200

Cos

Cot

59

1

2 3

51

9 10

742 .68412 .93797

11 12 13

434 852 455 906 476 .93961 497 .94016

50 49 48 47 46 45 44

43 42 41

40 39 38 37 36 35 34 33 32 31 30 29

28 27

26 25 24 23 22 21

20

472 452

19 18 17 16

.73432

16

413 393 373 353

14 13 12 11

1.0786 .73333 .0780 314 .0774 294 .0768 274 .0761 254

10 9 8

Tan 47°

Sin

Tan

.68200 .93252 22 306 242 360 264 415

58 57 56 55 54 53 52

511 491

1.0755 .73234 .0749 215 .0742 088 195 .0736 143 175 197 .0730 155 .93252 1.0724 .73135

Sin

60

.73531

.68093 .92980 115 .93034

136 157 179

44°

43° '

7

6 5

4 3 2 1

'

4 6 6 7

8

14

15 16 17 18 19

2Jv3

469

.68306 .93524

327 349 370 391

578 633 688

.68518 .94071 539 125 561 180

582 603

235 290

Cot

.07

1

.0705 .0699

1.0661 .72937 917 .0655 .0649 897 .0643 877 .0637 857

1.0630 .72837 817 .0624 .0618 797 .0612 777 757 .0606 1.0599 .72737 .0593 717 .0587 697

21

645 400 666 455 688 510 .0581 .0575 709 565 .68730 .94620 1.0569 751 676 .0562 77'? 731 .0550 793 786 .0550 841 814 .0544 .68835 .94896 1.0538 857 .94952 .0532 878 .95007 .0526 899 062 .0519 920 118 .0513 .68941 .95173 1.0507 962 229 .0501 .68983 284 .0495 .69004 340 .0489 025 395 .0483 .69046 .95451 1.0477 067 .0470 506 088 562 .0464 109 618 .0458 673 .0452 130 .69151 .95729 1.0446 172 785 .0440 .0434 193 841 .042S 214 897 235 .95952 .0422 .69256 .96008 1.0416 277 064 .0410 298 120 .0404 319 176 .0398 340 232 .0392 .69361 .96288 1.0385 344 382 .0379 .0373 403 400 424 457 .0367 445 513 .0361 .69466 .96569 1.0355

31

32 33 34 35 36 37 38 39 40 41 42 43

44 45 46 47 48 49 60 51

52 53 54

55 56 57 58 59 60

Cos

Cot

096 076 056

1.0692 .73036 .0686 .73016 .0680 .72996 .0674 976 .0668 957

.68624 .94345

Tan 46°

'

Cos

1.0724 .73135 .0717 116

20 22 23 24 25 26 27 28 29 30

('

677 657 .72637

617 597 577 557 .72537

517 497 477 457 .72437

417 397 377 357

60 59 58 57 56 66 54 53 52 51

60 49 48 47 46 46 44 43 42

Sin

6 6 7 8 9

.69570 .96850 591 907 612 .96963 633 .97020

1.0325 .71833 .0319 813 792 .0313 .0307 772

752

51

10

.69675 .97133 696 189

1.0295 .71732 .0289 711 .0283 691 .0277 671 .0271 650

50 49 48 47 46 45 44 43 42

4

11

12 13 14

15

41

16 17 18 19

40

20

39 38 37 36 35 34 33 32

21

22 23

24

654

717 737 758

625 681

076

246 302 359

.69779 .97416

800 821 842 862

472 529 586 643

.69883 .97700

904 925 946 966

756 813 870 927

.69987 .97984 .70008 .98041

30 29

30

.70091 .98270

28

32 33 34 36 36

27

26 25

31

317 297 277 257

19 18 17 16

.72236

15

45

216

14 13 12 11

46 47 48 49 60

10 9

095 075 055

8

.72035 .72015 .71995

6 4 3 2

7 6

1

.71934 Sin

4S7 508 529 549

25 26 27 28 29

31

'

Cos

58 57 56 65 54 53 52

3

.72337

974 954

Cot

738 794

i

2

24 23 22 21 20

196 176 156 .72136 116

Tan

.69466 .90569

1.0355 .719:it 91 .0349 .0343 B94 .0337 .0331 853

6

37

38 39 40 41

42 43 44

51 52 53 54

65 56 57 58 59 60

029 049 070

098 155 213

1

.0301

1.0265 .71630 .0259 610 .0253 590 .0247 569 .0241 549 1.0235 .71529 .0230 508 .0224 488 .0218 468 .0212 447 1.0206 .71427 .0200 407 .0194 386 .0188 366 .0182 345

1.0176 .71325 112 327 .0170 305 132 384 .0164 284 153 441 .0158 264 174 499 .0152 243 .70195 .98556 1.0147 .71223 215 613 .0141 203 671 .0135 236 182 .0129 257 728 162 277 .0123 141 786 .70298 .98843 1.0117 .71121 .0111 319 901 100 .0105 339 .98958 080 360 .99016 .0099 059 381 073 .0094 039 .70401 .99131 1.0088 .71019 422 189 .0082 .70998 247 .0076 978 443 957 463 304 .0070 484 362 .0064 937 .99420 .70916 .70505 1.0058 525 478 .0052 896 546 536 .0047 875 567 .0041 855 594

587

652

.70608 .99710

628 768 649 826 670 884 690 .99942 .70711 1.0000

Cos

.0035 834 1.0029 .70813 .0023 793 772 .0017 752 .0012 731 .0006 1.0000 .70711

Tan

Cot

45°

Sin

60 59

41

40 39 38 37 36 36 34 33 32 31

30 29 28 27 26 25 24 23

22 21

20 19 18 17 16

15 14 13 12 11

10 9

8 7 6 6

4 3 2 1

'

INDEX

A hand (see St nation bands) Acceleration of body parts. 64 I7t>

Acetylcholine esterase (AChase), 171 Action current. 106 Action potential (se* Potential, action) Act in -filaments

(see

also

104

Analysis of segmental velocities effective rotation of spine when pelvis in motion, 70 overhand throw, 66, 68, 69, 72, 73

68

>:iing to illustrate. t^4. 67,

Acetylcholine (ACh), 171, action of. 171

with gravity board and three BCBles,

Negative. IS2 Positive. 167

Sliding

displacement of hand and forearm, 66 angular displacement of trunk due to

theory). 134. 136. 153. 154. 155. 156 in

is

angular

filament

--viation with tropomyosin laxed muscle. 143

Centripetal acceleration, 57

re-

pelvic

and spinal

rotation, 66, 70, 71

calculations of linear velocities resulting from angular displacements, 68 for determining amount of

method

Circuits,

neural,

combination

gamma

188,

L90 191,

L93

repeating,

191

189,

of, 191

loop, 203

convergent,

divergent,

Golgi tendon organ circuits, 208-209 reverberating, 175, 191 spindle afferent, 200, 203 two-neuron circuit, 190-191, Circular motion angular acceleration, 56-57

in contraction. 142

protraction at sternoclavicular joint,

centripetal acceleration, 57

in myofibril. 132

68. 72

radial acceleration, 57

subunits oi (F-aetin. G-actin), 132, 133 Activation. oi muscle. 140. 144, 158 Active state. 13S. 149. 150. 155. 159, 187 decay in fast vs. slow muscle fibers, 159 in activation oi muscle. 135 in development of overt tension. 149, 150 in muscle twitch. 147-148 in post-tetanic potentiation. 150 in tetanic contraction. 150 study by quick release, 148 study by quick stretch. 148 Actomyosin. 132. 135. 136, 140. 143. 159

Adaptation (accommodation). 149 decline of generator potential in sensory neurons, 181-182

in,

182

131.

134,

condensation of. to produce ATP, 138 in muscle metabolism. 136-138 in oxygen debt. 138 in regulation of muscle metabolism, 139

Adenosine triphosphate (ATP).

131,

Co-contraction, 30, 200, 202

Archimedes principle, 110 ATP (see Adenosine triphosphate) ATPase (see also Adenosine triphosphate, 135, 155, 159

correlation with speed of contraction, 155 in contraction, 144

slow muscle fibers, 158-159 in hydrolysis of ATP, 135-136 in relaxation, 143, 144 Available energy, 138, 151 in fast vs.

segment, 162, 170, 172, 174, 176

muscle fibers. 131 muscle metabolism. 136-138 oxygen debt, 138 regulation of muscle metabolism, 139

in relaxation. 143, 144 production of by condensation of ADP, 138 from food compounds, 136, 138

from Krebs cycle and ETS, 137

Backswing Golgi tendon organ function spindle function

in,

in,

224

224, 226

muscle), 137 in size of, 32 Bridges (cross-bridges) 133, 148, 150, 153, 154, 155, 156, 157 action of, in contraction, 142-143 heavy meromysin and, 132, 142 in development of tension, 148 in myofibrils, 131-132 in post-tetanic potentiation, 150 in tetanic contraction, 150 relation to economy of energy, 156 relation to increased muscle strength, 157 relation to shortening speed, 155 Buoyancy, 110 centers of, 110

synapse, 175

location

velocity

201

167

of,

Connective

tissues, in muscle, 128, 129

Contractile components, 147, 148, 150, 151 Contractile elements (see also Contractile components), 128, 129

Contraction chemistry

of,

of,

of velocity of contraction on,

154-159 nature of, 134-144 contraction time, 150 excitation-contraction

coupling,

139-

140 excitation of muscle, 134

filament theory

slicing

of,

142-143

relaxation, 143-144

speed of (see also Contraction, velocity of),

154-155,

187

as a scalar quantity, 155

chemical reaction rates as factors

in,

155

isometric contraction, 155

in

in relation to load, 155

intrinsic shortening speed, 155, 159 of,

isotonic,

by the segmental method, 105-106 fractions of body, using tables

in,

of, in

saltatory, 166

150-152

eccentric,

Center of gravity, 103

Adenosine diphosphate)

Afterpotential(s). 167. 187

polarity

166

166-168

151-152, 153, 156, 159, 227 isometric, 151, 152, 153, 155, 156, 204

Cation(s), 162

Adjustment to load; 201. 203, 226 Electromyograms of, 228, 229

tsee

of,

types

rephosphorylation of by CP, 137

ADP

in,

characteristics

effect

in glycolysis. 137

gamma neurons in, 203 primary afferents of spindle

action) 162, 163, 164, 166

action current

134-139 147-159 effect of muscle length on, 151-154 effect of stimulus on, 147-151

Bony prominences, change

high energy bands in. 134 hydrolysis of. by ATPase, 135-136 in contraction, 134-135, 144 in excitation. 144

Conditioned response(s), 204, 219 in sports, 226 Conduction, nerve (see also Potential,

magnitude

Biological oxidation (see also Metabolism, 139,

140. 141 as energy source. 134-137

in

Citric acid cycle, 137

Anion(s), 162

initial

and distance, 56

velocity

Circulation in muscle, 129

hillock, 161, 162, 171

135. 136, 137. 139. 141

in

the

Axon(s)

of receptor cells. 182

Adenosine diphosphate (ADP).

in

throw,

Axolemma, 162

of joint receptors. 209

in

tangential acceleration, 57-58

68-72 determining wrist extension, amount of, 70-71 Anatomical position, 6, 6 Angular momentum, 54

underhand

in overlap. 142. 143

Swearingen method, 105 with gravity board and single 103-104

257

of,

105

scale,

151,

152,

153,

155,

156,

204

velocity of (see also Contraction, speed of), 138, 155-158, 159, 224 as a vector quantity, 155 in eccentric, isometric and

contraction, 156

• i

258

Index Endurance,

continued continued influence of load upon, 156

Contraction

velocity of

influence upon development of force,

156 in relation to

economy

of

movement,

156

maximal, 156 156-157 optimal, Q 10 of, 157 Contraction strength increased by impulses from joint receptors, 224 Contracture, 138, 149 Coordination of movement, controlled by proprioceptive feedback. 237 Creatine phosphate (CP), 131, 137-138, 141 Critical level, 171, 172, 173, 175, 176

Cross-bridges (see Bridges) Cutaneous feedback, in timing and transitions, 225

Cutaneous

proprioceptors,

210-212,

225,

importance to spotter in gymnastics, 225 Cutaneous stimuli from contact with environment, 225 from contact with equipment used, 225 from contact with other body parts, 225 Cytochromes, 137

Energy cost

Fiber(s)

effect of

muscle length

in negative vs.

relation to contractile velocity,

in

effects

neurons, 198 index, 196, 200

components,

147,

148,

149,

150,

155, 159

in isotonic contraction, 156 in production of overt tension, 149, 150 in relaxation, in tension in

143, 144

development. 135

work production, 129

Equilibrating forces, 98-101 determination of force acting on distal end of humerus, 98, 100 determination of rotatory and secondary components biceps, 100 determination of TMF, 98, 100 Equilibrium, 107-111

Electromyograms (EMG), 226, 228, 229 motor unit potentials, 185, 186 of adjustment to load, 201, 228, 229 Electromyography of reciprocal inhibition, 30 Electron transport system (ETS) ATP production in, 137, 141 electron cascade, 138

enzymes processes

EMG

of, in,

137 137

Electromyogram) Endoplasmic reticulum, 161 (see

of neuron, 161

sarcoplasmic reticulum compared with, 133

all-or-none response, 147, 185

arrangement in muscles, 128, 154 chemical composition of, 130-131 extrafusal,

in

on, 111

space orientation, 111

length of in stretched

111

Equilibrium reflexes (see also Labyrinthine proprioceptors), 214, 218

Excitation-contraction

in,

muscle.

Force(s)

88 as effort, 39 resistance,

as

139-140

139-142

movement,

223, 224, 225

39

components, positive and negative, 97 composition of, 88 graphical, by parallelogram, 91 graphical, by polygon, 92 concurrent, 87, 88, 89, 91 graphical composition, 91-92

composition, by parallelograms, 91-92 graphical composition, by polygons, 92 graphical composition of hip abductors, 91-92 mathematical composition, solution, 95-96, 97-98 graphical

209, 223, 224, 225,

227, 231

by disinhibition, 173 presynaptic, 174 synaptic, 171-172 (see Flavin

contracted

acting on the bodv, external and internal.

Excitation time, 149 Exteroceptors, 180, 211 172,

vs.

mean lengths in frog sarcomere, 154 Flavin adenine dinucleotide (FAD), 137

as objective in

169 coupling,

calcium in, 139-140 sarcoplasmic reticulum sodium in, 140 T system in, 140

159

142-143

110-111

coupling,

131-132,

132

unstable, 108

168,

muscle,

molecular subunits of, 132 myosin (see also Myosin), 132, 142, 154 overlap of, in contraction (see also sliding filament theory), 135, 142, 143, 153, 154 theory of sliding, 142-143

Equilibrium in water, 110-111 dynamic. 111 proprioceptive system, dependency

FAD

207

(phasic, white),

actin (see also Actin), 132, 142, 154 arrangement in the myofibril, 131-132 bonding between (see also Bridges),

under dynamic conditions, 109-110 under static conditions, 107-108

Facilitation,

196,

157-159 response to repetitive stimulation, 149-151 response to single pulse, 147-149 slow (tonic, red), 157-159 striation bands of, 130 nerve (see Neurons) fast

Filaments

parallel vs. series,

128 passive stretch of, 153 Elasticity, 128, 129. 159

197, 207

percapsular, 196

excita-

of receptors, 180-181

1

196,

muscle

tory)

-secretion Elastic

156

Excitation of muscle, 130, 134, 144, 151 of nerve, 162, 164-165, 166

gamma

195,

intracapsular, 196

the vertical floater, 110, 111

in synaptic inhibition, 173, 175

Dynamics,

intrafusal,

in,

EPP (see Potential, endplate) EPSP (see Potential, postsynaptic,

static (floating),

Dendrites, 161, 162, 163, 168, 170, 176, 189

Dynamic

111

153 positive work, 151

visual feedback,

Dale-Feldberg law, 171 Degrees of freedom at joints, 3

inhibition,

Feedforward, 174

over a minimal base, 109-110 neutral, 108 potential energy and, 107-108 Stable, 107

225 muscle belly, 225

coordination of arms and legs, 225 174 positive, 164, 204, 212 spindle, 204 visual use in learning water ballet skills, in

over a changing base, 110

effect of pressure over bone,

on action of

libers,

dynamic

Cutaneous reflexes

Drug

slow muscle

dynamic

227 contribution to specific reflexes, 211 in equilibrium, 212 in righting reflexes, 211, 212 morphology of, 210 reflexes of, 211, 212, 224, 231

effect of pressure over

in fast vs.

159 Energy. 81-83 conservation of (law), 81 kinetic, 81 potential, 81 related to equilibrium, 107-108

muscle

adenine dinucleotide)

Fascicles (fasciculi), 128, 184

external, 151

Fatigue

force-velocity relation,

slow muscle fibers, 159 of synapses, 175 Feedback, 193, 206, 207, 209, 219, 223, 225, 231 cutaneous in timing of performance, 225

eccentric contraction, 151 in isometric contraction, 151 in isotonic contraction, 151

in fast vs.

racilitative

from joint receptors, 224 Irom joint receptors, importance

in

155-156

in

internal, 151

mathematical composition, solu92-93 resolution of, 88-89 biceps, mathematical solution, 95 graphical, 89, 90 parallel,

tion,

Index Peace coupler, Force velocit) curs g

vv

96,

in iioi^i

magnitude

o!

relation

Golgi tendon effect on. 224

organ

delayed,

reset,

Gamma Gamma

in,

reciprocal.

224

acid

(GABA),

171

202, 203, 204, 205, 219, 226,

motor (fusimotar)

neurons

(see

138

metabolism

of,

by biological oxidation.

Glycogen. 137, muscle, 137

in

muscle

in

post-tetanic

167.

194.

neurons

of.

shape of planet. 85-86 universal, law of. 85 variation with altitude and latitude, 86 Gravity boards. 104 Gravity distance (table!. 239 GTO (see Golgi tendon organ) of.

54

Half-relaxation time, 150

Heat in

heat:

muscle.

138-139 shortening,

activiation.

relaxation. 138 recover,- heat. 138-139

Meromyosin, heavy

uniaxial, 3 fibers,

164-165 159

vertical, 19, 8, 9, 10

movements, 6-27 abduction-adduction, 12-15

Joint

potential. 162-164

165-

15-16 15-16

at the shoulder, 15,

of fingers

131,

162,

163,

164,

in action potential generation,

and

toes, 15

at acromioclavicular joints, 27 165,

164-165

in excitation-contraction coupling, 140

potential, 162-164

in recovery of resting potential,

165-

166

at sternoclavicular joints

elevation-depression, 24, 25-26 protraction-retraction, 24, 26

circumduction, 15, 17 flexion, approximation of volar or ventral surfaces,

6,

12

flexion-extension,

6,

the ankle, 12, 14

coordination centers, 190 function of the neuron. 188-190 neural circuits, 189, 190-191

at

neuromuscular, function, 219

of elbow,

152

(movements of

the shoulder girdle), 24-27, 25, 26 circumduction, 27

synaptic transmission, 170

Integration

horizontal,

12

plantar

15.

and

dorsiflexion,

18 fingers

wrist,

and

toes,

12

of hip, 12, 13

of limbs, labyrinthine receptors in, 217 Intensity-duration curve, 149 Interneurons, 189 inhibitory', 172 Interoceptors, 180 IPSP (see Potential, postsynaptic, inhibi-

of shoulder, 12, 14 23-24, 24

inversion-eversion,

compromise axes for, 23, 11 movements of the scapula, 27 neutral or midposition of forearm, 20 of the axial skeleton. 20

flexion-extension,

Irradiation of impulses from joint receptors,

224

peripheral control. 194 Zone (see Striation bands)

band (see Striation bands) Immersion technique for determining

of the wrist, 9

167. 171, 172

in

column

20(13)

4, 11,

of the shoulder. 8 of the shoulder girdle, 8

172 synaptic transmission, 170

membrane

//

head). //

at the hip, 15,

sodium (Na T ),

foot 4,

of the axial skeleton (vertebral

164, 165,

in synaptic inhibition,

in

and

of the ankle

176

in relaxation, 143

in

10

19, 27, 8, 9,

of the phalanges, 9

135,

143

slow muscle

membrane

longitudinal,

multiaxial, 3

of the knee,

172

131,

tory)

I

Hypotheses, neurophysiologies! central control. 194

H

potential, 162-164

relaxation,

degrees of freedom, 3

of the elbow, 9

166

affect of

production

membrane

biaxial, 3

of the forearm, 9 of the hip, 10

in recovery of resting potential,

autogenic inhibition by. 209. 224. 231 comparison with spindle. 207. 208 examples of reflexes. 209 in voluntary movement. 209 reflexes, follow through, effect on, 224 Gradients chemical 164. 165 concentration, of ions. 165 Gravitation

Gyration, radius

131.

150

144 172

in action potential generation,

in

distinguishing characteristics. 208 effects on muscles. 208-209

16

Joint axes

and

potentiation,

143,

(CI),

in fast vs.

208

metacarpophalangeal, 3, L2 metatarsophalangeal, 3 pivot, 20 radioulnar, 20. 23, 23 shoulder (glenohumerall. .(,

131

fiber.

167, 172, 182

137

tendon organ(s) (GTO).

(see

176

144,

142,

muscle, 135-136 135-136, 144 in excitation-contraction coupling, 139-140 in excitation of muscle, 140, 144 in fast vs. slow muscle fibers, 158

in

207-209, 224. 231

HMM

135.

contraction.

magnesium (Mg* + ),

in.

12

12

wrist. 3. 12 131.

potassium (K*), 131, 162, 163,

ATP

3,

within the forearm. 20, 23, 23

159

in synaptic inhibition,

substrate synthesis of

knee,

sternoclavicular, 3 tarsometatarsal, 23

166 fibers,

calcium (Ca ++ ),

in

158

139,

oxygen debt. 138

initial

171,

Ions

chloride

Glycolysis. 137, 138, 139. 141 ATP production in. 137. 141

afferent

muscle

in relaxation,

aerobic phase. 137 138 anaerobic phase. 137

Golgi

173,

172. 2(H). 205. 217. 221

gradients.

in last

in

loop, 203, 204

Spindle innervation, efferent) Generator potential, ISO. 181, 185

in

172

in activation of

224

Glucose,

173

disinhibition, 173 recurrent (surround).

reflexes,

Ionic

bias.

172

1S3

Golgi tendon organ functions

Gamma-aminobutvnc

"I

'\

intervertebral,

sensory input, L82 action potential in, 171

postsynaptic,

155 15€

of,

Forward swing

Gamma

tendon organs, 209

in regulation ol

20.

21

lateral flexion (abduction), 20,

rotation,

20,

22

22

movements at and acromioclavic-

of the shoulder girdle (see

sternoclavicular

Joint(s)

acromioclavicular, 3 3, 12 atlantoaxial, 4

ular joints)

pronation-supination, 20-23. 23 radius, the moving bone, 23 radial and ulnar deviation of the hand

ankle.

I

seg-

mental weights, 105-106, 106 Inhibition. 223. 224. 226, 227, 231

atlantooccipital, 3

costovertebral, 3

at the wrist, 23, 24

autogenic (see also Golgi tendon organs). 209

elbow.

conscious, in motor learning, 225, 226, 231

hip, 3, 15

interphalangeal, 3

at hip, 19,

presynaptic, 173-174

intertarsal, 23

at shoulder, 19, 19

3,

12

glenohumeral

range (shoulder),

15

of,

6

medial (outward)

rotation,

(inward)

20

and

lateral

260 Joint

Index 209-210,

proprioceptors,

212,

224,

Labyrinth proprioceptors, 212-216

interconnections, 210 compression as stimulus, 209, 227, 231 contribution to kinesthetic perception,

in

neck

reflexes,

183

216-217

209 sensitivities of, 209 types of, 209, 210 locations

inertia

of,

traction as stimulus, 209, 231

from,

irradiation

receptors,

to

in-

crease contraction strength, 224 Joint types ball

and

socket, 3

Kinematic analysis, 1 Kinematic approach, 1 Kinematic chain, 4-5, 35,

4,

5

closed, 4-5, 4, 5

open, 5 Kinematics, 59

Kinematics mobility terminology (engineers), 59, 60

movement

of

momentum, 79

(Newton's

equipment used. 61

second

Kinetic analysis, 1 Kinetic analysis scales, 113-122 determination of compression forces at the acetabulum, 119-122

TMF

class, 39,

abductors and weight of

TMF

hamstrings and weight of lever, 119-121 resultant, size of forces, 121-122 determination of TMF abductors 117-119 determination of TMF hamstrings, 114— of,

113-114

Kinetic approach, 1 Knee jerk, 200-201 137, 139, 140

inhibition

225-226

anal-

4, 4,

5

Link boundaries and segmental centers of gravity, 233 (see Meromyosin, light)

line (see Striation

bands)

equilibrium reflexes, 214, 218, 231 post -rotatory movements, 214 in posture, 212,

216

labyrinthine righting reflexes, 212, 214, 215, 218, 226, 227 tonic labyrinthine reflexes (TLR), 212, 214. 217. 225. 227

187, 193, 219, 224, 225, 227

skill(s),

conscious

inhibition

labyrinthine reflexes

spindle and

synaptic inhibition

in,

conductance,

Motor

ion, 164, 165, 167

depolarization, 164

176

excitation, 166

216,

and

223 in,

membrane)

226

of,

150

slow, 186-188

"catch"

Meromyosin

(HMM), 132, (LMM), 132

218,

definition, 129, 184 fast

potential (see Potential,

in,

units, 150, 176, 184-188, 199

asynchrony of response

critical level, 164, 165, 170,

light

214,

in,

tendon organ reflexes

Motor system, 184-188

Metabolism

reflexes

223-224

active transport through, 163

heavy

of

225-226 definition, 223 feedback mechanisms and, 223 Golgi tendon organ reflexes and, 223 joint and cutaneous reflexes in, 224-225

reflexes, consciously inhibited, 225

permeability, 164

212-214

Motor

223, 231

effect of spindle reflexes on,

Membrane

semicircular canals, 214-216 acceleratory reflexes, 214

between acceleration of gravity, time, velocity and length of drop, 53

relation

neck reflexes in, 225 preparatory movements effect of Golgi tendon organ reflexes on, 223

Mass, 85

location, 212, 213

212, 213

deceleration, 52

225

Local excitatory state, 164, 165 Local sign, 212, 231

Maximal body length, 153 Mechanical advantage, 40

utricles,

constant acceleration due to gravity,

Motor endplates(s), 158 Motor learning, 193, 219, Motor nerve, 150

in body, 3-4, 4

in integration of limbs, 217

of.

49

acceleration, 52

rotatory, 49

conservation of (law), 79

Labyrinth(s)

re-

43

while standing on toes, 43, 44 Linear forces, definition, 86, 87 Linear momentum, 54, 55 Linear movement, rotational and, ogies between, 56, 55 Link(s), 3-4

M of,

tonic, 225

morphology

linear,

translatory, 49

in quiet standing,

137

Labyrinthine reflexes, 225 skills,

angular, 49, 51 curvilinear, 49

velocity, 52

LMM

Krebs cycle (see also Metabolism, muscle) of,

Motion

rectilinear, 49

Leverage

joint axes and,

coenzymes

40

third class, 39, 40

117 gravity line, location

resistance arm), 39

body, 43-45 when muscular tension becomes a sistance, 45

122

lever, 121

motor

arm and

in the

direction of resultant,

of, 139 arms, 41, 42 in body, 42 muscle, 42, 44, 45 short, advantages and disadvantages, 45-48 Moment of force, 39, 41, 54 Moment of inertia, 54 Momentum, 78-81 absorption of, 78-79 application of force and changes in, 78 change per unit time effect of, 78-79 conservation of, 79-81 conservation of (law), 79 receiving a force, effect of time consumed on, 78-79

regulation

52-53

parts (effort

of football punt, 60

cycle, 137, 140

Moment

77

first),

47

analysis, 59-63

data required, 59

in

Krebs

oxidative, 130, 159

lengthened by implement, 45, 47 lengthened by implement, suitable size and weight of, 48 shortening to decrease length of RMA, 48 shortening to increase speed of movement, 48 brachioradialis as effort in a seond or third class lever, 43-44, 45 first class, 39, 40 long, advantages and disadvantages, 45,

plane, 3

glycolysis, 137, 138, 139, 141

of mass, 79

body-

pivot, 3

from

momentum, 54

of angular

electron transport system (ETS), 137, 141

of energy, 81

77, 78 motion, Newton's, 77-78 universal gravitation, 85 Length-tension phenomenon (see Tension, tension-length relationship) Lengthening reaction, 202 Leverts), 39

hinge, 3

from

third), 77

mass and acceleration (Newton's second),

reflexes of, 209, 210, 226, 231

Joint

anaerobic, 137

and reaction (Newton's

conservation conservation conservation conservation

supporting reaction, 209

in signaling joint-position, 211

182,

aerobic, 137

225

Law(s) of action

167,

217, 218, 225

of,

in sport skills,

210

receptors,

muscle

examples

afferent

in positive

glucose (see also Glycolysis), 136-137

reflexes

225, 227

mechanism

in

homogeneity of composition 135, 142, 143, 144

motor

slow

units, 187

187

recruitment order types of, 187-188

of,

188

of,

186,

/t)Ui'\

not all-or none. 1S4 185, L86 response characteristics of, 185 186

secondary (compression stabilising and decompression dislocation), B9 where applied while or shortening lengthening, 36 36

129. 184

i

Movement

59 69

analysis

data required, 58 extremity m football punt. •

tn>

61

by

affected

from

Movement, voluntary,

191, 193, 194

884 Golgi tendon organ spindle innervation velocity of,

33,

187, 188,

biceps brachii. 127. 128 crureus. 1ST

of

attachment

moment arm,

158-159, 187.

peri merit >.

ex-

-

fibers.

157-159

fibers,

classification into three types.

159 correlation of his-

differences:

and chemical with physio-

tological

logical characteristics.

Muscle moment arms factors which affect length hip and knee, 36, 38

38

sibility.

pectoralis. 128

32

(contractility,

elasticity,

disten-

of

on

levers

184

joint

effect of lengths of

moment arms on

function, 38 of. 36,

38

components, 128

Muscle

29. 89-91,

90

stretch

(external),

223 lengthening eccentric i. 35 shortening (isotonic), 35 (

Muscle force as a resistance. 36

as effort, 35-36

components rotatory. 89

effect

of.

and

227

227, 231

postural reflexes, 226, 227 responses, adjustment to

load.

229

ory), 132, 142, 153, 154, 155, 156

and

standing long jump, 227 Neuromuscular junction, 130,

134, 184, 187

of, 176, 198 motor, 143, 150, 151, 168

alpha, 129, 186, 196, 199, 209 classification

of,

by

Erlanger

and

Gasser, 129, 167, 168 cross-union and hetero-innervation

experiments, 187

gamma

(see

Spindle

innervation,

efferent)

sarcoplasmic reticulum and T system, 133-134 Muscle tension (see Tension, developed) Muscle weakness, spindle reflexes used in correction of, 224 Myelin sheath, 161, 162 Myofibrils (see Muscle structure, ultramicroscopic) Myoglobin, 129 Myosin, 131 filaments (see also Sliding filament thein contraction

scale,

in,

in,

fast vs. slow, 186, 187

myofibril striation bands, 131, 142

Muscle contraction

practice exercises

chemical transmission at, 171 Neuron(s) collateral branches of, 162, 168, 171. 174. 191, 200 recurrent, 168, 171, 173 conduction velocity, 167

Myosin) 131-133

red and white. 29 spurt and shunt. 29 tonic and phasic. 29

reflexes inhibited

driving

myofibril filaments (see also Actin

classification

Neurons,

as a specialized synapse, 176

chemical composition of fiber, 130-131 histology of fiber, 129-130 neuromuscular junction, 130 ultramicroscopic, 131-134 chemical structure of myofibrils. 131-132 myofibrils. 131-134. 139

capacity,

and

epi-

Muscle amplitude. 36 contractors and expanders, 29

muscle (see also

spindle reflexes, 226 (endo-,

microscopic, 129-131

stapedius. 127. 128

also Spindle innervation,

efferent). 196-199, 207

skeletal

226, 228,

myotendinous junction, 129 tendon function, 129

soleus. 159. 187. 188

(.see

efferent (see also Spindle innervation,

simple

128-129 arrangement, 128 fiber bundles (fasciculi), 128 fiber size, 128

sartorius. 128

Muscle components.

127

irritability),

fiber,

phasic and tonic. 29 platysma, 184

mechanics

of,

fiber! s),

penniform. 29 peroneus. .-"

tibialis anterior. 159.

(TNR), 216. 217-218. 225, 231 with head dorsiflexed, 225 with head rotation. 225 with head ventriflexed, 225

tonic

motor, alpha), 129 Neuroglia, 161 Neurokinesiological analysis dive for height

127-129

elastic

216-217

reflexes active in, 227

properties

connective-tissues perimysia), 128 of, 38,

and limb move-

of trunks

righting. 214, 215, 231

to

Muscle shortening, effect moved. 45, 35, 46 Muscle structure

interosseus. 205 multijoint. mechanics

in

207. 218

of,

ments by, 225 motor skills, 225

afferent

38

gross,

reflexes, 216 218. 225

joint receptors in,

Muscle functional excursion. 36 Muscle insufficiency, active and passive,

flexor digitorum longus. 159. 187 gastrocnemius. 127. 128. 155, 184, gluteus maximus. 127. 128

infraspinatus. 12S

dinu

afferent), 196, 197, 207

Muscle

gracilis. 1ST

adenine

Nicotinamide adenine dinu

(see

examples

counteracting or neutralizing, 31

theories regarding interconversion, 187 first lumbrical. 1S4

188

Nicotinamide

(see

Nerve supply to neuromuscular spindle

long, 33

158-159

129

ion,

cleotide phosphate)

stabilizing. 31

and hetero-innervation

crass union

M

facilitation

35

conjoint, 31

(phasic)-slow (tonic). 195

fibers,

its

LMM)

illMM,

ol i

cleotide)

Neck

joint,

synergists. 31

deltoid. 127, 128

two

bony promi-

passage of tendon or muscle over bony prominences, 32 33, 33, 34 antagonists, 30 fixators. 32 movers, 30 prime movers. 30 stabilization vs. fixation, 32

antigravity-postural. 29

fast

to

13,

endinoua junct

.'i

affected bj

209 in. 803 204 156-156, 209 in,

Muscles) abdominal.

\l\

NADP distance

affected bv length of

ballistic. 283,

i

NAD

Muscle function, 29 32 affected bj attachment nences, 32

sca'u

usefulness, 64

ubunita

stretch, 142-143

in the myofibril, 132

in muscle stimulation, 150, 203, 204 phasic vs. tonic, 186, 187 pool(s), 184, 219 role in determining properties of motor units, 186-187 sensory, 129, 161 classification of, by Lloyd, 167, 168

Neuropil, 168

Nicotinamide adenine dinucleotide (NAD), 137

Nicotinamide adenine dinucleotide phosphate (NADP), 137 Nociceptors, 180

Norepinephrine (Nor-E), 171

262

Index labyrinthine (see Labyrinthine proprioceptors) muscle (see Spindle, neuromuscular and Golgi tendon organs)

Occlusion, 175 Oligodendrocytes, 161 Oscillation

damping in

201, 202, 212

of,

muscle, 150

Overlap (of muscle filaments), 135,

153, 154

in contraction,

PSP

142-143

Q l0 Parallel forces, definition, 86.

,

definition, 157

Quick Quick

righting (see Righting reflexes) (mvotatic), 201, 224, 226, 231

release, 148, 159

stretch

stretch, 148

Quiet standing, electromyogram

87

Perception sensory, 182, 183-184

"Phantom limb" phenomenon, 184 Positive supporting reaction, 205-207, 209

Planes of body, 6

Receptors) afferent neurons

63

as,

istry of) 143

cells, 179,

classification of, 179-180

180

Renshaw

cutaneous (see also Cutaneous proprioceptors), 210-212

7

PMA (see Projection moment arm)

Refractory period(s) of muscle, 150 of nerve, 166, 172 Relaxation, 127, 143-144, 147, 150, 231 chemistry of, 143-144 physical changes in, 143 Relaxing factor (see also Relaxation, chem-

179

midsagittal,

7

227,

half-relaxation time, 150

frontal (coronal), 6, 7

6,

of,

Radial acceleration, 57 Radian, 56 Radius of gyration, 54 Ranvier, nodes of, 161

kinesthetic, 204, 210, 225

transverse,

reaction, 212

209, 227, 231

24

6,

magnet

neck (see Neck reflexes)

postural (attitudinal), 212, 216, 226

Pacinian corpuscles, 183, 209, 210 Pain, 166, 167, 183 5,

227

labyrinthine, 210, 214, 218, 231

(see Potential, postsynaptic)

debt, 138

Pelvic girdle,

226-227

visual, joint,

placing reactions, 211, 212, 231 positive supporting reaction, 205-207,

132, 142, 143, 154

double, 142, 143, 153 single, 135, 143 153 Overt tension (see Tension, developed)

Oxygen

neck (see Neck reflexes) Pseudo-H zone (see also Striation bonds),

tonic labyrinthine, 227

cells, 171,

174

inhibition by, 173, 186

Response(s)

Post-tetanic potentiation (PTP) muscle, 150 synaptic, 175

definition. 179

all-or-none

Goldi tendon organs (see Golgi tendon

Potential

joint (see also Joint proprioceptors). 167,

motor unit, 184-185, 186 muscle, 147, 185 nerve, 168 conditioned in skills, 219, 223, 226, 231 of gamma neurons, 204 jerk, 200-201, 205 phasic (of spindle), 182, 195, 196, 197, 198, 199 examples of, 200-201

organs)

action, 151, 161, 162, 165, 170

conduction

in

of,

neurons,

166,

174 generation of. in neurons, 164, 165 in muscle, 134, 144 in T tubules, 140, 144 local, graded, in slow muscle fibers, 158 sodium theory of, 164 endplate (EPPK 176 equilibrium, 163 generator, 180, 181, 185

membrane,

labyrinthine (see Labyrinth proprioceptors)

neck (see neck reflexes) neuromuscular spindles neuromuscular) physiology of, 180-182

postural, 201 stretch, 153, 224

sensitivity

Primary movement objective of accuracy, 224 objective of strong force, 224

moment

88

ceptors)

negative aspects, 225 Proprioceptive system, dependency on, during water ballet skills, 111 Proprioceptors, 180, 194, 211 classification of, 194 conditioned responses in motor skills, 231

Cutaneous

propriocep-

tors)

of,

neck, 214, 215, 226, 227, 231 visual, 214, 215, 226, 227, 231

Receptor potential, 180. 181

Rigor. 138

Recruitment of motor units, 184

RNA

self regulating control of, 231

proprioceptors)

(see Ribonucleic acid)

Rope climbing, 36, 37 Rotational movement, 54-56

188

of sensory neurons, 181

Reflexes and synaptic inhibition of, 226 conscious inhibition of, in motor

skills.

225-226 equilibrium, 210, 214, 218, 231 extensor thrust, 207, 211, 227, 231

angular acceleration, 54 angular motion, 54 angular velocity, 54 linear and. analogies between, 56, 55 moment of force in, 54 moment of inertia, 54 torque in, 54

flexion (withdrawal), 211

Sarcolemma,

grasp, 212, 231

Sarcomere, 130,

in

motor

201

Ribonucleic acid (RNA), 134 Righting reflexes, 210, 211, 214, 215 body-on-body, 212, 214, 215, 231 body-on-head, 212, 214, 215, 231

Reciprocal inhibition, 30

of,

of,

vibration, 201

labyrinthine, 212, 214, 215, 216, 226, 227

179-180

extensor thrust, 227 226-227

129, 130, 134, 139, 144, 161 139,

140,

in

in stretched muscle, 142

labyrinthine, 226-227

of muscle fiber, 130 of myofibril, 142. 143

list of,

231

response, 227 stretch, 227

143,

153,

contracted muscle, 142-143

joint,

labyrinthine head righting, 227

142,

154, 155

skills

neck righting, 226, 227 neck righting, body following head

definition, 194

feedback, joim

179

68

Proprioceptive reflexes (see also Proprio-

(see

of,

types

of muscle fibers, 157

forces, finding

moment arm (PMA).

structure

order

in force analysis

equilibrium force (necessary arm), finding of, 88

cutaneous

167,

Cutaneous propriocep183, 210-212

anatomical basis, 182-183

receptor, 180, 181

of,

example also.

of sensory perception, 183-184

excitatory (EPSP), 171, 172, 174 inhibitory (IPSP), 172

Projection

(see tors),

165-166

two or more

tonic (of spindle), 182, 195, 197, 198, 199

181

specificity of, 182-183

postsynaptic (PSP), 169, 170

resultant of

of,

range, 183

resting, 162, 164, 169

Problems met

Spindles.

potential. 180, 181

165, 169, 170

of,

(see

sense organs, 178, 189

skin

ionic distribution in, 162-164

recovery

209-210

182, 183, 161,

Sacroplasmic

reticulum,

132-134,

140,

143, 144

differences from endoplasmic reticulum,

133 function

of,

140

263

Index contraction coupling,

in excitation

149 in fast \s slew muscle fibers. L58 Scalar quantity. 51

139

.uul

phasic

examples

159

dynamic

movement anahsis

of,

cells.

161

Segment, body,

4

body weight accord to somatotyp, of

IS

ins;

ntal mass, r egression equations determining

for

cntal velocities factors

determine

which

sequence

of

angular displacements of segments

of,

199, 207, 223,

198,

Si nai ion

231

in

index, L96, 200

functions

of,

joint actions

involved

of force exerted by. 105

tdavers, 235

Sensation (see also Perception). 182. 183184

Sensory input. 223 control by presynaptic inhibition. 174 regulation of. 181-182

summation

of,

181

threshold. 181

Sensory system. 179-184 Sensory unit. 180 Shortening reaction. 209 Skill(s).

Skin,

motor (see Motor

proprioceptors proprioceptors)

of

skills)

(see

Cutaneous

Sliding filament theory. 142-143 relation to tension-length curve,

Sodium pump.

154

Soma, neuron. 161 Specific gravity of

body segments. 237

Spindle(s)

207 innervation afferent (see Spindle innervation, afferent)

129,

in,

efferent (see Spindle innervation, efi

intrafusal fibers. 195, 196. 197. 207

bag. 195

chain. 195. 196 significance of tvpe of contraction in,

198-199 of.

205-207

neuromuscular. 129. 167. 182, 186, 194207. 209 structure. 195-196 duality of. 195. 196. 197, 209 tandem. 196

of inadequate stimuli, 165 spatial, 172, 175

synaptic, 175

temporal, 172, 175 unnecessary at neuromuscular junction, 176 Synapse(s) facilitation at, 171-172, 173 function of, 169-176 inhibition at, 172-174 properties of, 174-176 after-discharge, 175

nonlinearity of response, 175 polarity, 175

Spindle reflexes adjustment to load phasic response of primary afferents. 226 simple responses in, 226

summation, 175 synaptic delay, 174 threshold, 175 receptive site, 168, 170 structure of, 168, 169 subsynaptic membrane of, 168 transmission at, 169-171

motor skills back swing, 223, 226

forward swing, 223, 226 preparatory movements, 223 Spindle secondary afferents, decreased sensitivity of. 224 Spindle sensitivity, enhancement of, 223

types

Spindle innervation, afferent, 196, 199-201, 201-202. 207 of intrafusal fibers. 197. 207

173

electrical, 169

excitation-secretion coupling

in, 169 transmitter substance(s), 162. 169, 170171, 181 action upon subsynaptic membrane. 170 chemical, types of, 170-171

stabilized joints, 63

64

1

Steady state, in slow muscle fibers, 159 Stimulation (stimulus), 147-151 adequate, adequacy of, 148, 150 frequency of (see also Stimulation, repetitive), 150-151 effect upon muscle response, 150 in post-tetanic potentiation, 150 normal, in living animal, 150 repetitive, 149-151 muscle response to, 149-151 single pulse, 147-149 all-or-none response to, 147 characteristics of, 148-149 twitch response to, 147, 148-149 Stretch. 127, 152, 223

upon muscle

fiber striation bands.

142 effect

of, 168, 169,

Synaptic transmission, 162, 169-171 action of drugs on, 173, 175

data required, 63 scale, 63

effect

libers, 158

1

Subneural apparatus, 130 Substrate synthesis (see also Glycolysis). 137

204

vs. static, 198, 199,

of,

slow muscle

of input, 181

and pharmacological distinctions between. 197-198 neural circuits of, 199, 200, 203 innervation of intrafusal fibers, 197, 207 role of. in voluntary movement, 203-204

Statics.

fasl VS.

12 Stretched muscle, of muscle fiber. 130 of myofibril, 131, 136 Subcortical mechanisms, 194, 223

196,

electrical use of, 147

beta. 199. 205

instances of activity location of. 194-195

neurons,

204 effects on phasic acid tonic responses of spindles. 198, 199, 200 fast vs. slow, 197-198 types of, anatomical, physiological,

usefulness

comparison with Golgi tendon organs.

ferent

m

Static analysis

163, 165

bands

contraction, 142 L43 moderate, 112 L43

in excitation contraction coupling, 140

definition. 186

in

fibers to,

Summation, 184

202-204

conditioned response

dynamic

muscle

super-contracted, 143

197, 207. 223

methodology optimal sequence. 72, 74 value of determining. 72-74 Segmental weighus) immersion technique for determining. 105-106. 106 mathematical determination of. 106

moment

of,

(fusimotor)

slow

in

201 202

significance of, questions regarding. 204 196-199, Spindle innervation, efferent, 202 203. 205 by beta axons, 199, 205 effect upon primary afferent response to

gamma

72

t.i^t

strong, 143

stretch. 202

of, t>4

of

159

200 201

projection to cerebral cortex. 204 secondary, 196, 198, 201 202. 204, 205, 207. 224. 231

functions

in projectile skills

analysis

response, 153, 224

223,

Functions of, 199 201 neural circuits of. 2(H). 203

ol 63, 63

static analysis of, 63

Schwann

L99,

sensitivity

primary, 196,

Scale

responses,

tonic

23]

;

upon muscle response, 152-154

Dale-Feldberg law, 171 in autonomic nervous system, 171 quantal release of, 169, 172, 176 Synergy, 30-31

T

T

tubules,

system (see also Triads),

132-134 in excitation-contraction coupling, 140 in excitation of muscle, 144 Tangential acceleration, 57-58 Telodendria, neural, 161, 162, 163, 168 Tendon, 128, 129, 150 Tension developed (overt), 147. 148, 150, 155, 156

active state contractile effect of elastic

in, 147,

148

components

in. 147,

components

in.

147. 148, 149

in eccentric vs. isometric

and

internal, 197, 202

contraction, 152 influence of velocity upon, 156

passive, 129, 153

rate of

external, 197, 223, 227

148

muscle length on, 153

development

of, 156,

isotonic

159

264

Index

continued of, to muscle length. 152-154 elastic (see also Elastic components),

Tension

relation

147, 148, 153

Training, 157, 187

length relationship, 152-154. 223

curve

of, 153,

(transducers), 179, 181, Transduction 209 Transmitter substance (see Synaptic trans-

154, 159

153 optimal length, 153, 224 sliding filament theory in, 154 initial length, 152,

mission, transmitter substance) Transverse tubules (see T tubules) Tremor. 150 Triads (see also Sarcoplasmic reticulum). 133, 139-140

movement, 30 muscle, 128, 129, 142, 151 definition, 151

gradations of, 185 in shortening (isotonic) 155, 156 maximal. 153-154

contraction,

(tetanic

contraction),

138,

TLR

{see usee

Ton

.

Tori'-

ratio,

149,

Tonic neck

reflexes

Warm

(see also

effect

Work,

129, 134, 136. 138. 139, 151

components in. 129 equation for calculation of. 151 elastic

131,

in isotonic contraction. 151

negative, 151 output, relation to velocity, 156

Tropomyosin), 131, 132,

positive, 151

power and, 83

all-or-none, 147, 185

muscle. 138, 147. 150. 184. 185, 208

Z

seg-

on velocity of muscle

Twitch 212.

relaxa-

limb

Weight, 85

composition

144

(TLR),

up.

of

shortening. 157

(Appendix C), 241-255 Tropomyosin (see also Relaxation),

Troponin

muscle. 204, 212, 214, 223, 226 labyrinthine

Valsalva maneuver. 31 Vectors), 51 Vector quantity. 51, 155 Vertical floater. 110-111 Volumetric displacements ments. 105

-troponin system. 144 reflexes)

and

contract

response to single pulse. 147-148

tables of

reflexes)

214, 217, 218, 225

mathematical

(latent,

tion) of. 147

132, 159

150

Tonic labyrinthine

for

periods

of forces, 94

150, 185

Tetanus-twitch

Tricarboxylic acid cycle, 137 functions Trigonometric

needed

produced against resistance, 155 fttanus

Tonic neck reflexes (TNR), 212, 214, 216. 217-218, 225 examples of. 217, 218 Torque, 41, 54

line (disc) (see Striation

bands)