Vector mechanics for engineers statics - beer and johnston 8th edition
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Vector mechanics for engineers statics - beer and johnston 8th edition
Slide Content
erdinand P. Beer = E. Russell Johnston, Jr.
VECTOR MECHANICS
for ENGINEERS
Statics
Eighth Edition
Sn
VECTOR MECHANICS
for ENGINEERS
Statics
Eighth Edition
Sn
Eighth Edition
VECTOR MECHANICS
FOR ENGINEERS
Statics
FERDINAND P. BEER
Late of Leigh Unes
E. RUSSELL JOHNSTON, JR.
‘Unversity of Conneccut
ELLIOT A. EISENBERG
‘The Ponnstana Stato Unversity
ith te colaboran ot
David F Mazur
US. Coast Guard Academy
Mic E
Er Higher Education
Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco. Si. Louis
Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico Cty
Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto
VECTOR MECHANICS
FOR ENGINEERS
Statics
FERDINAND P. BEER
Late of Leigh Unes
E. RUSSELL JOHNSTON, JR.
‘Unversity of Conneccut
ELLIOT A. EISENBERG
‘The Ponnstana Stato Unversity
ith te colaboran ot
David F Mazur
US. Coast Guard Academy
Mic E
Er Higher Education
Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco. Si. Louis
Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico Cty
Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto
Daa a
[E Higher Education
VECTOR wt
Falla by MG. a nes nt of The Mer Compas, In, 122 Avene of he Ames New or NY 10020, Cora
een Compass a. Al hs reser. Ne pa of publi ma be ges or ste in fr or
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brary of Congres Cataloging Publication Data
Bex Fea Pre, 1915-200.
"or means o engoser as Ferdinand Beer: F Ross Jon Je
aR seer, David Marek — So
Pd nde,
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[E Higher Education
VECTOR wt
Falla by MG. a nes nt of The Mer Compas, In, 122 Avene of he Ames New or NY 10020, Cora
een Compass a. Al hs reser. Ne pa of publi ma be ges or ste in fr or
a et as or etal sem. ht de prie wre conse a The Mr il Compas Inc, cig, ha
nea ctor oc ltr rag rar o beac for dance karl,
(AMICS FOR ENGINEERS: STATICS, EIGHTH EDITION
Same ale, mg croi nd print component, may not be walle to customer nse the United Salen
hs bok erated on a ce pes
254567890 DoWwDOWw 0987
saya 7.0.07 T0576
OS
Per Sesame Jens
Senor Soma or Michel 5 Hacker
Decenal aor Kalen L We
Ee eng en Me a
‘Sour Projet Manor Kay J Bringer
Se Proton Super Sherry Kane
cia Meda Pod Chi Nela
Sor Deser: acid Wash
TSE Oe image Fate O Wes ThonpntCORBIS Back op), © chend CmmiavCORBIS Back nie
Ro Wa CORPIS Back (atom Mer my Moser
ead Foto Reseach Colt Carre K Barger
io Resear Sabine Doel!
Sipplemont Peer, Try L Kowa
paie TehBouT ork EX
10312 New Caledonia
Her RA may Wild, ON
‘ner Cap Enger ae ten fc withthe config ech of ty to be epee basic innate of rept
nel per caos and atmen we maining wood rem sac, Tis testa Ten the ak ad natura
erin art be eb tho oughta sting amd ate he setts, lo tote fsa pects and cs,
M Le Par inc an between the Noth Dalle Toy al the President George Bush Tupi so. à ae of harmony
SRS ithe sung bas Ea chee eg he carl ing and shape ofthe xa ad spring pie, The roba,
ena bi eae ol ing hh prt mn cae de. pd er
‘mera pert ad never teas dut ext rom pes,
‘The cis secon for is ok gs om page 605 and considered an nenn the cpr pag
brary of Congres Cataloging Publication Data
Bex Fea Pre, 1915-200.
"or means o engoser as Ferdinand Beer: F Ross Jon Je
aR seer, David Marek — So
Pd nde,
IBN SOS — ISBN 0-07-20787-X hardcopy al per),
VA rim 3 Means, Apple L Es He KM Joos, Rae (Eco Ravel) 1925-1 Te
Tas ar
Eee mons 72
ae
About the Authors
As publishers ofthe books by Ferd Beer and Russ Johnston, we are
often asked how they happened to write their books together with one
of them at Lehigh and the other atthe University of Connecticut,
The answer to this question i simple. Russ Johnston first tech“
Department of Civil Engineering and Me-
iy. There he met Ferd Beer, who had joined
that department two years earlier and was in charge of the courses in
mechanics.
Ferd was delighted to discover that the young man who had been
hired chiefly to teach graduate structural engineering courses was not
only willing but eager to help him reorganize the mech
Both believed that these courses should be taught fr
principles and that the various concepts involved would be best un
derstood and remembered by the students if they were presented to
them in a graphic way: Together they wrote lecture notes in statics
and dynamics, to which they later added problems they felt would ap-
peal to future engineers, and soon they produced the manuscript of
the frst edition of Mechanics for Engineers that was published in
June 1956.
The second edition of Mechanics for Engineers
tion of Vector Mechanics for Engineers found Ri
Worcester Polytechnic Institute
sity of Connecticut. In the meantime, both Fe
administrative responsibilities in their depart
volved in research, consulting, and supervising graduate sudents—
Ferd in the area of stochastic processes and random vibrations and
Russ in the area of elastic stability and structural analysis and design
However, their i proving the teaching of the basic me
ans comes ha not subsided. and they both tanght sections of
these courses as they kept revising their texts and began writing the
manuscript of the first edition of their Mechanics of Materials text.
Th collaboration qu hal ace
successful revi
Electric
Fund Award for excellence in the instruction of engineering students
by their respective regional sections of the American Society for
ngineering Education, and they both received the Distinguished
Educator Award from the Mechanics Division of the same society.
Starting in 2001, the New Mechanics Educator Award of the Me-
chanies Division has been named in honor of the Beer and Johnston
thor tea
As publishers ofthe books by Ferd Beer and Russ Johnston, we are
often asked how they happened to write their books together with one
of them at Lehigh and the other atthe University of Connecticut,
The answer to this question i simple. Russ Johnston first tech“
Department of Civil Engineering and Me-
iy. There he met Ferd Beer, who had joined
that department two years earlier and was in charge of the courses in
mechanics.
Ferd was delighted to discover that the young man who had been
hired chiefly to teach graduate structural engineering courses was not
only willing but eager to help him reorganize the mech
Both believed that these courses should be taught fr
principles and that the various concepts involved would be best un
derstood and remembered by the students if they were presented to
them in a graphic way: Together they wrote lecture notes in statics
and dynamics, to which they later added problems they felt would ap-
peal to future engineers, and soon they produced the manuscript of
the frst edition of Mechanics for Engineers that was published in
June 1956.
The second edition of Mechanics for Engineers
tion of Vector Mechanics for Engineers found Ri
Worcester Polytechnic Institute
sity of Connecticut. In the meantime, both Fe
administrative responsibilities in their depart
volved in research, consulting, and supervising graduate sudents—
Ferd in the area of stochastic processes and random vibrations and
Russ in the area of elastic stability and structural analysis and design
However, their i proving the teaching of the basic me
ans comes ha not subsided. and they both tanght sections of
these courses as they kept revising their texts and began writing the
manuscript of the first edition of their Mechanics of Materials text.
Th collaboration qu hal ace
successful revi
Electric
Fund Award for excellence in the instruction of engineering students
by their respective regional sections of the American Society for
ngineering Education, and they both received the Distinguished
Educator Award from the Mechanics Division of the same society.
Starting in 2001, the New Mechanics Educator Award of the Me-
chanies Division has been named in honor of the Beer and Johnston
thor tea
vi Ana th amer
Ferdinand P. Beer. Born in France and educated in France an
Switzerland, Ferd received an M.S. degree from the Sorbonne an
an ScD. degree in theoretical mechanies from the University of
Geneva. He eame to the United States ater serving in the French
army during the early for four years
at Williams College ms-MIT joint arts and engineering
program. Following his service at Williams College, Ferd joined the
Faculty of Lehigh University where he taught for thirty-seven years,
He held several positions, including University Distinguished Profes
sor and chairman of the Department of Mechanical
“Mechanics, and in 1995 Ferd was awarded an honorary Doctor of En-
gineering degree by Lehigh U
E, Russell Johnston, Jr. Born in Philadelphia, Russ holds a BS. de-
greein evil engineering from the University of Delaware and an Se.D.
{degree inthe field of structural engineering from the Massachusetts
Institute of Technology. He taught at Lehigh University and Worces-
ter Polytechnic Institute before joining the faculty ofthe University of
Connecticut where he held the position of chairman of the Depart-
ment of Cial Engineering and tanght for twentysi years, In 1991
Russ received the Outstanding Civil Engineer Award from the Con-
necticut Section ofthe American Society of Civil Engineers
Elliot R. Eisenberg. Elliot holds a B.S. degree in engineering and
an ME. degree, both from Comell University. He has focused his
‘scholarly activities on professional service and teaching, and he was
recognized for this work in 1992 when the American Society of Me-
‘Chanical Engineers awarded him the Ben C. Sparks Medal for his con-
tributions to mechanical engineering and mechanical engineering
technology education and for service to the American Society for
ducation. Elliot taught for thirty-two years, including.
Penn State where he was recognized with awards
advising
twenty.
for both teaching an
David F. Mazurek, David holds a B.S, degree in ocean engineer-
ing and an MS. degree in civil engineering from the Florida In
logy and a Ph.D. degree in civil engineering from the
University of Connecticut. He was employed by the Electric Boat
Division of General Dynamics Corporation and taught at Lafayette
to joining the U.S. Coast Guard Academy, where he
ice 1990, He has served on the Ameri
College pı
has been s
past fourteen years. His professiona
‘sts include bridge engineering, tll towers, structural fore
blastresistant design
Ferdinand P. Beer. Born in France and educated in France an
Switzerland, Ferd received an M.S. degree from the Sorbonne an
an ScD. degree in theoretical mechanies from the University of
Geneva. He eame to the United States ater serving in the French
army during the early for four years
at Williams College ms-MIT joint arts and engineering
program. Following his service at Williams College, Ferd joined the
Faculty of Lehigh University where he taught for thirty-seven years,
He held several positions, including University Distinguished Profes
sor and chairman of the Department of Mechanical
“Mechanics, and in 1995 Ferd was awarded an honorary Doctor of En-
gineering degree by Lehigh U
E, Russell Johnston, Jr. Born in Philadelphia, Russ holds a BS. de-
greein evil engineering from the University of Delaware and an Se.D.
{degree inthe field of structural engineering from the Massachusetts
Institute of Technology. He taught at Lehigh University and Worces-
ter Polytechnic Institute before joining the faculty ofthe University of
Connecticut where he held the position of chairman of the Depart-
ment of Cial Engineering and tanght for twentysi years, In 1991
Russ received the Outstanding Civil Engineer Award from the Con-
necticut Section ofthe American Society of Civil Engineers
Elliot R. Eisenberg. Elliot holds a B.S. degree in engineering and
an ME. degree, both from Comell University. He has focused his
‘scholarly activities on professional service and teaching, and he was
recognized for this work in 1992 when the American Society of Me-
‘Chanical Engineers awarded him the Ben C. Sparks Medal for his con-
tributions to mechanical engineering and mechanical engineering
technology education and for service to the American Society for
ducation. Elliot taught for thirty-two years, including.
Penn State where he was recognized with awards
advising
twenty.
for both teaching an
David F. Mazurek, David holds a B.S, degree in ocean engineer-
ing and an MS. degree in civil engineering from the Florida In
logy and a Ph.D. degree in civil engineering from the
University of Connecticut. He was employed by the Electric Boat
Division of General Dynamics Corporation and taught at Lafayette
to joining the U.S. Coast Guard Academy, where he
ice 1990, He has served on the Ameri
College pı
has been s
past fourteen years. His professiona
‘sts include bridge engineering, tll towers, structural fore
blastresistant design
Contents
Preface xii
List of Symbols xvi
1
INTRODUCTION
1
11 What le Mechanics? 2
12 Fundamental Concepts and Principles 2
13 Systems of Units 5
4.4 Conversion from One System of Unis to Another 10
4115 Method of Problem Solution 11
118 Numerical Accuracy 13,
E
STATICS OF PARTICLES
15
21 Introduction 16
Forces in a Plane 16
22 Force on a Parce. Resultant ol Tao Forces 16.
23 Vectors 17
24 Addiion of Vectors 18
25 Resultant of Several Concurrent Forces 20
28 Resolution of a Force into Components 21
2.7 Rectangular Components of a Force. Uni Vectors 27
28 Adlon of Forcos by Summing x and y Components 30
29 Equilbrim ofa Partie 35,
210. Newton Fist Law of Motion 36
2.11 Problems Involving the Equilorum of a Parle.
Free-Body Diagrams 36
Forces in Space 45
2.12 Rectangular Components of a Force in Space — 45
2:13 Force Defined by ls Magnitude and Two Points on lis
Line of Acton > 48
vii
Preface xii
List of Symbols xvi
1
INTRODUCTION
1
11 What le Mechanics? 2
12 Fundamental Concepts and Principles 2
13 Systems of Units 5
4.4 Conversion from One System of Unis to Another 10
4115 Method of Problem Solution 11
118 Numerical Accuracy 13,
E
STATICS OF PARTICLES
15
21 Introduction 16
Forces in a Plane 16
22 Force on a Parce. Resultant ol Tao Forces 16.
23 Vectors 17
24 Addiion of Vectors 18
25 Resultant of Several Concurrent Forces 20
28 Resolution of a Force into Components 21
2.7 Rectangular Components of a Force. Uni Vectors 27
28 Adlon of Forcos by Summing x and y Components 30
29 Equilbrim ofa Partie 35,
210. Newton Fist Law of Motion 36
2.11 Problems Involving the Equilorum of a Parle.
Free-Body Diagrams 36
Forces in Space 45
2.12 Rectangular Components of a Force in Space — 45
2:13 Force Defined by ls Magnitude and Two Points on lis
Line of Acton > 48
vii
vi
2.14 Addiion of Concurent Forces in Space 40
215 Equileium of a Parc in Space — 57
Review and Summary for Chapter2 64
Review Problems 87
Computer Problems — 69
3
RIGID BODIES: EQUIVALENT SYSTEMS OF FORCES
73
34 Introduction 74
32 Exforal and intemal Forces 74
33 Principie of Tansmissibity. Equivalent Forces — 76
24 Vector Product of Two Vectors 77
35 Vector Products Expressed in Terms of
Rectangular Components 78
38 Moment ofa Force about a Pont 81
37 — Varignon’s Thoorom 83
3.8 Roctanguiar Components of the Moment of Force 83
39 Sealar Product of Two Vectors. 98
3.10 Mixed Tple Product of Tree Vectors — 95
341 Moment of a Force about a Given Axis 97
312 Moment of a Couple — 107
313 Equivalent Couples 108
314 Addon of Coupes 110
3.15 Couples Can Be Represented by Vectors 110
316 Resolution of a Given Force Into a Force at O
and a Couple 111
3.17 Reduction of a System of Forces 10 One Force
and One Couple 122
Equivalent Systems of Forces 129
Equipolent Systems of Vectors 124
320 Further Reduction of a System of Forces 124
"321 Reduction ola System of Forces to a Wrench 127
Review and Summary for Chapter 3.146
Review Problems 151
Computer Problems 183
4
EQUILIBRIUM OF RIGID BODIES
187
4.1 Introduction — 158
(2 Froe-Body Diagram — 159
Equilibrium in Two Dimensions — 160
43 Reactions at Supports and Connections
fora Two-Dimensional Stucture — 160
44 Equilbrium of a Rigid Body in Two Dimensions 162
45. Statcally Indeterminate Reactions. Partial Constraints 164
48 Equilerium of aTwo-Force Body 183
47 — Equilrium of a Three-Force Body 184
Equilibrium in Three Dimensions 191
4.8. Equilorum of a Rigid Body in Three Dimensions — 191
2.14 Addiion of Concurent Forces in Space 40
215 Equileium of a Parc in Space — 57
Review and Summary for Chapter2 64
Review Problems 87
Computer Problems — 69
3
RIGID BODIES: EQUIVALENT SYSTEMS OF FORCES
73
34 Introduction 74
32 Exforal and intemal Forces 74
33 Principie of Tansmissibity. Equivalent Forces — 76
24 Vector Product of Two Vectors 77
35 Vector Products Expressed in Terms of
Rectangular Components 78
38 Moment ofa Force about a Pont 81
37 — Varignon’s Thoorom 83
3.8 Roctanguiar Components of the Moment of Force 83
39 Sealar Product of Two Vectors. 98
3.10 Mixed Tple Product of Tree Vectors — 95
341 Moment of a Force about a Given Axis 97
312 Moment of a Couple — 107
313 Equivalent Couples 108
314 Addon of Coupes 110
3.15 Couples Can Be Represented by Vectors 110
316 Resolution of a Given Force Into a Force at O
and a Couple 111
3.17 Reduction of a System of Forces 10 One Force
and One Couple 122
Equivalent Systems of Forces 129
Equipolent Systems of Vectors 124
320 Further Reduction of a System of Forces 124
"321 Reduction ola System of Forces to a Wrench 127
Review and Summary for Chapter 3.146
Review Problems 151
Computer Problems 183
4
EQUILIBRIUM OF RIGID BODIES
187
4.1 Introduction — 158
(2 Froe-Body Diagram — 159
Equilibrium in Two Dimensions — 160
43 Reactions at Supports and Connections
fora Two-Dimensional Stucture — 160
44 Equilbrium of a Rigid Body in Two Dimensions 162
45. Statcally Indeterminate Reactions. Partial Constraints 164
48 Equilerium of aTwo-Force Body 183
47 — Equilrium of a Three-Force Body 184
Equilibrium in Three Dimensions 191
4.8. Equilorum of a Rigid Body in Three Dimensions — 191
49
Foactions at Supports and Connections fora
‘Three-Dimensional Structure 191
Review and Summary for Chapter 4211
Review Problems 213
Computer Problems 215
510
sn
512
5
DISTRIBUTED FORCES: CENTROIDS
AND CENTERS OF GRAVITY
219
Introduction — 220
Areas and Lines — 220
‘Center of Gravity of a Two-Dimensional Body 220
Certroid of Areas and Lines 222
First Moments of Areas and Lines 229
‘Composite Plates and Wires 226
Determination of Centoids by Integration 238
Theorems of Pappus-Guldinus 238
Distrbuted Loads on Beams — 248
Forces on Submerged Suraces 249
Volumes 259
(Center of Gravity of a Treo-Dimensional Body
Gontoid of a Volume — 250
Composite Bodies 262
Determination of Centroids of Volumes by Integration
Review and Summary for Chapter 274
Roviow Problems 278
Computer Problems — 281
612
6
ANALYSIS OF STRUCTURES
284
Introduction — 285
Trusses 206
Dotintion of a Truss 205
Simple Tussos — 288
Analysis ol Trusses by tho Method of Jints 289
‘Joints under Special Loading Conditions 291
‘Space Trusses 293
‘Analysis of Trusses by the Method of Sections 309.
russes Made of Several Simple Tusses 304
Frames and Machines 315
‘Structures Containing Mutlorce Members 318
Analysis of a Frame 815
Frames Which Coase to Be Rigid When Detached
from Their Suppons 316
Machines — 331
Review and Summary for Chapter 6 343
Review Problems 346
Computer Problems 349
Foactions at Supports and Connections fora
‘Three-Dimensional Structure 191
Review and Summary for Chapter 4211
Review Problems 213
Computer Problems 215
510
sn
512
5
DISTRIBUTED FORCES: CENTROIDS
AND CENTERS OF GRAVITY
219
Introduction — 220
Areas and Lines — 220
‘Center of Gravity of a Two-Dimensional Body 220
Certroid of Areas and Lines 222
First Moments of Areas and Lines 229
‘Composite Plates and Wires 226
Determination of Centoids by Integration 238
Theorems of Pappus-Guldinus 238
Distrbuted Loads on Beams — 248
Forces on Submerged Suraces 249
Volumes 259
(Center of Gravity of a Treo-Dimensional Body
Gontoid of a Volume — 250
Composite Bodies 262
Determination of Centroids of Volumes by Integration
Review and Summary for Chapter 274
Roviow Problems 278
Computer Problems — 281
612
6
ANALYSIS OF STRUCTURES
284
Introduction — 285
Trusses 206
Dotintion of a Truss 205
Simple Tussos — 288
Analysis ol Trusses by tho Method of Jints 289
‘Joints under Special Loading Conditions 291
‘Space Trusses 293
‘Analysis of Trusses by the Method of Sections 309.
russes Made of Several Simple Tusses 304
Frames and Machines 315
‘Structures Containing Mutlorce Members 318
Analysis of a Frame 815
Frames Which Coase to Be Rigid When Detached
from Their Suppons 316
Machines — 331
Review and Summary for Chapter 6 343
Review Problems 346
Computer Problems 349
7
FORCES IN BEAMS AND CABLES
353
«71 introduction — 954
+72. Internal Forces in Members 354
Beams 961
«73. Various Types of Loading and Support 281
7.4 Shear and Bending Moment in a Beam 362
7,5 Shear and Bending-Moment Diagrams 364
:7:8 Relations among Load, Shear, and Bending Moment — 372
Cables — 383
“7.7. Cables with Concentrated Loads — 380
“7.8. Cables with Distributed Loads 384
“79 Parabolic Cable 385
“710. Catenary — 994
Review and Summary for Chapter 7 402
Review Problems 405
Computer Problems — 408
8
FRICTION
a
Introduction 412
“The Laws of Dry Friction. Coefficients af Ficion 412
Angles ol Fiction 415
Problems Involving Dry Friction 416
Wedges — 431
‘Square-Throaded Screws 431
Joumal Bearings. Axle Frcton — 440
‘Thrust Boarings. Disk Fiction — 442
Whee! Ficton.Roling Resistance 449
Bolt Fiction — 450
Review and Summary for Chapter 8 461
Review Problems 484
Computer Problems — 487
IOMENTS OF INERTIA
DISTRIBUTED FORCES:
an
91 Introduction 472
Moments of Inertia of Areas 473
‘Second Moment, or Moment of mor, of an Area 473
Determination of tho Moment of Inertia of an Area,
by Integration 474
Polar Moment of Inorta 475
Radius of Graton ofan Area 478
Paralo-Axis Theorem 483
Moments of Inertia of Composite Areas — 484
Product of Inoria 497
Principal Axes and Principal Moments of Inertia 498
FORCES IN BEAMS AND CABLES
353
«71 introduction — 954
+72. Internal Forces in Members 354
Beams 961
«73. Various Types of Loading and Support 281
7.4 Shear and Bending Moment in a Beam 362
7,5 Shear and Bending-Moment Diagrams 364
:7:8 Relations among Load, Shear, and Bending Moment — 372
Cables — 383
“7.7. Cables with Concentrated Loads — 380
“7.8. Cables with Distributed Loads 384
“79 Parabolic Cable 385
“710. Catenary — 994
Review and Summary for Chapter 7 402
Review Problems 405
Computer Problems — 408
8
FRICTION
a
Introduction 412
“The Laws of Dry Friction. Coefficients af Ficion 412
Angles ol Fiction 415
Problems Involving Dry Friction 416
Wedges — 431
‘Square-Throaded Screws 431
Joumal Bearings. Axle Frcton — 440
‘Thrust Boarings. Disk Fiction — 442
Whee! Ficton.Roling Resistance 449
Bolt Fiction — 450
Review and Summary for Chapter 8 461
Review Problems 484
Computer Problems — 487
IOMENTS OF INERTIA
DISTRIBUTED FORCES:
an
91 Introduction 472
Moments of Inertia of Areas 473
‘Second Moment, or Moment of mor, of an Area 473
Determination of tho Moment of Inertia of an Area,
by Integration 474
Polar Moment of Inorta 475
Radius of Graton ofan Area 478
Paralo-Axis Theorem 483
Moments of Inertia of Composite Areas — 484
Product of Inoria 497
Principal Axes and Principal Moments of Inertia 498
"9.10 Monrs Circle for Moments and Products of nerta 506.
Moments of Inertia of Masses 512
9.11 Moment of inertia of a Mass 512
9.12 Paralle-Axis Theorem 514
9.13 Moments of inertia of Thin Platos 515
9.18 Determination ofthe Moment of inera of a Three-Dimensional
Body by integration — 516
9.15 Moments ol Ineria of Composite Bodies 516
“0.16 Moment of Inertia ol a Body with Respect to an Arbitrary Axis
through O, Mass Products of Inertia 531
8:17. Ellpeoi of Inrta. Principal Axes of Inertia 332
"9.18 Determination of the Principal Axes and Principal Moments
of Inertia ofa Body of Arbivary Shape 524
Review and Summary for Chapter 9 545
Review Problems 551
‘Computer Problems 554
10
METHOD OF VIRTUAL WORK
+10. introduction 558
“102 Work ofa Force 558
+103 Principio of Virtual Work 581
“104 Applications ofthe Principle of Virtual Work 562.
“10.5 Real Machines. Mechanical Efciency 564
“108 Wok of a Force during a Finite Displacement 578
“107 Potential Energy 580
108 Potential Energy and Equilbrum 581
+109 Stability ol Equilibrium — 582
Review and Summary for Chapter 10 592
Review Problems 585
Computer Problems 507
Appendix
FUNDAMENTALS OF ENGINEERING EXAMINATION
601
Photo Credits 603
Index 605
Answers to Problems 611
Moments of Inertia of Masses 512
9.11 Moment of inertia of a Mass 512
9.12 Paralle-Axis Theorem 514
9.13 Moments of inertia of Thin Platos 515
9.18 Determination ofthe Moment of inera of a Three-Dimensional
Body by integration — 516
9.15 Moments ol Ineria of Composite Bodies 516
“0.16 Moment of Inertia ol a Body with Respect to an Arbitrary Axis
through O, Mass Products of Inertia 531
8:17. Ellpeoi of Inrta. Principal Axes of Inertia 332
"9.18 Determination of the Principal Axes and Principal Moments
of Inertia ofa Body of Arbivary Shape 524
Review and Summary for Chapter 9 545
Review Problems 551
‘Computer Problems 554
10
METHOD OF VIRTUAL WORK
+10. introduction 558
“102 Work ofa Force 558
+103 Principio of Virtual Work 581
“104 Applications ofthe Principle of Virtual Work 562.
“10.5 Real Machines. Mechanical Efciency 564
“108 Wok of a Force during a Finite Displacement 578
“107 Potential Energy 580
108 Potential Energy and Equilbrum 581
+109 Stability ol Equilibrium — 582
Review and Summary for Chapter 10 592
Review Problems 585
Computer Problems 507
Appendix
FUNDAMENTALS OF ENGINEERING EXAMINATION
601
Photo Credits 603
Index 605
Answers to Problems 611
Preface
OBJECTIVES
“The main objective ofa first course in mechanics should be to develop
in the engineering student the ability to analyze any problem in a
simple and logical manner and to apply to its solution a few, well-
‘understood, basic principles It is hoped that this text, designed for
the frst course in statics offered in the sophomore year, and the vol-
‘ume that follows, Vector Mechanics for Engineers: Dynamics, wil help
structor achieve this goal.
GENERAL APPROACH
Vector amass introduced eat inthe text and i sed the pre
Sentaton and discusion ofthe fundamental principles of mechas.
Victor methods ar also used to sche many problems, pricy
{hres menciona! problems whore thse toch
Fae al ee sce a ait, Ti oni ESE
Ras on the cores understanding lbs principles ol mechanics
Anden tir application to the sen of engineering problems and
ee Gare (GOL)
Practical Applications Are Introduced Early. One of t
characteristics of the approach used in these volumes is that m
‘chanics of particles is clearly separated from the
bodies. This approach makes it possible to co
‘applications at an early stage and to postpone the
more difficult concepts. For example:
+ In Statics, statics of particles is treated first (Chap. 2); alter the
rales of ait are introduced, the
principle of equilibrium of a particle is immediately applied to
practical situations involving only concurrent forces. The statics
Tis et sabe singe o
Dynamics, ih ton
Tin a pad te. Mechs for Engineer: Stat, aunk ton, the we of vector
Alga ine othe adn and subtraction of eco
2. Veto Mecha for Enger: Sis en
OBJECTIVES
“The main objective ofa first course in mechanics should be to develop
in the engineering student the ability to analyze any problem in a
simple and logical manner and to apply to its solution a few, well-
‘understood, basic principles It is hoped that this text, designed for
the frst course in statics offered in the sophomore year, and the vol-
‘ume that follows, Vector Mechanics for Engineers: Dynamics, wil help
structor achieve this goal.
GENERAL APPROACH
Vector amass introduced eat inthe text and i sed the pre
Sentaton and discusion ofthe fundamental principles of mechas.
Victor methods ar also used to sche many problems, pricy
{hres menciona! problems whore thse toch
Fae al ee sce a ait, Ti oni ESE
Ras on the cores understanding lbs principles ol mechanics
Anden tir application to the sen of engineering problems and
ee Gare (GOL)
Practical Applications Are Introduced Early. One of t
characteristics of the approach used in these volumes is that m
‘chanics of particles is clearly separated from the
bodies. This approach makes it possible to co
‘applications at an early stage and to postpone the
more difficult concepts. For example:
+ In Statics, statics of particles is treated first (Chap. 2); alter the
rales of ait are introduced, the
principle of equilibrium of a particle is immediately applied to
practical situations involving only concurrent forces. The statics
Tis et sabe singe o
Dynamics, ih ton
Tin a pad te. Mechs for Engineer: Stat, aunk ton, the we of vector
Alga ine othe adn and subtraction of eco
2. Veto Mecha for Enger: Sis en
of gd bodes ts considered in Caps 3 nd 4. In Chap, 3 the
Vector and scalar products of tro vectors are introduced and wed
15 deine the moment al force abouts pint and about nan.
ion ofthese new concepts allowed bya throng
and goons discusion of equal stone of forces leading
in Chap, to many patel applications involving he equ
tum a ig bodies under general force stems
the same dilo oben. The base concept
rk nd eneng; aed of
nd fit apie to
oki only paris Ths students can amaia he
selves with ro basic methods us in amis and ern
the epectveadntags before fang the dies aso
tied sith the mation of ig bodes.
"New Concepts Are introduced in Simple Terms. Since this
text is designed forthe frst course in statics, new concepts are pre-
ented in simple terms and every sep is explained in detail, On the
cl by discussing the broader aspects of the problems c
Sidered, and by stressing methods of general applicability, a def
ity of approach is achieved, For example, the concepts of par
tial constraints and statical indeterminacy are introduced early and
Fundamental Principles Are Placed in the Context of Sim-
ple Applications. The fact that mechanies is essentially a deduc-
ie science based on a few fundamental principles is stressed. Der:
ivations have been presented in their logical sequence and with all
the rigor warranted at this level. However, the learning process be-
ing largely induetice, simple applications are considered first. For
examples
+ The statics of particles precedes the staties of rigid bodies, and
problems involving internal forces are postponed until Chap. 6.
+ In Chap. 4, equilibrium problems involving only coplanar forces
idered first and solved by ordinary algebra, while prob
1g three-dimensional forces and requiring the full
use of vector algebra are discussed in the second part of the
chapter.
Free-Body Diagrams Are Used Both to Solve Equilibrium
Problems and to Express the Equivalence of Force Systems
Free-body diagrams are introduced early, and their importance is
phasized throughout the text. They are used not only to salve equilib-
rium problems but also to express the equivalence of two systems of
forces or, more generally, of two systems of vectors. The advantage of
this approach becomes apparent in the study ofthe dynamics of rigid
bodies, where it is used to solve three-dimensional as well as two-
imensional problems. By placing the emphasis on “free-body-diagram
equations” rather than on the standard algebraic equations of motion,
a more intuitive and more complete understanding of the fundamen
tal principles of dynamics can be achieved. This approach, which vas
first introduced in 1962 in the first edition of Vector Mechanics for
xl
Vector and scalar products of tro vectors are introduced and wed
15 deine the moment al force abouts pint and about nan.
ion ofthese new concepts allowed bya throng
and goons discusion of equal stone of forces leading
in Chap, to many patel applications involving he equ
tum a ig bodies under general force stems
the same dilo oben. The base concept
rk nd eneng; aed of
nd fit apie to
oki only paris Ths students can amaia he
selves with ro basic methods us in amis and ern
the epectveadntags before fang the dies aso
tied sith the mation of ig bodes.
"New Concepts Are introduced in Simple Terms. Since this
text is designed forthe frst course in statics, new concepts are pre-
ented in simple terms and every sep is explained in detail, On the
cl by discussing the broader aspects of the problems c
Sidered, and by stressing methods of general applicability, a def
ity of approach is achieved, For example, the concepts of par
tial constraints and statical indeterminacy are introduced early and
Fundamental Principles Are Placed in the Context of Sim-
ple Applications. The fact that mechanies is essentially a deduc-
ie science based on a few fundamental principles is stressed. Der:
ivations have been presented in their logical sequence and with all
the rigor warranted at this level. However, the learning process be-
ing largely induetice, simple applications are considered first. For
examples
+ The statics of particles precedes the staties of rigid bodies, and
problems involving internal forces are postponed until Chap. 6.
+ In Chap. 4, equilibrium problems involving only coplanar forces
idered first and solved by ordinary algebra, while prob
1g three-dimensional forces and requiring the full
use of vector algebra are discussed in the second part of the
chapter.
Free-Body Diagrams Are Used Both to Solve Equilibrium
Problems and to Express the Equivalence of Force Systems
Free-body diagrams are introduced early, and their importance is
phasized throughout the text. They are used not only to salve equilib-
rium problems but also to express the equivalence of two systems of
forces or, more generally, of two systems of vectors. The advantage of
this approach becomes apparent in the study ofthe dynamics of rigid
bodies, where it is used to solve three-dimensional as well as two-
imensional problems. By placing the emphasis on “free-body-diagram
equations” rather than on the standard algebraic equations of motion,
a more intuitive and more complete understanding of the fundamen
tal principles of dynamics can be achieved. This approach, which vas
first introduced in 1962 in the first edition of Vector Mechanics for
xl
xiv Poe
Engineers has now gained wide acceptance among mechanies teach:
‘ers in this country. Its, therefore, used in preference to the method
of dynamic equilibrium and tothe equations of motion in the solution
‘of all sample problems inthis book.
‘A Four-Color Presentation Uses Color to Distinguish Vectors.
Color has been used, not only to enhance the quality of the illustra
tions, but also to help students ds
ter. W
color sven chapter to repre-
o type. Throughout Statics, for example, red
is used exclusively to represent forces and couples, while position vec-
tors are shown in blue and dime black. This makes it easier
for the students to identify the forces acting on a given particle or
rig body and to follow sample problems and other
‘examples given in the text
A Careful Balance Between SI and US. Customary Units 6
Consistently Maintained. Because ofthe current tend In the
“American government and industry adopt the international yt
Fun (SE metro uni) the SL it ment frequently usd in me-
hanes ae introduced in Chap, and are used thoughout te text
Approsimatchy alt of the sample problems and 60 percent of the
Homework problems are stated in these unt, while the remainder
are in US. cstomary unis. The authors believe that this approach
wal best serve the need of students, who, as engineers, il ave to
ie conversant with both systems of nt.
Tl alo shou be recognized that using both ST and US, cu
tals mare than thease of conversion factors. Since
stem of us sn abu system bared onthe uit of
nd mass, whereas the US. customary stm isa gravitational
Based on the uns of timo, length, and force, different ap
p quie for the solution ol many problems. For exam.
Ple. when SE uns are wed, «body ls general pe bys mass
Expressed in klogruns; in most problems al tates i wl he neces:
si dtr the wih of th ty est nd a ll
ol calculation vil be required fr this purpose. On te ther
when US. customary unt are used, a body le peca hy w
Pounds and. In «mamis problems, an additional clean vil
breed to determine ts mus in logs (or 1-51), The author,
therefore, belle hat problem asiguments shuld include bth sy
tems of units
"The Instructor and Solutions Manual provides six diferent Hits
of sign der of problems stated nS units
ad in US. tts cn be selected. Iso desired, two come
plete lists of sigaments can alo be selected with upto 75 pero
Of the problems stated in S
Optional Sections Offer Advanced or Specialty Topics. A
large number of optio have been included. These sections
tare indicated by asterisks and thu
which form the core of the base staties course. They may be omitted.
without prejudice to the understanding ofthe rest of the tent
Engineers has now gained wide acceptance among mechanies teach:
‘ers in this country. Its, therefore, used in preference to the method
of dynamic equilibrium and tothe equations of motion in the solution
‘of all sample problems inthis book.
‘A Four-Color Presentation Uses Color to Distinguish Vectors.
Color has been used, not only to enhance the quality of the illustra
tions, but also to help students ds
ter. W
color sven chapter to repre-
o type. Throughout Statics, for example, red
is used exclusively to represent forces and couples, while position vec-
tors are shown in blue and dime black. This makes it easier
for the students to identify the forces acting on a given particle or
rig body and to follow sample problems and other
‘examples given in the text
A Careful Balance Between SI and US. Customary Units 6
Consistently Maintained. Because ofthe current tend In the
“American government and industry adopt the international yt
Fun (SE metro uni) the SL it ment frequently usd in me-
hanes ae introduced in Chap, and are used thoughout te text
Approsimatchy alt of the sample problems and 60 percent of the
Homework problems are stated in these unt, while the remainder
are in US. cstomary unis. The authors believe that this approach
wal best serve the need of students, who, as engineers, il ave to
ie conversant with both systems of nt.
Tl alo shou be recognized that using both ST and US, cu
tals mare than thease of conversion factors. Since
stem of us sn abu system bared onthe uit of
nd mass, whereas the US. customary stm isa gravitational
Based on the uns of timo, length, and force, different ap
p quie for the solution ol many problems. For exam.
Ple. when SE uns are wed, «body ls general pe bys mass
Expressed in klogruns; in most problems al tates i wl he neces:
si dtr the wih of th ty est nd a ll
ol calculation vil be required fr this purpose. On te ther
when US. customary unt are used, a body le peca hy w
Pounds and. In «mamis problems, an additional clean vil
breed to determine ts mus in logs (or 1-51), The author,
therefore, belle hat problem asiguments shuld include bth sy
tems of units
"The Instructor and Solutions Manual provides six diferent Hits
of sign der of problems stated nS units
ad in US. tts cn be selected. Iso desired, two come
plete lists of sigaments can alo be selected with upto 75 pero
Of the problems stated in S
Optional Sections Offer Advanced or Specialty Topics. A
large number of optio have been included. These sections
tare indicated by asterisks and thu
which form the core of the base staties course. They may be omitted.
without prejudice to the understanding ofthe rest of the tent
Among the topics covered in these additional sections are the re
duction of a system of forces to a wrench, applications to hydrostat-
les, shear and bending-moment diagrams for beams, equilibrium of
cables, products of inertia and Mohr’ circle, the determination ofthe
principal axes and the mass moments of inertia of a body of arbitrary
shape, and the method of virtual work. The sections on beams are es-
pecially useful when the course in statis is immediately followed by
à course in mechanics of materials, while the sections on the inertia
ional bodies are primarily intended for tu
ly in dynamics the three-dimensional motion
dite
no previous mathematical knowledge beyond Keller
ee
en
een
na te at
een
Feen
CHAPTER ORGANIZATION AND PEDAGOGICAL FEATURES:
Chapter introduction. Each chapter bins wth an introdus-
ton sect sting the purpose nd gon ofthe chapter and dessin
in simple terms the mater o be covered and its application tothe
Solution of engineering problems. Chapter outlines provide students
Sith a provi of chapter opi
(Chapter Lessons. The body ofthe tet is ded nto uns cach
consising of one or sever theory actions, one or sever sample prob.
aa large number of problems to be assigned. ach unit ore:
sponds o we dell topic and general cn be covered in one le
Da. Ina number of exes, bower the instractor vil nd desimble
to devote more than one lesson to a ge tpl. The Instructors and
Solutions Manual contain suggestions on the coverage ofeach eso.
Sample Problems. The sample problems are set up in much
these for that students wl use when sing the assigne prob.
Teme They thos serve the double purpose of amplifying the txt and
the type of neat, orderly work that students should
their ow solutions
skate
Solving Problems on Your Own. À section entitled Sing
Problems on Your Oun sled foreach lesan, betwee the sam
ple problems and the problems to be assigned. The purpose of these
Lions lt hep students organiza in tet nm mite pred
ing theory ofthe text and the solution methods ofthe sample prob
lems o that they can more successful solve the homework probe
tems. Also included in these sections are specie suggestions and
Strategies tht wil enable students lo more een, atack any
signed problems
duction of a system of forces to a wrench, applications to hydrostat-
les, shear and bending-moment diagrams for beams, equilibrium of
cables, products of inertia and Mohr’ circle, the determination ofthe
principal axes and the mass moments of inertia of a body of arbitrary
shape, and the method of virtual work. The sections on beams are es-
pecially useful when the course in statis is immediately followed by
à course in mechanics of materials, while the sections on the inertia
ional bodies are primarily intended for tu
ly in dynamics the three-dimensional motion
dite
no previous mathematical knowledge beyond Keller
ee
en
een
na te at
een
Feen
CHAPTER ORGANIZATION AND PEDAGOGICAL FEATURES:
Chapter introduction. Each chapter bins wth an introdus-
ton sect sting the purpose nd gon ofthe chapter and dessin
in simple terms the mater o be covered and its application tothe
Solution of engineering problems. Chapter outlines provide students
Sith a provi of chapter opi
(Chapter Lessons. The body ofthe tet is ded nto uns cach
consising of one or sever theory actions, one or sever sample prob.
aa large number of problems to be assigned. ach unit ore:
sponds o we dell topic and general cn be covered in one le
Da. Ina number of exes, bower the instractor vil nd desimble
to devote more than one lesson to a ge tpl. The Instructors and
Solutions Manual contain suggestions on the coverage ofeach eso.
Sample Problems. The sample problems are set up in much
these for that students wl use when sing the assigne prob.
Teme They thos serve the double purpose of amplifying the txt and
the type of neat, orderly work that students should
their ow solutions
skate
Solving Problems on Your Own. À section entitled Sing
Problems on Your Oun sled foreach lesan, betwee the sam
ple problems and the problems to be assigned. The purpose of these
Lions lt hep students organiza in tet nm mite pred
ing theory ofthe text and the solution methods ofthe sample prob
lems o that they can more successful solve the homework probe
tems. Also included in these sections are specie suggestions and
Strategies tht wil enable students lo more een, atack any
signed problems
xvi
Homework Problem Sets. Most of the problems are of a prac-
tical nature and should appeal to engineering students. They are pri
mary designed, trate the material presented in the
ts understand the principles of mechanies. The
to the portions of material they le
Tustate and are arranged in order of increasing difficult. Problems
equiring special attention ae indicated by asterisks. Answers to 70
percent af the problems are given at the end ofthe book. Problems
for which the answers are given are sot in straight type
wile problems for which no answer is given are set in italic
Chapter Review and Summary. Each with are
view and summary of the material covered in that chapter. Marginal
notes ar used t help students organize their review work, and eros-
references have been included to help them find the portions of ma-
terialsequiring their special attention
Review Problems. À set of review problems is included at the
. ofeach chapter. These problems provide students further oppor
ty o apply the most important concepts introduced in the chapter
Computer Problems. ach chapter includes a set of problems
designed to be solved with computational software. Many of these
problems are relevant to the design proces. In s
they may involve the ana
and loadings of th
positions of a mechanism which may require an iterative method of
solution. Developing the algorithm required to solve a gh
chanics problem will benefit students in two different wags: (1) it will
Help them gain a better understanding ofthe mechanics principles
involved (2) it will provide them with an opportunity to apply their
computer sills tothe solution of a meaning engineering problem.
ics, for example,
‘SUPPLEMENTS
ements package for both instructors and students
is available with the txt
Instructor's and Solutions Manual. The Instructors and So-
lutions Manual that accompanies the eighth edition features typeset,
‘one-per-page solutions to the homework problems. This manual also
features a numberof tables designed to assis instructors in creating a
schedule of assignments for their courses. The various topics covered
in the text are listed in Table I, and a suggested number of periods to
be spent on each topic is indicated. Table 11 provided a brief deserip-
tion of all groups a classification of the problems in
each group according to the units used. Sample lesson schedules are
shown in Tables IH, IV, and
McGraw-Hill’: ARIS—Assessment, Review, and Instruction
‘System. ARIS is a complete homework and course management
system for Vector Mechanics for Engineers: Staties and Dynamics. In
structors can create and share course materials and assignments with
‘other instructors, edit questions and algorithms, import their own con-
Homework Problem Sets. Most of the problems are of a prac-
tical nature and should appeal to engineering students. They are pri
mary designed, trate the material presented in the
ts understand the principles of mechanies. The
to the portions of material they le
Tustate and are arranged in order of increasing difficult. Problems
equiring special attention ae indicated by asterisks. Answers to 70
percent af the problems are given at the end ofthe book. Problems
for which the answers are given are sot in straight type
wile problems for which no answer is given are set in italic
Chapter Review and Summary. Each with are
view and summary of the material covered in that chapter. Marginal
notes ar used t help students organize their review work, and eros-
references have been included to help them find the portions of ma-
terialsequiring their special attention
Review Problems. À set of review problems is included at the
. ofeach chapter. These problems provide students further oppor
ty o apply the most important concepts introduced in the chapter
Computer Problems. ach chapter includes a set of problems
designed to be solved with computational software. Many of these
problems are relevant to the design proces. In s
they may involve the ana
and loadings of th
positions of a mechanism which may require an iterative method of
solution. Developing the algorithm required to solve a gh
chanics problem will benefit students in two different wags: (1) it will
Help them gain a better understanding ofthe mechanics principles
involved (2) it will provide them with an opportunity to apply their
computer sills tothe solution of a meaning engineering problem.
ics, for example,
‘SUPPLEMENTS
ements package for both instructors and students
is available with the txt
Instructor's and Solutions Manual. The Instructors and So-
lutions Manual that accompanies the eighth edition features typeset,
‘one-per-page solutions to the homework problems. This manual also
features a numberof tables designed to assis instructors in creating a
schedule of assignments for their courses. The various topics covered
in the text are listed in Table I, and a suggested number of periods to
be spent on each topic is indicated. Table 11 provided a brief deserip-
tion of all groups a classification of the problems in
each group according to the units used. Sample lesson schedules are
shown in Tables IH, IV, and
McGraw-Hill’: ARIS—Assessment, Review, and Instruction
‘System. ARIS is a complete homework and course management
system for Vector Mechanics for Engineers: Staties and Dynamics. In
structors can create and share course materials and assignments with
‘other instructors, edit questions and algorithms, import their own con-
create announcements and Aue dats fr assignments, ARIS
oti grading and reporting of cayo assigner
generated homework: quizes and tts. Other resources avalabe on
ANS include SMART tutorial, a homework problem bank, Lac
ture FowerPoins, and Images from the tert. Vt ss lo con
Deerjlnston for more nation onthe supplements asbl with
ths text
Hands-on Mechanics. Hands-on Mechanics is a website de-
signed for instructors who are interested in incorporating, three
aids into their lectures. Developed
sh a partnership between the MeGrav-Thll Engineering Team
and the Department of Civil and Mechanical Engineering at the
United States Miltary Academy at West Point, this website not only
provides detailed instructions for how to build 3D teaching tools using
a a lb or local hardware store
y where educators can share ideas, trado bes practices,
and submit their own demonstrations for posting on the site. Visit
mmiechanisscomn.
ACKNOWLEDGMENTS,
‘The authors wish to acknowledge the collaboration of David Mazurek:
to this eighth edition of Veetor Mechanics for Engineers and thank
him especially for his role in making the extensive problem set rev
sion possible.
À special thanks go to our collea
the solutions and answers ofall prob
prepared the solutions for the accompanying Instructor's and Sol
tion Manual: Yohannes Ketema of University of Minnesota; Amy
Mazurek of Williams Memorial Institute; David Oglesby of Univer-
sity of Missouri-Rolla; and Daniel W. Yannitell of Louisiana State
University
We are pleased to recognize Dennis Ormond and Michael
Haughey of Fine Line Illustrations of Farmingdale, New York, for
the artful illustrations which contribute so much t the effectiveness
of the text
‘The authors thank the many companies that provided photo-
graphs for this edition, We also wish to recognize the determined
‘lforts and patience of our photo researcher, Sabina Dowell.
“The authors also thank the members ofthe staff at MeGra-Hill
for their support and dedication during the preparation of this new
particularly wish to acknawledge the contributions of
soring Editor Michael Hackett, Developmental Editor
ite, and Senior Project Manager Kay Brimeyer.
al, the authors gratefully acknowledge the many helpful com
ments and suggestions offered by users of the previous editions of
Vector Mechanics for Engineers
E, Russell
Pres evil
oti grading and reporting of cayo assigner
generated homework: quizes and tts. Other resources avalabe on
ANS include SMART tutorial, a homework problem bank, Lac
ture FowerPoins, and Images from the tert. Vt ss lo con
Deerjlnston for more nation onthe supplements asbl with
ths text
Hands-on Mechanics. Hands-on Mechanics is a website de-
signed for instructors who are interested in incorporating, three
aids into their lectures. Developed
sh a partnership between the MeGrav-Thll Engineering Team
and the Department of Civil and Mechanical Engineering at the
United States Miltary Academy at West Point, this website not only
provides detailed instructions for how to build 3D teaching tools using
a a lb or local hardware store
y where educators can share ideas, trado bes practices,
and submit their own demonstrations for posting on the site. Visit
mmiechanisscomn.
ACKNOWLEDGMENTS,
‘The authors wish to acknowledge the collaboration of David Mazurek:
to this eighth edition of Veetor Mechanics for Engineers and thank
him especially for his role in making the extensive problem set rev
sion possible.
À special thanks go to our collea
the solutions and answers ofall prob
prepared the solutions for the accompanying Instructor's and Sol
tion Manual: Yohannes Ketema of University of Minnesota; Amy
Mazurek of Williams Memorial Institute; David Oglesby of Univer-
sity of Missouri-Rolla; and Daniel W. Yannitell of Louisiana State
University
We are pleased to recognize Dennis Ormond and Michael
Haughey of Fine Line Illustrations of Farmingdale, New York, for
the artful illustrations which contribute so much t the effectiveness
of the text
‘The authors thank the many companies that provided photo-
graphs for this edition, We also wish to recognize the determined
‘lforts and patience of our photo researcher, Sabina Dowell.
“The authors also thank the members ofthe staff at MeGra-Hill
for their support and dedication during the preparation of this new
particularly wish to acknawledge the contributions of
soring Editor Michael Hackett, Developmental Editor
ite, and Senior Project Manager Kay Brimeyer.
al, the authors gratefully acknowledge the many helpful com
ments and suggestions offered by users of the previous editions of
Vector Mechanics for Engineers
E, Russell
Pres evil
ABC,
À BC,
A
b
E
d
F
xvii
List of Symbols
Constant; radius: distance
Reactions at supports and connections
Points
Area
Width: distance
Constant
Controid
Distance
Base of natural logarithms,
Force; frietion force
Acceleration of gravity
Center of gravity; constant of gravit
Height; sag of cable
Unit vectors along coordinate axes
Moments of inertia
Centroidal moment of inertia
Products of ine
Polar
Centroidal radius of gyration
Length
Couple; momen
Moment about point O.
Moment resultant about point O
‘Magnitude of couple or moment; mass of earth
Moment about axis OL
Normal component of reaction
Origin of coordinates
Pressure
Force: vector
Force; vector
Position vector
À BC,
A
b
E
d
F
xvii
List of Symbols
Constant; radius: distance
Reactions at supports and connections
Points
Area
Width: distance
Constant
Controid
Distance
Base of natural logarithms,
Force; frietion force
Acceleration of gravity
Center of gravity; constant of gravit
Height; sag of cable
Unit vectors along coordinate axes
Moments of inertia
Centroidal moment of inertia
Products of ine
Polar
Centroidal radius of gyration
Length
Couple; momen
Moment about point O.
Moment resultant about point O
‘Magnitude of couple or moment; mass of earth
Moment about axis OL
Normal component of reaction
Origin of coordinates
Pressure
Force: vector
Force; vector
Position vector
Radius; distance; polar coordinate stot mba
Resultant force; resultant vector; reaction
Radius of earth
Position vector
h of arc; length of cable
Force; vector
‘Thickness
Force
Tension
Work
Vector product shearing force
Volume; potential energy shear
Load per unit length
Wade kel
Rectangular coordinates; distances
Rectangular coordinates of eentroid or center
of gravity
Angles
Specific weight
Elongation
Virtual displacen
Viral work
Unit vector along a
Efficiency
Angular coordinate;
Coeiicient of friction
Density
Angle of friction; angle
+ polar coordinate
xix
Resultant force; resultant vector; reaction
Radius of earth
Position vector
h of arc; length of cable
Force; vector
‘Thickness
Force
Tension
Work
Vector product shearing force
Volume; potential energy shear
Load per unit length
Wade kel
Rectangular coordinates; distances
Rectangular coordinates of eentroid or center
of gravity
Angles
Specific weight
Elongation
Virtual displacen
Viral work
Unit vector along a
Efficiency
Angular coordinate;
Coeiicient of friction
Density
Angle of friction; angle
+ polar coordinate
xix
Introduction
INTRODUCTION
14. What Is Mechanics?
12. Fundamental Concepts and
Principles
4.3. Systems of Unis
1.4. Conversion from One System of
Units to Another
1.5. Method of Problem Solution
1.5. Numerical Accuracy
1.1. WHAT IS MECHANICS?
Mechanics can be defined as that science which describes and pre
cts the conditions of rest or motion of bodies under the action of
forces. I is divided into three parts: mechanics of rigid bodies, me
chanics of deformable bodies, and mechanies of fluids
The mechanics of rigid bodies is subdivided into staties and dy
namics, the former dealing with bodies at rest, the latter with bodies.
in motion. In this part of the study of mechanics, bodies are assumed
to be perfectly rigid. Actual structures and machines, however, are
never absolutely rigid and deform under the loads to which they are
subjected. But these deformations are usually small and do not ap-
preciab affect the conditions of equilibrium or motion of the struc
ture under consideration, They are important, though, as far as the
resistance of the structure to führe is concerned and are studied in
mechanies of materials, which isa part of the mechanics of deformable
bodies. The third division of mechanics, the mechanies of Muids, is
subdivided into the study of incompressible fluids and of compress
be fluids. An important subdivision of the study of incompressible
fluids is hydraulics, which deals with problems involving water
Mechanics is a physical science, since it deals with the study of
physical phenomena. However, some associate mechanics with math
€ many consider it as an engineering subject, Both these
views are justified in part. Mechanics is the foundation of most eng
neering sciences and is an indispensable prerequisite to their study
However, it does not have the empiricism found in some engineerin
sciences, hat is, it does not rely on experience or observation al
by its rigor and the emphasis it places on deduetive reasoning it re
sembles mathematics But, again tis not an abstract or even a pure
science: mechanics isan applied science, The purpose of mechanics
is to explain and predict physical phenomena and thus to lay the foun
dations for en
neering appli
1.2. FUNDAMENTAL CONCEPTS AND PRINCIPLES
Although the study of mechanics goes back to the time of Aristotle
354-322 nc.) and Archimedes (257-212 nc), one has to wait until
Newton (1642-1727) to And a satisfatory formulation of its funda:
mental principes, These principles were later expressed in a mod
fied form by d'Alembert, Lagrange, and Hamilton, Their validity
remained unchallenged, however, until Einstein formulated his cor
relativity (1905). While its limitations have now been recognized,
neutonian mechanics still remains the basis of today’s €
“The basic concepts used in mechanics are space, time, mass, and
force, These concepts cannot be truly defined: they should be ac
‘cepted on the basis of our intuition and experience and used asa men
tal frame of reference for our study of mechanics,
The concept af space is associated with the notion of the position
of a point P The position of P can be dei
xl by three lengths mea-
sured from a certain reference point, or origin, in three given
directions. These lengths are known as the coordinates of P
14. What Is Mechanics?
12. Fundamental Concepts and
Principles
4.3. Systems of Unis
1.4. Conversion from One System of
Units to Another
1.5. Method of Problem Solution
1.5. Numerical Accuracy
1.1. WHAT IS MECHANICS?
Mechanics can be defined as that science which describes and pre
cts the conditions of rest or motion of bodies under the action of
forces. I is divided into three parts: mechanics of rigid bodies, me
chanics of deformable bodies, and mechanies of fluids
The mechanics of rigid bodies is subdivided into staties and dy
namics, the former dealing with bodies at rest, the latter with bodies.
in motion. In this part of the study of mechanics, bodies are assumed
to be perfectly rigid. Actual structures and machines, however, are
never absolutely rigid and deform under the loads to which they are
subjected. But these deformations are usually small and do not ap-
preciab affect the conditions of equilibrium or motion of the struc
ture under consideration, They are important, though, as far as the
resistance of the structure to führe is concerned and are studied in
mechanies of materials, which isa part of the mechanics of deformable
bodies. The third division of mechanics, the mechanies of Muids, is
subdivided into the study of incompressible fluids and of compress
be fluids. An important subdivision of the study of incompressible
fluids is hydraulics, which deals with problems involving water
Mechanics is a physical science, since it deals with the study of
physical phenomena. However, some associate mechanics with math
€ many consider it as an engineering subject, Both these
views are justified in part. Mechanics is the foundation of most eng
neering sciences and is an indispensable prerequisite to their study
However, it does not have the empiricism found in some engineerin
sciences, hat is, it does not rely on experience or observation al
by its rigor and the emphasis it places on deduetive reasoning it re
sembles mathematics But, again tis not an abstract or even a pure
science: mechanics isan applied science, The purpose of mechanics
is to explain and predict physical phenomena and thus to lay the foun
dations for en
neering appli
1.2. FUNDAMENTAL CONCEPTS AND PRINCIPLES
Although the study of mechanics goes back to the time of Aristotle
354-322 nc.) and Archimedes (257-212 nc), one has to wait until
Newton (1642-1727) to And a satisfatory formulation of its funda:
mental principes, These principles were later expressed in a mod
fied form by d'Alembert, Lagrange, and Hamilton, Their validity
remained unchallenged, however, until Einstein formulated his cor
relativity (1905). While its limitations have now been recognized,
neutonian mechanics still remains the basis of today’s €
“The basic concepts used in mechanics are space, time, mass, and
force, These concepts cannot be truly defined: they should be ac
‘cepted on the basis of our intuition and experience and used asa men
tal frame of reference for our study of mechanics,
The concept af space is associated with the notion of the position
of a point P The position of P can be dei
xl by three lengths mea-
sured from a certain reference point, or origin, in three given
directions. These lengths are known as the coordinates of P
‘To define an event, itis not sufficient to indicate its position in
space. The timo of the event should also be given.
‚The concept of mass is used to characterize and compare bodies.
funda jechanical experiments. Two bod-
, will be attracted by the earth in
ill also offer the same resistance to a change
in translational mot
A force represents the action of one body on another. It can be
‘exerted by actual contact or ata distance, as in the case of gravita-
tional forces and magnetic forces. A force is characterized by is point
of application, ts magnitude, and its direction; a force is represented
by a tector (See. 23).
stonian mechanics, space, time, and mass aro absolute con-
‘cepts, independent of each other. (This is not true in reatiostie me.
where the time of an event depends upon its position, and
ci he other hand,
below indicates that the resultant force acting on a body is related to
the mass of the body and to the manner in which its velocity varies
with
ions of rest or motion of particles and
rigid bodies in terms of the four basic concepts we have introduced.
By particle we mean a very small amount of matter which may be
assumed to oceupy a single point in space. A rigid body is a combi.
pig cd postions with
jechanies of particles is
Besides, the results
a large number of prob
Tems dealing with the conditions of rest or motion of actual bodies.
‘The study of elementary ic rests on six fundamental prin-
ciples based on experimental evidence.
The Parallelogram Law for the Addition of Forces. This
states that two forces acting on a particle may be replaced by a
o force, called their resultant, obtained by drawing the diagonal of
the parallelogram which has sides equal to the given forces (See. 2.2).
‘The Principle of Transmissibility. This states that th
tions of equilibrium or of motion of a rigid body will
changed i a force acting a a given point of the rigid body is replaced
bya force of the same magnitude and same direction, but acting at a
different point, provided that the two forces have the same line of
action (Sec. 33).
Nowton’s Three Fundamental Laws. Formulated by Sir Isaac
Newton in he latter part of the seventeenth century, these Laws can
be stated as follows
FIRST LAW. Ifthe resultant force acting on a particle is zero,
‘the particle will remain at rest (if originally at rest) or will move with
‘constant speed in a straight line (i originally in motion) (See. 2.10).
1.2 Fundamental Concept and Pit 3
space. The timo of the event should also be given.
‚The concept of mass is used to characterize and compare bodies.
funda jechanical experiments. Two bod-
, will be attracted by the earth in
ill also offer the same resistance to a change
in translational mot
A force represents the action of one body on another. It can be
‘exerted by actual contact or ata distance, as in the case of gravita-
tional forces and magnetic forces. A force is characterized by is point
of application, ts magnitude, and its direction; a force is represented
by a tector (See. 23).
stonian mechanics, space, time, and mass aro absolute con-
‘cepts, independent of each other. (This is not true in reatiostie me.
where the time of an event depends upon its position, and
ci he other hand,
below indicates that the resultant force acting on a body is related to
the mass of the body and to the manner in which its velocity varies
with
ions of rest or motion of particles and
rigid bodies in terms of the four basic concepts we have introduced.
By particle we mean a very small amount of matter which may be
assumed to oceupy a single point in space. A rigid body is a combi.
pig cd postions with
jechanies of particles is
Besides, the results
a large number of prob
Tems dealing with the conditions of rest or motion of actual bodies.
‘The study of elementary ic rests on six fundamental prin-
ciples based on experimental evidence.
The Parallelogram Law for the Addition of Forces. This
states that two forces acting on a particle may be replaced by a
o force, called their resultant, obtained by drawing the diagonal of
the parallelogram which has sides equal to the given forces (See. 2.2).
‘The Principle of Transmissibility. This states that th
tions of equilibrium or of motion of a rigid body will
changed i a force acting a a given point of the rigid body is replaced
bya force of the same magnitude and same direction, but acting at a
different point, provided that the two forces have the same line of
action (Sec. 33).
Nowton’s Three Fundamental Laws. Formulated by Sir Isaac
Newton in he latter part of the seventeenth century, these Laws can
be stated as follows
FIRST LAW. Ifthe resultant force acting on a particle is zero,
‘the particle will remain at rest (if originally at rest) or will move with
‘constant speed in a straight line (i originally in motion) (See. 2.10).
1.2 Fundamental Concept and Pit 3
4 bron
Photo 12. Wonin earth ot, people ae objets
fre said 19 De wolghdess even though De
raten Jrs aci appronmate 80% of
fat experenoed onthe surco of he earth THs
apparent condor wl bo esca in Chapter
2 un we ap Neon second law toe
‘maton à pares
SECOND LAW. Ifthe resultant force acting on a particle fs not
zero, the particle will have an acceleration proportional to the mag-
nitude of the resultant and in the direction ofthis resultant force.
As you will sce in Sec. 12.2, this law can be stated as
Fema an
re E, m, and a represent, respectively the resultant force acting,
the particle, the mass ofthe particle, and the acceleration of the
particle, expressed in a consistent system of units,
THIRD LAW. The forces of action and reaction betwee
in contact have the same magnitude, same line of ation,
sense (Sec. 6.1)
Nowton’s Law of Gravitation. ‘This stats that two particles of
mass M and m are mutually attracted with equal and opposite forces
Fand —F (Fig, 1.1) of magnitude F given by the formula
bodies
‘opposite
where r = distance between the two particles
G = universal constant called the constant of gravitation
Newton’ law of gravitation introduces the idea of an action exerted
ata distance and extends the range of application of Newton’ third
law: the action F and the reaction ~F in Fig. 1.1 are equal and op-
posite, and they have the same line of action,
A particular case of great importance is that ofthe attraction ofthe
yb on a particle located on its surface, The force F exerted by the
rth on the particle is then defined as the weight W of the particle.
Taking M equal to the mass of the earth, m equal to the mass of the
particle, and r equal to the radius ofthe earth, and introducing th
constant
eN]
as)
re
the magnitude W of the weight of a particle of mass m may be ex-
pressed ast
‘The value of R in formul
point considered; it also depends upo
hot try spherical. The value of g therefore varies with the positio
‘of the point considered. As long as the point actually remains on the
ace of the earth, it is sufficiently accurate in most engineering
‘computations to assume that g equals 9.81 mA? or 32.2 Ws"
YA more cut den ofthe weight Won tak no acount the ratio of
the cut
Photo 12. Wonin earth ot, people ae objets
fre said 19 De wolghdess even though De
raten Jrs aci appronmate 80% of
fat experenoed onthe surco of he earth THs
apparent condor wl bo esca in Chapter
2 un we ap Neon second law toe
‘maton à pares
SECOND LAW. Ifthe resultant force acting on a particle fs not
zero, the particle will have an acceleration proportional to the mag-
nitude of the resultant and in the direction ofthis resultant force.
As you will sce in Sec. 12.2, this law can be stated as
Fema an
re E, m, and a represent, respectively the resultant force acting,
the particle, the mass ofthe particle, and the acceleration of the
particle, expressed in a consistent system of units,
THIRD LAW. The forces of action and reaction betwee
in contact have the same magnitude, same line of ation,
sense (Sec. 6.1)
Nowton’s Law of Gravitation. ‘This stats that two particles of
mass M and m are mutually attracted with equal and opposite forces
Fand —F (Fig, 1.1) of magnitude F given by the formula
bodies
‘opposite
where r = distance between the two particles
G = universal constant called the constant of gravitation
Newton’ law of gravitation introduces the idea of an action exerted
ata distance and extends the range of application of Newton’ third
law: the action F and the reaction ~F in Fig. 1.1 are equal and op-
posite, and they have the same line of action,
A particular case of great importance is that ofthe attraction ofthe
yb on a particle located on its surface, The force F exerted by the
rth on the particle is then defined as the weight W of the particle.
Taking M equal to the mass of the earth, m equal to the mass of the
particle, and r equal to the radius ofthe earth, and introducing th
constant
eN]
as)
re
the magnitude W of the weight of a particle of mass m may be ex-
pressed ast
‘The value of R in formul
point considered; it also depends upo
hot try spherical. The value of g therefore varies with the positio
‘of the point considered. As long as the point actually remains on the
ace of the earth, it is sufficiently accurate in most engineering
‘computations to assume that g equals 9.81 mA? or 32.2 Ws"
YA more cut den ofthe weight Won tak no acount the ratio of
the cut
Te pe wen jt itil bina ith ne
of ou study of mechani asthe are needed. The study ofthe stt-
des of particles cared out in Chap. 2 wil be bax onthe paralelo:
gram Le of edn and on Newton's first law alone, The principle
run il be introduced in Chup- 3 as ve begin the study
fe sais of ig brs, and Neston’ third law in Chap. 6 a ve
Analyse the forcesexeredon each other bythe various member
ing structure In the st à, Newton second law
col gaia td vil then be shown
that Newton fist law a particular case of Newtons second low
(Sec. 12.2) and thatthe principle of transmis ould be derived
from the other principles and this eliminated (Sec. 103) the mean.
E Newtons fist and thd avs, the parallelogram law of
Aion. and the principle transmita wil prose swith the
sic! foundation forthe entre sty of the stat
rig Boris, and systems of rgd bodies.
‘As need erie, the si fundamental principles [ted above are
tased on experimental evidence, Except ot Newtons ft hw andthe
sib they are independent principles which a
hematicaly fom ach ther or rom any aber ce
mentay Phys principle. On these prnl pets most aho tr
cate store of ewonian mechan. For more thas two centres a
tremendous number of problems dealing with the conditions of rest
and mation of rgd bodies, deformable bodies, and Mids were solved
ty appli these fundamental principles. Many of the solutions ob-
tunel could be checked experimenta; ds proidng a further ver
fet ofthe principles from which ey were dived It as only in
de st century tht Newtons mechanis vas found at an the study
ofthe motion of atoms and in the study of he moto
fs, where st spp
the human or engineering,
sth the sped ight, Newton n
‘where velocities are small compared
anis has yet to be disproved.
13. SYSTEMS OF UNITS
Associated with the fo cepts introduced in the pre-
‘ceding section are the so-called kinetic units, that is, the units of
length, time, mass, and force. These units cannot be chosen inde-
pendently if Eq, (11) isto be satisfied. Three of the units may be
defined arbitra: they are then referred to as base units. The fourth
accordance with Eq. (1.1) and is
ts selected in this way are
referred to as a derived unit. Kinetic u
said to form a consistent system of units
International System of Units (SI Units}). In this system,
which will be in universal use when the United States co
conversion to SI the units of length, mass,
and time, and they are called, respectively, the meter (m), the klo-
‘gram (kg), and the second (s). All three are arbitrarily defined. The
second, which was originally chosen to represent 1/86 400 of the mean
Stand fr Spt Intemational tés (French)
13. aa ls 5)
of ou study of mechani asthe are needed. The study ofthe stt-
des of particles cared out in Chap. 2 wil be bax onthe paralelo:
gram Le of edn and on Newton's first law alone, The principle
run il be introduced in Chup- 3 as ve begin the study
fe sais of ig brs, and Neston’ third law in Chap. 6 a ve
Analyse the forcesexeredon each other bythe various member
ing structure In the st à, Newton second law
col gaia td vil then be shown
that Newton fist law a particular case of Newtons second low
(Sec. 12.2) and thatthe principle of transmis ould be derived
from the other principles and this eliminated (Sec. 103) the mean.
E Newtons fist and thd avs, the parallelogram law of
Aion. and the principle transmita wil prose swith the
sic! foundation forthe entre sty of the stat
rig Boris, and systems of rgd bodies.
‘As need erie, the si fundamental principles [ted above are
tased on experimental evidence, Except ot Newtons ft hw andthe
sib they are independent principles which a
hematicaly fom ach ther or rom any aber ce
mentay Phys principle. On these prnl pets most aho tr
cate store of ewonian mechan. For more thas two centres a
tremendous number of problems dealing with the conditions of rest
and mation of rgd bodies, deformable bodies, and Mids were solved
ty appli these fundamental principles. Many of the solutions ob-
tunel could be checked experimenta; ds proidng a further ver
fet ofthe principles from which ey were dived It as only in
de st century tht Newtons mechanis vas found at an the study
ofthe motion of atoms and in the study of he moto
fs, where st spp
the human or engineering,
sth the sped ight, Newton n
‘where velocities are small compared
anis has yet to be disproved.
13. SYSTEMS OF UNITS
Associated with the fo cepts introduced in the pre-
‘ceding section are the so-called kinetic units, that is, the units of
length, time, mass, and force. These units cannot be chosen inde-
pendently if Eq, (11) isto be satisfied. Three of the units may be
defined arbitra: they are then referred to as base units. The fourth
accordance with Eq. (1.1) and is
ts selected in this way are
referred to as a derived unit. Kinetic u
said to form a consistent system of units
International System of Units (SI Units}). In this system,
which will be in universal use when the United States co
conversion to SI the units of length, mass,
and time, and they are called, respectively, the meter (m), the klo-
‘gram (kg), and the second (s). All three are arbitrarily defined. The
second, which was originally chosen to represent 1/86 400 of the mean
Stand fr Spt Intemational tés (French)
13. aa ls 5)
now defined as the duration of 9 192 631 770 cycles of
to the transition between two levels of
33 atom. The meter, originally
tance from the equator to ether
pole, is now defined as 1 650 763.73 wavelengths of the orange-red
light corresponding to a certain transition in an atom of krypton-S6.
‚The kilogram, which is approximately equal to the mass of 0.001 mi
of water, is defined as the mass ofa platinum-ridium standard kept
at the International Bureau of Weights and Measures at Sèvres, near
Paris, France. The unit of force is a derived unit I is called the new-
ton (N) and is defined as the force which gives an acceleration of
Tis to a mass of 1 kg (Fig. 1.2). From Eq, (1.1) we write
LN = (1 KL mA?) = 1 kg mi? as)
depende
the kilogram, and the seco
may be used anywhere on the earth: they may even be used on an-
‘They will absays have the same significance,
weight of body, or the force of gravity exerted
should, like any other force, be expressed in newtons. Fr
it follows that the weight of a body of mass 1 kg (Fi
w
mg
= (1 kg)(981 ms?)
ISIN
‘Multiples and submultiples of the fundamental SI units may be
ned through the use of the prefixes defined in Table 1.1. The
multiples and submultiples of the units of length, mass, and force most
Frequently used in engincering are, respectively, the kilometer (km)
and the millimeter (mm); the megagramt (Mg) and the gram (g) and
the kilonewton (KN). According to Table 1.1, we have
1m = 100m 1m
1 Mg = 1000 kg
TKN = 1000 N
‘The conversion of these units into
respectively; can be effected by sin
places tothe right orto the lit. For
ters, one moves the decimal point three places to the right:
3.82 km = 3820 m
ters by moving the decimal
larly, 47.2 mm is converted into n
point three places to the left
2 mm = 0.0472 m
Hal brown asa mer ton
to the transition between two levels of
33 atom. The meter, originally
tance from the equator to ether
pole, is now defined as 1 650 763.73 wavelengths of the orange-red
light corresponding to a certain transition in an atom of krypton-S6.
‚The kilogram, which is approximately equal to the mass of 0.001 mi
of water, is defined as the mass ofa platinum-ridium standard kept
at the International Bureau of Weights and Measures at Sèvres, near
Paris, France. The unit of force is a derived unit I is called the new-
ton (N) and is defined as the force which gives an acceleration of
Tis to a mass of 1 kg (Fig. 1.2). From Eq, (1.1) we write
LN = (1 KL mA?) = 1 kg mi? as)
depende
the kilogram, and the seco
may be used anywhere on the earth: they may even be used on an-
‘They will absays have the same significance,
weight of body, or the force of gravity exerted
should, like any other force, be expressed in newtons. Fr
it follows that the weight of a body of mass 1 kg (Fi
w
mg
= (1 kg)(981 ms?)
ISIN
‘Multiples and submultiples of the fundamental SI units may be
ned through the use of the prefixes defined in Table 1.1. The
multiples and submultiples of the units of length, mass, and force most
Frequently used in engincering are, respectively, the kilometer (km)
and the millimeter (mm); the megagramt (Mg) and the gram (g) and
the kilonewton (KN). According to Table 1.1, we have
1m = 100m 1m
1 Mg = 1000 kg
TKN = 1000 N
‘The conversion of these units into
respectively; can be effected by sin
places tothe right orto the lit. For
ters, one moves the decimal point three places to the right:
3.82 km = 3820 m
ters by moving the decimal
larly, 47.2 mm is converted into n
point three places to the left
2 mm = 0.0472 m
Hal brown asa mer ton
Men Factor Proc Sym
1 000 000 000 000 = 1 ra a
1 000 000 000 = 10°. i G
1 000 000 = 10 mega M
kilo k
hectot h
delas de
deci a
cont e
sl m
iro a
ico BD
femto r
ato a
(ein be fee ref cent a pre wl ane ent
fee pont of Someter places the accent on he it abe, not
The un ofthese pres should be de, xcept or the measurement of areas
and vols and o the oneal wu of centimeter a for body and tion,
Using scien
= 3.82 10° m
172x107 m
inute (min) and the
‘The multiples of the unit of time are the
3600 5, these multi
hour (1): Since 1 min = 60 sand 1 h = 60 mi
ples cannot be converted as readily as the others
By using the appropriate multiple or submultiple of a given unit,
‘one can avoid writing very large or very small numbers. For example,
fone usually writes 427.2 km rather than 427 200 m, and 2.16 mm
rather than 0.002 16 m.t
Units of Area and Volume. The unit of areas the square me-
ter (m2), which represents the area ofa square of side 1 m; the
‘of volume is the cubic meter (m'). equal to the volume of a
side 1 m. In order to avoid exceedingly small or large nu
in the computation of areas and volumes, one uses.
units obtained by respectively squaring and cubing not only the mil
limeter but also two intermediate submultiples of the meter, namely,
the deeimeter (dn) and the centimeter (em). Since, by definition,
‘ie shold rte hat when nore han fou gs ao used ce side fe de
mal plot ers py nS nes in 47 200 mo M218 space ee
mas shal be wel pate the igs ato rope of he, The aed ee
Faso wth the conan edn place of decia) pb, wih coment mary
13 gym tunes 7,
1 000 000 000 000 = 1 ra a
1 000 000 000 = 10°. i G
1 000 000 = 10 mega M
kilo k
hectot h
delas de
deci a
cont e
sl m
iro a
ico BD
femto r
ato a
(ein be fee ref cent a pre wl ane ent
fee pont of Someter places the accent on he it abe, not
The un ofthese pres should be de, xcept or the measurement of areas
and vols and o the oneal wu of centimeter a for body and tion,
Using scien
= 3.82 10° m
172x107 m
inute (min) and the
‘The multiples of the unit of time are the
3600 5, these multi
hour (1): Since 1 min = 60 sand 1 h = 60 mi
ples cannot be converted as readily as the others
By using the appropriate multiple or submultiple of a given unit,
‘one can avoid writing very large or very small numbers. For example,
fone usually writes 427.2 km rather than 427 200 m, and 2.16 mm
rather than 0.002 16 m.t
Units of Area and Volume. The unit of areas the square me-
ter (m2), which represents the area ofa square of side 1 m; the
‘of volume is the cubic meter (m'). equal to the volume of a
side 1 m. In order to avoid exceedingly small or large nu
in the computation of areas and volumes, one uses.
units obtained by respectively squaring and cubing not only the mil
limeter but also two intermediate submultiples of the meter, namely,
the deeimeter (dn) and the centimeter (em). Since, by definition,
‘ie shold rte hat when nore han fou gs ao used ce side fe de
mal plot ers py nS nes in 47 200 mo M218 space ee
mas shal be wel pate the igs ato rope of he, The aed ee
Faso wth the conan edn place of decia) pb, wih coment mary
13 gym tunes 7,
the submultiples of
1 dm? = (1 dm)? = (107! m= 10° m
(Lem)? = (10° m? = 10 n
(107? m} = 10-®
and the submultiples ofthe unit of volume are
(1 dm)? = (10! m? = 10°
(Lem)? = (10°? m)? = 10*
mm = (1073 mj = 10° m?
à
€ unit of area are
1mm?
It should be noted that
sured, the cubic decim
the volume of a liquid is being mea-
(dm) is usually referred to as alter (L)
‘Other derived SI units used to measure the moment of a force,
the work of a force, eto, are shown in Table 1.2. While these units
‘will be introduced in later chapters as they are needed, we should
important rule at this time: When a derived unit is obtained
y dividing a base unit by another base unit, a prefix may be used in
the numerator ofthe derived unit but notin its denominator. For ex-
ample, the constant k of a spring which stretehes 20 mm under a loud
of 100 N will bo expressed as
100N _ 100 N
20 020 m
ke
5000 Nm or k= 5 RNY
but never as k = 5 N/mm.
‘Table 12, Principal SI Units Used in Mechanics
Samy nt Bol Formula
Acceleration Meter por second squared mst
Angle Radian md |
‘Angular acceleration Radian per second squared nds
‘Angular velocity Hada per second nds
Area. Square n m
Density Kilogram per eubie meter yim
Eneng! Joule j
Force Newton N
Frequency Hertz mz
npube Newton-second
Length Meter in
Mass Kilogram oak
Moment ofa force Neuton-meter Nom
Power war w pm
Pressure Pascal Pa Nit
Pascal Pa Na
Time Second A ‘
Meter per second ws
Cubic meter
Liter
Joue y
‘Supplementary wt (rhin = u rd = 367).
soe un
1 dm? = (1 dm)? = (107! m= 10° m
(Lem)? = (10° m? = 10 n
(107? m} = 10-®
and the submultiples ofthe unit of volume are
(1 dm)? = (10! m? = 10°
(Lem)? = (10°? m)? = 10*
mm = (1073 mj = 10° m?
à
€ unit of area are
1mm?
It should be noted that
sured, the cubic decim
the volume of a liquid is being mea-
(dm) is usually referred to as alter (L)
‘Other derived SI units used to measure the moment of a force,
the work of a force, eto, are shown in Table 1.2. While these units
‘will be introduced in later chapters as they are needed, we should
important rule at this time: When a derived unit is obtained
y dividing a base unit by another base unit, a prefix may be used in
the numerator ofthe derived unit but notin its denominator. For ex-
ample, the constant k of a spring which stretehes 20 mm under a loud
of 100 N will bo expressed as
100N _ 100 N
20 020 m
ke
5000 Nm or k= 5 RNY
but never as k = 5 N/mm.
‘Table 12, Principal SI Units Used in Mechanics
Samy nt Bol Formula
Acceleration Meter por second squared mst
Angle Radian md |
‘Angular acceleration Radian per second squared nds
‘Angular velocity Hada per second nds
Area. Square n m
Density Kilogram per eubie meter yim
Eneng! Joule j
Force Newton N
Frequency Hertz mz
npube Newton-second
Length Meter in
Mass Kilogram oak
Moment ofa force Neuton-meter Nom
Power war w pm
Pressure Pascal Pa Nit
Pascal Pa Na
Time Second A ‘
Meter per second ws
Cubic meter
Liter
Joue y
‘Supplementary wt (rhin = u rd = 367).
soe un
U.S. Customary Units. Most practicing Americ
‘commonly use a system in which the base units are the units of length,
force, and time. These units are, respectively, the foot (10, the pound
(ib), and the second (s). The second is the same as the corresponding
SI unit. The foot is defined as 0.3048 m. The pound is defined as the
(weight of a platinum standard, called the standard pound, which is kept
at the National Institute of Standards and Technology outside Wash.
ington, the mass of which is 0.453 592 43 kg, Since the weight of a body
depends upon the earths gravitational attraction, which varies with lo
itis specified that the standard pound should be placed at sea
ind at a latitude of 45° to properly define a force of 1 Ib. Clearly
the US. customary units do not form an absolute system of units. Be
cause oftheir dependence upon the gravitational attraction ofthe earth
they form a gravitational system of units.
‘While the standard pound also serves asthe unit of mass in com
mercial transactions inthe United Stats, it cannot be so used in en
gineering mechanics computations, since such a unit would not be
consistent with the base units defined in the preceding paragraph. In-
‘deed, when acted upon by a force of 1 Ib, that i, when subjected to
the force of gravity, the standard pound receives the acceleration of
gravity, g = 92.2 (Us (Fig, 1.4), not the unit acceleration required by
Eq, (1:1). The unit of mass consistent with the foot, the pound, and
the second is the mass which receives an acceleration of 1 fs? when
a force of IIb is applied to it (Fig, 1.5). This unit, sometimes called
a slug, can be derived from the equation F = ma after substituting
1 lb and 1 A/S for F and a, respectively. We write
1b = (1 slug) MS)
and obtain
Lb
1 slug =
Comparing Figs. 1.4 and 1.5, we conclude that the slug is a
times larger than the mass of the standard poun
The fact that in the U.S. customary system of units bodies are
characterized by their weight in pounds rather than by their mass in
slugs will be a convenience in the study of staics, where on
stantly deals with weights and other forces and only seldo
masses. However, in the study of dynamics, where forces, masses, and
accelerations are involved, the mass m of a body will be expressed in
slugs when its weight W is given in pounds. Recalling Eq. (14), we
write
where g isthe acceleration of
Other US. customary unis frequently encountered in engineer:
ing problems are the mie (mi), equal to 5280 ft; the inch (in. eq
to à fi and the kilopound (kip), equal to a force of 1000 Ib. The ton
is often used to represe 00 1 but, lke the pound, must
be converted into slugs in engineering computations
The conversion into feet, pounds, and seconds of quantities ex
pressed in other U.S. customary units is generally moro involved
wity (g = 32.2 Ms)
12 Smemeotuts g
Fig 15
Photo 1.8. The unt of mass i heen base unt
ssl based on à physical standard. Work 6 m
progress % ocios te standard wi one based
fr unchanging natural phenome,
‘commonly use a system in which the base units are the units of length,
force, and time. These units are, respectively, the foot (10, the pound
(ib), and the second (s). The second is the same as the corresponding
SI unit. The foot is defined as 0.3048 m. The pound is defined as the
(weight of a platinum standard, called the standard pound, which is kept
at the National Institute of Standards and Technology outside Wash.
ington, the mass of which is 0.453 592 43 kg, Since the weight of a body
depends upon the earths gravitational attraction, which varies with lo
itis specified that the standard pound should be placed at sea
ind at a latitude of 45° to properly define a force of 1 Ib. Clearly
the US. customary units do not form an absolute system of units. Be
cause oftheir dependence upon the gravitational attraction ofthe earth
they form a gravitational system of units.
‘While the standard pound also serves asthe unit of mass in com
mercial transactions inthe United Stats, it cannot be so used in en
gineering mechanics computations, since such a unit would not be
consistent with the base units defined in the preceding paragraph. In-
‘deed, when acted upon by a force of 1 Ib, that i, when subjected to
the force of gravity, the standard pound receives the acceleration of
gravity, g = 92.2 (Us (Fig, 1.4), not the unit acceleration required by
Eq, (1:1). The unit of mass consistent with the foot, the pound, and
the second is the mass which receives an acceleration of 1 fs? when
a force of IIb is applied to it (Fig, 1.5). This unit, sometimes called
a slug, can be derived from the equation F = ma after substituting
1 lb and 1 A/S for F and a, respectively. We write
1b = (1 slug) MS)
and obtain
Lb
1 slug =
Comparing Figs. 1.4 and 1.5, we conclude that the slug is a
times larger than the mass of the standard poun
The fact that in the U.S. customary system of units bodies are
characterized by their weight in pounds rather than by their mass in
slugs will be a convenience in the study of staics, where on
stantly deals with weights and other forces and only seldo
masses. However, in the study of dynamics, where forces, masses, and
accelerations are involved, the mass m of a body will be expressed in
slugs when its weight W is given in pounds. Recalling Eq. (14), we
write
where g isthe acceleration of
Other US. customary unis frequently encountered in engineer:
ing problems are the mie (mi), equal to 5280 ft; the inch (in. eq
to à fi and the kilopound (kip), equal to a force of 1000 Ib. The ton
is often used to represe 00 1 but, lke the pound, must
be converted into slugs in engineering computations
The conversion into feet, pounds, and seconds of quantities ex
pressed in other U.S. customary units is generally moro involved
wity (g = 32.2 Ms)
12 Smemeotuts g
Fig 15
Photo 1.8. The unt of mass i heen base unt
ssl based on à physical standard. Work 6 m
progress % ocios te standard wi one based
fr unchanging natural phenome,
10. uvas
Photo 14. Tre ngorance e ieudg unas na
deaatens cannot be over empassed. 1 was
{Gund thatthe 8125 milo Mare Ciao ror
{She fo go io ert around Mars because the
bree contactor had provided to navigation team
‘tm operating a nen U.S uns rater Pan
‘he se Ss
and requires greater attention than the corresponding operation in SI
units. If, for example, the magnitude of a velocity is given as e
30 mi, we convert i to fs as follows. First we write
Since we want to get rid of the unit miles and introduce instead the
nit feet, we should multiply the right-hand member of the equation
by an expression containing miles in the denominator and fect in the
‘numerator. But, since we do not want to change the value ofthe right
hand member the expression used should have a value equal to unity
280 1)/(1 mi) is such an expression. Operating in a
The quotient
similar way to transform the unit hour into seconds, we writ
of \ 1h |
(po mi yan)
Y mi )\ 36005 )
o= (ol
Carrying out the numerical computations and canceling out units
‘which appear in both the numerator and the denominator, we obtain
ñ
ohms
1.4. CONVERSION FROM ONE SYSTEM OF UNITS
TO ANOTHER
There ar
ST units a numerical result obtained in U.S. customary units or vice
versa. Because the unit of time is the same in both systems, only two
Kinetic base units need be converted. Thus, since all other kinetic
units can be derived from these base units, only two conversion fac
tors need be remembered
Units of Length. By definition the US. customary
length is
1h = 0308 Ls)
It follows that
1 mi = 5280 fe = 5280(0,3048 m) = 1609 m
1 mi = 1.609 km 19)
Also
Lin, = à R= (03048 m)
Lin, = 25.4 mm 1.10)
hat the U.S, customary unit of force
t of the standard pound (of mass
atitude of 45° (where g = 9.807 mvs")
Units of Force. Recall
(pound) is defined as the w
0.4536 kg) at sea level and ata
and using Eq, (1-4), we write
Photo 14. Tre ngorance e ieudg unas na
deaatens cannot be over empassed. 1 was
{Gund thatthe 8125 milo Mare Ciao ror
{She fo go io ert around Mars because the
bree contactor had provided to navigation team
‘tm operating a nen U.S uns rater Pan
‘he se Ss
and requires greater attention than the corresponding operation in SI
units. If, for example, the magnitude of a velocity is given as e
30 mi, we convert i to fs as follows. First we write
Since we want to get rid of the unit miles and introduce instead the
nit feet, we should multiply the right-hand member of the equation
by an expression containing miles in the denominator and fect in the
‘numerator. But, since we do not want to change the value ofthe right
hand member the expression used should have a value equal to unity
280 1)/(1 mi) is such an expression. Operating in a
The quotient
similar way to transform the unit hour into seconds, we writ
of \ 1h |
(po mi yan)
Y mi )\ 36005 )
o= (ol
Carrying out the numerical computations and canceling out units
‘which appear in both the numerator and the denominator, we obtain
ñ
ohms
1.4. CONVERSION FROM ONE SYSTEM OF UNITS
TO ANOTHER
There ar
ST units a numerical result obtained in U.S. customary units or vice
versa. Because the unit of time is the same in both systems, only two
Kinetic base units need be converted. Thus, since all other kinetic
units can be derived from these base units, only two conversion fac
tors need be remembered
Units of Length. By definition the US. customary
length is
1h = 0308 Ls)
It follows that
1 mi = 5280 fe = 5280(0,3048 m) = 1609 m
1 mi = 1.609 km 19)
Also
Lin, = à R= (03048 m)
Lin, = 25.4 mm 1.10)
hat the U.S, customary unit of force
t of the standard pound (of mass
atitude of 45° (where g = 9.807 mvs")
Units of Force. Recall
(pound) is defined as the w
0.4536 kg) at sea level and ata
and using Eq, (1-4), we write
W= mg
1 Mh = (0.4536 kg9.507 mvs?) = 4.448 kg = mA
or recalling Eq, (1.3),
nein au
Units of Mass. The U.S. customary
derived unit, Thus, using Eqs. (16), (18), an
it of mass (shu) is a
(LAD), we write
and, recalling Eq. (1.5),
A
Although it cannot be used as a consistent unit of mass, we recall that
the mass ofthe standard pound is, by definition,
4536 kg aus)
‘This constant may be used to determine the mass in SI units (slo
gras) of a body which has been characterized by its weight in US.
castomary units (pounds).
“To convert a derived US. customary unit into SE units, one sim-
is or divides by the appropriate conversion factors. For
Ho comer the moment ofa force which was fund to be
into SI units, we use formulas (1.10) and (1.11) and
1 pound mass
M= 47 Ib in, = 47(4:448 N)(254 mm)
= 5310 N - mm
‘The conversion factors given in this section may also be
convert a numerical result obtained in SI units into U.S. ex
nits. For example, if the moment of a force was for
M= 40 N + m, we write, following the procedure used in the last
paragraph of See. 13,
DEN BLY ETS
merical computations and canceling out units
in both the numerator and the denominator, we ob
2951b-f
Carrying
“which appea
‘The US. customary units most frequently used in mech:
listed in Table 1.3 with their ST equivalents
1.5. METHOD OF PROBLEM SOLUTION
You should approach a problem in mechanics as you would approach
¡gincering situation. By drawing on your own experience
y derstand and formulate the
problem. Once the problem has been clearly stated, however, there is
15 Memo ot Polen Sen 44
1 Mh = (0.4536 kg9.507 mvs?) = 4.448 kg = mA
or recalling Eq, (1.3),
nein au
Units of Mass. The U.S. customary
derived unit, Thus, using Eqs. (16), (18), an
it of mass (shu) is a
(LAD), we write
and, recalling Eq. (1.5),
A
Although it cannot be used as a consistent unit of mass, we recall that
the mass ofthe standard pound is, by definition,
4536 kg aus)
‘This constant may be used to determine the mass in SI units (slo
gras) of a body which has been characterized by its weight in US.
castomary units (pounds).
“To convert a derived US. customary unit into SE units, one sim-
is or divides by the appropriate conversion factors. For
Ho comer the moment ofa force which was fund to be
into SI units, we use formulas (1.10) and (1.11) and
1 pound mass
M= 47 Ib in, = 47(4:448 N)(254 mm)
= 5310 N - mm
‘The conversion factors given in this section may also be
convert a numerical result obtained in SI units into U.S. ex
nits. For example, if the moment of a force was for
M= 40 N + m, we write, following the procedure used in the last
paragraph of See. 13,
DEN BLY ETS
merical computations and canceling out units
in both the numerator and the denominator, we ob
2951b-f
Carrying
“which appea
‘The US. customary units most frequently used in mech:
listed in Table 1.3 with their ST equivalents
1.5. METHOD OF PROBLEM SOLUTION
You should approach a problem in mechanics as you would approach
¡gincering situation. By drawing on your own experience
y derstand and formulate the
problem. Once the problem has been clearly stated, however, there is
15 Memo ot Polen Sen 44
12
Table 1 ‘and Their SI Equivalents
Suan US Cusiemary Unt Si Eguren
Acceleration me 0.304 wise
inst 00254 ms?
Aros e 0.0020 m?
m 6452 mn?
Energy Ab 1356]
Force kip ass kN
7
Impulse thes
Length e
Mass
Moment of «force
Moment of inertia
Ofan area 04162 x 10
Of a mass 1356 hg m
Momentun AS ke: me
Power 1.356 W
457 W
Ss Pa
6.805 kPa
03048 mvs
00251 mv
04870 is
1.600 kn,
0.02852 mn
1630 em?
3785 L
02464 La
1356]
es
Velocity
vol
Liquids
for your particular fancy: The solution must be
bs onthe si fundamento principles stated in Sec. 12 or on theo.
‘roms dried fom them. Every step taken must be justified an tht
Fais Strict ries mus be followed wich lead to te soto
almost leving no rom for your nulo o Tek
img Aer an anger has een obtained it shouldbe checked, Here
gain, you may ell upon our common ense and perinal experience
Tino completly sali wi the result obtained, you should cre-
ful check your formulation ofthe problem, the validity of the meth:
ds used fr solution and the accuney of your computations
The statement ofa problem should be clara precise. It should
conti the given data and indicate what information is required. A
neat drang swing al quantities involved shouldbe included. Scp-
trate Gags should be drawn for all bodies im, dating
‘deal the forces acting on each body. These diagrams are known as
fredy diagrams and are described in detain Ses. 2.1 and 42.
Table 1 ‘and Their SI Equivalents
Suan US Cusiemary Unt Si Eguren
Acceleration me 0.304 wise
inst 00254 ms?
Aros e 0.0020 m?
m 6452 mn?
Energy Ab 1356]
Force kip ass kN
7
Impulse thes
Length e
Mass
Moment of «force
Moment of inertia
Ofan area 04162 x 10
Of a mass 1356 hg m
Momentun AS ke: me
Power 1.356 W
457 W
Ss Pa
6.805 kPa
03048 mvs
00251 mv
04870 is
1.600 kn,
0.02852 mn
1630 em?
3785 L
02464 La
1356]
es
Velocity
vol
Liquids
for your particular fancy: The solution must be
bs onthe si fundamento principles stated in Sec. 12 or on theo.
‘roms dried fom them. Every step taken must be justified an tht
Fais Strict ries mus be followed wich lead to te soto
almost leving no rom for your nulo o Tek
img Aer an anger has een obtained it shouldbe checked, Here
gain, you may ell upon our common ense and perinal experience
Tino completly sali wi the result obtained, you should cre-
ful check your formulation ofthe problem, the validity of the meth:
ds used fr solution and the accuney of your computations
The statement ofa problem should be clara precise. It should
conti the given data and indicate what information is required. A
neat drang swing al quantities involved shouldbe included. Scp-
trate Gags should be drawn for all bodies im, dating
‘deal the forces acting on each body. These diagrams are known as
fredy diagrams and are described in detain Ses. 2.1 and 42.
The fundamental principles of mechanics listed in Sec. 1.2 will
be used io write equations expressing the conditions of rest or motion
of the bodies considered. Each equation should be clearly related to
‘one of the free-body diagrams. You will then proceed to solve the
problem, observing strictly the usual rules of algebra and recording
‘neatly the various steps taken.
‘After the answer has been obtained, it should be carefully
checked. Mistakes in reasoning can often be detected by checking the
nits. For example, to determine the moment ofa force of 50 N about
‘point 0.60 m from is line of aetion, we would have written 12)
JM = Fd = (50 N)(0.60 m) = 30 N-m
The unit Nm obtained by multiplying newtons by
correct unit for the moment ofa force; I another
tained, we would have known that some mistake had been made.
Errors in computation will usually be found by substituting the
ves obtained into an equation which has not yet been
ng that the equation is satisfied. The importance of
tations in engineering cannot be overemphasized,
ers is the
had been ob-
1.5. NUMERICAL ACCURACY
‘The accuracy of the solution of a problem depends upon two items:
(1) the accuracy of the given data and (2) the accuracy of the com-
putations performed.
“The solution cannot be more accurate than the less accurate of
these two items. For example, i the loading of a bridge is known to
be 75,000 Ib with a possible error of 100 Ib either way, the relative
sures the degree of accuracy of he data is
Fb = 0,0013 = 0.13 percent
In computing the reaction at one of the bridge supports, it would then
be meaningless to record it as 14,322 Ib. The accuracy ofthe solution
‘eannot be greater than 0.13 percent, no matter how accurate the com:
úputations are, and the possible error in the answer may be as large as
{0.13/100)(14,322 Ib) = 20 lb, The answer should be properly recorded.
as 14,320 & 20 1b.
In engince
problems, the data are seldom known with an
“accuracy greater than 02 percent. I is therefore seldom justified
to write the answers to such problems with an accuracy greater than
02 percent. A practical rule is to use 4 figures to record numbers
beginning with a “17 and 3 figures in al other cases. Unless otherwise
indicated, he data given in a problem should be assumed known with
a comparable degree of accuracy: A force of 40 I, for example, should
be read 40.0 Ib, and a force of 15 Ib should be read 15:00 Ib.
Pocket electronic calculator are widely used by practicing engi-
neers and engineering students. The speed and accuracy of these cal
‘ulators facilitate the numerical computations in the solution of many
ms. However, students should not record more significant fig-
‘can be justified merely because they are easily obtained. As
noted above, an accuracy greater than 0.2 percent is seldom neces
in the solution of practical engineering problems.
16. romana Accor 13
be used io write equations expressing the conditions of rest or motion
of the bodies considered. Each equation should be clearly related to
‘one of the free-body diagrams. You will then proceed to solve the
problem, observing strictly the usual rules of algebra and recording
‘neatly the various steps taken.
‘After the answer has been obtained, it should be carefully
checked. Mistakes in reasoning can often be detected by checking the
nits. For example, to determine the moment ofa force of 50 N about
‘point 0.60 m from is line of aetion, we would have written 12)
JM = Fd = (50 N)(0.60 m) = 30 N-m
The unit Nm obtained by multiplying newtons by
correct unit for the moment ofa force; I another
tained, we would have known that some mistake had been made.
Errors in computation will usually be found by substituting the
ves obtained into an equation which has not yet been
ng that the equation is satisfied. The importance of
tations in engineering cannot be overemphasized,
ers is the
had been ob-
1.5. NUMERICAL ACCURACY
‘The accuracy of the solution of a problem depends upon two items:
(1) the accuracy of the given data and (2) the accuracy of the com-
putations performed.
“The solution cannot be more accurate than the less accurate of
these two items. For example, i the loading of a bridge is known to
be 75,000 Ib with a possible error of 100 Ib either way, the relative
sures the degree of accuracy of he data is
Fb = 0,0013 = 0.13 percent
In computing the reaction at one of the bridge supports, it would then
be meaningless to record it as 14,322 Ib. The accuracy ofthe solution
‘eannot be greater than 0.13 percent, no matter how accurate the com:
úputations are, and the possible error in the answer may be as large as
{0.13/100)(14,322 Ib) = 20 lb, The answer should be properly recorded.
as 14,320 & 20 1b.
In engince
problems, the data are seldom known with an
“accuracy greater than 02 percent. I is therefore seldom justified
to write the answers to such problems with an accuracy greater than
02 percent. A practical rule is to use 4 figures to record numbers
beginning with a “17 and 3 figures in al other cases. Unless otherwise
indicated, he data given in a problem should be assumed known with
a comparable degree of accuracy: A force of 40 I, for example, should
be read 40.0 Ib, and a force of 15 Ib should be read 15:00 Ib.
Pocket electronic calculator are widely used by practicing engi-
neers and engineering students. The speed and accuracy of these cal
‘ulators facilitate the numerical computations in the solution of many
ms. However, students should not record more significant fig-
‘can be justified merely because they are easily obtained. As
noted above, an accuracy greater than 0.2 percent is seldom neces
in the solution of practical engineering problems.
16. romana Accor 13
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