حل كتاب Vector Mechanics For Engineers Statics and Dynamics 11th Edition Solution Manual

حل كتاب Vector Mechanics For Engineers Statics and Dynamics 11th Edition Solution Manual
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Ferdinand P. Beer, E. Russell Johnston, David F. Mazurek, Brian P. Self
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11 نوفمبر 2021
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حل كتاب
Vector Mechanics For Engineers Statics and Dynamics 11th Edition Solution Manual
Twelfth Edition
Ferdinand P. Beer
Late of Lehigh University
E. Russell Johnston, Jr.
Late of University of Connecticut
David F. Mazurek
U.S. Coast Guard Academy
Phillip J. Cornwell
Rose-Hulman Institute of Technology
Brian P. Self
California Polytechnic State University—San Luis Obispo
Brief Contents
1 Introduction 1
2 Statics of Particles 16
3 Rigid Bodies: Equivalent Systems of Forces 83
4 Equilibrium of Rigid Bodies 170
5 Distributed Forces: Centroids and Centers of Gravity 232
6 Analysis of Structures 299
7 Internal Forces and Moments 368
8 Friction 431
9 Distributed Forces: Moments of Inertia 485
10 Method of Virtual Work 575
11 Kinematics of Particles 615
12 Kinetics of Particles: Newton’s Second Law 721
13 Kinetics of Particles: Energy and Momentum Methods 799
14 Systems of Particles 920
15 Kinematics of Rigid Bodies 982
16 Plane Motion of Rigid Bodies: Forces and Accelerations 1115
17 Plane Motion of Rigid Bodies: Energy and Momentum Methods 1192
18 Kinetics of Rigid Bodies in Three Dimensions 1279
19 Mechanical Vibrations 1350
Appendix: Fundamentals of Engineering Examination A1
Answers to Problems AN1
Index I1
Properties of Geometric Shapes I17
ix
Preface xv
Guided Tour xix
Digital Resources xxiii
Acknowledgments xxv
List of Symbols xxvii
1 Introduction 1
1.1 What is Mechanics? 2
1.2 Fundamental Concepts and Principles 3
1.3 Systems of Units 5
1.4 Converting between Two Systems of Units 10
1.5 Method of Solving Problems 11
1.6 Numerical Accuracy 15
2 Statics of Particles 16
2.1 Addition of Planar Forces 17
2.2 Adding Forces by Components 29
2.3 Forces and Equilibrium in a Plane 38
2.4 Adding Forces in Space 54
2.5 Forces and Equilibrium in Space 67
Review and Summary 76
Review Problems 80
3 Rigid Bodies: Equivalent Systems
of Forces 83
3.1 Forces and Moments 85
3.2 Moment of a Force about an Axis 105
3.3 Couples and Force-Couple Systems 119
3.4 Simplifying Systems of Forces 138
Review and Summary 162
Review Problems 167
Contentsx Contents
4 Equilibrium of Rigid Bodies 170
4.1 Equilibrium in Two Dimensions 173
4.2 Two Special Cases 199
4.3 Equilibrium in Three Dimensions 207
Review and Summary 227
Review Problems 229
5 Distributed Forces: Centroids and
Centers of Gravity 232
5.1 Planar Centers of Gravity and Centroids 234
5.2 Further Considerations of Centroids 250
5.3 Additional Applications of Centroids 262
5.4 Centers of Gravity and Centroids of Volumes 276
Review and Summary 293
Review Problems 297
6 Analysis of Structures 299
6.1 Analysis of Trusses 301
6.2 Other Truss Analyses 319
6.3 Frames 334
6.4 Machines 350
Review and Summary 363
Review Problems 365
7 Internal Forces and Moments 368
7.1 Internal Forces in Members 369
7.2 Beams 379
7.3 Relations Among Load, Shear, and Bending Moment 392
*7.4 Cables 407
*7.5 Catenary Cables 419
Review and Summary 426
Review Problems 429
8 Friction 431
8.1 The Laws of Dry Friction 433
8.2 Wedges and Screws 453
*8.3 Friction on Axles, Disks, and Wheels 462Contents xi
8.4 Belt Friction 471
Review and Summary 480
Review Problems 482
9 Distributed Forces: Moments of
Inertia 485
9.1 Moments of Inertia of Areas 487
9.2 Parallel-Axis Theorem and Composite Areas 497
*9.3 Transformation of Moments of Inertia 516
*9.4 Mohr’s Circle for Moments of Inertia 526
9.5 Mass Moments of Inertia 533
*9.6 Additional Concepts of Mass Moments of Inertia 553
Review and Summary 568
Review Problems 573
10 Method of Virtual Work 575
*10.1 The Basic Method 576
*10.2 Work, Potential Energy, and Stability 596
Review and Summary 610
Review Problems 613
11 Kinematics of Particles 615
11.1 Rectilinear Motion of Particles 617
11.2 Special Cases and Relative Motion 638
*11.3 Graphical Solutions 654
11.4 Curvilinear Motion of Particles 665
11.5 Non-Rectangular Components 692
Review and Summary 713
Review Problems 717
12 Kinetics of Particles:
Newton’s Second Law 721
12.1 Newton’s Second Law and Linear Momentum 723
12.2 Angular Momentum and Orbital Motion 767
*12.3 Applications of Central-Force Motion 778
Review and Summary 792
Review Problems 796
*Advanced or specialty topicsxii Contents
13 Kinetics of Particles: Energy and
Momentum Methods 799
13.1 Work and Energy 801
13.2 Conservation of Energy 830
13.3 Impulse and Momentum 858
13.4 Impacts 883
Review and Summary 910
Review Problems 916
14 Systems of Particles 920
14.1 Applying Newton’s Second Law and Momentum Principles
to Systems of Particles 922
14.2 Energy and Momentum Methods for a System of
Particles 940
*14.3 Variable Systems of Particles 956
Review and Summary 975
Review Problems 979
15 Kinematics of Rigid Bodies 982
15.1 Translation and Fixed-Axis Rotation 985
15.2 General Plane Motion: Velocity 1002
15.3 Instantaneous Center of Rotation 1023
15.4 General Plane Motion: Acceleration 1037
15.5 Analyzing Motion with Respect to a Rotating Frame 1056
*15.6 Motion of a Rigid Body in Space 1073
*15.7 Motion Relative to a Moving Reference Frame 1090
Review and Summary 1105
Review Problems 1111
16 Plane Motion of Rigid Bodies: Forces
and Accelerations 1115
16.1 Kinetics of a Rigid Body 1117
16.2 Constrained Plane Motion 1152
Review and Summary 1186
Review Problems 1188Contents xiii
17 Plane Motion of Rigid Bodies: Energy
and Momentum Methods 1192
17.1 Energy Methods for a Rigid Body 1194
17.2 Momentum Methods for a Rigid Body 1222
17.3 Eccentric Impact 1245
Review and Summary 1271
Review Problems 1275
18 Kinetics of Rigid Bodies in Three
Dimensions 1279
18.1 Energy and Momentum of a Rigid Body 1281
*18.2 Motion of a Rigid Body in Three Dimensions 1300
*18.3 Motion of a Gyroscope 1323
Review and Summary 1341
Review Problems 1346
19 Mechanical Vibrations 1350
19.1 Vibrations without Damping 1352
19.2 Free Vibrations of Rigid Bodies 1368
19.3 Applying the Principle of Conservation of Energy 1382
19.4 Forced Vibrations 1393
19.5 Damped Vibrations 1407
Review and Summary 1424
Review Problems 1429
Appendix: Fundamentals of Engineering Examination A1
Answers to Problems AN1
Index I1
Properties of Geometric Shapes I17
Index
A
Absolute acceleration, 1038
Absolute velocity, 1003
Acceleration
absolute, 1038
angular
constrained (plane) motion, 1154–1155
fixed-axis rotation, 987–988, 994
average, 618–619, 666
Coriolis
motion with respect to a rotating frame,
985, 1059–1060, 1066
three-dimensional (space) motion,
1091, 1097
two-dimensional (planar) motion,
1059–1060, 1066
of particles, 621–622
curvilinear motion and, 666–667
determining, 618–619
instantaneous, 618, 666
radial and transverse components of, 696
rectangular components of, 669–670
rectilinear motion and, 617–631
tangential and normal components of,
692–694, 702
relative, 671, 1037–1038, 1046–1047
of rigid bodies, 1037–1047
constrained (plane) motion, 1056–1057,
1170–1171
moving frames of reference, 1091–1092,
1097–1098
normal components, 1037–1039
plane motion, 1037–1047
tangential and normal components of,
693, 702
tangential components, 1037–1039
three-dimensional (space) motion, 1074,
1076, 1080–1081, 1091–1092,
1097–1098
two-dimensional (planar) motion, 994
vector polygons for determination of, 1038
two-dimensional (planar) motion, 987–988,
1037–1047
vector, 666–667, 669–670
Addition
of couples, 122
of forces, 4, 32–33
parallelogram law for, 4, 19
polygon rule for, 20
summing x and y components, 32–33
triangle rule for, 19–20
of vectors, 19–21
Amplitude, 1351, 1354, 1361
Analysis. See Structural analysis
Angles
Eulerian, 1323–1324, 1330
firing, 673, 678
formed by two vectors, 106, 112
of friction, 435–436
lead, 454
phase, 1354, 1361
of repose, 436
Angular acceleration
constrained (plane) motion, 1154–1155
fixed-axis rotation, 988, 994
Angular coordinate, 987, 994
Angular moment couple, 1223
Angular momentum
about a mass center, 926–928, 932, 1119,
1282–1283, 1291
central force and, 768–769, 773
conservation of, 769, 928, 946, 950
equations for, 924–925
Newton’s law of gravitation for, 769–770
orbital motion and, 767–773
of particle motion, 722, 767–773
in polar coordinates, 768
rate of change
of a particle, 767–768
of rigid bodies in plane motion, 1119
of three-dimensional rigid bodies,
1300–1301, 1312–1313
of a rigid body in plane motion, 1118–1119
of systems of particles, 924–928, 932, 946, 950
three-dimensional rigid bodies, 1281–1285,
1291, 1300–1301
about a fixed point, 1284–1285, 1291–1292
about a mass center, 1282–1283, 1291
inertia tensor, 1283
principle axes of inertia, 1283, 1291
reduction of particle moments, 1284
vector forms, 767
Angular velocity, 987, 994, 1005, 1046
Apogee, 781
Arbitrary shaped bodies, moments of inertia of,
555–557, 560
Area
centroid of common, 240
composite, 241
first moment of, 233, 237–240, 245, 251
integration
centroids determined by, 250–251
moments of inertia determined by,
488–489, 494
moment of inertia, 487–494, 497–509, 516–522
of common geometric shapes, 499
for composite areas, 497–509
for hydrostatic force system, 487, 508
polar moments, 489, 494
principle axis and moments, 517–519, 522
product of inertia, 516–517, 522
for a rectangular area, 488–489
second moments, 487–494
transformation of, 516–522
using same strip elements, 489
radius of gyration, 489–490
theorems of Pappus-Guldinus, 251–252
two-dimensional bodies, 233, 237–240, 245
units of, 7–8
Areal velocity, 769
Average acceleration, 618–619, 666
Average power, 807
Average velocity, 618, 665
Axel friction, 462–463, 467
Axes
moments of a force about, 84, 105–109
arbitrary point for, 109
given origin for, 107–109
mixed triple products, 106–107
scalar products, 105–106
neutral, 487
principle axis and moments of inertia
about the centroid, 519
of an area, 517–519, 522
for a body of arbitrary shape, 555–557, 560
ellipsoid of inertia, 554–556
of a mass, 555–557, 561
Axisymmetric body analysis, 1324, 1326–1327,
1331–1332
B
Balance, 1304
Ball and socket supports, 209
Basic units, 5
Beams
bending moment in, 380–381, 386, 392–401
centroids of, 262–263, 270
classification of, 379
loading conditions
concentrated, 262, 379
distributed, 262–263, 270, 379
internal forces and, 369, 379–387
uniformly distributed, 379
pure bending, 487
shear and bending moment diagrams for,
382, 387
shearing forces, 380–381, 386, 392–401
span, 379–380
Belt friction, 471–475
Bending moments, 371–374
beams, 380–381, 386, 392–401
diagrams for, 382, 387
external forces and, 381
internal forces as, 369, 371–374
shearing force relations with, 393
Binormal, 694
Body centrode, 1025
Body cone, 1074
Bracket supports, 209
C
Cables, 369, 407–414, 419–423
catenary, 419–423
internal forces of, 369, 407–414
parabolic, 409–410, 414I2 Index
Cables—Cont.
sag, 409
solutions for reactions, 413–414, 423
span, 409
supporting concentrated loads, 407–408, 413
supporting distributed loads, 408, 414, 419–423
supporting vertical loads, 407–408, 413
Cable supports, 174, 209
Catenary cables, 419–423
Center of force, 768
Center of gravity, 85
composite area, 241
composite bodies, 278
composite plate, 240–241
location of, 234–235
problem solving with, 240–249
three-dimensional bodies, 276–278, 285
two-dimensional bodies, 233–235
Center of pressure, 263
Central-force motion, 723, 768–769, 773,
778–786
angular momentum of a particle,
768–769, 773
applications of, 778–786
of particles, 723, 768–769, 773, 778–786
space mechanics, 779–782
eccentricity, 779–780
escape velocity, 780
gravitational force, 779–780
initial conditions, 780–781
Kepler’s laws of planetary motion, 782
periodic time, 781–782
trajectory of a particle, 778–779
Central impact, 883, 899
Centrifugal force, 1154
Centrodes, 1025
Centroid, 233, 235–237
of areas, 235–237, 240, 245, 250–251, 262–269
of common shapes, 239–240, 279, B3
distributed load problems using, 262–269
integration for determination of,
250–251, 279
of lines, 235–237, 240, 245
location of, 235–237, 245
theorems of Pappus-Guldinus, 251–252
three-dimensional bodies, 276–278
two-dimensional bodies, 235–237, 239–240
of volume, 276–278
Centroidal frame of reference, 926, 940
Centroidal rotation, 1120, 1134
Circle of friction, 463, 467
Circular orbits, 771–773
Coefficients
of critical damping, 1407, 1412
of friction, 434–435, 465, 467
impact analysis, 884–886, 888, 899–900
of restitution, 884–886, 888, 899–900
vibration analysis, 1407, 1412
of viscous damping, 1407
Collar bearings, 463–464
Commutative property, 88, 105
Complimentary function, 1394
Composite bodies, 278–280
center of gravity of, 278
centroid of, 278–280
mass moment of inertia of, 537–544, 560
Composite plates and wires, 240–244
Compound truss, 320
Compressibility of fluids, 2
Compression, deformation from, 87
Concentrated loads
beams, 262, 379
cables supporting, 407–408, 413
Concurrent forces
resultants, 21, 58
system reduction of, 140
Connections, 173–175. See also Support reactions
Conservation of angular momentum, 769, 946,
950, 1225, 1233
Conservation of energy
conservative forces, 832–835, 843
energy conversion and, 834
kinetic energy, 941, 950
in particle motion, 830–844
potential energy, 830–832, 843–844
principle of, 833–834
in rigid-body plane motion, 1197–1199, 1210
space mechanics applications, 834–835
in systems of particles, 941, 950
vibration applications of, 1382–1386
Conservation of momentum, 859
angular, 769, 946, 950, 1225, 1233
direct central impact and, 883–884, 899
linear, 859, 946, 950
oblique central impact and, 886, 900
particle motion, 859
rigid-body plane motion, 1225, 1233
systems of particles, 928, 932, 946, 950
Conservative forces
exact differential, 832
potential energy of, 599, 832–833
space mechanics applications, 834–835
work of, 832
Constant force, work of in rectilinear motion,
802, 818
Constant of gravitation, 770
Constrained (plane) motion, 1152–1171
acceleration, 1056–1057, 1170–1171
angular acceleration, 1154–1155
free-body and kinematic diagrams for,
1152–1153, 1170
moments about a fixed axis, 1154, 1170
noncentroidal rotation, 1153–1154, 1170
rolling, 1154–1155, 1171
sliding and, 1154–1155, 1171
system of rigid bodies, 1171
unbalanced rolling disk or wheel, 1155, 1171
Constraining forces, 173, 177–178, 208
completely constrained, 177
free-body diagram reactions, 173
improperly constrained, 178, 208
partially constrained, 177, 208
three-dimensional rigid bodies, 208
two-dimensional rigid bodies, 177–178
Conversion
of energy, 834
of units, 10–11
Conveyor belt, fluid stream diversion by,
957, 964
Coplanar forces
resultants, 20
system reduction of, 141–142
Coplanar vectors, 20
Coriolis acceleration, 985
motion with respect to a rotating frame, 985,
1059–1060, 1066
three-dimensional (space) motion, 1091, 1097
two-dimensional (planar) motion, 1059–1060,
1066
Coulomb friction. See Dry friction
Couples, 119–129
addition of, 122
angular moment, 1223
equivalent, 120–122
force-couple system resolution, 122
moment of, 119–120
work of, 578
Critical damping coefficient, 1407, 1412
Critically damped vibration, 1408, 1412
Cross products. See Vector products
Curvilinear motion of particles, 665–679
acceleration vectors, 666–667, 669–671
derivatives of vector functions, 667–669
firing angle, 673, 678
frame of reference, 669–671
position vectors, 665, 671
projectile motion, 669–670, 672–674, 678
rate change of a vector, 668–669
rectangular components, 669–670
relative-motion problems, 670–671, 675–678
rotation compared to, 984
two-dimensional problems, 678
velocity vectors, 665–666, 669–671
Cylindrical coordinates for radial and transverse
components, 696, 703
D
Damped circular frequency, 1408
Damped vibration, 1352, 1407–1418
critically, 1408, 1412
electrical analogs, 1411–1413
forced vibration, 1409–1411, 1413
free vibration, 1407–1408, 1412–1413
friction causes of, 1407
magnification factor, 1410–1411, 1413
overdamped, 1408, 1412
period of, 1408–1409
phase difference, 1410
underdamped, 1408, 1412
Damping factor, 1408
Deceleration, 619
Definite integrals, 621
Deformable bodies, mechanics of, 2
Deformation, 87
from impact, 883–884, 1245
internal forces and, 87
principle of transmissibility for prevention
of, 87
Degrees of freedom, 601, 640
Dependent motion of particles, 640, 646–647
Derived units, 5
Dick clutch, 467
Direct central impact, 883–886, 899
coefficient of restitution, 884–886, 899
conservation of momentum and, 883–884, 899
deformation from, 883–884
energy loss from, 885–886
perfectly elastic, 885
perfectly plastic, 885
period of restitution, 883–884
Direct impact, 883
Direction cosines, 55, 56Index I3
Direction of a force, 17, 31. See also Line of
action
Disk friction, 463–464, 467
Displacement
finite, 596–598, 802, 804
from mechanical vibration, 1351
of a particle, 577
vertical, 803
virtual, 579, 588
work of a force, 579, 588, 596–598, 801–804
Displacement vector, 665
Distributed forces, 232–298. See also Centroid
beam loads, 262–263, 270, 379
cables supporting loads, 408, 414, 419–423
concentrated load and, 262
integration methods for centroid location,
250–257, 279, 285
moments of inertia, 485–574
of areas, 487–494, 497–509
polar, 486, 489, 494
transformation of, 516–522
submerged surfaces, 263, 270–271
theorems of Pappus-Guldinus, 251–252
three-dimensional bodies, 276–284
center of gravity, 276–278, 285
centroid of volume location, 276–278, 285
composite bodies, 278–280
two-dimensional bodies, 233–244
center of gravity, 234–235, 245
centroid of area and line location,
235–237, 239–240, 245
composite plates and wires, 240–244
first moment of an area or line,
237–240, 245
planar elements, 233–244
Distributive property, 88
Dot product, 105. See also Scalar product
Double integration, 250, 280
Dry friction, 432–445
angles of, 435–436
coefficients of, 434–435
kinetic friction force, 433–434
laws of, 433
problems involving, 436–445
static friction force, 433–434
E
Eccentric impact, 883, 1245–1261
Eccentricity, 779–780
Efficiency
mechanical, 581–582, 808
overall, 808
power and, 807–808
Elastic force, 597–598, 603. See also Spring force
Elastic potential energy, 831
Electrical analogs, 1411–1413
Ellipsoid of inertia, 554–556
Elliptical orbits, 771–773
Elliptical trajectory, 779–780, 785–786
Elliptic integral, 1357
End bearings, 463–464
Energy and momentum methods, 799–919,
1192–1278
angular momentum, 1281–1285, 1291
conservation of energy
conservative forces, 832–835, 843
in particle motion, 830–844
potential energy, 830–832, 843–844
principle of, 833–834
in rigid-body plane motion, 1197–1199, 1210
space mechanics applications, 834–835
displacement, 801–804
efficiency and, 807–808
friction forces and, 807, 834
impact
conservation of energy and, 889–890, 900
direct central, 883–886, 899
eccentric, 1245–1261
oblique central, 886–888, 899–900
problems involving multiple kinetics
principles, 888–890
impulse and momentum
conservation of angular momentum,
1225, 1233
conservation of linear momentum, 859
impulse of a force, 858–859, 871–872
impulsive motion, 859–860
of particle motion, 858–872
of rigid-body plane motion, 1222–1233
kinetic energy
particle motion, 804–805, 819,
833–834, 843
rigid-body plane motion, 1196–1197, 1209
systems of particles, 940, 950
three-dimensional rigid-body motion,
1286–1287, 1292
particle motion, 799–919
power
from particle motion, 807–808, 819
from rigid-body plane motion, 1199, 1210
principle of impulse and momentum
particle motion, 800, 858–860, 871
rigid-body plane motion, 1222–1224,
1232–1233
three-dimensional rigid-body motion,
1285–1286, 1292
principle of work and energy
particle motion, 800, 804–807
rigid-body plane motion, 1194–1195, 1209
rigid-body plane motion, 1192–1278
systems of particles, 940–950
conservation of energy, 941, 950
conservation of momentum, 946, 950
impulse-momentum principle, 941–942
work-energy principle, 941
systems of rigid bodies, 1124, 1135, 1225, 1233
three-dimensional rigid-body motion,
1281–1292
work of a force
constant force in rectilinear motion,
802, 818
force of gravity, 803, 818
gravitational force, 804, 819
particle motion, 801–819
pin-connected members, 1210
rigid-body plane motion, 1195–1196, 1209
spring force, 803, 818, 1210
Energy conversion, 834
Energy loss from impact, 885–886, 900
Engineering examination, fundamentals of,
A1–A2
Equal and opposite vectors, 19
Equilibrium, 38
equations of, 38–39
force relations and, 17, 38–46
frame determinacy and, 335–336
free-body diagrams for, 39–40, 171–173
neutral, 600
Newton’s first law of motion and, 39
of a particle, 38–39, 67–75
three-dimensional (space) problems, 67–75
two-dimensional (planar) problems, 38–39
principle of transmissibility and, 4
of rigid bodies, 170–231
statically determinate reactions, 177
statically indeterminate reactions,
177–178, 208
support reactions, 173–175, 207–209
three-dimensional structures, 207–215
three-force body, 199–201
two-dimensional structures, 173–183
two-force body, 199
stable, 600–601
unstable, 600–601
virtual work conditions, 599–603
potential energy and, 599–600, 603
stability and, 600–601
Equipollent forces, 1120
Equipollent particles, 923
Equipollent systems, 139–140
Equivalent couples, 120–122
Equivalent systems of forces, 83–169
deformation and, 87
external, 85–86
internal, 85, 87
point of application, 85–86
principle of transmissibility and, 84, 86–87
reduction to force-couple system, 138–139
rigid bodies, 83–169
simplifying, 138–150
weight and, 85–86
Escape velocity, 780
Eulerian angles, 1323–1324, 1330
Euler’s equations for motion, 1073, 1301–1302
Euler’s theorem, 1073
Exact differential, 832
External forces
acting on a rigid body in plane motion,
1119–1120
acting on systems of particles, 921–924
equivalent systems and, 85–86
shear and bending moment conventions, 381
F
Finite rotation, 1074
Firing angle, 673, 678
First moment
of an area or line, 233, 237–240, 245
of volume, 277
Fixed-axis rotation
angular acceleration, 988, 994
angular coordinate, 987, 994
angular velocity, 987, 994
equations for, 989, 995
noncentroidal, 1154, 1170
rigid-body motion, 983–985
shaft balance, 1304
slab representation, 988–989, 994–995
three-dimensional motion analysis,
1303–1304, 1313–1314I4 Index
Fixed (bound) vector, 19
Fixed frame of reference, 670
Fixed point, motion about a, 984
acceleration, 1074, 1080
angular momentum, 1284–1285, 1291–1292
Euler’s theorem for, 1073
instantaneous axis of rotation, 1073–1074
plane motion analysis, 1073–1075, 1080
rate of change of angular momentum,
1300–1301, 1312
three-dimensional motion analysis,
1302–1303, 1313
velocity, 1075, 1080
Fixed supports, 174–175, 209
Fluid friction, 432–433
Fluids
compressibility of, 2
flow through a pipe, 957
stream diversion by a vane, 957, 964
Force. See also Distributed forces; Equivalent
systems of forces; Force systems
centrifugal, 1154
concept of, 3
concurrent, 21, 140
conservative, 598, 832–835, 843
constant in rectilinear motion, 802, 818
constraining, 173, 177–178, 208
conversion of units of, 10
coplanar, 20, 141–142
direction, 17, 31
elastic, 598
equilibrium and, 17, 38–46
equipollent, 1120
equivalent, 86
external, 921–924, 1119–1120
friction, 432–433, 807, 834, 1154
gravitational, 779–780
of gravity, 598, 803, 818, 830
impulsive, 859, 871
input (machines), 350
internal, 921–924
kinetic friction, 433–434
line of action (direction), 18, 57–58
magnitude, 17, 54, 57–58
output (machines), 350
parallel, 142–143
parallelogram law for addition of, 4
particle equilibrium and, 16–82
planar (two-dimensional), 17–53 (See also
Planar forces)
point of application, 17
of rigid-body plane motion, 1119–1124
scalar representation, 19, 20–21
sense of, 18
spring, 803, 818
static friction, 433–434
systems of particles, 921–924
three dimensions of space, 54–75
concurrent force resultants, 58
rectangular components, 54–57
scalar components, 54–55
unit vectors for, 56–57
vector representation, 18–21
weight, 4–5
work of, 577–579, 596–598, 801–819
Force-couple systems, 122
conditions for, 139
equipollent, 139–140
equivalent systems reduced to, 138–139
reactions equivalent to, 174–175
reducing a systems of forces into, 138–139
resolution of a given force into, 122, 123–124
resultant couples, 140–143
wrench, 143–144
Forced circular frequency, 1393
Forced frequency, 1394
Forced vibration, 1351
caused by periodic force, 1393, 1398
caused by simple harmonic motion, 1393, 1398
damped, 1385, 1409–1411
forced circular frequency, 1393
forced frequency, 1394
frequency ratio for, 1394
magnification factor, 1394–1395,
1410–1411, 1413
resonance of the system, 1395, 1398
undamped, 1393–1399
Force systems, 83–169
center of gravity, 85
concurrent, 138
coplanar, 141–142
couples, 119–129
equipollent, 139–140
equivalent, 86–87, 138–150
external forces, 85–86
force-couple, 122
internal forces, 86–87, 300–301
moment about an axis, 84
moment about a point, 84
parallel, 142–143
point of application, 85–86
position vectors defining, 90, 138
reducing into force-couple system, 138–139
resolution of a given force into force-couple
system, 122, 123–124
simplifying, 138–150
virtual work application to connected rigid
bodies, 579–581
weight and, 85–86
Force triangle, 40
Frame of reference, 669–671
centroidal, 926, 940
fixed, 670
general motion, 1091–1092, 1098
motion relative to, 670–671
moving, 670, 1090–1098
newtonian, 724
rate of change of a vector, 668, 1056–1057,
1066
relative position, velocity, and
acceleration, 671
rotating, 1056–1066, 1090–1091, 1097
three-dimensional particle motion,
1090–1091, 1097
translation of, 668, 671
Frames, 301, 334–339
collapse of without supports, 335–336
equilibrium of forces, 335–336
free-body diagrams of force members, 334–335
multi-force members, 301, 334
statically determinate and rigid, 336
statically indeterminate and nonrigid, 336
Free-body diagrams, 12
equilibrium
particle force, 39–40
rigid-body force, 171–173
frame analysis using, 334–335
machine analysis of members, 350, 353
particle motion, 727–728, 746
rigid-body constrained motion, 1152–1153,
1170
rigid-body plane motion, 1122–1124, 1134
truss analysis of joint forces, 304
two-dimensional problems, 39–40,
171–173
Free vector, 19
Free vibration, 1351
damped, 1407–1409, 1412–1413
pendulum motion, 1355–1357, 1361
of rigid bodies, 1368–1374
simple harmonic motion with, 1352–1361
undamped, 1352–1361, 1368–1374
Frequency, 1351
damped circular, 1408
forced, 1394
forced circular, 1393
natural, 1355, 1360
natural circular, 1353, 1361
units of, 1355
Frequency ratio, 1394
Friction, 431–484
angles of, 435–436
axel, 462–463, 467
belt, 471–475
circle of, 463
coefficients of, 434–435
disk, 463–464, 467
dry, 432–445
fluid, 432–433
forces of, 432–433
journal bearings, 462–463, 467
lubrication and, 433, 462
potential energy of, 834
rolling resistance, 464–465, 467
screws, 453–454, 457
sliding and, 1154
slipping and, 472–473, 475
thrust bearings, 462, 463–464, 467
vibration caused by, 1407
wedges, 453, 457
wheel, 464–465, 467
work of, 807
Frictionless pins, 173–174
Frictionless surface supports, 174, 209
Fundamental of Engineering Exam, A1–A2
G
General motion, 984
about a fixed point, 1075–1076, 1081
acceleration, 1076, 1081
relative to a moving frame of reference,
1091–1092, 1098
velocity, 1075, 1081, 1091–1092, 1098
General plane motion. See Plane motion
Gradient of the scalar function, 833
Graphical solutions, 654–657
Gravitational force, work of, 804, 819
Gravitational units, 8–9
Gravity (weight)
constant of, 770
force of, 803, 818
Newton’s law of, 769–770Index I5
potential energy with respect to, 598, 603,
830–831
work of, 597
Gyroscopes, 1323–1332
axisymmetric body analysis, 1324,
1326–1327, 1331–1332
Eulerian angles of, 1323–1324, 1330
steady precession of, 1324–1326, 1330–1331
three-dimensional analysis of motion,
1323–1332
H
Harmonic motion, 1352–1361
Homogeneous equation, 1393
Hydraulics, 2
Hydrostatic force system, 487, 508
Hyperbolic trajectory, 779–780, 785–786
I
Impacts
central, 883, 899
coefficient of restitution, 884–886, 888,
899–900, 1246
direct, 883
direct central, 883–886, 899
eccentric, 883, 1245–1261
energy loss from, 885–886, 900
line of, 883
oblique, 883
oblique central, 886–888, 899–900
particle motion, 883–900
problems involving multiple kinetics
principles, 888–890
rigid-body plane motion, 1245–1261
Impending motion, 434–435, 444
sliding, 1154
slip, 472
Impulse, 858–872
eccentric impact and, 1245–1261
of a force, 858–859, 871–872, 1261
linear, 858
momentum and, 858–872
principle of impulse and momentum, 800,
858–860, 871
time interval, 871
units of, 858–859
Impulse-momentum diagram, 859, 871,
886–888, 899, 1261
Impulse-momentum principle, 860, 941–942
Impulsive force, 859, 871, 1261
Impulsive motion, 859–860, 1261
Inertia, principle axes and moments of, 1283,
1291, 1312
Inertia tensor, 1283
Inextensible cord, work of forces exerted on,
1210
Infinitesimal rotation, 1074
Initial conditions, 621, 646
Input forces, 350, 581
Instantaneous acceleration, 618, 666
Instantaneous axis of rotation, 1073–1074
Instantaneous center of rotation, 985,
1023–1030
Instantaneous velocity, 618, 665
Integration
centroids determined by
of an area, 250–251
of volume, 280
definite integrals, 621
double, 250, 280
moments of inertia determined by
of an area, 488–489, 494
for a body of revolution, 537, 544
of a mass, 537, 544
for a rectangular area, 488–489
for a three-dimensional body, 537
using the same elemental strips, 489
motion determined by, 621–622
theorems of Pappus-Guldinus applied to,
251–252
triple, 280
Internal forces, 85, 368–430
acting on systems of particles, 921–924
axial forces as, 370, 371
beams, 369, 379–387
bending moments, 369, 371, 380–381, 386,
392–401
cables, 369, 407–414, 419–423
in compression, 87, 369
deformation and, 87
equivalent systems and, 85, 87
loadings, 379–380, 392–401, 407–408
in members, 369–374
principle of transmissibility for equilibrium
of, 87
relations among load, shear, and bending
moments, 392–401
rigid bodies, 85, 87
shear and bending moment diagrams for,
382, 387
shearing forces as, 369, 371, 380–381, 386,
392–401
structural analysis and, 300–301
in tension, 87, 369
International System of Units (SI), 5–7
J
Jet engines, steady stream of particles from,
958, 964
Joints under special loading conditions,
306–308
Journal bearings, axel friction of, 462–463, 467
K
Kepler’s laws of planetary motion, 782, 786
Kinematics, 616
Coriolis acceleration, 985, 1059–1060, 1066,
1091, 1097
degrees of freedom, 640
graphical solutions for, 654–657
initial conditions for, 621, 646
of particles, 615–720
curvilinear motion, 665–679
dependent motion, 640, 646–647
independent motion, 639–640, 646
non-rectangular components, 692–703
rectilinear motion, 617–631
relative motion, 638–647
solutions for motion problems, 631–632,
646–647
three-dimensional (space) motion, 694
two-dimensional (planar) motion,
692–694
uniform rectilinear motion, 638–639
of rigid bodies, 982–1114
acceleration of, 1037–1047, 1074, 1076,
1080–1081, 1091–1092, 1097–1098
general motion, 984, 1075–1076, 1081,
1091–1092, 1098
general plane motion, 984–985,
1002–1014, 1037–1047
instantaneous center of rotation, 985,
1023–1030
motion about a fixed point, 984,
1073–1075, 1080
moving frames, motion relative to,
1090–1098
rotating frames, motion relative to,
1056–1066
rotation about a fixed axis, 983–995
three-dimensional (space) motion, 985,
1073–1081, 1090–1098
translation, 983, 985–986, 995
two-dimensional (planar) motion,
983–1066
velocity of, 1002–1014, 1023–1030, 1075,
1080–1081, 1092, 1097–1098
Kinetic diagrams
particle motion, 727–728, 746
rigid-body constrained motion, 1152–1153,
1170
rigid-body plane motion, 1119–1120, 1134
Kinetic energy
of a particle
principle of conservation of energy,
833–834, 843
principle of work and energy, 804–805, 819
of rigid-body plane motion
body in translation, 1196, 1209
noncentroidal rotation, 1196–1197
of systems of particles
centroidal frame of reference for, 940
conservation of energy, 941, 950
loss of in collisions, 950
work-energy principle, 941
of three-dimensional rigid-body motion
with a fixed point, 1287, 1292
with respect to the mass center,
1286–1287, 1292
Kinetic friction force, 433–434
Kinetics, 616
free-body and kinetic diagrams for, 727–728,
746, 1122–1124, 1134
constrained motion, 1152–1153, 1170
Newton’s second law and, 727–728, 746
plane motion, 1119–1120, 1134
of particles, 721–798
angular momentum, 722, 767–773
central-force motion, 723, 768–769, 773,
778–786
energy and momentum methods, 799–919
Kepler’s laws of planetary motion, 782, 786
motion of, 721–798
multiple principles, problems involving,
888–890
Newton’s law of gravitation for, 769–770I6 Index
Kinetics—Cont.
Newton’s second law for, 723–747, 890
principle of impulse and momentum, 800,
858–860, 871, 890
principle of work and energy, 800,
804–807, 890
of rigid bodies, 1115–1191, 1279–1349
angular momentum of, 1118–1119
centroidal rotation, 1120, 1134
constrained (plane) motion, 1152–1171
forces of, 1119–1120
general plane motion, 1120–1121, 1134
noncentroidal rotation, 1153–1154, 1170
plane motion of, 1115–1191
principle of transmissibility and, 1121
rolling, 1154–1155, 1171
systems of, 1124, 1135
three-dimensional motion, 1279–1349
translation, 1120, 1134
Kinetic units, 5–6
L
Lead and lead angle, 454, 457
Length, conversion of units of, 10
Linear momentum
conservation of, 859, 928, 946, 950
equations for, 924–925, 932
particle motion, 859
systems of particles, 924–925, 928, 932,
946, 950
Linear momentum vector, 1223
Line of action, 18
force direction representation, 4, 18, 57–58
magnitude and, 3–4, 17
moment of a force, 91
particles, 17–18, 56–58
planar (two dimensional) force, 18
reactions with known, 173
rigid bodies, 91, 173–175
three-dimensional (space) force, 56–58
unit vector along, 56–57
Line of impact, 883
Lines
centroid of common, 240
first moment of, 233, 237–240, 245
two-dimensional bodies, 233, 237–240, 245
Loading conditions
beams, 262–263, 270, 369, 379–387
cables, 369, 407–414, 419–423
center of pressure, 263
centroid of the area, 262–269
concentrated, 262, 379, 407–408, 413
distributed, 262–270, 379, 408, 414, 419–423
relations with shear and bending moments,
392–401
submerged surfaces, 263, 270–271
uniformly distributed, 379
Lubrication, friction and, 433, 462
M
Machines
free-body diagrams of members, 350, 353
input forces, 350
mechanical efficiency of, 581–582
multi-force members, 301, 334
output forces, 350
structural analysis of, 301, 334, 350–352
Magnification factor
damped vibration, 1410–1411, 1413
undamped vibration, 1394–1395
Magnitude, speed as, 618, 665
Magnitude of a force
force characteristics, 3–4, 17–18
line of action and, 4, 18, 57–58
moments of a force, 90–93
particles, 17–18, 54, 57–58
reactions with unknown direction and,
173–174
rigid bodies, 90–93
units of, 70
vector characteristics, 54
Mass
concept of, 3
conversion of units of, 10–11
gain and loss effects on thrust, 959, 964–965
moments of inertia, 487, 533–544
of common geometric shapes, 499, 538
of composite bodies, 537–544, 560
integration used to determine, 537, 544
parallel-axis theorem for, 534–535, 543
of simple mass, 533–534
of thin plates, 536–537
principle axis and moments, 555–557, 561
product of inertia, 553–554, 560
Mass center
angular momentum about
rigid bodies in plane motion, 1119
rigid bodies in three-dimensional motion,
1282–1283, 1291
systems of particles, 926–928, 932
center of gravity compared to, 925
centroidal frame of reference, 926
equations for, 925–926, 932
projectile motion and, 926
systems of particles, 921, 925–928, 932
Mechanical efficiency of machines,
581–582, 808
Mechanical energy, 833–834
Mechanical vibration, 1351. See also Vibration
conservation of energy applications,
1382–1386
electrical analogs, 1411–1413
pendulum motion, 1355–1357, 1361
approximate solution, 1355–1356
exact solution, 1356–1357
oscillations, 1356, 1361
of rigid bodies, 1368–1374
system displacement as, 1351
Mechanics
conversion of units, 10–11
of deformable bodies, 2
of fluids, 2
fundamental concepts and principles, 3–5
method of solving problems, 11–13
newtonian, 3
numerical accuracy, 15
of particles, 3–4
relativistic, 3
of rigid bodies, 2, 4
role of statics and dynamics in, 2
study of, 2–3
systems of units, 5–11
Members
axial forces in, 370, 371
free-body diagrams of, 334–335
internal forces in, 369–374
machine analysis of, 350, 353
multi-force, 301, 334, 370–371
redundant, 320
shearing force in, 371
two-force, 301, 371
zero-force, 307
Method of joints, 304–306
Method of sections, 319–326
Mixed triple products of vectors, 106–107
Mohr’s circle for moments of inertia, 526–530
Moment arm, 91. See also Line of action
Moments of a force
about an axis, 84, 105–109
angles formed by two vectors, 106, 112
arbitrary point for, 109
given origin for, 107–109
mixed triple products, 106–107
perpendicular distance between lines, 108,
112–113
projection of a vector for, 106, 112
scalar products, 105–106
about a point, 84, 90–93
line of action (moment arm), 91
magnitude of, 90–93
position vector of, 90
rectangular components of, 93–94
right-hand rule for, 90
three-dimensional problems, 93–94, 99
two-dimensional problems, 92–94, 99
Varignon’s theorem for, 93
vector products, 90
of a couple, 119–120
Moments of inertia, 485–574
of arbitrary shaped bodies, 555–557, 560
of areas, 487–494, 497–509
of common geometric shapes, 499, B4
for composite areas, 497–509
for hydrostatic force system, 487, 508
integration used to determine, 488–489, 494,
537, 544
of masses, 487, 533–544
of common geometric shapes, 499,
538, B4
for composite bodies, 537–544, 560
of simple mass, 533–534
for thin plates, 536–537, 543
Mohr’s circle for, 526–530
neutral axis, 487
parallel-axis theorem for, 497–509, 517,
534–535, 543, 554
polar, 486, 489, 494
radius of gyration, 489–490, 494, 534
second moment as, 486–487, 494
transformation of, 516–522, 553–561
ellipsoid of inertia, 554–556
mass product of inertia, 553–554, 560
principle axis and moments, 517–519, 522,
555–557, 561
product of inertia, 516–517, 522
unit-related errors, 543
Momentum, 858–872. See also Principle of
impulse and momentum
angular, 924–928, 932
angular couple, 1223Index I7
conservation of, 859, 883–884, 899, 928, 932
direct central impact and, 883–884, 899
impulse and, 858–872
impulsive force of, 859, 871
linear, 883–884, 924–925, 932
linear vector, 1223
particle motion, 858–872
rigid-body plane motion, 1222–1224,
1232–1233
systems of particles, 922–932
total, 859, 871–872
Motion, 39, 85–86
equations of
Euler’s, 1073, 1301–1302
particle kinetics, 727–729, 746
radial and transverse components,
730, 746
rectangular components, 728–729, 746
rigid-body kinetics, 1117–1118, 1120, 1134,
1280
rotational, 1117
scalar form, 1120
tangential and normal components,
730, 746
translational, 1117
external forces and, 85–86
free-body and kinetic diagrams for,
727–728, 746
impending, 434–435, 444, 472
kinematics of a particle, 615–720
curvilinear, 665–679
dependent, 640, 646–647
determination of a particle, 621–622
independent, 639–640, 646
initial conditions for, 631
integration for determination of, 621–622
projectile, 670, 678
rectilinear, 617–631
relative, 638–647
kinematics of rigid bodies, 982–1114
about a fixed point, 984, 1073–1075, 1080
acceleration of, 1037–1047, 1076, 1081,
1091–1092, 1097–1098
general, 984, 1075–1076, 1081, 1091–1092,
1098
instantaneous center of rotation, 985,
1023–1030
moving frames, relative to, 1090–1098
plane, 984–985, 1002–1014, 1037–1047
rotating frames, relative to, 1056–1066
rotation about a fixed axis, 983–995
three-dimensional (space), 985,
1073–1081, 1090–1098
translation, 983, 985–986, 995
two-dimensional (planar), 983–1066
velocity of, 1002–1014, 1023–1030, 1075,
1081, 1092, 1097–1098
kinetics of a particle, 721–798
angular momentum, 722, 767–773
central force, 723, 768–769, 773, 778–786
Newton’s second law for, 723–747
orbital, 767–773
kinetics of rigid bodies, 1115–1191
centroidal rotation, 1120, 1134
constrained, 1152–1171
general plane, 1120–1121, 1134
noncentroidal rotation, 1153–1154, 1170
plane, 1115–1191
rolling, 1154–1155, 1171
sliding, 1154–1155, 1171
three-dimensional (space), 1279–1349
Newton’s first law of, 4, 39
relative, 437
rotation, 86
slipping, 472–473, 475
space mechanics, 779–782, 834–835
under a conservative central force,
834–835
under a gravitational force, 779–780
gyroscopes, 1323–1332
trajectories, 779–782, 786
translation, 86
weight and, 85–86
Moving frame of reference, 670–671
acceleration of, 1091–1092, 1097–1098
Coriolis acceleration, 1091, 1097
in general motion, 1091–1092, 1098
rigid-body motion relative to, 1090–1098
rotating frame, 1090–1091, 1097
three-dimensional particle motion,
1090–1091, 1097
velocity of, 1090–1092, 1097–1098
Multi-force members, 301, 334, 370–371.
See also Frames; Machines
N
Natural circular frequency, 1353, 1361
Natural frequency, 1355, 1360
Neutral axis, 487
Neutral equilibrium, 600, 603
Newtonian frame of reference, 724
Newtonian mechanics, 3
Newton’s laws, 4–5
first law of motion, 4, 39
gravitation, 4–5, 769–770
motion, 4, 39
particles in equilibrium and, 39
second law of motion, 4, 723–747
application of, 731–745
equations of motion, 727–729, 746
free-body and kinetic diagrams for,
727–728, 746
linear momentum and, 723–747
mass and, 723
of multiple forces, 724
radial and transverse components, 730, 746
rectangular components, 728–729, 746
statement of, 723
systems of particles, 922–924
tangential and normal components,
730, 746
third law of motion, 4, 301
Noncentroidal rotation
about a fixed axis, 1154, 1170
of a body in constrained motion, 1153–1154,
1170
kinetic energy of a body in, 1196–1197
principle of impulse and momentum for,
1224
Nonhomogeneous equation, 1393
Nonimpulsive forces, 1261
Nonrigid truss, 320
Normal components. See Tangential and normal
components
Numerical accuracy, 15
Nutation, rate of, 1323
O
Oblique impact, 883
central impact, 886–888, 899–900
coefficient of restitution, 888, 900
conservation of momentum and, 886, 900
impulse-momentum diagrams for,
886–888, 899
Orbital motion, 767–773. See also Angular
momentum
Oscillations, 1356, 1361, 1369
Osculating plane, 694
Output forces, 350, 581
Overdamped vibration, 1408, 1412
Over rigid truss, 320
P
Pappus-Guldinus, theorems of, 251–252
Parabolic cables, 409–410, 414
Parabolic trajectory, 670, 779–780, 785–786
Parallel-axis theorem
composite area application of, 497–509
for mass moments of inertia, 534–535, 543
for mass product of inertia, 554
for moments of inertia of an area, 497–509
for product of inertia, 517
Parallel forces, reduction of system of, 142–143
Parallelogram law, 4
addition of forces, 4
addition of two vectors, 19
resultant of two forces, 18
Particle moments, reduction of in
three-dimensional motion, 1284
Particles, 3–4. See also Systems of particles
direction of a force, 17, 31
displacement of, 577
equipollent, 923
kinematics of, 615–720
curvilinear motion, 665–679
dependent motion, 640, 646–647
independent motion, 639–640, 646
non-rectangular components, 692–703
radial and transverse components,
694–696, 698–701, 703
rectilinear motion, 617–631
relative motion, 639–640
relative to a rotating frame, 1090–1091, 1097
solutions for motion problems, 631–632,
646–647
tangential and normal components
three-dimensional (space) motion,
694, 1090–1091, 1097
two-dimensional (planar) motion, 692–694
uniform rectilinear motion, 638–639
kinetics of, 721–798
angular momentum, 722, 767–773
central-force motion, 723, 768–769, 773,
778–786
linear momentum, 722
mass, 722
Newton’s second law for, 723–747
resultant of forces, 722I8 Index
Particles—Cont.
line of action (direction), 18, 57–58
magnitude of force, 17, 54, 57–58
mechanics of, 3–4
resultant of forces, 17–18, 21
scalars for force representation, 19, 20–21,
29–30
statics of, 16–82
three-dimensional (space) problems, 54–75
adding forces in space, 54–66
concurrent force resultants, 58
direction cosines for, 55, 56
equilibrium of, 38–46
force defined by magnitude and two
points, 57–58
rectangular components, 54–57
two-dimensional (planar) problems, 17–53
adding forces by components, 29–37
concurrent force resultants, 21
equilibrium of, 38–46
free-body diagrams, 39–40
Newton’s first law of motion for, 39
planar forces in, 17–28
rectangular components, 29–31
resolving several forces into two
components, 32–33
unit vectors for, 29–31, 56–57
vectors for force representation, 18–21, 29
Path-independent forces. See Conservative
forces
Pendulum motion
approximate solution, 1355–1356
exact solution, 1356–1357
impact, 889–890
oscillations, 1356, 1361
vibration, 1355–1357, 1361
Perfectly elastic impact, 885
Perfectly plastic impact, 885
Perigee, 781
Periodic function, 1353
Periodic time, 781–782, 786
Period of restitution, 883–884, 1245
Period of vibration, 1354
correction factor for, 1357
damped vibration, 1408–1409
free vibration equation for, 1354
time intervals as, 1351, 1354–1355
undamped vibration, 1352–1355, 1360
Perpendicular distance between lines, 108,
112–113
Phase angle, 1354, 1361
Phase difference, 1410
Pin-connected members, work of forces exerted
on, 1210
Pin supports, 173–174, 209
Pipes, fluid flow through, 957
Pitch, 454, 457
Planar forces, 17–53
equilibrium of, 38–46
line of action, 18
magnitude of, 17
parallelogram law for, 18
rectangular components, 29–31
resolution into components, 21
resultant of several concurrent forces, 21
resultant of two forces, 18
scalar components, 29–30
scalar representation of, 19, 20–21
summing x and y components, 32–33
unit vectors for, 29–31
vector representation of, 18–21
Plane motion, 984
acceleration of, 1037–1047
absolute, 1038
normal components, 1037–1039
relative, 1037–1039, 1046–1047
tangential components, 1037–1039
analysis of, 1002–1003, 1039
diagrams for rotation and translation,
1002–1003, 1014, 1037–1038, 1046
equations of, 1117–1118, 1134
free-body diagrams for, 1122–1124, 1134
instantaneous center of rotation, 985,
1023–1030
kinetic diagrams for, 1119–1120, 1134
particles in, 692–694
rigid bodies in, 983–1066, 1115–1278
angular momentum of, 1118–1119
centroidal rotation, 1120, 1134
constrained, 1152–1171
energy and momentum methods, 1192–1278
forces of, 1119–1120
general, 1120–1121, 1134
principle of transmissibility and, 1121
systems of, 1124, 1135
translation, 1120, 1134
rotating frames of reference, 1057–1060
velocity of, 1002–1014
absolute, 1003
angular, 1005
instantaneous center of zero, 1023
relative, 1003–1005
Plates
center of gravity for, 240–241
circular, 537
composite, 240–241
mass moment of inertia for, 536–537
rectangular, 537
thin, 536–537, 543
Point of application, 17, 85–86
Polar coordinates
angular momentum of particle motion in, 768
radial and transverse components, 694–696,
698–701, 703
Polar moment of inertia, 486, 489, 494
Position coordinate, 617–618
Position relative to frame of reference, 671
Position vector, 90, 138, 665
Potential energy
conservation of energy, 830–832, 843–844
of conservative forces, 832–833
determination of, 830–832
elastic, 831
equations of, 598–599
equilibrium and, 599–600, 603
friction forces and, 834
gravitational, 830–831
with respect to gravity (weight), 598, 603
of spring forces, 598, 603, 831–832
virtual work and, 576, 598–599, 603
work of, 830
Potential function, 832
Power
average, 807
efficiency and, 807–808, 819
from particle motion, 807–808, 819
rate of work as, 807–808, 819
from rigid-body plane motion, 1199, 1210
units of, 807–808
Precession
rate of, 1323
steady, 1324–1326, 1330–1331
Principal normal, 694
Principle axes of inertia, 1283, 1291, 1312
Principle axis and moments of inertia
about the centroid, 519
of an area, 517–519, 522
for a body of arbitrary shape, 555–557, 560
ellipsoid of inertia, 554–556
of a mass, 555–557, 561
Principle of conservation of energy, 833–834
Principle of impulse and momentum
noncentroidal rotation, 1224
particle motion, 800, 858–860, 871
rigid-body plane motion, 1222–1224,
1232–1233
three-dimensional rigid-body motion,
1285–1286, 1292
Principle of transmissibility, 4, 84, 86–87
equivalent forces of, 86–87
rigid-body applications, 84, 86–87
for rigid-body plane motion, 1121
sliding vectors from, 84, 86
Principle of virtual work, 576, 579–581, 588
application of, 579–581
virtual displacement, 579, 588
Principle of work and energy
particle motion, 800, 804–807
rigid-body plane motion, 1194–1195, 1209
Problems, 11–13
error detection, 13
force triangle, 40
free-body diagrams for, 12, 39–40
methods for solving, 11–13
SMART method for solving, 11–12
solution basis, 11–13
space diagram for, 39
Product of inertia, 516–517, 522
Projectile motion, 670, 672–674, 678
Projection of a vector, 106, 112
Pure bending, 487
Q
Quadratic surface equation, 554
R
Radial and transverse components
acceleration in, 696
cylindrical coordinates for, 696, 703
equations of motion, 730, 746
particle motion analysis using, 694–696,
698–701, 703
polar coordinates for, 694–696, 698–701, 703
velocity in, 696
Radial direction, 694, 703
Radius of gyration, 489–490, 494, 534
Rate of change
of angular momentum
fixed-point motion, 1300–1301, 1312
particles, 767–768Index I9
rotational motion, 1301, 1313
three-dimensional rigid-bodies,
1300–1301, 1313
in polar coordinates, 768
rotating frames of reference, 1056–1057
of a vector, 668–669, 767, 1056–1057
Reactions, 173
constraining forces, 173, 177–178
equilibrium of rigid bodies and, 173–175,
207–209
equivalent to force and couple, 174–175
free-body diagrams showing, 173
with known line of action, 173
support, 173–175, 207–209
three-dimensional structures, 207–209
two-dimensional structures, 173–176
with unknown direction and magnitude,
173–174
Rectangular components
curvilinear motion of, 669–670
equations of motion, 728–729, 746
moments of a force, 93–94
of particles, 29–31, 54–57, 669–670,
728–729, 746
planar (two-dimensional) forces, 29–31
of rigid bodies, 89–90, 93–94
space (three-dimensional) forces, 54–57
unit vectors for, 29–31, 56–57
vector products, 89–90
Rectilinear motion of particles, 617–631
acceleration, 618–619, 631
constant force in, 802, 818
deceleration, 619
determination of, 621–630
graphical solutions for, 654–657
initial conditions for, 621
position coordinate, 617–618
speed (magnitude), 618
uniform, 638–639
uniformly accelerated, 638–653
velocity, 618
work of constant force in, 803, 818
Redundant members, 320
Relative acceleration, 671, 1037–1038,
1046–1047
Relative motion, 437
curvilinear solution to problems, 670–671,
675–679
dependent motion, 640, 646–647
independent motion, 639–640, 646
of particles, 638–647
Relative position, 671
plane motion, 1003–1005
variable systems of particles, 958–959
Relativistic mechanics, 3
Resonance, 1395, 1398
Resultant couples, 140–143
Resultant of forces, 17–18
concurrent, 21, 58
parallelogram law for, 18
particle statics, 17–18, 21
planar forces, 18, 21
of several concurrent forces, 21
statics and, 17
three-dimensional (space), 58
of two forces, 18
Revolution, mass moment of inertia for a body
of, 537, 544
Right-handed triad, 88–89
Right-hand rule, 88, 90
Rigid bodies, 84. See also Systems of rigid bodies
constraining forces, 173, 177–178, 208
couples, 119–129
energy and momentum methods, 1192–1278
conservation of angular momentum,
1225, 1233
conservation of energy, 1197–1199, 1210
eccentric impact, 1245–1261
kinetic energy, 1196–1197, 1209
noncentroidal rotation, 1196–1197, 1224
power, 1199, 1210
principle of impulse and momentum,
1222–1224, 1232–1233
principle of work and energy, 1194–1195,
1209
systems, analysis of, 1197, 1210, 1225
work of forces, 1195–1196, 1209
equilibrium of, 170–231
statically determinate reactions, 177
statically indeterminate reactions, 177–178
support reactions for, 173–175, 207–209
three-dimensional structures, 207–215
three-force body, 199–201
two-dimensional structures, 173–183
two-force body, 199
equivalent systems of forces, 83–169
center of gravity, 85
deformation and, 87
external forces, 85–86
internal forces, 85, 87
point of application, 85–86
reduction to force-couple system,
138–139
simplifying, 138–150
weight and, 85–86
force-couple systems
equipollent, 139–140
equivalent systems reduced to, 138–139
reducing a systems of forces into, 138–139
resolution of force into, 122, 123–124
resultant couples, 140–143
wrench, 143–144
free-body diagrams for, 171–173
free vibration of, 1368–1374
kinematics of, 982–1114
acceleration of, 988, 994, 1037–1047,
1074, 1076, 1080–1081, 1091–1092,
1097–1098
in general motion, 984, 1075–1076, 1081,
1091–1092, 1098
general plane motion, 984–985,
1002–1014, 1037–1047
instantaneous center of rotation, 985,
1023–1030
motion about a fixed point, 984,
1073–1075, 1080
rotating frames, motion relative to,
1056–1066, 1090–1091, 1097
rotation about a fixed axis, 983–995
three-dimensional (space) motion, 985,
1073–1081, 1090–1098
translation, 983, 985–986, 995
two-dimensional (planar) motion,
983–1066
velocity of, 1002–1014, 1075, 1080–1081,
1092, 1097–1098
kinetics of, 1115–1191
angular momentum of, 1118–1119
centroidal rotation, 1120, 1134
constrained plane motion, 1152–1171
forces of, 1119–1120
general plane motion, 1120–1121, 1134
plane motion, 1115–1191
principle of transmissibility and, 1121
systems of, 1124, 1135
three-dimensional (space) motion,
1279–1349
translation, 1120, 1134
mechanics of, 2, 4
moment of a force about an axis, 84
moment of a force about a point, 84
moments
of a couple, 119–129
of a force about an axis, 105–109
of a force about a point, 90–93
principle of transmissibility and, 4, 84, 86–87
reactions, 173–175
rectangular components, 89–90, 93–94
scalar (dot) products, 105–106
sliding vector representation, 19, 84
vector products, 87–89, 105–109
virtual work application to systems of
connected, 579–581
Rigid truss, 302
Rocker supports, 172–174
Roller supports, 173
Rolling
angular acceleration, 1154–1155
resistance, 464–465, 467
sliding and, 1154–1155, 1171
unbalanced disk or wheel, 1155, 1171
Rotating frames of reference
Coriolis acceleration, 1059–1060, 1066
plane motion of a particle relative to,
1057–1060
rate of change of a vector, 1056–1057, 1066
rigid-body motion relative to, 1056–1066,
1090–1091, 1097
three-dimensional particle motion,
1090–1091, 1097
Rotation, 86, 983
about a fixed axis
angular acceleration, 988, 994
angular coordinate, 987, 994
angular velocity, 987, 994
equations for, 989, 995
noncentroidal, 1154, 1170
rate of change of angular momentum,
1301, 1313
rigid-body motion, 983–985
slab representation, 988–989, 994–995
centrifugal force, 1154
centroidal, 1120, 1134
curvilinear translation compared to, 984
finite, 1074
force as, 86
infinitesimal, 1074
instantaneous axis of, 1073–1074
instantaneous center of, 985, 1023–1030
motion about a fixed point, 1073–1075
noncentroidal
of a body in constrained motion,
1153–1154, 1170
kinetic energy of a body in, 1196–1197I10 Index
Rotation—Cont.
plane motion diagrams, 1002–1003, 1014,
1037–1038, 1046
uniform, 1154
Rotational equation of motion, 1117
Rough surface supports, 174, 209
S
Sag, 409
Scalar product
of vector functions, 668
of vectors, 105–106
Scalars, 19
particle force representation, 19
product of vector and, 20–21
rectangular force components, 29–30, 54–55
Screws, 453–454, 457
friction and, 453–454, 457
lead and lead angle, 454, 457
pitch, 454, 457
self-locking, 454
square threaded, 453–454, 457
Self-locking screws, 454
Shaft rotation, balance of, 1304
Shearing forces, 371–374
beams, 369, 371, 380–381, 386, 392–401
bending moment relations with, 393
diagrams for, 382, 387
external forces and, 381
internal forces as, 369, 371–374
load relations with, 392–393
Simple truss, 301–304, 308
Slab representation for fixed-axis rotation,
988–989, 994–995
Sliding motion, 1154–1155, 1171
Sliding vectors, 19, 84, 86
Slipping, belt friction and, 472–473, 475
Slipstream, 958
SMART method for solving problems, 11–12
Space, concept of, 3. See also Threedimensional problems
Space centrode, 1025
Space cone, 1074
Space diagram, 39
Space mechanics
conservation of energy, 834–835
under a conservative central force, 834–835
eccentricity, 779–780
escape velocity, 780
gravitational force, 779–780
gyroscopes, 1323–1332
analysis of motion, 1323–1332
axisymmetric body analysis, 1324,
1326–1327, 1331–1332
Eulerian angles of, 1323–1324, 1330
steady precession of, 1324–1326,
1330–1331
initial conditions, 780–781
Kepler’s laws of planetary motion,
782, 786
periodic time, 781–782, 786
projectile motion, 670, 672–674, 678
thrust, 958–959, 964–965
trajectories, 779–782, 786
Space truss, 308
Span, 379–380, 409
Speed (magnitude), 618, 665
Spin, rate of, 1323
Spring force
potential energy of, 831–832
virtual work of, 803, 818, 1210
work of elastic force, 597–598, 603
Square threaded screws, 453–454
Stable equilibrium, 600–601, 603
Statically determinate reactions, 177, 336
Statically indeterminate reactions, 177–178,
208, 336
Static friction force, 433–434
Statics
of particles, 16–82
resultant of forces, 17–18, 21
role of in mechanics, 2
state of equilibrium, 17
Steady-state vibration, 1385, 1398
Steady stream of particles, 956–958, 964
fan flow, 958
fluid flow through a pipe, 957
fluid stream diversion by a vane, 957, 964
helicopter blade flow, 958
jet engine flow, 958
units for, 957
Structural analysis, 299–367
frames, 301, 334–339
internal force reactions, 300–301
machines, 301, 334, 350–352
multi-force members, 301, 334
Newton’s third law for, 301
trusses, 301–310, 319–326
two-force members, 301
virtual work applications, 579–581
zero-force members, 307
Structures
analysis of, 299–367
equilibrium of, 173–183, 207–215
statically determinate reactions, 177, 336
statically indeterminate reactions, 177–178,
208, 336
three-dimensional, 207–215
two-dimensional, 173–183
Submerged surfaces, distributed forces on, 263,
270–271
Support reactions, 173–175, 207–209
fixed, 174–175
frictionless pins, 173–174
of one unknown and one direction, 173–174
rollers and rockers, 172–174
static determinacy and, 336
three-dimensional structures, 207–209, B2
two-dimensional structures, 173–175, B1
Symmetry, planes of, 280
Systems of particles, 920–981
conservation of energy in, 941, 950
conservation of momentum in, 928, 932,
946, 950
energy and momentum methods for, 940–950
impulse-momentum principle, 941–942
kinetic energy, 940, 950
work-energy principle, 941
external and internal forces acting on,
921–924
mass center of, 921, 925–928, 932
momentum in, 922–932
angular, 924–928, 932, 946, 950
linear, 924–925, 928, 932, 946, 950
Newton’s second law for, 922–924
variable, 956–965
fluid flow, 957
fluid stream diversion, 957, 964
mass gain and loss, 959, 964–965
relative velocity, 957, 959
steady stream of particles, 956–958, 964
thrust, 958, 959, 964
Systems of rigid bodies
constrained (plane) motion of, 1171
plane motion of, 1124
principle of impulse and momentum for,
1225, 1233
principle of work and energy for, 1124
Systems of units, 5–11
converting between, 10–11
International System of Units (SI), 5–7
U.S. customary units, 8–9, 12
T
Tangential and normal components
acceleration in, 693, 702, 1037–1039
equations of motion, 730, 746
particle analysis using, 692–694,
697–698, 702
rigid-body analysis using, 1037–1039
three-dimensional (space) motion, 694
two-dimensional (planar) motion, 692–694,
1037–1039
Tension, deformation from internal forces of, 87
Theorems
Euler’s, 1073
Pappus-Guldinus, 251–252
parallel-axis, 497–509, 517, 534–535, 543
Varignon’s, 93
Theory of relativity, 3
Thin plates, mass moment of inertia for,
536–537, 543
Three-dimensional bodies, 276–284
center of gravity, 276–278, 285
centroid of volume location, 276–278, 285
composite bodies, 278–280
Three-dimensional (space) motion
about a fixed point
analysis of, 1302–1303, 1313
angular momentum of, 1284–1285,
1291–1292
instantaneous axis of rotation, 1073–1074
about a mass center, 1282–1283, 1291
angular momentum in
about a fixed point, 1284–1285, 1291–1292
about a mass center, 1282–1283, 1291
inertia tensor, 1283
principle axes of inertia, 1283, 1291, 1312
rate of change of, 1300–1301, 1313
reduction of particle moments, 1284
of rigid bodies, 1281–1285, 1291,
1300–1304, 1312–1313
energy and momentum in, 1281–1292
angular momentum, 1281–1285, 1291
kinetic energy, 1286–1287, 1292
principle of impulse and momentum for,
1285–1286, 1292
equations and principles for, 1280–1281
Euler’s equations for, 1073, 1301–1302
general, 1075–1076, 1081, 1091–1092, 1097Index I11
gyroscopes, 1323–1332
axisymmetric body analysis, 1324,
1326–1327, 1331–1332
Eulerian angles of, 1323–1324, 1330
steady precession of, 1324–1326,
1330–1331
kinematics of, 694, 985, 1073–1081,
1090–1098
kinetics of, 1279–1349
of particles, 694, 1090–1091, 1097
relative to moving frame of reference,
1090–1098
of rigid bodies, 985, 1073–1081, 1090–1098,
1279–1349
rotation about a fixed axis, 1303–1304,
1313–1314
solutions for problems, 1300–1314
Three-dimensional (space) problems, 54–75
adding forces in, 54–66
concurrent force resultants, 58
direction cosines for, 55, 56
equilibrium in, 67–75, 207–215
forces in, 54–75
line of action, 56–58
magnitude of force, 57–58
moments of a force about a point, 93–94, 99
particles, 54–75
rectangular components, 54–57
rigid bodies, 93–94, 99, 207–215
support reactions, 207–209
unit vector for, 56–57
Three-force body, equilibrium of, 199–201
Thrust
fluid flow causing, 958, 964
mass gain and loss required for rockets,
959, 965
units for, 959
Thrust bearings, disk friction of, 462,
463–464, 467
Time, concept of, 3
Time interval
impulse of a force, 871
period of damped vibration, 1408–1409
period of undamped vibration, 1352–1355,
1360
Trajectory
central-force motion and, 778–786
elliptical, 779–780, 785–786
hyperbolic, 779–780, 785–786
parabolic, 670, 779–780, 785–786
of a particle, 778–779
periodic time, 781–782, 786
of space mechanics, 779–782, 786
Transient vibration, 1394. See also Free
vibration
Translation, 86, 983
curvilinear motion and, 668
external forces from plane motion,
1120, 1134
force as, 86
kinetic diagrams for, 1120, 1134
kinetic energy of a body in, 1196
plane motion diagrams for, 1002–1003, 1014,
1037–1038, 1046
rigid body in, 985–986, 995, 1120, 1134
Translational equation of motion, 1117
Transmissibility. See Principle of
transmissibility
Transverse components. See Radial and
transverse components
Transverse direction, 694, 703
Triangle rule for addition of vectors, 19–20
Triple integration, 280
Trusses, 301–310, 319–326
analysis of, 301–310, 319–326
compound, 320
free-body joint diagrams, 304
joints under special loading conditions,
306–308
method of joints, 304–306
method of sections, 319–326
nonrigid, 320
over rigid, 320
redundant members, 320
rigid, 302
simple, 301–304, 308
space, 308
two-force members, 301
zero-force members, 307
Two-dimensional bodies, 233–244
center of gravity, 234–235, 245
centroid of area and line location, 235–237,
239–240, 245
composite plates and wires, 240–244
first moment of an area or line, 237–240, 245
planar elements, 233–244
Two-dimensional (planar) motion
of particles, 692–694
of rigid bodies, 983–1066
Two-dimensional (planar) problems
equilibrium in, 173–183
moments of a force, 92–94, 99
rigid-body structures, 173–183
statically determinate reactions, 177
statically indeterminate reactions, 177–178
support reactions, 173–175
Two-force body, equilibrium of, 199
Two-force members, 301, 371. See also Trusses
U
Unbalanced rolling disk or wheel, 1155, 1171
Undamped vibration
forced vibration, 1352, 1393–1399
free vibration, 1352–1361, 1368–1374
simple harmonic motion, 1352–1361
Underdamped vibration, 1408, 1412
Uniformly accelerated rectilinear motion, 638–653
Uniformly distributed loads, 379
Uniform rectilinear motion of particles, 638–639
Uniform rotation, 1154
Units, 5–11
of area and volume, 7–8
basic, 5
converting between systems, 10–11
derived, 5
of energy, 577
of force, conversion of, 10
of frequency, 1355
gravitational, 8–9
of impulse, 858–859
International System of Units (SI), 5–7
kinetic, 5–6
of length, conversion of, 10
of mass, conversion of, 10–11
of power, 807–808
quantity equivalents of SI and U.S.
customary, 12
SI abbreviations (formulas) of, 8
SI prefixes, 7
for steady stream of particles, 957
systems of, 5–11
for thrust, 959
of work, 802
Unit vectors, 29–31, 56–57
Universal joint supports, 209
Unstable equilibrium, 600–601, 603
U.S. customary units, 8–9, 12
V
Vanes, fluid stream diversion by, 957, 964
Variable systems of particles, 956–965
fluid flow, 957
fluid stream diversion, 957, 964
mass gain and loss, 959, 965
relative velocity, 958–959
steady stream of particles, 956–958, 964
thrust, 958, 959, 964–965
Varignon’s theorem, 93
Vector products, 87–89, 105–109
commutative property and, 88
distributive property and, 88
mixed triple products, 106–107
moment of force about a given axis, 105–109
moment of force about a point, 87–89
rectangular components of, 89–90
right-hand rule for, 88, 90
of scalar products, 105–106
triple product, 988
of vector functions, 668
Vectors, 18–21
acceleration, 666–667, 669–670
addition of, 19–21
addition of, parallelogram law for, 19
addition of, polygon rule for, 20
addition of, triangle rule for, 19–20
angle formed by, 106
angular momentum of particles as, 767
coplanar, 20
curvilinear motion and, 665–679
derivatives of functions, 667–669
displacement, 665
equal and opposite, 19
fixed (bound), 19
force addition using, 18–21, 54–57
frame of reference, 668
free, 19
function, 665–666
linear momentum, 1223
mixed triple products, 106–107
moments of a force, 90, 105–109
about a given axis, 105–109
about a point, 90
negative, 19–20
particle force representation, 18–21
planar forces, 18–21
position, 90, 138, 665
product of scalar and, 20–21
projection of, 106
rate of change of, 668–669, 1056–1057, 1066
rectangular force components, 29–31, 669–670I12 Index
Vectors—Cont.
rigid-body representation, 84, 86
sliding, 19, 84, 86
subtraction of, 19–20
three-dimensional forces, 55–57
unit, 29–31, 56–57
velocity, 665–666, 669–670
Velocity
absolute, 1003
angular, 987, 994, 1005
areal, 769
average, 618, 665
curvilinear motion and, 665–666
determining, 618
escape, 780
general motion, 1075, 1081, 1091–1092, 1098
instantaneous, 618, 665
instantaneous center at zero, 1023
instantaneous center of rotation for,
1023–1030
moving frames of reference, 1090–1092,
1097–1098
plane motion, 1002–1014
radial and transverse components of, 696
rectangular components of, 669–670
rectilinear motion and, 618
relative, 671, 958–959, 1005
rotating frame of reference, 1090, 1097
speed (magnitude), 618, 665
three-dimensional (space) motion, 1075,
1080–1081, 1090–1092, 1097–1098
two-dimensional (planar) motion, 1002–1014
variable systems of particles, 958–959
vector, 665–666, 669–670
Vibration, 1350–1432
amplitude, 1351, 1354, 1361
conservation of energy applications,
1382–1386
damped, 1352, 1407–1418
forced, 1351, 1393–1399, 1409–1411
free, 1352–1361, 1368–1374, 1407–1409,
1412–1413
frequency, 1351, 1353, 1355, 1361, 1393–1394
oscillations, 1356, 1361, 1369
period, 1351, 1354, 1360, 1408–1409
periodic function, 1353
phase angle, 1354, 1361
of rigid bodies, 1368–1374
simple harmonic motion, 1352–1361
steady-state, 1385, 1398
transient, 1394
undamped, 1352–1361, 1368–1374, 1393–1394
Virtual work, 575–614
displacement of a particle, 577
equilibrium conditions, 599–603
mechanical efficiency of machines, 581–582
method of, 576–588
potential energy and, 576, 598–599, 603
principle of, 576, 579–581, 588
application to systems of connected rigid
bodies, 579–581
virtual displacement, 579, 588
work
of a couple, 578
during finite displacement, 596–598
of a force, 577–579, 596–598
input, 581
output, 581
virtual, 579, 588
Viscosity. See Fluid friction
Volume, units of, 7–8
W
Wedges, 453, 457
Weight, 4–5
center of gravity, 85
external force as, 85–86
as a force, 4–5
gravity and, 597–598, 603
point of application, 85
potential energy effected by, 598, 603
rigid-body motion and, 85–86
work of, 597
Wheel friction, 464–465, 467
Work
of a couple, 578
during finite displacement, 596–598
of a force, 577–579, 596–598
constant in rectilinear motion, 802, 818
gravitational, 804, 819
of gravity, 803, 818, 830–831
for particle motion, 801–819
for pin-connected members, 1210
for potential energy, 830–832
principle of work and energy, 800,
804–807, 1194–1195, 1209
for rigid-body plane motion, 1194–1195,
1209–1210
of a spring, 597–598, 803, 818, 1210
input, 581
output, 581
virtual, 579, 588
of a weight (gravity), 597
Work-energy principle for systems of
particles, 941
Wrench, reduction of force-couple forces into,
143–144
Z
Zero-force members, 307

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