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 موضوع: كتاب Vector Mechanics For Engineers الثلاثاء 05 أكتوبر 2021, 12:41 am  

أخواني في الله أحضرت لكم كتاب Vector Mechanics For Engineers 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 RoseHulman Institute of Technology Brian P. Self California Polytechnic State University—San Luis Obispo
و المحتوى كما يلي :
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 ForceCouple 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 ParallelAxis 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 NonRectangular 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 CentralForce 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 FixedAxis 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 fixedaxis rotation, 987–988, 994 average, 618–619, 666 Coriolis motion with respect to a rotating frame, 985, 1059–1060, 1066 threedimensional (space) motion, 1091, 1097 twodimensional (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 threedimensional (space) motion, 1074, 1076, 1080–1081, 1091–1092, 1097–1098 twodimensional (planar) motion, 994 vector polygons for determination of, 1038 twodimensional (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 fixedaxis 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 threedimensional rigid bodies, 1300–1301, 1312–1313 of a rigid body in plane motion, 1118–1119 of systems of particles, 924–928, 932, 946, 950 threedimensional 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 PappusGuldinus, 251–252 twodimensional 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 threedimensional bodies, 276–278, 285 twodimensional bodies, 233–235 Center of pressure, 263 Centralforce 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 PappusGuldinus, 251–252 threedimensional bodies, 276–278 twodimensional 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 rigidbody 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 rigidbody 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 freebody 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 freebody diagram reactions, 173 improperly constrained, 178, 208 partially constrained, 177, 208 threedimensional rigid bodies, 208 twodimensional 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 threedimensional (space) motion, 1091, 1097 twodimensional (planar) motion, 1059–1060, 1066 Coulomb friction. See Dry friction Couples, 119–129 addition of, 122 angular moment, 1223 equivalent, 120–122 forcecouple 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 relativemotion problems, 670–671, 675–678 rotation compared to, 984 twodimensional 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 PappusGuldinus, 251–252 threedimensional bodies, 276–284 center of gravity, 276–278, 285 centroid of volume location, 276–278, 285 composite bodies, 278–280 twodimensional 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 rigidbody 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 rigidbody plane motion, 1222–1233 kinetic energy particle motion, 804–805, 819, 833–834, 843 rigidbody plane motion, 1196–1197, 1209 systems of particles, 940, 950 threedimensional rigidbody motion, 1286–1287, 1292 particle motion, 799–919 power from particle motion, 807–808, 819 from rigidbody plane motion, 1199, 1210 principle of impulse and momentum particle motion, 800, 858–860, 871 rigidbody plane motion, 1222–1224, 1232–1233 threedimensional rigidbody motion, 1285–1286, 1292 principle of work and energy particle motion, 800, 804–807 rigidbody plane motion, 1194–1195, 1209 rigidbody plane motion, 1192–1278 systems of particles, 940–950 conservation of energy, 941, 950 conservation of momentum, 946, 950 impulsemomentum principle, 941–942 workenergy principle, 941 systems of rigid bodies, 1124, 1135, 1225, 1233 threedimensional rigidbody 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 pinconnected members, 1210 rigidbody 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 freebody diagrams for, 39–40, 171–173 neutral, 600 Newton’s first law of motion and, 39 of a particle, 38–39, 67–75 threedimensional (space) problems, 67–75 twodimensional (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 threedimensional structures, 207–215 threeforce body, 199–201 twodimensional structures, 173–183 twoforce 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 forcecouple 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 Fixedaxis rotation angular acceleration, 988, 994 angular coordinate, 987, 994 angular velocity, 987, 994 equations for, 989, 995 noncentroidal, 1154, 1170 rigidbody motion, 983–985 shaft balance, 1304 slab representation, 988–989, 994–995 threedimensional 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 threedimensional 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 (twodimensional), 17–53 (See also Planar forces) point of application, 17 of rigidbody 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 Forcecouple 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 forcecouple, 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 forcecouple system, 138–139 resolution of a given force into forcecouple 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 threedimensional particle motion, 1090–1091, 1097 translation of, 668, 671 Frames, 301, 334–339 collapse of without supports, 335–336 equilibrium of forces, 335–336 freebody diagrams of force members, 334–335 multiforce members, 301, 334 statically determinate and rigid, 336 statically indeterminate and nonrigid, 336 Freebody diagrams, 12 equilibrium particle force, 39–40 rigidbody force, 171–173 frame analysis using, 334–335 machine analysis of members, 350, 353 particle motion, 727–728, 746 rigidbody constrained motion, 1152–1153, 1170 rigidbody plane motion, 1122–1124, 1134 truss analysis of joint forces, 304 twodimensional 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 threedimensional 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 rigidbody 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 Impulsemomentum diagram, 859, 871, 886–888, 899, 1261 Impulsemomentum 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 threedimensional body, 537 using the same elemental strips, 489 motion determined by, 621–622 theorems of PappusGuldinus 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 nonrectangular components, 692–703 rectilinear motion, 617–631 relative motion, 638–647 solutions for motion problems, 631–632, 646–647 threedimensional (space) motion, 694 twodimensional (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 threedimensional (space) motion, 985, 1073–1081, 1090–1098 translation, 983, 985–986, 995 twodimensional (planar) motion, 983–1066 velocity of, 1002–1014, 1023–1030, 1075, 1080–1081, 1092, 1097–1098 Kinetic diagrams particle motion, 727–728, 746 rigidbody constrained motion, 1152–1153, 1170 rigidbody 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 rigidbody 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 workenergy principle, 941 of threedimensional rigidbody motion with a fixed point, 1287, 1292 with respect to the mass center, 1286–1287, 1292 Kinetic friction force, 433–434 Kinetics, 616 freebody 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 centralforce 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 threedimensional 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 threedimensional (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 twodimensional 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 freebody diagrams of members, 350, 353 input forces, 350 mechanical efficiency of, 581–582 multiforce 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 parallelaxis 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 threedimensional 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 freebody diagrams of, 334–335 internal forces in, 369–374 machine analysis of, 350, 353 multiforce, 301, 334, 370–371 redundant, 320 shearing force in, 371 twoforce, 301, 371 zeroforce, 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 righthand rule for, 90 threedimensional problems, 93–94, 99 twodimensional 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 parallelaxis 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 unitrelated 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 rigidbody 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 rigidbody kinetics, 1117–1118, 1120, 1134, 1280 rotational, 1117 scalar form, 1120 tangential and normal components, 730, 746 translational, 1117 external forces and, 85–86 freebody 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 threedimensional (space), 985, 1073–1081, 1090–1098 translation, 983, 985–986, 995 twodimensional (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 threedimensional (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 rigidbody motion relative to, 1090–1098 rotating frame, 1090–1091, 1097 threedimensional particle motion, 1090–1091, 1097 velocity of, 1090–1092, 1097–1098 Multiforce 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 freebody 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 impulsemomentum 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 PappusGuldinus, theorems of, 251–252 Parabolic cables, 409–410, 414 Parabolic trajectory, 670, 779–780, 785–786 Parallelaxis 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 threedimensional 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 nonrectangular 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 threedimensional (space) motion, 694, 1090–1091, 1097 twodimensional (planar) motion, 692–694 uniform rectilinear motion, 638–639 kinetics of, 721–798 angular momentum, 722, 767–773 centralforce 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 threedimensional (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 twodimensional (planar) problems, 17–53 adding forces by components, 29–37 concurrent force resultants, 21 equilibrium of, 38–46 freebody 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 Pathindependent 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 Pinconnected 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 freebody 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 rigidbody 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 rigidbody plane motion, 1222–1224, 1232–1233 threedimensional rigidbody motion, 1285–1286, 1292 Principle of transmissibility, 4, 84, 86–87 equivalent forces of, 86–87 rigidbody applications, 84, 86–87 for rigidbody 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 rigidbody plane motion, 1194–1195, 1209 Problems, 11–13 error detection, 13 force triangle, 40 freebody 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 fixedpoint motion, 1300–1301, 1312 particles, 767–768Index I9 rotational motion, 1301, 1313 threedimensional rigidbodies, 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 freebody diagrams showing, 173 with known line of action, 173 support, 173–175, 207–209 threedimensional structures, 207–209 twodimensional 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 (twodimensional) forces, 29–31 of rigid bodies, 89–90, 93–94 space (threedimensional) 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 threedimensional (space), 58 of two forces, 18 Revolution, mass moment of inertia for a body of, 537, 544 Righthanded triad, 88–89 Righthand 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 threedimensional structures, 207–215 threeforce body, 199–201 twodimensional structures, 173–183 twoforce 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 forcecouple system, 138–139 simplifying, 138–150 weight and, 85–86 forcecouple 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 freebody 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 threedimensional (space) motion, 985, 1073–1081, 1090–1098 translation, 983, 985–986, 995 twodimensional (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 threedimensional (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 rigidbody motion relative to, 1056–1066, 1090–1091, 1097 threedimensional 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 rigidbody 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 selflocking, 454 square threaded, 453–454, 457 Selflocking 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 fixedaxis 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 Steadystate 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 multiforce members, 301, 334 Newton’s third law for, 301 trusses
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