**Vector Mechanics For EngineersTwelfth EditionFerdinand P. BeerLate of Lehigh UniversityE. Russell Johnston, Jr.Late of University of ConnecticutDavid F. MazurekU.S. Coast Guard AcademyPhillip J. CornwellRose-Hulman Institute of TechnologyBrian P. SelfCalifornia Polytechnic State University—San Luis ObispoBrief Contents1 Introduction 12 Statics of Particles 163 Rigid Bodies: Equivalent Systems of Forces 834 Equilibrium of Rigid Bodies 1705 Distributed Forces: Centroids and Centers of Gravity 2326 Analysis of Structures 2997 Internal Forces and Moments 3688 Friction 4319 Distributed Forces: Moments of Inertia 48510 Method of Virtual Work 57511 Kinematics of Particles 61512 Kinetics of Particles: Newton’s Second Law 72113 Kinetics of Particles: Energy and Momentum Methods 79914 Systems of Particles 92015 Kinematics of Rigid Bodies 98216 Plane Motion of Rigid Bodies: Forces and Accelerations 111517 Plane Motion of Rigid Bodies: Energy and Momentum Methods 119218 Kinetics of Rigid Bodies in Three Dimensions 127919 Mechanical Vibrations 1350Appendix: Fundamentals of Engineering Examination A1Answers to Problems AN1Index I1Properties of Geometric Shapes I17ixPreface xvGuided Tour xixDigital Resources xxiiiAcknowledgments xxvList of Symbols xxvii1 Introduction 11.1 What is Mechanics? 21.2 Fundamental Concepts and Principles 31.3 Systems of Units 51.4 Converting between Two Systems of Units 101.5 Method of Solving Problems 111.6 Numerical Accuracy 152 Statics of Particles 162.1 Addition of Planar Forces 172.2 Adding Forces by Components 292.3 Forces and Equilibrium in a Plane 382.4 Adding Forces in Space 542.5 Forces and Equilibrium in Space 67Review and Summary 76Review Problems 803 Rigid Bodies: Equivalent Systemsof Forces 833.1 Forces and Moments 853.2 Moment of a Force about an Axis 1053.3 Couples and Force-Couple Systems 1193.4 Simplifying Systems of Forces 138Review and Summary 162Review Problems 167Contentsx Contents4 Equilibrium of Rigid Bodies 1704.1 Equilibrium in Two Dimensions 1734.2 Two Special Cases 1994.3 Equilibrium in Three Dimensions 207Review and Summary 227Review Problems 2295 Distributed Forces: Centroids andCenters of Gravity 2325.1 Planar Centers of Gravity and Centroids 2345.2 Further Considerations of Centroids 2505.3 Additional Applications of Centroids 2625.4 Centers of Gravity and Centroids of Volumes 276Review and Summary 293Review Problems 2976 Analysis of Structures 2996.1 Analysis of Trusses 3016.2 Other Truss Analyses 3196.3 Frames 3346.4 Machines 350Review and Summary 363Review Problems 3657 Internal Forces and Moments 3687.1 Internal Forces in Members 3697.2 Beams 3797.3 Relations Among Load, Shear, and Bending Moment 392*7.4 Cables 407*7.5 Catenary Cables 419Review and Summary 426Review Problems 4298 Friction 4318.1 The Laws of Dry Friction 4338.2 Wedges and Screws 453*8.3 Friction on Axles, Disks, and Wheels 462Contents xi8.4 Belt Friction 471Review and Summary 480Review Problems 4829 Distributed Forces: Moments ofInertia 4859.1 Moments of Inertia of Areas 4879.2 Parallel-Axis Theorem and Composite Areas 497*9.3 Transformation of Moments of Inertia 516*9.4 Mohr’s Circle for Moments of Inertia 5269.5 Mass Moments of Inertia 533*9.6 Additional Concepts of Mass Moments of Inertia 553Review and Summary 568Review Problems 57310 Method of Virtual Work 575*10.1 The Basic Method 576*10.2 Work, Potential Energy, and Stability 596Review and Summary 610Review Problems 61311 Kinematics of Particles 61511.1 Rectilinear Motion of Particles 61711.2 Special Cases and Relative Motion 638*11.3 Graphical Solutions 65411.4 Curvilinear Motion of Particles 66511.5 Non-Rectangular Components 692Review and Summary 713Review Problems 71712 Kinetics of Particles:Newton’s Second Law 72112.1 Newton’s Second Law and Linear Momentum 72312.2 Angular Momentum and Orbital Motion 767*12.3 Applications of Central-Force Motion 778Review and Summary 792Review Problems 796*Advanced or specialty topicsxii Contents13 Kinetics of Particles: Energy andMomentum Methods 79913.1 Work and Energy 80113.2 Conservation of Energy 83013.3 Impulse and Momentum 85813.4 Impacts 883Review and Summary 910Review Problems 91614 Systems of Particles 92014.1 Applying Newton’s Second Law and Momentum Principlesto Systems of Particles 92214.2 Energy and Momentum Methods for a System ofParticles 940*14.3 Variable Systems of Particles 956Review and Summary 975Review Problems 97915 Kinematics of Rigid Bodies 98215.1 Translation and Fixed-Axis Rotation 98515.2 General Plane Motion: Velocity 100215.3 Instantaneous Center of Rotation 102315.4 General Plane Motion: Acceleration 103715.5 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 1090Review and Summary 1105Review Problems 111116 Plane Motion of Rigid Bodies: Forcesand Accelerations 111516.1 Kinetics of a Rigid Body 111716.2 Constrained Plane Motion 1152Review and Summary 1186Review Problems 1188Contents xiii17 Plane Motion of Rigid Bodies: Energyand Momentum Methods 119217.1 Energy Methods for a Rigid Body 119417.2 Momentum Methods for a Rigid Body 122217.3 Eccentric Impact 1245Review and Summary 1271Review Problems 127518 Kinetics of Rigid Bodies in ThreeDimensions 127918.1 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 1323Review and Summary 1341Review Problems 134619 Mechanical Vibrations 135019.1 Vibrations without Damping 135219.2 Free Vibrations of Rigid Bodies 136819.3 Applying the Principle of Conservation of Energy 138219.4 Forced Vibrations 139319.5 Damped Vibrations 1407Review and Summary 1424Review Problems 1429Appendix: Fundamentals of Engineering Examination A1Answers to Problems AN1Index I1Properties of Geometric Shapes I17IndexAAbsolute acceleration, 1038Absolute velocity, 1003Accelerationabsolute, 1038angularconstrained (plane) motion, 1154–1155fixed-axis rotation, 987–988, 994average, 618–619, 666Coriolismotion with respect to a rotating frame,985, 1059–1060, 1066three-dimensional (space) motion,1091, 1097two-dimensional (planar) motion,1059–1060, 1066of particles, 621–622curvilinear motion and, 666–667determining, 618–619instantaneous, 618, 666radial and transverse components of, 696rectangular components of, 669–670rectilinear motion and, 617–631tangential and normal components of,692–694, 702relative, 671, 1037–1038, 1046–1047of rigid bodies, 1037–1047constrained (plane) motion, 1056–1057,1170–1171moving frames of reference, 1091–1092,1097–1098normal components, 1037–1039plane motion, 1037–1047tangential and normal components of,693, 702tangential components, 1037–1039three-dimensional (space) motion, 1074,1076, 1080–1081, 1091–1092,1097–1098two-dimensional (planar) motion, 994vector polygons for determination of, 1038two-dimensional (planar) motion, 987–988,1037–1047vector, 666–667, 669–670Additionof couples, 122of forces, 4, 32–33parallelogram law for, 4, 19polygon rule for, 20summing x and y components, 32–33triangle rule for, 19–20of vectors, 19–21Amplitude, 1351, 1354, 1361Analysis. See Structural analysisAnglesEulerian, 1323–1324, 1330firing, 673, 678formed by two vectors, 106, 112of friction, 435–436lead, 454phase, 1354, 1361of repose, 436Angular accelerationconstrained (plane) motion, 1154–1155fixed-axis rotation, 988, 994Angular coordinate, 987, 994Angular moment couple, 1223Angular momentumabout a mass center, 926–928, 932, 1119,1282–1283, 1291central force and, 768–769, 773conservation of, 769, 928, 946, 950equations for, 924–925Newton’s law of gravitation for, 769–770orbital motion and, 767–773of particle motion, 722, 767–773in polar coordinates, 768rate of changeof a particle, 767–768of rigid bodies in plane motion, 1119of three-dimensional rigid bodies,1300–1301, 1312–1313of a rigid body in plane motion, 1118–1119of systems of particles, 924–928, 932, 946, 950three-dimensional rigid bodies, 1281–1285,1291, 1300–1301about a fixed point, 1284–1285, 1291–1292about a mass center, 1282–1283, 1291inertia tensor, 1283principle axes of inertia, 1283, 1291reduction of particle moments, 1284vector forms, 767Angular velocity, 987, 994, 1005, 1046Apogee, 781Arbitrary shaped bodies, moments of inertia of,555–557, 560Areacentroid of common, 240composite, 241first moment of, 233, 237–240, 245, 251integrationcentroids determined by, 250–251moments of inertia determined by,488–489, 494moment of inertia, 487–494, 497–509, 516–522of common geometric shapes, 499for composite areas, 497–509for hydrostatic force system, 487, 508polar moments, 489, 494principle axis and moments, 517–519, 522product of inertia, 516–517, 522for a rectangular area, 488–489second moments, 487–494transformation of, 516–522using same strip elements, 489radius of gyration, 489–490theorems of Pappus-Guldinus, 251–252two-dimensional bodies, 233, 237–240, 245units of, 7–8Areal velocity, 769Average acceleration, 618–619, 666Average power, 807Average velocity, 618, 665Axel friction, 462–463, 467Axesmoments of a force about, 84, 105–109arbitrary point for, 109given origin for, 107–109mixed triple products, 106–107scalar products, 105–106neutral, 487principle axis and moments of inertiaabout the centroid, 519of an area, 517–519, 522for a body of arbitrary shape, 555–557, 560ellipsoid of inertia, 554–556of a mass, 555–557, 561Axisymmetric body analysis, 1324, 1326–1327,1331–1332BBalance, 1304Ball and socket supports, 209Basic units, 5Beamsbending moment in, 380–381, 386, 392–401centroids of, 262–263, 270classification of, 379loading conditionsconcentrated, 262, 379distributed, 262–263, 270, 379internal forces and, 369, 379–387uniformly distributed, 379pure bending, 487shear and bending moment diagrams for,382, 387shearing forces, 380–381, 386, 392–401span, 379–380Belt friction, 471–475Bending moments, 371–374beams, 380–381, 386, 392–401diagrams for, 382, 387external forces and, 381internal forces as, 369, 371–374shearing force relations with, 393Binormal, 694Body centrode, 1025Body cone, 1074Bracket supports, 209CCables, 369, 407–414, 419–423catenary, 419–423internal forces of, 369, 407–414parabolic, 409–410, 414I2 IndexCables—Cont.sag, 409solutions for reactions, 413–414, 423span, 409supporting concentrated loads, 407–408, 413supporting distributed loads, 408, 414, 419–423supporting vertical loads, 407–408, 413Cable supports, 174, 209Catenary cables, 419–423Center of force, 768Center of gravity, 85composite area, 241composite bodies, 278composite plate, 240–241location of, 234–235problem solving with, 240–249three-dimensional bodies, 276–278, 285two-dimensional bodies, 233–235Center of pressure, 263Central-force motion, 723, 768–769, 773,778–786angular momentum of a particle,768–769, 773applications of, 778–786of particles, 723, 768–769, 773, 778–786space mechanics, 779–782eccentricity, 779–780escape velocity, 780gravitational force, 779–780initial conditions, 780–781Kepler’s laws of planetary motion, 782periodic time, 781–782trajectory of a particle, 778–779Central impact, 883, 899Centrifugal force, 1154Centrodes, 1025Centroid, 233, 235–237of areas, 235–237, 240, 245, 250–251, 262–269of common shapes, 239–240, 279, B3distributed load problems using, 262–269integration for determination of,250–251, 279of lines, 235–237, 240, 245location of, 235–237, 245theorems of Pappus-Guldinus, 251–252three-dimensional bodies, 276–278two-dimensional bodies, 235–237, 239–240of volume, 276–278Centroidal frame of reference, 926, 940Centroidal rotation, 1120, 1134Circle of friction, 463, 467Circular orbits, 771–773Coefficientsof critical damping, 1407, 1412of friction, 434–435, 465, 467impact analysis, 884–886, 888, 899–900of restitution, 884–886, 888, 899–900vibration analysis, 1407, 1412of viscous damping, 1407Collar bearings, 463–464Commutative property, 88, 105Complimentary function, 1394Composite bodies, 278–280center of gravity of, 278centroid of, 278–280mass moment of inertia of, 537–544, 560Composite plates and wires, 240–244Compound truss, 320Compressibility of fluids, 2Compression, deformation from, 87Concentrated loadsbeams, 262, 379cables supporting, 407–408, 413Concurrent forcesresultants, 21, 58system reduction of, 140Connections, 173–175. See also Support reactionsConservation of angular momentum, 769, 946,950, 1225, 1233Conservation of energyconservative forces, 832–835, 843energy conversion and, 834kinetic energy, 941, 950in particle motion, 830–844potential energy, 830–832, 843–844principle of, 833–834in rigid-body plane motion, 1197–1199, 1210space mechanics applications, 834–835in systems of particles, 941, 950vibration applications of, 1382–1386Conservation of momentum, 859angular, 769, 946, 950, 1225, 1233direct central impact and, 883–884, 899linear, 859, 946, 950oblique central impact and, 886, 900particle motion, 859rigid-body plane motion, 1225, 1233systems of particles, 928, 932, 946, 950Conservative forcesexact differential, 832potential energy of, 599, 832–833space mechanics applications, 834–835work of, 832Constant force, work of in rectilinear motion,802, 818Constant of gravitation, 770Constrained (plane) motion, 1152–1171acceleration, 1056–1057, 1170–1171angular acceleration, 1154–1155free-body and kinematic diagrams for,1152–1153, 1170moments about a fixed axis, 1154, 1170noncentroidal rotation, 1153–1154, 1170rolling, 1154–1155, 1171sliding and, 1154–1155, 1171system of rigid bodies, 1171unbalanced rolling disk or wheel, 1155, 1171Constraining forces, 173, 177–178, 208completely constrained, 177free-body diagram reactions, 173improperly constrained, 178, 208partially constrained, 177, 208three-dimensional rigid bodies, 208two-dimensional rigid bodies, 177–178Conversionof energy, 834of units, 10–11Conveyor belt, fluid stream diversion by,957, 964Coplanar forcesresultants, 20system reduction of, 141–142Coplanar vectors, 20Coriolis acceleration, 985motion with respect to a rotating frame, 985,1059–1060, 1066three-dimensional (space) motion, 1091, 1097two-dimensional (planar) motion, 1059–1060,1066Coulomb friction. See Dry frictionCouples, 119–129addition of, 122angular moment, 1223equivalent, 120–122force-couple system resolution, 122moment of, 119–120work of, 578Critical damping coefficient, 1407, 1412Critically damped vibration, 1408, 1412Cross products. See Vector productsCurvilinear motion of particles, 665–679acceleration vectors, 666–667, 669–671derivatives of vector functions, 667–669firing angle, 673, 678frame of reference, 669–671position vectors, 665, 671projectile motion, 669–670, 672–674, 678rate change of a vector, 668–669rectangular components, 669–670relative-motion problems, 670–671, 675–678rotation compared to, 984two-dimensional problems, 678velocity vectors, 665–666, 669–671Cylindrical coordinates for radial and transversecomponents, 696, 703DDamped circular frequency, 1408Damped vibration, 1352, 1407–1418critically, 1408, 1412electrical analogs, 1411–1413forced vibration, 1409–1411, 1413free vibration, 1407–1408, 1412–1413friction causes of, 1407magnification factor, 1410–1411, 1413overdamped, 1408, 1412period of, 1408–1409phase difference, 1410underdamped, 1408, 1412Damping factor, 1408Deceleration, 619Definite integrals, 621Deformable bodies, mechanics of, 2Deformation, 87from impact, 883–884, 1245internal forces and, 87principle of transmissibility for preventionof, 87Degrees of freedom, 601, 640Dependent motion of particles, 640, 646–647Derived units, 5Dick clutch, 467Direct central impact, 883–886, 899coefficient of restitution, 884–886, 899conservation of momentum and, 883–884, 899deformation from, 883–884energy loss from, 885–886perfectly elastic, 885perfectly plastic, 885period of restitution, 883–884Direct impact, 883Direction cosines, 55, 56Index I3Direction of a force, 17, 31. See also Line ofactionDisk friction, 463–464, 467Displacementfinite, 596–598, 802, 804from mechanical vibration, 1351of a particle, 577vertical, 803virtual, 579, 588work of a force, 579, 588, 596–598, 801–804Displacement vector, 665Distributed forces, 232–298. See also Centroidbeam loads, 262–263, 270, 379cables supporting loads, 408, 414, 419–423concentrated load and, 262integration methods for centroid location,250–257, 279, 285moments of inertia, 485–574of areas, 487–494, 497–509polar, 486, 489, 494transformation of, 516–522submerged surfaces, 263, 270–271theorems of Pappus-Guldinus, 251–252three-dimensional bodies, 276–284center of gravity, 276–278, 285centroid of volume location, 276–278, 285composite bodies, 278–280two-dimensional bodies, 233–244center of gravity, 234–235, 245centroid of area and line location,235–237, 239–240, 245composite plates and wires, 240–244first moment of an area or line,237–240, 245planar elements, 233–244Distributive property, 88Dot product, 105. See also Scalar productDouble integration, 250, 280Dry friction, 432–445angles of, 435–436coefficients of, 434–435kinetic friction force, 433–434laws of, 433problems involving, 436–445static friction force, 433–434EEccentric impact, 883, 1245–1261Eccentricity, 779–780Efficiencymechanical, 581–582, 808overall, 808power and, 807–808Elastic force, 597–598, 603. See also Spring forceElastic potential energy, 831Electrical analogs, 1411–1413Ellipsoid of inertia, 554–556Elliptical orbits, 771–773Elliptical trajectory, 779–780, 785–786Elliptic integral, 1357End bearings, 463–464Energy and momentum methods, 799–919,1192–1278angular momentum, 1281–1285, 1291conservation of energyconservative forces, 832–835, 843in particle motion, 830–844potential energy, 830–832, 843–844principle of, 833–834in rigid-body plane motion, 1197–1199, 1210space mechanics applications, 834–835displacement, 801–804efficiency and, 807–808friction forces and, 807, 834impactconservation of energy and, 889–890, 900direct central, 883–886, 899eccentric, 1245–1261oblique central, 886–888, 899–900problems involving multiple kineticsprinciples, 888–890impulse and momentumconservation of angular momentum,1225, 1233conservation of linear momentum, 859impulse of a force, 858–859, 871–872impulsive motion, 859–860of particle motion, 858–872of rigid-body plane motion, 1222–1233kinetic energyparticle motion, 804–805, 819,833–834, 843rigid-body plane motion, 1196–1197, 1209systems of particles, 940, 950three-dimensional rigid-body motion,1286–1287, 1292particle motion, 799–919powerfrom particle motion, 807–808, 819from rigid-body plane motion, 1199, 1210principle of impulse and momentumparticle motion, 800, 858–860, 871rigid-body plane motion, 1222–1224,1232–1233three-dimensional rigid-body motion,1285–1286, 1292principle of work and energyparticle motion, 800, 804–807rigid-body plane motion, 1194–1195, 1209rigid-body plane motion, 1192–1278systems of particles, 940–950conservation of energy, 941, 950conservation of momentum, 946, 950impulse-momentum principle, 941–942work-energy principle, 941systems of rigid bodies, 1124, 1135, 1225, 1233three-dimensional rigid-body motion,1281–1292work of a forceconstant force in rectilinear motion,802, 818force of gravity, 803, 818gravitational force, 804, 819particle motion, 801–819pin-connected members, 1210rigid-body plane motion, 1195–1196, 1209spring force, 803, 818, 1210Energy conversion, 834Energy loss from impact, 885–886, 900Engineering examination, fundamentals of,A1–A2Equal and opposite vectors, 19Equilibrium, 38equations of, 38–39force relations and, 17, 38–46frame determinacy and, 335–336free-body diagrams for, 39–40, 171–173neutral, 600Newton’s first law of motion and, 39of a particle, 38–39, 67–75three-dimensional (space) problems, 67–75two-dimensional (planar) problems, 38–39principle of transmissibility and, 4of rigid bodies, 170–231statically determinate reactions, 177statically indeterminate reactions,177–178, 208support reactions, 173–175, 207–209three-dimensional structures, 207–215three-force body, 199–201two-dimensional structures, 173–183two-force body, 199stable, 600–601unstable, 600–601virtual work conditions, 599–603potential energy and, 599–600, 603stability and, 600–601Equipollent forces, 1120Equipollent particles, 923Equipollent systems, 139–140Equivalent couples, 120–122Equivalent systems of forces, 83–169deformation and, 87external, 85–86internal, 85, 87point of application, 85–86principle of transmissibility and, 84, 86–87reduction to force-couple system, 138–139rigid bodies, 83–169simplifying, 138–150weight and, 85–86Escape velocity, 780Eulerian angles, 1323–1324, 1330Euler’s equations for motion, 1073, 1301–1302Euler’s theorem, 1073Exact differential, 832External forcesacting on a rigid body in plane motion,1119–1120acting on systems of particles, 921–924equivalent systems and, 85–86shear and bending moment conventions, 381FFinite rotation, 1074Firing angle, 673, 678First momentof an area or line, 233, 237–240, 245of volume, 277Fixed-axis rotationangular acceleration, 988, 994angular coordinate, 987, 994angular velocity, 987, 994equations for, 989, 995noncentroidal, 1154, 1170rigid-body motion, 983–985shaft balance, 1304slab representation, 988–989, 994–995three-dimensional motion analysis,1303–1304, 1313–1314I4 IndexFixed (bound) vector, 19Fixed frame of reference, 670Fixed point, motion about a, 984acceleration, 1074, 1080angular momentum, 1284–1285, 1291–1292Euler’s theorem for, 1073instantaneous axis of rotation, 1073–1074plane motion analysis, 1073–1075, 1080rate of change of angular momentum,1300–1301, 1312three-dimensional motion analysis,1302–1303, 1313velocity, 1075, 1080Fixed supports, 174–175, 209Fluid friction, 432–433Fluidscompressibility of, 2flow through a pipe, 957stream diversion by a vane, 957, 964Force. See also Distributed forces; Equivalentsystems of forces; Force systemscentrifugal, 1154concept of, 3concurrent, 21, 140conservative, 598, 832–835, 843constant in rectilinear motion, 802, 818constraining, 173, 177–178, 208conversion of units of, 10coplanar, 20, 141–142direction, 17, 31elastic, 598equilibrium and, 17, 38–46equipollent, 1120equivalent, 86external, 921–924, 1119–1120friction, 432–433, 807, 834, 1154gravitational, 779–780of gravity, 598, 803, 818, 830impulsive, 859, 871input (machines), 350internal, 921–924kinetic friction, 433–434line of action (direction), 18, 57–58magnitude, 17, 54, 57–58output (machines), 350parallel, 142–143parallelogram law for addition of, 4particle equilibrium and, 16–82planar (two-dimensional), 17–53 (See alsoPlanar forces)point of application, 17of rigid-body plane motion, 1119–1124scalar representation, 19, 20–21sense of, 18spring, 803, 818static friction, 433–434systems of particles, 921–924three dimensions of space, 54–75concurrent force resultants, 58rectangular components, 54–57scalar components, 54–55unit vectors for, 56–57vector representation, 18–21weight, 4–5work of, 577–579, 596–598, 801–819Force-couple systems, 122conditions for, 139equipollent, 139–140equivalent systems reduced to, 138–139reactions equivalent to, 174–175reducing a systems of forces into, 138–139resolution of a given force into, 122, 123–124resultant couples, 140–143wrench, 143–144Forced circular frequency, 1393Forced frequency, 1394Forced vibration, 1351caused by periodic force, 1393, 1398caused by simple harmonic motion, 1393, 1398damped, 1385, 1409–1411forced circular frequency, 1393forced frequency, 1394frequency ratio for, 1394magnification factor, 1394–1395,1410–1411, 1413resonance of the system, 1395, 1398undamped, 1393–1399Force systems, 83–169center of gravity, 85concurrent, 138coplanar, 141–142couples, 119–129equipollent, 139–140equivalent, 86–87, 138–150external forces, 85–86force-couple, 122internal forces, 86–87, 300–301moment about an axis, 84moment about a point, 84parallel, 142–143point of application, 85–86position vectors defining, 90, 138reducing into force-couple system, 138–139resolution of a given force into force-couplesystem, 122, 123–124simplifying, 138–150virtual work application to connected rigidbodies, 579–581weight and, 85–86Force triangle, 40Frame of reference, 669–671centroidal, 926, 940fixed, 670general motion, 1091–1092, 1098motion relative to, 670–671moving, 670, 1090–1098newtonian, 724rate of change of a vector, 668, 1056–1057,1066relative position, velocity, andacceleration, 671rotating, 1056–1066, 1090–1091, 1097three-dimensional particle motion,1090–1091, 1097translation of, 668, 671Frames, 301, 334–339collapse of without supports, 335–336equilibrium of forces, 335–336free-body diagrams of force members, 334–335multi-force members, 301, 334statically determinate and rigid, 336statically indeterminate and nonrigid, 336Free-body diagrams, 12equilibriumparticle force, 39–40rigid-body force, 171–173frame analysis using, 334–335machine analysis of members, 350, 353particle motion, 727–728, 746rigid-body constrained motion, 1152–1153,1170rigid-body plane motion, 1122–1124, 1134truss analysis of joint forces, 304two-dimensional problems, 39–40,171–173Free vector, 19Free vibration, 1351damped, 1407–1409, 1412–1413pendulum motion, 1355–1357, 1361of rigid bodies, 1368–1374simple harmonic motion with, 1352–1361undamped, 1352–1361, 1368–1374Frequency, 1351damped circular, 1408forced, 1394forced circular, 1393natural, 1355, 1360natural circular, 1353, 1361units of, 1355Frequency ratio, 1394Friction, 431–484angles of, 435–436axel, 462–463, 467belt, 471–475circle of, 463coefficients of, 434–435disk, 463–464, 467dry, 432–445fluid, 432–433forces of, 432–433journal bearings, 462–463, 467lubrication and, 433, 462potential energy of, 834rolling resistance, 464–465, 467screws, 453–454, 457sliding and, 1154slipping and, 472–473, 475thrust bearings, 462, 463–464, 467vibration caused by, 1407wedges, 453, 457wheel, 464–465, 467work of, 807Frictionless pins, 173–174Frictionless surface supports, 174, 209Fundamental of Engineering Exam, A1–A2GGeneral motion, 984about a fixed point, 1075–1076, 1081acceleration, 1076, 1081relative to a moving frame of reference,1091–1092, 1098velocity, 1075, 1081, 1091–1092, 1098General plane motion. See Plane motionGradient of the scalar function, 833Graphical solutions, 654–657Gravitational force, work of, 804, 819Gravitational units, 8–9Gravity (weight)constant of, 770force of, 803, 818Newton’s law of, 769–770Index I5potential energy with respect to, 598, 603,830–831work of, 597Gyroscopes, 1323–1332axisymmetric body analysis, 1324,1326–1327, 1331–1332Eulerian angles of, 1323–1324, 1330steady precession of, 1324–1326, 1330–1331three-dimensional analysis of motion,1323–1332HHarmonic motion, 1352–1361Homogeneous equation, 1393Hydraulics, 2Hydrostatic force system, 487, 508Hyperbolic trajectory, 779–780, 785–786IImpactscentral, 883, 899coefficient of restitution, 884–886, 888,899–900, 1246direct, 883direct central, 883–886, 899eccentric, 883, 1245–1261energy loss from, 885–886, 900line of, 883oblique, 883oblique central, 886–888, 899–900particle motion, 883–900problems involving multiple kineticsprinciples, 888–890rigid-body plane motion, 1245–1261Impending motion, 434–435, 444sliding, 1154slip, 472Impulse, 858–872eccentric impact and, 1245–1261of a force, 858–859, 871–872, 1261linear, 858momentum and, 858–872principle of impulse and momentum, 800,858–860, 871time interval, 871units of, 858–859Impulse-momentum diagram, 859, 871,886–888, 899, 1261Impulse-momentum principle, 860, 941–942Impulsive force, 859, 871, 1261Impulsive motion, 859–860, 1261Inertia, principle axes and moments of, 1283,1291, 1312Inertia tensor, 1283Inextensible cord, work of forces exerted on,1210Infinitesimal rotation, 1074Initial conditions, 621, 646Input forces, 350, 581Instantaneous acceleration, 618, 666Instantaneous axis of rotation, 1073–1074Instantaneous center of rotation, 985,1023–1030Instantaneous velocity, 618, 665Integrationcentroids determined byof an area, 250–251of volume, 280definite integrals, 621double, 250, 280moments of inertia determined byof an area, 488–489, 494for a body of revolution, 537, 544of a mass, 537, 544for a rectangular area, 488–489for a three-dimensional body, 537using the same elemental strips, 489motion determined by, 621–622theorems of Pappus-Guldinus applied to,251–252triple, 280Internal forces, 85, 368–430acting on systems of particles, 921–924axial forces as, 370, 371beams, 369, 379–387bending moments, 369, 371, 380–381, 386,392–401cables, 369, 407–414, 419–423in compression, 87, 369deformation and, 87equivalent systems and, 85, 87loadings, 379–380, 392–401, 407–408in members, 369–374principle of transmissibility for equilibriumof, 87relations among load, shear, and bendingmoments, 392–401rigid bodies, 85, 87shear and bending moment diagrams for,382, 387shearing forces as, 369, 371, 380–381, 386,392–401structural analysis and, 300–301in tension, 87, 369International System of Units (SI), 5–7JJet engines, steady stream of particles from,958, 964Joints under special loading conditions,306–308Journal bearings, axel friction of, 462–463, 467KKepler’s laws of planetary motion, 782, 786Kinematics, 616Coriolis acceleration, 985, 1059–1060, 1066,1091, 1097degrees of freedom, 640graphical solutions for, 654–657initial conditions for, 621, 646of particles, 615–720curvilinear motion, 665–679dependent motion, 640, 646–647independent motion, 639–640, 646non-rectangular components, 692–703rectilinear motion, 617–631relative motion, 638–647solutions for motion problems, 631–632,646–647three-dimensional (space) motion, 694two-dimensional (planar) motion,692–694uniform rectilinear motion, 638–639of rigid bodies, 982–1114acceleration of, 1037–1047, 1074, 1076,1080–1081, 1091–1092, 1097–1098general motion, 984, 1075–1076, 1081,1091–1092, 1098general plane motion, 984–985,1002–1014, 1037–1047instantaneous center of rotation, 985,1023–1030motion about a fixed point, 984,1073–1075, 1080moving frames, motion relative to,1090–1098rotating frames, motion relative to,1056–1066rotation about a fixed axis, 983–995three-dimensional (space) motion, 985,1073–1081, 1090–1098translation, 983, 985–986, 995two-dimensional (planar) motion,983–1066velocity of, 1002–1014, 1023–1030, 1075,1080–1081, 1092, 1097–1098Kinetic diagramsparticle motion, 727–728, 746rigid-body constrained motion, 1152–1153,1170rigid-body plane motion, 1119–1120, 1134Kinetic energyof a particleprinciple of conservation of energy,833–834, 843principle of work and energy, 804–805, 819of rigid-body plane motionbody in translation, 1196, 1209noncentroidal rotation, 1196–1197of systems of particlescentroidal frame of reference for, 940conservation of energy, 941, 950loss of in collisions, 950work-energy principle, 941of three-dimensional rigid-body motionwith a fixed point, 1287, 1292with respect to the mass center,1286–1287, 1292Kinetic friction force, 433–434Kinetics, 616free-body and kinetic diagrams for, 727–728,746, 1122–1124, 1134constrained motion, 1152–1153, 1170Newton’s second law and, 727–728, 746plane motion, 1119–1120, 1134of particles, 721–798angular momentum, 722, 767–773central-force motion, 723, 768–769, 773,778–786energy and momentum methods, 799–919Kepler’s laws of planetary motion, 782, 786motion of, 721–798multiple principles, problems involving,888–890Newton’s law of gravitation for, 769–770I6 IndexKinetics—Cont.Newton’s second law for, 723–747, 890principle of impulse and momentum, 800,858–860, 871, 890principle of work and energy, 800,804–807, 890of rigid bodies, 1115–1191, 1279–1349angular momentum of, 1118–1119centroidal rotation, 1120, 1134constrained (plane) motion, 1152–1171forces of, 1119–1120general plane motion, 1120–1121, 1134noncentroidal rotation, 1153–1154, 1170plane motion of, 1115–1191principle of transmissibility and, 1121rolling, 1154–1155, 1171systems of, 1124, 1135three-dimensional motion, 1279–1349translation, 1120, 1134Kinetic units, 5–6LLead and lead angle, 454, 457Length, conversion of units of, 10Linear momentumconservation of, 859, 928, 946, 950equations for, 924–925, 932particle motion, 859systems of particles, 924–925, 928, 932,946, 950Linear momentum vector, 1223Line of action, 18force direction representation, 4, 18, 57–58magnitude and, 3–4, 17moment of a force, 91particles, 17–18, 56–58planar (two dimensional) force, 18reactions with known, 173rigid bodies, 91, 173–175three-dimensional (space) force, 56–58unit vector along, 56–57Line of impact, 883Linescentroid of common, 240first moment of, 233, 237–240, 245two-dimensional bodies, 233, 237–240, 245Loading conditionsbeams, 262–263, 270, 369, 379–387cables, 369, 407–414, 419–423center of pressure, 263centroid of the area, 262–269concentrated, 262, 379, 407–408, 413distributed, 262–270, 379, 408, 414, 419–423relations with shear and bending moments,392–401submerged surfaces, 263, 270–271uniformly distributed, 379Lubrication, friction and, 433, 462MMachinesfree-body diagrams of members, 350, 353input forces, 350mechanical efficiency of, 581–582multi-force members, 301, 334output forces, 350structural analysis of, 301, 334, 350–352Magnification factordamped vibration, 1410–1411, 1413undamped vibration, 1394–1395Magnitude, speed as, 618, 665Magnitude of a forceforce characteristics, 3–4, 17–18line of action and, 4, 18, 57–58moments of a force, 90–93particles, 17–18, 54, 57–58reactions with unknown direction and,173–174rigid bodies, 90–93units of, 70vector characteristics, 54Massconcept of, 3conversion of units of, 10–11gain and loss effects on thrust, 959, 964–965moments of inertia, 487, 533–544of common geometric shapes, 499, 538of composite bodies, 537–544, 560integration used to determine, 537, 544parallel-axis theorem for, 534–535, 543of simple mass, 533–534of thin plates, 536–537principle axis and moments, 555–557, 561product of inertia, 553–554, 560Mass centerangular momentum aboutrigid bodies in plane motion, 1119rigid bodies in three-dimensional motion,1282–1283, 1291systems of particles, 926–928, 932center of gravity compared to, 925centroidal frame of reference, 926equations for, 925–926, 932projectile motion and, 926systems of particles, 921, 925–928, 932Mechanical efficiency of machines,581–582, 808Mechanical energy, 833–834Mechanical vibration, 1351. See also Vibrationconservation of energy applications,1382–1386electrical analogs, 1411–1413pendulum motion, 1355–1357, 1361approximate solution, 1355–1356exact solution, 1356–1357oscillations, 1356, 1361of rigid bodies, 1368–1374system displacement as, 1351Mechanicsconversion of units, 10–11of deformable bodies, 2of fluids, 2fundamental concepts and principles, 3–5method of solving problems, 11–13newtonian, 3numerical accuracy, 15of particles, 3–4relativistic, 3of rigid bodies, 2, 4role of statics and dynamics in, 2study of, 2–3systems of units, 5–11Membersaxial forces in, 370, 371free-body diagrams of, 334–335internal forces in, 369–374machine analysis of, 350, 353multi-force, 301, 334, 370–371redundant, 320shearing force in, 371two-force, 301, 371zero-force, 307Method of joints, 304–306Method of sections, 319–326Mixed triple products of vectors, 106–107Mohr’s circle for moments of inertia, 526–530Moment arm, 91. See also Line of actionMoments of a forceabout an axis, 84, 105–109angles formed by two vectors, 106, 112arbitrary point for, 109given origin for, 107–109mixed triple products, 106–107perpendicular distance between lines, 108,112–113projection of a vector for, 106, 112scalar products, 105–106about a point, 84, 90–93line of action (moment arm), 91magnitude of, 90–93position vector of, 90rectangular components of, 93–94right-hand rule for, 90three-dimensional problems, 93–94, 99two-dimensional problems, 92–94, 99Varignon’s theorem for, 93vector products, 90of a couple, 119–120Moments of inertia, 485–574of arbitrary shaped bodies, 555–557, 560of areas, 487–494, 497–509of common geometric shapes, 499, B4for composite areas, 497–509for hydrostatic force system, 487, 508integration used to determine, 488–489, 494,537, 544of masses, 487, 533–544of common geometric shapes, 499,538, B4for composite bodies, 537–544, 560of simple mass, 533–534for thin plates, 536–537, 543Mohr’s circle for, 526–530neutral axis, 487parallel-axis theorem for, 497–509, 517,534–535, 543, 554polar, 486, 489, 494radius of gyration, 489–490, 494, 534second moment as, 486–487, 494transformation of, 516–522, 553–561ellipsoid of inertia, 554–556mass product of inertia, 553–554, 560principle axis and moments, 517–519, 522,555–557, 561product of inertia, 516–517, 522unit-related errors, 543Momentum, 858–872. See also Principle ofimpulse and momentumangular, 924–928, 932angular couple, 1223Index I7conservation of, 859, 883–884, 899, 928, 932direct central impact and, 883–884, 899impulse and, 858–872impulsive force of, 859, 871linear, 883–884, 924–925, 932linear vector, 1223particle motion, 858–872rigid-body plane motion, 1222–1224,1232–1233systems of particles, 922–932total, 859, 871–872Motion, 39, 85–86equations ofEuler’s, 1073, 1301–1302particle kinetics, 727–729, 746radial and transverse components,730, 746rectangular components, 728–729, 746rigid-body kinetics, 1117–1118, 1120, 1134,1280rotational, 1117scalar form, 1120tangential and normal components,730, 746translational, 1117external forces and, 85–86free-body and kinetic diagrams for,727–728, 746impending, 434–435, 444, 472kinematics of a particle, 615–720curvilinear, 665–679dependent, 640, 646–647determination of a particle, 621–622independent, 639–640, 646initial conditions for, 631integration for determination of, 621–622projectile, 670, 678rectilinear, 617–631relative, 638–647kinematics of rigid bodies, 982–1114about a fixed point, 984, 1073–1075, 1080acceleration of, 1037–1047, 1076, 1081,1091–1092, 1097–1098general, 984, 1075–1076, 1081, 1091–1092,1098instantaneous center of rotation, 985,1023–1030moving frames, relative to, 1090–1098plane, 984–985, 1002–1014, 1037–1047rotating frames, relative to, 1056–1066rotation about a fixed axis, 983–995three-dimensional (space), 985,1073–1081, 1090–1098translation, 983, 985–986, 995two-dimensional (planar), 983–1066velocity of, 1002–1014, 1023–1030, 1075,1081, 1092, 1097–1098kinetics of a particle, 721–798angular momentum, 722, 767–773central force, 723, 768–769, 773, 778–786Newton’s second law for, 723–747orbital, 767–773kinetics of rigid bodies, 1115–1191centroidal rotation, 1120, 1134constrained, 1152–1171general plane, 1120–1121, 1134noncentroidal rotation, 1153–1154, 1170plane, 1115–1191rolling, 1154–1155, 1171sliding, 1154–1155, 1171three-dimensional (space), 1279–1349Newton’s first law of, 4, 39relative, 437rotation, 86slipping, 472–473, 475space mechanics, 779–782, 834–835under a conservative central force,834–835under a gravitational force, 779–780gyroscopes, 1323–1332trajectories, 779–782, 786translation, 86weight and, 85–86Moving frame of reference, 670–671acceleration of, 1091–1092, 1097–1098Coriolis acceleration, 1091, 1097in general motion, 1091–1092, 1098rigid-body motion relative to, 1090–1098rotating frame, 1090–1091, 1097three-dimensional particle motion,1090–1091, 1097velocity of, 1090–1092, 1097–1098Multi-force members, 301, 334, 370–371.See also Frames; MachinesNNatural circular frequency, 1353, 1361Natural frequency, 1355, 1360Neutral axis, 487Neutral equilibrium, 600, 603Newtonian frame of reference, 724Newtonian mechanics, 3Newton’s laws, 4–5first law of motion, 4, 39gravitation, 4–5, 769–770motion, 4, 39particles in equilibrium and, 39second law of motion, 4, 723–747application of, 731–745equations of motion, 727–729, 746free-body and kinetic diagrams for,727–728, 746linear momentum and, 723–747mass and, 723of multiple forces, 724radial and transverse components, 730, 746rectangular components, 728–729, 746statement of, 723systems of particles, 922–924tangential and normal components,730, 746third law of motion, 4, 301Noncentroidal rotationabout a fixed axis, 1154, 1170of a body in constrained motion, 1153–1154,1170kinetic energy of a body in, 1196–1197principle of impulse and momentum for,1224Nonhomogeneous equation, 1393Nonimpulsive forces, 1261Nonrigid truss, 320Normal components. See Tangential and normalcomponentsNumerical accuracy, 15Nutation, rate of, 1323OOblique impact, 883central impact, 886–888, 899–900coefficient of restitution, 888, 900conservation of momentum and, 886, 900impulse-momentum diagrams for,886–888, 899Orbital motion, 767–773. See also AngularmomentumOscillations, 1356, 1361, 1369Osculating plane, 694Output forces, 350, 581Overdamped vibration, 1408, 1412Over rigid truss, 320PPappus-Guldinus, theorems of, 251–252Parabolic cables, 409–410, 414Parabolic trajectory, 670, 779–780, 785–786Parallel-axis theoremcomposite area application of, 497–509for mass moments of inertia, 534–535, 543for mass product of inertia, 554for moments of inertia of an area, 497–509for product of inertia, 517Parallel forces, reduction of system of, 142–143Parallelogram law, 4addition of forces, 4addition of two vectors, 19resultant of two forces, 18Particle moments, reduction of inthree-dimensional motion, 1284Particles, 3–4. See also Systems of particlesdirection of a force, 17, 31displacement of, 577equipollent, 923kinematics of, 615–720curvilinear motion, 665–679dependent motion, 640, 646–647independent motion, 639–640, 646non-rectangular components, 692–703radial and transverse components,694–696, 698–701, 703rectilinear motion, 617–631relative motion, 639–640relative to a rotating frame, 1090–1091, 1097solutions for motion problems, 631–632,646–647tangential and normal componentsthree-dimensional (space) motion,694, 1090–1091, 1097two-dimensional (planar) motion, 692–694uniform rectilinear motion, 638–639kinetics of, 721–798angular momentum, 722, 767–773central-force motion, 723, 768–769, 773,778–786linear momentum, 722mass, 722Newton’s second law for, 723–747resultant of forces, 722I8 IndexParticles—Cont.line of action (direction), 18, 57–58magnitude of force, 17, 54, 57–58mechanics of, 3–4resultant of forces, 17–18, 21scalars for force representation, 19, 20–21,29–30statics of, 16–82three-dimensional (space) problems, 54–75adding forces in space, 54–66concurrent force resultants, 58direction cosines for, 55, 56equilibrium of, 38–46force defined by magnitude and twopoints, 57–58rectangular components, 54–57two-dimensional (planar) problems, 17–53adding forces by components, 29–37concurrent force resultants, 21equilibrium of, 38–46free-body diagrams, 39–40Newton’s first law of motion for, 39planar forces in, 17–28rectangular components, 29–31resolving several forces into twocomponents, 32–33unit vectors for, 29–31, 56–57vectors for force representation, 18–21, 29Path-independent forces. See ConservativeforcesPendulum motionapproximate solution, 1355–1356exact solution, 1356–1357impact, 889–890oscillations, 1356, 1361vibration, 1355–1357, 1361Perfectly elastic impact, 885Perfectly plastic impact, 885Perigee, 781Periodic function, 1353Periodic time, 781–782, 786Period of restitution, 883–884, 1245Period of vibration, 1354correction factor for, 1357damped vibration, 1408–1409free vibration equation for, 1354time intervals as, 1351, 1354–1355undamped vibration, 1352–1355, 1360Perpendicular distance between lines, 108,112–113Phase angle, 1354, 1361Phase difference, 1410Pin-connected members, work of forces exertedon, 1210Pin supports, 173–174, 209Pipes, fluid flow through, 957Pitch, 454, 457Planar forces, 17–53equilibrium of, 38–46line of action, 18magnitude of, 17parallelogram law for, 18rectangular components, 29–31resolution into components, 21resultant of several concurrent forces, 21resultant of two forces, 18scalar components, 29–30scalar representation of, 19, 20–21summing x and y components, 32–33unit vectors for, 29–31vector representation of, 18–21Plane motion, 984acceleration of, 1037–1047absolute, 1038normal components, 1037–1039relative, 1037–1039, 1046–1047tangential components, 1037–1039analysis of, 1002–1003, 1039diagrams for rotation and translation,1002–1003, 1014, 1037–1038, 1046equations of, 1117–1118, 1134free-body diagrams for, 1122–1124, 1134instantaneous center of rotation, 985,1023–1030kinetic diagrams for, 1119–1120, 1134particles in, 692–694rigid bodies in, 983–1066, 1115–1278angular momentum of, 1118–1119centroidal rotation, 1120, 1134constrained, 1152–1171energy and momentum methods, 1192–1278forces of, 1119–1120general, 1120–1121, 1134principle of transmissibility and, 1121systems of, 1124, 1135translation, 1120, 1134rotating frames of reference, 1057–1060velocity of, 1002–1014absolute, 1003angular, 1005instantaneous center of zero, 1023relative, 1003–1005Platescenter of gravity for, 240–241circular, 537composite, 240–241mass moment of inertia for, 536–537rectangular, 537thin, 536–537, 543Point of application, 17, 85–86Polar coordinatesangular momentum of particle motion in, 768radial and transverse components, 694–696,698–701, 703Polar moment of inertia, 486, 489, 494Position coordinate, 617–618Position relative to frame of reference, 671Position vector, 90, 138, 665Potential energyconservation of energy, 830–832, 843–844of conservative forces, 832–833determination of, 830–832elastic, 831equations of, 598–599equilibrium and, 599–600, 603friction forces and, 834gravitational, 830–831with respect to gravity (weight), 598, 603of spring forces, 598, 603, 831–832virtual work and, 576, 598–599, 603work of, 830Potential function, 832Poweraverage, 807efficiency and, 807–808, 819from particle motion, 807–808, 819rate of work as, 807–808, 819from rigid-body plane motion, 1199, 1210units of, 807–808Precessionrate of, 1323steady, 1324–1326, 1330–1331Principal normal, 694Principle axes of inertia, 1283, 1291, 1312Principle axis and moments of inertiaabout the centroid, 519of an area, 517–519, 522for a body of arbitrary shape, 555–557, 560ellipsoid of inertia, 554–556of a mass, 555–557, 561Principle of conservation of energy, 833–834Principle of impulse and momentumnoncentroidal rotation, 1224particle motion, 800, 858–860, 871rigid-body plane motion, 1222–1224,1232–1233three-dimensional rigid-body motion,1285–1286, 1292Principle of transmissibility, 4, 84, 86–87equivalent forces of, 86–87rigid-body applications, 84, 86–87for rigid-body plane motion, 1121sliding vectors from, 84, 86Principle of virtual work, 576, 579–581, 588application of, 579–581virtual displacement, 579, 588Principle of work and energyparticle motion, 800, 804–807rigid-body plane motion, 1194–1195, 1209Problems, 11–13error detection, 13force triangle, 40free-body diagrams for, 12, 39–40methods for solving, 11–13SMART method for solving, 11–12solution basis, 11–13space diagram for, 39Product of inertia, 516–517, 522Projectile motion, 670, 672–674, 678Projection of a vector, 106, 112Pure bending, 487QQuadratic surface equation, 554RRadial and transverse componentsacceleration in, 696cylindrical coordinates for, 696, 703equations of motion, 730, 746particle motion analysis using, 694–696,698–701, 703polar coordinates for, 694–696, 698–701, 703velocity in, 696Radial direction, 694, 703Radius of gyration, 489–490, 494, 534Rate of changeof angular momentumfixed-point motion, 1300–1301, 1312particles, 767–768Index I9rotational motion, 1301, 1313three-dimensional rigid-bodies,1300–1301, 1313in polar coordinates, 768rotating frames of reference, 1056–1057of a vector, 668–669, 767, 1056–1057Reactions, 173constraining forces, 173, 177–178equilibrium of rigid bodies and, 173–175,207–209equivalent to force and couple, 174–175free-body diagrams showing, 173with known line of action, 173support, 173–175, 207–209three-dimensional structures, 207–209two-dimensional structures, 173–176with unknown direction and magnitude,173–174Rectangular componentscurvilinear motion of, 669–670equations of motion, 728–729, 746moments of a force, 93–94of particles, 29–31, 54–57, 669–670,728–729, 746planar (two-dimensional) forces, 29–31of rigid bodies, 89–90, 93–94space (three-dimensional) forces, 54–57unit vectors for, 29–31, 56–57vector products, 89–90Rectilinear motion of particles, 617–631acceleration, 618–619, 631constant force in, 802, 818deceleration, 619determination of, 621–630graphical solutions for, 654–657initial conditions for, 621position coordinate, 617–618speed (magnitude), 618uniform, 638–639uniformly accelerated, 638–653velocity, 618work of constant force in, 803, 818Redundant members, 320Relative acceleration, 671, 1037–1038,1046–1047Relative motion, 437curvilinear solution to problems, 670–671,675–679dependent motion, 640, 646–647independent motion, 639–640, 646of particles, 638–647Relative position, 671plane motion, 1003–1005variable systems of particles, 958–959Relativistic mechanics, 3Resonance, 1395, 1398Resultant couples, 140–143Resultant of forces, 17–18concurrent, 21, 58parallelogram law for, 18particle statics, 17–18, 21planar forces, 18, 21of several concurrent forces, 21statics and, 17three-dimensional (space), 58of two forces, 18Revolution, mass moment of inertia for a bodyof, 537, 544Right-handed triad, 88–89Right-hand rule, 88, 90Rigid bodies, 84. See also Systems of rigid bodiesconstraining forces, 173, 177–178, 208couples, 119–129energy and momentum methods, 1192–1278conservation of angular momentum,1225, 1233conservation of energy, 1197–1199, 1210eccentric impact, 1245–1261kinetic energy, 1196–1197, 1209noncentroidal rotation, 1196–1197, 1224power, 1199, 1210principle of impulse and momentum,1222–1224, 1232–1233principle of work and energy, 1194–1195,1209systems, analysis of, 1197, 1210, 1225work of forces, 1195–1196, 1209equilibrium of, 170–231statically determinate reactions, 177statically indeterminate reactions, 177–178support reactions for, 173–175, 207–209three-dimensional structures, 207–215three-force body, 199–201two-dimensional structures, 173–183two-force body, 199equivalent systems of forces, 83–169center of gravity, 85deformation and, 87external forces, 85–86internal forces, 85, 87point of application, 85–86reduction to force-couple system,138–139simplifying, 138–150weight and, 85–86force-couple systemsequipollent, 139–140equivalent systems reduced to, 138–139reducing a systems of forces into, 138–139resolution of force into, 122, 123–124resultant couples, 140–143wrench, 143–144free-body diagrams for, 171–173free vibration of, 1368–1374kinematics of, 982–1114acceleration of, 988, 994, 1037–1047,1074, 1076, 1080–1081, 1091–1092,1097–1098in general motion, 984, 1075–1076, 1081,1091–1092, 1098general plane motion, 984–985,1002–1014, 1037–1047instantaneous center of rotation, 985,1023–1030motion about a fixed point, 984,1073–1075, 1080rotating frames, motion relative to,1056–1066, 1090–1091, 1097rotation about a fixed axis, 983–995three-dimensional (space) motion, 985,1073–1081, 1090–1098translation, 983, 985–986, 995two-dimensional (planar) motion,983–1066velocity of, 1002–1014, 1075, 1080–1081,1092, 1097–1098kinetics of, 1115–1191angular momentum of, 1118–1119centroidal rotation, 1120, 1134constrained plane motion, 1152–1171forces of, 1119–1120general plane motion, 1120–1121, 1134plane motion, 1115–1191principle of transmissibility and, 1121systems of, 1124, 1135three-dimensional (space) motion,1279–1349translation, 1120, 1134mechanics of, 2, 4moment of a force about an axis, 84moment of a force about a point, 84momentsof a couple, 119–129of a force about an axis, 105–109of a force about a point, 90–93principle of transmissibility and, 4, 84, 86–87reactions, 173–175rectangular components, 89–90, 93–94scalar (dot) products, 105–106sliding vector representation, 19, 84vector products, 87–89, 105–109virtual work application to systems ofconnected, 579–581Rigid truss, 302Rocker supports, 172–174Roller supports, 173Rollingangular acceleration, 1154–1155resistance, 464–465, 467sliding and, 1154–1155, 1171unbalanced disk or wheel, 1155, 1171Rotating frames of referenceCoriolis acceleration, 1059–1060, 1066plane motion of a particle relative to,1057–1060rate of change of a vector, 1056–1057, 1066rigid-body motion relative to, 1056–1066,1090–1091, 1097three-dimensional particle motion,1090–1091, 1097Rotation, 86, 983about a fixed axisangular acceleration, 988, 994angular coordinate, 987, 994angular velocity, 987, 994equations for, 989, 995noncentroidal, 1154, 1170rate of change of angular momentum,1301, 1313rigid-body motion, 983–985slab representation, 988–989, 994–995centrifugal force, 1154centroidal, 1120, 1134curvilinear translation compared to, 984finite, 1074force as, 86infinitesimal, 1074instantaneous axis of, 1073–1074instantaneous center of, 985, 1023–1030motion about a fixed point, 1073–1075noncentroidalof a body in constrained motion,1153–1154, 1170kinetic energy of a body in, 1196–1197I10 IndexRotation—Cont.plane motion diagrams, 1002–1003, 1014,1037–1038, 1046uniform, 1154Rotational equation of motion, 1117Rough surface supports, 174, 209SSag, 409Scalar productof vector functions, 668of vectors, 105–106Scalars, 19particle force representation, 19product of vector and, 20–21rectangular force components, 29–30, 54–55Screws, 453–454, 457friction and, 453–454, 457lead and lead angle, 454, 457pitch, 454, 457self-locking, 454square threaded, 453–454, 457Self-locking screws, 454Shaft rotation, balance of, 1304Shearing forces, 371–374beams, 369, 371, 380–381, 386, 392–401bending moment relations with, 393diagrams for, 382, 387external forces and, 381internal forces as, 369, 371–374load relations with, 392–393Simple truss, 301–304, 308Slab representation for fixed-axis rotation,988–989, 994–995Sliding motion, 1154–1155, 1171Sliding vectors, 19, 84, 86Slipping, belt friction and, 472–473, 475Slipstream, 958SMART method for solving problems, 11–12Space, concept of, 3. See also Threedimensional problemsSpace centrode, 1025Space cone, 1074Space diagram, 39Space mechanicsconservation of energy, 834–835under a conservative central force, 834–835eccentricity, 779–780escape velocity, 780gravitational force, 779–780gyroscopes, 1323–1332analysis of motion, 1323–1332axisymmetric body analysis, 1324,1326–1327, 1331–1332Eulerian angles of, 1323–1324, 1330steady precession of, 1324–1326,1330–1331initial conditions, 780–781Kepler’s laws of planetary motion,782, 786periodic time, 781–782, 786projectile motion, 670, 672–674, 678thrust, 958–959, 964–965trajectories, 779–782, 786Space truss, 308Span, 379–380, 409Speed (magnitude), 618, 665Spin, rate of, 1323Spring forcepotential energy of, 831–832virtual work of, 803, 818, 1210work of elastic force, 597–598, 603Square threaded screws, 453–454Stable equilibrium, 600–601, 603Statically determinate reactions, 177, 336Statically indeterminate reactions, 177–178,208, 336Static friction force, 433–434Staticsof particles, 16–82resultant of forces, 17–18, 21role of in mechanics, 2state of equilibrium, 17Steady-state vibration, 1385, 1398Steady stream of particles, 956–958, 964fan flow, 958fluid flow through a pipe, 957fluid stream diversion by a vane, 957, 964helicopter blade flow, 958jet engine flow, 958units for, 957Structural analysis, 299–367frames, 301, 334–339internal force reactions, 300–301machines, 301, 334, 350–352multi-force members, 301, 334Newton’s third law for, 301trusses, 301–310, 319–326two-force members, 301virtual work applications, 579–581zero-force members, 307Structuresanalysis of, 299–367equilibrium of, 173–183, 207–215statically determinate reactions, 177, 336statically indeterminate reactions, 177–178,208, 336three-dimensional, 207–215two-dimensional, 173–183Submerged surfaces, distributed forces on, 263,270–271Support reactions, 173–175, 207–209fixed, 174–175frictionless pins, 173–174of one unknown and one direction, 173–174rollers and rockers, 172–174static determinacy and, 336three-dimensional structures, 207–209, B2two-dimensional structures, 173–175, B1Symmetry, planes of, 280Systems of particles, 920–981conservation of energy in, 941, 950conservation of momentum in, 928, 932,946, 950energy and momentum methods for, 940–950impulse-momentum principle, 941–942kinetic energy, 940, 950work-energy principle, 941external and internal forces acting on,921–924mass center of, 921, 925–928, 932momentum in, 922–932angular, 924–928, 932, 946, 950linear, 924–925, 928, 932, 946, 950Newton’s second law for, 922–924variable, 956–965fluid flow, 957fluid stream diversion, 957, 964mass gain and loss, 959, 964–965relative velocity, 957, 959steady stream of particles, 956–958, 964thrust, 958, 959, 964Systems of rigid bodiesconstrained (plane) motion of, 1171plane motion of, 1124principle of impulse and momentum for,1225, 1233principle of work and energy for, 1124Systems of units, 5–11converting between, 10–11International System of Units (SI), 5–7U.S. customary units, 8–9, 12TTangential and normal componentsacceleration in, 693, 702, 1037–1039equations of motion, 730, 746particle analysis using, 692–694,697–698, 702rigid-body analysis using, 1037–1039three-dimensional (space) motion, 694two-dimensional (planar) motion, 692–694,1037–1039Tension, deformation from internal forces of, 87TheoremsEuler’s, 1073Pappus-Guldinus, 251–252parallel-axis, 497–509, 517, 534–535, 543Varignon’s, 93Theory of relativity, 3Thin plates, mass moment of inertia for,536–537, 543Three-dimensional bodies, 276–284center of gravity, 276–278, 285centroid of volume location, 276–278, 285composite bodies, 278–280Three-dimensional (space) motionabout a fixed pointanalysis of, 1302–1303, 1313angular momentum of, 1284–1285,1291–1292instantaneous axis of rotation, 1073–1074about a mass center, 1282–1283, 1291angular momentum inabout a fixed point, 1284–1285, 1291–1292about a mass center, 1282–1283, 1291inertia tensor, 1283principle axes of inertia, 1283, 1291, 1312rate of change of, 1300–1301, 1313reduction of particle moments, 1284of rigid bodies, 1281–1285, 1291,1300–1304, 1312–1313energy and momentum in, 1281–1292angular momentum, 1281–1285, 1291kinetic energy, 1286–1287, 1292principle of impulse and momentum for,1285–1286, 1292equations and principles for, 1280–1281Euler’s equations for, 1073, 1301–1302general, 1075–1076, 1081, 1091–1092, 1097Index I11gyroscopes, 1323–1332axisymmetric body analysis, 1324,1326–1327, 1331–1332Eulerian angles of, 1323–1324, 1330steady precession of, 1324–1326,1330–1331kinematics of, 694, 985, 1073–1081,1090–1098kinetics of, 1279–1349of particles, 694, 1090–1091, 1097relative to moving frame of reference,1090–1098of rigid bodies, 985, 1073–1081, 1090–1098,1279–1349rotation about a fixed axis, 1303–1304,1313–1314solutions for problems, 1300–1314Three-dimensional (space) problems, 54–75adding forces in, 54–66concurrent force resultants, 58direction cosines for, 55, 56equilibrium in, 67–75, 207–215forces in, 54–75line of action, 56–58magnitude of force, 57–58moments of a force about a point, 93–94, 99particles, 54–75rectangular components, 54–57rigid bodies, 93–94, 99, 207–215support reactions, 207–209unit vector for, 56–57Three-force body, equilibrium of, 199–201Thrustfluid flow causing, 958, 964mass gain and loss required for rockets,959, 965units for, 959Thrust bearings, disk friction of, 462,463–464, 467Time, concept of, 3Time intervalimpulse of a force, 871period of damped vibration, 1408–1409period of undamped vibration, 1352–1355,1360Trajectorycentral-force motion and, 778–786elliptical, 779–780, 785–786hyperbolic, 779–780, 785–786parabolic, 670, 779–780, 785–786of a particle, 778–779periodic time, 781–782, 786of space mechanics, 779–782, 786Transient vibration, 1394. See also FreevibrationTranslation, 86, 983curvilinear motion and, 668external forces from plane motion,1120, 1134force as, 86kinetic diagrams for, 1120, 1134kinetic energy of a body in, 1196plane motion diagrams for, 1002–1003, 1014,1037–1038, 1046rigid body in, 985–986, 995, 1120, 1134Translational equation of motion, 1117Transmissibility. See Principle oftransmissibilityTransverse components. See Radial andtransverse componentsTransverse direction, 694, 703Triangle rule for addition of vectors, 19–20Triple integration, 280Trusses, 301–310, 319–326analysis of, 301–310, 319–326compound, 320free-body joint diagrams, 304joints under special loading conditions,306–308method of joints, 304–306method of sections, 319–326nonrigid, 320over rigid, 320redundant members, 320rigid, 302simple, 301–304, 308space, 308two-force members, 301zero-force members, 307Two-dimensional bodies, 233–244center of gravity, 234–235, 245centroid of area and line location, 235–237,239–240, 245composite plates and wires, 240–244first moment of an area or line, 237–240, 245planar elements, 233–244Two-dimensional (planar) motionof particles, 692–694of rigid bodies, 983–1066Two-dimensional (planar) problemsequilibrium in, 173–183moments of a force, 92–94, 99rigid-body structures, 173–183statically determinate reactions, 177statically indeterminate reactions, 177–178support reactions, 173–175Two-force body, equilibrium of, 199Two-force members, 301, 371. See also TrussesUUnbalanced rolling disk or wheel, 1155, 1171Undamped vibrationforced vibration, 1352, 1393–1399free vibration, 1352–1361, 1368–1374simple harmonic motion, 1352–1361Underdamped vibration, 1408, 1412Uniformly accelerated rectilinear motion, 638–653Uniformly distributed loads, 379Uniform rectilinear motion of particles, 638–639Uniform rotation, 1154Units, 5–11of area and volume, 7–8basic, 5converting between systems, 10–11derived, 5of energy, 577of force, conversion of, 10of frequency, 1355gravitational, 8–9of impulse, 858–859International System of Units (SI), 5–7kinetic, 5–6of length, conversion of, 10of mass, conversion of, 10–11of power, 807–808quantity equivalents of SI and U.S.customary, 12SI abbreviations (formulas) of, 8SI prefixes, 7for steady stream of particles, 957systems of, 5–11for thrust, 959of work, 802Unit vectors, 29–31, 56–57Universal joint supports, 209Unstable equilibrium, 600–601, 603U.S. customary units, 8–9, 12VVanes, fluid stream diversion by, 957, 964Variable systems of particles, 956–965fluid flow, 957fluid stream diversion, 957, 964mass gain and loss, 959, 965relative velocity, 958–959steady stream of particles, 956–958, 964thrust, 958, 959, 964–965Varignon’s theorem, 93Vector products, 87–89, 105–109commutative property and, 88distributive property and, 88mixed triple products, 106–107moment of force about a given axis, 105–109moment of force about a point, 87–89rectangular components of, 89–90right-hand rule for, 88, 90of scalar products, 105–106triple product, 988of vector functions, 668Vectors, 18–21acceleration, 666–667, 669–670addition of, 19–21addition of, parallelogram law for, 19addition of, polygon rule for, 20addition of, triangle rule for, 19–20angle formed by, 106angular momentum of particles as, 767coplanar, 20curvilinear motion and, 665–679derivatives of functions, 667–669displacement, 665equal and opposite, 19fixed (bound), 19force addition using, 18–21, 54–57frame of reference, 668free, 19function, 665–666linear momentum, 1223mixed triple products, 106–107moments of a force, 90, 105–109about a given axis, 105–109about a point, 90negative, 19–20particle force representation, 18–21planar forces, 18–21position, 90, 138, 665product of scalar and, 20–21projection of, 106rate of change of, 668–669, 1056–1057, 1066rectangular force components, 29–31, 669–670I12 IndexVectors—Cont.rigid-body representation, 84, 86sliding, 19, 84, 86subtraction of, 19–20three-dimensional forces, 55–57unit, 29–31, 56–57velocity, 665–666, 669–670Velocityabsolute, 1003angular, 987, 994, 1005areal, 769average, 618, 665curvilinear motion and, 665–666determining, 618escape, 780general motion, 1075, 1081, 1091–1092, 1098instantaneous, 618, 665instantaneous center at zero, 1023instantaneous center of rotation for,1023–1030moving frames of reference, 1090–1092,1097–1098plane motion, 1002–1014radial and transverse components of, 696rectangular components of, 669–670rectilinear motion and, 618relative, 671, 958–959, 1005rotating frame of reference, 1090, 1097speed (magnitude), 618, 665three-dimensional (space) motion, 1075,1080–1081, 1090–1092, 1097–1098two-dimensional (planar) motion, 1002–1014variable systems of particles, 958–959vector, 665–666, 669–670Vibration, 1350–1432amplitude, 1351, 1354, 1361conservation of energy applications,1382–1386damped, 1352, 1407–1418forced, 1351, 1393–1399, 1409–1411free, 1352–1361, 1368–1374, 1407–1409,1412–1413frequency, 1351, 1353, 1355, 1361, 1393–1394oscillations, 1356, 1361, 1369period, 1351, 1354, 1360, 1408–1409periodic function, 1353phase angle, 1354, 1361of rigid bodies, 1368–1374simple harmonic motion, 1352–1361steady-state, 1385, 1398transient, 1394undamped, 1352–1361, 1368–1374, 1393–1394Virtual work, 575–614displacement of a particle, 577equilibrium conditions, 599–603mechanical efficiency of machines, 581–582method of, 576–588potential energy and, 576, 598–599, 603principle of, 576, 579–581, 588application to systems of connected rigidbodies, 579–581virtual displacement, 579, 588workof a couple, 578during finite displacement, 596–598of a force, 577–579, 596–598input, 581output, 581virtual, 579, 588Viscosity. See Fluid frictionVolume, units of, 7–8WWedges, 453, 457Weight, 4–5center of gravity, 85external force as, 85–86as a force, 4–5gravity and, 597–598, 603point of application, 85potential energy effected by, 598, 603rigid-body motion and, 85–86work of, 597Wheel friction, 464–465, 467Workof a couple, 578during finite displacement, 596–598of a force, 577–579, 596–598constant in rectilinear motion, 802, 818gravitational, 804, 819of gravity, 803, 818, 830–831for particle motion, 801–819for pin-connected members, 1210for potential energy, 830–832principle of work and energy, 800,804–807, 1194–1195, 1209for rigid-body plane motion, 1194–1195,1209–1210of a spring, 597–598, 803, 818, 1210input, 581output, 581virtual, 579, 588of a weight (gravity), 597Work-energy principle for systems ofparticles, 941Wrench, reduction of force-couple forces into,143–144ZZero-force members, 307**

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