Fundamentals of Biomechanics Equilibrium, Motion, and Deformation - Fourth Edition Nihat O¨ zkaya David Goldsheyder Margareta Nordin Project Editor: Dawn Leger

Appendix C: Calculus 447 Compare this with Eq. (C.47) to observe that: a ¼ 1 b ¼ 2 c ¼ À8 Substitute a, b, and c into Eq. (C.48): x ¼ Àð2Þ Æ qffiÀffiffi2ffiffi2ffiffiÁffiffiffiÀffiffiffiffiffi4ffiffiðffiffi1ffiffiffiÞffiðffiffiÀffiffiffiffi8ffiffiÞffiffi ¼ À2 pffiffiffiffiffi ¼ À2 Æ 6 2ð1Þ Æ 36 2 2 Consider the plus sign: x ¼ À2 þ 6 ¼ 4 ¼ 2 2 2 Consider the minus sign: x ¼ À2 À 6 ¼ À8 ¼ À4 2 2 Note that x ¼ 2 and x ¼ À4 are the “roots” of the quadratic equation. The original equation can now be expressed as: ðx À 2Þðx þ 4Þ ¼ 0 C.6 Exercise Problems Problem C.1 Show that slopes of the graph of the function Y ¼ 1 À X2 at X ¼ À2, X ¼ 0; and X ¼ 2 are 4, 0, and À4, respectively. Problem C.2 Evaluate the derivatives of the following functions with respect to X. FUNCTIONS ANSWERS Y ¼ 5x2 Y ¼ À4:5xÀ2 10x Y ¼ 3 sin x 9 x3 Y ¼ ð2 þ 8 sin xÞ 3 cos x Y ¼ X2 sin X 8 cos x Y¼ X 2X sin X þ X2 cos X cos X cos X þ X sin X Y¼ cos X cos 2X X ÀX sin X À cos X X2 (continued)

448 Appendix C: Calculus FUNCTIONS ANSWERS pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 Àffiffiffiffiffi3ffiffiXffiffiffiffi2ffiffiffiffi Y ¼ X À X3 2 X À X3 p2ffiffiÀffiffiffiffiffi1ffiffi0ffiffixffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2x À 5x2 Y ¼ 2x À 5x2 Problem C.3 Show that the integral of the function Y ¼ cos X with respect to X between X ¼ 0 and X ¼ 180 is 2. (Examine the graph of the function before evaluating the integral.) Problem C.4 Evaluate the following integrals. INTEGRALS ANSWERS ð 5x + c0 5dx 4x2 + c0 ð 8xdx ð À6 cos x þ c0 6 sin xdx ð X2 þ cos X þ c0 ðX À sin XÞdX 2 ð  1  1 1 X2 X À dX X þ þ c0 ð2  1  0.5 1 X2 À dX 1 4ð5 1 ð cos X þ sin XÞdX 0 ð4 112 6x2dx 2 ð2 ð6 144 2x2dx þ 2x2dx 02 Problem C.5 Show that the roots of the quadratic equation x2 À 7.5x ¼ 4 are x ¼ 8 and x ¼ À0.5.

Index1 A pendulum, 208, 208f torque and angular acceleration, Abductor gait, 128 polar coordinates, 205, 205f 239–240, 239f, 240f Acceleration position and displacement, work and power angular, 64, 206–207, 239–240, 205–206, 205f, 206t angular displacements, 248–249 239f, 240f position vs. time, 213, 214, 214f definition, 248 range of motion, 209 knee extension, 250, 250f average, 152 relative motion, 220–222, 220f work–energy theorem, 249 constant, 155–156, 156f rotational motion definition, 6, 145 Angular motion, 64 normal, 217 about fixed axis, 217–218, Ankle, 133–134 tangential, 217 217f, 218f Antagonist muscle, 105 vector Apparent yield strength, 295 constant acceleration, 219–220 Area, 10t radial/centripetal, 217 shoulder abduction, 211–212, 211f Area moment of inertia, 347 tangential and normal simple harmonic motion, 208f, Articular cartilage, 103, 381–382, component, 217, 217f 209–210 381f, 382f Acromioclavicular joint, 113 uniform circular motion, 219 Atlantoaxial joint, 117 Actin and myosin, 104, 380 velocity, 206, 210, 210f, 211, 211f Atlantooccipital joint, 116 Angular acceleration, 10t, 64 Angular kinetics Axis Angular kinematics mass moment of inertia, 240–242 of motion centroidal, 241 acceleration, 206–207 neutral, 339, 347 air resistance, 208, 210f arms, gymnast vs. angular rotational motion about, 217–218, amplitude, 209 position, 237, 237f coordinate transformation, 216 217f, 218f damped oscillation1, 210, 211f circular end region, ski jump, dimensions and units, 207–208 238, 238f B displacement, velocity, and Balance method, 89f free-body diagrams, 194, 194f Ballistic pendulum, 266, 267f acceleration vs. time, 214f, 216 gymnast, on high bar, 234, 234f Beam(s) flexion-extension test, 214, 214f net force, 233 frequency, 209 rotational motion, 233f, 234 cantilever, 57, 80, 80f inverted pendulum, 214 speed vs. angular position, hinged to ground, with roller linear and angular quantities, 235, 235f on top, 94, 358 218–219 parallel-axis theorem, 242, 242f hinged to wall, with cable support, linkage systems radius of gyration, 242 rotational kinetic energy, 76, 76f circular motion, 223, 225f L-shaped double pendulum, 223, 223f 247–248, 247f single pendulum, 222, 223f segmental motion analysis scalar analysis of, 83–84, 83f tangential velocity and normal vector analysis of, 84–85, 84f angular displacement on wedge and roller support, 72–73 acceleration, 224f, 225 and time, 243 Bending uniform circular motion, bench test, 353, 353f dynamic model, 243 cantilever beams, 346, 346f 219, 223f knee extension, 244–247, 244f centroids and neutral axes, 347, 347f unit vectors, 224–226, 225f two-dimensional (planar) motion neutral/equilibrium position, 208 analysis, 243–244 velocity and displacement, 244 1 Note: Page numbers followed by f indicate figures; those followed by t indicate table. 449 # Springer International Publishing Switzerland 2017 N. O¨ zkaya et al., Fundamentals of Biomechanics, DOI 10.1007/978-3-319-44738-4

450 Index Bending (cont.) Kelvin-Voight model, statically determinate and element analyses, 352, 352f 365–366, 365f indeterminate systems, 281 femur, 353 flexural stress distribution, 347, 348f linearly elastic material, stress and strain, 283–284, 283f, 284f flexure formula, 347 363, 363f Degrees of freedom, 223 free-body diagram, 350–351, 350f Diarthrodial joints, 31, 103, 103f, 110, material elements, stress Maxwell model, 366–367, 366f components, 351, 351f standard solid model, 121, 381 method of section, 344 Dimensional analysis, 7–9 negative shear forces, 345f, 346 367–368, 367f Dynamics, 4 normal and shear force, 349, 349f stress and strain, 363 parameters involved definitions, Biomechanics, 5–6 acceleration, 145 349, 349f Bone definitions, 143 positive shear forces, 345, 345f composition, 373, 373f differential and integral shear, and moment diagram, fractures, 377 344, 345f mechanical properties, 374–376, calculus, 147 shear stress distribution, 349, 349f distance and displacement, straight beam, 344, 344f 374f–376f stress distribution, 351, 351f structural integrity, 376–377, 377f 145, 145f tension and compression, 347, 347f Built-in member, 72t friction, 29 three and four-point bending, Built-in structures inertia and momentum, 146 344, 344f cantilever beam, 80, 80f kinematics and kinetics, 143 three-point bending apparatus, L-shaped beam, 82–84, 82f, 83f linear, angular, and general 350, 350f C motions, 144, 144f Bending effects, 39, 39f Cable-pulley systems, 78–80 particle concept, 146–147 Biaxial and triaxial stresses Cannonball, 167 reference frames and coordinate Cantilever beams, 80 block loading, 323, 323f Center of gravity systems, 147, 147f cubical material element, 320, 320f speed and velocity, 145 first order tensors, 322 determinations of, 88–99 vector algebra, 147 normal and shear components, of system, 91 Dynamometer, 214, 214f Center of mass, 27, 88 322, 322f Centroidal axis, 241 E plane stress components, 322, 322f Collinear forces, 26 Elastic deformations, 295–297, 295f, second order tensors, 322 Compressive force, 25 stress–strain relationships, 321, 321f Concentrated load, 27 296f superposition method, 321, 321f Concurrent forces, 26, 26f Elasto-plastic collision, 268–270, tensile/compressive, 321 Constitutive equations, 297 Biological tissues Coplanar force system, 25 268f, 269f articular cartilage, 381–382, moments due to, 66f Elbow net moment, 42–43, 42f 381f, 382f scalar method, 48 arm, 108, 108f bone Coulomb theory, 331 bones, 107, 107f Couple, 47, 47f brachialis and brachioradialis composition, 373, 373f Couple moment, 47 fractures, 377 Crash test, 263, 263f muscles, 111 mechanical properties, 374–376, Cross (vector) product, 48 diarthrodial joint, 110 forearm, 108, 108f 374f–376f D humeroradial and humeroulnar structural integrity, 376–377, 377f D’Alembert principle, 285 characteristics of, 371–373, 373f Damped oscillations, 210 joints, 107 elasticity vs. viscoelasticity, Deformable body mechanics injuries of, 108 joint reaction force, 108–109, 110f, 134 369–371, 369f–371f applied forces muscles, 107, 107f skeletal muscles, 379–381, 380f and deformations, 282 parallel force system, 109 springs and dashpots, 364–365, 364f pronation and supination tendons and ligaments, 378–379, internal forces and moments, 282–283 movements, 108 378f, 379f proximal radioulnar joint, 107 time-dependent material response, mathematics, 285 rotational and stabilizing/sliding procedure, 284–285 368–369, 368f rigid body mechanics, 281 components, 110, 110f viscoelasticity, 363–364, 363f–364f three-muscle system, 111, 111f Endurance limit, 334–335 definition, 363 Energy, 12t empirical models, 365 Engineering mechanics, 6 Equation(s) equilibrium, 70–71 of motion, 63

Index 451 Equilibrium, 7 Frictional forces, 29–31, 29f, 30f, 30t impact and collisions, 264–265, 265f conditions for, 65–67 Frontal plane, 92 linear definition of, 65 rotational, 65 G average force and an average static, 65 Glenohumeral articulation, 112 time, 256, 256f translational, 65 Golfer’s elbow syndrome, 108 Gravitational acceleration, 26t definition, 255 Equilibrium equations, 70–71 Gravitational force, 26–27, 26t force vs. time curve, 256, 256f Equilibrium systems Gravity, center of impulse-momentum analysis of, 68–69 determinations of, 88–93, 89f, 90f, theorem, 255 built-in structures 91f, 92f, 92t impulsive forces and impulsive cantilever beam, 80, 80f leg, 92, 92f motions, 256 L-shaped beam, 82–84, 82f, 83f Gravity line, 89 scalar and vector additions, cable-pulley systems, 78–80 Greek alphabet, 13t center of gravity determinations in, 257, 257f H units, 257, 257t 88–99 Hammering, 64f vector quantity, 255 conditions for, 65–67 Hamstring muscle, 122 one-dimensional collisions constraints and reactions in, 71, 72t Hip elastoplastic collision, definition, 63 equations on, 69–71 bony structure, 122 268–269, 268f free-body diagrams, 67–68, 68f femoral head, 121, 122 perfectly elastic collision, 265, friction in, 86–88, 86f–88f forces, 122, 124f simply supported structures, 71–78, free-body diagram 267, 267f perfectly inelastic collision, 73f, 74f, 76f–78f abductor gait, 128 traction devices, 78–80, 79f carrying load, hand, 126, 126f, 266, 266f Equivalent force, 27 rigid bodies in plane motion, External forces, 24–25 127, 127f center of gravity, 126–127, 126f 275–276, 275f F concurrent system, 125, 125f two-dimensional collisions First law, Newton’s, 7, 63 gravitational force, 127 Flexion–extension test, 214–216, 214f leg, 123–124, 124f conservation of momentum, 270 Force(s), 10t, 15 muscle and joint reaction forces, elastic collision, 273, 273f Force vector oblique central impact, pool 125, 125f collinear, 26, 26f pelvis, 124, 124f balls, 270, 270f compressive, 25, 25f three-force system, 125, 126f target and cue ball, 270–273 concurrent, 26, 26f translational equilibrium, Impulse-momentum theorem, 255 coplanar, 25 Inertia definition, 23 125–126 area moment, 347, 350, 353 dimension and units of, 23–24 pelvis, 121, 121f mass moment of, 64 distributed force systems and single-leg stance, 122, 123f Internal forces, 24–25 Hooke’s law, 297, 297f International System of Units (SI), 10t pressure, 27–28, 27f, 28f, 28t Humeroradial joint, 107 conversion of, 11t–12t equivalent, 27 Humeroulnar joint, 107, 107f external, 24–25 Hysteresis loop, 299–300, 299f K frictional, 29–31, 29f, 30f, 30t Kelvin-Voight model, 365–366, 365f gravitational/weight, 26–27, I Kinematics, 4 Impulse and momentum Kinetic friction, 29 26t, 27f Kinetics internal, 24–25 angular, 273–274 normal, 25, 25f applications angular (see Angular kinetics) parallel, 26, 26f definition, 4 properties of, 23, 23f ball hits, floor and bounces, linear, 181–202 (see also Linear reactive, 71 258–259, 258f resultant, 23, 23f kinetics) tangential, 25, 25f crash test, 263, 263f rigid bodies, plane motion, tensile, 25, 25f force platform, 261, 261f translation, 47–48, 48f force vs. time, athlete, 275–276, 275f Free body diagrams, 67–68, 67f, 68f Knee Friction 261–262, 261f coefficient of, 30, 30t soccer player, 259, 259f exercises, 129, 129f kinetic and static, 29–30 basic equations, 274, 274t, 275 forces acting, lower leg, 129, 130f systems with, 86–88, 86f, 87f, 88f conservation of linear lever arms, 130, 130f muscles, 129, 129f momentum, 264 patella, 131–132, 131f, 132f patellofemoral joint, 128, 128f, 131 rotational and translatory components, 131, 131f tibial plateau, 131 tibiofemoral joint, 128, 131f

452 Index L free-body diagram, 194–196, vector product, 48–53, 48f, 49f Le Syste`me International d’Unite´s 194f, 195f in wrench and bolt, 40, 40f Moment arm, 39 (SI), 9 pendulum, 195, 195f Motion(s) Ligamentous capsule, 103, 103f ski jumper, 195, 195f angular, 64 Linear kinematics velocity and displacement, 197 from coplanar force, 66f equations of motion, 181–183, 182f equations of, 63 athletics applications mechanical work linear, 64 center of gravity, 170, 170f, 171, constant force, 188–189, 188f, 189f Multiaxial deformations and stress 171f, 174–175, 174f scalar product, 189–190 diver, 174–175, 173f varying force, 189, 189f analyses long jumper, 170, 170f Newton’s second law allowable stress and safety factor, shot-putter, 172, 172f trajectory, 171, 172f of motion, 143 332–333 potential energy, 190–191, 191f bending biaxial motion power, 192, 193t with constant acceleration, procedure, 185–187 bench test, 353f, 353f 166–167 translational motion, 183–185 cantilever beams, 346, 346f two-and three-dimensional work and energy methods, centroids and neutral axes, linear movements, 163 187–188 347, 347f dimensions and units, 153–154, 154t work–energy theorem, 191 element analyses, 352, 352f displacement, velocity, and Linear motion, 64 femur, 353 Load intensity, 28 flexural stress distribution, acceleration, 151–153, 151f, Longitudinal plane, 92 152f, 153t L-shaped beams 347, 348f measured and derived quantities, scalar analysis of, 82–84, 83f flexure formula, 347 154–155 vector analysis of, 84–85, 84f free-body diagram, 350–351, 350f position, velocity, and acceleration Lubricants, 31 method of section, 344 vectors, 163–165, 163f, 164f negative shear forces, 345f, 346 projectile motion M normal and shear force, 349, 349f cannonball fired, 167, 167f Mathematics, 12–13 parameters involved definitions, gravity and air resistance, 167 Maximum distortion energy theory, motion trajectory, 167 349, 349f speed of release, 167 331–332 positive shear forces, 345, 345f time lapses, 169 Maxwell model, 366–367, 366f shear, and moment diagram, trajectory, 168, 168f Mechanical work velocity vector, 168, 168f 344, 345f uniaxial motion, 151 constant force, 188–189, 188f, 189f shear stress distribution, car speed, 157–161 definition, 188 constant acceleration, scalar product, 189–190 349, 349f 161–162, 161f varying force, 189, 189f straight beam, 344, 344f with constant acceleration, Mechanics, 3–5 stress distribution, 351, 351f 155–156 Mises-Hencky theory, 332 tension and compression, displacement and acceleration vs. Mohr’s circle time, 160, 160f material element, 329–330, 347, 347f free fall, 162, 162f three-point bending apparatus, impact speed, 163 329f, 330f relative position vs. time, maximum shear stress, 326 350, 350f 157, 157f plane stress element, 327, 327f biaxial and triaxial stresses skier speed, 161, 161f positive and negative stresses, 327 speed vs. time, 157, 158, 158f principal stresses, 327 block loading, 323, 323f uniformly accelerated Moment(s), 10t cubical material element, motion, 162 couple, 47 Linear kinetics definitions, 39 320, 320f conservation of energy principle dimension and, 40 first order tensors, 322 in, 191 direction of, 39–40 normal and shear components, dimension and units, 192, 192t fine points on, 41–42, 41f energy methods in, applications of magnitude of, 39 322, 322f acceleration, 196–197 net/resultant, 42–47 plane stress components, block, 193–194, 193f opposite, 41, 41f conservation, 196 resolution of forces and, 41, 41f 322, 322f equations and formulas, 193, 193t under sliding force, 41, 41f second order tensors, 322 stress–strain relationships, 321, 321f superposition method, 321, 321f tensile/compressive, 321 combined loading, 354–355 Coulomb/Tresca theory, 331 fatigue and endurance, 334–335, 334f, 335f materials strength, 333

Index 453 maximum distortion energy theory, Newton’s third law, 7 knee extension, 244–245, 244f 331–332 Normal force, 25 two-dimensional (planar) motion Notation, 15 maximum normal stress theory, 332 analysis, 243–244 maximum shear stress theory, O velocity and displacement, 244 Offset method, 295, 295f Shoulder 331, 331f One-dimensional collisions acromioclavicular joint, 113 Mohr’s circle arm, 113, 113f elasto-plastic collision, 268–270, bony structure, 113 material element, 329–330, 268f, 269f glenohumeral articulation, 112 329f, 330f injuries, 116 perfectly elastic collision, joint reaction force, 114–115 maximum shear stress, 326 267–268, 267f mechanical model, 113, 114f plane stress element, 327, 327f muscles, 113, 113f positive and negative perfectly inelastic collision, scapular movements, 113 266–267, 266f sternoclavicular joint, 113 stresses, 327 Simply supported structures, 71–78, principal stresses, 327 Osteoporosis, 376 normal and shear stresses, 73f–78f P SI units, 10t 331, 331f Parallel-axis theorem, 242, 242f Skeletal muscles, 379–381, 380f Poisson’s ratio, 319–320, 319f Parallel force, 26, 26f Spinal column principal stresses, 326–327, 326f Patellofemoral joint, 128–129, 128f safety and reliability, 330 Perfectly elastic collision, athlete’s body, 120, 120f stress concentration, 335–337, atlantoaxial joint, 117 267–268, 267f atlantooccipital joint, 116–117 335f, 336f Perfectly inelastic collision, free-body diagram, 120, 120f stress transformation, 325, 325f head and the neck, 117, 117f torsion 266–267, 266f injuries, 117 Plastic deformations, 297–298 joint reaction force, 119, 119f angle of twist, 338, 338f Poisson’s ratio, 319–320, 320f ligaments and muscles, 117, 117f circular shaft, 338, 338f Power, 10t sacrum, 120–121 definition, 337 Pressure, 10t three-force system, 117–118, formula and stresses, Pulley traction, 79f 117f–118f 338–339, 338f R vertebral column, 116, 116f fractured bone, 341, 341f Radioulnar joint, proximal, 107 weight lifter, 119, 119f longitudinal and transverse Radius of gyration, 242 Standard solid model, 367–368, 367f Reaction board method, 90, 90f Statically determinate systems, 70–71 planes, 339 Reactions, 71, 72t Statically indeterminate systems, 71 material element, 343, 343f Rectangular/Cartesian coordinate Static equilibrium, 65 pendulum, 340–341 Static friction, 29 plane perpendicular, centerline system, 147 Statics, 4 Resultant moment, 42–47 assumptions and limitations, cuts, 338, 338f Right-hand rule, 40, 40f polar moment of inertia, 338 106–107 shear stress, 338, 338f in Cartesian coordinate directions, basic considerations, 105 solid circular cylinder, 343, 343f 49, 49f mechanics spiral fracture pattern, 340, 340f standard torsion testing machine, couple-moment, 47, 82 ankle, 133–134 Rotational effect, 39, 39f elbow (see Elbow) 340, 340f Rotational equilibrium, 43, 65 hip (see Hip) stress trajectories, 343, 343f knee (see Knee) torque vs. angle of twist, shoulder (see Shoulder) spinal column (see Spinal 341–342, 341f torque vs. angular displacement, column) skeletal joints, 103–104 341, 341f skeletal muscles, 104–105 von Mises yield/Mises-Hencky Sternoclavicular joint, 113 Strain, 292. See also Stress and strain theory, 332 average shear N S axial, 293, 293f Necking, 298, 298f Sagittal plane, 92 definitions, 295 Net moment, 42–47 Scalars, 13 Neutral axis, 339, 340, 347 Secondary dimensions, 8 Newton’s first law, 7 Second law, Newton’s, 63 Newton’s laws of mechanics, 63–65 Segmental motion analysis Newton’s second law, 7 Newton’s second law of motion, 181 angular displacement and time, 243 dynamic model, 243

454 Index Strain (cont.) Supplementary units, 10 U elastic, 299, 299f Synovial fluid, 31, 103 Uniaxial motion, 151 normal, 293, 293f Synovial membrane, 103 plastic, 297–298, 297f Systems of units, 9–10 car speed, 158–159 shear, 293, 293f constant acceleration, 161, 161f simple, 292–294, 293f T with constant acceleration, tensile, 302, 302f Tangential force, 25 Tensile force, 25 155–156 Stress(es), 10t Third law, Newton’s, 64–65 displacement and acceleration vs. average normal stress, 291 Three-force systems, 70 conventional stresses, 298, 298f Tibiofemoral joint, 128–129, 128f time, 161, 160f definitions, 296 Torque, 10t free fall, 162, 162f normal stress, 291, 291f impact speed, 163 shear, 291, 292, 292f definition of, 39 relative position vs. time, 157, 157f simple, 291–292, 291f, 292t dimension and units of, 40, 41t skier speed, 161, 161f tensile, 302–304, 303f, 304f Torsion speed vs. time, 157, 158, 157f true/actual, 298 angle of twist, 338, 338f uniformly accelerated motion, 162 circular shaft, 338, 338f Unit(s) Stress and strain average shear definition, 337 conversion of, 11–12 bone specimens, 304, 304f formula and stresses, of force, 23–24 deflections, 308, 308f systems of, 9–10 elastic deformations, 295–297, 338–339, 338f 295f–296f fractured bone, 341, 341f V fixation device, 305, 305f longitudinal and transverse Vectors, 13 free-body diagram, 307, 307f Velocity, 10t Hooke’s law, 297, 297f planes, 339 Viscoelasticity human cortical bone tissue, material element, 343, 343f 304, 306f pendulum, 340–341 definition, 363 hysteresis loop in, 299–300, 299f plane perpendicular, centerline empirical models, 365 load-elongation diagrams, 290, 290f Kelvin-Voight model, loading configurations, 289 cuts, 338, 338f material behavior, in idealized, polar moment of inertia, 338 365–366, 365f 300–301, 301f shear stress, 338, 338f linearly elastic material, 363, 363f mechanical properties of materials, solid circular cylinder, 343, 343f Maxwell model, 366–367, 366f 301–302, 301t, 302f spiral fracture pattern, 340, 340f standard solid model, 367–368, 367f necking, 176–177, 176f–177f standard torsion testing machine, stress and strain, 363 plastic deformations, 297–298, 297f Volume, 10t statically indeterminate system, 340, 340f 307, 307f stress trajectories, 343, 343f W strain hardening, 299, 299f torque vs. angle of twist, Weight, 26–27 stress–strain diagrams, 294–295, Work, 10t 294f–295f, 300, 300f 341–342, 341f Work–energy theorem, 191, 249 tensile strain and average tensile torque vs. angular displacement, stress, 302, 302f Y uniaxial tension test, 303–304, 303f 341, 341f Young’s modulus, 295 work done, 299, 299f Traction devices, 78–79 Translational equilibrium, 65 Transverse plane, 92 Tresca theory, 331 Two-force systems, 70