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ENGINEERING MECHANICS DYNAMICS FOURTEENTH EDITION This page intentionally left blank ENGINEERING MECHANICS DYNAMICS FOURTEENTH EDITION R C HIBBELER Hoboken Boston Columbus San Francisco New York Indianapolis London Toronto Sydney Singapore Tokyo Montreal Dubai Madrid Hong Kong Mexico City Munich Paris Amsterdam Cape Town Library of Congress Cataloging-in-Publication Data on File Vice President and Editorial Director, ECS: Marcia Horton Senior Editor: Norrin Dias Editorial Assistant: Michelle Bayman Program and Project Management Team Lead: Scott Disanno Program Manager: Sandra L Rodriguez Project Manager: Rose Kernan Cover Designer: Black Horse Designs Art Editor: Gregory Dulles Senior Digital Producer: Felipe Gonzalez Operations Specialist: Maura Zaldivar-Garcia Product Marketing Manager: Bram Van Kempen Field Marketing Manager: Demetrius Hall Marketing Assistant: Jon Bryant Cover Image: Alan Schein Photography/Corbis © 2016 by R.C Hibbeler Published by Pearson Prentice Hall Pearson Education, Inc Hoboken, New Jersey 07030 All rights reserved No part of this book may be reproduced or transmitted in any form or by any means, without permission in writing from the publisher Pearson Prentice Hall™ is a trademark of Pearson Education, Inc The author and publisher of this book have used their best efforts in preparing this book These efforts include the development, research, and testing of the theories and programs to determine their effectiveness The author and publisher shall not be liable in any event for incidental or consequential damages with, or arising out of, the furnishing, performance, or use of these programs Pearson Education Ltd., London Pearson Education Australia Pty Ltd., Sydney Pearson Education Singapore, Pte Ltd Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Inc., Toronto Pearson Educación de Mexico, S.A de C.V Pearson Education—Japan, Tokyo Pearson Education Malaysia, Pte Ltd Pearson Education, Inc., Hoboken, New Jersey Printed in the United States of America 10 ISBN-10: 0133915387 ISBN-13: 9780133915389 To the Student With the hope that this work will stimulate an interest in Engineering Mechanics and provide an acceptable guide to its understanding This page intentionally left blank PREFACE The main purpose of this book is to provide the student with a clear and thorough presentation of the theory and application of engineering mechanics To achieve this objective, this work has been shaped by the comments and suggestions of hundreds of reviewers in the teaching profession, as well as many of the author’s students New to this Edition Preliminary Problems This new feature can be found throughout the text, and is given just before the Fundamental Problems The intent here is to test the student’s conceptual understanding of the theory Normally the solutions require little or no calculation, and as such, these problems provide a basic understanding of the concepts before they are applied numerically All the solutions are given in the back of the text Expanded Important Points Sections Summaries have been added which reinforces the reading material and highlights the important definitions and concepts of the sections Re-writing of Text Material Further clarification of concepts has been included in this edition, and important definitions are now in boldface throughout the text to highlight their importance End-of-the-Chapter Review Problems All the review problems now have solutions given in the back, so that students can check their work when studying for exams, and reviewing their skills when the chapter is finished New Photos The relevance of knowing the subject matter is reflected by the real-world applications depicted in the over 30 new or updated photos placed throughout the book These photos generally are used to explain how the relevant principles apply to real-world situations and how materials behave under load New Problems There are approximately 30% new problems that have been added to this edition, which involve applications to many different fields of engineering VII VIII PREFACE Hallmark Features Besides the new features mentioned above, other outstanding features that define the contents of the text include the following Organization and Approach Each chapter is organized into well-defined sections that contain an explanation of specific topics, illustrative example problems, and a set of homework problems The topics within each section are placed into subgroups defined by boldface titles The purpose of this is to present a structured method for introducing each new definition or concept and to make the book convenient for later reference and review Chapter Contents Each chapter begins with an illustration demonstrating a broad-range application of the material within the chapter A bulleted list of the chapter contents is provided to give a general overview of the material that will be covered Emphasis on Free-Body Diagrams Drawing a free-body diagram is particularly important when solving problems, and for this reason this step is strongly emphasized throughout the book In particular, special sections and examples are devoted to show how to draw free-body diagrams Specific homework problems have also been added to develop this practice Procedures for Analysis A general procedure for analyzing any mechanical problem is presented at the end of the first chapter Then this procedure is customized to relate to specific types of problems that are covered throughout the book This unique feature provides the student with a logical and orderly method to follow when applying the theory The example problems are solved using this outlined method in order to clarify its numerical application Realize, however, that once the relevant principles have been mastered and enough confidence and judgment have been obtained, the student can then develop his or her own procedures for solving problems Important Points This feature provides a review or summary of the most important concepts in a section and highlights the most significant points that should be realized when applying the theory to solve problems Fundamental Problems These problem sets are selectively located just after most of the example problems They provide students with simple applications of the concepts, and therefore, the chance to develop their problem-solving skills before attempting to solve any of the standard problems that follow In addition, they can be used for preparing for exams, and they can be used at a later time when preparing for the Fundamentals in Engineering Exam Conceptual Understanding Through the use of photographs placed throughout the book, theory is applied in a simplified way in order to illustrate some of its more important conceptual features and instill the physical meaning of many of the terms PREFACE used in the equations These simplified applications increase interest in the subject matter and better prepare the student to understand the examples and solve problems Homework Problems Apart from the Fundamental and Conceptual type problems mentioned previously, other types of problems contained in the book include the following: r Free-Body Diagram Problems Some sections of the book contain introductory problems that only require drawing the free-body diagram for the specific problems within a problem set These assignments will impress upon the student the importance of mastering this skill as a requirement for a complete solution of any equilibrium problem r General Analysis and Design Problems The majority of problems in the book depict realistic situations encountered in engineering practice Some of these problems come from actual products used in industry It is hoped that this realism will both stimulate the student’s interest in engineering mechanics and provide a means for developing the skill to reduce any such problem from its physical description to a model or symbolic representation to which the principles of mechanics may be applied Throughout the book, there is an approximate balance of problems using either SI or FPS units Furthermore, in any set, an attempt has been made to arrange the problems in order of increasing difficulty except for the end of chapter review problems, which are presented in random order r Computer Problems An effort has been made to include some problems that may be solved using a numerical procedure executed on either a desktop computer or a programmable pocket calculator The intent here is to broaden the student’s capacity for using other forms of mathematical analysis without sacrificing the time needed to focus on the application of the principles of mechanics Problems of this type, which either can or must be solved using numerical procedures, are identified by a “square” symbol (᭿) preceding the problem number The many homework problems in this edition, have been placed into two different categories Problems that are simply indicated by a problem number have an answer and in some cases an additional numerical result given in the back of the book An asterisk (*) before every fourth problem number indicates a problem without an answer Accuracy As with the previous editions, apart from the author, the accuracy of the text and problem solutions has been thoroughly checked by four other parties: Scott Hendricks, Virginia Polytechnic Institute and State University; Karim Nohra, University of South Florida; Kurt Norlin, Bittner Development Group; and finally Kai Beng, a practicing engineer, who in addition to accuracy review provided suggestions for problem development IX INDEX Horsepower (hp), unit of, 204 Hyperbolic functions, 682 I Impact, 266–272, 314–315, 544–547, 557 central, 266–268, 268, 314–315 coefficient of restitution (e), 267–269, 315, 544–547, 557 conservation of momentum, 267, 269–272, 314–315 deformation and, 266–272, 544–547 eccentric, 544–547, 557 elastic, 268 energy loss from, 268, 270 kinetics of a particle, 266–272, 314–315 line of impact, 266, 269, 314–315, 544 oblique, 266, 269, 315 plastic (inelastic), 268 procedures for analysis of, 269 restitution from, 266–269, 544 rigid-body planar motion, 544–547, 557 separation of contact points due to, 546 Impulse, 236–317, 516–559, 604, 640 angular, 284–289, 296, 315, 523–524 conservation of angular momentum and, 286 conservation of linear momentum and, 254–255 control volumes, 295–304, 315 diagrams, 239–240 equations of motion, 238–239 external forces, 240, 254 graphical representation of, 238, 314 impact and, 266–272, 314–315, 544–547 internal forces, 255–256 kinetics of a particle, 236–317 linear, 237–244, 296, 314, 523–524 magnitude of, 238 momentum and, 236–317, 516–559 principle of momentum and, 237–244, 284–289, 295–299, 314–315, 523–530, 556, 604, 640 procedures for analysis of, 241, 255, 286, 525 propulsion and, 300–304, 315 restitution, 266, 545 rigid-body planar motion, 516–559 steady flow and, 295–299, 315 three-dimensional rigid bodies, 604, 640 Impulsive forces, 254–255 Inertia (I), 409–417, 456–457, 469, 591–596, 604, 640 acceleration (a) and, 409–417, 456–457, 469 angular acceleration (a) and, 409 angular momentum (H) and, 604 arbitrary axis, moment of about, 595 composite bodies, 415 equations of motion and, 456–457 integration of, 410–411, 592 mass moments of, 409–417 moment of, 409–417, 456–457, 469, 592, 592–597, 640 parallel-axis theorem, 414–415, 593 parallel-plane theorem, 594 principle axes of, 594–595, 603 procedure for analysis of, 411 product of, 592–593, 640 radius of gyration, 415 resistance of body to acceleration, 409 rigid-body planar motion and, 409–417, 456–457, 469 tensor, 594–595 three-dimensional rigid-body motion, 591–596, 640 volume elements for integration of, 410–411 Inertial reference frames, 116–117, 175, 423–426, 473–474, 612–613 angular momentum (H), 601–602 equations of motion, 116–117, 175, 423–426, 612–613 force vector, 116 kinetic energy, 473–474 kinetics of a particle, 116–117, 175 rigid-body planar motion, 423–426, 473 rotational motion, 424–425 slab in, 473–474 symmetry of, 423–426 three-dimensional rigid-body motion, 612–613 translational motion, 423 751 Infinitesimal rotation, 563 Instantaneous acceleration, 7, 35 Instantaneous center (IC), 360–366, 405, 456 centrode, 362 circular motion and, 360–366, 405 general plane motion, 456 location of, 361–366 moment equation about, 456 procedure for analysis of, 362 zero velocity, 360–366, 405, 456 Instantaneous velocity, 6, 34 Integral equations, 683 Integration of equations, 21, 410–411, 592, 604–605 erratic motion, 21 kinetic energy, 604–605 moment of inertia, 410–411, 592 Internal energy, 187 Internal force, 118–119, 282, 424–425 Internal impulses, 254–255 K Kepler’s laws, 170 Kinematics, 2–111, 318–407, 560–589 See also Planar motion continuous motion, 5–14 coordinates for, 36–38, 56–58, 71–78, 107–108, 569–571 curvilinear motion, 34–40, 56–62, 71–78, 107–108, 320–321 cylindrical components, 71–78, 108 cylindrical (r, u, z) coordinates, 74 dependent motion analysis, 85–90, 109 erratic motion, 20–25 fixed-axis rotation, 320, 322–329, 404 fixed-point rotation, 561–568, 589 graphs for solution of, 20–25, 106 normal (n) axes, 56–62, 108 particles and, 2–111 planar, 318–407 polar coordinates, 71–73 procedures for analysis of, 9, 38, 42, 59, 86, 92, 327, 338, 349, 362, 375, 394, 581 projectile motion, 41–45, 107 radial (r) coordinate, 71–73 rectangular (x, y, z) coordinates, 36–40, 107 752 INDEX Kinematics (continued) rectilinear, 5–15, 20–25, 106, 320–321 relative-motion analysis, 91–95, 109, 346–352, 373–380, 389–397, 405, 578–585, 589 rigid bodies, 318–407, 560–589 rotating axes, 389–397, 405, 564–568, 578–585, 589 rotation and, 320, 322–329, 338–341, 346–352, 404 sign conventions for, 5–7 tangential (t) axes, 56–62, 108 three-dimensional motion, 560–589 time derivative, 564–568 translating axes, 91–95, 109, 389–397, 578–585, 589 translating-rotating systems, 564–568 translation and, 320–321, 338–341, 346–352, 404–405 transverse (u) coordinate, 71–73 Kinetic diagram, 116 Kinetic energy, 184–185, 213, 217–218, 232, 473–476, 480–486, 511, 604–607, 640–641 conservation of, 217–218 center of mass (G) for, 605 fixed-point O for, 605 general plane motion and, 475, 511 integration for, 604–605 particles, 184–185, 213, 217–218, 232 potential energy and, 213, 217–218 principle of work and energy, 184–185, 232, 480–486, 605, 640–641 procedure for analysis of, 481 rigid-body planar motion and, 473–476, 480–486, 511 rotation about a fixed axis and, 475, 511 slab in inertial reference for, 473–474 system of bodies, 476 three-dimensional rigid-body motion, 604–607, 640–641 translation for, 475, 511 Kinetics, 3, 112–177, 178–235, 236–317, 408–471, 472–515, 516–559, 590–641 See also Planar motion; Space mechanics acceleration (a) and, 112–177, 408–471 angular momentum (H), 280–289, 315, 518–522, 523–524, 540– 543, 556–557, 601–604, 640 central-force motion, 164–170, 175 conservation of energy, 217–221, 233 conservation of momentum, 254–260, 286, 314, 540–543, 557 conservative forces and, 213–221, 233 control volumes, 295–304, 315 cylindrical (r, u, z) coordinates, 152–156, 175 efficiency (e) and, 204–207, 233 energy (E) and, 178–235, 472–515, 604–607 equations of motion, 114–126, 138– 143, 152–156, 423–431, 441–447, 456–461, 469, 612–621, 641 force (F) and, 112–177, 179–183, 213–221, 232–233, 408–471 free-body diagrams for, 116–117, 175, 423–428 gyroscopic motion, 615–616, 626–631, 641 impact and, 266–272, 314–315, 544–547, 557 impulse and momentum, 236–317, 516–559, 640 inertia (I), 409–417, 456–457, 469, 591–596, 640 inertial reference frame for 116–117, 175 linear momentum, 517, 520–522, 540–543 mass moments of inertia, 409–417, 469 Newton’s laws and, 113–115, 175 normal (n) coordinates, 138–143, 175 particles, 112–177, 178–235, 236–317 planar motion, 408–471, 472–515, 516–559 power (P), 204–207, 233 principle of, principle of impulse and momentum, 523–530, 640 principle of work and energy, 184–192, 232, 605, 640–641 procedures for analysis of, 120–121, 139, 153, 185, 205, 218, 241, 255, 269, 286, 297, 411, 428, 443, 457, 481, 525, 616 propulsion, 300–304, 315 rectangular (x, y, z) coordinates, 120–126, 175, 602–603, 614–616, 641 rigid-bodies, 408–471, 472–515, 516–559, 590–641 rotation and, 424–425, 441–447, 469, 520, 556 steady flow, 295–299, 315 tangential (t) coordinates, 138–143, 175 three-dimensional rigid bodies, 590–641 torque-free motion, 632–635, 641 trajectories, 165–170, 175 translation and, 423, 426–431, 469, 520, 556 work (U) and, 178–235, 472–515, 605, 640–641 L Line of action, 361, 425 Line of impact, 266, 269, 314, 544 Linear impulse and momentum, 237– 244, 254–260, 314, 517, 520–522, 523–524, 540–543, 556–557 conservation of momentum, 254–260, 540–543, 557 diagrams for, 239–241 external force and, 240 fixed-axis rotation and, 520, 556 force (F) and, 237–244 impulsive forces and, 254–255 general plane motion and, 521, 556 kinetics of a particle, 237–244, 254–260, 314 principle of impulse and, 237–244, 523–524, 556 procedures for analysis of, 241, 255, 541 rigid-body planar motion, 517, 520–522, 540–543 systems of particles, 240–244, 254–260, 314 translation and, 520, 556 vector, 238 INDEX M Magnification factor (MF), 664–665, 671 Magnitude, 5–7, 34, 36–37, 56–58, 72–73, 108, 238, 280, 322, 361, 373, 390, 392, 441–442, 478, 512 acceleration (a), 7, 37, 57–58, 73, 373, 392, 441–442 angular displacement and, 322 angular momentum (H), 280 average speed, constant, 478, 512 couple moment (M), work of and, 478, 512 curvilinear motion and, 34, 36–37, 56–58, 72–73, 108 distance as, fixed-axis rotation and, 441–442 graphical representation of, 238 impulse, 238 instantaneous center (IC) location from, 361 position vector (r) and, 36 rectilinear kinematics and, 5–7 relative-motion analysis and, 373, 390, 392 rotating axes, changes in motion from, 390, 392 rotation, changes in motion from, 322 speed as, 6, 34, 36, 57–58, 72 time rate of change of, 58 velocity (v), 6, 34, 36–37, 56, 72, 361, 390 Mass (m), 113–115, 118–119, 296–297, 300–304, 315, 409–417, 517–522 center (G) of, 119, 518–519 continuity of, 297 control volumes and, 296–297, 300–304, 315 equations of motion and, 114–115, 118–119 gain of, 301–302, 315 gravitational attraction and, 114–115 loss of, 300–301, 315 moments (M) of inertia (I), 409–417 momentum and, 517–522 particle body, 113–115 propulsion and, 300–304, 315 Newton’s laws and, 113–115 rigid-body planar motion, 409–417, 517–522 steady flow of fluid systems and, 296–297, 315 system of particles and, 118–119 Mass flow, 296–297, 300–302 Mathematical expressions, 682–683 Maximum deformation, 266 Mechanical efficiency, 204–205 Mechanical energy, 217–221 See also Conservation of energy Mechanics, study of, Moment arm, 410 Moment of inertia, 409–417, 442–443, 456–457, 469, 592–597, 640 acceleration (a) and, 409–417, 442–443, 456–457, 469 arbitrary axis, about, 597 body resistance to acceleration, 409 composite bodies, 415 disk elements, 411 equations of motion and, 442–443, 456–461 fixed-axis rotation, 442–443 force (F) and, 456–457 integration of, 410–411, 592 mass, 409–417 parallel-axis theorem for, 414–415, 593 parallel-plane theorem for, 594 principal, 594, 640 procedure for analysis of, 411 radius of gyration for, 415 rigid-body planar motion, 409–417, 442–443, 456–461, 469 shell elements, 411 slipping and, 456 three-dimensional rigid-body motion, 592–597, 640 volume elements for integration of, 410–411 Moments, work of a couple, 478–479, 512 Momentum, 236–317, 516–559, 601–604, 640 angular (H), 280–289, 296, 315, 518–522, 523–524, 540–543, 556–557, 601–604, 640 conservation of, 254–260, 267, 269–272, 286, 314, 540–543, 557 control volumes, 295–304, 315 753 diagrams, 234 equations of, 239 fixed-axes rotation and, 520 general plane motion and, 521 impact (eccentric) and, 266–272, 314–315, 544–547, 557 impulse and, 236–317, 516–559 kinetics of a particle, 236–317 linear (L), 237–244, 254–260, 296, 314, 517, 520–522, 523–524, 540–543, 556–557 moments of force and, 281–283 principle of impulse and, 237–244, 284–289, 295–299, 314–315, 523–530, 556, 604, 640 procedures for analysis of, 241, 255, 269, 286, 525, 541 propulsion and, 300–304, 315 rigid-body planar motion, 516–559 steady flow and, 295–299, 315 systems of particles, 240–244, 254–260, 282, 314 three-dimensional rigid bodies, 601–604, 640 translation and, 520 vector form, 238 N Natural frequency (vn), 644, 646–647, 657–660, 680 energy conservation and, 657–660 procedures for analysis of, 647, 658 undamped free vibration, 644, 646–647, 680 Newton’s laws, 113–116, 175 body mass and weight from, 115 equation of motion, 114, 175 first law of motion, 116 gravitational attraction, 114–115 kinetics of particles and, 113–115, 175 second law of motion, 113–115, 175 static equilibrium and, 116 Nonconservative force, 213 Nonimpulsive forces, 254 Nonrigid bodies, principle of work and energy for, 186 Normal (n) coordinates, 56–62, 138–143, 175, 325–326, 441–442 acceleration (a) and, 57–58, 325–326, 441–442 754 INDEX Normal (n) coordinates (continued) circular motion components, 325–326 curvilinear motion components, 56–62 equations of motion and, 138–143, 175 particle kinetics, 138–143, 175 planar motion and, 56 procedure for analysis of, 59 rigid-body planar motion, 325–326, 441–442 rotation about a fixed axis, 325–326, 441–442 three-dimensional motion, 58 velocity (v) and, 56 Normal (N) force, 152 Nutation, 626 O Oblique impact, 266, 269, 315 Orbit, trajectory and, 158–170 Osculating plane, 56 Overdamped vibration systems, 668 P Parabolic path, 168 Parallel-axis theorem, 414–415, 593 Parallel-plane theorem, 594 Particles, 2–111, 112–177, 178–235, 236–317 acceleration (a), 7–8, 35, 37, 57–58, 73, 92, 106, 112–177 angular momentum (H) of, 280–289, 314 central-force motion of, 164–170, 175 conservation of energy, 217–221, 233 conservation of angular momentum, 286, 315 conservation of linear momentum, 254–260, 267, 269–272, 315 conservative forces and, 213–221, 233 continuous motion of, 5–14 control volume, 295–304, 315 coordinates for, 36–38, 56–58, 71–78, 107–109, 120–126, 138–143, 152–156 curvilinear motion of, 34–40, 56–62, 71–78, 107–108 dependent motion analysis, 85–90, 109 deformation of, 186–187, 266–272 displacement (Δ), 5, 34 energy (E) and, 178–235 equations of motion, 114–126, 138–143, 152–156, 164–165, 175 erratic motion of, 20–25, 106 force (F) and, 112–177, 179–183, 213–221, 233 free-body diagrams, 116–117, 175 gravitational attraction (G), 114–115, 165–166 hodographs, 35 impact, 266–272, 314–315 impulse and momentum of, 236–317 inertial reference frame, 116–117, 175 kinematics of, 2–111 kinetic energy of, 184–185, 213, 217–218 kinetics of, 112–177, 178–235, 236–317 mass (m), 113–115 Newton’s second law of motion, 113–115, 175 planar motion of, 56–58 position (s), 5, 8, 34, 36, 72, 91, 106 position-coordinate equations, 85–90 potential energy of, 213–221 power (P) and, 204–207, 233 principle of work and energy for, 184–192, 233 principles of impulse and momentum, 237–244, 284–289 procedures for analysis of, 9, 38, 42, 58, 74, 86, 92, 120–121, 139, 153, 185, 205, 218, 241, 255, 269, 286, 297 projectile motion of, 41–45, 107 propulsion of, 300–304, 315 rectilinear kinematics of, 5–14, 20–25, 106 relative motion analysis, 91–95, 109 speed (magnitude), 6, 34, 36, 37, 72 system of, 118–119, 186–192, 240–244, 254–260, 314 three-dimensional motion of, 58 time derivatives, 74, 86 translating axes, two particles on, 91–95, 109 velocity (v), 6–8, 34–37, 56, 72, 91, 106 work (U) and, 178–235 Path of motion, 164–165 Perigee, 169 Period of deformation, 266 Period of vibration, 646 Periodic force, 663–666 Periodic support displacement, 665 Phase angle (K), 647 Pinned-end members, 346–352, 373–380 acceleration (a) and, 373–380 relative-motion analysis of, 346–352, 373–380 velocity (v) and, 346–352 Planar motion, 56–58, 318–407, 408–471, 472–515, 516–559 absolute (dependent) motion analysis, 338–341, 404 acceleration (a) and, 57–58, 321, 323, 325–326, 373–380, 392–393, 404–405, 408–471 angular motion and, 322–323, 404 conservation of energy, 496–501, 513 conservation of momentum, 540–543, 557 couple moment (M) in, 478–479, 512 curvilinear, 56–58 displacement, 322, 324, 346 energy (E) and, 472–515 equations of motion for, 423–431, 441–447, 456–461, 469 fixed-axis rotation, 320, 322–329, 404, 441–447, 469, 520, 556 force (F) and, 408–471, 476–479, 512 general, 320, 338–397, 404–405, 456–461, 469, 521, 556 impact (eccentric), 544–547, 557 impulse and momentum, 516–559 instantaneous center of zero velocity, 360–366, 405, 456 kinematics, 56–58, 318–407 kinetic energy and, 473–476, 480–481, 511 kinetics, 408–471, 472–515, 516–559 moment of inertia (I) for, 409–417, 442–443, 456–457, 469 normal component (n) coordinates, 56–58, 441–442 INDEX position (r) and, 321, 322, 324, 346, 389 potential energy (V) of, 496–501, 513 principles of impulse and momentum, 523–530, 556 principle of work and energy, 480–486, 513 procedures for analysis of, 327, 338, 349, 362, 375, 394, 411, 428, 443, 457, 481, 498, 525, 541 relative-motion analysis, 346–352, 373–380, 389–397, 405 rigid bodies, 318–407, 408–471, 472–515 rotation and, 320, 322–329, 338– 341, 346–352, 373–380, 404–405, 424–425, 441–447, 469, 520 rotating axes, 389–397, 405 tangential component (t) coordinates, 56–58, 441–442 translation, 320–321, 338–341, 338– 341, 346–352, 373–380, 404–405, 423, 426–431, 469, 520, 556 velocity (v) and, 56, 321, 322, 324, 346–352, 360–366, 390–391, 404–405 work (U) and, 472–515 Plastic (inelastic) impact, 268 Polar coordinates, 71–73, 108 Position (s), 5, 8, 20–22, 34, 36, 72, 85–90, 91–95, 106, 109, 321, 322, 324, 346, 389, 579 absolute dependent motion and, 85–90 angular (u), 322 continuous motion and, 5, coordinate, curvilinear motion and, 34, 36, 72 dependent-motion analysis and, 85–90, 109 displacement (Δ) from changes of, 5, 322, 324 erratic motion and, 20–22 graphs of variables, 20–22 kinematics of particles and, 5, 8, 34, 72, 91–95 magnitude and, 36 planar kinematics of rigid bodies and, 321, 322, 324, 346, 389 position-coordinate equations, 85–90, 109 rectangular components, 36 rectilinear kinematics and, 5, 8, 20–22, 106 relative-motion analysis and, 91–95, 109, 346, 389, 579 rotating axes, 389, 579 rotation about fixed axis, 322, 324, 346 three-dimensional rigid-body motion, 579 time (t), as a function of, translating axes, 91–95, 579 translation and, 321, 346 vectors (r), 34, 36, 72, 91, 321, 346, 389 velocity (v) as a function of, 8, 91 Position coordinate origin (O), Potential energy (V), 213–221, 233, 496–501, 513 conservation of energy and, 217–221, 233, 496–501, 513 conservative forces and, 213–216, 233, 496–501, 513 elastic, 214, 233, 496, 513 equations for conservation of, 497 gravitational, 213–214, 233, 496, 513 kinetic energy and, 213, 217–218 particles, 213–216, 233 potential function for, 215–216 procedure for analysis of, 218, 498 rigid-body planar motion, 496–501, 513 spring force and, 213–216, 233, 496, 513 weight (W), displacement of, 213, 215–216, 233, 496 work (U) and, 213–216 Power (P), 204–207, 233 efficiency (e) and, 204–207, 233 energy (E) and, 204–207, 233 procedure for analysis of, 205 units of, 204 Power-flight trajectory, 167 Power-series expansions, 682 Precession, 626, 633–634 Principal moments of inertia, 594, 640 Principle axes of inertia (I), 594–595, 603 755 Principle of work and energy, 184–192, 232–233, 480–486, 513, 605, 640–641 deformation and, 186–187 equation for, 184, 232 kinetic energy and, 184–185, 232, 480–486, 513, 605, 640–641 kinetics of particles, 184–192, 232–233 procedures for analysis using, 185, 481 rigid-body planar motion, 480–486, 513 three-dimensional rigid bodies, 605, 640–641 systems of particles, 186–192 units of, 184 work of friction caused by sliding, 187 Principles of impulse and momentum, 237–244, 284–289, 295–299, 314–315, 523–530, 556, 604, 640 angular, 284–289, 296, 315, 523–530, 556 diagrams for, 238–239 external forces, 240 kinetics of particles, 237–244, 284–289, 314–315 linear, 237–244, 296, 314, 523–530, 556 procedures for analysis using, 241, 286, 525 steady flow and, 295–299 systems of particles, 240–244 three-dimensional rigid-body motion, 604, 640 Problem solving procedure, Product of inertia, 592–593, 640 Projectile motion, 41–45, 107 horizontal, 41 particle kinematics and, 41–45, 107 procedure for analysis of, 42 vertical, 41 Propulsion, 300–304, 315 See also Control volume Q Quadratic formula, 682 756 INDEX R Radial component (vr), 72 Radial coordinate (r), 71–73 Radius of curvature (W), 56 Radius of gyration, 415 Rectangular (x, y, z) coordinates, 36–40, 107, 120–126, 175, 602–603, 614–616, 641 angular momentum components, 602–603 curvilinear motion, 36–40, 107 dot notation for, 36–37 equations of motion and, 120–126, 175, 614–616, 641 kinematics of a particle, 36–40, 107 kinetics of a particle, 120–126, 175 procedures for analysis using, 38, 120–121 three-dimensional rigid-plane motion and, 602–603, 614–616, 641 Rectilinear kinematics, 5–15, 20–25, 106 acceleration (a), 7–8, 20–22, 106 continuous motion, 5–15 displacement (Δ), erratic motion, 20–25 graphs for solution of, 20–25, 106 particles and, 5–15, 20–25, 106 position (s), 5, 8, 20–22, 106 procedure for analysis of, sign conventions for, 5–7 time (t) and, 8, 20–21, 106 velocity (v), 6–8, 20–22, 106 Rectilinear translation, 320–321, 404, 426–427, 469 Reference frames, 91–95, 116–117, 175, 322–329, 346–352, 404–405, 423–426, 564–568 angular motion and, 322–324 axis of rotation, 564 circular path, 324–326 coordinating fixed and translating axes, 346–352, 405 equations of motion and, 116–117, 175, 423–426 fixed, 91–95, 322–329, 404, 564–568 inertial, 116–117, 175, 423 kinetics of particles, 116–117, 175 relative-motion analysis, 346–352 relative motion of particles using, 91–95 rigid-body planar motion, 423–425 rotation about fixed axis, 322–329 rotational motion, 424–425 three-dimensional rigid-body motion, 564–568 time derivative from, 564–568 translational motion, 423 translating, 91–95 translating-rotating systems, 564–568 symmetry of, 423–426 Relative acceleration, 92, 405 Relative-motion analysis, 91–95, 109, 346–352, 360–366, 373–380, 389–397, 405, 578–585, 589 acceleration (a) and, 92, 373–380, 392–393, 405, 580 circular motion, 347–348, 360–366, 373–375, 405 coordinating fixed and translating reference frames, 346–352, 373–380, 405 displacement and, 346 instantaneous center (IC) of zero velocity, 360–366, 405 kinematics of a particle, 91–95, 109 pinned-end members, 346–352, 373–380 position vectors (r) and, 91, 346, 389, 579 procedures for analysis using, 92, 349, 375, 394, 581 rigid-body planar motion, 346–352, 360–366, 373–380, 389–397, 405 rotating axes, 389–397, 405, 578–585, 589 rotation and, 346–352, 373–380, 405 three-dimensional rigid-body motion, 578–585, 589 translating axes, 91–95, 109, 346– 352, 373–380, 405, 578–585, 589 translating coordinate system for, 578 velocity (v) and, 91, 346–352, 360–366, 390–391, 405, 579 Relative velocity, 91, 347–348, 405 Resonance, 665, 681 Restitution, 266–269, 544–547 angular velocity (v) and, 544–547 coefficient (e) of, 267–269, 544–547 deformation from impact, 266–269, 544–547 eccentric impact and, 544–547 impulse, 266, 545 period of, 266, 544 rigid-body planar motion, 544–547 Resultant force, 116, 187, 281–282 Retrograde precession, 634 Right-hand rule, 280, 322, 324 Rigid bodies, 186, 318–407, 408–471, 472–515, 516–559, 560–589, 590–641 absolute (dependent) motion analysis, 338–341, 402 acceleration (a) and, 321, 322, 325–326, 373–380, 392–393, 404–405, 408–471, 580 angular motion, 322–323, 327, 561–564 circular motion, 324–327, 347–348, 360–366, 373–375, 404–405 conservation of energy, 496–501, 513 conservation of momentum, 540–543, 557 couple moment (M) in, 478–479, 512 displacement (Δ) of, 322, 324, 477–478, 512 energy (E) and, 472–515 equations of motion for, 421–431, 441–447, 456–461, 469, 612–621, 641 fixed-axis rotation, 320, 322–329, 404, 441–447, 469, 475, 511, 556 fixed-point rotation, 561–568, 589, 602, 605 force (F) and, 408–471, 476–479, 512 free-body diagrams for, 423–428 general plane motion, 320, 338–352, 456–461, 469, 475, 511, 521, 556 gyroscopic motion, 626–631, 641 impact (eccentric), 544–547, 557 impulse and momentum, 516–559, 601–604, 640 INDEX inertia and, 591–596, 640 instantaneous center (IC) of zero velocity, 360–366, 405, 456 kinematics of, 318–407, 560–589 kinetic energy and, 473–476, 511, 604–607, 640–641 kinetics of, 408–471, 472–515, 516–559, 590–641 moments of inertia (I) for, 409–417, 442–443, 456–457, 469 planar motion, 318–407, 408–471, 472–515 position (r), 321, 322, 324, 389, 579 potential energy (V) of, 496–501, 513 principle of impulse and momentum, 523–530 principle of work and energy, 186, 480–486, 513 procedures for analysis of, 327, 338, 349, 362, 375, 394, 428, 443, 457, 481, 498, 525, 541, 581, 616 relative-motion analysis, 346–352, 373–380, 389–397, 405, 578–585, 589 rotating axes, 389–397, 405, 578–585, 589 rotation of, 320, 322–329, 338–331, 346–352, 373–380, 404–405, 424–425, 441–447, 469, 475, 511, 520, 556 systems of particles and, 186, 476 three-dimensional, 560–589, 590–641 time derivatives for, 564–568, 589 torque-free motion, 632–635, 641 translating axes, 389–397, 405, 578–585, 589 translation of, 320–321, 338–341, 346–352, 373–380, 404–405, 423, 426–431, 469, 475, 511, 520, 556 velocity (v), 321, 322, 324, 346–352, 360–366, 390–391, 404–405, 579 work (U) and, 472–515 zero velocity, 360–366, 405 Rotating axes, 389–397, 405, 564–568, 578–585, 589 acceleration (a) of, 392–393, 580 axis of rotation, 564 Coriolis acceleration of, 393, 405 fixed reference frame, 564–568 magnitude changes and, 390, 392 position vectors (r) for, 389, 579 procedure for analysis of, 394, 581 relative-motion analysis for, 389–397, 405, 578–585, 589 three-dimensional motion and, 564–568, 578–585, 589 time derivatives for, 564–568 translating-rotating systems, 564–568 velocity (v) of, 390–391, 579 Rotation, 320, 322–329, 338–341, 346– 352, 360–366, 404–405, 424–425, 441–447, 469, 475, 511, 520, 556, 561–571, 589, 612–616, 641 absolute (dependent) motion analysis, 338–341, 404 acceleration (a) and, 323, 325–326, 424–425, 441–447 angular motion and, 322–323, 327, 563–568 circular motion and, 324–327, 360–366, 404–405 displacement and, 322, 324, 346 equations of motion for, 424–425, 441–447, 469, 612–616, 641 Euler’s theorem for, 562 finite, 562 fixed-axis, 320, 322–329, 404, 441–447, 469, 475, 511, 520, 612–616 fixed-point, 561–568, 589 force (F) and, 424–425, 441–447, 469 general three-dimensional motion, 564–568 impulse and momentum of, 520, 556 infinitesimal, 563 instantaneous axis of, 563–564 instantaneous center of zero velocity, 360–366, 405 kinetic energy and, 475, 511 line of action, 361, 425, 442 moment of inertia of, 442–443 position and, 322, 324, 346 procedures for analysis of, 327, 338, 349, 362, 616 relative-motion analysis, 346–352, 405 right-hand rule for, 322, 324 757 rigid-body planar motion and, 320, 322–329, 338–341, 346– 352, 404–405, 424–425, 441–447, 469, 475, 511, 520, 556 symmetrical spinning axes, 615–616 symmetry of reference frames for, 424–425 three-dimensional rigid bodies, 561–568, 589, 612–616, 641 time derivatives for, 564–568, 589 translation and, 338–341, 346–352 velocity (v) and, 322, 324, 346–352, 360–366 S s–t (position–time) graphs, 20–21 Scalar formulation of angular momentum, 280, 285 Separation of contact points after impact, 546 Shell elements, moment of inertia for, 411 Simple harmonic motion, 644, 680 Sliding, 187, 389 relative-motion analysis for, 389 work of friction by, 187 Slipping, 348, 374, 456, 477, 512 circular motion and, 348, 374 equations of motion and, 456 forces that no work, 477, 512 general plane motion, 456 relative-motion analysis and, 348, 374 rigid-body planar motion, 477, 512 zero velocity and, 348, 477 Space cone, 634 Space mechanics, 164–170, 175, 300–304, 315, 591–596, 632–635, 641 central-force motion and, 164–170, 175 circular orbit, 168 control volume of particles, 300–304, 315 elliptical orbit, 169–170 free-flight trajectory, 166 inertia (I) and, 591–596 Kepler’s laws, 170 kinetics of particles and, 164–170, 175 758 INDEX Space mechanics (continued) mass flow, 300–302 parabolic path, 168 power-flight trajectory, 167 propulsion, 300–304, 315 three-dimensional rigid-body motion and, 591–596, 632–635, 641 thrust, 300–301 torque-free motion, 632–635, 641 trajectories, 165–170, 175 Speed, 6, 32, 57-58 See also Magnitude Spheres, fixed-point rotation and, 563, 589 Spin, 626, 633 Spinning axes, equations of motion for, 615–616 Spring force, 121, 182–183, 213–216, 232–233, 477, 496, 512–513, 644 conservation of energy and, 496, 513 conservative force of, 213–216 displacement by, 477 elastic potential energy and, 214, 233, 496, 513 equations of motion for, 121 particle kinetics, 121, 182–183, 213–216, 232–233 rigid-body planar motion, 477, 496, 512–513 vibrations and, 644 weight and, 215–216 work of, 182–183, 213–216, 232, 477, 496, 512 Static equilibrium, 116 Statics, study of, Steady flow, 295–299, 315 angular impulse and momentum, 296 closed volume, 295 control volume, 295, 315 fluid streams, 295–299 linear impulse and momentum, 296 mass flow, 296–297 principles of impulse and momentum for, 295–299, 315 procedure for analysis of, 297 volumetric flow (discharge), 297 Steady-state vibration, 670 Symmetrical spinning axes, see Gyroscopic motion Systems, 118–119, 186–192, 218, 240–244, 282, 476, 564–568, 589 angular momentum of, 282 center of mass (G), 119 conservation of energy, 218 conservative forces and, 218 deformation in bodies, 186–187 equations of motion for, 118–119 external forces, 118–119, 240 fixed rotating, 564–568 internal forces, 118–119 kinetic energy and, 476 nonrigid bodies, 186 particle kinetics, 118–119, 186–192, 240–244, 282 potential energy (V) and, 218 principle of impulse and momentum for, 240–244 principle of work and energy for, 186–192 rigid bodies, 186, 476, 564–568, 589 sliding and, 187 time derivatives for, 564–568, 589 translating-rotating, 564–568, 589 work of friction and, 187 T Tangential (t) coordinates, 56–62, 138–143, 175, 325–326, 440–441 acceleration (a) and, 57–58, 325–326, 440–441 circular motion components, 325–326 curvilinear motion components, 56–62 equations of motion and, 138–143, 175, 440–441 particle kinetics, 138–143, 175 planar motion and, 56 procedure for analysis of, 59 rigid-body planar motion, 325–326, 440–441 rotation about a fixed axis, 325–326, 440–441 three-dimensional motion, 58 velocity (v) and, 56 Tangential force, 152–153, 175 Three-dimensional motion, 58, 560–589, 590–641 angular, 561–564 angular momentum of, 601–604, 629, 640 curvilinear, 58 equations of motion for, 612–621, 641 Euler’s equations for, 614–615 fixed-point rotation, 561–568, 589, 626–631 frames of reference for, 564–568 general motion of, 569–571, 589 gyroscopic motion, 615–616, 626–631, 641 inertia, moments and products of, 591–596, 640 inertial coordinates for, 601–602 kinematics of, 58, 560–589 kinetic energy of, 604–607, 640–641 kinetics of, 590–641 particles, 58 principle of impulse and momentum, 604, 640 principle of work and energy of, 605, 640–641 procedures for analysis of, 581, 616 rectangular (x, y, z) coordinates, 602–603, 614–616, 641 relative-motion analysis of, 578–585, 589 rotating axes, 564–568, 578–585, 589 time derivatives for, 564–568, 589 torque-free motion, 632–635, 641 translating axes, 578–585 translating coordinate systems for, 569–571 translating-rotating systems, 564–568, 589 Thrust, 300–301 Time (t), 8, 20–21, 106, 170, 646 continuous motion and, cycle, 646 erratic motion and, 20–21 graphs of variables, 20–21, 106 orbital revolution, 170 period, 646 position (s), as a function of, rectilinear kinematics and, 8, 20–21, 106 velocity (v) as a function of, vibration and, 646 INDEX Time derivatives, 74, 86, 108–109, 564–568, 589 absolute dependent motion, 86, 109 curvilinear motion, 74, 108 fixed-point rotation, 564–568, 589 three-dimensional motion, 564–568, 589 translating-rotating systems, 564–568, 589 Time-differential equations, 338 Torque-free motion, 632–635, 641 Trajectories, 165–170, 175 circular orbit, 168 eccentricity of, 166–167, 175 elliptical orbit, 169–170 free-flight, 166 gravitational attraction and, 165–166 parabolic path, 168 power-flight, 167 Translating axes, 91–95, 109, 346–352, 373–380, 405, 564–568, 578–585, 589 acceleration (a), 92, 373–380, 580 coordinates for, 91 fixed reference frame, 91–95 kinematics of particles, 91–95, 109 observers, 91–92, 109 position vectors (r) for, 91, 346, 579 procedures for analysis of, 92, 338, 349, 375, 581 relative-motion analysis of, 91–95, 109, 346–352, 373–380, 405, 578–585, 589 rigid-body planar motion, 91–95, 109, 346–352, 373–380, 405, 564–568, 578–585 rotation and, 338–341, 346–352, 373–380, 404 three-dimensional rigid bodies, 564–568, 578–585, 589 time derivatives for systems, 564–568 translating-rotating systems, 564–568, 589 velocity (v) of, 91, 346–352, 405, 579 Translating coordinate systems, 569–571, 578, 589 Translation, 320–321, 338–341, 389– 397, 404–405, 423, 426–431, 469, 475, 511, 520, 556, 612, 641 absolute (dependent) motion analysis, 338–341, 404 acceleration (a) and, 321, 392–393, 404 circular motion and, 347–348 coordinate system axes, 346–352, 404 curvilinear, 320–321, 404, 427, 469 displacement and, 346 equations of motion for, 423, 426–431, 469, 612, 641 impulse and momentum, 520, 556 kinetic energy and, 475, 511 paths of, 320 position vectors (r), 321, 389 procedures for analysis using, 394, 428, 616 rectilinear, 320–321, 404, 426–427, 469 relative-motion analysis, 389–397, 405 rigid-body planar motion, 320– 321, 338–341, 389–397, 404–405, 423, 426–431, 469, 475, 511, 520, 556, 641 rotating axes with, 389–397, 405 symmetry of reference frames for, 423 three-dimensional rigid-body motion, 612, 641 velocity (v) and, 321, 390–391, 404 Transverse component (vu), 72 Transverse coordinate (u), 71–73 Trigonometric identities, 682 U Unbalanced force, 113–114 Undamped vibrations, 643–651, 663–666, 680 forcing frequency (vu) for, 663–665, 680 forced, 663–666, 680 free, 643–651, 680 natural frequency (vn) for, 644, 646–647, 680 periodic force and, 663–666 periodic support displacement of, 665 procedure for analysis of, 647 Underdamped vibration systems, 669 Unit vectors, 684 759 V v–s (velocity–position) graphs, 22 v–t (velocity–time) graphs, 20–21 Variable force, work of, 180, 476 Vector analysis, 684–688 Vector formulation of angular momentum, 280, 285 Vector functions, 685 Vector quantity, particle position and displacement as, 5, 36 Velocity (v), 6–8, 20–22, 34–37, 56, 72, 91, 106, 164, 168, 321, 322, 324, 346–352, 360–366, 390–391, 404–405, 477, 544–547, 563–564, 579, 626–628 absolute, 91, 347 acceleration (a) and, 7–8 angular (v), 72, 322, 544–547, 563, 626–628 areal, 164 average, 6, 34 central-force motion and, 164, 168 circular motion and, 324, 347–348 constant, continuous motion and, 6–8 curvilinear motion and, 34–37, 56, 72 cylindrical components and, 72 eccentric impact and, 544–547 erratic motion and, 20–22 escape, 168 fixed-point rotation and, 322, 324, 404, 563–564, 626–628 forces doing no work, 477 graphs of variables, 20–22, 106 gyroscopic motion and, 626–628 instantaneous, 6, 34 instantaneous center (IC) of zero, 360–366, 405 kinematics of particles and, 6–8, 20–22, 34–37, 56, 72, 91, 106 magnitude of, 6, 34, 36–37, 56, 72, 390, 404 normal component (n) coordinates, 56, 404 position (s), as a function of, procedures for analysis of, 349, 362 radial component (vr), 72 rectangular components and, 36–37 760 INDEX Velocity (v) (continued) rectilinear kinematics and, 6–8, 20–22, 106 relative, 91, 347 relative-motion analysis and, 91, 346–352, 390–391, 405, 579 rigid-body planar motion, 321, 322, 324, 346–352, 360–366, 390–391, 404–405, 544–547 rotating axis, 390–391, 405, 579 rotation and, 322, 324, 346–352, 404–405 sign convention for, slipping and, 348, 477 speed (magnitude), 6, 34, 36, 72 tangential component (t) coordinates, 56, 72, 404 three-dimensional rigid-body motion, 563–564, 579, 626–628 time (t), as a function of, time derivative and, 564 translating axes and, 91, 346–352, 405, 579 translation and, 321, 390–391, 404 transverse component (vV), 72 zero, 348, 360–366, 405, 477 Vertical displacement ()), 477 Vertical projectile motion, 41–45 Vibrations, 642–681 amplitude of, 645–646 critically damped systems, 668 cycle, 646 damped, 643, 667–672, 681 displacement and, 644–651 electrical circuit analogs and, 673, 681 energy methods for conservation of, 657–660, 680 equilibrium position, 644–646 forced, 643, 663–666, 670–672, 680–681 forcing frequency (vu), 663–666, 680 free, 643–651, 667–669, 680–681 frequency (f), 644, 646–647, 663, 669 magnification factor (MF) for, 664–665, 671 natural frequency (vn), 644, 646–647, 657–660, 680 overdamped systems, 668 period, 646 periodic force and, 663–666 periodic support displacement of, 665 phase angle (K), 646 procedures for analysis of, 647, 658 resonance, 665, 681 simple harmonic motion of, 644 undamped forced, 663–666, 680 undamped free, 643–651, 680 underdamped systems, 666 viscous damped, 667–672, 681 Viscous damping force, 667, 681 Viscous vibration, 667–672, 681 coefficient of damping, 667 critically damped systems, 668 damped, 667–672, 681 damping force, 667 forced, 670–672, 681 free, 667–669, 681 overdamped systems, 668 steady-state, 670 underdamped systems, 669 Volume elements, integration of moments of inertia using, 410–411 Volumetric flow (discharge), 297 W Watt (W), unit of, 204 Weight (W), 115, 181, 213, 215–217, 232–233, 477, 496, 512 conservation of energy and, 217, 233, 496 conservative forces and displacement of, 213, 215–216, 233 constant, 213 gravitational attraction and, 115 gravitational potential energy and, 213, 496 potential energy (V) and, 213, 215–216, 496 spring force and, 215–216 vertical displacement (Δ) of, 477 work (U) of a, 181, 213, 215–216, 232, 477, 496, 512 Work (U), 178–235, 472–515, 605, 640–641 conservation of energy and, 217–221, 233, 496–501, 513 conservative forces and, 213–216, 233 constant force, 181, 232, 476, 512 couple moment (M), of a, 478–479, 512 deformation and, 186–187 displacement and (Δ), 179–180, 477–478, 512 energy (E) and, 178–235, 472–515, 605 external, 187 force (F) as, 179–183, 186–192, 232–233, 476–479, 512 friction caused by sliding, 187 internal, 187 kinetic energy and, 605 kinetics of a particle, 178–235 potential energy (V) and, 213–216, 496–501, 513 potential function and, 215–216 principle of energy and, 184–192, 232–233, 480–486, 513, 605, 640–641 procedures for analysis of, 185, 481, 498 rigid-body planar motion, 472–515 slipping and, 477 spring force as, 182–183, 213–216, 232, 477, 512 system of particles, 186–192 three-dimensional rigid bodies, 605, 640–641 units of, 180 variable force, of a, 180, 476 weight (W) as, 181, 213, 215–216, 232, 477, 512 zero velocity and (no work), 477 Z Zero velocity, 348, 360–366, 405, 456, 477 general plane motion, 456 instantaneous center (IC) of, 360–366, 405 relative-motion analysis, 348 slipping (no work) and, 348, 477 Fundamental Equations of Dynamics KINEMATICS Particle Rectilinear Motion Variable a Constant a = ac dv v = v0 + act a= dt ds s = s0 + v0t + 12 act v = dt a ds = v dv v2 = v20 + 2ac(s - s0) Equations of Motion Particle ⌺F = ma Rigid Body ⌺Fx = m(aG)x (Plane Motion) ⌺Fy = m(aG)y ⌺MG = IGa or ⌺MP = ⌺(mk)P Principle of Work and Energy T + ⌺U1 - = T Kinetic Energy Particle T = 21mv2 Rigid Body (Plane Motion) T = 1mvG2 + 1IG v2 Particle Curvilinear Motion x, y, z Coordinates r, u, z Coordinates # ## # # ## ax = x vx = x Work vr = r ar = r - r u # ## # ## # # Variable force vy = y ay = y vu = r u au = r u + 2r u # ## vz = z az = z n, t, b Coordinates # v = s # vz = z ## az = z dv # at = v = v ds [1 + (dy >dx)2]3>2 v an = r = r d2y >dx Relative Motion vB = vA + vB/A aB = aA + aB/A Rigid Body Motion About a Fixed Axis Variable a Constant a = ac dv a = v = v + ac t dt du v = u = u0 + v0t + 21act dt v dv = a du v2 = v20 + 2ac(u - u0) For Point P s = ur v = vr at = ar an = v2r Relative General Plane Motion—Translating Axes vB = vA + vB>A (pin) aB = aA + aB>A (pin) Relative General Plane Motion—Trans and Rot Axis vB = vA + ⍀ * rB>A + (vB>A )xyz Mass Moment of Inertia I = Parallel-Axis Theorem Radius of Gyration r2 dm L I = IG + md2 I k = Am UF = F cos u ds L (Fc cos u) ⌬ s - W ⌬y - 21 ks 22 - 21 ks 21 M⌬u Constant force UF = Weight UW = Spring Us = Couple moment UM = Power and Efficiency Pout Uout dU e = = P= = F#v dt Pin Uin Conservation of Energy Theorem T1 + V = T2 + V Potential Energy V = V g + V e, where V g = {W y, V e = + 21 ks Principle of Linear Impulse and Momentum Particle mv1 + ⌺ L F dt = mv2 F dt = m(vG)2 L Conservation of Linear Momentum ⌺(syst mv)1 = ⌺(syst mv)2 (vB)2 - (vA)2 Coefficient of Restitution e = (vA)1 - (vB)1 Principle of Angular Impulse and Momentum Rigid Body Particle # aB = aA + ⍀ * rB>A + ⍀ * (⍀ * rB>A ) + 2⍀ * (vB>A )xyz + (aB>A )xyz KINETICS m(vG)1 + ⌺ (HO)1 + ⌺ MO dt = (HO)2 L where HO = (d)(mv) (HG)1 + ⌺ Rigid Body (Plane motion) MG dt = (HG)2 L where HG = IGv (HO)1 + ⌺ MO dt = (HO)2 L where HO = IOv Conservation of Angular Momentum ⌺(syst H)1 = ⌺(syst H)2 SI Prefixes Multiple Exponential Form Prefix SI Symbol 000 000 000 109 giga G 000 000 106 mega M 000 103 kilo k 0.001 10−3 milli m 0.000 001 10−6 micro μ 0.000 000 001 10−9 nano n Submultiple Conversion Factors (FPS) to (SI) Quantity Unit of Measurement (FPS) Equals Unit of Measurement (SI) Force lb Mass slug 14.59 kg ft 0.3048 m Length Conversion Factors (FPS) ft = 12 in (inches) mi (mile) = 5280 ft kip (kilopound) = 1000 lb ton = 2000 lb 4.448 N Geometric Properties of Line and Area Elements Centroid Location Centroid Location y y L ϭ 2ur r u C u r x u Area Moment of Inertia A ϭ ur u C r sin u u x r sin u u Ix = r (u – sin 2u) Iy = r (u + sin 2u) Circular sector area Circular arc segment y A ϭ 14 pr L ϭ p–2 r r L ϭ pr 2r — p C C 3p Quarter circle area y (a ϩ b) C h x 4r — Quarter and semicircle arcs a C r A ϭ –12 h πr 16 Iy = πr 16 Ix = 4r — 3p r x –1 a ϩ b a ϩb b Trapezoidal area πr Iy = πr4 Ix = 4r — 3p r h r A ϭ p—– C x Semicircular area y b a C A ϭ pr2 Aϭ 23– ab Ix = πr4 r 3– 5a x C Iy = πr4 –3 b Semiparabolic area Circular area — 10 b C –3 a A ϭ bh y Aϭ — ab b h a x C b Exparabolic area Ix = bh 12 Iy = hb 12 Ix = bh 36 Rectangular area a b A ϭ –12 bh C ab A= — — a C h b Parabolic area Triangular area x 1– 3h Center of Gravity and Mass Moment of Inertia of Homogeneous Solids z z r Vϭ 4–3 π r V ϭ πr h r h – G G x x Cylinder m(3r2 + h2) Ixx = Iyy = 12 z Sphere Ixx = Iyy = Izz = mr2 y h – y Izz = mr2 z V ϭ –π r h V ϭ –23 pr G r y 3– r x z Izz = r x mr2 h y Cone m(4r2 + h2) Ixx = Iyy = 80 Hemisphere Ixx = Iyy = 0.259 mr2 h – G Izz = mr2 10 z z' G G r y a b y x x Thin plate Thin Circular disk 1 Ixx = Iyy = mr2 Izz = mr2 Iz¿z¿ = mr2 2 z Ixx = mb2 12 Iyy = ma2 12 z Izz = m(a2 + b2) 12 l r G G x y x y l y¿ x¿ Thin ring Ixx = Iyy = mr2 Izz = mr2 Slender Rod ml Ixx = Iyy = 12 Ix¿x¿ = Iy¿y¿ = ml Iz¿z¿ = .. .ENGINEERING MECHANICS DYNAMICS FOURTEENTH EDITION This page intentionally left blank ENGINEERING MECHANICS DYNAMICS FOURTEENTH EDITION R C HIBBELER Hoboken Boston Columbus... for each valuepack is as follows: r Engineering Mechanics: Dynamics with Study Pack: ISBN: 0134116658 r Engineering Mechanics: Dynamics Plus MasteringEngineering with Pearson eText — Access... into simpler steps r Dynamics Study Pack This supplement contains chapter-by-chapter study materials and a Free-Body Diagram Workbook r Video Solutions Complete, step-by-step solution 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