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(BQ) Part 1 book Vector mechanics for engineers has contents: Introduction; statics of particles, equilibrium of rigid bodies, analysis of structures, forces in beams and cables, friction, distributed forces - ixmoments of inertia; distributed forces - centroids and centers of gravity,...and other contents. Beer Johnston VECTOR MECHANICS FOR ENGINEERS VECTOR MECHANICS FOR ENGINEERS MD DALIM #999860 12/18/08 CYAN MAG YELO BLK ISBN 978-0-07-352940-0 MHID 0-07-352940-0 Part of ISBN 978-0-07-727555-6 MHID 0-07-727555-1 Ninth Edition www.mhhe.com BEER | JOHNSTON | MAZUREK | CORNWELL | EISENBERG Ninth Edition bee29400_fm_i-xxiv.indd Page i 12/18/08 3:39:27 PM user-s172 /Volumes/204/MHDQ078/work%0/indd%0 NINTH EDITION VECTOR MECHANICS FOR ENGINEERS Statics and Dynamics Ferdinand P Beer Late of Lehigh University E Russell Johnston, Jr University of Connecticut David F Mazurek U.S Coast Guard Academy Phillip J Cornwell Rose-Hulman Institute of Technology Elliot R Eisenberg The Pennsylvania State University bee29400_fm_i-xxiv.indd Page ii 12/18/08 3:39:28 PM user-s172 /Volumes/204/MHDQ078/work%0/indd%0 VECTOR MECHANICS FOR ENGINEERS: STATICS & DYNAMICS, NINTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2010 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2007, 2004, and 1997 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper QPV/QPV ISBN 978–0–07–352940–0 MHID 0–07–352940–0 Global Publisher: Raghothaman Srinivasan Senior Sponsoring Editor: Bill Stenquist Director of Development: Kristine Tibbetts Developmental Editor: Lora Neyens Senior Marketing Manager: Curt Reynolds Senior Project Manager: Sheila M Frank Senior Production Supervisor: Sherry L Kane Senior Media Project Manager: Tammy Juran Designer: Laurie B Janssen Cover/Interior Designer: Ron Bissell (USE) Cover Image: ©John Peter Photography/Alamy Lead Photo Research Coordinator: Carrie K Burger Photo Research: Sabina Dowell Supplement Producer: Mary Jane Lampe Compositor: Aptara®, Inc Typeface: 10.5/12 New Caledonia Printer: Quebecor World Versailles, KY The credits section for this book begins on page 1291 and is considered an extension of the copyright page Library of Congress Cataloging-in-Publication Data Vector mechanics for engineers Statics and dynamics / Ferdinand Beer [et al.] — 9th ed p cm Includes index ISBN 978–0–07–352940–0 (combined vol : hc : alk paper) — ISBN 978–0–07–352923–3 (v — “Statics” : hc : alk paper) — ISBN 978–0–07–724916–8 (v — “Dynamics” : hc : alk paper) Mechanics, Applied Vector analysis Statics Dynamics I Beer, Ferdinand Pierre, 1915– TA350.B3552 2009 620.1905—dc22 2008047184 www.mhhe.com bee29400_fm_i-xxiv.indd Page iii 12/18/08 3:39:28 PM user-s172 /Volumes/204/MHDQ078/work%0/indd%0 About the Authors As publishers of the books by Ferd Beer and Russ Johnston we are often asked how they happened to write their books together with one of them at Lehigh and the other at the University of Connecticut The answer to this question is simple Russ Johnston’s first teaching appointment was in the Department of Civil Engineering and Mechanics at Lehigh University There he met Ferd Beer, who had joined that department two years earlier and was in charge of the courses in mechanics Ferd was delighted to discover that the young man who had been hired chiefly to teach graduate structural engineering courses was not only willing but eager to help him reorganize the mechanics courses Both believed that these courses should be taught from a few basic principles and that the various concepts involved would be best understood and remembered by the students if they were presented to them in a graphic way Together they wrote lecture notes in statics and dynamics, to which they later added problems they felt would appeal to future engineers, and soon they produced the manuscript of the first edition of Mechanics for Engineers that was published in June 1956 The second edition of Mechanics for Engineers and the first edition of Vector Mechanics for Engineers found Russ Johnston at Worcester Polytechnic Institute and the next editions at the University of Connecticut In the meantime, both Ferd and Russ assumed administrative responsibilities in their departments, and both were involved in research, consulting, and supervising graduate students—Ferd in the area of stochastic processes and random vibrations and Russ in the area of elastic stability and structural analysis and design However, their interest in improving the teaching of the basic mechanics courses had not subsided, and they both taught sections of these courses as they kept revising their texts and began writing the manuscript of the first edition of their Mechanics of Materials text Their collaboration spanned more than half a century and many successful revisions of all of their textbooks, and Ferd’s and Russ’s contributions to engineering education have earned them a number of honors and awards They were presented with the Western Electric Fund Award for excellence in the instruction of engineering students by their respective regional sections of the American Society for Engineering Education, and they both received the Distinguished Educator Award from the Mechanics Division of the same society Starting in 2001, the New Mechanics Educator Award of the Mechanics Division has been named in honor of the Beer and Johnston author team Ferdinand P Beer Born in France and educated in France and Switzerland, Ferd received an M.S degree from the Sorbonne and an Sc.D degree in theoretical mechanics from the University of Geneva He came to the United States after serving in the French army during iii bee29400_fm_i-xxiv.indd Page iv 12/18/08 3:39:29 PM user-s172 iv About the Authors /Volumes/204/MHDQ078/work%0/indd%0 the early part of World War II and taught for four years at Williams College in the Williams-MIT joint arts and engineering program Following his service at Williams College, Ferd joined the faculty of Lehigh University where he taught for thirty-seven years He held several positions, including University Distinguished Professor and chairman of the Department of Mechanical Engineering and Mechanics, and in 1995 Ferd was awarded an honorary Doctor of Engineering degree by Lehigh University E Russell Johnston, Jr Born in Philadelphia, Russ holds a B.S degree in civil engineering from the University of Delaware and an Sc D degree in the field of structural engineering from the Massachusetts Institute of Technology He taught at Lehigh University and Worcester Polytechnic Institute before joining the faculty of the University of Connecticut where he held the position of Chairman of the Civil Engineering Department and taught for twenty-six years In 1991 Russ received the Outstanding Civil Engineer Award from the Connecticut Section of the American Society of Civil Engineers David F Mazurek David holds a B.S degree in ocean engineering and an M.S degree in civil engineering from the Florida Institute of Technology and a Ph.D degree in civil engineering from the University of Connecticut He was employed by the Electric Boat Division of General Dynamics Corporation and taught at Lafayette College prior to joining the U.S Coast Guard Academy, where he has been since 1990 He has served on the American Railway Engineering and Maintenance of Way Association’s Committee 15—Steel Structures for the past eighteen years His professional interests include bridge engineering, tall towers, structural forensics, and blast-resistant design Phillip J Cornwell Phil holds a B.S degree in mechanical engineering from Texas Tech University and M.A and Ph.D degrees in mechanical and aerospace engineering from Princeton University He is currently a professor of mechanical engineering at Rose-Hulman Institute of Technology where he has taught since 1989 His present interests include structural dynamics, structural health monitoring, and undergraduate engineering education Since 1995, Phil has spent his summers working at Los Alamos National Laboratory where he is a mentor in the Los Alamos Dynamics Summer School and does research in the area of structural health monitoring Phil received an SAE Ralph R Teetor Educational Award in 1992, the Dean’s Outstanding Scholar Award at Rose-Hulman in 2000, and the Board of Trustees Outstanding Scholar Award at Rose-Hulman in 2001 Elliot R Eisenberg Elliot holds a B.S degree in engineering and an M.E degree, both from Cornell University He has focused his scholarly activities on professional service and teaching, and he was recognized for this work in 1992 when the American Society of Mechanical Engineers awarded him the Ben C Sparks Medal for his contributions to mechanical engineering and mechanical engineering technology education and for service to the American Society for Engineering Education Elliot taught for thirty-two years, including twenty-nine years at Penn State where he was recognized with awards for both teaching and advising bee29400_fm_i-xxiv.indd Page v 12/18/08 3:39:29 PM user-s172 /Volumes/204/MHDQ078/work%0/indd%0 Contents Preface xv List of Symbols xxiii Introduction 1.1 1.2 1.3 1.4 1.5 1.6 What Is Mechanics? Fundamental Concepts and Principles Systems of Units Conversion from One System of Units to Another Method of Problem Solution 11 Numerical Accuracy 13 Statics of Particles 2.1 Introduction 10 14 16 Forces in a Plane 16 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 Force on a Particle Resultant of Two Forces 16 Vectors 17 Addition of Vectors 18 Resultant of Several Concurrent Forces 20 Resolution of a Force into Components 21 Rectangular Components of a Force Unit Vectors 27 Addition of Forces by Summing x and y Components 30 Equilibrium of a Particle 35 Newton’s First Law of Motion 36 Problems Involving the Equilibrium of a Particle Free-Body Diagrams 36 Forces in Space 45 2.12 Rectangular Components of a Force in Space 45 2.13 Force Defined by Its Magnitude and Two Points on Its Line of Action 48 2.14 Addition of Concurrent Forces in Space 49 2.15 Equilibrium of a Particle in Space 57 Review and Summary 64 Review Problems 67 Computer Problems 70 v bee29400_fm_i-xxiv.indd Page vi 12/18/08 3:39:29 PM user-s172 vi Contents /Volumes/204/MHDQ078/work%0/indd%0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 *3.21 Rigid Bodies: Equivalent Systems of Forces 72 Introduction 74 External and Internal Forces 74 Principle of Transmissibility Equivalent Forces 75 Vector Product of Two Vectors 77 Vector Products Expressed in Terms of Rectangular Components 79 Moment of a Force about a Point 81 Varignon’s Theorem 83 Rectangular Components of the Moment of a Force 83 Scalar Product of Two Vectors 94 Mixed Triple Product of Three Vectors 96 Moment of a Force about a Given Axis 97 Moment of a Couple 108 Equivalent Couples 109 Addition of Couples 111 Couples Can Be Represented by Vectors 111 Resolution of a Given Force into a Force at O and a Couple 112 Reduction of a System of Forces to One Force and One Couple 123 Equivalent Systems of Forces 125 Equipollent Systems of Vectors 125 Further Reduction of a System of Forces 126 Reduction of a System of Forces to a Wrench 128 Review and Summary 146 Review Problems 151 Computer Problems 154 Equilibrium of Rigid Bodies 4.1 4.2 Introduction 158 Free-Body Diagram 159 156 Equilibrium in Two Dimensions 160 4.3 4.4 4.5 4.6 4.7 Reactions at Supports and Connections for a Two-Dimensional Structure 160 Equilibrium of a Rigid Body in Two Dimensions Statically Indeterminate Reactions Partial Constraints 164 Equilibrium of a Two-Force Body 181 Equilibrium of a Three-Force Body 182 162 bee29400_fm_i-xxiv.indd Page vii 12/18/08 3:39:30 PM user-s172 /Volumes/204/MHDQ078/work%0/indd%0 Contents Equilibrium in Three Dimensions 189 4.8 4.9 Equilibrium of a Rigid Body in Three Dimensions Reactions at Supports and Connections for a Three-Dimensional Structure 189 189 Review and Summary 210 Review Problems 213 Computer Problems 216 5.1 Distributed Forces: Centroids and Centers of Gravity 218 Introduction 220 Areas and Lines 5.2 5.3 5.4 5.5 5.6 5.7 *5.8 *5.9 220 Center of Gravity of a Two-Dimensional Body 220 Centroids of Areas and Lines 222 First Moments of Areas and Lines 223 Composite Plates and Wires 226 Determination of Centroids by Integration 236 Theorems of Pappus-Guldinus 238 Distributed Loads on Beams 248 Forces on Submerged Surfaces 249 Volumes 258 5.10 Center of Gravity of a Three-Dimensional Body Centroid of a Volume 258 5.11 Composite Bodies 261 5.12 Determination of Centroids of Volumes by Integration 261 Review and Summary 274 Review Problems 278 Computer Problems 281 6.1 Analysis of Structures Introduction 284 286 Trusses 287 6.2 6.3 6.4 *6.5 *6.6 6.7 *6.8 Definition of a Truss 287 Simple Trusses 289 Analysis of Trusses by the Method of Joints 290 Joints under Special Loading Conditions 292 Space Trusses 294 Analysis of Trusses by the Method of Sections 304 Trusses Made of Several Simple Trusses 305 vii bee29400_fm_i-xxiv.indd Page viii 12/18/08 3:39:30 PM user-s172 viii /Volumes/204/MHDQ078/work%0/indd%0 Frames and Machines 316 Contents 6.9 Structures Containing Multiforce Members 316 6.10 Analysis of a Frame 316 6.11 Frames Which Cease to Be Rigid When Detached from Their Supports 317 6.12 Machines 331 Review and Summary 345 Review Problems 348 Computer Problems 350 *7.1 *7.2 Forces in Beams and Cables Introduction 354 Internal Forces in Members 352 354 Beams 362 *7.3 *7.4 *7.5 *7.6 Various Types of Loading and Support 362 Shear and Bending Moment in a Beam 363 Shear and Bending-Moment Diagrams 365 Relations among Load, Shear, and Bending Moment *7.7 *7.8 *7.9 *7.10 Cables with Concentrated Loads 383 Cables with Distributed Loads 384 Parabolic Cable 385 Catenary 395 Cables 383 Review and Summary 403 Review Problems 406 Computer Problems 408 Friction 410 8.1 8.2 Introduction 412 The Laws of Dry Friction Coefficients of Friction 412 8.3 Angles of Friction 415 8.4 Problems Involving Dry Friction 416 8.5 Wedges 429 8.6 Square-Threaded Screws 430 *8.7 Journal Bearings Axle Friction 439 *8.8 Thrust Bearings Disk Friction 441 *8.9 Wheel Friction Rolling Resistance 442 *8.10 Belt Friction 449 Review and Summary 460 Review Problems 463 Computer Problems 467 373 bee29400_fm_i-xxiv.indd Page ix 12/18/08 3:39:30 PM user-s172 9.1 Distributed Forces: Moments of Inertia Introduction /Volumes/204/MHDQ078/work%0/indd%0 Contents 470 472 Moments of Inertia of Areas 473 9.2 9.3 Second Moment, or Moment of Inertia, of an Area 473 Determination of the Moment of Inertia of an Area by Integration 474 9.4 Polar Moment of Inertia 475 9.5 Radius of Gyration of an Area 476 9.6 Parallel-Axis Theorem 483 9.7 Moments of Inertia of Composite Areas 484 *9.8 Product of Inertia 497 *9.9 Principal Axes and Principal Moments of Inertia 498 *9.10 Mohr’s Circle for Moments and Products of Inertia 506 Moments of Inertia of a Mass 9.11 9.12 9.13 9.14 9.15 *9.16 *9.17 *9.18 512 Moment of Inertia of a Mass 512 Parallel-Axis Theorem 514 Moments of Inertia of Thin Plates 515 Determination of the Moment of Inertia of a Three-Dimensional Body by Integration 516 Moments of Inertia of Composite Bodies 516 Moment of Inertia of a Body with Respect to an Arbitrary Axis through O Mass Products of Inertia 532 Ellipsoid of Inertia Principal Axes of Inertia 533 Determination of the Principal Axes and Principal Moments of Inertia of a Body of Arbitrary Shape 535 Review and Summary 547 Review Problems 553 Computer Problems 555 10 Method of Virtual Work *10.1 *10.2 *10.3 *10.4 *10.5 *10.6 *10.7 *10.8 *10.9 Introduction 558 Work of a Force 558 Principle of Virtual Work 561 Applications of the Principle of Virtual Work 562 Real Machines Mechanical Efficiency 564 Work of a Force during a Finite Displacement 578 Potential Energy 580 Potential Energy and Equilibrium 581 Stability of Equilibrium 582 Review and Summary 592 Review Problems 595 Computer Problems 598 556 ix bee29400_ch09_470-555.indd Page 542 542 11/26/08 7:14:08 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 9.153 through 9.156 Distributed Forces: Moments of Inertia A section of sheet steel mm thick is cut and bent into the machine component shown Knowing that the density of steel is 7850 kg/m3, determine the mass products of inertia Ixy, Iyz, and Izx of the component y 180 mm y 400 mm 200 mm x z x 225 mm 400 mm 300 mm 225 mm z Fig P9.153 Fig P9.154 y y 195 mm 225 mm r = 135 mm z 150 mm 350 mm x z Fig P9.155 x Fig P9.156 9.157 and 9.158 Brass wire with a weight per unit length w is used to form the figure shown Determine the mass products of inertia Ixy, Iyz, and Izx of the wire figure y y a 2a a a z 2a x 2a a x a z Fig P9.157 a Fig P9.158 bee29400_ch09_470-555.indd Page 543 11/26/08 7:14:09 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 Problems 9.159 The figure shown is formed of 1.5-mm-diameter aluminum wire Knowing that the density of aluminum is 2800 kg/m3, determine the mass products of inertia Ixy, Iyz, and Izx of the wire figure y 180 mm 250 mm x 300 mm z y Fig P9.159 R1 9.160 Thin aluminum wire of uniform diameter is used to form the R2 figure shown Denoting by m9 the mass per unit length of the wire, determine the mass products of inertia Ixy, Iyz, and Izx of the wire figure x z 9.161 Complete the derivation of Eqs (9.47), which express the parallel- axis theorem for mass products of inertia Fig P9.160 9.162 For the homogeneous tetrahedron of mass m shown, (a) determine by direct integration the mass product of inertia Izx, (b) deduce Iyz and Ixy from the result obtained in part a y y b z a c A x a Fig P9.162 h O 9.163 The homogeneous circular cylinder shown has a mass m Deter- mine the mass moment of inertia of the cylinder with respect to the line joining the origin O and point A that is located on the perimeter of the top surface of the cylinder z Fig P9.163 x 543 bee29400_ch09_470-555.indd Page 544 544 11/26/08 Distributed Forces: Moments of Inertia 7:14:09 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 9.164 The homogeneous circular cone shown has a mass m Determine the mass moment of inertia of the cone with respect to the line joining the origin O and point A y O z a a 3a x 3a A Fig P9.164 9.165 Shown is the machine element of Prob 9.141 Determine its mass moment of inertia with respect to the line joining the origin O and point A y 40 mm 40 mm 20 mm A 20 mm O 80 mm 60 mm z x 40 mm Fig P9.165 9.166 Determine the mass moment of inertia of the steel fixture of Probs 9.145 and 9.149 with respect to the axis through the origin that forms equal angles with the x, y, and z axes 9.167 The thin bent plate shown is of uniform density and weight W Determine its mass moment of inertia with respect to the line joining the origin O and point A y A a O a a z Fig P9.167 x bee29400_ch09_470-555.indd Page 545 11/26/08 7:14:10 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 Problems 9.168 A piece of sheet steel of thickness t and specific weight g is cut and bent into the machine component shown Determine the mass moment of inertia of the component with respect to the line joining the origin O and point A y 9.169 Determine the mass moment of inertia of the machine component of Probs 9.136 and 9.155 with respect to the axis through the origin characterized by the unit vector L (24i 8j k)/9 a A 9.170 through 9.172 For the wire figure of the problem indicated, determine the mass moment of inertia of the figure with respect to the axis through the origin characterized by the unit vector L (23i 6j 2k)/7 9.170 Prob 9.148 9.171 Prob 9.147 9.172 Prob 9.146 O 2a x z Fig P9.168 9.173 For the rectangular prism shown, determine the values of the ratios b/a and c/a so that the ellipsoid of inertia of the prism is a sphere when computed (a) at point A, (b) at point B y c a c b b a B z A x Fig P9.173 L 9.174 For the right circular cone of Sample Prob 9.11, determine the y value of the ratio a/h for which the ellipsoid of inertia of the cone is a sphere when computed (a) at the apex of the cone, (b) at the center of the base of the cone 9.175 For the homogeneous circular cylinder shown, of radius a and length L, determine the value of the ratio a/L for which the ellipsoid of inertia of the cylinder is a sphere when computed (a) at the centroid of the cylinder, (b) at point A 9.176 Given an arbitrary body and three rectangular axes x, y, and z, prove that the mass moment of inertia of the body with respect to any one of the three axes cannot be larger than the sum of the mass moments of inertia of the body with respect to the other two axes That is, prove that the inequality Ix # Iy Iz and the two similar inequalities are satisfied Further, prove that Iy $ 12 Ix if the body is a homogeneous solid of revolution, where x is the axis of revolution and y is a transverse axis z L L A a x Fig P9.175 545 bee29400_ch09_470-555.indd Page 546 546 11/27/08 Distributed Forces: Moments of Inertia 9:26:25 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 9.177 Consider a cube of mass m and side a (a) Show that the ellipsoid of inertia at the center of the cube is a sphere, and use this property to determine the moment of inertia of the cube with respect to one of its diagonals (b) Show that the ellipsoid of inertia at one of the corners of the cube is an ellipsoid of revolution, and determine the principal moments of inertia of the cube at that point 9.178 Given a homogeneous body of mass m and of arbitrary shape and three rectangular axes x, y, and z with origin at O, prove that the sum Ix Iy Iz of the mass moments of inertia of the body cannot be smaller than the similar sum computed for a sphere of the same mass and the same material centered at O Further, using the result of Prob 9.176, prove that if the body is a solid of revolution, where x is the axis of revolution, its mass moment of inertia Iy about a transverse axis y cannot be smaller than 3ma2 /10, where a is the radius of the sphere of the same mass and the same material *9.179 The homogeneous circular cylinder shown has a mass m, and the diameter OB of its top surface forms 45° angles with the x and z axes (a) Determine the principal mass moments of inertia of the cylinder at the origin O (b) Compute the angles that the principal axes of inertia at O form with the coordinate axes (c) Sketch the cylinder, and show the orientation of the principal axes of inertia relative to the x, y, and z axes y O a z a B x Fig P9.179 9.180 through 9.184 For the component described in the problem indicated, determine (a) the principal mass moments of inertia at the origin, (b) the principal axes of inertia at the origin Sketch the body and show the orientation of the principal axes of inertia relative to the x, y, and z axes *9.180 Prob 9.165 *9.181 Probs 9.145 and 9.149 *9.182 Prob 9.167 *9.183 Prob 9.168 *9.184 Probs 9.148 and 9.170 bee29400_ch09_470-555.indd Page 547 11/26/08 7:14:11 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 REVIEW AND SUMMARY In the first half of this chapter, we discussed the determination of the resultant R of forces DF distributed over a plane area A when the magnitudes of these forces are proportional to both the areas DA of the elements on which they act and the distances y from these elements to a given x axis; we thus had DF ky DA We found that the magnitude of the resultant R is proportional to the first moment Qx ey dA of the area A, while the moment of R about the x axis is proportional to the second moment, or moment of inertia, Ix ey2 dA of A with respect to the same axis [Sec 9.2] The rectangular moments of inertia Ix and Iy of an area [Sec 9.3] were obtained by evaluating the integrals Ix #y dA Iy #x dA Rectangular moments of inertia (9.1) These computations can be reduced to single integrations by choosing dA to be a thin strip parallel to one of the coordinate axes We also recall that it is possible to compute Ix and Iy from the same elemental strip (Fig 9.35) using the formula for the moment of inertia of a rectangular area [Sample Prob 9.3] y y y x dIx = dIy = dA y dx x2 y dx y r x x O A x dx Fig 9.35 Fig 9.36 The polar moment of inertia of an area A with respect to the pole O [Sec 9.4] was defined as JO #r dA Polar moment of inertia (9.3) where r is the distance from O to the element of area dA (Fig 9.36) Observing that r2 x2 y2, we established the relation JO Ix Iy (9.4) 547 bee29400_ch09_470-555.indd Page 548 548 11/26/08 Distributed Forces: Moments of Inertia Radius of gyration 7:14:12 PM user-s173 The radius of gyration of an area A with respect to the x axis [Sec 9.5] was defined as the distance kx, where Ix K2x A With similar definitions for the radii of gyration of A with respect to the y axis and with respect to O, we had kx Parallel-axis theorem /Volumes/204/MHDQ076/work%0/indd%0 Ix BA ky Iy BA C B' d A A' Fig 9.37 JO BA (9.5–9.7) The parallel-axis theorem was presented in Sec 9.6 It states that the moment of inertia I of an area with respect to any given axis AA9 (Fig 9.37) is equal to the moment of inertia I of the area with respect to the centroidal axis BB9 that is parallel to AA9 plus the product of the area A and the square of the distance d between the two axes: I I Ad B kO (9.9) This formula can also be used to determine the moment of inertia I of an area with respect to a centroidal axis BB9 when its moment of inertia I with respect to a parallel axis AA9 is known In this case, however, the product Ad should be subtracted from the known moment of inertia I A similar relation holds between the polar moment of inertia JO of an area about a point O and the polar moment of inertia JC of the same area about its centroid C Letting d be the distance between O and C, we have JO JC Ad (9.11) Composite areas The parallel-axis theorem can be used very effectively to compute the moment of inertia of a composite area with respect to a given axis [Sec 9.7] Considering each component area separately, we first compute the moment of inertia of each area with respect to its centroidal axis, using the data provided in Figs 9.12 and 9.13 whenever possible The parallel-axis theorem is then applied to determine the moment of inertia of each component area with respect to the desired axis, and the various values obtained are added [Sample Probs 9.4 and 9.5] Product of inertia Sections 9.8 through 9.10 were devoted to the transformation of the moments of inertia of an area under a rotation of the coordinate axes First, we defined the product of inertia of an area A as Ixy # xy dA (9.12) and showed that Ixy if the area A is symmetrical with respect to either or both of the coordinate axes We also derived the parallelaxis theorem for products of inertia We had Ixy Ix¿y¿ x y A (9.13) where Ix¿y¿ is the product of inertia of the area with respect to the centroidal axes x9 and y9 which are parallel to the x and y axis and x and y are the coordinates of the centroid of the area [Sec 9.8] bee29400_ch09_470-555.indd Page 549 11/27/08 9:26:30 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 y y' Review and Summary x' q x O Fig 9.38 In Sec 9.9 we determined the moments and product of inertia Ix9, Iy9, and Ix9y9 of an area with respect to x9 and y9 axes obtained by rotating the original x and y coordinate axes through an angle u counterclockwise (Fig 9.38) We expressed Ix9, Iy9, and Ix9y9 in terms of the moments and product of inertia Ix, Iy, and Ixy computed with respect to the original x and y axes We had Ix¿ Iy¿ Ix¿y¿ Ix Iy Ix Iy Ix Iy 2 Ix Iy Ix Iy cos 2u Ixy sin 2u (9.18) cos 2u Ixy sin 2u (9.19) sin 2u Ixy cos 2u (9.20) The principal axes of the area about O were defined as the two axes perpendicular to each other, with respect to which the moments of inertia of the area are maximum and minimum The corresponding values of u, denoted by um, were obtained from the formula tan 2um 2Ixy Ix Iy ; B a Ix Iy Principal axes (9.25) Ix Iy The corresponding maximum and minimum values of I are called the principal moments of inertia of the area about O; we had Imax,min Rotation of axes Principal moments of inertia b I2xy (9.27) We also noted that the corresponding value of the product of inertia is zero The transformation of the moments and product of inertia of an area under a rotation of axes can be represented graphically by drawing Mohr’s circle [Sec 9.10] Given the moments and product of inertia Ix, Iy, and Ixy of the area with respect to the x and y coordinate axes, we Mohr’s circle 549 bee29400_ch09_470-555.indd Page 550 550 11/26/08 7:14:14 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 Distributed Forces: Moments of Inertia y y' I xy b Ix Ix' X' Imin q O qm x' x B O X 2q qm A C Ixy Ix'y' Ix, Iy –I x y –Ix'y' Y a Y' Iy Iy' Imax Fig 9.39 plot points X (Ix, Ixy) and Y (Iy, Ixy) and draw the line joining these two points (Fig 9.39) This line is a diameter of Mohr’s circle and thus defines this circle As the coordinate axes are rotated through u, the diameter rotates through twice that angle, and the coordinates of X9 and Y9 yield the new values Ix9, Iy9, and Ix9y9 of the moments and product of inertia of the area Also, the angle um and the coordinates of points A and B define the principal axes a and b and the principal moments of inertia of the area [Sample Prob 9.8] Moments of inertia of masses The second half of the chapter was devoted to the determination of moments of inertia of masses, which are encountered in dynamics in problems involving the rotation of a rigid body about an axis The mass moment of inertia of a body with respect to an axis AA9 (Fig 9.40) was defined as # I r2 dm A' where r is the distance from AA9 to the element of mass [Sec 9.11] The radius of gyration of the body was defined as k5 r1 Δm1 Δm r2 r3 Δm A Fig 9.40 (9.28) I Bm (9.29) The moments of inertia of a body with respect to the coordinates axes were expressed as # (y z ) dm I # (z x ) dm I # (x y ) dm Ix y z 2 2 2 (9.30) bee29400_ch09_470-555.indd Page 551 11/26/08 7:14:15 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 We saw that the parallel-axis theorem also applies to mass moments of inertia [Sec 9.12] Thus, the moment of inertia I of a body with respect to an arbitrary axis AA9 (Fig 9.41) can be expressed as I I md Review and Summary 551 Parallel-axis theorem (9.33) where I is the moment of inertia of the body with respect to the centroidal axis BB9 which is parallel to the axis AA9, m is the mass of the body, and d is the distance between the two axes A' d B' A' t A' B' C G r t b B C' A A a C' B A B Fig 9.41 B' C Fig 9.42 Fig 9.43 The moments of inertia of thin plates can be readily obtained from the moments of inertia of their areas [Sec 9.13] We found that for a rectangular plate the moments of inertia with respect to the axes shown (Fig 9.42) are IBB9 121 mb2 IAA9 121 ma2 ICC9 IAA9 IBB9 121 m(a2 b2) Moments of inertia of thin plates (9.39) (9.40) while for a circular plate (Fig 9.43) they are IAA9 IBB9 14 mr ICC9 IAA9 IBB9 12 mr (9.41) (9.42) When a body possesses two planes of symmetry, it is usually possible to use a single integration to determine its moment of inertia with respect to a given axis by selecting the element of mass dm to be a thin plate [Sample Probs 9.10 and 9.11] On the other hand, when a body consists of several common geometric shapes, its moment of inertia with respect to a given axis can be obtained by using the formulas given in Fig 9.28 together with the parallel-axis theorem [Sample Probs 9.12 and 9.13] Composite bodies In the last portion of the chapter, we learned to determine the moment of inertia of a body with respect to an arbitrary axis OL which is drawn through the origin O [Sec 9.16] Denoting by lx, ly, Moment of inertia with respect to an arbitrary axis bee29400_ch09_470-555.indd Page 552 552 11/26/08 7:14:15 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 lz the components of the unit vector L along OL (Fig 9.44) and introducing the products of inertia Distributed Forces: Moments of Inertia Ixy # xy dm Iyz # yz dm Izx # zx dm (9.45) we found that the moment of inertia of the body with respect to OL could be expressed as IOL Ixl2x Iyl2y Izl2z 2Ixylxly 2Iyzlylz 2Izxlzlx y' y x' L p q O y (9.46) dm O x r x z z z' Fig 9.44 Ellipsoid of inertia Principal axes of inertia Principal moments of inertia Fig 9.45 By plotting a point Q along each axis OL at a distance OQ 1IOL from O [Sec 9.17], we obtained the surface of an ellipsoid, known as the ellipsoid of inertia of the body at point O The principal axes x9, y9, z9 of this ellipsoid (Fig 9.45) are the principal axes of inertia of the body; that is, the products of inertia Ix9y9, Iy9z9, Iz9x9 of the body with respect to these axes are all zero There are many situations when the principal axes of inertia of a body can be deduced from properties of symmetry of the body Choosing these axes to be the coordinate axes, we can then express IOL as IOL Ix9l2x9 Iy9l2y9 Iz9l2z (9.50) where Ix9, Iy9, Iz9 are the principal moments of inertia of the body at O When the principal axes of inertia cannot be obtained by observation [Sec 9.17], it is necessary to solve the cubic equation K3 (Ix Iy Iz)K2 (IxIy IyIz Iz Ix I2xy I2yz I2zx)K (IxIyIz IxI2yz IyI2zx IzI2xy 2IxyIyzIzx) (9.56) We found [Sec 9.18] that the roots K1, K2, and K3 of this equation are the principal moments of inertia of the given body The direction cosines (lx)1, (ly)1, and (lz)1 of the principal axis corresponding to the principal moment of inertia K1 are then determined by substituting K1 into Eqs (9.54) and solving two of these equations and Eq (9.57) simultaneously The same procedure is then repeated using K2 and K3 to determine the direction cosines of the other two principal axes [Sample Prob 9.15] bee29400_ch09_470-555.indd Page 553 11/26/08 7:14:16 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 REVIEW PROBLEMS 9.185 Determine by direct integration the moments of inertia of the y shaded area with respect to the x and y axes y2 = mx 9.186 Determine the moments of inertia and the radii of gyration of the shaded area shown with respect to the x and y axes b y y1 = kx2 x a y = b΄1 − xa1/2΅ Fig P9.185 y b y = 2b − cx2 x a Fig P9.186 9.187 Determine the moment of inertia and the radius of gyration of the shaded area shown with respect to the y axis 2b b 9.188 Determine the moments of inertia of the shaded area shown with y = kx2 respect to the x and y axes 9.189 Determine the polar moment of inertia of the area shown with respect to (a) point O, (b) the centroid of the area 9.190 To form an unsymmetrical girder, two L76 76 6.4-mm angles Fig P9.187 y and two L152 102 12.7-mm angles are welded to a 16-mm steel plate as shown Determine the moments of inertia of the combined section with respect to its centroidal x and y axes y x a a a O L76 × 76 × 6.4 a a x Fig P9.188 54 mm 54 mm Semiellipse 36 mm 540 mm O C 16 mm x 18 mm Fig P9.189 L152 × 102 × 12.7 Fig P9.190 553 bee29400_ch09_470-555.indd Page 554 554 11/26/08 of the L5 3 12 -in angle cross section shown with respect to the centroidal x and y axes y 0.746 in in /Volumes/204/MHDQ076/work%0/indd%0 9.191 Using the parallel-axis theorem, determine the product of inertia Distributed Forces: Moments of Inertia 7:14:19 PM user-s173 L5 × × 9.192 For the L5 3 12 -in angle cross section shown, use Mohr’s circle to determine (a) the moments of inertia and the product of inertia with respect to new centroidal axes obtained by rotating the x and y axes 30° clockwise, (b) the orientation of the principal axes through the centroid and the corresponding values of the moments of inertia in 9.193 A piece of thin, uniform sheet metal is cut to form the machine x C 1.74 in component shown Denoting the mass of the component by m, determine its mass moment of inertia with respect to (a) the x axis, (b) the y axis y in in B A Fig P9.191 and P9.192 a B' A' a a a z y 0.76 m Fig P9.195 x Fig P9.193 and P9.194 9.194 A piece of thin, uniform sheet metal is cut to form the machine 0.48 m z a component shown Denoting the mass of the component by m, determine its mass moment of inertia with respect to (a) the axis AA9, (b) the axis BB9, where the AA9 and BB9 axes are parallel to the x axis and lie in a plane parallel to and at a distance a above the xz plane 9.195 A 2-mm thick piece of sheet steel is cut and bent into the machine x component shown Knowing that the density of steel is 7850 kg/m3, determine the mass moment of inertia of the component with respect to each of the coordinate axes 9.196 Determine the mass moments of inertia and the radii of gyration of the steel machine element shown with respect to the x and y axes (The density of steel is 7850 kg/m3.) y 44 70 120 120 44 70 40 20 20 x z Dimensions in mm Fig P9.196 bee29400_ch09_470-555.indd Page 555 11/26/08 7:14:20 PM user-s173 /Volumes/204/MHDQ076/work%0/indd%0 COMPUTER PROBLEMS 9.C1 Write a computer program that, for an area with known moments and product of inertia Ix, Iy, and Ixy, can be used to calculate the moments and product of inertia Ix9, Iy9, and Ix9y9 of the area with respect to axes x9 and y9 obtained by rotating the original axes counterclockwise through an angle u Use this program to compute Ix9, Iy9, and Ix9y9 for the section of Sample Prob 9.7 for values of u from to 90° using 5° increments 9.C2 Write a computer program that, for an area with known moments and product of inertia Ix, Iy, and Ixy, can be used to calculate the orientation of the principal axes of the area and the corresponding values of the principal moments of inertia Use this program to solve (a) Prob 9.89, (b) Sample Prob 9.7 wn dn C3 y 9.C3 Many cross sections can be approximated by a series of rectangles as x C shown Write a computer program that can be used to calculate the moments of inertia and the radii of gyration of cross sections of this type with respect to horizontal and vertical centroidal axes Apply this program to the cross sections shown in (a) Figs P9.31 and P9.33, (b) Figs P9.32 and P9.34, (c) Fig P9.43, (d) Fig P9.44 d2 C2 w2 d1 C1 9.C4 Many cross sections can be approximated by a series of rectangles as w1 shown Write a computer program that can be used to calculate the products of inertia of cross sections of this type with respect to horizontal and vertical Fig P9.C3 and P9.C4 centroidal axes Use this program to solve (a) Prob P9.71, (b) Prob P9.75, (c) Prob 9.77 y 9.C5 The area shown is revolved about the x axis to form a homogeneous solid of mass m Approximate the area using a series of 400 rectangles of the form bcc9b9, each of width Dl, and then write a computer program that can be used to determine the mass moment of inertia of the solid with respect to the x axis Use this program to solve part a of (a) Sample Prob 9.11, (b) Prob 9.121, assuming that in these problems m kg, a 100 mm, and h 400 mm y = kx n c 9.C6 A homogeneous wire with a weight per unit length of 0.04 lb/ft is used b l1 to form the figure shown Approximate the figure using 10 straight line segments, and then write a computer program that can be used to determine the mass moment of inertia Ix of the wire with respect to the x axis Use this Fig P9.C5 program to determine Ix when (a) a in., L 11 in., h in., (b) a in., L 17 in., h 10 in., (c) a 5 in., L 25 in., h in y c' Δl d x b' l2 y = h(1 – ax ) *9.C7 Write a computer program that, for a body with known mass moments and products of inertia Ix, Iy, Iz, Ixy, Iyz, and Izx, can be used to calculate the principal mass moments of inertia K1, K2, and K3 of the body at the origin Use this program to solve part a of (a) Prob 9.180, (b) Prob 9.181, (c) Prob 9.184 *9.C8 Extend the computer program of Prob 9.C7 to include the computation of the angles that the principal axes of inertia at the origin form with the coordinate axes Use this program to solve (a) Prob 9.180, (b) Prob 9.181, (c) Prob 9.184 L–a 10 h x a L Fig P9.C6 555 bee29400_ch10_556-599.indd Page 556 11/28/08 3:15:08 PM user-s172 The method of virtual work is particularly effective when a simple relation can be found among the displacements of the points of application of the various forces involved This is the case for the scissor lift platform being used by workers to gain access to a highway bridge under construction 556 /Volumes/204/MHDQ076/work%0/indd%0 ... bee29400_fm_i-xxiv.indd Page xi 12 /18 /08 3:39:30 PM user-s172 13 13 .1 13.2 13 .3 13 .4 13 .5 13 .6 *13 .7 13 .8 13 .9 13 .10 13 .11 13 .12 13 .13 13 .14 13 .15 /Volumes/204/MHDQ078/work%0/indd%0 Kinetics of Particles: Energy... Momentum 11 06 17 .11 Impulsive Motion 11 19 17 .12 Eccentric Impact 11 19 Review and Summary 11 35 Review Problems 11 39 Computer Problems 11 42 18 Kinetics of Rigid Bodies in Three Dimensions 11 44 *18 .1 *18 .2... Several Particles 618 Graphical Solution of Rectilinear-Motion Problems Other Graphical Methods 6 31 630 Curvilinear Motion of Particles 6 41 11. 9 11 .10 11 .11 11 .12 11 .13 11 .14 Position Vector,
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