Vista previa parcial del texto
¡Descarga tema 2 ejercicio y más Ejercicios en PDF de Ingeniería Mecánica solo en Docsity!
PROPRIETARY 4 This Manual is the proprietary property of Tho Me and protected by copyright and other state and federal user agrees to fhe following restrictions, and if he.recip only to authorized prolessors aud instructors Tor: Use 1 affiliated textbook, No other use or distribution ob No “part of this Manual may be úeproduced, di played. or means, clectronic or otherwise, without the prior.y Instructor's and Era e Manda Volume 1, Cl apters 2: to accompany VECTOR. MECHANICS | FOR ENGINEERS BEER | JOHNSTON | MAZUREK | EISENBERG h Es The MeGraw-HiK Compomes [EY Higher Education Justructorós and Solutions Manual, Vaínme Í to accompany VECTOR MECHANICS POR ENGINEERS, STATICS, NINTH EDITION Ferdinand P. Beer, E. Russeíl Johnston, dr., David F. Macurek, and Elliot Eisenberg Published by McGraw-Ilil [ligher Education, an impriot of The MeGraw-Hill Companies, Ine., 1221 Aveuuc oFtho Americas, New Ynuk, NY 10920. Copyright 3) 2010, 2007, 2004, and 1997 by 1he McGrav-Hill Companies, Ine, All siglus reserved The contents, ot parts thercol, may be reproduced in print form solely for classroom use wito VECTOR MECHANICS FOR ENGINEERS, STATICS, NINTH EDITION provided such repreductions bear cupyriglt notice, bat say not he reproduced in any other foria or for any other purpose without the prior written consent of he MeGraw-Hill Companies, Inc., jncluding, but not limited to, ia any network or other elzctronic storage or transmission, or broadcast for distance leuroing. This book is printed on acidice paper 1234567890 CCW/CCW DY ISBN; 978-0-07-7249 18-2 MHID: 0-07-7249:8-6 wew.mhho, com TABLE OF CONTENTS TO THE INSTRUCTOR coccion rr v DESCRIPTION OF THE MATERIAL CONTAINED IN VECTOR MECHANICS FOR ENGINEERS: STATICS, NINTH EDITION coccion vii TABLE L LIST OF THE TOPICS COVERED IN VECTOR MECHANICS FOR ENGINEERS: STATIOS ocacion immer xiv TABEE Il CLASSIFICATION AND DESCRIPTION OF PROBLEMS coccncoincoocinnocicnnocoos xv TABLE Ml; SAMPLE ASSIGNMENT SCHEDULE FOR A COURSE IN STATICS (50% of Problems in SI Units and 50% of Problems in U.S, Customary Units). TABLE IV: SAMPLE ASSIGNMENT SCHEDULE FOR A COURSE IN STATICS (75% of Problems in SI Units and 25% of Problems ia U.S. Customary Units). TABLE V: SAMPLE ASSIGNMENT SCHEDULE FOR A COMBINED COURSE TN STATICS AND DYNAMICS (50% of Problems in SÍ Units and $0% of Problems in U.S. Customary Units) oniccnninannnonococaconcncanonno casco rncesrronesr oy cor coranan AXX PROBLEM SOLUTIONS coccancconcicinnircor nora ore rrmnrnnramrass 1 ii TO THE INSTRUCTOR As indicated in its preface, Vector Mechanics for Engincers: Statics is designed for the first course in stalics offered in he sophomore year of college. New concepts have, therefore, been presented in simple terms and every step has been explained in dotail, llowever, becanse of the large numbcr of oplional sections which have heen included and the maturity of approach which has been achieved, this text can also be used to teach a course which will challenge the more advanced student. The texl hás been divided into units, each corresponding to a well-delincd topic and consisting of one or several theory sections, one or several Sample Problems, a section entitled Sotving Problems on Your Own. and a large number of problems fo be assigned. To assist instructors in making up a schedule of assignments thal will best fit their classes, the various topics coverod in the toxt have been listed in Table Y and a suggested number of periods to be spent on each topic has been indicated. Both a minimum and a maximum number of periods havo bccn suggested, and the topics which form the standard basic course in statics bave been separated from thase which aro optional. The tolal number of periods required to teach the basic material varies from 26 to 39, while covering the entire text would require from 41 to 65 periods. If allowance is made for the time spent lor review and exams, it is seen that this text is equally suitable for teaching a basic statics course to studenis with limited preparation (since this can be done in 39 periods or less) and for teaching a more complete stafics course to advanced students (since 41 periods or more are necessary to cover Ue entire text). In most instances, of course, the instructor will want to include some, but not all, of the additional materíal presented in the text in addition, ít is noted that lbe lexi is suitable Por icaching an abridged course in statics which can be used as an introduction lo lho stady of dynamics (see Tahle T). The problems have been grouped according to ths portions of material hoy iNastrate and have hcen arranged in order of increasing difficulty, with problems requiring special attention jedicated by asterisks. We note thal, in most cases, problems have been arranged in groups of six or more, all problems of the same group being closely related. This means that instructors will casily find additional problems to amplily a particular point which they may havo brought up in discussing a problem assigned for homework. A group of problems designed to be solved with computational softwaro can be found at the end of each chapter. Solutions tor these problems, including analyses of the problems and problem solutions and output for the most widely used computational programs, pre provided at the instructor s edition of fhe texts websile: http://www. mhhe.com/beerjohnston. To assist in the preparation of homework assignments, Table II provides a brief description of all groups of problems and a classification of the problems in each group according to the units used. 1t should also be noted thal the answers to all problems are given at the end of tho text, except lor ose with a number in italíc. Becanse of the large number of problems available in both systems ot units, the instructor bus the choice of assigning problems using ST units and problems using U.S. customary units in whatever proportion is found to be most desirable for a given class. To illustrato this point, sample lesson schedules are shown in Tables UL 1V, and V, together with various alternative lists of assigned homework problems, Half of the problems in cach af the six lists suggested in Table 1 and Table Y are stated in Sl units and half in U.S. customary units. On the other hand, 75% of the problems in the four lists suggested in Table IV arc stated in Si units and 25% in U.S. customary units. Since the approach used in this text differs in a number of respecis Irom the approach usod in other books, instructors will be well advi- sed to read the preface to Vector Mechanics for Engineers, in which the authors have outlined their general philosophy. In addition, instructors will find in the following pages a description, chapter by chapter, of the more vi significant features of this toxt, H is hoped that this matorial will help instructors in organizing their courses to best fit the needs of their students, The authors wish to acknowledse and thank Amy Mazurek of Williams Memorial Institute for her careful preparation of the solutions contained in (his manual. E, Russell Johnston, Jr. David Mazurek Elliot R Eisenberg In (he carly sections of Chap. 2 the following basic topics are presented: the equilibrium of a particle, Newton's first law, and the concept of (he Iree-body diagram, Theso first sections provide a review of the methods of plane trigonometry and familiarize the students with the proper use of a calculator, A general procedure Tor the solution of problems ¡nvolving concurrent forces is given: when a problem involves only three forces, the use of a force triangle and a trigonometric salution is preferred; when a problem involves more than three forces, the forces should be resolved into rectangular components and the equations EF+— 0, EA) = 0 should be uscd. The second part of Chap. 2 deals with forces in space and with he equilibrium of particlos in space. Unit vectors are used and forces are expressed in the form F = fi + Aj + Fi= FA, where i, j, and k are (he unit veclors directed respectively along tho x, 7 axes, and 4 js the unit vector directed along the líne of action of EF. and z Note [hat since lhis chapter deals only with particles or bodies which can be considered as particles, problems involving compression members have been postponed wilh only a lew exccplions until Chap. 4, where students will learn ta bandle rigid-body problems in a uniform fashion and will pot be lempted to erronsously assume hal lorces are concurront or that reactions are directed along members. Ti should be observed that when ST units are used a body is generally specified by is mass expressed in kilograms, ho weight of the bady, hawever, should he expressed in newtons. Therefore, in many equilibrium problems involving SÍ units, an additional calculation 18 requirod bolore a ficc-body diagram can be drawn (compare the example in Sec. 2.11 and Sample Probs. 2.5 and 2.9). This apparent disadvantage of the SI system of units, when viii compared to the U.S. customary units, will be offset in dynamics, where the mass of a body exprossed in kilograms can be entered directly into the equation E — ma, whereas with U.S. customary units the mass of the body must first be determined in lb - st (or slugs) from is weight in pounds. Chapter 3 Rigíd Bodies: Equivalent Systems of Forces The principle of transmissibility is presented as the basic assumption of the stalics ul rigid bodies, However, il is pointed out thal this principle can be derived from Newton s three laws of motion (see Sec. 16,5 of Dyramics). The vector product is then introduced and used to define the moment of a force about a point. The convenience of using the determinant form (Egs. 3.19 and 3,21) to express the moment of a force about a point should be noted. The sealar product and the mixed triple product are introduced and used lo define the moment of a Force aboul an axis. Again, the convenience of using the determinant form (Egs. 3.43 and 3.46) should be noted, The amount of time which should bo assigned ta this part of the chapter will depend on the extent to which vector algebra has been considered and used in prerequisite matbematics and physics courses. lt is felt that, even with no previous knowledge of vector algebra, a maximum of four periods is adequale (seo Table D. la Sces. 3.12 through 3.15 couples are introduced, and it is proved that couples are equivalent if they have (he same moment, While (his fundamental property of couples is often taken for granted, the authors believe that its rigorous and logical proof is necessary if rigor and logic are lo be demanded of the students in the solution of their mechanics problems. In Sections 3.16 through 3.20, the concept of equivalent systems of forces is carefully presented. This concept is made moro intuilive through the extensive use of free-body-diagram equations (see Figs. 3.39 through 3.46). Note that the moment of a lorec is cilher not shown or is represented by a green vector (Figs, 3,12 and 3.27). A red vector with the symbal 7 is used only lo represent a couple, that is, an actual system consisting ol two forces (Figs, 3.38 through 3.46). Section 3.21 is optional; it introduces the concept of a wrench and shows how lhe mosí general system of Íorces in space ean be reduced to this combination of a Toreo and a couple with the same line of action. Since one of the purposes of Chap. 3 is to familiarizo students with the fundamental operations of vector algebra, students should be encouraged to solve all problems in this chapter (two-dimensional as well as (hree- dimensional) using the methods of vector algebra. However, many students may be expected to develop solutions of their own, particularly in the casc of two-dimcusional problems, based on the direct computation of the moment of a force about a given point as he produel of tho magnitude of (he force and the perpendicular distance to tho point considered. Such alternative solutions may occasionally be indicated by the instructor (as in Sample Prob. 3,9), who may then wish to compare the solutions of the sample problems of this chapter with the solutions of the same sample problems given in Chaps, 3 and 4 of the parallel text Mechanics for bagineers. Mi should be pointed out that in later chapters the use of vector products will generally be reserved for the solution of (hrce-dimensional problems. Chapter 4 Equilibrium of Rigid Bodies In the first part of this chapter, problems involving the equilibrium of tigid bodies in two dimensions are considered and solved using ordinary algebra, while problems involving Ihree dimensions and requiring (he full usc of vector algebra are discussed in the second part of the chapter. Particular emphasis is placed on the correct drawing and usc Of free-body diagrams and on the lypos of reactions produced by various supports and connections (see Pigs. 4. and 4.10). Note that a distinction is made between hinges used in pairs and binges uscd alone; in the first case the reactions consist only of force components, while in the second case the reactions may, if necessary, include couples. For a rigid body in two dimensions, it is shown (Sec. 4.4) that no more than three independent equations can be wrillen for a given free body, so that a problem involving the equilibrium of a single vrigid body can be solved for no more than three unknowns. lt is also shown thal dt is possible to choose equilibrium equations containing only one unknown to avoid the necessity of solving simultanevus equations. Section 4,5 introduces the concepts of statical indoterminacy and partial constraints. Sections 4.6 and 4.7 are devoted to the equilibrium of two- and threo- force bodies; it is shown how theso concepts can be used to simplify the solution of certain problems. This topic is presented only after lhe general case ol equilibritn of a rigid body to lessen the possibility of students misusing, this particular method of solution. The equilibriun of a rigid body in three dimensions is considered with full emphasis placed on the free-body diagram. While the tool of vector algebra is freely used to simplify (he computations involved, vector algebra does not, and indeed cannot, replace ibe free-body diagram as the focal point of an equilibrium problem. Therefore, the solution of every sample problem in this section begins with a reference fo the drawing of a free-body diagram. Emphasis is also (Sec. 6.7) should be used (a) if only the forces in a few members are desired, or (6) if the truss is not a simple truss and if the solulion of simultaneous equations is to be avoíded (for example, Fink truss). Students should be urged to draw a separate iree-body diagram for each section uscd, The frec body obtained should he emphasized by shading and the intersected members should be removed and replaced by (1 [orces they exerted on the free body. Ht is shown that, through a judicious choice of equilibrium equations, the force in any given member can be obtained in most cases by solving a single equation. Section 6.8 is optional; it deals with the trusses obtained by combining several simple trusses and discusses ihe statical determinacy of such structures as well as the completeness of their constraints. Structures involving multiforce members are separated into frames and machines. Frames aro designed lo support loads, while machines are designed to transmit and modify forces. H is shown that while some frames remain rigid aller they have been detached from their supports, othors will collapse (Soc, 6.11), la the latter case, the equations obtained by considering the entire frame as a free body provide necessary but not suflicient conditions for the equilibrium of the frame. 1t is then necessary to dismember the frame and lo consider the equilibrium of its component parts in order to determino the reactions at the external supports. The same procedure is necessary with most machines in order to determine he vutpul force Q from the input force P or inversely (Soc. 6.12). Students should be urgod to resolve a lorco of unknown magnitude and direction into two components but to represent a force of known direction by a single unknown, namely its magnitude. While this rule may sometimos result in slightly more complicated aríthmetic, it has the advantage of matching the numbers xi of equations and unknowns and thus makes il possible for students to know at any time dyring the computations what is known and what is yot to be determincd. Chapter 7 Forces in Beams and Cables This chapter consists of Íivo groups of sections, all of which are optional, The first three groups deal with forces in beams and the last Iwo groups wiih forces in cables. Most likcly the instructor will not have time to cover the entire chapter and will have to choose between beams and cables. Section 7.2 defines the internal forces in a member. While these forces are limited to tension or compression in a siraighl lwo-foree member, they include a shcaring force and a bending couple in the ease of multiforce members or curved two-force members. Problems in this section do nof make usc of sign conventions for shear and bending moment and answers should specify which part of the member is used as (he free body. ín Secs. 73 through 7.5 the usual sign conventions are introduced and shear and bending-moment diagrams are drawn. All problems in these scctions should be solvcd by drawing the free-body diagrams of the various portions of the beams. The relations among load, sbear, and bending moment are introduced in See. 7.6. Problems in this section should be solved by evaluating areas under load and shear curves or by formal integration (as in Probs, 7.87 and 7.88). Some instructors may feel that the special methods used in this section detract frorn (he unily achieved in Ue rest ol the lext through the usc of the free-body diagram, and they may wish to omit Sec. 7.6. Others will feel that the study of shear and bending- mnoment diagrams is incomplete wilhout this section, and they will want to include it, “ho latter view is particularly justified when tho course in statics is immediately followed by a course in mechanics ol materials, Sections 7.7 through 7.9 are devoted to cables, first with concentrated loads and then with distributed loads, In both cases, the anal is based on frec-body diagrams. “The differential-equation approach is considered in the last problems of this group (Probs. 7.124 ihrough 7.126). Section 7.10 is devoted lo catenarios and requires the use of hyperbolic finctions. Chapter 8 Priction 'Phis chapter not only introduces the general topic of friction but also provides an opportunity for students lo consolidate their knowledge of the methods of analysis presented in Chaps. 2, 3, 4, and 6. It is recommended that each course in staties include at least a portion ol this chapter. a The first group ef sections (Secs. 8.1 through 8.4) is devoted to the presentation of the laws ol' dry lriction and to their application to varicus prablems, "he different cases which can be encountered are illustrated by diagrams in Figs. 8.2, 8,3, and 8,4. Particular emphasis is placed on the fact that no relation exists between the friction force and the normal force except when motion is impending or when motion actually taking place. Following the general procedure outlined in Chap. 2, problems involving only three forces are solved by a force triangle, while problems involving more than thrce forces arc solwed by summing x and y: components. ln the first ase the reaction of the surface of contact is should be represented by (he resultant R of the friction force and normal force, while in the second case 11 should be resolved into its components F and N. xi Spcctal applications of friction are considered in Secs. 8.5 through 8.10. Thoy are divided into the following groups: wedges and screws (Sccs. 8,5 and 8.6); axte and disk friction, rolling resistance (Secs. 8.7 through 8.9), belt friction (Sec. 8.10). The sections on axle and disk friction and on rolling resistance arc not essential to the understanding of the rest af tho text and thus may he omitted. Chapter 9 Distributed Furces Moments of Inertia The purpose of Sec. 9.2 is to give motivation to the study of moments of inertia of areas. Two examples are considered: one deals witb tho pure bending of a beam and the other with the hydrostatic forces exerted on a submerged circular gate. It is shown in each case that thu solution of the problem rcduces to the computation of ihe moment of inertia of an area. The other sections in the first assigament are devoted lo Ur definition and the computation of rectangular moments of inertía, polar moments of inertia, and the corresponding radii of gyration. Il is shown how Ue same differential element can be uscd to determine the moment of inertia of an area about each of the two coordinate axes. Seclions 9.6 and 9.7 introduce lhe parallcl- axis fhcorem and His application to the determination Of moments of inertia of composite areas, Particular emphasis is placed on the proper use of the parallol-axis thcorcm (sce Sample Prab. 9,5). Sections 9.8 through 9.10 are optional; they are devoted to products of inertia and lo the delermination of principal axcs of incrtia. Sections 9.11 through 9.18 deal with the moments of inertia of masses. Particular emphasis is placed on tho moments of inortia of thin platos (Sec. 9.13) and on the use of these plates as differential elements in the computation of momenis of inertia VABLE E LISCOF DUE FOPHOS COVERED IN (4 ¿TOR MECHANICS FOR ENGIA RSE STATICS Sections Topics Basic Course Suggested Number of Porivds Additional Abridged Course lo be Topics uscd as an introduction ta dynamics? L ENTRODUCTION LELé-—— Thismaterial may be used for the first or far later reference 2. STATICS OF PARTICLES fanment 1-6 Addirion and Resolution of Forees 0.5-1 7-8 —— Rectangular Components 0.5-1 9-11 Fguikibiiom afa Particle 1 2.12 14. Forces in Space , 15 Equilibrium in S 1 5. RIGID BODIES: EQUIVALENT SYSTEMS OF FORC 3.1.8 Vestar Product; Mowient of a Force abort a Poial 12 12 39-11 alar Product; Moment ol'a Force about an Axis 12 1-2 3.12-16 Comples 1 í 3.17-20 — Tquivalent Systems of Forces 15 5) +321 Reduction ofa Wrsock 051 4. EQUILIBRIUM OT RIGID BODI 4,4 Equilibrium in Two Dim 152 1.52 45 Judeternáinate Resetivns: Partial Consteainls 051 4.61 fwu- end Hlirce-Voree Budies 1 4,8-9 —— Exuilibrium in Three Dimensions 2 $. CENTROMS AND CENTERS OF GRAVITY 5.1.5 Centruids and First Moments of Areus and 4 jnes 1-2 5.6-7 ntroids by Integration 1-2 +5.8-9 — Beams and Submerged Surfaces 11.5 4.J0-12 — Centroids of Values 12 6. ANALYSIS OF SIRUCTURES 6.64 Trusses hy Method of Joints 1-15 +65 Joints under Special Lvading Cond 02341, +66 Space Trusses 051 lyusses by Method ol Secnons 12 Combinod Trassos 0.2505 Frames 23 12 Machines 12 Y5 15 7. FORCES IN DEAMS AND CABL 7.1.2 Internal Forces in Members +75 Shcac an Moment Diagramas by FR Diagram 6 Sheot and Moment Diagramas by Integration 7.7-9 Cables with Concentrated Lozds; Parabolio Cable 7.40 Catenary 4 ERICTION 3.14 Laws of Friction aad Applications 1-2 1-2 8.56 — Wedges and Serows 1 + riction, Relling Resistance 12 3. 1 9, MOMENTS OF IN 9.1--5 1 9.6 7 Cormposile Ar 12 *9,8-9 — Productsol ln 12 +9.10 Mohr's Cirele 1 9.1115 Momenis of Inertia af M: 12 +9.46 18 Mass Products ol fuertis Principal Ases aud Principal 12 Moments of inertis 10. METHOD OF VIRTUAL WORK 10.14 Principle of Vintoal Work 10.5 Mechanical Eflicioney 10.6 Potential Energy: Stability Total Number el Periods 26 39 1421 + A sample assienment schedule for a cose ie dymonnes inclodiag fhás nivimam anvownt el introdectors material in stes is given Table V. 105 recommended Ural a more complete stas course, such us He one outlined in Tables 4 and IV of this manual, be used in curricula which include tc study of mochanics of marcriais, Mass moments of ineria have at been included in the hasic staties enrse since Mis maserial is oftee tanght in dynamics. ES xiv TABLE 1; CLASSIFICATION AND DESCRIPTION OF PROBLEMS Probiem Number* SI Units Units Problem Description CHAPTER 2: STATICS ON PARTICLIS FORCES IN A PLANH Resultant of concurrent forcos 21,4 22,3 ersphica racrbod 27,8 25,6 low of sines 2.9, 10 241,12 2.13 2.14 special problems 2,17, 18 2.15, 16 laws of cosines and sines 2.19, 20 Rectangular components of force 221,24 simple problems 2.24, 27 more advanced problems 2.28, 29 2,32, 34 2 Resullant by EF. =0, EF. =0 2.35, 36 2. 2.39, 40 2 Select force so that resultant has a given direction Equilibium. Free-Body Diagram 2,43, 44 2.47, 48 cquilibrium of 3 forces 2.43, 46 2,51, 52 2.49, 50 equilibrjum of 4 forces , 56 2.53, 54 37, 60 2.58, 59 find parameter to satisiy specified conditions 82 2.63, 64 2.65, 66 2,67, 68 special problems 2.69, 70 FORCES IN SPACE Rectaugular compouneals of a force in space given £, 8 and é, find components and direction angles relations between components and direction angles direction of force defited by two poirus on ls line of uction 2.93, 94 resultant of to or fhree forces 2,97, 98 Equilibrium of a particle in space 2,99, 100 2,103, 104 low applicd lo rec cables, introductory problems 2.101, 102 2.107, 108 2.105, 406 intermediate problems 2.415, 112 2.109, 110 2.115.116 2.113, 114 advanced problems 2.117, 118 2,119,120 * Problems vohich do nat involve any specific system af tits have been indicated hy undertining their mimbes- Answers are not given to problems with a number set la italic type. xv TABLE H: CLASSIFICATION AND DESCRIPTION OF PROBLEMS (CONTINUED) Problem Number” SI Unita U.S. Units Problem Description 3.101, 102 3,104 Equivalent force-cowple systems 3.03 3.107 3.105, 106 Finding the resultant of parallel forces: two dimensions 3,00, 002 3,108, 109 Finding lho resultant and és line ofactiora bwo dimensions 3,115,116 3.110, 113 3.118 3.114, 117 3,149, 120 3,121, 22 Reducing a Urrec- dimensional system el forces lo u single forse-couple system 3,124, 125 3.123, 126 3.127, 128 3.129, 130 Finding the resultant of parallel forces: three dimensions 43.131, +132 Keducing threc-dimensional systems of forces or forces and comples to a wrench 13.133, *135 43,134, +136 axis of wrench ís parallel to a coordinate axis oc passes through O 13,137 Turce-couple system parallel Lo he coordinate axes 3.139, 140 3.138 general, theco-dimensional case 13,141 3.142 special cases where the wrench reduces to a síngle force 13,142, *J44 43.145, 1146 special, more advanced problerns 3.147, 148 3.149, 151 Review problems 3.150, 154 3,152, 153 3.155, 157 3.156, 158 3,C1, 04 Computer problems 5 1: TQUILIRRIUM OF RIGID RODIES EQUILIBRIUM IN TWO DIMENSIONS Parallel forces Parallel forces, find range of values of loads to satisfy multiple criteria Rigid bodies wiúh one reaction of vnknowo direction and oue ol know dir Rigid bodies with rre reactions of known direction 4.39, 4) 4.44, 45 Rigid bodies with a couple included in the reactions 4.47, 48 4.52, 54 Find position of rigid body in eguilibriam 4.57, 58 4.60 Partial constraints, stutical indelermínacy > Problems which do not involve any specific systora of units have been indicated by undertining their number. Answers are not given ta problems with a number set ín italie type, xvii ABLA 1h CLASSIFICATION AND DESCRIPTION OF PROBLEMS (CONTINUED) Problem Number? SI Units 0,5, Units Problem Description Three-force bodies 4,63, 64 simple gcometry, solution of'a right triangle required 4.65, 68 simple geometry, frame includes a two-force member 4.71,74 more fuvolved geometry 4.75,81 4.82 4.86, 87 find position of equilibrium 4,89, 90 EQUILIBRIUM IN THREE DIMENSIONS 491,94 Rigid bodies with two hinges along a coordimuato axis and an additional reaction parallel to another ceordinate axis 497, 98 4.99, 100 Rigid bodies supported by three vertical wires or by vertical reactions 4,101, 102 4,103, 104 4.106, 107 4.105, 108 Derrick and hoom problems invelving unknown tension in two cables 4.109, 130 47,112 4,113, 114 4.117, 118 Rigid bodies with bo hinges ulong a coordinate axis and an additional 4,115, 116 ion not parallel to a coordinate axis 4.119, 4.120, 121 Probleras involving couples as part of the reaction at a hinge 4,123,124 Advanced problems . 4.127, 128 4,131,132 4.135, 136 4,133, 134 Problems involving taking moments aboul an oblique line passing 4.140, 141 4,137, 138 through two supports 4.139 4.142, 143 4.144 146 Review problems 4.145, 149 4,147,148 4,150, 151 4.152, 153 4,02, 05 401,03 Computer problems 4.06 4,04 CHAPTER 5: DISTRIBUTED FORCES: CENTROIDS AND CENTERS OF GRAVITY Centroid of an arca formed by combining 3.1,2 rectangles and triangles 56,9 rectangles, triangles, and portions el circular arvas triangles, portioos of círcular or elliptical areas, and areas of analytical functions NV Derive capression for location ef centroid 5.18 Find ratio of dimensions so that centraid is at a given polnt 521,22 First moment oFás area Center of gravivy of a wire figure Equilibrinm of wire figutes Find dimension to maximize distance lo centroid + Problems which do not involve any specific sysiem of units have beon indicatod by underlinina their number. Answers are not given to problems with a number set in italic type. aiii 
