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# Vector Mechanics for Engineers - Statics 9E Solutions part 1of2

(Parte **1** de 15)

Instructor’s and Solutions Manual to accompany

Vector Mechanics for Engineers, Statics

Ninth Edition

Volume 1, Chapters 2–5

Ferdinand P. Beer Late of Lehigh University

E. Russell Johnston, Jr |

University of Connecticut

United States Coast Guard Academy |

David F. Mazurek

Elliot Eisenberg The Pennsylvania State University

Prepared by

Amy Mazurek Williams Memorial Institute

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Instructor’s and Solutions Manual, Volume 1 to accompany VECTOR MECHANICS FOR ENGINEERS, STATICS, NINTH EDITION Ferdinand P. Beer, E. Russell Johnston, Jr., David F. Mazurek, and Elliot Eisenberg

Published by McGraw-Hill Higher Education, an imprint of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © 2010, 2007, 2004, and 1997 by The McGraw-Hill Companies, Inc. All rights reserved.

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TO THE INSTRUCTOR | v |

VECTOR MECHANICS FOR ENGINEERS: STATICS, NINTH EDITION | vii |

VECTOR MECHANICS FOR ENGINEERS: STATICS | xiv |

TABLE I: CLASSIFICATION AND DESCRIPTION OF PROBLEMS | xv |

(50% of Problems in SI Units and 50% of Problems in U.S. Customary Units) | xxviii |

(75% of Problems in SI Units and 25% of Problems in U.S. Customary Units) | xxxix |

of Problems in U.S. Customary Units) | x |

IN STATICS AND DYNAMICS (50% of Problems in SI Units and 50% PROBLEM SOLUTIONS ...............................................................................................................1

As indicated in its preface, Vector Mechanics for Engineers: Statics is designed for the first course in statics offered in the sophomore year of college. New concepts have, therefore, been presented in simple terms and every step has been explained in detail. However, because of the large number of optional sections which have been 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 text has been divided into units, each corresponding to a well-defined topic and consisting of one or several theory sections, one or several Sample Problems, a section entitled Solving Problems on Your Own, and a large number of problems to be assigned. To assist instructors in making up a schedule of assignments that will best fit their classes, the various topics covered in the text have been listed in Table I and a suggested number of periods to be spent on each topic has been indicated. Both a minimum and a maximum number of periods have been suggested, and the topics which form the standard basic course in statics have been separated from those which are optional. The total 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 for review and exams, it is seen that this text is equally suitable for teaching a basic statics course to students with limited preparation (since this can be done in 39 periods or less) and for teaching a more complete statics course to advanced students (since 41 periods or more are necessary to cover the entire text). In most instances, of course, the instructor will want to include some, but not all, of the additional material presented in the text. In addition, it is noted that the text is suitable for teaching an abridged course in statics which can be used as an introduction to the study of dynamics (see Table I).

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The problems have been grouped according to the portions of material they illustrate and have been arranged in order of increasing difficulty, with problems requiring special attention indicated by asterisks. We note that, 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 easily find additional problems to amplify a particular point which they may have brought up in discussing a problem assigned for homework. A group of problems designed to be solved with computational software can be found at the end of each chapter. Solutions for these problems, including analyses of the problems and problem solutions and output for the most widely used computational programs, are provided at the instructor s edition of the text’s website:

To assist in the preparation of homework assignments, Table I provides a brief description of all groups of problems and a classification of the problems in each group according to the units used. It should also be noted that the answers to all problems are given at the end of the text, except for those with a number in italic. Because of the large number of problems available in both systems of units, the instructor has the choice of assigning problems using SI units and problems using U.S. customary units in whatever proportion is found to be most desirable for a given class. To illustrate this point, sample lesson schedules are shown in Tables I, IV, and V, together with various alternative lists of assigned homework problems. Half of the problems in each of the six lists suggested in Table I and Table V are stated in SI units vi and half in U.S. customary units. On the other hand, 75% of the problems in the four lists suggested in Table IV are stated in SI units and 25% in U.S. customary units.

Since the approach used in this text differs in a number of respects from the approach used 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 significant features of this text. It is hoped that this material will help instructors in organizing their courses to best fit the needs of their students. The authors wish to acknowledge and thank Amy Mazurek of Williams Memorial Institute for her careful preparation of the solutions contained in this manual.

E. Russell Johnston, Jr. David Mazurek Elliot R Eisenberg vii

DESCRIPTION OF THE MATERIAL CONTAINED IN VECTOR MECHANICS FOR ENGINEERS: STATICS, Ninth Edition

Chapter 1 Introduction

The material in this chapter can be used as a first assignment or for later reference. The six fundamental principles listed in Sec. 1.2 are introduced separately and are discussed at greater length in the following chapters. Section 1.3 deals with the two systems of units used in the text. The SI metric units are discussed first. The base units are defined and the use of multiples and submultiples is explained. The various SI prefixes are presented in Table 1.1, while the principal SI units used in statics and dynamics are listed in Table 1.2. In the second part of Sec. 1.3, the base U.S. customary units used in mechanics are defined, and in Sec. l.4, it is shown how numerical data stated in U.S. customary units can be converted into SI units, and vice versa. The SI equivalents of the principal U.S. customary units used in statics and dynamics are listed in Table 1.3.

The instructor’s attention is called to the fact that the various rules relating to the use of SI units have been observed throughout the text. For instance, multiples and submultiples (such as kN and m) are used whenever possible to avoid writing more than four digits to the left of the decimal point or zeros to the right of the decimal point. When 5-digit or larger numbers involving SI units are used, spaces rather than commas are utilized to separate digits into groups of three (for example, 20 0 km). Also, prefixes are never used in the denominator of derived units; for example, the constant of a spring which stretches 20 m under a load of 100 N is expressed as 5 kN/m, not as 5 N/m.

In order to achieve as much uniformity as possible between results expressed respectively in SI and U.S. customary units, a center point, rather than a hyphen, has been used to combine the symbols representing U.S. customary units (for example, 10 lb · ft); furthermore, the unit of time has been represented by the symbol s, rather than sec, whether SI or U.S. customary units are involved (for example, 5 s, 50 ft/s, 15 m/s). However, the traditional use of commas to separate digits into groups of three has been maintained for 5-digit and larger numbers involving U.S. customary units.

Chapter 2 Statics of Particles

This is the first of two chapters dealing with the fundamental properties of force systems. A simple, intuitive classification of forces has been used: forces acting on a particle (Chap. 2) and forces acting on a rigid body (Chap. 3).

Chapter 2 begins with the parallelogram law of addition of forces and with the introduction of the fundamental properties of vectors. In the text, forces and other vector quantities are always shown in bold-face type. Thus, a force F (boldface), which is a vector quantity, is clearly distinguished from the magnitude F (italic) of the force, which is a scalar quantity. On the blackboard and in handwritten work, where bold-face lettering is not practical, vector quantities can be indicated by underlining. Both the magnitude and the direction of a vector quantity must be given to completely define that quantity. Thus, a force F of magnitude F = 280 lb, directed upward to the right at an angle of 25° with the horizontal, is indicated as F = 280 lb 25° when printed or as F = 280 lb 25° when handwritten. Unit vectors i and j are introduced in Sec. 2.7, where the rectangular components of forces are considered.

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In the early sections of Chap. 2 the following basic topics are presented: the equilibrium of a particle, Newton’s first law, and the concept of the free-body diagram. These 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 for the solution of problems involving concurrent forces is given: when a problem involves only three forces, the use of a force triangle and a trigonometric solution is preferred; when a problem involves more than three forces, the forces should be resolved into rectangular components and the equations ΣFx = 0, ΣFy = 0 should be used.

The second part of Chap. 2 deals with forces in space and with the equilibrium of particles in space. Unit vectors are used and forces are expressed in the form F = Fxi + Fyj + Fzk = Fλ, where i, j, and k are the unit vectors directed respectively along the x, y, and z axes, and λ is the unit vector directed along the line of action of F.

Note that since this chapter deals only with particles or bodies which can be considered as particles, problems involving compression members have been postponed with only a few exceptions until Chap. 4, where students will learn to handle rigid-body problems in a uniform fashion and will not be tempted to erroneously assume that forces are concurrent or that reactions are directed along members.

It should be observed that when SI units are used a body is generally specified by its mass expressed in kilograms. The weight of the body, however, should be expressed in newtons. Therefore, in many equilibrium problems involving SI units, an additional calculation is required before a free-body diagram can be drawn (compare the example in Sec. 2.1 and Sample Probs. 2.5 and 2.9). This apparent disadvantage of the SI system of units, when compared to the U.S. customary units, will be offset in dynamics, where the mass of a body expressed in kilograms can be entered directly into the equation F = ma, whereas with U.S. customary units the mass of the body must first be determined in lb · s2/ft (or slugs) from its weight in pounds.

Chapter 3 Rigid Bodies: Equivalent Systems of Forces

The principle of transmissibility is presented as the basic assumption of the statics of rigid bodies. However, it is pointed out that this principle can be derived from Newton s three laws of motion (see Sec. 16.5 of Dynamics). 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 (Eqs. 3.19 and 3.21) to express the moment of a force about a point should be noted. The scalar product and the mixed triple product are introduced and used to define the moment of a force about an axis. Again, the convenience of using the determinant form (Eqs. 3.43 and 3.46) should be noted. The amount of time which should be assigned to this part of the chapter will depend on the extent to which vector algebra has been considered and used in prerequisite mathematics and physics courses. It is felt that, even with no previous knowledge of vector algebra, a maximum of four periods is adequate (see Table I).

In Secs. 3.12 through 3.15 couples are introduced, and it is proved that couples are equivalent if they have the same moment. While this 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 to be demanded of the students in the solution of their mechanics problems.

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In Sections 3.16 through 3.20, the concept of equivalent systems of forces is carefully presented. This concept is made more intuitive through the extensive use of free-body-diagram equations (see Figs. 3.39 through 3.46). Note that the moment of a force is either not shown or is represented by a green vector (Figs. 3.12 and 3.27). A red vector with the symbol is used only to represent a couple, that is, an actual system consisting of two forces (Figs. 3.38 through 3.46). Section 3.21 is optional; it introduces the concept of a wrench and shows how the most general system of forces in space can be reduced to this combination of a force and a couple with the same line of action.

Since one of the purposes of Chap. 3 is to familiarize students with the fundamental operations of vector algebra, students should be encouraged to solve all problems in this chapter (two-dimensional as well as threedimensional) using the methods of vector algebra. However, many students may be expected to develop solutions of their own, particularly in the case of two-dimensional problems, based on the direct computation of the moment of a force about a given point as the product of the magnitude of the force and the perpendicular distance to the 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 Engineers. It should be pointed out that in later chapters the use of vector products will generally be reserved for the solution of three-dimensional problems.

Chapter 4 Equilibrium of Rigid Bodies

In the first part of this chapter, problems involving the equilibrium of rigid bodies in two dimensions are considered and solved using ordinary algebra, while problems involving three dimensions and requiring the full use of vector algebra are discussed in the second part of the chapter. Particular emphasis is placed on the correct drawing and use of free-body diagrams and on the types of reactions produced by various supports and connections (see Figs. 4.1 and 4.10). Note that a distinction is made between hinges used in pairs and hinges used 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 written for a given free body, so that a problem involving the equilibrium of a single rigid body can be solved for no more than three unknowns. It is also shown that it is possible to choose equilibrium equations containing only one unknown to avoid the necessity of solving simultaneous equations. Section 4.5 introduces the concepts of statical indeterminacy and partial constraints. Sections 4.6 and 4.7 are devoted to the equilibrium of two- and threeforce bodies; it is shown how these concepts can be used to simplify the solution of certain problems. This topic is presented only after the general case of equilibrium of a rigid body to lessen the possibility of students misusing this particular method of solution.

The equilibrium 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 the computations involved, vector algebra does not, and indeed cannot, replace the 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 to the drawing of a free-body diagram. Emphasis is also placed on the fact that the number of unknowns and the number of equations must be equal if a structure is to be statically determinate and completely constrained.

Chapter 5 Distributed Forces: Centroids and Centers of Gravity

Chapter 5 starts by defining the center of gravity of a body as the point of application of the resultant of the weights of the various particles forming the body. This definition is then used to establish the concept of the centroid of an area or line. Section 5.4 introduces the concept of the first moment of an area or line, a concept fundamental to the analysis of shearing stresses in beams in a later study of mechanics of materials. All problems assigned for the first period involve only areas and lines made of simple geometric shapes; thus, they can be solved without using calculus.

Section 5.6 explains the use of differential elements in the determination of centroids by integration. The theorems of Pappus-Guldinus are given in Sec. 5.7. Sections 5.8 and 5.9 are optional; they show how the resultant of a distributed load can be determined by evaluating an area and by locating its centroid. Sections 5.10 through 5.12 deal with centers of gravity and centroids of volumes. Here again the determination of the centroids of composite shapes precedes the calculation of centroids by integration.

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