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dynamics is the analysis of forces on structures in static equilibrium, properties of static forces, static moments, couples and resultant, conditions for equilibrium (dynamics), static friction, centroids, and area moments of inertia. Not only does this site have over 60 hours of streaming video, but we also have dynamics tutors to help you with your dynamics questions. dynamics is the foundation of all of the engineering classes. Getting a firm understanding of dynamics is key to your dynamics class as well as other classes that use dynamics in the analysis such as strength of materials.
dynamics has never been easier than with streaming video. Watch and listen as your instructor guides you step-by-step to mastering dynamics. Learning dynamics using our technique makes you feel like you are in a classroom. We have over 60 hours (330 problems) of streaming video for dynamics. Many of our videos are free, just look for the Free icon next to the lesson. We also have online tutors to answer your dynamics questions. This is the place to go if you really want to learn dynamics!
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Learning dynamics dynamics: dynamics Analysis of forces on structures in dynamics equilibrium, properties of forces, moments, couples and resultant, conditions for dynamics equilibrium, friction, centroids, and area moments of inertia. dynamics is one of the first hardcore engineering courses that engineering students take. The key to mastering dynamics is solving many problems so that after you have done enough dynamics problems you relize that most dynamics problems are the same, just a slight twist between the different dynamics problems. dynamics has never been easier than with streaming video. Visit our site: www.yourotherteacher.com/dynamics.htm Watch and listen as your instructor guides you step-by-step to mastering dynamics. Learning dynamics using our technique makes you feel like you are in a classroom. We have over 40 hours of streaming video for dynamics. We also have online tutors to answer you dynamics questions. This is the place to go if you really want to learn dynamics! Come see us at: www.yourotherteacher.com/dynamics.htm Following is the preface from the dynamics book by Beer and Johnson and is the book about dynamics in my opinion and does a good job describing what dynamics is. The main objective of a first course in mechanics should be to develop in the engineering student the ability to analyze any problem in a simple and logical manner and to apply to its solution a few, well-understood, basic principles. This text is designed for the first courses in dynamics and dynamics offered in the sophomore or junior year, and it is hoped that it will help the instructor achieve this goal. Vector analysis is introduced early in the text and used throughout the presentation of dynamics and dynamics. This approach leads to more concise derivations of the fundamental principles of mechanics. It also results in simpler solutions of three-dimensional problems in dynamics, and makes it possible to analyze many advanced problems in kinematics and kinetics, which could not be solved by scalar methods. The emphasis in this text, however, remains on the correct understanding of the principles of mechanics and on their application to the solution of engineering problems, and vector analysis is presented chiefly as a convenient tool. One of the characteristics of the approach used in this book is that the mechanics of particles has been clearly separated from the mechanics of rigid bodies. This approach makes it possible to consider simple practical applications at an early stage and to postpone the introduction of the more difficult concepts. In dynamics, for example, the dynamics of particles is treated first (Chap. 2); after the rules of addition and subtraction of vectors have been introduced, the principle of equilibrium of a particle is immediately applied to practical situations involving only concurrent forces. The dynamics of rigid bodies is considered in Chaps. 3 and 4. In Chap. 3, the vector and scalar products of two vectors are introduced and used to define the moment of a force about a point and about an axis. The presentation of these new concepts is followed by a thorough and rigorous discussion of equivalent systems of forces leading, in Chap. 4, to many practical applications involving the equilibrium of rigid bodies under general force systems. In dynamics, the same division is observed. The basic concepts of force, mass, and acceleration, of work and energy, and of impulse and momentum are introduced and first applied to problems involving only particles. Thus, students may familiarize themselves with the three basic methods used in dynamics and learn their respective advantages before facing the difficulties associated with the motion of rigid bodies. Since this text is designed for first courses in dynamics and dynamics, new concepts have been presented in simple terms and every step explained in detail. On the other hand, by discussing the broader aspects of the problems considered, and by stressing methods of general applicability, a definite maturity of approach has been achieved. For example, the concepts of partial constraints and of statical indeterminacy are introduced early in the text and used throughout dynamics. In dynamics, the concept of potential energy is discussed in the general case of a conservative force. Also, the study of the plane motion of rigid bodies has been designed to lead naturally to the study of their general motion in space. This is true in kinematics as well as in kinetics, where the principle of equivalence of external and effective forces is applied directly to the analysis of plane motion, thus facilitating the transition to the study of three-dimensional motion. The fact that mechanics is essentially a deductive science based on a few fundamental principles has been stressed. Derivations have been presented in their logical sequence and with all dynamics, mechanics of materials rigor warranted at this level. However, the learning process being largely inductive, simple applications have been considered first. Thus the dynamics of particles precedes dynamics, mechanics of materials dynamics of rigid bodies, and problems involving internal forces are postponed until Chap. 6. Also, in Chap. 4, equilibrium problems involving only coplanar forces are considered first and solved by ordinary algebra, while problems involving three-dimensional forces and requiring the full use of vector algebra are discussed in the second part of dynamics, mechanics of materials chapter. Again, the dynamics of particles precedes the dynamics of rigid bodies; and, in dynamics, mechanics of materials latter, the fundamental principles of kinetics are first applied to the solution of two-dimensional problems, which can be more easily visualized by dynamics, mechanics of materials student (Chaps. 16 and 17), while three-dimensional problems are postponed until Chap. 18. The sixth edition of Vector Mechanics for Engineers retains the unified presentation of the principles of kinetics which characterized the previous editions. The concepts of linear and angular momentum are introduced in Chap. 12 so that Newton's second law of motion may be presented not only in its conventional form F=ma, but also as a law relating, respectively, dynamics, mechanics of materials sum of the forces acting on a particle and the sum of their moments to the rates of change of the linear and angular momentum of dynamics, mechanics of materials particle. This makes possible an earlier introduction of the principle of conservation of angular momentum and a more meaningful discussion of the motion of a particle under a central force (Sec. 12.9). More importantly, this approach can be readily extended to the study of the motion of a system of particles (Chap. 14) and leads to a more dynamics, videos, dynamics tutors, dynamics help concise and unified treatment of dynamics, mechanics of materials kinetics of rigid bodies in two and three dimensions (Chaps. 16 through 18). Free-body diagrams are introduced early, and their importance is emphasized throughout the text. They are used not only to solve equilibrium problems but also to express the equivalence of two systems of forces or, more generally, of two systems of vectors. The advantage of this approach becomes apparent in the study of dynamics, mechanics of materials dynamics of rigid bodies, where it is used to solve three-dimensional as well as two-dimensional problems. By placing the emphasis on "free-body-diagram equations" rather than on the standard algebraic equations of motion, a more intuitive and more complete understanding of the fundamental principles of dynamics can be achieved. This approach, which was first introduced in 1962 in the first edition of Vector Mechanics for Engineers, has now gained wide acceptance among mechanics teachers in this country. It is, therefore, used in preference to dynamics, mechanics of materials method of dynamic equilibrium and to the equations of motion in the solution of all sample problems in this edition. Color has been used, not only to enhance the quality of the illustrations, but also to help students distinguish among dynamics, mechanics of materials various types of vectors they will encounter. While there was no intention to "color code" this text, the same color was used in any given chapter to represent vectors of the same type. Throughout dynamics, for example, red is used exclusively to represent forces and couples, while position vectors are shown in blue and dimensions in black. This makes it easier for the students to identify the forces acting on a given particle or rigid body and to follow dynamics, mechanics of materials discussion of sample problems and other examples given in the text. In the chapters on kinetics, red is used again for forces and couples, as well as for effective forces. Red is also used to represent impulses and momenta in free-body-diagram equations, while green is used for velocities, and dynamics, videos, dynamics tutors, dynamics help blue for accelerations. In the two chapters on kinematics, which do not involve any forces, blue, green, and red are used, respectively, for displacements, velocities, and accelerations. Because of the current trend among American engineers to adopt dynamics, mechanics of materials international system of units (SI metric units), the SI units most frequently used in mechanics have been introduced in Chap. 1 and are used throughout the text. Approximately half of the sample problems and 57 percent of the homework problems are stated in these units, while dynamics, mechanics of materials remainder are in U.S. customary units. The authors believe that this approach will best serve the need of the students, who, as engineers, will have to be conversant with both systems of units. It also should be recognized that using both SI and U.S. customary units entails more than the use of dynamics, videos, dynamics tutors, dynamics help conversion factors. Since dynamics, mechanics of materials SI system of units is an absolute system based on the units of time, length, and mass, whereas the U.S. customary system is a gravitational system based on the units of time, length, and force, different approaches are required for the solution of many problems. For example, when SI units are used, a body is generally specified by its mass expressed in kilograms; in most problems of dynamics it will be necessary to determine the weight of dynamics, mechanics of materials body in newtons, and an additional calculation will be required for this purpose. On the other hand, when U.S. customary units are used, a body is specified by its weight in pounds and, in dynamics problems, an additional calculation will be required to determine its mass in slugs (or lb• s2 /ft). The authors, therefore, believe that problem assignments should include both systems of units. A sufficient number of problems of each type are provided so that six different lists of assignments can be selected with an equal number of problems stated in SI units and in U.S. customary units. If so desired, two complete lists of assignments can also be selected with up to 75 percent of the problems stated in SI units. A large number of optional sections have been included. These sections are indicated by asterisks and may thus easily be distinguished from those which form the core of the basic mechanics course. They may be omitted without prejudice to dynamics, mechanics of materials understanding of the rest of the text. Among the topics covered in these additional sections in the dynamics portion of the text are dynamics, mechanics of materials reduction of a system of forces to a wrench, applications to hydrodynamics, shear and bending-moment diagrams for beams, equilibrium of cables, products of inertia and Mohr's circle, mass products of inertia and principal axes of inertia for three-dimensional bodies, and the method of virtual work. An optional section on the determination of the principal axes and moments of inertia of a body of arbitrary shape has also been included in this new edition (Sec. 9.18). The sections on beams are especially useful when the course in dynamics is immediately followed by a course in mechanics of materials, while dynamics, mechanics of materials sections on the inertia properties of three-dimensional bodies are primarily intended for the students who will later study in dynamics the three-dimensional motion of rigid bodies. The topics covered in these sections in the dynamics portion of dynamics, mechanics of materials text include graphical methods for the solution of rectilinear-motion problems, the trajectory of a particle under a central force, the deflection of fluid streams, problems involving jet and rocket propulsion, the kinematics and kinetics of rigid bodies in three dimensions, damped mechanical vibrations, and electrical analogues. These topics will be found of particular interest when dynamics is taught in the junior year. The material presented in dynamics, mechanics of materials text and most of the problems require no previous mathematical knowledge beyond algebra, trigonometry, and elementary calculus, and all the elements of vector algebra necessary to the understanding of the text have been carefully presented in Chaps. 2 and 3. However, special problems have been included, which make use of a more advanced knowledge of calculus, and certain sections, such as Secs. 19.8 and 19.9 on damped vibrations, should be assigned only if the students possess dynamics, mechanics of materials proper mathematical background. In the portions of the text using elementary calculus, a greater emphasis has been placed on the correct understanding and application of the concepts of differentiation and integration than on the nimble manipulation of mathematical formulas. In this connection, it should be mentioned that dynamics, mechanics of materials determination of the centroids of composite areas precedes the calculation of centroids by integration, thus making it possible to establish the concept of moment of area firmly before introducing the use of integration. The presentation of numerical solutions takes into dynamics, videos, dynamics tutors, dynamics help account dynamics, mechanics of materials universal use of calculators by engineering students and instructions on the proper use of calculators for the solution of typical dynamics problems have been included in Chap. 2. Each chapter begins with an introductory section setting the purpose and goals of the chapter and describing in simple terms the material to be covered and its application to dynamics, mechanics of materials solution of engineering problems. The body of the text has been divided into units, each consisting of one or several theory sections, one or several sample problems, and a large number of problems to be assigned. Each unit corresponds to a well-defined topic and generally can be covered in one lesson. In a number of cases, however, the instructor will find it desirable to devote more than one lesson to a given topic. Each chapter ends with a review and summary of the material covered in that chapter. Marginal notes are used to help students organize their review work, and cross-references have been included to help them find the portions of material requiring their special attention. The sample problems are set up in much dynamics, mechanics of materials same form that students will use when solving the assigned problems. They thus serve the double purpose of amplifying the text and demonstrating the type of neat and orderly work that students should cultivate in their own solutions. A section entitled Solving Problems on Your Own has been added to each lesson, between the sample problems and dynamics, mechanics of materials problems to be assigned. The purpose of these new sections is to help students organize in their own minds the preceding theory of the text and the solution methods of the sample problems so that they may more successfully solve dynamics, mechanics of materials homework problems. Also included in these sections are specific suggestions and strategies which will enable the students to more efficiently attack any assigned problems. Most of the problems are of a practical nature and should appeal to engineering students. They are primarily designed, however, to illustrate the material presented in the text and to help students understand the basic principles of mechanics. dynamics, mechanics of materials problems have been grouped according to the portions of material they illustrate and have been arranged in order of increasing difficulty. Problems requiring special attention have been indicated by asterisks. Answers to 70 percent of the problems are given at the end of the book. Problems for which no answer is given are indicated by a number set in italic. The inclusion in the engineering curriculum of instruction in computer programming and dynamics, mechanics of materials increasing availability of personal computers or mainframe terminals on most campuses make it now possible for engineering students to solve a number of challenging dynamics problems. At one time these problems would have been considered inappropriate for an undergraduate course because of the large number of computations their solutions require. In this new edition of Vector Mechanics for Engineers, a group of problems designed to be solved with a computer has been included to the review problems at the end of each chapter. Many of these problems are relevant to the design process. In dynamics, they may involve the analysis of a structure for various configurations and loadings of dynamics, mechanics of materials structure, or the determination of the equilibrium positions of a given mechanism which may require an iterative method of solution. In dynamics, they may involve the determination of the motion of a particle under various initial conditions, the kinematic or kinetic analysis of mechanisms in successive positions, or dynamics, mechanics of materials numerical integration of various equations of motion. Developing the algorithm required to solve a given mechanics problem will benefit the students in two different ways: (1) it will help them gain a better understanding of the mechanics principles involved; (2) it will provide them with an opportunity to apply the skills acquired in their computer programming course to the solution of a meaningful engineering problem. The authors wish to acknowledge dynamics, mechanics of materials collaboration of Professors Elliot Eisenberg and Robert Sarubbi to this sixth edition of Vector Mechanics for Engineers and thank them especially for contributing many new and challenging problems. The authors also gratefully acknowledge the many helpful comments and suggestions offered by the users of the previous editions of Mechanics for Engineers and of Vector Mechanics for Engineers. Ferdinand P. Beer E. Russell Johnston, Jr. MECHANICS FOR ENGINEERS: dynamics, Fourth Edition -------------------------------------------------------------------------------- Authors: Ferdinand P. Beer, Lehigh University E. Russell Johnston, Jr., University of Connecticut Request a Review Copy Visit the Book Site! ISBN: 0-07-004580-1 Description: ©1987 / Hardcover / 496 pages Publication Date: September 1986 -------------------------------------------------------------------------------- Overview This scalar-based introductory dynamics text, ideally suited for engineering technology programs, provides first-rate treatment of rigid bodies without vector mechanics. This edition provides an extensive selection of new problems and end-of-chapter summaries. dynamics, mechanics of materials text brings the careful dynamics, videos, dynamics tutors, dynamics help presentation of content, unmatched levels of accuracy, and attention to detail that have made Beer and Johnston texts the standard for excellence in engineering mechanics education. -------------------------------------------------------------------------------- Features Precision, accuracy and the appropriate math level for Engineering Techology courses. Two-color graphics are used throughout the book. Sample Problems help students work through the solution of typical engineering problems. These also help students prepare for dynamics, mechanics of materials chapter problems they'll be assigned. -------------------------------------------------------------------------------- Supplements Instructor's Solutions Manual: dynamics / 0-07-004581-X -------------------------------------------------------------------------------- Table of Contents 1 Introduction 2 dynamics of Particles 3 dynamics of Rigid Bodies in Two Dimensions 4 dynamics of Rigid Bodies in Three Dimensions 5 Distributed Forces: Centroids and Centers of Gravity 6 Analysis dynamics, videos, dynamics tutors, dynamics help of Structures 7 Forces in Beams and Cables 8 Friction 9 Distributed Forces: Moments of Inertia 10 Method of Virtual Work Index Answers to Even-Numbered Problems Description For second-year Introductory courses taught in departments of Mechanical, Civil, Aerospace, General, and Engineering Mechanics. More than just a book, this text is part of a system to teach engineering mechanics, a system comprised of three components: 1) this core principles book, 2) algorithmic problem material available online, and 3) a course management system to track and monitor student progress. By using this system, instructors and their students benefit from increased flexibility in the ability to assign and grade problems, and the ability to make sure each student works a “unique” version of a problem, all coming at a lower price and in a smaller package. Table of Contents dynamics 1. Introduction. Engineering and Mechanics. Learning Mechanics. Fundamental Concepts. Units. 2. Vectors. Vector Operations and Definitions. Scalars and Vectors. Rules for Manipulating Vectors. Cartesian Components. Components in Two Dimensions. Components in Three Dimensions. Products of Vectors. Dot Products. Cross Products. Mixed Triple Products. 3. Forces. Types of Forces. Equilibrium and Free-Body Diagrams. Two-Dimensional Force Systems. Three-Dimensional Force Systems. 4. Systems of Forces and Moments. Two-Dimensional Description of the Moment. The Moment Vector. Moment of a Force about a Line. dynamics, videos, dynamics tutors, dynamics help Couples. Equivalent Systems. Representing Systems by Equivalent Systems. 5. Objects in Equilibrium. The Equilibrium Equations. Two-Dimensional Applications. Statically Indeterminate Objects. Three-Dimensional Applications. Two-Force and Three-Force Members. 6. Structures In Equilibrium. Trusses. The Method of Joints. dynamics, mechanics of materials Method of Sections. Space Trusses. Frames and Machines. 7. Centroids and Centers of Mass. Centroids. Centroids of Areas. Centroids of Composite Areas. Distributed Loads. Centroids of Volumes and Lines. The Pappus-Guldinus Theorems. Centers of Mass. Definition of the Center of Mass. Centers of Mass of Composite Objects. 8. Moments of Inertia. Areas. Definitions. Parallel-Axis Theorems. Rotated and Principal Axes. Masses. Simple Objects. Parallel-Axis Theorem. 9. Friction. Theory of Dry Friction. Applications. 10. Internal Forces and Moments. Beams. Axial Force, Shear Force, and Bending Moment. Shear Force and Bending Moment Diagrams. Relations between Distributed Load, Shear Force, and Bending Moment. Cables. Loads Distributed Uniformly Along Straight Lines. Loads Distributed Uniformly Along Cables. Discrete Loads. Liquids and Gases. Pressure and the Center of Pressure. Pressure in a Stationary Liquid. 11. Virtual Work and Potential Energy. Virtual Work. Potential Energy. Dynamics 1. Introduction. Engineering and Mechanics. Learning Mechanics. Fundamental Concepts. Units. 2. Motion of a Point. Position, Velocity, and Acceleration. Straight-Line Motion. Curvilinear Motion. Relative Motion. 3. Force, Mass, and Acceleration. Newton's Second Law. Equation of Motion for the Center at Mass. Inertial Reference Frames. Applications. Orbital Mechanics. 4. Energy Methods. Work and Kinetic Energy. Principle of Work and Energy. Work and Power. Work Done by Particular Forces. Potential Energy Conservation of Energy. Conservative Forces. 5. Momentum Methods. Principle of Impulse and Momentum. Conservation of Linear Momentum. Impacts. Angular Momentum. Mass Flows. 6. Planar Kinematics of Rigid Bodies. Rigid Bodies and Types of Motion. Rotation about a Fixed Axis. General Motions: Velocities. General Motions: Accelerations. Sliding Contacts. Moving Reference Frames. 7. Planar Dynamics of Rigid Bodies. Preview of the Equations of Motion. Momentum Principles for a System of Particles. Derivation of dynamics, mechanics of materials Equations of Motion. Applications. Appendix: Moments of Inertia. 8. Energy and Momentum In Rigid Body Dynamics. Principle of Work and Energy. Work and Potential Energy. Power. Principles of Impulse and Momentum. Impacts. 9. Three-Dimensional Kinematics and Dynamics of Rigid Bodies. Kinematics. Angular Momentum. Moments and Products of Inertia. Euler's Equations. Eulerian Angles. 10. Vibrations. Conservative Systems. Damped Vibrations. Forced Vibrations. Description For introductory dynamics and dynamics courses found in mechanical engineering, civil engineering, aeronautical engineering, and engineering mechanics departments. This best-selling text offers a concise and thorough presentation of engineering mechanics theory and application. The material is reinforced with numerous examples to illustrate principles and imaginative, well-illustrated problems of varying degrees of difficulty. The text is committed to developing students' problem-solving skills and includes pedagogical features that have made Hibbeler synonymous with excellence in the field. The Ninth Edition has been updated to offer insightful new problems, improved examples, and a stronger dynamics, videos, dynamics tutors, dynamics help supplement package. Table of Contents dynamics Edition 1. General Principles. Mechanics. Fundamental Concepts. Units of Measurement. The International System of Units. Numerical Calculations. 2. Force Vectors. Scalars and Vectors. Vector Operations. Vector Addition of Forces. Addition of a System of Coplanar Forces. Cartesian Vectors. Addition and Subtraction of Cartesian Vectors. Position Vectors. Force Vector Directed Along a Line. Dot Product. 3. Equilibrium of a Particle. Condition for dynamics, mechanics of materials Equilibrium of a Particle. The Free-Body Diagram. Coplanar Force Systems. Three-Dimensional Force Systems. 4. Force System Resultants. Moment of a Force—Scalar Formation. Cross Product. Moment of a Force—Vector Formulation. Principle of Moments. Moment of a Force About a Specified Axis. Moment of a Couple. Equivalent System. Resultants of a Force and Couple System. Further Reduction of a Force and Couple System. Reduction of a Simple Distributed Loading. 5. Equilibrium of a Rigid Body. Conditions for Rigid-Body Equilibrium. Equilibrium in Two Dimensions. Free-Body Diagrams. Equations of Equilibrium. Two- and Three-Force Members. Equilibrium in Three Dimensions. Free-Body Diagrams. Equations of Equilibrium. Constraints for a Rigid Body. 6. Structural Analysis. Simple Trusses. The Method of Joints. Zero-Force Members. The Method dynamics, videos, dynamics tutors, dynamics help of Sections. Space Trusses. Frames and Machines. 7. Internal Forces. Internal Forces Developed in Structural Members. Shear and Moment Equations and Diagrams. Relations Between Distributed Load, Shear, and Moment. Cables. 8. Friction. Characteristics of Dry Friction. Problems Involving Dry Friction. Wedges. Frictional Forces on Screws. Frictional Forces on Flat Belts. Frictional Forces on Collar Bearings, Pivot Bearings, and Disks. Frictional Forces on Journal Bearings. Rolling Resistance. 9. Center of Gravity and Centroid. Center of Gravity and Center of Mass for a System of Particles. Center of Gravity, Center of Mass, and Centroid for a Body. Composite Bodies. Theorems of Pappus and Guldinus. Resultant of a General Distributed Force System. Fluid Pressure. 10. Moments of Inertia. Definitions of Moments of Inertia for Areas. Parallel-Axis Theorem for an Area. Radius of Gyration of an Area. Moments of Inertia for an Area by Integration. Moments of Inertia for Composite Areas. Product of Inertia for an Area. Moments of Inertia for an Area About Inclined Axes. Mohr's Circle for Moments of Inertia. Mass Moment of Inertia. 11. Virtual Work. Definition of Work and Virtual Work. Principle of Virtual Work for a Particle and a Rigid Body. Principle of Virtual Work for a System of Connected Rigid Bodies. Conservative Forces. Potential Energy. Potential Energy Criterion for Equilibrium. Stability of Equilibrium. Appendixes. A. Mathematical Expressions. B. Numerical and Computer Analysis. Answers. Index. Dynamics Edition 12. Kinematics of a Particle. Introduction. Rectilinear Kinematics: Continuous Motion. Rectilinear Kinematics: Erratic Motion. General Curvilinear Motion. Curvilinear Motion: Rectangular Components. Motion of a Projectile. Curvilinear Motion: Normal and Tangential Components. Curvilinear Motion: Cylindrical Components. Absolute Dependent Motion Analysis of Two Particles. Relative-Motion Analysis of Two Particles Using Translating Axes. 13. Kinetics of a Particle: Force and Acceleration. Newton's Laws of Motion. The Equation of Motion. Equation of Motion for a System of Particles. Equations of Motion: Rectangular Coordinates. Equations of Motion: Normal and Tangential Coordinates. Equations of Motion: Cylindrical Coordinates. Central-Force Motion and Space Mechanics. 14. Kinetics of a Particle: Work and Energy. The Work of a Force. Principle of Work and Energy. Principle of Work and Energy for a System of Particles. Power and Efficiency. Conservative Forces and Potential Energy. Conservation of Energy. 15. Kinetics of a Particle: Impulse and Momentum. Principle of Linear Impulse and Momentum. Principle of Linear Impulse and Momentum for a System of Particles. Conservation of Linear Momentum for a System of Particles. Impact. Angular Momentum. Relation Between Moment of a Force and Angular Momentum. Angular Impulse and Momentum Principles. Steady Fluid Streams. Propulsion with Variable Mass. REVIEW 1: KINEMATICS AND KINETICS OF A PARTICLE. 16. Planar Kinematics of a Rigid Body. Rigid-Body Motion. Translation. Rotation About a Fixed Axis. Absolute General Plane Motion Analysis. Relative-Motion Analysis: Velocity. Instantaneous Center of Zero Velocity. Relative-Motion Analysis: Acceleration. Relative-Motion Analysis Using Rotating Axes. 17. Planar Kinetics of a Rigid Body: Force and Acceleration. Moment of Inertia. Planar Kinetic Equations of Motion. Equations of Motion: Translation. Equations of Motion: Rotation About a Fixed Axis. Equations of Motion: General Plane Motion. 18. Planar Kinetics of a Rigid Body: Work and Energy. Kinetic Energy. The Work of a Force. dynamics, mechanics of materials Work of a Couple. Principle of Work and Energy. Conservation of Energy. 19. Planar Kinetics of a Rigid Body: Impulse and Momentum. Linear and Angular Momentum. Principle of Impulse and Momentum. Conservation of Momentum. Eccentric Impact. REVIEW 2: PLANAR KINEMATICS AND KINETICS OF A RIGID BODY. 20. Three-Dimensional Kinematics of a Rigid Body. Rotation About a Fixed Point. The Time Derivative of a Vector Measured from a Fixed and Translating-Rotating System. General Motion. Relative-Motion Analysis Using Translating and Rotating Axes. 21. Three-Dimensional Kinetics of a Rigid Body. Moments and Products of Inertia. Angular Momentum. Kinetic Energy. Equations of Motion. Gyroscopic Motion. Torque-Free Motion. 22. Vibrations. Undamped Free Vibration. Energy Methods. Undamped Forced Vibration. Viscous Damped Free Vibration. Viscous Damped Forced Vibration. Electrical Circuit Analogs. Appendixes. A. Mathematical Expressions. B. Numerical and Computer Analysis. C. Vector Analysis. D. Review for the Fundamentals of Engineering Examination. Answers to Selected Problems. Index. dynamics is the study of the conditions under which mechanical and other systems remain in a configuration ("state") which does not change with time. dynamics theory is based on five axioms: 1. A rigid body acted upon by two forces is in a state of static equilibrium if and only if the two forces are of dynamics, mechanics of materials same intensity, lie along the same line of action, and are oriented in opposite directions along the line. 2. If a system of two forces in equilibrium is added to or extracted from a given system of forces, the way that the system of forces acts on a rigid body undergoes no change. 3. The resultant of two forces acting at dynamics, mechanics of materials same material point is equal to the vector sum of the two forces. The line of the resulting force's action contains the material point. This axiom obeys dynamics, mechanics of materials principle of vector summation. 4. Two interacting bodies react on each other with two forces of equal intensity, and along the same line of dynamics, videos, dynamics tutors, dynamics help action, but in opposite directions along the line. This axiom is also known as principle of action and reaction. 5. If a deformable body is in a state of static equilibrium, it would also be in static equilibrium if the body were rigid. This axiom is also known as the principle of solidification. 1.1 dynamics , what is it good for ? NextSec Let's say you as engineer are presented with all the design details of a certain structure and your task is to make sure that this structure does not break under influence of dynamics, mechanics of materials design load ( that includes a certain safety factor ). no material is wasted Here a structure can be just about anything of interest, the chair you are sitting on, a simple tower for a power line or a bridge over a river. Often a structure consists of many, sometimes thousands of members (parts) which are connected ( glued , welded, rivetted etc.) to each other. Each individual member is subject to break. So, how do you make sure that none does ? Well, even if the design is already given to you this task can be difficult enough. It can be roughly broken down into two steps Determine the forces acting on each member (this is the EMch 11 stuff ). Determine whether the member under consideration can withstand dynamics, mechanics of materials calculated forces with whatever safety margin is appropriate (this is EMch 13 stuff and things coming after that). Here is a sample problem, take a peek. If this problem does not make much sense to you now you may want to read Chapter 1 in its entirety first and then come back to it. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 1.2 Types of forces PrecSec NextSec A force is the action of one body onto another. We distinguish : Contact or support forces which occur whenever two bodies are in physical contact with each other. They always depend on the details of how the two bodies are connected to each other. You and the chair you are sitting on would be a nice example, your ear being attached to your head and not falling off would be another. Field Forces which may be due to gravitation, electro-static,electro-magnetic or nuclear interactions. No contact between the involved bodies is necessary. Actually, dynamics, mechanics of materials distance between two such bodies can be rather large in comparison to size of the bodies themselves (earth and moon for example). Warning : Often there are forces of both types acting simultaneously between two bodies. My example : Take a car sitting outside your house. The car is pulled towards the center of the earth by gravity (we call that force commonly "weight"). This force is constantly present even if the car drives over a hump at high speed and its wheels are not touching dynamics, mechanics of materials ground momentarily. On the other hand (when the wheels are touching the ground) the earth's surface is pushing upwards onto the car at dynamics, mechanics of materials points where the wheels and earth's surface are touching. You can think of this force as a kind of resistance force the earth's surface is exerting onto the wheels, trying to prevent (resist) the wheels from penetrating the earth. If dynamics, mechanics of materials surface is made out of a soft material (like mud or sand) which has not much resistance the car's wheels will actually sink into the ground until enough resistance is encountered. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 1.3 Action and Re-action PrecSec NextSec We just looked at what kind of forces are acting on a car sitting on the ground and argued that the earth exerts not only a gravitational pull onto the car but that dynamics, mechanics of materials ground is pushing upwards against the wheels of the car. But what does the car do to the earth ? Well, the contact force is easy, dynamics, mechanics of materials car is pushing down onto the earth's surface and might even make a dent into it. Obviously, the force the car exerts onto the ground and the force from dynamics, mechanics of materials ground onto the wheels are in opposite direction (one upwards one downwards in this example). How about the gravitational force and with that all other field forces ? If I told you that the car is actually pulling the earth towards its own center of mass, you would probably ask me to prove that, wouldn't you ? Well, let me ask YOU to make up a thought experiment which clearly shows that the gravitational forces between car and earth are oppositely directed. Here is my thought experiment. Later we will learn that ALL forces acting between any two bodies come in pairs and are exactly equal in value but exactly opposite in direction. This law is often called Newton's Third Law. The interaction between different bodies is not always easy to see but is of utmost importance, so here is an example you definitely should have a look at. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 1.4 What a force can do to a body PrecSec NextSec A force acting on a rigid body can in general cause dynamics, mechanics of materials body to undergo two different types of motion. The body starts to accelerate The body will start to rotate Whether the body actually moves/rotates depends on all the forces acting on the given body. In dynamics we don't want either to happen and therefore : dynamics : Study of bodies in equilibrium The demand of bodies neither to translate ( = motion of its center of mass if you will) nor to rotate ( = spinning around its center of mass ) will lead us to different types of equations relating the forces ( in magnitude and direction ) acting on a body to each other. Once we have enough equation we will solve for the unknown forces and we are done with dynamics, mechanics of materials EMch11-step of the designing process. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 1.5 Graphical representation of forces PrecSec Figure 1.5a Graphical Representation of Forces In Figure 1.5a we have three forces acting on the box-like body. Each force is represented by an arrow, pointing in the direction the force is acting and having a length equal to the magnitude of dynamics, mechanics of materials force. Often the head or the tail of this arrow is placed at the point where the force is acting on the body. This point is then called dynamics, mechanics of materials point of attachment. Next to each arrow we have a symbol ( F1 etc. ) which we use later in equations or inside the text to indicate which of several forces we are talking about. The letters F and subscript 1,2 etc. are completely arbitrary although often used. Other letters which you find in textbooks to represent forces are P, Q, and R. The arrow above each symbol is a reminder for us that a force is a quantity of a certain value and direction. It is the same notation we use in vector algebra and vector calculus, because as we will shortly see, the physical quantity of a force is as far as mathematical properties are concerned equal to a vector. NOTE : In Figure 1.5a we do not show the bodies which are exerting dynamics, mechanics of materials shown forces F1 etc. Well, maybe now you'd like to have another peek at the afore mentioned sample problem. 2.1 What is a particle ? NextSec A particle is body whose rotational aspects are not of interest at the moment. This is the case of dynamics, videos, dynamics tutors, dynamics help course when the body itself is extremely small so that we think of it as a single point at which all its mass is concentrated. Often we take a large body and shift all its mass to its center of mass and just look at the motion (or dynamics, mechanics of materials lack thereof) of the center of mass. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 2.2 The general problem PrecSec NextSec Here is the general idea as to what we are interested in for the moment. Figure 2.2a Multiple Forces on a Particle Assume you have given a small body A with several forces as shown in Figure 2.2a. Two of the forces, let's say Fa and Fb, are given, that means their values (in Newton for example) and their directions are known. How can we determine how big dynamics, mechanics of materials third force, Fc must be in order for the particle A to remain at rest ? This question is very representative of the entire subject of dynamics. Some forces ( we refer to them often as the loads ) are known dynamics, videos, dynamics tutors, dynamics help while other forces are unknown but of interest to us. Below we will take the position of an experimentalist and discover the governing laws ourselves. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 2.3 A simple 2-force experiment PrecSec NextSec Figure 2.3a 2-force Experiment Well, having 2 forces given and looking for dynamics, mechanics of materials third might more than we can chew for the time being. So, let's step back a little. In Figure 2.3a a body A is acted upon by two forces. Assume that the force F4 is given. What must the magnitude and direction of force F3 be, in order for the body A to remain at rest ? The experimentalist will tell us immediately : The forces F3 and F4 must be exactly equal in value and exactly opposite in direction. Going back to the business of action/reaction : can you now think of an experiment which clearly demonstrates that dynamics, mechanics of materials force you are exerting on the chair you are sitting on is equal and opposite to that the chair is exerting on you ? TopSec TopChapt NextSec -------------------------------------------------------------------------------- 2.4 Parallelogram Law PrecSec NextSec Figure 2.4a 3-force Experiment Turning now back to the original 3-force problem from secion 4.2. obvious where things are going. So bear with me. In the Figure 2.4a assume that the forces Fa and Fb are given. Question : What is dynamics, mechanics of materials magnitude and direction of the force Fc such that the particle A remains at rest? After some experimenting one would find out that body A remains at rest whenever the forces Fa and Fb form the sides of a parallelogram the diagonal of which is equal Figure 2.4b 3-force Experiment and opposite to Fc of Figure 2.4a. We call this diagonal force dynamics, mechanics of materials resultant R (in this case of Fa and Fb ) It has the same effect on the body A as the original forces Fa and Fb namely to counter-balance Fc in the same way F4 did in the 2-force experiment of section 2.3. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 2.5 Resultant of two forces PrecSec NextSec Finding dynamics, mechanics of materials resultant of a given set of forces is one of the many tools we will use over and over again. Figure 2.5a Resultant of 2 forces Here are two equations relating the various quantities in Figure 2.5a to each other : Note that the angle is measured between the resultant R and the force appearing first on dynamics, mechanics of materials right hand side of equation (2.5b). Above equations are directly related to the Law of Sines and Cosines . How about you try to derive them yourself ? We need lots of trigonometry in the weeks to come and this would be an exercise as good as any. Speaking of trigonometry. Can you derive an equation for the tan() ? You have to be careful when applying equation 2.5b because it gives you always two possible values for the angle because in the ambiguity in determination of an angle with only its cosine is given. (example cos() = 0.5 has as solutions =60 and =300 degrees) The rule is here that the resultant R has to lie always inbetween dynamics, mechanics of materials given forces F1 and F2, hence you choose such that : 0 < < You definitely should look at Problem 2.5a. An important special case occurs when F1 and F2 are equal in magnitude and the angle between them approaches 180o. Maybe you can visualize the outcome from Figure 2.5a. Alternatively, from Equation 2.5a we obtain that R=0. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 2.6 Resultant of several forces forces PrecSec NextSec Figure 2.6a 3-force on a particle As an example let's look at Figure 2.6a where three forces act on a small body A. How can we find their resultant, that is that force which has the same action on body A as the three given forces Fa, Fb, and Fc? In principle this is not too difficult dynamics, videos, dynamics tutors, dynamics help : We could first replace for example Fa and Fb by a resultant, say Rab, using Equation 2.5a to find its value and 2.5b to find the orientation of Rab. Now we have only two forces left, Rab and Fc. Again we can find their resultant by using Equations 2.5a and 2.5b. Obviously, this concept can be expanded to as many forces as we like. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 2.7 Summary and statement of equilibrium PrecSec NextSec There are several items we have worked out in this chapter : The resultant of two forces is that force which has the same effect on a body as the original two forces. Its value and direction can be dynamics, videos, dynamics tutors, dynamics help determined using Equations 2.5a and 2.5b, respectively. A set of forces acting on a particle can be reduced to a single resultant by replacing pairs of forces by their resultant. The resultant of two equal and oppositely directed forces is zero. A particle remains at rest under influence of two forces if these two forces are equal in value and opposite in direction. Here is an alternative statement concerning dynamics, mechanics of materials equilibrium of a particle : A particle remains at rest if the resultant of ALL forces acting on it is equal to zero. In essence this is a special case of Newton's 2nd law : F = m a where m is the mass of the particle, F is the resultant of ALL forces acting on the particle, and a is the corresponding acceleration (change of velocity) of the particle. In dynamics we want a=0 which requires that F=0. TopSec TopChapt NextSec 3.1 Orthogonal components of forces NextSec The determination of dynamics, mechanics of materials resultant of three or more forces using strictly the Parallelogram Law in the form of Equations 2.5a and 2.5b is somewhat tedious and in the long run almost useless. We need better tools !!! Figure 3.1a 3 forces on a particle Let's have a look at Figure 3.1a. Three forces , F1, F2, and F3 are shown acting on a particle A . Also shown is an orthogonal coordinate system whose axes I labelled x and y. The location of its origin and the alignment of its axes with the borders of the figure are arbitrary choices of mine. Our task is now to develop a more efficient way to determine dynamics, mechanics of materials magnitude and direction of the resultant of the three forces shown. The trick we will be employing is the following. We interpret each of the three forces in Figure 3.1a as the resultant of two forces, one aligned Figure 3.1b Force F1 and its components with the x- the other with dynamics, mechanics of materials y-axis as shown in Figure 3.1b for the force F1. We call these two new forces the x- and y-component of the force F1. Their values can be easily calculated if the magnitude (the absolute value) of F1 and its orientation (the angle alpha) are known. Note that the angle alpha is measured between the positive x-axis and the force in counterclockwise direction. Also, depending on dynamics, mechanics of materials value of the angle alpha one or both of the components might have a negative value, indicating that that component is pointing in the direction of the minus x-axis for example. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 3.2 Determination of resultant of forces PrecSec NextSec In Figure 3.1a we now replace the force F1 by its x- and y-component and repeat this step for the two other forces involved. The result is that we have replaced dynamics, mechanics of materials original three forces by six new forces, of which three are aligned with the x-axis and three with the y-axis of our coordinate system. The final step is then to add the three force components in the x-direction (no sweat here, that would be just adding/subtracting numbers) to get the x-component of the resultant. The y-component of dynamics, mechanics of materials resultant is obtained in similar fashion. Formally we write this as : Once we have these components we can determine the magnitude of the resultant and the angle beta between the resultant and the x-axis : Equation 3.2d has always two solutions for the angle beta. If for given Rx and R your calculator gives beta=30o for example then beta=180-30=150 is a solution as well. But which value is correct, 30 or 150o ? The answer to this question can be found by looking at the signs (+/-) of dynamics, mechanics of materials components Rx and Ry which informs in which quadrant of the unit circle your resultant R lies. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 3.3 Resultant of forces, a sample case PrecSec NextSec The Equations 3.1a through 3.2d together form a set of equations which will follow you throughout this course and beyond (together with the statement of equilibrium of particles ). When applying these equations it is extremely important to know about the sign (plus/minus) conventions which go along with the cos() and sin() function used in the Equations 3.2a and 3.2b. Of course in dynamics we don't make up our own rules but follow strictly the rules of trigonometry. Here is a short sample case I would recommend you read carefully. To some of you it might seem silly to harp on sign conventions. However, in practical engineering applications not observing the correct sign amounts often to the difference between a well designed structure and a failing structure with possible loss of human life and/or millions of dollars. So, here is dynamics, mechanics of materials problem as depicted in Figure 3.3a. Originally you know only the magnitudes ( 400 N , 350 N, 600 N, and 100 N ) and the orientations (angles 50, 70, 30, and 15 degrees) of the four forces. Our task is to find the resultant of these four forces, that is to find that single force which has the same action on particle A as the four given forces. If the labels ( F1, F2, F3, and F4) are not given, you must label them. I furthermore entered already an x-y coordinate system. If it is not given, you must make a choice. I aligned dynamics, mechanics of materials x-axis with the line a-a. Figure 3.3a Find resultant of four forces After these preliminary steps the real work begins. Here are my calculations for the x-components of the four forces and then the x-component of the resultant : Three points of interest : The angle used as argument of the cosine function is always determined by going on an arc from dynamics, mechanics of materials positive x-axis in counter-clockwise direction towards the force of which you want to determine the x-component. Two of the forces have negative x-components ( cos(110) is negative as is cos(210) ). A negative value of the x-component means that the x- component is pointing in the direction of the minus x-axis. The plus/minus sign of dynamics, mechanics of materials obtained x-components has to be entered when calculating the x-component of the resultant. Here Rx comes out to be negative itself, meaning that the combined action of the forces is to pull to the left in minus x-direction (forgetting at the moment about what happens in the y-direction). Please, do determine dynamics, mechanics of materials value of the y-component of the resultant yourself. My results are displayed in Figure 3.3b. Figure 3.3b Resultant of four forces The action of the four original forces becomes now clear. They will pull the body A to the left and upwards. Also, please check out whether I got the magnitude and angle for the resultant right. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 3.4 Three-dimensional aspects PrecSec NextSec Although this course will mostly deal with 2-dimensional structures and forces ( planar cases ) from time to time we will dive into dynamics, mechanics of materials three-dimensional space. The concept of orthogonal components we looked at in the preceeding chapters will be employed again, but the direction of a force in 3-D is harder to describe and therefore the calculations of the component of a 3-D force is a little bit more involved. In the dynamics, videos, dynamics tutors, dynamics help Figures 3.4a and 3.4b I show the same force F embedded in identical x-y-z coordinate systems. Also, the dashed line 0B lies in both Figures in dynamics, mechanics of materials x-y plane. The direction of the force F is expressed in different ways though. Figure 3.4a Force in 3-D For Figure 3.4a : Given is the magnitude of the force and the values of the angle beta and gamma. We then determine the three components of the force by : Figure 3.4b Force in 3-D For Figure 3.4b dynamics, mechanics of materials magnitude and the angle alphax, alphay, and alphaz are given. Now the three components are evaluated according to : On the other hand, to determine the components of the resultant of several forces Equation 3.2a to 3.2c have to be modified only slightly : And the magnitude of a force (like the resultant) is calculated according to : TopSec TopChapt NextSec -------------------------------------------------------------------------------- 3.5 Summary,Re-statement of Equilibrium, and Free-Body-Diagram PrecSec NextSec The concept of orthogonal components of forces greatly facilitates dynamics, mechanics of materials determination of resultants of multiple forces. Its usefulness will become much more apparent and wide-spread in subsequent chapters. The choice of the orientation of the coordinate system (x-y or x-y-z) is arbitrary and does not affect the final outcome of your calculations. In order to determine the components of a force only simple trigonometric functions ( cosine and sine ) are needed, the sign (plus/minus) conventions for angles greater than 90o have to be obeyed. The value of the component of a force (and later that of a force) can be positive, negative, or zero. If dynamics, videos, dynamics tutors, dynamics help negative, it indicates that the action of that force component is directed in dynamics, mechanics of materials negative direction of the corresponding coordinate axis. By the magnitude of a force (or force component) we mean the value of it stripped of its sign. In mathematics we know this also as the absolute value. In section 2.7 we concluded that in order for a particle to remain at rest the resultant of all forces acting on the particle has to be zero. But if the resultant itself is zero then all its components are zero!!! Hence, dynamics, mechanics of materials statement of equilibrium of a particle can be re-phrased : A particle remains at rest if the sum of the x-components and the sum of the y-components and the sum of the z-components of ALL forces acting on it, are zero. or in equation form : Taking into account ALL of the forces acting on a particular body is not always an easy task. A particularly useful tool is the Free-Body-Diagram The Free-Body-Diagram ( F.B.D ) of a body is a rough sketch of a body which includes ALL forces acting on that body from dynamics, mechanics of materials outside. Apart from the difficulty of realizing which forces are acting on a given body and what can be said about each of these forces much confusion arises from some sign conventions which are unfortunately necessary. 4.1 Vectors, what are they anyway ? NextSec Well, that question is not so easy to answer without getting totally carried away. If you search Encyclopedia Britannica Online for "vector AND mathematics" or for "vector AND physics" you can find the phrase a branch of mathematics that deals with quantities that have both magnitude and direction . From that point of view forces fall under this category. There is a little bit more to it -- that is not everything having direction and magnitude can be treated as a vector as far as applying to it the mathematics of vector algebra and vector analysis is concerned. One test your quantity has to pass is for example whether your quantities can be added to each other like we did when we wanted to find the resultant of forces using Equations 3.2a and 3.2b or Equations 3.4c and 3.4d and whether doing so makes sense (in the case of forces from the view point of physics). We will see in the next chapter that not only forces but also location vectors -- which are vectors pointing from one point in space to another -- are full-fledged vectors. And so are velocity and acceleration ( EMch 12 stuff). TopSec TopChapt NextSec -------------------------------------------------------------------------------- 4.2 Addition and subtraction PrecSec NextSec In section 3.2 and 3.4 we dealt with dynamics, mechanics of materials subject of finding the resultant of two or more forces : In the language of mathematics we address the forces , .... etc. as vectors and use the notation : If we have numerical values available for the components we would write : The arrow on top of the symbol merely indicates that this quantity is a vector. Many textbooks use bold/italics symbols instead, like F1 because the arrows are costly to print. On dynamics, mechanics of materials WWW I am in a similar situation and will use the arrow mode and bold/italics interchangeably. Using this notation, Equation 3.4c (just above) is then written as The mathematician would read this as : Add the vector and to obtain the vector and Equation 3.4c states the rule as to how to do that : Adding two vectors to each other results in a new vector the x-component of which is determined by the adding the x-components of the given vectors. (Same for the y- and z- component) The physicist and we would say : The forces and together have the same action on a given body as the force . To obtain one force from the others we perform the same mathematical operations, that is use Equation 3.4c. How about subtraction of vectors ? Formally we would write : but how would we determine the components of if the components of and are given ? Well, a requirement one could put forward here would be dynamics, mechanics of materials following : If you subtract first from a vector a vector to obtain an intermediate vector and than ADD to that, you should come back to . dynamics, mechanics of materials only way how this can be accomplished by using the following rule : Subtracting two vectors from each other results in a new vector the x-component of which is determined by subtracting the x-components of the given vectors. (Same for the y- and z- component) TopSec TopChapt NextSec -------------------------------------------------------------------------------- 4.3 Magnitude PrecSec NextSec There is not much to say here. We encountered the magnitude of force before and calculated it using Equations 3.2c (for 2-dimensional cases) and 3.4d (for 3-dimensional cases). Just as a reminder : If the components of a vector are given, its magnitude (which is always non-negative) is calculated according to : TopSec TopChapt NextSec -------------------------------------------------------------------------------- 4.4 Multiplication by a scalar PrecSec NextSec A scalar is a single number which can be negative, zero, or positive. If we wish to multiply a given vector, say by a scalar, say "a", we obtain a new vector, say . We formally write : and dynamics, mechanics of materials rule which goes along with that is the following : Multiplying a vector by a scalar results in a new vector the x-component of which is determined by multiplying the the x-component of the given vector by the scalar. (Same for the y- and z- component) At least for the case of dynamics, mechanics of materials scalar being a positive integer this rule makes a lot of sense because it ties in nicely with the addition of vectors : whether you add a given vector 4 times to itself or whether you multiply the vector by 4 should give identical results : Speaking in terms of forces : 4 times is a force which is 4 times as strong as and pointing in the same direction as . Similarily, in the following equation the left side and right side give identical results : Speaking in terms of forces : -1 times is a force which has the same magnitude as but is pointing exactly in opposite direction. TopSec TopChapt NextSec -------------------------------------------------------------------------------- 4.5 Unit vectors PrecSec NextSec If F denotes the magnitude of a vector calculated according to Equation 4.3a then we can of course multiply dynamics, mechanics of materials vector by the scalar 1/F where F=magnitude of . The result is a vector which has a special property and therefore is given in many textbooks a special symbol, the greek lower case lambda : Because the magnitude F (and with that 1/F) is never negative the vector is pointing in the same direction as the vector itself. The unique property of is that its magnitude is always 1 (one) regardless of what values (and units) dynamics, mechanics of materials components of are (except if they are all zero). Because of this property we call a unit vector. Among the infinitely many unit vectors which one can calculate there are a few worth mentioning. If in Equation 4.5a the vector has a positive x-component but zero y- and z-component, that is if it is pointing along the positive x-axis, then is pointing along the positive x-axis as well, still having the length one. In this special case we often (and this is almost universal) give that unit vector the symbol . The analogs for dynamics, mechanics of materials y- and z-directions are given the symbols and , respectively. The unit vectors , , and can be used to present any arbitrary vector with components Fx, Fy, and Fz in an alternative form : This way of writing out a vector will come in handy later on in the course, for now it is one example of how addition of vectors