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Engineering Mechanics Dynamics Engineering Mechanics Volume 2 Dynamics Seventh Edition J. L. Meriam L. G. Kraige Virginia Polytechnic Institute and State . Engineering Mechanics Dynamics by J.L. Meriam, L.G. caite.info Hassan Muhammad. Loading Preview. Sorry, preview is currently unavailable. You can. Sorry, this document isn't available for viewing at this time. In the meantime, you can download the document by clicking the 'Download' button above.

This repeated transition of thought between the physical and the mathematical is required in the analysis of every problem. If the particle is slowing down, the particle is said to be decelerating. Visibility Others can see my Clipboard. Mechanics by its very na- ture is geometrical, and students should bear this in mind as they review their mathematics. Wallace Grant.

The answers to all problems have been provided in a special section at the end of the textbook. In recognition of the need for emphasis on SI units, there are approximately two prob- lems in SI units for every one in U.

This apportionment between the two sets of units permits anywhere from a 50—50 emphasis to a percent SI treatment. A notable feature of the Seventh Edition, as with all previous editions, is the wealth of interesting and important problems which apply to engineering design.

Illustrations In order to bring the greatest possible degree of realism and clarity to the illustrations, this textbook series continues to be produced in full color. Whenever possible, mechanisms or objects which commonly have a cer- tain color will be portrayed in that color. All of the fundamental elements of technical illus- tration which have been an essential part of this Engineering Mechanics series of textbooks have been retained. Special Features While retaining the hallmark features of all previous editions, we have incorporated these improvements: These diagrams are well integrated with the time-order form of the impulse-momentum equations.

All new problems have been independently solved in order to ensure a high degree of accuracy. Preface ix 9. Organization The logical division between particle dynamics Part I and rigid-body dynamics Part II has been preserved, with each part treating the kinematics prior to the kinetics. In Chapter 1, the fundamental concepts necessary for the study of dynamics are established. Chapter 2 treats the kinematics of particle motion in various coordinate systems, as well as the subjects of relative and constrained motion.

Chapter 3 on particle kinetics focuses on the three basic methods: The special topics of impact, central-force motion, and relative motion are grouped together in a special applications section Section D and serve as optional material to be assigned according to instructor preference and available time. With this arrangement, the attention of the stu- dent is focused more strongly on the three basic approaches to kinetics.

Chapter 4 on systems of particles is an extension of the principles of motion for a single particle and develops the general relationships which are so basic to the modern compre- hension of dynamics. In Chapter 5 on the kinematics of rigid bodies in plane motion, where the equations of relative velocity and relative acceleration are encountered, emphasis is placed jointly on solution by vector geometry and solution by vector algebra. This dual approach serves to reinforce the meaning of vector mathematics.

In Chapter 6 on the kinetics of rigid bodies, we place great emphasis on the basic equations which govern all categories of plane motion. Special emphasis is also placed on forming the direct equivalence between the actual applied forces and couples and their and resultants. In this way the versatility of the moment principle is em- phasized, and the student is encouraged to think directly in terms of resultant dynamics effects.

For students who later pursue more advanced work in dynamics, Chapter 7 will provide a solid foundation. Gyroscopic motion with steady precession is treated in two ways. With this treatment, the student can understand the gyroscopic phenomenon of steady precession and can handle most of the engineering problems on gyroscopes without a detailed study of three-dimensional dynamics.

The second approach employs the more general momentum equations for three-dimensional rotation where all components of momentum are ac- counted for. Chapter 8 is devoted to the topic of vibrations. This full-chapter coverage will be espe- cially useful for engineering students whose only exposure to vibrations is acquired in the basic dynamics course. Moments and products of inertia of mass are presented in Appendix B. Appendix C con- tains a summary review of selected topics of elementary mathematics as well as several nu- merical techniques which the student should be prepared to use in computer-solved problems.

Useful tables of physical constants, centroids, and moments of inertia are con- tained in Appendix D. Supplements The following items have been prepared to complement this textbook: Instructor Lecture Resources The following resources are available online at www. There may be additional resources not listed. A complete online learning system to help prepare and present lectures, assign and manage homework, keep track of student progress, and customize your course content and delivery.

See the description in front of the book for more information, and the website for a demonstration. Talk to your Wiley representative for details on setting up your Wiley- Plus course. Major use of animation, concise review of the theory, and numerous sample problems make this tool extremely useful for student self-review of the material. Acknowledgments Special recognition is due Dr.

Hale, formerly of Bell Telephone Laboratories, for his continuing contribution in the form of invaluable suggestions and accurate checking of the manuscript.

Hale has rendered similar service for all previous versions of this entire series of mechanics books, dating back to the s.

Preface xi These include Scott L. Hendricks, Saad A. Ragab, Norman E. Dowling, Michael W. Hyer, Michael L. Madi- gan, and J. Wallace Grant. Jeffrey N. The following individuals listed in alphabetical order provided feedback on recent editions, reviewed samples of the Seventh Edition, or otherwise contributed to the Seventh Edition: Michael Ales, U.

The talented illustrators of Precision Graphics continue to maintain a high standard of illustration excellence. In addition to providing patience and support for this project, my wife Dale has managed the prepara- tion of the manuscript for the Seventh Edition and has been a key individual in checking all stages of the proof. In addition, both my daughter Stephanie Kokan and my son David Kraige have contributed problem ideas, illustrations, and solutions to a number of the prob- lems over the past several editions.

I am extremely pleased to participate in extending the time duration of this textbook series well past the sixty-year mark. In the interest of providing you with the best possible educational materials over future years, I encourage and welcome all comments and sugges- tions. Please address your comments to kraige vt. Blacksburg, Virginia Preface xiii Stocktrek Images, Inc.

The study of dynamics in engineer- ing usually follows the study of statics, which deals with the effects of forces on bodies at rest. Dynamics has two distinct parts: A thorough comprehension of dynamics will provide one of the most useful and powerful tools for analysis in engineering. History of Dynamics Dynamics is a relatively recent subject compared with statics. The beginning of a rational understanding of dynamics is credited to Galileo — , who made careful observations concerning bodies in free fall, motion on an inclined plane, and motion of the pendulum.

Galileo was continually under severe criticism for refusing to accept the established beliefs of his day, such as the philoso- phies of Aristotle which held, for example, that heavy bodies fall more rapidly than light bodies. Although his mathe- matical description was accurate, he felt that the concept of remote transmission of gravitational force without a supporting medium was an absurd notion. Applications of Dynamics Only since machines and structures have operated with high speeds and appreciable accelerations has it been necessary to make calculations based on the principles of dynamics rather than on the principles of statics.

The rapid technological developments of the present day require increasing application of the principles of mechanics, particularly dy- namics. Students with interests in one or more of these and many other activities will constantly need to apply the fundamental principles of dynamics. They are summarized here along with additional comments of special relevance to the study of dynamics. Space is the geometric region occupied by bodies. Position in space is determined relative to some geometric reference system by means of linear and angular measurements.

The basic frame of reference for the laws of Newtonian mechanics is the primary inertial system or astro- nomical frame of reference, which is an imaginary set of rectangular axes assumed to have no translation or rotation in space.

Cajori, University of California Press, For most engineering problems involving machines and structures which remain on the surface of the earth, the corrections are extremely small and may be neglected. For these problems the laws of mechanics may be applied directly with mea- surements made relative to the earth, and in a practical sense such mea- surements will be considered absolute. Time is a measure of the succession of events and is considered an absolute quantity in Newtonian mechanics.

Mass is the quantitative measure of the inertia or resistance to change in motion of a body. Mass may also be considered as the quantity of matter in a body as well as the property which gives rise to gravita- tional attraction. Force is the vector action of one body on another. The properties of forces have been thoroughly treated in Vol.

A particle is a body of negligible dimensions. When the dimensions of a body are irrelevant to the description of its motion or the action of forces on it, the body may be treated as a particle. A rigid body is a body whose changes in shape are negligible com- pared with the overall dimensions of the body or with the changes in po- sition of the body as a whole.

For this purpose, then, the treatment of the airplane as a rigid body is an accept- able approximation. On the other hand, if we need to examine the inter- nal stresses in the wing structure due to changing dynamic loads, then the deformation characteristics of the structure would have to be exam- ined, and for this purpose the airplane could no longer be considered a rigid body.

Vector and scalar quantities have been treated extensively in Vol. Scalar quantities are printed in lightface italic type, and vectors are shown in boldface type. Thus, V denotes the scalar magnitude of the vector V. It is important that we use an identifying mark, such as an underline V, for all handwritten vectors to take the place of the boldface designation in print.

We assume that you are familiar with the geometry and algebra of vectors through previous study of statics and mathematics. Mechanics by its very na- ture is geometrical, and students should bear this in mind as they review their mathematics.

In addition to vector algebra, dynamics re- quires the use of vector calculus, and the essentials of this topic will be developed in the text as they are needed. Dynamics involves the frequent use of time derivatives of both vec- tors and scalars. As a notational shorthand, a dot over a symbol will fre- quently be used to indicate a derivative with respect to time.

In modern terminology they are: Law I. A particle remains at rest or continues to move with uniform velocity in a straight line with a constant speed if there is no unbal- anced force acting on it. Law II. The acceleration of a particle is proportional to the resul- tant force acting on it and is in the direction of this force.

The forces of action and reaction between interacting bod- ies are equal in magnitude, opposite in direction, and collinear. The third law constitutes the principle of action and reaction with which you should be thoroughly familiar from your work in statics.

However, numerical conversion from one system to the other will often be needed in U.

Both formulations are equally correct when applied to a particle of constant mass. To become familiar with each system, it is necessary to think directly in that system. Familiarity with the new system cannot be achieved simply by the conversion of numerical re- sults from the old system.

Charts comparing selected quantities in SI and U. The four fundamental quantities of mechanics, and their units and symbols for the two systems, are summarized in the following table: The U. In the U. The SI system is termed an absolute system because the standard for the base unit kilogram a platinum-iridium cylinder kept at the In- ternational Bureau of Standards near Paris, France is independent of the gravitational attraction of the earth.

On the other hand, the U. This distinction is a fundamental difference be- tween the two systems of units. Thus, for each system we have from Eq. In SI units, the kilogram should be used exclusively as a unit of mass and never force. Unfortunately, in the MKS meter, kilogram, sec- ond gravitational system, which has been used in some countries for many years, the kilogram has been commonly used both as a unit of force and as a unit of mass. In order to avoid the confusion which would be caused by the use of two units for mass slug and lbm , in this textbook we use almost exclusively the unit slug for mass.

This practice makes dynamics much simpler than if the lbm were used. In addition, this approach allows us to use the symbol lb to always mean pound force. Professional organizations have established detailed guidelines for the consistent use of SI units, and these guidelines have been followed throughout this book.

The most essential ones are summarized inside the front cover, and you should observe these rules carefully. Except for some spacecraft applications, the only gravitational force of appreciable magnitude in engineering is the force due to the attraction of the earth. It was shown in Vol. Because the gravitational attraction or weight of a body is a force, it should always be expressed in force units, newtons N in SI units and pounds force lb in U.

Effect of Altitude The force of gravitational attraction of the earth on a body depends on the position of the body relative to the earth. If the earth were a perfect homogeneous sphere, a body with a mass of exactly 1 kg would be attracted to the earth by a force of 9. Thus the variation in gravita- tional attraction of high-altitude rockets and spacecraft becomes a major consideration.

Every object which falls in a vacuum at a given height near the sur- face of the earth will have the same acceleration g, regardless of its mass. This result can be obtained by combining Eqs.

This com- bination gives where me is the mass of the earth and R is the radius of the earth. The variation of g with altitude is easily determined from the gravi- tational law. If g0 represents the absolute acceleration due to gravity at sea level, the absolute value at an altitude h is where R is the radius of the earth.

Effect of a Rotating Earth The acceleration due to gravity as determined from the gravita- tional law is the acceleration which would be measured from a set of axes whose origin is at the center of the earth but which does not ro- tate with the earth. Because the earth rotates, the acceleration of a freely falling body as measured from a position at- tached to the surface of the earth is slightly less than the absolute value.

Accurate values of the gravitational acceleration as measured rela- tive to the surface of the earth account for the fact that the earth is a rotating oblate spheroid with flattening at the poles.

The reason for the small difference is that the earth is not exactly ellipsoidal, as assumed in the formulation of the In- ternational Gravity Formula.

The formula is based on an ellipsoidal model of the earth and also ac- counts for the effect of the rotation of the earth. The absolute acceleration due to gravity as determined for a nonro- tating earth may be computed from the relative values to a close approxi- mation by adding 3. The variation of both the absolute and the relative values of g with latitude is shown in Fig.

The values of 9. Apparent Weight The gravitational attraction of the earth on a body of mass m may be calculated from the results of a simple gravitational experiment. The body is allowed to fall freely in a vacuum, and its absolute acceleration is measured. If the gravitational force of attraction or true weight of the body is W, then, because the body falls with an absolute acceleration g, Eq. The difference is due to the rotation of the earth.

The ratio of the apparent weight to the appar- ent or relative acceleration due to gravity still gives the correct value of mass. The apparent weight and the relative acceleration due to gravity are, of course, the quantities which are measured in experiments con- ducted on the surface of the earth. Thus, a di- mension is different from a unit. The principle of dimensional homogene- ity states that all physical relations must be dimensionally homogeneous; that is, the dimensions of all terms in an equation must be the same.

It is customary to use the symbols L, M, T, and F to stand for length, mass, time, and force, respectively. In SI units force is a derived quantity and from Eq. We can derive the following expression for the velocity v of a body of mass m which is moved from rest a horizontal distance x by a force F: You should perform a dimensional check on the answer to every problem whose solution is carried out in symbolic form. This description, which is largely mathematical, en- ables predictions of dynamical behavior to be made.

A dual thought process is necessary in formulating this description. It is necessary to think in terms of both the physical situation and the corresponding mathematical description.

This repeated transition of thought between the physical and the mathematical is required in the analysis of every problem. You should recognize that the mathematical formulation of a physical problem represents an ideal and limiting description, or model, which approximates but never quite matches the actual physical situation. In Art. We assume therefore, that you are familiar with this approach, which we summarize here as applied to dynamics.

Approximation in Mathematical Models Construction of an idealized mathematical model for a given engi- neering problem always requires approximations to be made. Some of these approximations may be mathematical, whereas others will be physical. For instance, it is often necessary to neglect small distances, angles, or forces compared with large distances, angles, or forces. An interval of mo- tion which cannot be easily described in its entirety is often divided into small increments, each of which can be approximated.

As another example, the retarding effect of bearing friction on the motion of a machine may often be neglected if the friction forces are small compared with the other applied forces. Thus, the type of assumptions you make depends on what infor- mation is desired and on the accuracy required.

You should be constantly alert to the various assumptions called for in the formulation of real problems. The ability to understand and make use of the appropriate assumptions when formulating and solving engi- neering problems is certainly one of the most important characteristics of a successful engineer.

Along with the development of the principles and analytical tools needed for modern dynamics, one of the major aims of this book is to provide many opportunities to develop the ability to formulate good mathematical models. Strong emphasis is placed on a wide range of practical problems which not only require you to apply theory but also force you to make relevant assumptions.

Application of Basic Principles The subject of dynamics is based on a surprisingly few fundamental concepts and principles which, however, can be extended and applied over a wide range of conditions. The study of dynamics is valuable partly be- cause it provides experience in reasoning from fundamentals. This experi- ence cannot be obtained merely by memorizing the kinematic and dynamic equations which describe various motions.

It must be obtained through ex- posure to a wide variety of problem situations which require the choice, use, and extension of basic principles to meet the given conditions. At times a single particle or a rigid body is the system to be isolated, whereas at other times two or more bodies taken together con- stitute the system. Development of good habits in formulating problems and in representing their solutions will be an invaluable asset. Each solution should proceed with a logical se- quence of steps from hypothesis to conclusion.

The following sequence of steps is useful in the construction of problem solutions. Formulate the problem: Develop the solution: The arrangement of your work should be neat and orderly. This will help your thought process and enable others to understand your work. The discipline of doing orderly work will help you to develop skill in prob- lem formulation and analysis.

This diagram consists of a closed out- line of the external boundary of the system. All bodies which contact and exert forces on the system but are not a part of it are removed and replaced by vectors representing the forces they exert on the isolated system.

In this way, we make a clear distinction between the action and reaction of each force, and all forces on and external to the system are accounted for. We assume that you are familiar with the technique of drawing free-body diagrams from your prior work in statics. Numerical versus Symbolic Solutions In applying the laws of dynamics, we may use numerical values of the involved quantities, or we may use algebraic symbols and leave the answer as a formula.

When numerical values are used, the magnitudes of all quantities expressed in their particular units are evident at each stage of the calculation. This approach is useful when we need to know the magnitude of each term. The symbolic solution, however, has several advantages over the numerical solution: The use of symbols helps to focus attention on the connection between the physical situation and its related mathematical description. A symbolic solution enables you to make a dimensional check at every step, whereas dimensional homogeneity cannot be checked when only numerical values are used.

We can use a symbolic solution repeatedly for obtaining answers to the same problem with different units or different numerical values. Thus, facility with both forms of solution is essential, and you should practice each in the problem work. In the case of numerical solutions, we repeat from Vol.

Solution Methods Solutions to the various equations of dynamics can be obtained in one of three ways. Obtain a direct mathematical solution by hand calculation, using ei- ther algebraic symbols or numerical values. We can solve the large majority of the problems this way. Obtain graphical solutions for certain problems, such as the deter- mination of velocities and accelerations of rigid bodies in two- dimensional relative motion.

Solve the problem by computer. A number of problems in Vol. They ap- pear at the end of the Review Problem sets and were selected to illustrate the type of problem for which solution by computer offers a distinct advantage. The choice of the most expedient method of solution is an important aspect of the experience to be gained from the problem work.

We em- phasize, however, that the most important experience in learning me- chanics lies in the formulation of problems, as distinct from their solution per se. Perform calculations using SI and U. Express the law of gravitation and calculate the weight of an object.

Discuss the effects of altitude and the rotation of the earth on the acceleration due to gravity. Apply the principle of dimensional homogeneity to a given physical relation. Describe the methodology used to formulate and solve dynamics problems. Determine its weight under these conditions in both pounds and newtons. The shuttle is in a circular orbit at an altitude of miles above the surface of the earth. Determine the weight of the module in both pounds and newtons under these conditions.

Kraige Free Download. Khurmi, J. Introduction to Dynamics. Kinematics of Particles. Kinetics of Particles. Kinetics of Systems of Particles. Plane Kinematics of Rigid Bodies. Plane Kinetics of Rigid Bodies. Vibration and Time Response. Appendix A: Area Moments of Inertia.

Appendix B: Mass Moments of Inertia. Appendix C: Selected Topics of Mathematics. Appendix D: Useful Tables Index Problem Answers.

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