Schaum's Outline of Engineering Mechanics Dynamics

By E. W. Nelson, Charles L. Best, W. G. McLean and Merle C. Potter

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Modified to conform to the current curriculum, Schaum's Outline of Engineering Mechanics: Dynamics complements these courses in scope and sequence to help you understand its basic concepts. The book offers extra practice on topics such as rectilinear motion, curvilinear motion, rectangular components, tangential and normal components, and radial and transverse components. You’ll also get coverage on acceleration, D'Alembert's Principle, plane of a rigid body, and rotation. Appropriate for the following courses: Engineering Mechanics; Introduction to Mechanics; Dynamics; Fundamentals of Engineering.

Features:

• 765 solved problems
• Additional material on instantaneous axis of rotation and Coriolis' Acceleration
• Support for all the major textbooks for dynamics courses

Topics include: Kinematics of a Particle, Kinetics of a Particle, Kinematics of a Rigid Body, Kinetics of a Rigid Body, Work and Energy, Impulse and Momentum, Mechanical Vibrations

• Language: English
• Publisher: McGraw-Hill Education
• Release date: Aug 27, 2010
• ISBN: 9780071713610

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Preface

This book is designed to supplement standard texts, primarily to assist students of engineering and science in acquiring a more thorough knowledge and proficiency in dynamics, the course that follows statics in the mechanics sequence. It is based on the authors’ conviction that numerous solved problems constitute one of the best means for clarifying and fixing in mind of basic principles. While this book will not mesh precisely with any one text, the authors feel that it can be a very valuable adjunct to all.

The previous editions of this book have been very favorably received. This edition incorporates SI units only. This eliminates the problems encountered when mixing units and allows students to focus on the subject being studied.

The authors attempt to use the best mathematical tools available to students at the sophomore level. Thus the vector approach is applied in those chapters where its techniques provide an elegance and simplicity in theory and problems. On the other hand, we have not hesitated to use scalar methods elsewhere, since they provide entirely adequate solutions to many of the problems. Chapter 1 is a complete review of the minimum number of vector definitions and operations necessary for the entire book, and applications of this introductory chapter are made throughout the book.

Chapter topics correspond to material usually covered in a standard introductory dynamics course. Each chapter begins with statements of pertinent definitions and principles. The text material is followed by graded sets of solved and supplementary problems. The solved problems serve to illustrate and amplify the theory, present methods of analysis, provide practical examples, and bring into sharp focus those fine points that enable the student to apply the basic principles correctly and confidently. Numerous derivations of formulas are included among the solved problems. The many supplementary problems serve as a review of the material covered in each chapter.

In the first edition the authors gratefully acknowledged their indebtedness to Paul B. Eaton and J. Warren Gillon. In the second edition the authors received helpful suggestions and criticism from Charles L. Best and John W. McNabb. Also in that edition Larry Freed and Paul Gary checked the solutions to the problems. In the third and fourth editions, computer solutions were added to numerous problems; these solutions have been eliminated in this sixth edition since several software packages have been developed that allow students to perform such solutions. For the fifth edition the authors thank William Best for checking the solutions to the new problems and reviewing the added new material. For typing the manuscripts of the third and fourth editions we are indebted to Elizabeth Bullock.

E. W. NELSON

C. L. BEST

W. G. MCLEAN

M. C. POTTER

About the Authors

E. W. NELSON graduated from New York University with a B.S.M.E. and an M.Adm.E. He taught mechanical engineering at Lafayette College and later joined the engineering organization of the Western Electric Company (now Lucent Technologies). Retired from Western Electric, he is currently a Fellow of the American Society of Mechanical Engineers. He is a registered Professional Engineer and a member of Tau Beta Pi and Pi Tau Sigma.

CHARLES L. BEST (deceased) was Emeritus Professor of Engineering at Lafayette College. He held a B.S. in mechanical engineering from Princeton, an M.S. in mathematics from Brooklyn Polytechnic Institute, and a Ph.D. in applied mechanics from Virginia Polytechnic Institute. He is coauthor of two books on engineering mechanics and coauthor of another book on FORTRAN programming for engineering students. He was a member of Tau Beta Pi.

W. G. McLEAN (deceased) was Emeritus Director of Engineering at Lafayette College. He held a B.S.E.E. from Lafayette College, an Sc.M. from Brown University, and an honorary Eng.D. from Lafayette College. Professor McLean is the coauthor of two books on engineering mechanics, was past president of the Pennsylvania Society of Professional Engineers, and was active in the codes and standards committees of the American Society of Mechanical Engineers. He was a registered Professional Engineer and a member of Phi Beta Kappa and Tau Beta Pi.

MERLE C. POTTER has engineering degrees from Michigan Technological University and the University of Michigan. He has coauthored Statics, Strength of Materials, Fluid Mechanics, The Mechanics of Fluids, Thermodynamics for Engineers, Thermal Sciences, Differential Equations, Advanced Engineering Mathematics, and Jump Start the HP-48G in addition to numerous exam review books. His research involved fluid flow stability and energy-related topics. He has received numerous awards, including the ASME’s 2008 James Harry Potter Gold Medal. He is Professor Emeritus of Mechanical Engineering at Michigan State University.

Contents

CHAPTER 1 Vectors

1.1 Definitions

1.2 Addition of Two Vectors

1.3 Subtraction of a Vector

1.4 Zero Vector

1.5 Composition of Vectors

1.6 Multiplication of Vectors by Scalars

1.7 Orthogonal Triad of Unit Vectors

1.8 Position Vector

1.9 Dot or Scalar Product

1.10 The Cross or Vector Product

1.11 Vector Calculus

1.12 Dimensions and Units

CHAPTER 2 Kinematics of a Particle

2.1 Kinematics

2.2 Rectilinear Motion

2.3 Curvilinear Motion

2.4 Units

CHAPTER 3 Dynamics of a Particle

3.1 Newton’s Laws of Motion

3.2 Acceleration

3.3 D’Alembert’s Principle

3.4 Problems in Dynamics

CHAPTER 4 Kinematics of a Rigid Body in Plane Motion

4.1 Plane Motion of a Rigid Body

4.2 Translation

4.3 Rotation

4.4 Instantaneous Axis of Rotation

4.5 Coriolis’ Acceleration

CHAPTER 5 Dynamics of a Rigid Body in Plane Motion

5.1 Vector Equations of Plane Motion

5.2 Scalar Equations of Plane Motion

5.3 Summary of the Equations

5.4 Translation of a Rigid Body

5.5 Rotation of a Rigid Body

5.6 Center of Percussion

5.7 Inertia Force Method for Rigid Bodies

CHAPTER 6 Work and Energy

6.1 Work

6.2 Special Cases

6.3 Power

6.4 Efficiency

6.5 Kinetic Energy of a Particle

6.6 Work-Energy Relations for a Particle

6.7 Kinetic Energy of a Rigid Body in Translation

6.8 Kinetic Energy of a Rigid Body in Rotation

6.9 Kinetic Energy of a Body in Plane Motion

6.10 Potential Energy

6.11 Work-Energy Relations for a Rigid Body

6.12 Conservation of Energy

CHAPTER 7 Impulse and Momentum

7.1 Impulse-Momentum Relation for a Particle

7.2 Impulse-Momentum Relation for an Assemblage of Particles

7.3 Angular Momentum

7.4 Relative Angular Momentum

7.5 Corresponding Scalar Equations

7.6 Units

7.7 Conservation of Momentum

7.8 Conservation of Angular Momentum

7.9 Impact

7.10 Variable Mass

CHAPTER 8 Mechanical Vibrations

8.1 Definitions

8.2 Degrees of Freedom

8.3 Simple Harmonic Motion

8.4 Multicomponent Systems

8.5 Units

APPENDIX A SI Units

Base Units

Supplementary Units

Derived Units with Special Names and Symbols

Derived Units without Special Names

SI Prefixes

Conversion Factors

APPENDIX B Second Moments of Areas and Mass Moments of Inertia

Index

CHAPTER 1

Vectors

1.1 Definitions

Scalar quantities .

Vector quantities possess both magnitude and direction;* examples are force, displacement, and velocity. A vector is represented by an arrow at the given angle. The head of the arrow indicates the sense, and the length represents the magnitude of the vector. The symbol for a vector is shown in print in boldface type, such as P. The magnitude is represented by |P| or P. .

A free vector may be moved anywhere in space provided it maintains the same direction and magnitude.

A sliding vector may be applied at any point along its line of action. By the principle of transmissibility, the external effects of a sliding vector remain the same.

A bound or fixed vector must remain at the same point of application.

A unit vector is a vector one unit in length. It is represented by i, n.

The negative of a vector P is the vector −P that has the same magnitude and angle but is of the opposite sense.

The resultant of a system of vectors is the least number of vectors that will replace the given system.

1.2 Addition of Two Vectors

(a) The parallelogram law states that the resultant R of two vectors P and Q is the diagonal of the parallelogram for which P and Q are adjacent sides. All three vectors P, Q, and R are concurrent as shown in Fig. 1-1(a). P and Q are also called the components of R.

Fig. 1-1 The components of a vector.

(b) If the sides of the parallelogram in Fig. 1-1(a) are perpendicular, the vectors P and Q are said to be rectangular components of the vector R. The rectangular components are illustrated in Fig. 1-1(b). The magnitudes of the rectangular components are given by

(c) Triangle law. Place the tail end of either vector at the head end of the other. The resultant is drawn from the tail end of the first vector to the head end of the other. The triangle law follows from the parallelogram law because opposite sides of the parallelogram are free vectors, as shown in Fig. 1-2.

Fig. 1-2 The triangle law.

(d.

1.3 Subtraction of a Vector

Subtraction of a vector is accomplished by adding the negative of the vector:

Note also that

1.4 Zero Vector

A zero vector . This is also called a null vector.

1.5 Composition of Vectors

Composition of vectors is the process of determining the resultant of a system of vectors. A vector polygon is drawn by placing the tail end of each vector in turn at the head end of the preceding vector, as shown in Fig. 1-3. The resultant is drawn from the tail end of the first vector to the head end (terminus) of the last vector. As will be shown later, not all vector systems reduce to a single vector. Since the order in which the vectors are drawn is immaterial, it can be seen that for three given vectors P, Q, and S,

Fig. 1-3 Composition of a vector.

Equation (3) may be extended to any number of vectors.

1.6 Multiplication of Vectors by Scalars

(a) The product of vector P and scalar m is a vector mP whose magnitude is |m| times as great as the magnitude of P and that is similarly or oppositely directed to P, depending on whether m is positive or negative.

(b) Other operations with scalars m and n are

1.7 Orthogonal Triad of Unit Vectors

An orthogonal triad of unit vectors i, j, and k is formed by drawing unit vectors along the x, y, and z axes, respectively. A right-handed set of axes is shown in Fig. 1-4.

Fig. 1-4 Unit vectors i, j, k.

A vector P is written as

where Pxi, Pyj, and Pzk are the vector components of P along the x, y, and z axes, respectively, as shown in Fig. 1-5.

Fig. 1-5 Vector components of P.

.

1.8 Position Vector

The position vector r of a point (x, y, z) in space is written

(see Fig. 1-6).

Fig. 1-6 The position vector r.

1.9 Dot or Scalar Product

The dot or scalar product of two vectors P and Q, is a scalar quantity and is defined as the product of the magnitudes of the two vectors and the cosine of their included angle θ (see Fig. 1-7). Thus,

Fig. 1-7 The included angle θ between two vectors.

The following laws hold for dot products, where m is a scalar:

Since i, j, and k are orthogonal,

, then

The magnitudes of the vector components of P along the rectangular axes can be written

since, e.g.,

Similarly, the magnitude of the vector component of P along any line L , where eL is the unit vector along the line L. (Some authors use u as the unit vector.) Figure 1-8 shows a plane through the tail end A of vector P and a plane through the head B, both planes being perpendicular to line L. The planes intersect line L at points C and D. The vector CD is the component of P along L.

Fig. 1-8 The component of P along a line.

Applications of these principles can be found in Problems 1.15 and 1.16.

1.10 The Cross or Vector Product

The cross or vector product of two vectors P and Q, written P × Q, is a vector R is normal to the plane of P and Q and points in the direction of advance of a right-handed screw when turned in the direction from P to Q through the smaller included angle θ. Thus if e , the cross product can be written

(not commutative).

Fig. 1-9 The cross product of two vectors.

The following laws hold for cross products, where m is a scalar:

Since i, j, and k are orthogonal,

, then

For proof of this cross-product determination, see Problem 1.12.

1.11 Vector Calculus

(a) Differentiation of a vector P that varies with respect to a scalar quantity such as time t is performed as follows.

; that is, P is a function of time t. A change ΔP in P as time changes from t to t + Δt is

Then

, where Px, Py, and Pz are functions of time t, we have

The following operations are valid:

(b) Integration of a vector P that varies with respect to a scalar quantity, such as time t; that is, P is a function of time t. Then

1.12 Dimensions and Units

In the study of mechanics, the characteristics of a body and its motion can be described in terms of a set of fundamental quantities called dimensions. In the United States, engineers have been accustomed to a gravitational system using the dimensions of force, length, and time. Most countries throughout the world use an absolute system in which the selected dimensions are mass, length, and time. There is a growing trend to use this second system in the United States.

Both systems derive from Newton’s second law of motion, which is often written as

where R is the resultant of all forces acting on a particle, a is the acceleration of the particle, and m is the constant of proportionality called the mass.

The International System (SI)

In the International System (SI),* the unit of mass is the kilogram (kg), the unit of length is the meter (m), and the unit of time is the second (s). The unit of force is the newton (N) and is defined as the force that will accelerate a mass of one kilogram one meter per second squared (m/s²). Thus,

A mass of 1 kg falling freely near the surface of the earth has an acceleration of gravity g that varies from place to place. In this book we assume an average value of 9.80 m/s². Thus the force of gravity acting on a 1-kg mass becomes

Of course, problems in statics involve forces; but, in a problem, a mass given in kilograms is not a force. The gravitational force acting on the mass, referred to as the weightacting on it.

In solving statics problems, the mass may not be mentioned. It is important to realize that the mass in kilograms is a constant for a given body. On the surface of the moon, this same given mass will have acting on it a force of gravity approximately one-sixth of that on the earth.

The student should also note that, in SI, the millimeter (mm) is the standard linear dimension unit for engineering drawings. Centimeters are tolerated in SI and can be used to avoid the many zeros required when using millimeters. Further, a space should be left between the number and unit symbol, for example, 2.85 mm, not 2.85mm. When using five or more figures, space them in groups of 3 starting at the decimal point as 12 830 000. Do not use commas in SI. A number with four figures can be written without the space unless it is in a column of quantities involving five or more figures.

Tables of SI units, SI prefixes, and conversion factors for the modern metric system (SI) are included in Appendix A. In this sixth edition, all the problems are in SI units.

SOLVED PROBLEMS

1.1. In a plane, find the resultant of a 300-N force at 30° and a −250-N force at 90°, using the parallelogram method. Refer to Fig. 1-10(a). Also, find the angle α between the resultant and the y axis. (Angles are always measured counterclockwise from the positive x axis.)

Fig. 1-10

SOLUTION

Draw a sketch of the problem, not necessarily to scale. The negative sign indicates that the 250-N force acts along the 90° line downward toward the origin. This is equivalent to a positive 250-N force along the 270° line, according to the principle of transmissibility.

As in Fig. 1-10(b), place the tail ends of the two vectors at a common point. Complete the parallelogram. Consider the triangle, one side of which is the y axis, in Fig. 1-10(b). The sides of this triangle are R, 250, and 300. The angle between the 250 and 300 sides is 60°. Applying the law of cosines gives

Now applying the law of sines, we get

Note: If the forces and angles are drawn to scale, the magnitude of R and the angle α could be measured from the drawing.

1.2. Use the triangle law and solve Problem 1.1 (see Fig. 1-11).

Fig. 1-11

SOLUTION

It is immaterial which vector is chosen first. Take the 300-N force. To the head of this vector attach the tail end of the 250-N force. Sketch the resultant from the tail end of the 300-N force to the head end of the 250-N force. Using the triangle shown, the results are the same as in Problem 1.1

1.3. The resultant of two forces in a plane is 400 N at 120°, as shown in Fig. 1-12. One of the forces is 200 N at 20°. Determine the missing force F and the angle α.

Fig. 1-12

SOLUTION

Select a point through which to draw the resultant and the given 200-N force. Draw the force connecting the head ends of the given force and the resultant. This represents the missing force F.

The result is obtained by the laws of trigonometry. The angle between R and the 200-N force is 100°, and hence, by the law of cosines, the unknown force F is

Then, by the law of sines, the angle α is found:

1.4. In a plane, subtract 130 N at 60° from 280 N at 320° (see Fig. 1-13).

Fig. 1-13

SOLUTION

To the 280-N, 320° force add the negative of the 130-N, 60° force. The resultant is found as follows:

The law of sines allows us to find α:

Thus, R makes an angle of −62.9° with the x axis.

1.5. Determine the resultant of the following coplanar system of forces: 26 N at 10°; 39 N at 114°; 63 N at 183°; 57 N at 261° (see Fig. 1-14).

Fig. 1-14

SOLUTION

This problem can be solved by using the idea of rectangular components. Resolve each force in Fig. 1-14 into x and y components. Since all the x components are collinear, they can be added algebraically, as can the y components. Now, if the x components and y components are added, the two sums form the x and y components of the resultant. Thus,

1.6. In Fig. 1-15 the rectangular component of the force F is 10 N in the direction of OH. The force F acts at 60° to