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(In S.I. Units)

[Strictly as per the Latest SyllabusPrescribed by B.P.U.T., Odisha]

By

Dr. I.S. GUJRALB.E. (Hons), M.E. (Thermal), Ph. D

Fellow Institution of Engineers (India)Formerly Principal

Hitkarini College of Engg. & Technology, JabalpurMadhya Pradesh

Presently, Director,Shri Ram Institute of Science and Technology,

Jabalpur (M.P.)

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Contents

Chapters Pages

1. SYSTEM OF FORCES ......................................................................................................... 1

1.1. Introduction ................................................................................................................ 11.2. Fundamental Concepts .............................................................................................. 11.3. Fundamental Principles ............................................................................................. 21.4. Units ............................................................................................................................ 41.5. Systems of Forces ....................................................................................................... 41.6. Law of Parallelogram of Forces ................................................................................. 71.7. Triangle Law of Forces ............................................................................................... 71.8. Polygon Law of Forces ................................................................................................ 91.9. Lami’s Theorem .......................................................................................................... 9

1.10. Resolution and Composition of Forces .................................................................... 101.11. Law of Action and Reaction ..................................................................................... 231.12. Equilibrium of Collinear Forces............................................................................... 231.13. Equilibrium of Concurrent Coplanar Forces .......................................................... 231.14. Free-Body Diagrams ................................................................................................. 241.15. Equations of Equilibrium ......................................................................................... 251.16. Analysis of Connected Bodies .................................................................................. 261.17. Introduction .............................................................................................................. 301.18. Moment of Force ....................................................................................................... 301.19. Varignon’s Theorem.................................................................................................. 321.20. Couple ........................................................................................................................ 331.21. Resolution of a Given Force into a Force Acting at

a Given Point and a Couple...................................................................................... 351.22. Wrench ...................................................................................................................... 351.23. Equivalent Couples ................................................................................................... 361.24. Addition of Couples ................................................................................................... 361.25. Resultant of a System of Coplanar Forces .............................................................. 371.26. Equivalent Systems of Coplanar Forces ................................................................. 381.27. The x and y Intercepts of the Resultant .................................................................. 39

Exercise ...................................................................................................................... 46

2. COPLANAR PARALLEL FORCE SYSTEMS .................................................................... 55

2.1. Introduction .............................................................................................................. 552.2. Resultant of Parallel Forces ..................................................................................... 55

2.3. Resolution of a Force into a Force and a Couple .................................................... 572.4. Properties of a Couple .............................................................................................. 582.5. Distributed Forces in a Plane .................................................................................. 68

Exercise ...................................................................................................................... 75

3. CENTROIDS AND MOMENTS OF INERTIA ..................................................................... 79

3.1. Introduction .............................................................................................................. 793.2. Centre of Gravity of a Flat Plate ............................................................................. 793.3. Centroids of Areas and Lines ................................................................................... 803.4. Centroids by Integration .......................................................................................... 823.5. Moment of Inertia ................................................................................................... 1083.6. Units and Sign of Moment of Inertia..................................................................... 1093.7. Parallel Axes Theorem ........................................................................................... 1093.8. Perpendicular Axes Theorem ................................................................................. 1103.9. Parallel-Axis and Perpendicular-Axis Theorem Combined ................................. 111

3.10. Radius of Gyration .................................................................................................. 1123.11. Moment of Inertia by Integration .......................................................................... 113

Exercise .................................................................................................................... 137

4. FRICTION ......................................................................................................................... 140

4.1. Introduction ............................................................................................................ 1404.2. Types of Friction ..................................................................................................... 1404.3. Laws of Friction ...................................................................................................... 1414.4. Theory of Dry Friction ............................................................................................ 1424.5. Angle of Friction ..................................................................................................... 1424.6. Angle of Repose ....................................................................................................... 1434.7. Cone of Friction ....................................................................................................... 1434.8. Equilibrium of Ladder ............................................................................................ 1584.9. The Wedge ............................................................................................................... 162

Exercise .................................................................................................................... 165

5. TRUSSES ......................................................................................................................... 169

5.1. Introduction ............................................................................................................ 1695.2. A Perfect or Rigid Truss ......................................................................................... 1695.3. Relation between Number of Members and Number of Joints ........................... 1705.4. Assumptions ............................................................................................................ 1725.5. Tensile and Compressive Members ....................................................................... 1725.6. Truss Nomenclature ............................................................................................... 1735.7. Nature of Forces in Different Members of a Truss ............................................... 1735.8. Methods of Analysing Trusses ............................................................................... 1745.9. Method of Sections .................................................................................................. 194

Exercise .................................................................................................................... 200

Chapters Pages

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Chapters Pages

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6. VIRTUAL WORK .............................................................................................................. 205

6.1. Definition and Meaning of Work ........................................................................... 2056.2. Principle of Virtual Work ....................................................................................... 2076.3. Application of Virtual Work ................................................................................... 2076.4. Procedure for Analysis using Principle of Virtual Work ...................................... 2086.5. Principle of Virtual Work of Mult-Degree of Freedom Systems .......................... 208

Exercise .................................................................................................................... 223

7. KINEMATICS OF RECTILINEAR MOTION ..................................................................... 227

7.1. Introduction ............................................................................................................ 2277.2. Position, Velocity, and Acceleration ...................................................................... 2287.3. Nature of Problems in Rectilinear Motion ............................................................ 2297.4. Uniform Rectilinear Motion ................................................................................... 2307.5. Uniformly Accelerated Rectilinear Motion ........................................................... 2317.6. Normal and Tangential Components of Acceleration .......................................... 2417.7. Projectile Motion or Trajectory .............................................................................. 244

Exercise .................................................................................................................... 256

8. DYNAMICS OF RECTILINEAR MOTION ........................................................................ 259

8.1. Introduction ............................................................................................................ 2598.2. Equations of Motion for Rectilinear Motion.......................................................... 2598.3. D’Alembert’s Principle or Equations of Dynamic Equilibrium ............................ 2608.4. Types of Problems to be Solved.............................................................................. 2618.5. Constant Force Acting on a Particle ...................................................................... 2628.6. Variable Force Acting on a Particle ....................................................................... 2648.7. Motion of a Lift ....................................................................................................... 2678.8. Motion Along a Rough Inclined Plane ................................................................... 2718.9. Motion of Connected Bodies ................................................................................... 273

Exercise .................................................................................................................... 281

9. DYNAMICS OF A PARTICLE-CURVILINEAR MOTION ................................................. 284

9.1. Introduction ............................................................................................................ 2849.2. Equations of Motion in Cartesian Co-ordinates ................................................... 2849.3. Equations of Motion in Tangential and Normal Components of Acceleration ... 2859.4. Equations of Motion in Polar Co-ordinates ........................................................... 285

Exercise .................................................................................................................... 298

10. CENTRE OF GRAVITY AND MASS MOMENT OF INERTIA .......................................... 302

10.1. Centre of Gravity .................................................................................................... 30210.2. Centre of Gravity of a Right Solid Circular Cone ................................................. 30310.3. Centre of Gravity of a Thin Hollow Circular Right Cone .................................... 30410.4. Centre of Gravity of Solid Hemi-sphere ................................................................ 306

10.5. Centre of Gravity of a Thin Hollow Hemi-sphere................................................. 30610.6. Centre of Gravity of the Solid Right Circular Cylinder ....................................... 30710.7. Mass Moment of Inertia ......................................................................................... 30910.8. Perpendicular Axis Theorem ................................................................................. 31010.9. Parallel Axes Theorem ........................................................................................... 310

10.10. Radius of Gyration.................................................................................................. 31110.11. Units of Mass Moment of Inertia .......................................................................... 31110.12. Combined Parallel and Perpendicular Axes Theorem ......................................... 31210.13. Notations for Mass Moment of Inertia .................................................................. 31210.14. Mass Moment of Inertia of a Thin Plate ............................................................... 31210.15. Mass Moment of Inertia of Thin Rectangular Plate ............................................ 31310.16. Mass Moment of Inertia of a Thin Circular Plate ................................................ 31310.17. Mass Moment of Inertia of Solid Sphere .............................................................. 31410.18. Mass Moment of Inertia of a Thin Long Rod ........................................................ 316

Exercise .................................................................................................................... 317

11. CURVILINEAR MOTION AND ROTATION OF RIGID BODIES ..................................... 319

11.1. Introduction ............................................................................................................ 31911.2. Motion of Translation ............................................................................................. 32011.3. Curvilinear Translation ......................................................................................... 32911.4. Differential Equations of Curvilinear Motions .................................................... 33011.5. Conical Pendulum................................................................................................... 33211.6. Moment of Momentum in Curvilinear Motion ..................................................... 33411.7. Work-Energy Equation in Curvilinear Motion..................................................... 33611.8. Introduction to Rotation of Rigid Bodies .............................................................. 33811.9. Equation of Motion for Rigid Body Rotation ........................................................ 341

11.10. Moment of Momentum or Angular Momentum ................................................... 34111.11. Centroidal and Non-Centroidal Rotation ............................................................. 34211.12. Rotation under the Action of a Constant Moment ............................................... 34311.13. Equation of Dynamic Equilibrium in Motion of

Rotation about a Fixed Axis ................................................................................... 34311.14. Workdone by a Couple and Kinetic Energy of Rotation ...................................... 35011.15. Angular Impulse and Angular Momentum .......................................................... 35111.16. Freely Rolling Body ................................................................................................ 35211.17. Plane Motion of a Rigid Body ................................................................................ 35311.18. Motion of a Vehicle Rolling Down the Road ......................................................... 35811.19. Motion of a Vehicle Going Round a Curve ............................................................ 36111.20. Banking a Curve ; Super-Elevation ...................................................................... 36411.21. Compound Pendulum ............................................................................................. 36811.22. Centre of Percussion ............................................................................................... 369

Chapters Pages

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11.23. Flywheels................................................................................................................. 37011.24. Circumferential or Hoop Stress due to Rotation .................................................. 372

Exercise .................................................................................................................... 373

12. IMPACT OR COLLISION OF ELASTIC BODIES ............................................................ 379

12.1. Introduction ............................................................................................................ 37912.2. Definitions ............................................................................................................... 37912.3. Direct Central Impact ............................................................................................ 38212.4. Oblique Impact........................................................................................................ 38312.5. Impact Against Fixed Plane ................................................................................... 38412.6. Loss of Kinetic Energy............................................................................................ 384

Exercise .................................................................................................................... 396Index ....................................................................................................................... 399

Chapters Pages

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PrefaceThe subject “ENGINEERING MECHANICS” is taught in the very first year of the

undergraduate Engineering curriculum in almost all universities of the world. An engineeringstudent starts learning to develop his capability to model an actual problem into an engineeringproblem and obtains its solution by using the fundamental principles and concepts of EngineeringMechanics Engineering problems, to a great extent, require the drawing of the free-body diagramsof the entire body and also of a small portion cut from it. Students are, therefore, advised tolearn drawing of the free-body diagrams. The entire book is divided into two portions-Staticsand Dynamics.

In the book SI system of units have been used. Analytical methods are emphasizedthroughout this book but the graphical and also the semi-graphical methods have not beenneglected. Most of the fundamentals needed to learn the basics of Engineering Mechanics havebeen discussed in the First Chapter.

Chapter on Trusses has been discussed at length. Here, certain so called “checks” arepresented which a student is advised to use to ascertain the correctness of his solution. If the“check” fails, the student should hurriedly go through his solution to correct the mistake. Avery large number of solved examples and the unsolved problems have been given for practice.The method of sections has also been discussed at length.

Separate chapters on Coplanar Parallel Force Systems; Friction; Centre of Gravity &Moments of inertia of Plane Areas have been included. Standard cases of areas dealing withcentroids and moment of inertia have been given in tabular forms for ready reference.

A separate chapter on Virtual Work has been given. The concept of virtual work has beenpresented in a very simple fashion.

Chapters on Dynamics include: Kinematics of Rectilinear Motion; Dynamics of RectilinearMotion; Dynamics of Particle-Curvilinear Motion; Center of Gravity and Mass Moment of Inertia;Curvilinear Motion and Motion of Rotation of Rigid Bodies; and Impact or Collision of Elasticbodies.

This book has been written to help the students in understanding the fundamentals ofthe subject and to develop their ability to deal any situation.

Although this book has been written for students of Odisha Technical University but itis hoped that the material presented in this book will be of value to the students and teachersof all Indian universities as well. It will help students in preparing for competitive examinationssuch as GATE, GRE, IES, IAS, PSC AMIE and other public sector examinations.

Many standard books on the subject have been used in preparation of this book and theauthor thankfully acknowledges the same. The valuable suggestions from learned teachers ofthe subject would help the author in improving the quality of this book.

I also acknowledge the moral support from my wife Smt. Kuldip Kaur and my all childrenfor bearing with me during the long period of writing of this book.

—Author

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1

1System of Forces

PART A—FUNDAMENTAL PRINCIPLES AND RELATIONS

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Mechanics is that branch of science which deals with the state of rest or motion ofbodies under the action of forces. No one subject plays a greater role in the engineering analy-sis and application than mechanics. Modern research and advancements in the fields of sta-bility, strength and design of structures and machines, vibration, robotics, rockets, missiles,aeroplane and spacecraft design, automobiles, automatic control, fluid flow, engine perform-ance, electrical machines and apparatus, transmission towers, superstructures, heavy earthmoving machines, locomotives, metro railways, super sonic aircrafts; molecular, atomic andsubatomic behaviour, etc., are highly dependent on the basic principles of mechanics. A thor-ough and clear understanding of this subject is an essential requirement for work in these andmany other subjects, not mentioned above.

It is divided into three parts: Mechanics of Rigid bodies, Mechanics of DeformableBodies and Mechanics of Fluids. Mechanics of rigid bodies is sub-divided into Statics andDynamics. Statics deals with bodies at rest, while Dynamics deals with bodies in motion. Inthe study of mechanics bodies are assumed to be perfectly rigid. Actual machines and structuresare not absolutely rigid and deform under the given loads but these deformations are negligiblysmall and therefore do not affect their conditions of equilibrium or motion.

The second division of mechanics is the Mechanics of Deformable Bodies, which is fur-ther subdivided into Strength of Materials, Theory of Elasticity, Theory of Plasticity. The thirddivision of mechanics is the Mechanics of Fluid, which is further subdivided into Mechanics ofIncompressible Fluids and Mechanics of Compressible Fluids.

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Certain concepts and definitions are basic to the study of mechanics and they should beunderstood before starting the subject. The basics used in mechanics are space, time, mass,force, particle and rigid body.

(a) Space. The concept of space is associated with the notion of the position of a body.Space is the geometric region occupied by bodies whose positions are described by linear orangular measurements relative to a co-ordinate system. For three dimensional problems threeindependent lengths (or co-ordinates) measured with a reference point (known as origin) areneeded. For two-dimensional problems only two co-ordinates will be required. These co-ordinatesare called the Cartesian co-ordinates or the rectangular co-ordinates. Sometime angular

2 ENGINEERING MECHANICS

co-ordinates are used. They are known as cylindrical co-ordinates for three dimensionalproblems and polar co-ordinates for two-dimensional problems. There are many otherco-ordinate systems used for different problems.

(b) Time. Time is the measure of happening of an event. It is a basic quantity in thestudy of dynamics.

(c) Mass. Mass is the measure of the quantity of matter in a body. In Statics mass is alsothe property of a body by which it experiences force of mutual attraction to other bodies inspace. Mass is also the measure of the inertia of a body. And, inertia is the resistance to thechange of velocity of a body. In SI units, unit of mass is kilogram (kg).

(d) Force. Force is the action of one body on another. A force tends to move a body in thedirection of its action. It may be exerted either by actual contact or from a distance, as in thecase of gravitational forces and magnetic forces. A force is described by:

(i) its point of application,(ii) its magnitude, and

(iii) its direction.These quantities are called specifications of a force. A force, having magnitude and

direction, is a vector quantity. In SI system of units force is measured in newtons and itssymbol is N.

(e) Particle. A body of negligible dimensions is called a particle, Mathematically, aparticle is a body whose dimensions approach zero so that it may be analysed as a point mass.Forces acting on a particle are called concurrent forces.

(f) Rigid Body. A rigid body may be defined as a definite amount of matter the parts ofwhich are fixed in position relative to one another. Physical bodies, such as we deal with in thedesign of engineering structures and machine parts, are never absolutely rigid but they deformslightly under the action of load which they are to carry. But, these deformations are verysmall and do not affect the equilibrium of the rigid body under study in mechanics. Thedeformations of bodies are studied in subjects like Mechanics of Deformable Bodies, also knownas Strength of Materials, Theory of Elasticity, Theory of Plasticity, etc.

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The study of mechanics rests on six fundamental principles based on experimental evi-dence. They will be discussed at length in subsequent sections as they are required. Theseprinciples are stated below for reference only:

(1) The Parallelogram Law of Forces. It states that the resultant of two forces isrepresented by the diagonal of the parallelogram formed as the sides representing these forces.

The Three Laws of Newton. Sir Isaac Newton (1642–1727) was the first to satisfac-torily formulate the fundamental principles of mechanics. Slightly reworded to use the mod-ern terminology, these laws are:

(2) First Law of Motion. A particle remains at rest or continues to move in a straightline with a uniform velocity if there is no unbalanced resultant force acting on it.

(3) Second Law of Motion. The acceleration is proportional to the resultant forceacting on a particle and is in the direction of this force.

Expressed mathematically, it may be stated as:

F = ma ...(1.1)

where F is the resultant force acting on the particle of mass m and a is the resultingacceleration.

SYSTEM OF FORCES 3

(4) Third Law of Motion. The forces of action and reaction between the bodies areequal in magnitude, opposite in direction and collinear.

(5) The Principle of Transmissibility. This states that the external effect of a forceon a rigid body is the same for all points of application of a force along its line of action, i.e., itis independent of its point of application. Its internal effect definitely change with the pointof application of the force.

Force P, in Fig. 1.3.1(a) will result the sliding of the block whether the force pulls theblock at A or pushes at B. The local internal effects at A and B will, however, be quite different.

PA B

(a)

P A B P

C D

(b) (c)

P A B P

C D

P

Fig. 1.3.1. Principle of transmissibility is for (a) and (b) but not for (c)

It should be noted that the principle of transmissibility applies to the external effect ofa force on the same rigid body. Observe that since the supports at C and D have changed inFigs. (b) and (c) therefore the principle of transmissibility will not hold good although the forceP acts along its line of action through points A and B. The support at D is changed from ahinge at (b) to a roller at (c), this changes the external condition of equilibrium of the body.

(6) Newton’s Law of Gravitation. This law statesthat the two masses M and m situated at a distance r aremutually attracted with equal and opposite forces (Fig. 1.3.2)

F and F′ of magnitude F given by the formula:

F = G . Mm

r2...(1.2)

where G = Universal constant known as the constant ofgravitation.

By experiments the constant of gravitation is found tobe G = 6.673 × 10–11 m3/(kgs2).

The mutual forces F obey the law of action and reaction since they are equal and oppo-site and are directed along the line joining the centres of the particles M and m as shown inFig. 1.3.2.

The gravitational attraction of the earth on a given body is called the weight of thebody. This force exits whether the body is at rest or in motion. In SI units the unit of weight isNewton (N). For a body of mass m near the surface of earth, the gravitational attraction on thebody may be calculated by using equation (1.1). If the gravitational force or weight has amagnitude W, then since the body falls with an acceleration g, equation (1.2) gives

W = mg ...(1.3)

The weight is in newtons (N) and mass in kilograms (kg) and g in m/s2. The standardvalue of g = 9.81 m/s2 is generally used.

F¢

F

M

m

r

Fig. 1.3.2. Newton’s law ofgravitation


In equation (1.2), let ME = mass of earth in kg and r = radius of earth in metres (m),we get

F = G (M )E×r2 × m = gm,

where, g = G(M )E

r2 = 9.81 × m

kg skgm2

3

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As discussed is Sec. 1.2, Mechanics deals with four fundamental quantities—length,mass force and time. The units used to measure these quantities can not be chosen independentlybecause they must be consistent with the Newton’s second law of motion, expressed byeqn. (1.1). In the present time the modern Metric System (SI) of units is used. In past, theFoot-Pound-Second (FPS) system of units were used. In some countries Metre-Kilogram-Second(MKS) units are used. Four fundamental quantities and units and their symbols for thesethree systems are given in Table 1.1, below.

Table 1.1. Different Systems of Units

S.No. Quantity Dimensional SI units MKS units FPS units

symbol Basic Symbol Basic Symbol Basic Symbolunits units units

1. Length L metre m metre m foot ft2. Mass M kilogram kg kilogram kgm slugs slugs

mass3. Time T second s second sec second sec4. Force F newton N kilogram kgf Pound lb

force

PART B—CONCURRENT COPLANAR FORCE SYSTEMS

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In the study of mechanics various systems of forces are encountered. When all forcesmeet at one point, they are known as concurrent forces. When all concurrent forces are lying inone plane they belong to coplanar concurrent force system. This force system is illustrated inFig. 1.5.2 (b), in which all forces are situated in the xy plane and meet at a point A.

If the concurrent forces are situated in different planes in space, they belong to con-current spatial forces. In Fig. 1.5.2(e) forces F1 and F3 are situated in xz plane; forces F1and F4 are situated in yz plane; forces F2 and F5 do not lie in any plane. All these forces meetat the corner C of the cube.

If all forces acting at a point lying in any plane are acting along the same straight linethey form coplanar collinear force system, as shown in Fig. 1.5.2(a).

SYSTEM OF FORCES 5

Force systems

Coplanar force systems Spatial force systems

1.2.3.

4.

Collinear force systemConcurrent force systemNon-concurrent parallelforce systemNon-concurrent (non-parallel)force system

1.2.3.

Spatial concurrent force systemSpatial parallel force systemSpatial non-concurrent (non-parallel)force system

Fig. 1.5.1. Various force systems

Various force systems listed in Fig. 1.5.1 are illustrated in Fig. 1.5.2 below.

A

F3 F1 F2

y

xO

(i)

A

F3

F1

F2

y

xO

(ii)

(a) Coplanar collinear force systems

A

F3

F1

F2

y

xO

F4

(b) Coplanar concurrentforce system

F3F1 F2

y

xO

F4

(i)

F3

F1

F2

y

xO

F4

(ii)

(c) Coplanar parallel force system

y

xO

F4F1

F3 F2

C B

D A

(i)

y

xO

F4

F1

F3

F2

(ii)

AB

(d) Coplanar non-concurrent force system


y

z

x

A B

E

FG

D

O

F3

F2

F1

F5F4

C

(e) S concurrent systempatial

y

z

x

A B

E

FG

D

O

C

F1

F2F3

F4 F5

F6

(f) S arallel systempatial p

z

x

y

A B

E

FG

D

O

C

F1F2 F3

F4F5

(g) S on-concurrent systempatial n

Fig. 1.5.2. Illustration of various force systems

Figure 1.5.1 gives classification of different systems of forces. Various force systems aredivided into (a) coplanar force systems, and (b) spatial force systems. Each system of forces isfurther sub-divided into concurrent force systems; parallel force systems and non-concurrent(non-parallel) force systems.

In Fig. 1.5.2 (c) all forces are parallel to each other and are situated in the xy-plane andhence are known as coplanar parallel force system. In Fig. 1.5.2 (f) all forces are parallel toeach other but they are lying in different planes. Forces F1, F2 and F6 are lying in yz-plane;forces F4 and F5 are situated in xy-plane and force F3 acting along the y-axis is situated in bothxy- and yz-planes. The force system shown in Fig. 1.5.2 (f) is known as spatial parallel forcesystem.

Coplanar non-concurrent force system is shown in Fig. 1.5.2 (d) in which forcesare situated in xy plane, but they donot act through the same point.

Spatial non-concurrent force system is illustrated in Fig. 1.5.2 (g). Here all forcesare situated in different planes and also act through different points.

The above force systems, their applications, equilibrium etc., will be discussed insubsequent sections of this book.

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