�������������������

�����������

�������

(In S.I. Units)Second Edition

By

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

Fellow Institution of Engineers (India) Principal

Hitkarini College of Engg. & Technology, JabalpurMadhya Pradesh

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

Jabalpur (M.P.)

UNIVERSITY SCIENCE PRESS(An Imprint of Laxmi Publications Pvt. Ltd.)

An ISO 9001:2008 Company

BENGALURU ● CHENNAI ● COCHIN ● GUWAHATI ● HYDERABADJALANDHAR ● KOLKATA ● LUCKNOW ● MUMBAI ● RANCHI ● NEW DELHI

BOSTON (USA) ● ACCRA (GHANA) ● NAIROBI (KENYA)

ENGINEERING MECHANICS

© by Laxmi Publications (P) Ltd. All rights reserved including those of translation into other languages. In accordance with the Copyright (Amendment) Act, 2012, no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise. Any such act or scanning, uploading, and or electronic sharing of any part of this book without the permission of the publisher constitutes unlawful piracy and theft of the copyright holder’s intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained from the publishers.

Printed and bound in India Typeset at Goswami Associates, Delhi

First Edition : 2008; Second Edition : 2011; Reprint : 2016ISBN 978-93-80856-66-7

Limits of Liability/Disclaimer of Warranty: The publisher and the author make no representation or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties. The advice, strategies, and activities contained herein may not be suitable for every situation. In performing activities adult supervision must be sought. Likewise, common sense and care are essential to the conduct of any and all activities, whether described in this book or otherwise. Neither the publisher nor the author shall be liable or assumes any responsibility for any injuries or damages arising here from. The fact that an organization or Website if referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers must be aware that the Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read.

All trademarks, logos or any other mark such as Vibgyor, USP, Amanda, Golden Bells, Firewall Media, Mercury, Trinity, Laxmi appearing in this work are trademarks and intellectual property owned by or licensed to Laxmi Publications, its subsidiaries or affiliates. Notwithstanding this disclaimer, all other names and marks mentioned in this work are the trade names, trademarks or service marks of their respective owners.

Published in india by

UNIVERSITY SCIENCE PRESS(An Imprint of Laxmi Publications Pvt. Ltd.)

An ISO 9001:2008 Company113, GOLDEN HOUSE, DARYAGANJ, NEW DELHI - 110002, INDIA Telephone : 91-11-4353 2500, 4353 2501 Fax : 91-11-2325 2572, 4353 2528 C—www.laxmipublications.com info@laxmipublications.com Printed at:

& Bengaluru 080-26 75 69 30

& Chennai 044-24 34 47 26, 24 35 95 07

& Cochin 0484-237 70 04, 405 13 03

& Guwahati 0361-254 36 69, 251 38 81

& Hyderabad 040-27 55 53 83, 27 55 53 93

& Jalandhar 0181-222 12 72

& Kolkata 033-22 27 43 84

& Lucknow 0522-220 99 16

& Mumbai 022-24 91 54 15, 24 92 78 69

& Ranchi 0651-220 44 64

Bran

ches

Contents

Chapters Pages

1. SYSTEM OF FORCES ................................................................................................. 1–101

1.1. Mechanics .................................................................................................................... 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 ..................................................................................... 251.12. Equilibrium of Collinear Forces .............................................................................. 261.13. Equilibrium of Concurrent Coplanar Forces .......................................................... 261.14. Free-body Diagrams ................................................................................................. 261.15. Equations of Equilibrium ......................................................................................... 281.16. Statically Determinate and Indeterminate Problems ........................................... 291.17. Analysis of Connected Bodies .................................................................................. 321.18. Introduction .............................................................................................................. 351.19. Moment of Force ....................................................................................................... 361.20. Varignon’s Theorem ................................................................................................. 371.21. Couple ........................................................................................................................ 391.22. Resolution of a Given Force into a Force Acting at

a Given Point and a Couple ..................................................................................... 401.23. Wrench ...................................................................................................................... 411.24. Equivalent Couples .................................................................................................. 411.25. Addition of Couples .................................................................................................. 421.26. Resultant of a System of Coplanar Forces .............................................................. 431.27. Equivalent Systems of Coplanar Forces ................................................................. 441.28. The x and y Intercepts of the Resultant .................................................................. 441.29. Reactions at Supports of Beams .............................................................................. 541.30. Types of Beams ......................................................................................................... 551.31. Types of Loads .......................................................................................................... 58

(v)

1.32. Methods of Finding Reaction at Supports of Beams .............................................. 591.33. Introduction .............................................................................................................. 831.34. Graphical Method ..................................................................................................... 841.35. Space Diagram .......................................................................................................... 841.36. Bow’s Notation .......................................................................................................... 851.37. To Determine Resultant of the Given Force System .............................................. 851.38. To Determine Reactions of the Beam ...................................................................... 861.39. Graphical Conditions of Equilibrium ...................................................................... 90

2. TRUSSES ................................................................................................................. 102–168

2.1. Introduction ............................................................................................................ 1022.2. A Perfect or Rigid Truss ......................................................................................... 1022.3. Relation between Number of Members and Number of Joints ........................... 1032.4. Assumptions ............................................................................................................ 1052.5. Tensile and Compressive Members ....................................................................... 1052.6. Truss Nomenclature ............................................................................................... 1062.7. Nature of Forces in Different Members of a Truss .............................................. 1062.8. Methods of Analysing Trusses ............................................................................... 1072.9. Method of Sections .................................................................................................. 144

2.10. Graphical Method ................................................................................................... 1512.11. Semi-graphical Method .......................................................................................... 151

3. CENTROIDS AND MOMENTS OF INERTIA ........................................................... 169–269

3.1. Introduction ............................................................................................................ 1693.2. Centre of Gravity of a Flat Plate ........................................................................... 1693.3. Centroids of Areas and Lines ................................................................................. 1703.4. Centroids of Symmetrical Figures ......................................................................... 1723.5. Centroids by Integration ........................................................................................ 1743.6. Moment of Inertia ................................................................................................... 2093.7. Units and Sign of Moment of Inertia .................................................................... 2103.8. Parallel Axes Theorem ........................................................................................... 2103.9. Perpendicular Axes Theorem................................................................................. 212

3.10. Parallel-axis and Perpendicular-axis Theorem Combined .................................. 2133.11. Radius of Gyration.................................................................................................. 2143.12. Moments of Inertia by Integration ........................................................................ 2153.13. Product of Inertia ................................................................................................... 2403.14. Product of Inertia is Zero if the Area has One or Two Axes of Symmetry ......... 2403.15. Parallel Axes Theorem for Product of Inertia ...................................................... 2413.16. Moment of Inertia with Respect to Inclined Axes ................................................ 2543.17. Mohr’s Circle for Moment of Inertia ...................................................................... 258

Chapters Pages

(vi)

4. FRICTION ................................................................................................................. 270–339

4.1. Introduction ............................................................................................................ 2704.2. Types of Friction ..................................................................................................... 2704.3. Laws of Friction ...................................................................................................... 2714.4. Theory of Dry Friction ............................................................................................ 2724.5. Angle of Friction ..................................................................................................... 2724.6. Angle of Repose ....................................................................................................... 2734.7. Cone of Friction ....................................................................................................... 2734.8. Motion on a Rough Inclined Plane ........................................................................ 2744.9. Equilibrium of Ladder ............................................................................................ 304

4.10. The Wedge ............................................................................................................... 3074.11. Screw Jack .............................................................................................................. 3114.12. (a) Conditions of Maximum Efficiency for Square Threads for Raising Load ... 3154.12. (b) Condition for ηmax for V-threads for Raising Load .......................................... 3164.13. Journal Bearings, Axle Friction ............................................................................ 3234.14. Thrust Bearings, Disk Friction.............................................................................. 3244.15. Pivots and Collars ................................................................................................... 3254.16. Mean Effective Radius of Bearing Surface ........................................................... 330

5. TRANSMISSION OF POWER ................................................................................. 340–381

5.1. Introduction ............................................................................................................ 3405.2. Types of Belt Drives ............................................................................................... 3405.3. Types of Belts .......................................................................................................... 3425.4. Types of Pulleys ...................................................................................................... 3425.5. Velocity Ratio .......................................................................................................... 3435.6. Length of Belt ......................................................................................................... 3475.7. Direction of Rotation of Driven Pulley .................................................................. 3505.8. Power Transmitted by a Belt Drive ...................................................................... 3505.9. Ratio of Belt Tensions for Flat Belt ....................................................................... 351

5.10. Tension Ratio for V-belt or Rope ........................................................................... 3535.11. Centrifugal Tension ................................................................................................ 3555.12. Total Tensions in Belt Drive .................................................................................. 3565.13. Condition for Transmission of Maximum Power .................................................. 3565.14. Initial Tension......................................................................................................... 3585.15. Angle of Lap of Contact .......................................................................................... 3595.16. Effect of Initial Tension on Maximum Power ....................................................... 369

6. LIFTING MACHINES................................................................................................ 382–413

6.1. Introduction ............................................................................................................ 3826.2. Definitions ............................................................................................................... 382

Chapters Pages

(vii)

6.3. An Ideal Machine .................................................................................................... 3826.4. Efficiency of a Lifting Machine .............................................................................. 3836.5. Ideal Machine ......................................................................................................... 3836.6. Friction Losses ........................................................................................................ 3846.7. Law of the Machine or Performance ..................................................................... 3846.8. Reversibility of a Machine ..................................................................................... 3866.9. Pulleys ..................................................................................................................... 390

6.10. Weston Differential Pulley Block (Fig. 6.10) ........................................................ 3966.11. Simple Wheel and Axle (Fig. 6.11) ........................................................................ 3976.12. Wheel and Differential Axle (Fig. 6.12) ................................................................ 3986.13. Winch Crabs ............................................................................................................ 4006.14. Single Purchase Winch Crab (Fig. 6.14) ............................................................... 4006.15. Double Purchase Winch Crab (Fig. 6.15) ............................................................. 4016.16. Worm and Worm Wheel (Fig. 6.16) ....................................................................... 4046.17. Screw Jack (Fig. 6.17) ............................................................................................ 4056.18. Differential Screw Jack (Fig. 6.18) ........................................................................ 408

7. KINEMATICS ............................................................................................................ 414–450

7.1. Introduction ............................................................................................................ 4147.2. Position, Velocity, and Acceleration ...................................................................... 4157.3. Nature of Problems in Rectilinear Motion ............................................................ 4167.4. Uniform Rectilinear Motion ................................................................................... 4177.5. Uniformly Accelerated Rectilinear Motion ........................................................... 4187.6. Normal and Tangential Components of Acceleration .......................................... 4317.7. Projectile Motion or Trajectory .............................................................................. 433

8. SHEAR FORCE AND BENDING MOMENT DIAGRAMS ....................................... 451–548

8.1. Beams ...................................................................................................................... 4518.2. Types of Supports ................................................................................................... 4518.3. The Types of Beams................................................................................................ 4528.4. Types of Loads ........................................................................................................ 4568.5. Equations of Equilibrium Used for Finding Support Reactions ......................... 4578.6. Shear Force and Bending Moment ....................................................................... 4598.7. Shear Force and Bending Moment Diagrams ...................................................... 4628.8. Some Standard Cases ............................................................................................. 4638.9. Relation between Rate of Loading, Shear Force and Bending Moment ............. 472

9. KINETICS OF A PARTICLE-RECTILINEAR MOTION ........................................... 549–579

9.1. Introduction ............................................................................................................ 5499.2. Equations of Motion for Rectilinear Motion ......................................................... 5499.3. D’Alembert’s Principle or Equations of Dynamic Equilibrium ........................... 550

Chapters Pages

(viii)

9.4. Types of Problems to be Solved ............................................................................. 5529.5. Constant Force Acting on a Particle ..................................................................... 5529.6. Variable Force Acting on a Particle ...................................................................... 5559.7. Motion of a Lift ....................................................................................................... 5599.8. Motion Along a Rough Inclined Plane................................................................... 5639.9. Motion of Connected Bodies ................................................................................... 568

10. KINETICS OF A PARTICLE-CURVILINEAR MOTION ........................................... 580–599

10.1. Introduction ............................................................................................................ 58010.2. Equations of Motion in Cartesian Co-ordinates ................................................... 58010.3. Equations of Motion in Tangential and Normal Components of Acceleration .. 58110.4. Equations of Motion in Polar Co-ordinates .......................................................... 581

11. CENTRE OF GRAVITY AND MASS MOMENT OF INERTIA ................................. 600–625

11.1. Centre of Gravity .................................................................................................... 60011.2. Centre of Gravity of a Right Solid Circular Cone ................................................ 60111.3. Centre of Gravity of a Thin Hollow Circular Right Cone .................................... 60211.4. Centre of Gravity of Solid Hemi-sphere ................................................................ 60411.5. Centre of Gravity of a Thin Hollow Hemi-sphere ................................................ 60411.6. Centre of Gravity of the Solid Right Circular Cylinder ....................................... 60511.7. Mass Moment of Inertia ......................................................................................... 60911.8. Perpendicular-axis Theorem .................................................................................. 60911.9. Parallel Axes Theorem ........................................................................................... 609

11.10. Radius of Gyration.................................................................................................. 61011.11. Units of Mass-moment of Inertia .......................................................................... 61111.12. Combined Parallel and Perpendicular Axes Theorem ......................................... 61111.13. Notations for Mass moment of Inertia .................................................................. 61111.14. Mass Moment of Inertia of a Thin Plate ............................................................... 61111.15. Mass Moment of Inertia of Thin Rectangular Plate ............................................ 61211.16. Moment of Inertia of a Thin Circular Plate .......................................................... 61211.17. Mass Moment of Inertia of Solid Sphere .............................................................. 61311.18. Mass Moment of Inertia of a Thin Hollow Sphere ............................................... 61411.19. Moment of Inertia of a Right Solid Cone .............................................................. 61511.20. M.I. About Axis Through Vertex and Perpendicular to

the Axis of Symmetry ............................................................................................. 61711.21. Moment of Inertia of a Thin Hollow Cone ............................................................ 61811.22. Moment of Inertia of a Thin Long Rod.................................................................. 62011.23. Mass Moment of Inertia of Hollow Cylinder ........................................................ 62111.24. Moment of Inertia of a Paraboloid of Revolution ................................................. 623

Chapters Pages

(ix)

12. KINETICS OF RIGID BODIES ................................................................................. 626–695

12.1. Introduction ............................................................................................................ 62612.2. Motion of Translation ............................................................................................. 62712.3. Curvilinear Translation ......................................................................................... 63912.4. Differential Equations of Curvilinear Motions .................................................... 64012.5. Conical Pendulum................................................................................................... 64312.6. Moment of Momentum in Curvilinear Motion ..................................................... 64512.7. Work-energy Equation in Curvilinear Motion ..................................................... 64612.8. Introduction to Rotation of Rigid Bodies .............................................................. 64812.9. Equation of Motion for Rigid Body Rotation ........................................................ 651

12.10. Moment of Momentum or Angular Momentum ................................................... 65212.11. Centroidal and Non-centroidal Rotation............................................................... 65212.12. Rotation Under the Action of a Constant Moment .............................................. 65312.13. Equation of Dynamic Equilibrium in Motion of

Rotation About a Fixed Axis .................................................................................. 65412.14. Work Done by a Couple and Kinetic Energy of Rotation .................................... 66412.15. Angular Impulse and Angular Momentum .......................................................... 66412.16. Freely Rolling Body ................................................................................................ 66512.17. Plane Motion of a Rigid Body ................................................................................ 66612.18. Motion of a Vehicle Rolling Down the Road ......................................................... 67512.19. Motion of a Vehicle Going Round a Curve ............................................................ 67812.20. Banking a Curve ; Super-elevation ....................................................................... 68112.21. Compound Pendulum ............................................................................................. 68512.22. Centre of Percussion ............................................................................................... 68612.23. Flywheels ................................................................................................................ 68712.24. Circumferential or Hoop Stress Due to Rotation ................................................. 689

13. WORK-ENERGY METHOD ..................................................................................... 696–749

13.1. Introduction ............................................................................................................ 69613.2. Work-energy Equation for Translation ................................................................. 69613.3. Workdone by a Constant Force .............................................................................. 69713.4. Workdone by a Variable Force ............................................................................... 69813.5. Workdone by the Force of Gravity ......................................................................... 70013.6. Work of Spring Force .............................................................................................. 70013.7. Workdone by a Fluid Under Constant Pressure .................................................. 70213.8. Workdone by a Gas Under Variable Pressure ...................................................... 70213.9. Energy of a Particle ................................................................................................ 703

13.10. Principle of Conservation of Energy...................................................................... 70313.11. Power and Efficiency .............................................................................................. 730

Chapters Pages

(x)

14. IMPULSE-MOMENTUM METHOD .......................................................................... 750–773

14.1. Introduction ............................................................................................................ 75014.2. Linear Impulse-momentum Method ..................................................................... 75014.3. Angular Momentum ............................................................................................... 75114.4. Alternative Symbols Used ...................................................................................... 75314.5. Dynamic Action of a Jet ......................................................................................... 765

15. COLLISION OF ELASTIC BODIES ......................................................................... 774–796

15.1. Introduction ............................................................................................................ 77415.2. Definitions ............................................................................................................... 77415.3. Direct Central Impact ............................................................................................ 77715.4. Oblique Impact........................................................................................................ 77815.5. Impact Against Fixed Plane ................................................................................... 77915.6. Loss of Kinetic Energy............................................................................................ 779

16. VIRTUAL WORK ...................................................................................................... 797–835

16.1. Definition and Meaning of Work ........................................................................... 79716.2. Principle of Virtual Work ....................................................................................... 79916.3. Applications of Virtual Work ................................................................................. 79916.4. Procedure for Analysis Using Principle of Virtual Work ..................................... 80016.5. Principle of Virtual Work for Multi-Degree of Freedom Systems ....................... 800

Index ............................................................................................................... 836–841

Chapters Pages

(xi)

Preface to the Second EditionThe author gratefully acknowledge the healthy criticism and encouraging appreciations

for the experienced and the worthy teachers and the large number of engineerging students ofthe country.

To further strengthen the text, the book has been thoroughly revised. The salient fea-tures of this revised editions are:

• Several figures have been corrected.• Typographic errors have been thoroughly edited.• A new chapter on Virtual Work has been added.The author hopes that this edition of the book will enhance its utility for students and

teachers of Engineering Mechanics of all universities.Suggestions from the readers for further upgrading the book will be appreciated and

thankfully acknowledged.

—Author

Preface to the First EditionThe subject “Engineering Mechanics” is taught in the very first year of undergraduate

Engineering curriculam in almost all universities in the world. A student learns to develop hiscapability to model an actual problem into an engineering problem and obtains its solution byusing the fundamental laws of mechanics. Engineering problems, to a great extent, require thedrawing of the free body diagrams of the entire body and also of a small portion cut from it.Students are advised to learn drawing of the free-body diagrams. The entire book is dividedinto two portions—Statics and Dynamics.

Throughout this book, SI systems of units have been used. Analytical methods are em-phasized throughout this book but the graphical and semi-graphical methods have not beenneglected. Most of the fundamentals needed to learn the basics of Engineering Mechanics havebeen discussed in First Chapter. Majority of the force systems, a student encounters in his life,have also been 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 confirm the correctness of his solution. If the“Check” fails, the student should hurriedly go through his solution to locate the mistake. A verylarge number of solved examples and the unsolved problems have been given for practice. Thegraphical method has been given in a very systematic fashion.

Separate chapters have been included on Friction, Centre of Gravity and Moment Iner-tia of Plane Area and the solid bodies. Standard cases have been given in tabular forms forready reference.

(xiii)

A separate chapter on Shear Force and Bending Moment Diagrams has been given. Anovel concept has been introduced which reduces labour and time of students for solving aproblem. The concept of Non-concurrent Coplanar force systems is used to draw the ShearForce Diagrams. Also, the basic approach of changing (or rotating) reference frames is used.

Chapters on Lifting Machines and on Transmission of Power have been given. Thoroughpreparation of chapters mentioned so far will ensure success in examination. These chaptersform part of Statics.

Chapter on Dynamics includes : Kinematics of Particle, Kinematics of Rigid Bodies, Rec-tilinear and Curvilinear Motion of Particles. Special methods for solving problems on Dynamicshave been discussed in chapters on Work-Energy Equations ; Impulse-Moment Method andCollision of Elastic Bodies.

This book has been written to help students in understanding the fundamentals of thesubject to develop their ability to deal with any situation.

It is hoped that the material presented in this book will be of value to the students andthe teachers as well. It will also help students in preparing for competitive examinations suchas GATE, GRE, UPSC, PSC, AMIE and other public sector utilities.

Many standard books have been used in preparation of this book and the author thankfullyacknowledges the same. I also thank my colleagues and the students of many colleges affiliatedto the Rajiv Gandhi Technical University, Bhopal, Pt. Ravishankar University, Raipur,Nagpur University, Amravati University, Pune University, Bangalore, Punjab University etc.,for giving their valuable suggestions which helped in the improvement of the quality of thebook.

I also acknowledge the moral support from my wife Smt. Kuldip Kaur and all my chil-dren for bearing with me during the long period of writing of this book. They also helped me inreading the proofs.

—Author

(xiv)

1

1System of Forces

PART A—FUNDAMENTAL PRINCIPLES AND RELATIONS

������������

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.

����� ����������������

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

Engineering Mechanics

Publisher : Laxmi Publications ISBN : 9789380856667 Author : Dr I S Gujral

Type the URL : http://www.kopykitab.com/product/10290

Get this eBook

40%OFF