a textbook of engineering mechanics€¦ · engineering mechanics [in s.i. units] [for degree,...

A TEXTBOOK OF

ENGINEERING MECHANICS

[In S.I. Units]

[For Degree, B.E./B.Tech. 1st Year]

By

Dr. R.K. BANSALB.Sc. Engg. (Mech.), M. Tech., Hons. (I.I.T., Delhi),

Ph.D., M.I.E. (India)Formerly

Professor in Mechanical EngineeringDepartment of Mechanical Engineering

Delhi College of EngineeringDelhi

AND

SANJAY BANSALB.E. (Computer)

LAXMI PUBLICATIONS (P) LTDBANGALORE CHENNAI COCHIN GUWAHATI HYDERABAD

JALANDHAR KOLKATA LUCKNOW MUMBAI RANCHI

NEW DELHI BOSTON, USA

Published by :

LAXMI PUBLICATIONS (P) LTD113, Golden House, Daryaganj,

New Delhi-110002Phone : 011-43 53 25 00

Fax : 011-43 53 25 [email protected]

Compiled by : Smt. Nirmal Bansal

Sixth Edition : 2013 ; Fifth Edition : 2008 ;Reprint : 2006, 2007, 2009, 2010, 2011 ;

Reprint : Oct. 2004, June 2005 ; Reprint : April 2003, March 2004 ;Fourth Edition : 2002 ; Reprint : 1997, 1998, 1999, 2000, 2001 ;

Third Edition : 1996 ; Second Edition : 1994 ; Reprint : 1992, 1993 ; First Edition : 1991

OFFICES

Bangalore 080-26 75 69 30 Jalandhar 0181-222 12 72Chennai 044-24 34 47 26 Kolkata 033-22 27 43 84Cochin 0484-237 70 04, 405 13 03 Lucknow 0522-220 99 16Guwahati 0361-251 36 69, 251 38 81 Mumbai 022-24 91 54 15, 24 92 78 69Hyderabad 040-24 65 23 33 Ranchi 0651-220 47 64

EEM-0552-550-ENGG MECHANICS-BAN C—15226/08/01Typeset at : Excellent Graphics, Delhi. Printed at : Sanjeev Offset Printers, Delhi.

All rights reserved with the Publisher and Authors. No part of this publicationmay be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, recording or otherwisewithout the prior written permission of authors and the publisher.

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ContentsChapters Pages

PART I—STATICS

1. Fundamentals of Engineering Mechanics 1—261.1. Introduction ... 11.2. Definitions ... 11.3. System of Units ... 51.4. Trigonometric Formulae and Expressions ... 71.5. Differentiation and Integration ... 81.6. Resolution of a Force ... 151.7. Resolution of a Number of Coplanar Forces ... 161.8. Moment of a Force ... 191.9. Laws of Mechanics ... 21

Highlights ... 23Exercise 1 ... 24

2. Coplanar Collinear and Concurrent Forces 27—462.1. Introduction ... 272.2. Classification of a Force System ... 272.3. Resultant of Several Forces ... 292.4. Resultant of Coplanar Forces ... 292.5. Resultant of Collinear Coplanar Forces ... 292.6. Resultant of Concurrent Coplanar Forces ... 31


3. Coplanar Parallel Forces 47—653.1. Introduction ... 473.2. Moment of a Force ... 473.3. Principle of Moments (Or Varignon’s Principle) ... 493.4. Types of Parallel Forces ... 513.5. Resultant of Two Parallel Forces ... 513.6. Resolution of a Force into a Force and a Couple ... 563.7. General Case of Parallel Forces in a Plane ... 583.8. Equivalent System ... 60


4. Conditions of Equilibrium 66—964.1. Introduction ... 66

4.2. Principle of Equilibrium ... 664.3. Force Law of Equilibrium ... 674.4. Action and Reaction ... 764.5. Free Body Diagram ... 77


5. Support Reactions 97—1205.1. Introduction ... 975.2. Types of Supports ... 975.3. Types of Loading ... 985.4. Methods for Finding out the Reactions of a Beam ... 995.5. Problems on Simple Supported Beams ... 1015.6. Problems on Overhanging Beams ... 1055.7. Problems on Roller and Hinged Supported Beams ... 1065.8. Problems when Beams are Subjected to Couples ... 115


6. Analysis of Perfect Frames 121—1626.1. Introduction ... 1216.2. Types of Frames ... 1226.3. Assumptions Made in Finding out the Forces in a Frame ... 1236.4. Reactions of Supports of a Frame ... 1236.5. Analysis of a Frame ... 1236.6. Method of Sections ... 1456.7. Graphical Method ... 154


7. Centre of Gravity and Moment of Inertia 163—2327.1. Centre of Gravity ... 1637.2. Centroid ... 1637.3. Centroid or Centre of Gravity of Simple Plane Figures ... 1637.4. Centre of Gravity of Plane Figures by the Method of Moments ... 1637.5. Important Points ... 1657.6. Area Moment of Inertia ... 1897.7. Radius of Gyration ... 1907.8. Theorem of the Perpendicular Axis ... 1907.9. Theorem of Parallel Axis ... 1917.10. Determination of Moment of Inertia ... 1927.11. Polar Moment of Inertia ... 2087.12. Product of Inertia ... 2097.13. Principal Axes ... 2107.14. Principal Moments of Inertia ... 2127.15. Mass Moment of Inertia ... 219

Chapters Pages

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Chapters Pages

7.16. Derivation of Mass Moment of Inertia for Rectangle Section, Prism,Sphere etc. from First Principal ... 220Highlights ... 227Exercise 7 ... 228

8. Friction 233—2928.1. Introduction ... 2338.2. Definitions ... 2338.3. Types of Friction ... 2358.4. Laws of Solid or Coulomb Friction ... 2358.5. Angle of Repose ... 2428.6. Equilibrium of a Body Lying on a Rough Inclined Plane ... 2428.7. Analysis of Ladder Friction ... 2618.8. Analysis of Wedge Friction ... 2698.9. Friction in Bearings ... 2738.10. Friction in Flat Bearing ... 2758.11. Friction in Conical Pivot Bearing ... 2788.12. Truncated Conical Pivot ... 2808.12(A) Friction in Collared Bearing ... 283


9. Lifting Machines 293—3439.1. Introduction ... 2939.2. Definitions ... 2939.3. Reversible and Irreversible Machine ... 2949.4. Condition for Irreversibility of a Machine ... 2959.5. Law of a Machine ... 2989.6. Friction of a Machine Expressed in Terms of Actual Effort (P) ... 3019.7. Friction of a Machine Expressed in Terms of Load ... 3029.8. Important Lifting Machines ... 306


10. Virtual Work 344—38310.1. Introduction ... 34410.2. Work ... 34410.3. Virtual Work ... 34510.4. Principle of Virtual Work ... 34510.5. Units of Work ... 34510.6. Forces to be Omitted while Applying the Principle of Virtual Work ... 34510.7. Important Points to be Remembered while Applying the Principle

of Virtual Work ... 34510.8. Uses of the Principle of Virtual Work ... 346

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10.9. Virtual Work Done by Moment (Or Torque) ... 377Highlights ... 380Exercise 10 ... 381

11. Belts, Ropes and Chain Drives 384—42911.1. Introduction ... 38411.2. Open Flat Belt Drive ... 38411.3. Cross Belt Drive ... 39111.4. Compound Belt Drive ... 39111.5. Length of Belt ... 39311.6. Ratio of Belt Tensions ... 40011.7. Power Transmitted by Belt ... 40311.8. Centrifugal Tension ... 40511.9. Maximum Power Transmitted by a Belt ... 40611.10. Initial Tension in the Belt ... 41111.11. V-Belt Drive and Rope-Drive ... 41711.12. Chain Drive ... 42411.13. Relative Advantages and Disadvantages of Chain and Belt (Or Rope)

Drives ... 425Highlights ... 425Exercise 11 ... 427

PART II—DYNAMICS

12. Kinematics of Linear Motion 433—46712.1. Introduction ... 43312.2. Velocity ... 43312.3. Acceleration ... 43312.4. Equations of Motion in a Straight Line Under Uniform Acceleration ... 43412.5. Points to be Remembered ... 44312.6. Rectilinear Motion Under Variable Acceleration ... 45412.7. Distance Travelled by a Body when Velocity-Time Graph is Given ... 464


13. Kinematics of Curvilinear Motion, Circular Motion, Rotationand Translation 468—51813.1. Introduction ... 46813.2. Angular Velocity ... 46813.3. Angular Acceleration ... 46913.4. Equations of Motions along a Circular Path ... 46913.5. Types of Rigid Body Motion ... 47713.6. Velocity and Acceleration for Combined Motion of Translation and

Rotation of Rigid Bodies ... 48413.7. Analysis of Four Bar Mechanism ... 494

Chapters Pages

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13.8. General Plane Motion ... 49813.9. Kinematics of Plane Motion ... 501


14. Projectiles 519—54814.1. Introduction ... 51914.2. Terms Used with the Projectiles ... 51914.3. Equation for the Path of a Projectile ... 51914.4. Motion of a Body Thrown Horizontally from a Given Height into the Air ... 53214.5. Projectile on an Inclined Plane ... 538


15. Kinetics of Rigid Bodies and Laws of Motion 549—63615.1. Introduction ... 54915.2. Newton’s Laws for Linear Motion ... 54915.3. Definitions ... 54915.4. Newton’s First Law of Motion ... 55015.5. Newton’s Second Law of Motion ... 55015.6. Unit of Force ... 55115.7. Motion on an Inclined Smooth Surface ... 55815.8. Motion on an Inclined Rough Surface ... 55815.9. Analysis of Lift Motion ... 56615.10. Analysis of the Motion of Two Bodies Connected by a String ... 57215.11. Analysis of the Motion of Two Bodies Connected by a String when

One Body is Lying on a Horizontal Surface and other is Hanging Free ... 58115.12. Analysis of the Motion of Two Bodies Connected by a String when

One Body is Lying on Inclined Plane and the other is Hanging Free in Air ... 58915.13. Momentum and Angular Momentum (Or Moment of Momentum) ... 59515.14. (A) Moment of Inertia of a Circular Section ... 59615.14. (B) Moment of Inertia of Solid Cylinder ... 59715.14. (C) Moment of Inertia of a Solid Sphere ... 59715.15. Radius of Gyration (K) ... 59815.16. Laws for Rotary Motion ... 60015.17. Kinetic Energy due to Rotation ... 60215.18. Total Kinetic Energy of a Body ... 60215.19. Rotation due to a Weight W Attached to One End of a String Passing

Over a Pulley of Weight W0 ... 60715.20. Rotation Due to Weights Attached to the Two Ends of a String,

Which Passes Over a Rough Pulley of Weight W0 ... 61015.21. D’ Alembert’s Principle and Dynamic Equilibrium ... 61915.22. D’ Alembert’s Principle Applicable to Rotary Motion ... 62215.23. Equations of Motion for Plane Motion of Rigid Body ... 625


Chapters Pages

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16. Simple Harmonic Motion and Mechanical Vibrations 637—67116.1. Introduction ... 63716.2. Definitions of Some Terms Used with S.H.M. ... 64016.3. Oscillation of the Bodies Having Simple Harmonic Motion ... 64916.4. Gain or Loss of Oscillations Due to Change in G or 1 for a Pendulum ... 66016.5. Free Vibration ... 662


17. Collision of Elastic Bodies 672—69717.1. Introduction ... 67217.2. Some Definition ... 67217.3. Types of Impacts ... 67217.4. Co-efficient of Restitution ... 67517.5. Loss of Kinetic Energy During Impact ... 68417.6. Impact of a Body on a Fixed Plane ... 68917.7. Direct Impact of a Body on a Fixed Plane ... 68917.8. Indirect Impact of a Body on a Fixed Plane ... 692


18. Work, Power and Energy 698—73018.1. Introduction ... 69818.2. Work ... 69818.3. Units of Work ... 69818.4. Power ... 70218.5. Work Done by a Torque ... 70618.6. Power Developed by a Torque ... 70618.7. Energy ... 70718.8. Mechanical Energy ... 70718.9. Law of Conservation of Energy ... 71118.10. Energy Lost by a Body Falling on Another Body and to Calculate the

Resistance Offered the Ground ... 719Highlights ... 727Exercise 18 ... 728

19. Forces in Space 731—75019.1. Introduction ... 73119.2. Notation of a Vector ... 73119.3. Components of a Force ... 73219.4. Unit Vector ... 73219.5. Components of a Force when Two Points on its Line of Action are Given ... 73319.6. Position Vector of a Given Point ... 73319.7. Important Vector Operations ... 73919.8. Moment of a Force ... 742

Chapters Pages

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20. Beams (Shear Force and Bending Moment) 751—80720.1. Introduction ... 75120.2. Shear Force and Bending Moment Diagrams ... 75120.3. Types of Beams ... 75120.4. Types of Load ... 75220.5. Sign Conventions for Shear Force and Bending Moment ... 75320.6. Important Points for Drawing Shear Force and Bending Moment Diagrams... 75420.7. Shear Force and Bending Moment Diagrams for a Cantilever with a

Point Load at the Free End ... 75520.8. Shear Force and Bending Moment Diagrams for a Cantilever with a

Uniformly Distributed Load ... 75820.9. Shear Force and Bending Moment Diagrams for a Cantilever Carrying a

Gradually Varying Load ... 76620.10. Shear Force and Bending Moment Diagrams for a Simply Supported

Beam with a Point Load at Mid-Point ... 76820.11. Shear Force and Bending Moment Diagrams for a Simply Supported Beam

with an Eccentric Point Load ... 77020.12. Shear Force and Bending Moment Diagrams for a Simply Supported

Beam Carrying a Uniformly Distributed Load ... 77220.13. Shear Force and Bending Moment Diagrams for a Simply Supported

Beam Carrying a Uniformly Varying Load from Zero at Each End to w PerUnit Length at the Centre ... 780

20.14. Shear Force and B.M. Diagrams for a Simply Supported Beam Carrying aUniformly Varying Load from Zero at One End to w Per Unit Length at theother End ... 782

20.15. Shear Force and Bending Moment Diagrams for Over-Hanging Beams ... 78620.16. S.F. and B.M. Diagrams for Beams Carrying Inclined Load ... 79520.17. Shear Force and Bending Moment Diagrams for Beams Subjected

to Couples ... 80020.18. Relations between Load, Shear Force and Bending Moment ... 803


Chapters Pages

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PREFACE TO THE SIXTH EDITION

The popularity of the fifth edition and reprints of the textbook of Engineering Mechanicsamongst the students and the teachers of the various Universities in the country, has promptedthe bringing out of the sixth edition of the book so soon. The sixth edition has been thoroughlyrevised and brought upto date. More articles have been added in the Laws of Motion and Work,power and energy chapters. A chapter on Shear Force and Bending Moment has been added inthis edition to complete the curriculum requirements of many Universities. The chapters ofCentre of Gravity, Moment of Inertia and Collision of Elastic Bodies have been thoroughlyrevised and explained in more details. In the Friction chapter, friction of flat pivot, conicaltruncated and collored thrust bearings have been added. A large number of problems fromdifferent B.E. degree examinations of Indian Universities and other examining bodies havebeen selected and have been solved at proper places in this edition. The suggestions made bystudents and teachers have been incorporated in this edition.

In this edition, the concept of General Plane Motion, kinematics and equations of planemotion are explained in simple and easy-to-follow language.

With these additions, it is hoped that the book will be quite useful for the students ofdifferent branches of Engineering at various Engineering Institutions.

We express our sincere thanks to our colleagues, friends, students and the teachers ofdifferent Indian Universities for their valuable suggestions and recommending the books totheir students.

Suggestions for the improvement of this book are most welcome and would be incorpo-rated in the next edition with a view to make the book more useful.

—AUTHORS

PREFACE TO THE FIRST EDITION

Authors are glad to present the book entitled, ‘Engineering Mechanics’ to the engineer-ing students of Mechanical, Civil, Electrical, Electronics, Aeronautical and Chemical and alsoto the students preparing for A.M.I.E. Examination of Institution of Engineers (India). Thecourse-contents have been planned in such a way that the general requirements of all engi-neering students are fulfilled.

During long experience of teaching, the authors have observed that the students facedifficulty in understanding clearly the basic principles, fundamental concepts and theory with-out adequate solved problems along with the text. To meet this very basic requirement of thestudents, a large number of the questions taken from different Universities and the A.M.I.E.Examinations have been solved along with the text in S.I. units at proper places.

The book is written in a simple and easy-to-follow language, so that even an averagestudent can grasp the subject by self-study. At the end of each chapter highlights, theoreticalquestions and many unsolved numerical problems with answers are given for the students tosolve them.

Authors are thankful to colleagues, friends and students who encouraged us to writethis book. We are grateful to the Institution of Engineers (India), various Universities of India

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and those authorities whose work have been consulted and gave us a great help in preparingthis book.

We express our appreciation and gratefulness to our publisher, Shri R.K. Gupta(a Mechanical Engineer) for his most co-operative, painstaking attitude and untiring effort forbringing out the book in a short period.

Mrs. Nirmal Bansal deserves special credit as she not only provided an ideal atmosphereat home for book writing but also gave inspiration and valuable suggestions.

Though every care has been taken in checking the manuscripts and proof reading, yetclaiming perfection is very difficult. We shall be very grateful to the readers and users of thisbook for pointing any mistake that might have crept in. Suggestions for improvement are mostwelcome and would be incorporated in the next edition with a view to make the book moreuseful.

—AUTHORS

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PART I

STATICS

A Textbook Of Engineering Mechanics

Publisher : Laxmi Publications ISBN : 9788131804094 Author : R K Bansal

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