A Text Book on

ENGINEERING MECHANICS

for

GATEPSUs & Other Competitive Exams

Dr. U.C. Jindal

M.Tech, Ph.D.

Former Professor & Head of the Department

Department of Mechanical Engineering

Delhi College of Engineering, Delhi

Publications

MADE EASY Publications

Corporate Office: 44-A/4, Kalu Sarai (Near Hauz Khas Metro Station), New Delhi-110016E-mail: infomep@madeeasy.inContact: 011-45124660, 8860378007

Visit us at: www.madeeasypublications.org

Engineering Mechanics for GATE, PSUs and Other Competitive Exams© Copyright by MADE EASY Publications.All rights are reserved. No part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photo-copying, recording or otherwise), without the prior written permission of the above mentioned publisher of this book.

First Edition : 2015Second Edition : 2016Reprint : 2017

© All rights reserved by MADE EASY PUBLICATIONS. No part of this book may be reproduced or utilized in any form without the written permission from the publisher.

Publications

At the outset I must thank Mr. B. Singh (Chairman and Managing Director of MADE EASY Group)

who has very kindly given me an opportunity to serve the student community through the

publication of this book on Engineering Mechanics.

The present book has been designed as a self study book taking into account the severe

shortage of technical teachers in engineering colleges and technical institutions.

The text in the book is well explained through examples supplemented by self explanatory

illustrations, exercises supplemented by hints, key points to remember, thought provoking

multiple choice questions, special problems so that a student can learn this basic subject

in the shortest possible time.

The book covers all the syllabi in Engineering Mechanics of GATE, PSUs, all the universities,

IITs, NITs, deemed universities. Students appearing in competitive examinations and other

competitive examinations will find the book as an asset to them. The book also serves the

purpose of AMIE students.

The book will greatly help the students who could not grasp the subject in the class room.

Any suggestion for the improvement in the text of the book will be thankfully acknowledged.

Dr. U. C. Jindal

Author

Preface

(v)

Chapter 1Vector Quantities in Mechanics ........................... 1

1.1 Introduction .............................................................................1

1.2 Position Vector ........................................................................1

1.3 Displacement Vector .............................................................2

1.4 Force Vector ..............................................................................3

1.5 Moment of a Force about a Point .....................................4

1.6 Varignon’s Theorem ...............................................................5

1.7 Moment of a Force about an Axis ....................................7

1.8 Couple and its Moment .......................................................8

1.9 Component of a Couple about a Line ..........................10

Problems .................................................................................11

Practice Problems.................................................................12

Multiple Choice Questions ...............................................12

Explanations ...........................................................................13

Chapter 2Equivalent Coplanar Force Systems .................. 15

2.1 Introduction ...........................................................................15

2.2 Resultant of a Force by an Equivalent Couple

and a Force at any other Point ........................................17

2.3 Resultant of a Coplanar Force System ..........................17

2.4 Resultant of a Parallel Forces Systems ..........................19

2.5 System of Distributed Forces ...........................................20

Problems .................................................................................21

Practice Problems.................................................................23

Multiple Choice Questions ...............................................24

Explanations ...........................................................................26

Chapter 3Equilibrium of Rigid Bodies ............................... 28

3.1 Introduction ...........................................................................28

3.2 Support Reactions ...............................................................29

3.3 Free body Diagrams ............................................................30

3.4 Conditions of Equilibrium .................................................31

3.5 System of Concurrent Coplanar Forces ........................33

3.6 Coplanar Force Systems .....................................................34

3.7 Equilibrium of a Two Force Body ....................................35

3.8 Equilibrium of a Three Force Body .................................36

3.9 Lami’s Theorem .....................................................................37

3.10 Two Parallel Forces ...............................................................38

3.11 Several Parallel Forces in a Plane ....................................40

Problems .................................................................................41

Practice Problems.................................................................45

Special Problems ..................................................................46

Multiple Choice Questions ...............................................47

Explanations ...........................................................................50

Chapter 4Plane Trusses ....................................................... 534.1 Introduction ...........................................................................53

4.2 Types of Plane Trusses ........................................................54

4.3 Determination of Support Reactions ............................54

4.4 Determination of Forces in Members of the Truss ...56

4.5 Method of Joints...................................................................56

4.6 Method of Sections .............................................................58

Practice Problems.................................................................63

Multiple Choice Questions ...............................................64

Explanations ...........................................................................69

Engineering MechanicsContents

(vi)(vi)

Chapter 5Centroid and Center of Gravity.......................... 73

5.1 Introduction ...........................................................................73

5.2 Location of Centroid ...........................................................73

5.3 Centroid of Simple Sections .............................................75

5.4 Centre of Gravity of 3 Dimensional Bodies .................78

5.5 Theorems of Pappus-Guldinus ........................................81

Multiple Choice Questions ...............................................83

Explanations ...........................................................................85

Chapter 6Friction ............................................................ 86

6.1 Introduction ...........................................................................86

6.2 Laws of Dry Friction or Coulomb’s Friction .................87

6.3 Angle of Friction ...................................................................87

6.4 Block on a Horizontal Plane ..............................................88

6.5 Two Mating Blocks ...............................................................89

6.6 Angle of Repose ....................................................................90

6.7 A Force Applied on a Body Lying on an Inclined Plane ....90

6.8 Bodies on Inclined Plane Connected Through a Link ..91

6.9 Ladder ......................................................................................93

6.10 Friction in Wheels .................................................................94

Problems .................................................................................95

Practice Problems.................................................................97

Multiple Choice Questions ...............................................98

Explanations ........................................................................101

Chapter 7Application of Friction in Machines ................ 104

7.1 Introduction ........................................................................104

7.2 Screw Threads ....................................................................104

7.3 Flat Belt Drive......................................................................106

7.4 V-Belt and Rope Drive ...................................................... 107

7.5 Rolling Resistance ............................................................. 108

Practice Problems..............................................................110

Multiple Choice Questions ............................................ 110

Explanations ........................................................................111

Chapter 8Second Moment of Area (Moment of Inertia).........112

8.1 Introduction ........................................................................112

8.2 Rectangular Section ......................................................... 112

8.3 Triangular Section ............................................................. 113

8.4 Circular Section .................................................................. 113

8.5 Polar Moment of Inertia .................................................. 114

8.6 Parallel Axes Theorem ..................................................... 115

8.7 Principal Axes and Product of Inertia ......................... 116

8.8 Parallel Axes Theorem for Product of Inertia ........... 117

Multiple Choice Questions ............................................ 119

Explanations ........................................................................121

Chapter 9Virtual Work ...................................................... 122

9.1 Introduction ........................................................................122

9.2 Virtual Work .........................................................................122

9.3 Work Done by a Couple .................................................. 123

9.4 Principle of Virtual Work Applied to Beams ............. 124

9.5 Potential Energy and Stability ...................................... 125

9.6 Multiple Degrees of Freedom ...................................... 128

Problems ..............................................................................130

Practice Problems..............................................................132

Multiple Choice Questions ............................................ 134

Explanations ........................................................................135

Chapter 10Rectilinear Motion ............................................ 137

10.1 Introduction ........................................................................137

10.2 Rectilinear Motion with Constant Acceleration ..... 138

10.3 Motion Under Gravity ...................................................... 139

10.4 Acceleration as a Function of Time ............................. 140

Problems ..............................................................................141

Practice Problems..............................................................142

Multiple Choice Questions ............................................ 143

Explanations ........................................................................144

(vii)

Chapter 11Curvilinear Motion ........................................... 146

11.1 Introduction ........................................................................146

11.2 Plane Curvilinear Motion ................................................ 146

11.3 Rectangular Coordinates ................................................ 147

11.4 Tangential and Normal Co-ordinates ......................... 148

Problems ..............................................................................149

Practice Problems..............................................................150

Multiple Choice Questions ............................................ 152

Explanations ........................................................................153

Chapter 12Relative Velocity ............................................... 154

12.1 Introduction ........................................................................154

12.2 Motion Relative to a Frame in Translation ................ 154

12.3 Absolute and Relative Velocity in Plane Motion .... 156

Problems ..............................................................................157

Practice Problems..............................................................160

Multiple Choice Questions ............................................ 160

Explanations ........................................................................162

Chapter 13Motion of Rotation ........................................... 163

13.1 Introduction ........................................................................163

13.2 Rotation about a Fixed Axis ........................................... 163

13.3 Various Types of Motion of Rotation .......................... 165

13.4 Instantaneous Centre of Rotation ............................... 166

Problems ..............................................................................169

Practice Problems..............................................................170

Multiple Choice Questions ............................................ 171

Explanations ........................................................................172

Chapter 14Kinetics of a Particle ......................................... 173

14.1 Introduction ........................................................................173

14.2 Newton’s Second Law of Motion ................................. 173

14.3 Types of Problems ............................................................. 174

14.4 Motion of Two Bodies in Contact ................................ 177

14.5 Tangential and Normal Forces in Curvilinear Motion ....178

14.6 Motion of a Bicycle Rider Along a Circular Track .... 179

Problems ..............................................................................180

Practice Problems..............................................................182

Multiple Choice Questions ............................................ 183

Explanations ........................................................................184

Chapter 15Dynamics of Rigid Bodies in Translation ........ 186

15.1 Introduction ........................................................................186

15.2 D’ Alembert’s Principle .................................................... 187

Practice Problems..............................................................192

Multiple Choice Questions ............................................ 193

Explanations ........................................................................194

Chapter 16Mass Moment of Inertia ................................... 195

16.1 Introduction ........................................................................195

16.2 Parallel-Axes Theorem .....................................................196

Multiple Choice Questions ............................................ 199

Explanations ........................................................................200

Chapter 17Dynamics of Rigid Bodies in Plane Motion ..... 201

17.1 Introduction ........................................................................201

Problems ..............................................................................203

Practice Problems..............................................................206

Multiple Choice Questions ............................................ 207

Explanations ........................................................................208

Chapter 18Rotation about Fixed Axis ................................ 210

18.1 Introduction ........................................................................210

18.2 Rotation about Centroidal Axis .................................... 212

18.3 Inertia Force Method ....................................................... 213

18.4 Centre of Percussion ........................................................ 214

Practice Problems..............................................................216

Multiple Choice Questions ............................................ 217

Explanations ........................................................................218

(viii)

Chapter 19Work and Energy .............................................. 220

19.1 Introduction ........................................................................220

19.2 Work done by a Force during Sliding Motion ......... 222

19.3 Work done by a Moment during Motion of Rotation .....223

19.4 Change of Potential Energy in a Spring .................... 224

19.5 Rolling Motion under Gravity ....................................... 225

Practice Problems..............................................................227

Multiple Choice Questions ............................................ 228

Explanations ........................................................................230

Chapter 20Impulse and Momentum .................................. 232

20.1 Introduction ........................................................................232

20.2 Angular Impulse and Angular Momentum Principle ......234

20.3 Pile and Pile Hammer .......................................................235

Problems ..............................................................................237

Practice Problems..............................................................238

Multiple Choice Questions ............................................ 239

Explanations ........................................................................240

Chapter 21Impact : Collision of Elastic Bodies .................. 241

21.1 Introduction ........................................................................241

21.2 Coefficient of Restitution................................................242

21.3 Impact of a Body on another Stationery Body ....... 244

Problems ..............................................................................245

Practice Problems..............................................................247

Multiple Choice Questions ............................................ 248

Explanations ........................................................................250

nnnn