ENGINEERING MECHANICS ENGINEERING MECHANICS (In S.I. Units) For B.E./B. Tech. 1st YEAR [ANNA UNIVERSITY, TAMILNADU] By Dr. R.K. BANSAL B.Sc. Engg. (Mech.), M. Tech., Hons. (I.I.T., Delhi), Ph.D., M.I.E. (India) Formerly, Professor in Mechanical Engineering Department of Mechanical Engineering Delhi College of Engineering Delhi Presently, Dean (Academics) Northern India Engineering College, New Delhi AND SANJAY BANSAL B.E. (Computer) LAXMI PUBLICATIONS (P) LTD BANGALORE N CHENNAI N COCHIN N GUWAHATI N HYDERABAD JALANDHAR N KOLKATA N LUCKNOW N MUMBAI N PATNA RANCHI N NEW DELHI Compiled by : Smt. Nirmal Bansal © All rights reserved with the Publishers. 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 without the prior written permission of the publisher. Published by : LAXMI PUBLICATIONS (P) LTD 113, Golden House, Daryaganj, New Delhi-110002 Phone : 011-43 53 25 00 Fax : 011-43 53 25 28 www.laxmipublications.com [email protected] Price : ` 350.00 Only. First Edition : 2005 ; Second Edition : 2007 ; Reprint : 2008, 2009, 2010, 2011 ; Third Edition : 2012 OFFICES Bangalore 080-26 75 69 30 Kolkata 033-22 27 43 84 Chennai 044-24 34 47 26 Lucknow 0522-220 99 16 Cochin 0484-237 70 04, 405 13 03 Mumbai 022-24 91 54 15, 24 92 78 69 Guwahati 0361-251 36 69, 251 38 81 Patna 0612-230 00 97 Hyderabad 040-24 65 23 33 Ranchi 0651-221 47 64 Jalandhar 0181-222 12 72 EEM-0638-350-ENGG MECHANICS (TN)–BAN C—2179/010/09 Typeset at : Excellent Graphics, Delhi. Printed at : Ajit Printers Contents Chapters Pages 1. Basics and Statics of Particles 1—95 1.1. Introduction ... 1 1.2. Units and Dimensions ... 5 1.3. Laws of Mechanics ... 14 1.4. Lame’s Theorem ... 17 1.5. Parallelogram and Triangular Law of Forces ... 17 1.6. Vectors ... 17 1.7. Vector Operations ... 20 1.8. Resolution and Composition of a Force ... 25 1.9. Coplanar Forces ... 31 1.10. Resultant of Coplanar Forces ... 32 1.11. Equilibrium of a Particle ... 57 1.12. Equilibrium of a Rigid Body ... 58 1.13. Forces in Space ... 69 1.14. Equilibrium of a Particle in Space ... 85 1.15. Equivalent Systems of Forces ... 86 1.16. Principle of Transmissibility ... 88 1.17. Single Equivalent Force ... 88 Highlights ... 88 Exercise-1 ... 90 2. Equilibrium of Rigid Bodies 96—180 2.1. Introduction ... 96 2.2. Free Body Diagram ... 97 2.3. Types of Supports and Their Reactions ... 114 2.4. Requirements of Stable Equilibrium ... 115 2.5. Moments and Couples ... 116 2.6. Moment of a Force about a Point and about an Axis ... 116 2.7. Vectorial Representation of Moments and Couples ... 117 2.8. Varignon’s Theorem (or Principle of Moments) ... 119 2.9. Equilibrium of Rigid Bodies in Two Dimensions and in Three Dimensions ... 121 2.10. Methods for Finding Out the Reactions of a Beam ... 122 2.11. Problems for Equilibrium of Rigid Bodies in Two-Dimension and Three Dimensions ... 123 2.12. Analysis of Pin-Jointed Plane Trusses (or Frames) ... 139 Highlights ... 172 Exercise-2 ... 173 3. Properties of Surfaces and Solids 181—245 3.1. Introduction ... 181 3.2. Determination of First Moment of Area and Centroid of Sections—Rectangle, Circle, Triangle by Integration ... 183 ( v ) ( vi ) Chapters Pages 3.3. Centroid of T-section, I-section, Angle-section, Hollow-section etc. ... 198 3.4. Centroid of Volume ... 203 3.5. Second Moment of Area (or Area Moment of Inertia) ... 206 3.6. Theorem of the Perpendicular Axis ... 207 3.7. Theorem of Parallel Axis ... 208 3.8. Determination of Second Moment of Area (or Area Moment of Inertia) of Plane Area like Rectangle, Triangle, Circle etc. from Integration ... 209 3.9. Moment of Inertia of T-section, I-section, Angle-section, Hollow-section etc. By using Standard Formula ... 216 3.10. Polar Moment of Inertia ... 223 3.11. Product of Inertia ... 224 3.12. Principal Axes ... 225 3.13. Principal Moments of Inertia ... 227 3.14. Mass Moment of Inertia ... 234 3.15. Derivation of Mass Moment of Inertia for Rectangle Section, Prism, Sphere etc. From First Principal ... 235 Highlights ... 241 Exercise-3 ... 242 4. Dynamics of Particles 246—418 4.1. Introduction ... 246 4.2. Velocity ... 246 4.3. Acceleration ... 246 4.4. Relationship of Velocity, Acceleration and Displacement ... 247 4.5. Relative Motion ... 276 4.6. Curvilinear Motion ... 280 4.7. Equations of Motions Along a Circular Path ... 281 4.8. Newton’s Laws ... 288 4.9. Momentum and Angular Momentum (or Moment of Momentum) ... 328 4.10. Laws for Rotary Motion ... 334 4.11. Work Energy Equation of Particles ... 354 4.12. Impulse and Momentum ... 383 4.13. Impact of Elastic Bodies ... 384 Highlights ... 405 Exercise-4 ... 409 5. Friction and Elements of Rigid Body Dynamics 419—512 5.1. Frictional Force ... 419 5.2. Limiting Force of Friction and Definitions of Certain Terms ... 419 5.3. Laws of Coulomb Friction ... 421 5.4. Simple Contact Friction ... 421 5.5. Rolling Resistance ... 455 5.6. Belt Friction ... 457 5.7. Transmission of Power Through Belts ... 459 5.8. Translation, Rotation and General Plane Motion ... 473 ( vii ) Chapters Pages 5.9. Velocity and Acceleration for Combined Motion of Translation and Rotation of Rigid Bodies ... 478 5.10 General Plane Motion ... 485 5.11 Kinematics of Plane Motion ... 488 5.12 Equations of Motion for Plane Motion of a Rigid Body ... 502 Highlights ... 506 Exercise-5 ... 508 Question Bank 513—561 Model Question Paper 562—564 Preface to the Third Edition Authors are glad to present the Third Edition of the book entitled, ‘Engineering Mechanics’ to the engineering students of all disciples of Anna University, Tamilnadu. The course contents are planned in such a way that the book covers the complete course of first year students of Anna University, Tamilnadu according to the revised syllabus. The Third Edition contains the following five units : Unit I : Basics and Statics of Particles. Unit II : Equilibrium of Rigid Bodies. Unit III : Properties of Surfaces and Solids. Unit IV : Dynamics of Particles. Unit V : Friction and Elements of Rigid Body Dynamics. The book is written in a simple and easy-to-follow language, so that even an average student can grasp the subject by self-study. At the end of each chapter highlights, theoretical questions and many unsolved numerical problems with answers are given for the students to solve them. Mrs. Nirmal Bansal deserves special credit as she not only provided an ideal atmos- phere at home for book writing but also gave inspiration and valuable suggestions. Though every care has been taken in checking the manuscript and proofreading, yet claiming perfection is very difficult. We shall be very grateful to the readers and users of this book for pointing any mistake that might have crept in. Suggestions for improvement are most welcome and would be incorporated in the next edition with a view to make the book more useful. —AUTHORS SYLLABUS (Anna University, Tamilnadu) GE1X04 ENGINEERING MECHANICS 3 1 0 100 (Common to B.E. (Civil), B. Tech (Chemical / Textile / Textile Tech (Textile Chemistry) Polymer Tech / Biotech / Petroleum Engg / Food Tech / Rubber and Plastics Tech)) OBJECTIVE At the end of this course the student should be able to understand the vectorial and scalar representation of forces and moments, static equilibrium of particles and rigid bodies both in two dimensions and also in three dimensions. Further, he should understand the principle of work and energy. He should be able to comprehend the effect of friction on equilibrium. He should be able to understand the laws of motion, the kinematics of motion and the interrelationship. He should also be able to write the dynamic equilibrium equation. All these should be achieved both conceptually and through solved examples. UNIT I : BASICS AND STATICS OF PARTICLES 12 Introduction—Units and Dimensions—Laws of Mechanics—Lame’s theorem, Parallelogram and triangular Law of forces—Vectors—Vectorial representation of forces and moments—Vector operations : additions, subtractions, dot product, cross product—Coplanar Forces—Resolution and Composition of forces—Equilibrium of a particle—Forces in space—Equilibrium of a particle in space—Equivalent systems of forces—Principle of transmissibility—Single equivalent force. UNIT II : EQUILIBRIUM OF RIGID BODIES 12 Free body diagram—Types of supports and their reactions—requirements of stable equilibrium—Moments and Couples—Moment of a force about a point and about an axis—Vectorial representation of moments and couples—Scalar components of a moment—Varignon’s theorem— Equilibrium of Rigid bodies in two dimensions—Equilibrium of Rigid bodies in three dimensions— Examples. UNIT III : PROPERTIES OF SURFACES AND SOLIDS 12 Determination of Areas and Volumes—First moment of area and the Centroid of sections— Rectangle, circle, triangle from integration—T-section, I-section, Angle section, Hollow section by using standard formula—Second and product moments of plane area—Rectangle, triangle, circle from integration—T-section, I-section, Angle section, Hollow section by using standard formula— Parallel axis theorem and perpendicular axis theorem—Polar moment of inertia—Principal mo- ments of inertia of plane areas—Principal axes of inertia—Mass moment of inertia—Derivation of mass moment of inertia for rectangular section, prism, sphere from first principle—Relation to area moments of inertia. UNIT IV : DYNAMICS OF PARTICLES 12 Displacements, Velocity and acceleration, their relationship—Relative motion—Curvilinear motion—Newton’s law—Work Energy Equation of particles—Impulse and Momentum—Impact of elastic bodies. UNIT V : FRICTION AND ELEMENTS OF RIGID BODY DYNAMICS 12 Frictional force—Laws of Coulomb friction—Simple contact friction—Rolling resistance— Belt friction.