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Applied Mechanics A TEXTBOOK OF APPLIED MECHANICS A TEXTBOOK OF APPLIED MECHANICS (Including Laboratory Practicals) S.I. UNITS By R.K. RAJPUT M.E. (Heat Power Engg.) Hons.–Gold Medallist ; Grad. (Mech. Engg. & Elect. Engg.) ; M.I.E. (India) ; M.S.E.S.I. ; M.I.S.T.E. ; C.E. (India) Principal (Formerly), Punjab College of Information Technology PATIALA (Punjab) LAXMI PUBLICATIONS (P) LTD BANGALORE l CHENNAI l COCHIN l GUWAHATI l HYDERABAD JALANDHAR l KOLKATA l LUCKNOW l MUMBAI l PATNA RANCHI l NEW DELHI 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] © 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. Price : Rs. 250.00 Only. Third Edition: 2011 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 EAM-0791-250-ATB APPLIED MECH-RAJ C—689/09/09 Typeset at : Goswami Associates, Delhi Printed at : Mehra Offset Printer, Delhi CONTENTS Chapters Pages INTRODUCTION TO S.I. UNITS AND CONVERSION FACTORS (i)–(v) Part I—APPLIED MECHANICS—THEORY 1. BASIC CONCEPTS 3–8 1.1. Introduction to Mechanics ... 3 1.2. Basic Definitions ... 4 1.3. Rigid Body ... 5 1.4. Scalar and Vector Quantities ... 5 1.5. Fundamental Units and Derived Units ... 5 1.6. Systems of Units ... 6 Highlights ... 7 Objective Type Questions ... 8 Exercises ... 8 2. LAWS OF FORCES 9–50 2.1. Force ... 9 2.2. Units of Force ... 9 2.3. Characteristics of a Force ... 10 2.4. Representation of Forces ... 10 2.5. Classification of Forces ... 10 2.6. Force Systems ... 11 2.7. Free Body Diagrams ... 12 2.8. Transmissibility of a Force ... 13 2.9. Particle ... 14 2.10. Resultant Force ... 14 2.11. Component of a Force ... 14 2.12. Principle of Resolved Parts ... 15 2.13. Laws of Forces ... 15 2.14. Resultant of Several Coplanar Concurrent Forces ... 17 2.15. Equilibrium Conditions for Coplanar Concurrent Forces ... 27 2.16. Lami’s Theorem ... 27 Highlights ... 47 Objective Type Questions ... 48 Exercises ... 49 Theoretical Questions ... 49 Unsolved Examples ... 49 3. MOMENTS 51–86 3.1. Moments ... 51 3.2. Clockwise and Anti-clockwise Moments ... 52 (v) (vi) 3.3. Principle of Moments ... 52 3.4. Equilibrium Conditions for Bodies Under Coplanar Non-concurrent Forces ... 53 3.5. Varignon’s Theorem ... 54 3.6. Parallel Forces ... 55 3.7. Graphical Method for Finding the Resultant of any Number of Like or Unlike Parallel Forces ... 55 3.8. Couple ... 56 3.9. Properties of a Couple ... 57 3.10. Engineering Applications of Moments ... 58 3.11. Resultant of a Coplanar, Non-concurrent Non-parallel Force System ... 63 Highlights ... 82 Objective Type Questions ... 82 Exercises ... 83 Theoretical Questions ... 83 Unsolved Examples ... 83 4. CENTRE OF GRAVITY AND CENTROID 87–123 4.1. Centre of Gravity of a Body ... 87 4.2. Determination of Centre of Gravity ... 88 4.3. Centroid ... 88 4.4. Positions of Centroids of Plane Geometrical Figures ... 89 4.5. Positions of Centre of Gravity of Regular Solids ... 90 4.6. (a) Centroids of Composite Areas ... 91 4.6. (b) Centre of Gravity of Simple Solids ... 91 4.7. Areas and Volumes—Centroid Method ... 92 4.8. Centre of Gravity in a Few Simple Cases ... 93 Highlights ... 117 Objective Type Questions ... 117 Exercises ... 118 Theoretical Questions ... 118 Unsolved Examples ... 118 5. MOMENT OF INERTIA 124–150 5.1. Moment of Inertia ... 124 5.2. Theorem of Parallel Axes ... 126 5.3. Theorem of Perpendicular Axes ... 126 5.4. Radius of Gyration of the Section ... 127 5.5. Moment of Inertia of Laminae of Different Shapes ... 128 Highlights ... 146 Objective Type Questions ... 147 Exercises ... 147 Theoretical Questions ... 147 Unsolved Examples ... 147 (vii) 6. FRICTION 151–187 6.1. Concept of Friction ... 151 6.2. Characteristics of Frictional Force ... 152 6.3. Types of Friction ... 152 6.4. Static and Dynamic Friction ... 152 6.5. Limiting Friction ... 153 6.6. Laws of Friction ... 153 6.7. Angle of Friction ... 154 6.8. Co-efficient of Friction ... 154 6.9. Angle of Repose ... 154 6.10. Cone of Friction ... 155 6.11. Motion of Body on Horizontal Plane ... 155 6.12. Motion up an Inclined Plane ... 156 6.13. Motion Down an Inclined Plane ... 157 6.14. Screw Friction ... 172 6.15. Screw Jack ... 172 Highlights ... 184 Objective Type Questions ... 185 Exercises ... 185 Theoretical Questions ... 185 Unsolved Examples ... 186 7. RECTILINEAR MOTION 188–214 7.1. Concept of Motion ... 188 7.2. Definitions ... 188 7.3. Displacement-Time Graphs ... 190 7.4. Velocity-Time Graphs ... 190 7.5. Equations of Motion Under Uniform Acceleration ... 191 7.6. Distance Covered in nth Second by a Body Moving with Uniform Acceleration ... 192 7.7. Motion Under Gravity ... 201 7.8. Some Hints on the use of Equations of Motion ... 201 Highlights ... 211 Objective Type Questions ... 212 Exercises ... 213 Theoretical Questions ... 213 Unsolved Examples ... 213 8. LAWS OF MOTION 215–246 8.1. Introduction ... 215 8.2. Momentum ... 215 8.3. Newton’s First Law of Motion ... 215 8.4. Newton’s Second Law of Motion ... 215 8.5. Newton’s Third Law of Motion ... 216 8.6. Gravitational and Absolute Units of Force ... 217 8.7. Law of Conservation of Momentum ... 218 8.8. Impulse and Impulsive Force ... 219 (viii) 8.9. D’alembert’s Principle ... 227 8.10. Motion of a Lift ... 228 8.11. Motion of two Bodies Connected by a String Passing over a Smooth Pulley ... 230 8.12. Motion of two Bodies Connected at the Edge of a Horizontal Surface ... 233 8.13. Motion of two Bodies Connected by a String One End of which is Hanging Free and the other Lying on a Rough Inclined Plane ... 237 8.14. Motion of Two Bodies Connected Over Rough Inclined Planes ... 240 Highlights ... 243 Objective Type Questions ... 243 Unsolved Examples ... 244 9. WORK, POWER AND ENERGY 247–268 9.1. Concept of Work ... 247 9.2. Units of Work ... 248 9.3. Graphical Representations of Work ... 248 9.4. Power ... 249 9.5. Law of Conservation of Energy ... 251 Highlights ... 265 Objective Type Questions ... 266 Unsolved Examples ... 267 10. SIMPLE MACHINES 269–308 10.1. General Concept of a Machine ... 269 10.2. Important Definitions ... 269 10.3. Relation between M.A., V.R. and η ... 270 10.4. Concept of Friction in a Machine ... 270 10.5. Condition for Reversibility of a Machine ... 271 10.6. Non-reversible/Irreversible or Self-locking Machine ... 272 10.7. Law of a Machine ... 272 10.8. Maximum Mechanical Advantage and Efficiency ... 273 10.9. Some Lifting Machines ... 280 10.10. Simple Wheel and Axle ... 281 10.11. Wheel and Differential Axle ... 283 10.12. Pulleys ... 285 10.13. Weston’s Differential Pulley Block ... 292 10.14. Worm and Worm Wheel ... 295 10.15. Single Purchase Crab Winch ... 297 10.16. Double Purchase Crab Winch ... 300 10.17. Simple Screw Jack ... 303 10.18. Differential Screw Jack ... 303 Highlights ... 304 Objective Type Questions ... 306 Exercises ... 306 Theoretical Questions ... 306 Unsolved Examples ... 307 (ix) 11. CIRCULAR AND CURVILINEAR MOTION 309–330 11.1. Introduction and Definitions ... 309 11.2. Equations of Angular Motion ... 310 11.3. Equations of Linear Motion and Angular Motion ... 311 11.4. Relation between Linear and Angular Motion ... 311 11.5. Centrifugal and Centripetal Force ... 312 11.6. Motion of a Cyclist Round a Curve ... 317 11.7. Motion of a Vehicle on a Level Curved Path ... 319 11.8. Motion of a Vehicle on a Banked Circular Track ... 321 11.9. Super Elevation for Railways ... 324 Highlights ... 327 Objective Type Questions ... 328 Exercises ... 329 Theoretical Questions ... 329 Unsolved Examples ... 329 12. SIMPLE STRESSES AND STRAINS 331–366 12.1. Classification of Loads ... 331 12.2. Stress ... 331 12.3. Simple Stress ... 332 12.4. Strain ... 333 12.5. Hooke’s Law ... 334 12.6. Mechanical Properties of Metals ... 335 12.7. Tensile Test ... 336 12.8. Strain Hardening (or Work Hardening) ... 340 12.9. Poisson’s Ratio ... 340 12.10. Relations between the Elastic Modulii ... 341 12.11. Stresses Induced in Compound Ties or Struts ... 344 12.12. Thermal Stresses and Strains ... 344 Highlights ... 360 Objective Type Questions ... 361 Theoretical Questions ... 363 Unsolved Examples ... 363 Part II—EXPERIMENTS Experiment No. 1 ... 369 Experiment No. 2 ... 371 Experiment No. 3 ... 373 Experiment No. 4 ... 375 Experiment No. 5 ... 379 Experiment No. 6 ... 381 Experiment No. 7 ... 383 Experiment No. 8 ... 385 Experiment No. 9 ... 387 Experiment No. 10 ... 389 PREFACE TO THE THIRD EDITION This 2011-edition of the book “Applied Mechanics” has the following features: The book has been thoroughly revised and the following additions have been incorporated to make it still more useful for the readers, from examination point of view: • It has been completely converted into S.I. Units. • “Selected Questions from recent Examination Papers—With Solutions” have been added at the end of each chapter. Any suggestions for the improvement of this book will be thankfully acknowledged and incorporated in the next edition.
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