CONTENTS Preface xi 1 Forces and Moments1 1.1 Introduction 1 1.2 Scalars, Vectors and Tensors 2 1.3 Fundamental Concepts and Operations 2 1.3.1 Force 2 1.3.2 Couple and moment 2 1.3.3 Resultant 2 1.3.4 Parallelogram law 3 1.3.5 Triangle law 4 1.4 Systems of Forces 4 1.4.1 Coplanar forces 4 1.4.2 Collinear forces 4 1.4.3 Coplanar concurrent forces 5 1.4.4 Parallel forces 6 1.4.5 Non-concurrent, non-parallel forces 6 1.4.6 Distributed forces 7 1.5 Moment of a Force 11 1.5.1 Varignon's theorem of moments 11 1.5.2 Moment of a couple 12 1.6 Equilibrium of Force Systems 12 1.6.1 Equilibrium equations for different coplanar force systems 13 1.7 Analysis of Spatial Force Systems 14 1.7.1 Concurrent -riatial forces 14 1.7.2 Parallel spatial forces 14 1.7.3 Non-concurrent, non-parallel spatial forces 15 1.8 Action, Reaction and Free-Body Diagram 15 Exercises 30 2. Friction 46 2.1 Introduction 46 2.2 Theory of Friction 46 2.3 General Concepts of Friction 46 2.4 Laws of Friction 48 2.5 Analysis of Friction of Wedges 60 2.6 Belt Friction 63 2.6.1 Analysis of a belt drive 63 2.7 Transmission of Power through Belt Drives 65 2.7.1 Open-belt drive 65 2.7.2 Crossed-belt drive 68 2.7.3 Compound-belt drive 69 2.8 Transmission of Power through Rope and V-Belt Drives 72 2.9 Centrifugal Tension 74 2.10 Initial Tension in the Belt 75 2.11Computation of Power Transmitted by Belt and Rope Drives 75 2.11.1Computation of maximum power transmitted by belt or rope drives 75 2.12 Lifting Machines 80 2.12.1Basic definitions 80 2.12.2 Ideal machine 81 2.12.3 Ideal effort 81 2.12.4 Ideal load 81 2.12.5 Frictional losses 81 2.12.6 Reversible and irreversible machines 82 2.12.7 The law of a machine 82 2.12.8 Maximum mechanical advantage 82 2.12.9Maximum efficiency 83 2.13 Principle of a Screw Jack 84 2.13.1 Simple screw jack 85 2.13.2 Differential screw jack 87 2.14 Pulley systems 91 2.14.1Single-pulley system 91 2.14.2 Multiple-pulley system 91 2.14.3 First system of pulleys 92 2.14.4 Second system of pulley 92 2.14.5 Third system of pulley 93 2.14.6 Weston differential pulley block 95 2.15 Other Lifting Machines 97 2.15.1 Simple wheel and axle 97 2.15.2 Differential wheel and axle 98 2.15.3 Worm and worm-wheel arrangement 98 2.15.4 Single-purchase crab winch 99 2.15.5 Dobule-purchase crab winch 100 Exercises 102 3. Centre of Gravity and Moment of Intertia 121 3.1Introduction 3.2 Centre of Gravity of a Body 3.3 Location of Centroids 3.3.1 Centroids of a volume 3.3.2 Centroid of an area 3.3.3 Centroid of a line 3.4 Theorems of Pappus and Guldinus 124 3.4.1 First theorem 124 3.4.2 Second theorem 125 3.5 Centroids of Standard Geometric Configurations 140 3.6 Area Moment of Inertia 143 3.6.1 Polar moment of inertia: Perprendicular axis theorem 143 3.6.2 Radius of gyration 144 3.6.3 Transfer of moment of inertia from one axis to another: Parallel axis theorem 144 3.7 Area Moments of Inertia of Standard Geometric Configurations 152 3.8 Mass Moment of Inertia 152 3.8.1 Mass radius of gyration 154 3.8.2 Transfer formula for mass moment of inertia 155 3.8.3 Relation between area MI and mass MI 155 3.9 Mass Moments of Inertia of Standard Geometric Bodies 160 Exercises 162 4. Kinematics 178 4.1 Introduction 178 4.2 Rectilinear Motion 178 4.2.1 Displacement 179 4.2.2 Distance 179 4.2.3 Velocity 179 4.2.4 Acceleration and retardation 179 4.3 Graphical Representation 180 4.3.1 Displacement-time curve (x - t curve) 180 4.3.2 Velocity-time curve (v - t curve) 180 4.3.3 Acceleration-time curve (a - t curve) 181 4.4 Motion with Uniform Acceleration 181 4.5 Motion with Variable Acceleration 182 4.6 Curvilinear Motion 190 4.7 Position Vector, Velocity and Acceleration, 190 4.7.1 Position Vectors 190 4.7.2 Velocity 190 4.7.3 Acceleration 191 4.8 Components of Motion 191 4.8.1 Rectangular components of velocity 191 4.8.2 Rectangular components of acceleration 192 4.8.3 Normal and tangential components of acceleration 193 4.9 Circular Motion 195 4.9.1 Angular velocity and angular acceleration 195 4.9.2 Relation between linear velocity and angular velocity 195 4.9.3 Relation between linear acceleration and angular acceleration 195 4.9.4 Equation of motion along a circular path 196 4.10 Projectiles 200 4.11 Motion of a body projected with an Inclination 200 4.11.1 Equation of trajectory 200 4.11.2 Time, height and range of projection 201 4.11.3 Angle of projection 202 4.12 Motion of a Body Projected Horizontally 202 4.13 Motion of a Projectile Up or Down an Inclined Plane 202 4.14 Kinematics of a Rigid Body 212 4.14.1 Absolute and relative velocities in plane motion 213 4.15 Instantaneous Centre in Plane Motion 214 4.15.1 Location of instantaneous centre in special cases 215 Exercises 222 5. Kinetics 240 5.1 Introduction 240 5.2 Kinetics of a Particle 240 5.2.1D' Alembert's Principle 241 5.3 Motion in a Curved Path 252 5.3.1Motion of vehicle on a level curved path 253 5.3.2 Motion of a vehicle on a banked curved path 254 5.4 Work and Power 256 5.4.1 Work 256 5.4.2 Enegy 256 5.4.3 Power 257 5.4.4 Conservative force 257 5.5 Principle of Conservation of Energy 257 5.6 Principle of Work-Energy of a Particle 262 5.7 Kinetics of a Rigid Body 266 5.7.1 Kinetic energy of a rigid body 266 5.7.2 Kinetic energy in translation 266 5.7.3 Kinetic energy in rotation 266 5.7.4 Work done on a rigid body 267 5.7.5 Forces neglected in work-energy analysis of rigid body motion 267 5.8 Principle of Work-Energy for a Rigid Body 268 5.8.1Principle of conservation of energy for a rigid body 268 5.9 Impulse and Momentum 272 5.9.1 Law of conservation of momentum 272 5.10 Angular Momentum 273 5.10.1 Conservation of angular momentum 274 5.11 Rotary Motion 274 5.12 Collision of Bodies 278 5.12.1 Types of impact 279 5.12.2 Analysis of direct impact 279 5.12.3 Analysis of indirect impact 280 5.12.4 Coefficient of restitution 281 Exercises 287 6. Additional Topics in Dynamics 304 6.1 Introduction 304 6.2 Simple Harmonic Motion (SHM) 304 6.3 Difinitions of SHM 305 6.3.1 Amplitude 305 6.3.2 Time period 305 6.3.3 Frequency 305 6.4 Important Relations of SHM 305 6.4.1 Damping 307 6.5 Mechanical Vibrations 309 6.6 Pendulum Motion 317 6.7 Simple Pendulum 317 6.7.1 Beat of a pendulum 318 6.7.2 Second's pendulum 318 6.7.3 Gain or loss of a simple pendulum 318 6.8 Compound Pendulum 319 6.9 Torsion pendulum 320 6.10 Theory of Virtual Work 323 6.11 Applications of the Principle of Virtual Work 323 6.11.1 Application to beams 323 6.11.2 Application to framed structures 326 6.11.3 Application to lifting machines 330 Exercises 331 Index 346