Mechanics

Mechanics Meets The Requirement For An Ideal Text On Mechanics For Undergraduate Students. The Book Gives The Readers A Better Understanding Of Topics Like Rectiline Motion, Conservation Of Energy And Equation Of Motion. Provides A Good Number Of Examples With Good Use Real Time Illustration And Exercises For Practice And Challenge. The Book Comprehensively Covers Of Newton's Law Of Motion, Conservation Laws Of Momentum, Energy And Law Of Gravitation And Includes 180 Worked Out Examples And 185 End Of Chapter Exercises. Cover......Page 1 Contents......Page 6 Preface......Page 14 1.1 Mechanics, the Science of Motion......Page 20 1.2 Time Evolution of Coordinates......Page 22 1.3 Galileo’s Law of Inertia, Newton’s First Law of Motion......Page 23 1.4 Experimental Verification of the First Law......Page 25 1.5 Inertial Frame of Reference......Page 26 1.6 In Search of Conservation Laws......Page 28 1.7 Measure of Inertia, Inertial Mass......Page 29 2.1 Displacement-Time Graph......Page 32 2.2 Velocity of a Particle......Page 35 2.3 Acceleration......Page 38 2.4 Simple Harmonic Motion......Page 42 2.5 Worked out Examples I......Page 43 2.6 Obtaining V,T and X,T Relations from Areas of A-T and V-T Graphs......Page 46 2.7 Standard Kinematical Relations for Constant Acceleration......Page 48 2.8 Velocity and Displacement for a Harmonically Varying Acceleration......Page 49 2.9 Worked Out Examples II......Page 50 Exercises......Page 52 3.1 Knowing Vectors by their Properties......Page 54 3.2 Is Vector Just a Directed Straight line?......Page 57 3.3 Mathematical Representation of Vectors......Page 58 3.4 The Displacement Vector......Page 60 3.5 Magnitude of a Vector......Page 62 3.6 Radius Vector as a Function of Time......Page 63 3.7 The Velocity Vector......Page 65 3.8 Infinitesimal Displacement, Line Element, Speed......Page 69 3.9 Acceleration......Page 70 3.10 Worked Out Examples I......Page 73 3.11 Centripetal Acceleration in Uniform Circular Motion......Page 76 3.12 Combination of Normal and Tangential Accelerations in Non-Uniform Circular Motion......Page 78 3.13 Worked Out Examples II......Page 80 3.14.1 Component of a Vector in a Given Direction; Scalar Product......Page 81 3.14.2 Vector Product is an Axial Vector......Page 84 3.15.1 Volume as a Triple Product......Page 86 3.15.2 Determinant of a Cross Product and of a Scalar Triple Product......Page 87 3.16 Worked Out Examples III......Page 89 Summary......Page 90 Exercises......Page 92 4.1 Galilean Transformation......Page 94 4.2 Momentum in One Dimension. Definition of Mass......Page 95 4.3 Conservation of Linear Momentum......Page 97 4.4 Invariance of Momentum Conservation Under Galilean Transformation......Page 98 4.5.1 Example 1: Velocity of a Large Block After Being Hit by Bullets......Page 100 4.5.2 Example 2: Recoil Velocity of a Cannon......Page 101 4.6 Propulsion of a Rocket......Page 103 4.7 Worked Out Examples. Set I......Page 106 4.8 When there is a Flow of Momentum......Page 107 4.9 Momentum Conservation from a Comoving Frame of Reference......Page 109 Summary......Page 111 Exercises......Page 112 5.1 How a Force Alters the Momentum of a Particle......Page 114 5.2 Equations of Motion, and how to Solve them......Page 115 5.3 Can the Second Law be Applicable to Extended Objects?......Page 119 5.4 Forces of Nature we Shall Reckon with......Page 120 5.5.1 Velocity-independent Nature of the Force of Gravity......Page 123 5.5.2 Stone Thrown from the Top of a Tower......Page 125 5.5.3 Motion of a Projectile......Page 126 5.5.4 Equation of a Parabola......Page 129 5.6 Worked Out Examples. Set I......Page 130 5.7 Motion Against Resistive Forces, Dry Friction......Page 139 5.8 Worked Out Problems. Set II......Page 141 5.9.1 Aerodynamic Drag, Terminal Velocity......Page 142 5.9.2 Example of Terminal Velocity – Millikan’s Experiment to Find Electronic Charge......Page 146 5.10 Worked Out Problems. Set III......Page 148 5.11 Dynamics of a Spring Mass System......Page 151 5.11.1 Motion Under a Pure Spring Force......Page 153 5.11.2 Spring Force in Combination with Gravity......Page 154 5.11.3 Vertical Oscillation of a Ship......Page 155 5.12 Worked Out Problems. Set IV......Page 156 5.13 Simple Harmonic Motion in Two Perpendicular Directions, Lissajous Figures......Page 158 5.14.1 The Equation of Motion......Page 162 5.14.2 Horizontal Propulsion of a Jet Plane......Page 164 5.14.3 Vertical Propulsion of a Rocket......Page 165 5.14.4 A Raindrop Falling through the Atmosphere......Page 166 5.15 Worked Out Problems. Set V......Page 167 5.16 Motion Under Electromagnetic Forces......Page 169 5.16.1 Charged Particle in a Uniform Electric Field......Page 171 5.17 Worked Out Problems. Set VI......Page 174 5.18 The Second Lawapplied to Uniform Circular Motion......Page 175 5.18.1 Example 1: Banking of Road Surface for Fast Moving Vehicles......Page 176 5.18.2 Example 2: Conical Pendulum......Page 178 5.18.3 Example 3: A Satellite in a Circular Orbit......Page 179 5.18.4 Example 4: Calculation of Bohr Radius......Page 180 5.19 Worked Out Examples. Set VII......Page 181 5.20 Geometrical Structure of the Second Law Exemplified by Force Perpendicular to Velocity......Page 187 5.21.1 Cyclotron Frequency......Page 190 5.21.2 General Solution: Helical Motion......Page 191 5.22 Simple Pendulum......Page 195 5.22.1 Complete Equations of Motion......Page 196 5.22.2 Simplified Solution. 1st-Order Approximation......Page 197 5.22.4 Exact Expressions for T, ac , at......Page 198 5.22.5 2nd-Order Approximation......Page 199 Summary......Page 200 Exercises......Page 203 6.1 Newton and the Apple and the Moon......Page 206 6.2.1 Motion of Planets as Seen from Earth – Geocentric View of the Greek School......Page 208 6.2.3 Geocentric Path of Venus from the Copernican model......Page 210 6.2.4 Geocentric Path of Mars from the Copernican Model......Page 214 6.2.5 Calculation of the Periods of the Planets by Copernicus......Page 217 6.2.6 Calculation of the Orbital Radii of the Planets by Copernicus......Page 219 6.3 Kepler’s Struggle with Mars......Page 222 6.4.1 Kepler’s Laws of Planetary Motion......Page 225 6.5 The Law of Universal Gravitation......Page 229 6.6.1 The Force of Gravitation between Two Point Objects in Vector Notation......Page 232 6.6.2 Principle of Superposition......Page 233 6.7.1 A Source and a Test Particle......Page 235 6.7.2 Gravitational Field Lines......Page 237 6.8.1 Finding the Field by Volume and Surface Integration......Page 238 6.8.2 An Elementary Introduction to the Spherical Coordinate System......Page 239 6.8.3 Field Due to a Spherical Shell of Uniform Surface Mass Density......Page 242 6.8.4 Field Due to a Spherically Symmetrical Distribution of Matter......Page 245 6.8.5 Force of Interaction between Two Spherically Symmetric Mass Distributions......Page 246 6.9.1 Relationship between Orbital Radius and the Period of Revolution......Page 249 6.9.2 Geostationary Satellites......Page 251 6.10.1 Examples of Free Fall......Page 253 6.10.2 Tidal Deformation......Page 255 6.10.3 Tidal Force......Page 256 6.10.4 Tidal Forces on a Sphere......Page 258 6.11 Summary......Page 261 6.12 Worked Out Problems......Page 262 6.13 Appendix 6A: Explaining the Null Field Inside a Spherical Shell......Page 268 Exercises......Page 270 7.1 A Slide Show on Newton’s Third Law of Motion......Page 272 7.2.1 What is a Freebody Diagram?......Page 279 7.2.2 The Golden Rules for Understanding Newton’s Laws of Motion......Page 280 7.3 Further Examples of FBDS......Page 283 7.4 Every Real Force Has a Parent......Page 288 7.5 Worked Out Problems......Page 289 Exercises......Page 297 8.1 Work and Kinetic Energy......Page 300 8.2 Example of Work–Work Done by the Uniform Force of Gravity – Near the Earth’s Surface......Page 303 8.3 Example of Work–Work Done by the Inverse Square Force of Gravity......Page 305 8.4 Power – the Rate of Doing Work......Page 307 8.5 Example of Work – the Spring Mass System......Page 309 8.6 Example of Work – Work Done by Electrostatic Forces......Page 311 8.7 Conservative and Non-Conservative Forces......Page 314 8.8 The Concept of Potential Energy – Example Spring......Page 316 8.9 Potential Energy in General......Page 320 8.11 Energy Conservation in a Spring Mass System......Page 325 8.13 Energy Conservation of a Particle Freely Falling Under the Gravitational Pull of the Earth (or the Sun)......Page 328 8.13.1 Example 1. An Object Thrown Upward with Velocity v0......Page 330 8.13.2 Example 2. Velocity of a Particle Falling Vertically from a Height h Above the Surface of the Earth......Page 331 8.14 Energy Conservation of a Charged Particle Moving in an Electrostatic Field......Page 332 8.15 Work and Energy in Rocket Propulsion......Page 334 8.17 Worked Out Problems......Page 338 Exercises......Page 352 9.1 Plane Polar Co-Ordinate System......Page 354 9.2.1 The Velocity v(r, θ)......Page 357 9.2.3 Derivatives of the Base Vectors......Page 358 9.2.4 Acceleration a(r, θ)......Page 359 9.3.1 Orbital Angular Momentum, an Axial Vector......Page 360 9.3.3 Conservation of Orbital Angular Momentum under Central Forces......Page 362 9.3.4 Kepler’s 2nd Law: Conservation of Areal Velocity......Page 363 9.4.1 Radial and Transverse Components of the Equation of Motion......Page 364 9.4.2 Motion Under a Central Force – the First Integrals of the Equation of Motion......Page 365 9.4.3 Second Integrals of the Equation of Motion; Effective Potential for Radial Motion......Page 367 9.5 Motion Under an Inverse-Square-Law Attractive Force......Page 371 9.6 Classification of Trajectories in an Inverse-Square-Law Field–Kepler’s 1st Law of Planetary Orbit......Page 374 9.7 Kepler’s Third Law of Planetary Orbits......Page 378 9.8 A Closer Look at Planetary (Satellite) Orbits......Page 379 9.9 The Parabolic Trajectory of a Projectile is Part of an Elliptical Orbit......Page 381 9.10 Motion Under an Inverse-Square-Law Repulsive Force......Page 385 9.11.1 What is a Conic Section......Page 389 9.11.2 Plotting Conic Sections using “Gnuplot”.......Page 390 9.11.3 Polar and Cartesian Equations of an Ellipse from a Special Property......Page 391 9.11.4 Polar and Cartesian Equations of a Hyperbola from a Special Property......Page 393 9.11.5 Polar Equations for the Asymptotes......Page 395 9.11.6 A Simplified View of a Hyperbola......Page 396 9.12 Summary......Page 397 9.13 Worked Out Problems......Page 399 Exercises......Page 415 10.1 Normal and Tangential Accelerations......Page 418 10.2 Effect of Force Acting Over a Displacement–General Case......Page 421 10.3 Evaluation of the Line Integral......Page 422 10.4 Example of Evaluation of W–Work Done by the Force of Gravity......Page 425 10.5 Example of Evaluation of W–Work Done by the Induced Electric Force......Page 429 10.6 Relationship of Work to Kinetic Energy......Page 433 10.7 Potential Energy of a Particle in a Conservative Field......Page 436 10.9.1 Bob Connected to a Fixed Point with a String......Page 438 10.9.2 Roller Coaster......Page 440 10.9.3 Cart Rolling Down a Chute Having the Shape of a Conic Section......Page 441 10.9.4 Cart Rolling Down a Dome having the Shape of a Conic Section......Page 444 10.10 The Example of a Betatron......Page 448 10.11.1 Using Cartesian coordinates......Page 451 10.11.2 Using Polar Coordinates......Page 452 10.11.3 Simplest Examples: Circle, Parabola......Page 453 10.11.4 Example: A General Conic Section......Page 454 10.12 Summary of Important Formulas......Page 456 10.13 Worked Out Problems......Page 457 Exercises......Page 459 11.1.1 Pressure and Density......Page 460 11.1.2 Pressure is Isotropic......Page 462 11.2.1 Atmospheric Pressure and Hydrostatic Pressure......Page 464 11.2.2 Important Corollaries of the Pressure Equation......Page 466 11.3.2 U-tube Manometer......Page 468 11.3.4 Hydraulic Devices......Page 469 11.4 Archimedes’ Principle......Page 470 11.5 Worked Out Examples. Set I......Page 472 11.6.1 Equation of Continuity in Steady Flow......Page 474 11.6.2 Bernoulli’s Principle......Page 476 11.6.3 Simple Applications of Bernoulli’s Principle. Flow Through Pipe, Venturi Meter, Pitot Tube, Forces on Airfoil, Cricket Ball......Page 477 11.7 Poiseulle’s Law......Page 482 11.9 Worked Out Examples. Set II......Page 484 Exercises......Page 486 12.1 Linear Bulk Motion of a System of Particles–Conservation of Momentum......Page 488 12.2 Rotational Motion of a System of Particles–Conservation of Angular Momentum......Page 491 12.3.2 For a Continuous Mass Distribution......Page 494 12.3.3 Useful Theorems......Page 497 12.4 Worked Out Examples II......Page 499 12.5 Breaking up Dynamical Variables with a Component in the CM Frame......Page 501 12.5.1 Angular Mometum and Kinetic Energy of a System of Particles......Page 502 12.5.2 Torque of a Set of Forces Acting on a System of Particles......Page 503 12.6.1 Moment of a Force and the Angular Momentum of a Particle about an Axis......Page 504 12.6.2 Angular Velocity......Page 507 12.6.3 Equation of Rotational Motion about a Fixed Axis......Page 508 12.6.4 Kinetic Energy of Rotation about a Fixed Axis......Page 509 12.6.5 Moment of Inertia. Worked Out Examples III......Page 510 12.6.6 Moment of Inertia. Useful Theorems......Page 513 12.6.7 Moment of Inertia.Worked Out Examples IV......Page 515 12.6.8 Table of Moments of Inertia......Page 517 12.7.1 Flywheel Spun by a Falling Weight......Page 519 12.7.2 Wheel Rolling Down a Slope......Page 521 12.7.3 Compund Pendulum......Page 523 12.8 Angular Momentum is a Vector Quantity......Page 526 12.9 The Amazing Gyroscope, the Spinning Top......Page 532 12.9.1 Angular Momentum of a Spinning Top......Page 533 12.9.2 Qualitative Understanding of Precession......Page 534 12.9.3 Calculation of the Rate of Precession......Page 537 12.9.4 Worked Out Examples V: Rotational Motion of a Rigid Body......Page 538 12.10 The 2-Body Problem......Page 540 12.10.1 Reduced Mass, Motion of the Reduced Particle......Page 541 12.10.3 Worked Out examples VI......Page 544 12.11.1 Impulse, Elastic and Inelastic Collisions......Page 547 12.11.2 Examples of Simple Collision......Page 548 Summary......Page 550 Exercises......Page 552 13.1.1 Transformation of Acceleration......Page 554 13.1.2 Origin of Pseudo Force......Page 556 13.2.1 Transformation of the Rate of Change of a Vector from a Rotating Frame to a Fixed Frame......Page 557 13.2.3 Origin of Other Pseudo Forces. Cetrifugal Force and Coriolis Force......Page 559 13.3.1 Effective Force of Gravity......Page 562 13.3.2 Deflection of a Falling Particle......Page 564 13.3.3 Foucault Pendulum......Page 565 13.4 Principle of Equivalence, Inertial Force and Pseudo-Gravity......Page 568 13.5 Worked Out Problems......Page 570 Summary......Page 575 Exercises......Page 576 14.1 Event Point in Space-Time......Page 578 14.2.1 Search for the Absolute Frame......Page 580 14.2.2 Michelson-Morley Experiment......Page 582 14.3.1 Einstein’s Postulates......Page 584 14.3.2 Relativity of Simultaneity......Page 585 14.3.3 Time Dilation......Page 586 14.3.4 Length Contraction......Page 589 14.3.5 Lorentz Transformation......Page 592 14.3.6 Simple Applications of Lorentz Transformation......Page 596 14.4 Worked Out Examples I......Page 597 14.5 Relativistic Mechanics......Page 602 14.5.1 The Velocity Addition Formula......Page 603 14.5.2 A New Definition of Momentum......Page 608 14.5.3 Force......Page 609 14.5.4 Energy......Page 613 14.5.5 Energy– Momentum Conservation Law......Page 617 14.6 Worked Out Examples II......Page 620 Summary......Page 622 Exercises......Page 623 A.1 Why Should a Beam Bend?......Page 624 A.2 Hook’s Law......Page 625 A.3 Bending Moment......Page 626 A.4 The Flexure Formula......Page 628 A.5 Deflection of Beams......Page 630 A.6 Conclusion......Page 632 B.1 To Start with......Page 634 B.2 Example 1: The Parabolic Trajectory of a Projectile......Page 635 B.4 Example 3: Displacement, Velocity and Acceleration of a Piston Driven by a Crank Wheel......Page 637 B.5 Example 4: Plotting Polar Equations of Conic Sections......Page 638 B.6 Example 5: Precessing Ellipse......Page 639 B.7 Example 6: Plotting the G-Path of MARS......Page 640 B.8 Example 7: Lissajous Figures......Page 641 Bibliography......Page 642 Index......Page 646 Mechanics meets the requirement for an ideal text on Mechanics for undergraduate students. The book gives the readers a better understanding of topics like Rectilinear Motion, Conservation of Energy and Equation of Motion, provides a good number of examples with good use of real-time illustrations and exercises for practice and challenge. The book comprehensively covers topics like Newton's Law of Motion, Conservation Laws of momentum, Energy and Law of Gravitation and includes 180 worked-out examples and 185 end-of-chapter exercises