book

Mechanics

Name: Mechanics
Author: Somnath Datta
ISBN: 9788131773734

by Somnath Datta

June 2012

Intermediate to advanced

644 pages

18h 11m

English

Pearson India

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Cover
Title Page
Contents
Dedication
Preface
1 ■ Introduction
1.1 Mechanics, the Science of Motion1.2 Time Evolution of Coordinates1.3 Galileo’s Law of Inertia, Newton’s First Law of Motion1.4 Experimental Verification of the First Law1.5 Inertial Frame of Reference1.6 In Search of Conservation Laws1.7 Measure of Inertia, Inertial Mass
2 ■ Velocity and Acceleration in Rectilinear Motion
2.1 Displacement-Time Graph2.2 Velocity of a Particle2.3 Acceleration2.4 Simple Harmonic Motion2.5 Worked Out Examples I2.6 Obtaining, T and X, Trelations from Areas of A-T and V-T Graphs2.7 Standard Kinematical Relations for Constant Acceleration2.8 Velocity and Displacement for a Harmonically Varying Acceleration2.9 Worked Out Examples II■ Summary■ Exercises
3 ■ Vectors in Physics. Velocity and Acceleration as Vectors
3.1 Knowing Vectors by their Properties3.2 Is Vector Just a Directed Straight Line?3.3 Mathematical Representation of Vectors3.4 The Displacement Vector3.5 Magnitude of a Vector3.6 Radius Vector as a Function of Time3.7 The Velocity Vector3.8 Infinitesimal Displacement, Line Element, Speed3.9 Acceleration3.10 Worked Out Examples I3.11 Centripetal Acceleration in Uniform Circular Motion3.12 Combination of Normal and Tangential Accelerations in Non-uniform Circular Motion3.13 Worked Out Examples II3.14 Multiplication of Two Vectors3.15 Multiplication of Three Vectors3.16 Worked Out Examples III■ Summary■ Exercises
4 ■ Conservation of Momentum
4.1 Galilean Transformation4.2 Momentum in One Dimension. Definition of Mass4.3 Conservation of Linear Momentum4.4 Invariance of Momentum Conservation Under Galilean Transformation4.5 Illustrative Examples of Momentum Conservation4.6 Propulsion of a Rocket4.7 Worked Out Examples. Set I4.8 When There is a Flow of Momentum4.9 Momentum Conservation from a Comoving Frame of Reference■ Summary■ Exercises
5 ■ Newton’s Second Law of Motion
5.1 How a Force Alters the Momentum of a Particle5.2 Equations of Motion, and How to Solve Them5.3 Can the Second Law be Applicable to Extended Objects?5.4 Forces of Nature We Shall Reckon With5.5 Motion Under Gravity Near the Surface of the Earth5.6 Worked Out Examples. Set I5.7 Motion Against Resistive Forces, Dry Friction5.8 Worked Out Problems. Set II5.9 Motion Against Resistive Forces, Fluid Friction5.10 Worked Out Problems. Set III5.11 Dynamics of a Spring Mass System5.12 Worked Out Problems. Set IV5.13 Simple Harmonic Motion in Two Perpendicular Directions, Lissajous Figures5.14 The Second Law Applied to a System of Varying Mass5.15 Worked Out Problems. Set V5.16 Motion Under Electromagnetic Forces5.17 Worked Out Problems. Set VI5.18 The Second Law Applied to Uniform Circular Motion5.19 Worked Out Examples. Set VII5.20 Geometrical Structure of the Second Law Exemplified by Force Perpendicular to Velocity5.21 Motion of a Charged Particle Moving in a Uniform Magnetic Field5.22 Simple Pendulum■ Summary■ Exercises

6 ■ The Law of Universal Gravitation
Part I: A Brief History of Gravitation6.1 Newton and the Apple and the Moon6.2 Heliocentric Model of Copernicus6.3 Kepler’s Struggle with Mars6.4 Kepler’s Third Law - Key to Inverse Square6.5 The Law of Universal GravitationPart II: Gravitational Field6.6 The Gravitational Force Between Two Extended Objects6.7 Gravitational Field6.8 Direct Computation of the Gravitational Field6.9 Satellites in Circular Orbits6.10 Free Fall and Tidal Acceleration6.11 Summary6.12 Worked Out Problems6.13 Appendix 6A: Explaining the Null Field Inside a Spherical Shell■ Exercises
7 ■ Newton’s Third Law of Motion
7.1 A Slide Show on Newton's Third Law of Motion7.2 Free Body Diagrams7.3 Further Examples of FBDs7.4 Every Real Force has a Parent7.5 Worked Out ProblemsExercises
8 ■ Work and Energy in One Dimensional Motion
8.1 Work and Kinetic Energy8.2 Example of Work – Work Done by the Uniform Force of Gravity – Near the Earth’s Surface8.3 Example of Work – Work Done by the Inverse Square Force of Gravity8.4 Power – The Rate of Doing Work8.5 Example of Work – The Spring Mass System8.6 Example of Work – Work Done by Electrostatic Forces8.7 Conservative and Non-Conservative Forces8.8 The Concept of Potential Energy – Example Spring8.9 Potential Energy in General8.10 Total Energy of a Particle in a Conservative Field8.11 Energy Conservation in a Spring Mass System8.12 Concept of a Potential Well8.13 Energy Conservation of a Particle Freely Falling Under the Gravitational Pull of the Earth (or the Sun)8.14 Energy Conservation of a Charged Particle Moving in an Electrostatic Field8.15 Work and Energy in Rocket Propulsion8.16 Summary of Important Formulas8.17 Worked Out Problems■ Exercises
9 ■ Motion Under Central Forces
9.1 Plane Polar Co-ordinate System9.2 Velocity and Acceleration of a Particle in the Polar System9.3 Orbital Angular Momentum9.4 Equations of Motion in the Polar Coordinate System9.5 Motion Under an Inverse-Square-Law Attractive Force9.6 Classification of Trajectories in an Inverse-Square-Law Field – Kepler’s 1st Law of Planetary Orbit9.7 Kepler’s Third Law of Planetary Orbits9.8 A Closer Look at Planetary (Satellite) Orbits9.9 The Parabolic Trajectory of a Projectile is Part of an Elliptical Orbit9.10 Motion Under an Inverse-Square-Law Repulsive Force9.11 Appendix 9A: Conic Sections – Ellipse, Parabola, Hyperbola9.12 Summary9.13 Worked Out Problems■ Exercises
10 ■ Work and Energy in 3-Dimensional Motion
10.1 Normal and Tangential Accelerations10.2 Effect of Force Acting Over a Displacement – General Case10.3 Evaluation of the Line Integral10.4 Example of Evaluation of W – Work Done by the Force of Gravity10.5 Example of Evaluation of W – Work Done by the Induced Electric Force10.6 Relationship of Work to Kinetic Energy10.7 Potential Energy of a Particle in a Conservative Field10.8 Total Energy of a Particle in a Conservative Field10.9 How Energy Determines the Forces of Constraint10.10 The Example of a Betatron10.11 Appendix 10A: How to Calculate the Radius of Curvature10.12 Summary of Important Formulas10.13 Worked Out Problems■ Exercises
11 ■ Ideal Fluid at Rest and in Motion
11.1 Stresses Inside a Fluid11.2 Hydrostatic Pressure Equation11.3 Hydrostatic Devices11.4 Archimedes’ Principle11.5 Worked Out Examples. Set I11.6 Elementary Mechanics of Fluid Flow11.7 Poiseulle’s Law11.8 Summary of Important Formulas11.9 Worked Out Examples. Set II■ Exercises
12 ■ Motion of a System of Particles. Rigid Body Rotating about a Fixed Axis
12.1 Linear Bulk Motion of a System of Particles – Conservation of Momentum12.2 Rotational Motion of a System of Particles – Conservation of Angular Momentum12.3 Examples of Centre of Mass. Worked Out examples I12.4 Worked Out Examples II12.5 Breaking Up Dynamical Variables With a Component in the CM Frame12.6 Rotation of a Rigid Body About a Fixed Axis12.7 Simplest Examples of Rigid Body Dynamics12.8 Angular Momentum is a Vector Quantity12.9 The Amazing Gyroscope, the Spinning Top12.10 The 2-Body Problem12.11 Collision of Two Particles■ Summary■ Exercises
13 ■ Accelerating and Rotating Frames of Reference
13.1 Linearly Accelerating Frame of Reference13.2 Rotating Frame of Reference13.3 Pseudo Forces Near the Earth’s Surface13.4 Principle of Equivalence, Inertial Force and Pseudo-Gravity13.5 Worked Out Problems■ Summary■ Exercises
14 ■ Relativistic Mechanics
14.1 Event Point in Space-Time14.2 Historical Background14.3 The Postulates of Relativity and Consequences14.4 Worked Out Examples I14.5 Relativistic Mechanics14.6 Worked Out Examples II■ Summary■ Exercises
Appendix A ■ Beam Bending and Deflection Formulas
A.1 Why Should a Beam Bend?A.2 Hook’s LawA.3 Bending MomentA.4 The Flexure FormulaA.5 Deflection of BeamsA.6 Conclusion
Appendix B ■ Instructions for Gnuplot
B.1 To Start WithB.2 Example 1: The Parabolic Trajectory of a ProjectileB.3 Example 2: Gun Intercepting a BombB.4 Example 3: Displacement, Velocity and Acceleration of a Piston Driven by a Crank WheelB.5 Example 4: Plotting Polar Equations of Conic SectionsB.6 Example 5: Processing EllipseB.7 Example 6: Plotting the G-Path of MarsB.8 Example 7: Lissajous Figures
Notes
Chapter 1
Chapter 2
Chapter 3
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Bibliography
Acknowledgements
Copyright

Content preview from Mechanics

Chapter 9 Motion Under Central Forces

Central forces enjoy a special place in physics. The most prominent forces in nature, namely, the force of gravitation between two spherically symmtric massive bodies (e.g., a star and its planet) and the electrostatic force (Coulomb force) between two point charges in slow motion ¹ are central forces.

By the term “central force” we mean a force F(r) acting on a particle P which is (a) always along the line CP joining P to some centre C (Fig. 9.1a), and which is (b) a function only of the radial distance r measured fron the centre C. The function F(r) is independent of the angle θ that the radius vector r makes with some arbitray direction in space.

FIGURE 9.1 Examples of a central force: (a) A schematic ...

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Mechanics

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Chapter 9

Motion Under Central Forces

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