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Mechanics

Preface

Purpose of This Book (trial)
Approach and Philosophy (trial)
Mathematical Requirements (trial)
Importance of Classical Mechanics
How to Use This Book (trial)
Final Note (trial)

Introduction to Mechanics

What is Mechanics? (trial)
Importance of Mechanics (trial)
Branches of Mechanics (trial)
Idealizations in Mechanics (trial)
Space and Time (trial)
Reference Frames (trial)
Physical Quantities and Units (trial)
Dimensions (trial)
Dimensional Analysis (trial)
Mathematical Description of Motion (trial)
Visualization of Motion (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Mathematical Foundations of Mechanics

Introduction (trial)
Scalars and Vectors (trial)
Representation of Vectors (trial)
Vector Addition (trial)
Dot Product (Scalar Product) (trial)
Cross Product (Vector Product) (trial)
Unit Vectors (trial)
Direction Cosines (trial)
Differentiation of Vectors (trial)
Units and Dimensions (trial)
Dimensional Analysis (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Kinematics of Particles

Introduction (trial)
Position of a Particle (trial)
Displacement (trial)
Velocity (trial)
Acceleration (trial)
Motion with Constant Acceleration (trial)
Derivation of Equations of Motion (trial)
Graphical Representation of Motion (trial)
Projectile Motion (trial)
Important Results in Projectile Motion (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Newton’s Laws of Motion

Introduction (trial)
Concept of Force (trial)
Types of Forces (trial)
Newton’s First Law (trial)
Newton’s Second Law (trial)
Newton’s Third Law (trial)
Free Body Diagrams (FBD) (trial)
Friction (trial)
Tension (trial)
Motion on a Horizontal Surface (trial)
Motion on an Inclined Plane (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Work and Energy

Introduction (trial)
Work (trial)
Special Cases of Work (trial)
Work-Energy Theorem (trial)
Potential Energy (trial)
Conservative and Non-Conservative Forces (trial)
Conservation of Mechanical Energy (trial)
Power (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Systems of Particles and Linear Momentum

Introduction (trial)
Linear Momentum (trial)
Newton’s Second Law in Momentum Form (trial)
Impulse (trial)
System of Particles (trial)
Total Momentum of a System (trial)
Center of Mass (trial)
Motion of Center of Mass (trial)
Conservation of Linear Momentum (trial)
Collisions (trial)
Types of Collisions (trial)
One-Dimensional Collision (trial)
Coefficient of Restitution (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Rotational Motion and Angular Momentum

Introduction (trial)
Angular Variables (trial)
Relation Between Linear and Angular Quantities (trial)
Rotational Kinematics (trial)
Torque (trial)
Moment of Inertia (trial)
Newton’s Second Law for Rotation (trial)
Angular Momentum (trial)
Conservation of Angular Momentum (trial)
Rotational Kinetic Energy (trial)
Rolling Motion (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Gravitation

Introduction (trial)
Newton’s Law of Universal Gravitation (trial)
Gravitational Field (trial)
Acceleration Due to Gravity (trial)
Variation of Gravity (trial)
Gravitational Potential Energy (trial)
Gravitational Potential (trial)
Escape Velocity (trial)
Orbital Motion (trial)
Kepler’s Laws of Planetary Motion (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Oscillations

Introduction (trial)
Periodic Motion (trial)
Simple Harmonic Motion (SHM) (trial)
Solution of SHM (trial)
Velocity and Acceleration (trial)
Key Features of SHM (trial)
Energy in SHM (trial)
Graphical Representation (trial)
Mass-Spring System (trial)
Simple Pendulum (trial)
Damped Oscillations (trial)
Forced Oscillations and Resonance (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Non-Inertial Frames

Introduction (trial)
Inertial vs Non-Inertial Frames (trial)
Need for Fictitious Forces (trial)
Fictitious Force in Linear Acceleration (trial)
Example: Accelerating Elevator (trial)
Rotating Frames (trial)
Centrifugal Force (trial)
Coriolis Force (trial)
Motion in Rotating Frame (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Lagrangian Mechanics

Introduction (trial)
Generalized Coordinates (trial)
Degrees of Freedom (trial)
Constraints (trial)
The Lagrangian (trial)
Principle of Least Action (trial)
Euler-Lagrange Equation (trial)
Physical Meaning (trial)
Example: Free Particle (trial)
Example: Simple Harmonic Oscillator (trial)
Example: Simple Pendulum (trial)
Small Angle Approximation (trial)
Advantages of Lagrangian Mechanics (trial)
Conserved Quantities (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Hamiltonian Mechanics

Introduction (trial)
Generalized Coordinates and Momenta (trial)
The Hamiltonian (trial)
Hamilton’s Equations of Motion (trial)
Physical Meaning (trial)
Phase Space (trial)
Example: Free Particle (trial)
Example: Simple Harmonic Oscillator (trial)
Conservation of Energy (trial)
Canonical Variables (trial)
Advantages of Hamiltonian Mechanics (trial)
Hamiltonian for Multiple Particles (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Central Force Motion

Introduction (trial)
Definition of Central Force (trial)
Properties of Central Forces (trial)
Torque in Central Force (trial)
Conservation of Angular Momentum (trial)
Motion in Plane Polar Coordinates (trial)
Equation of Motion (trial)
Conservation of Angular Momentum (trial)
Effective Potential (trial)
Total Energy (trial)
Orbital Equation (trial)
Inverse Square Law Force (trial)
Kepler’s Laws Revisited (trial)
Circular Orbits (trial)
Stability of Circular Orbits (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Rigid Body Dynamics

Introduction (trial)
Types of Motion of a Rigid Body (trial)
Center of Mass Motion (trial)
Rotation About a Fixed Axis (trial)
Angular Momentum of Rigid Body (trial)
Kinetic Energy of Rigid Body (trial)
Parallel Axis Theorem (trial)
Perpendicular Axis Theorem (trial)
Rolling Motion (trial)
Energy in Rolling Motion (trial)
Angular Momentum About Different Points (trial)
Equilibrium of Rigid Body (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Small Oscillations and Normal Modes

Introduction (trial)
Equilibrium and Stability (trial)
Small Oscillations Approximation (trial)
Equation of Motion (trial)
Frequency of Small Oscillations (trial)
Systems with Two Degrees of Freedom (trial)
Coupled Oscillations (trial)
Normal Modes (trial)
Solving for Normal Modes (trial)
Example: Two Coupled Masses (trial)
Matrix Formulation (trial)
Eigenvalue Problem (trial)
Energy in Coupled Systems (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Introduction to Nonlinear Dynamics and Chaos

Introduction (trial)
Linear vs Nonlinear Systems (trial)
Characteristics of Nonlinear Systems (trial)
Phase Space Representation (trial)
Fixed Points (Equilibrium Points) (trial)
Stability Analysis (trial)
Example: Nonlinear Oscillator (trial)
Chaos (trial)
Sensitive Dependence (trial)
Example: Double Pendulum (trial)
Strange Attractors (trial)
Logistic Map (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Variational Principles and Applications

Introduction (trial)
The Concept of Action (trial)
Principle of Stationary Action (trial)
Variation of a Function (trial)
Derivation of Euler-Lagrange Equation (trial)
Euler-Lagrange Equation (trial)
Physical Interpretation (trial)
Example: Free Particle (trial)
Example: Harmonic Oscillator (trial)
Example: Motion in Gravitational Field (trial)
Generalized Coordinates (trial)
Cyclic Coordinates (trial)
Noether’s Theorem (Introductory Idea) (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Appendix: Mathematical Methods for Mechanics

Introduction (trial)
Functions and Graphs (trial)
Limits (trial)
Derivatives (trial)
Physical Interpretation (trial)
Integrals (trial)
Vector Algebra (trial)
Coordinate Systems (trial)
Differential Equations (trial)
Series Expansion (trial)
Dimensional Analysis (trial)
Trigonometric Identities (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Appendix: Problem-Solving Techniques in Mechanics

Introduction (trial)
General Strategy for Problem Solving (trial)
Choosing the Right Method (trial)
Free Body Diagrams (FBD) (trial)
Common Mistakes (trial)
Dimensional Checking (trial)
Estimation Techniques (trial)
Problem Types (trial)
Worked Example (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Appendix: Units, Constants, and Reference Data

Introduction (trial)
SI System of Units (trial)
Important Physical Constants (trial)
Prefixes (trial)
Unit Conversions (trial)
Dimensional Formulae (trial)
Useful Mathematical Constants (trial)
Approximation Values (trial)
Scientific Notation (trial)
Significant Figures (trial)
Error Estimation (trial)
Worked Example (trial)
Worked Example (trial)
Conceptual Insights (trial)
Summary (trial)
Practice Problems (trial)

Conclusion and Outlook

Introduction (trial)
Key Themes of the Book (trial)
From Classical to Modern Physics (trial)
Importance of Classical Mechanics (trial)
Applications of Mechanics (trial)
Skills Developed (trial)
Suggested Next Steps (trial)
Final Thoughts (trial)
Closing Summary (trial)
Practice Reflection (trial)

Mechanics → Central Force Motion → Total Energy

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Total Energy

$$ E = \frac{1}{2}m\dot{r}^2 + U_{\text{eff}} $$