Introduction to Mechanics of Particles and Systems

This book is based on the author’s lecture notes for his Introductory Newtonian Mechanics course at the Hellenic Naval Academy. In order to familiarize students with the use of several basic mathematical tools, such as vectors, differential operators and differential equations, it first presents the elements of vector analysis that are needed in the subsequent chapters. Further, the Mathematical Supplement at the end of the book offers a brief introduction to the concepts of differential calculus mentioned. The main text is divided into three parts, the first of which presents the mechanics of a single particle from both the kinetic and the dynamical perspectives. The second part then focuses on the mechanics of more complex structures, such as systems of particles, rigid bodies and ideal fluids, while the third part consists of 60 fully solved problems. Though chiefly intended as a primary text for freshman-level physics courses, the book can also be used as a supplemental (tutorial) resource for introductory courses on classical mechanics for physicists and engineers Preface 5 Contents 6 1 Vectors 9 1.1 Basic Notions 9 1.2 Rectangular Components of a Vector 11 1.3 Position Vectors 14 1.4 Scalar (“Dot”) Product of Two Vectors 16 1.5 Vector (“Cross”) Product of Two Vectors 17 References 19 2 Kinematics 20 2.1 Rectilinear Motion 20 2.2 Special Types of Rectilinear Motion 22 2.3 Curvilinear Motion in Space 24 2.4 Change of Speed 26 2.5 Motion with Constant Acceleration 27 2.6 Tangential and Normal Components 28 2.7 Circular Motion 34 2.8 Relative Motion 36 References 38 3 Dynamics of a Particle 39 3.1 The Law of Inertia 39 3.2 Momentum, Force, and Newton’s 2nd and 3rd Laws 40 3.3 Force of Gravity 45 3.4 Frictional Forces 45 3.5 Systems with Variable Mass 48 3.6 Tangential and Normal Components of Force 50 3.7 Angular Momentum and Torque 51 3.8 Central Forces 56 References 58 4 Work and Energy 59 4.1 Introduction 59 4.2 Work of a Force 60 4.3 Kinetic Energy and the Work-Energy Theorem 63 4.4 Potential Energy and Conservative Forces 64 4.5 Conservation of Mechanical Energy 67 4.6 Examples of Conservative Forces 69 4.7 Kinetic Friction as a Non-Conservative Force 74 References 74 5 Oscillations 75 5.1 Simple Harmonic Motion (SHM) 75 5.2 Force in SHM 78 5.3 Energy Relations 79 5.4 Oscillations of a Mass-Spring System 80 5.5 Oscillation of a Pendulum 84 5.6 Differential Equation of SHM 85 References 87 6 Systems of Particles 88 6.1 Center of Mass of a System of Particles 88 6.2 Newton’s Second Law and Conservation of Momentum 90 6.3 Angular Momentum of a System of Particles 92 6.4 Kinetic Energy of a System of Particles 95 6.5 Total Mechanical Energy of a System of Particles 98 6.6 Collisions 100 References 104 7 Rigid-Body Motion 105 7.1 Rigid Body 105 7.2 Center of Mass of a Rigid Body 106 7.3 Revolution of a Particle About an Axis 109 7.4 Angular Momentum of a Rigid Body 115 7.5 Rigid-Body Equations of Motion 117 7.6 Moment of Inertia and the Parallel-Axis Theorem 120 7.7 Conservation of Angular Momentum 123 7.8 Equilibrium of a Rigid Body 127 7.9 Kinetic and Total Mechanical Energy 129 7.10 Rolling Bodies 131 7.11 The Role of Static Friction in Rolling 134 7.12 Gyroscopic Motion 135 References 136 8 Elementary Fluid Mechanics 137 8.1 Ideal Fluid 137 8.2 Hydrostatic Pressure 138 8.3 Fundamental Equation of Hydrostatics 140 8.4 Units of Pressure 142 8.5 Communicating Vessels 144 8.6 Pascal’s Principle 146 8.7 Archimedes’ Principle 147 8.8 Dynamics of the Submerged Body 149 8.9 Equilibrium of a Floating Body 150 8.10 Fluid Flow 152 8.11 Equation of Continuity 153 8.12 Bernoulli’s Equation 155 8.13 Horizontal Flow 157 Reference 158 Appendix A Composition of Forces Acting in Space 159 Appendix B Some Theorems on the Center of Mass 165 Appendix C Table of Moments of Inertia 170 Appendix D Principal Axes of Rotation 171 Appendix E Variation of Pressure in the Atmosphere 174 Appendix F Proof of Bernoulli’s Equation 176 Solved Problems 179 Mathematical Supplement 256 1. Differential of a Function 256 2. Differential Operators 258 3. Geometrical Significance of the Differential 259 4. Derivative of a Composite Function 260 5. Differential Equations 260 Basic Integrals 263 Index 265