The Fundamentals of Newtonian Mechanics: For an Introductory Approach to Modern Physics

Preface
Contents
1 Physical Quantities and Units of Measurement
	1.1 Physics and the Scientific Method
	1.2 Measurable Quantities and the International System
		1.2.1 Distances
		1.2.2 Times
		1.2.3 Masses
	1.3 Measuring Physical Quantities
		1.3.1 Direct and Indirect Measurements
		1.3.2 The Distance Ladder
		1.3.3 Mass Measurements Out of Range for Scales
		1.3.4 From Pico to Tera
	1.4 The Concept of Space and Time
		1.4.1 The Euclidean Space
		1.4.2 Newtonian Space and Time
		1.4.3 Four-Dimensional Spacetime (*)
		1.4.4 Cosmological Time
	1.5 Time Synchronization and Dissemination
	1.6 Time Measurements Without Periodic Phenomena (*)
		1.6.1 The Law of Radioactive Decay
		1.6.2 Radiocarbon Dating
	1.7 Dimensional Analysis
	1.8 Questions and Exercises
2 Vectors and Operations with Vectors
	2.1 Introduction: Scalars, Vectors and Tensors
	2.2 Cartesian Coordinate Systems
		2.2.1 Right-Handed Coordinate System and Right-Hand Rule
	2.3 Representation of Vectors
		2.3.1 Vectors in Intrinsic Representation
		2.3.2 Vectors in Cartesian Representation
		2.3.3 Unit Vectors in Cartesian Representation
	2.4 Product of a Scalar and a Vector
	2.5 Sum and Difference of Vectors
	2.6 Scalar Product
		2.6.1 Scalar Product in Intrinsic Representation
		2.6.2 Scalar Product in Cartesian Representation
	2.7 Vector Product Between Vectors
		2.7.1 Vector Product in Intrinsic Representation
		2.7.2 Vector Product in Cartesian Representation
	2.8 Area and Volume in Vector Spaces (*)
	2.9 The Equals Are Not All Equal
	2.10 Laws and Principles, Physics and Mathematics
	2.11 Questions
3 Kinematics of the Particle
	3.1 Introduction and Definitions
	3.2 Uniform and Uniformly Accelerated Motion
	3.3 Equation of Motion, Velocity and Acceleration
		3.3.1 Equation of Motion
		3.3.2 Velocity, Definition
		3.3.3 Acceleration, Definition
	3.4 The Direct Kinematics Problem
		3.4.1 Infinitesimal Displacement and Infinitesimal Path
		3.4.2 Composition of Motions and Trajectory
	3.5 The Inverse Kinematics Problem
	3.6 Uniform and Non-uniform Circular Motion
		3.6.1 Uniform Circular Motion
		3.6.2 The Cartesian Representation
		3.6.3 Non-uniform Circular Motion
	3.7 Polar Plane Coordinates
		3.7.1 Circular Motion with Co-moving Unit Vectors
		3.7.2 Definition of Polar and Cylindrical Coordinates
	3.8 The Intrinsic Coordinates
	3.9 Poisson's Rules for Moving Unit Vectors
	3.10 Motion on any Trajectory (*)
	3.11 Questions and Exercises
4 Forces and the Dynamics of the Particle
	4.1 Introduction
	4.2 Measuring Forces
		4.2.1 Weight and Anthropomorphic ``effort''
		4.2.2 The Dynamometer
		4.2.3 The Weight is a Force
	4.3 The Vector Nature of Forces
	4.4 Constraint Forces
	4.5 Contact Friction Between Solids
	4.6 Newton's First Law of Motion
	4.7 Newton's Second Law of Motion
	4.8 Kinematic Effects of Some Forces
		4.8.1 Weight
		4.8.2 Fall with the Presence of Viscous Friction
	4.9 Harmonic Oscillators
		4.9.1 The Simple Pendulum
		4.9.2 The Elastic Force and the Harmonic Oscillator
	4.10 What Do We Know Today About Forces
		4.10.1 Fundamental Interactions
		4.10.2 Force Fields
		4.10.3 The Arrow of Time
	4.11 Questions and Exercises
5 Frames of Reference in Relative Motion
	5.1 Galilean Relativity Principle
	5.2 Inertial Frames of Reference
	5.3 Non-inertial Frames of Reference
	5.4 Pseudo-forces in Non-inertial Frames
		5.4.1 Dynamics in an Accelerated Vehicle
		5.4.2 Dynamics in a Rotating Frame
		5.4.3 Earth's Drag Acceleration
	5.5 Coriolis Force in Guglielmini's Experiment (*)
	5.6 The Coriolis Force on the Motion of Terrestrial Fluids
	5.7 Generalized Principles of Relativity
	5.8 Questions and Exercises
6 Work and Energy
	6.1 Introduction
	6.2 Definition of Work
		6.2.1 Work in Vectors' Intrinsic Representation
		6.2.2 Work in Cartesian Coordinates
		6.2.3 Measurement Units of Work
	6.3 The Work-Energy Theorem
	6.4 Examples of Calculating the Work of a Force
		6.4.1 Work of the Weight Force
		6.4.2 Work of the Elastic Force
		6.4.3 Work of the Dynamic Friction
		6.4.4 Anthropomorphic Work (``work'' in Common Parlance)
	6.5 Conservative and Non-conservative Forces
	6.6 Potential Energy
	6.7 Mechanical Energy and Its Conservation
		6.7.1 Definition of Mechanical Energy
		6.7.2 Mechanical Energy with Non-conservative Forces
	6.8 Exact Differentials and Potential Energy
		6.8.1 Force is the Negative Gradient of Potential Energy
	6.9 Differential Operators: Divergence, Curl and Laplacian (*)
		6.9.1 Divergence
		6.9.2 Curl
		6.9.3 Laplacian
		6.9.4 Schwarz's Theorem
	6.10 Conservative Forces: Null Curl
	6.11 Central Forces are Conservative
	6.12 Power
	6.13 Towards the Principle of Energy Conservation
	6.14 Questions and Exercises
7 Dynamics of Mechanical Systems
	7.1 Mass Distribution: Discrete and Continuous Systems
		7.1.1 Systems of Material Points
		7.1.2 Continuous Systems
	7.2 Degrees of Freedom
	7.3 Center of Mass
		7.3.1 The Center of Mass of a System of Points
		7.3.2 The Center of Mass of a Continuous Body
	7.4 Torque
	7.5 Linear Momentum and Angular Momentum
		7.5.1 Linear Momentum of a System
		7.5.2 Angular Momentum of a System
	7.6 Conservation of Linear Momentum
		7.6.1 Euler's First Law
	7.7 Conservation of Angular Momentum
		7.7.1 Euler's Second Law
	7.8 Newton's Third Law of Motion
		7.8.1 Comment on Newton's Laws
	7.9 Properties of the Center of Mass Frame
		7.9.1 Linear Momentum P' in the C.M. Frame
		7.9.2 Intrinsic Angular Momentum (Spin) L' in the C.M. Frame
		7.9.3 Euler's Second Law in the C.M. Frame
		7.9.4 Kinetic Energy T' in the C.M. Frame
	7.10 Equilibrium of a Rigid Body
	7.11 Questions and Exercises
8 Collisions and Decays
	8.1 Introduction
	8.2 Impulsive Forces
	8.3 Elastic Collision Between Two Bodies
		8.3.1 The One-Dimensional Case
		8.3.2 Two-Dimensional Case and Constraining Forces
		8.3.3 Let Us Reflect on the Nature of the Constraining Forces
	8.4 Collisions in the Center-of-Mass Frame
		8.4.1 One-Dimensional Case in the Center-of-Mass Frame
	8.5 Motion of a Rocket
	8.6 Energy-Mass Conservation in Collisions and Decays
	8.7 Conservation Laws in Two or Three Body Decays
		8.7.1 Alpha Decay (Two-Body Process) (*)
		8.7.2 Beta Decay (Three-Body Process) (*)
	8.8 Partially Elastic Collisions
	8.9 Questions and Exercises
9 Considerations on Vectors
	9.1 Isotropy and Homogeneity of the Universe
	9.2 Translation of Reference Frames
	9.3 Rotation of Reference Frames
	9.4 Reflexion of Reference Frames
	9.5 Not all Triplets are Vectors
	9.6 Polar and Axial Vectors
		9.6.1 How Can Axial Vectors be Distinguished
		9.6.2 Scalar and Pseudoscalar Quantities
		9.6.3 Pseudoscalar Quantities in Nuclear Phenomena
10 Newton's Law of Gravitation
	10.1 Introduction
	10.2 Pre-Galilean Astronomical Measurements
		10.2.1 Earth's Radius
		10.2.2 Earth-Moon Distance
		10.2.3 Earth-Sun Distance
	10.3 The Apple and the Moon
	10.4 Law of Universal Gravitation
		10.4.1 The Inverse of the Square of the Distance
	10.5 Inertial Mass and Gravitational Mass
	10.6 Gravitational Potential Energy
		10.6.1 Potential Energy of the Weight Force
		10.6.2 Classical Limits of Relativity and Quantum Mechanics
	10.7 Escape Speed From a Celestial Body of Mass M
		10.7.1 Event Horizon
	10.8 Measurement of G: The Torsion Balance
	10.9 Spherical Coordinates (*)
		10.9.1 Line, Surface and Volume Elements
		10.9.2 Gravitational Potential Energy: Second Method
	10.10  Mass in the Center of the Sphere (*)
	10.11 Questions
11 Motions in Gravitational Fields
	11.1 Historical Introduction
	11.2 Kepler's Empirical Laws
	11.3 The Two-Body System
	11.4 Angular Momentum, Kepler's 1st and 2nd Laws
		11.4.1 Kepler's First Law, Plane Orbits
		11.4.2 Kepler's Second Law
	11.5 Kepler's Third Law
	11.6 Mechanical Energy of the Two-Body System
		11.6.1 Qualitative Solutions for the Effective Potential
	11.7 Kepler's First Law, Elliptic Orbits (*)
		11.7.1 Conic Functions
		11.7.2 First Integral
		11.7.3 Eccentricity Versus Energy and Angular Momentum
		11.7.4 Elliptic Solutions: Semi-axes as a Function of E, L
		11.7.5 Degeneracy in Classical and Quantum Physics
	11.8 Kepler's Third Law, Revised
		11.8.1 The Black Hole in the Center of the Galaxy
	11.9 Two Neutron Stars
	11.10 Triumphs and Falls of Newtonian Theory
	11.11 Gravitational Indications for Dark Matter
	11.12 Questions and Exercises
12 Dynamics of Rigid Bodies
	12.1 Introduction
	12.2 Angular Momentum and Angular Velocity
	12.3 Moment of Inertia of Rigid Bodies
		12.3.1 Moment of Inertia of a Cylinder and Hollow Cylinder
		12.3.2 Moment of Inertia of a Rod
		12.3.3 Moment of Inertia of a Rod on a Disk
	12.4 Conservation of Angular Momentum and Angular Velocity
	12.5 An Application of the II Euler's Law
	12.6 Huygens-Steiner Theorem for Moments of Inertia
	12.7 Center of Gravity
		12.7.1 Physical Pendulum
		12.7.2 Torsion Pendulum
	12.8 Moment of Inertia Tensor (*)
	12.9 Rotational Kinetic Energy
		12.9.1 Body Rolling Without Sliding
		12.9.2 The Motion of the Wheel
	12.10 The Motion of the Spinning Top (*)
	12.11 Questions and Summary Exercise
13 Considerations on Energy
	13.1 Work Needed to Build a System
	13.2 Potential Energy of a Bound Spherical System
		13.2.1 Age of the Sun
		13.2.2 Energy Conservation in Stellar Gravitational  Collapse (*)
	13.3 Gravitational Field and Potential
	13.4 First Integral from Energy Conservation
		13.4.1 Application to a Falling Particle
	13.5 Motion in a Potential Energy Field
		13.5.1 Stable and Unstable Equilibrium
		13.5.2 Permitted and Forbidden Regions
	13.6 More on Harmonic Motion (*)
		13.6.1 Complex Numbers
		13.6.2 Harmonic Oscillator with Complex Numbers
		13.6.3 Mechanical Energy of the Harmonic Oscillator
	13.7 Damped Harmonic Oscillator (*)
		13.7.1 Overdamped Motion
		13.7.2 Discussion on Mechanical Energy and Developments
	13.8 Developments and Problems of Classical Mechanics
		13.8.1 Lagrangian and Hamiltonian Formalisms
		13.8.2 Determinism in Newtonian Mechanics
	13.9 Epilogue
	13.10 Questions and Exercises
Appendix  Appendix: Numerical Solution  of the Exercises
Index