# Physics for scientists and engineers with modern physics / Douglas C. Giancoli

##### Giancoli, Douglas C.

Material type: Book; Format: print Publisher: Upper Saddle River, N.J. : Pearson Education Internacional, 2008Edition: 4th ed.Description: pag. var. : il. ; 28 cm + Online resource.ISBN: 0-13-607480-4.Subject(s): FísicaItem type | Home library | Call number | Status | Loan | Date due | Barcode | Item holds | Course reserves |
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Monografías | 02. BIBLIOTECA CAMPUS PUERTO REAL | 53/GIA/phy (Browse shelf) | Checked out | PREST. LIBROS | 31/01/2019 | 3742363513 |
FÍSICA II GRADO EN MATEMÁTICAS Asignatura actualizada 2017-2018 |

#### Enhanced descriptions from Syndetics:

For the calculus-based General Physics course primarily taken by engineers and science majors (including physics majors). This long-awaited and extensive revision maintains Giancoli's reputation for creating carefully crafted, highly accurate and precise physics texts. Physics for Scientists and Engineers combines outstanding pedagogy with a clear and direct narrative and applications that draw the student into the physics. The new edition also features an unrivaled suite of media and on-line resources that enhance the understanding of physics. This book is written for students. It aims to explain physics in a readable and interesting manner that is accessible and clear, and to teach students by anticipating their needs and difficulties without oversimplifying. Physics is a description of reality, and thus each topic begins with concrete observations and experiences that students can directly relate to. We then move on to the generalizations and more formal treatment of the topic. Not only does this make the material more interesting and easier to understand, but it is closer to the way physics is actually practiced.

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For the calculus-based General Physics course primarily taken by engineers and science majors (including physics majors). This long-awaited and extensive revision maintains Giancoli's reputation for creating carefully crafted, highly accurate and precise physics texts. Physics for Scientists and Engineers combines outstanding pedagogy with a clear and direct narrative and applications that draw the student into the physics. The new edition also features an unrivaled suite of media and on-line resources that enhance the understanding of physics. This book is written for students. It aims to explain physics in a readable and interesting manner that is accessible and clear, and to teach students by anticipating their needs and difficulties without oversimplifying. Physics is a description of reality, and thus each topic begins with concrete observations and experiences that students can directly relate to. We then move on to the generalizations and more formal treatment of the topic. Not only does this make the material more interesting and easier to understand, but it is closer to the way physics is actually practiced.

### Table of contents provided by Syndetics

**Applications List**(p. xii)**Preface**(p. xiv)**Available Supplements and Media**(p. xxii)**Notes to Students (and Instructors) on the Format**(p. xxiv)**Color Use: Vectors, Fields, and Symbols**(p. xxv)**Chapter 1 Introduction, Measurement, Estimating****1-1 The Nature of Science****1-2 Models, Theories, and Laws****1-3 Measurement and Uncertainty Significant Figures****1-4 Units, Standards, and the SI System****1-5 Converting Units****1-6 Order of Magnitude: Rapid Estimating****1-7 Dimensions and Dimensional Analysis****Summary****Questions****Problems****General Problems****Chapter 2 Describing Motion: Kinematics in One Dimension****2-1 Reference Frames and Displacement****2-2 Average Velocity****2-3 Instantaneous Velocity****2-4 Acceleration****2-5 Motion at Constant Acceleration****2-6 Solving Problems****2-7 Freely Falling Objects****2-8 Variable Acceleration Integral Calculus****2-9 Graphical Analysis and Numerical Integration****Summary****Questions****Problems****General Problems****Chapter 3 Kinematics in Two or Three Dimensions Vectors****3-1 Vectors and Scalars****3-2 Addition of Vectors-Graphical Methods****3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar****3-4 Adding Vectors by Components****3-5 Unit Vectors****3-6 Vector Kinematics****3-7 Projectile Motion****3-8 Solving Problems Involving Projectile Motion****3-9 Relative Velocity****Summary****Questions****Problems****General Problems****Chapter 4 Dynamics: Newton's Laws of Motion****4-1 Force****4-2 Newton's First Law of Motion****4-3 Mass****4-4 Newton's Second Law of Motion****4-5 Newton's Third Law of Motion****4-6 Weight-the Force of Gravity and the Normal Force****4-7 Solving Problems with Newton's Laws: Free-Body Diagrams****4-8 Problem Solving-A General Approach****Summary****Questions****Problems****General Problems****Chapter 5 Using Newton's Laws: Friction, Circular Motion, Drag Forces****5-1 Applications of Newton's Laws Involving Friction****5-2 Uniform Circular Motion-Kinematics****5-3 Dynamics of Uniform Circular Motion****5-4 Highway Curves: Banked and Unbanked****5-5 Nonuniform Circular Motion****5-6 Velocity-Dependent Forces: Drag and Terminal Velocity****Summary****Questions****Problems****General Problems****Chapter 6 Gravitation and Newton's6 Synthesis****6-1 Newton's Law of Universal Gravitation****6-2 Vector Form of Newton's Law of Universal Gravitation****6-3 Gravity Near the Earth's Surface Geophysical Applications****6-4 Satellites and "Weightlessness"****6-5 Kepler's Laws and Newton's Synthesis****6-6 Gravitational Field****6-7 Types of Forces in Nature****6-8 Principle of Equivalence Curvature of Space Black Holes****Summary****Questions****Problems****General Problems****Chapter 7 work and Energy****7-1 Work Done by a Constant Force****7-2 Scalar Product of Two Vectors****7-3 Work Done by a Varying Force****7-4 Kinetic Energy and the Work-Energy Principle****Summary****Questions****Problems****General Problems****Chapter 8 Conservation of Energy****8-1 Conservative and Nonconservative Forces****8-2 Potential Energy****8-3 Mechanical Energy and Its Conservation****8-4 Problem Solving Using Conservation of Mechanical Energy****8-5 The Law of Conservation of Energy****8-6 Energy Conservation with Dissipative Forces: Solving Problems****8-7 Gravitational Potential Energy and Escape Velocity****8-8 Power****8-9 Potential Energy Diagrams Stable and Unstable Equilibrium****Summary****Questions****Problems****General Problems****Chapter 9 Linear Momentum****9-1 Momentum and Its Relation to Force****9-2 Conservation of Momentum****9-3 Collisions and Impulse****9-4 Conservation of Energy and Momentum in Collisions****9-5 Elastic Collisions in One Dimension****9-6 Inelastic Collisions****9-7 Collisions in Two or Three Dimensions****9-8 Center of Mass (CM)****9-9 Center of Mass and Translational Motion****9-10 Systems of Variable Mass Rocket Propulsion****Summary****Questions****Problems****General Problems****Chapter 10 Rotational Motion****10-1 Angular Quantities****10-2 Vector Nature of Angular Quantities****10-3 Constant Angular Acceleration****10-4 Torque****10-5 Rotational Dynamics Torque and Rotational Inertia****10-6 Solving Problems in Rotational Dynamics****10-7 Determining Moments of Inertia****10-8 Rotational Kinetic Energy****10-9 Rotational Plus Translational Motion Rolling****10-10 Why Does a Rolling Sphere Slow Down?****Summary****Questions****Problems****General Problems****Chapter 11 Angular Momentum General Rotation****11-1 Angular Momentum-Object Rotating About a Fixed Axis****11-2 Vector Cross Product Torque as a Vector****11-3 Angular Momentum of a Particle****11-4 Angular Momentum and Torque for a System of Particles General Motion****11-5 Angular Momentum and Torque for a Rigid Object****11-6 Conservation of Angular Momentum****11-7 The Spinning Top and Gyroscope****11-8 Rotating Frames of Reference Inertial Forces****11-9 The Coriolis Effect****Summary****Questions****Problems****General Problems****Chapter 12 Static Equilibrium Elasticity and Fracture****12-1 The Conditions for Equilibrium****12-2 Solving Statics Problems****12-3 Stability and Balance****12-4 Elasticity Stress and Strain****12-5 Fracture****12-6 Trusses and Bridges****12-7 Arches and Domes****Summary****Questions****Problems****General Problems****Chapter 13 Fluids****13-1 Phases of Matter****13-2 Density and Specific Gravity****13-3 Pressure in Fluids****13-4 Atmospheric Pressure and Gauge Pressure****13-5 Pascal's Principle****13-6 Measurement of Pressure Gauges and the Barometer****13-7 Buoyancy and Archimedes' Principle****13-8 Fluids in Motion Flow Rate and the Equation of Continuity****13-9 Bernoulli's Equation****13-10 Applications of Bernoulli's Principle: Torricelli, Airplanes, Baseballs, TIA****13-11 Viscosity****13-12 Flow in Tubes: Poiseuille's Equation, Blood Flow****13-13 Surface Tension and Capillarity****13-14 Pumps, and the Heart****Summary****Questions****Problems****General Problems****Chapter 14 Oscillations****14-1 Oscillations of a Spring****14-2 Simple Harmonic Motion****14-3 Energy in the Simple Harmonic Oscillator****14-4 Simple Harmonic Motion Related to Uniform Circular Motion****14-5 The Simple Pendulum****14-6 The Physical Pendulum and the Torsion Pendulum****14-7 Damped Harmonic Motion****14-8 Forced Oscillations Resonance****Summary****Questions****Problems****General Problems****Chapter 15 Wave Motion****15-1 Characteristics of Wave Motion****15-2 Types of Waves: Transverse and Longitudinal****15-3 Energy Transported by Waves****15-4 Mathematical Representation of a Traveling Wave****15-5 The Wave Equation****15-6 The Principle of Superposition****15-7 Reflection and Transmission****15-8 Interference****15-9 Standing Waves Resonance****15-10 Refraction****15-11 Diffraction****Summary****Questions****Problems****General Problems****Chapter 16 Sound****16-1 Characteristics of Sound****16-2 Mathematical Representation of Longitudinal Waves****16-3 Intensity of Sound: Decibels****16-4 Sources of Sound: Vibrating Strings and Air Columns****16-5 Quality of Sound, and Noise Superposition****16-6 Interference of Sound Waves Beats****16-7 Doppler Effect****16-8 Shock Waves and the Sonic Boom****16-9 Applications: Sonar, Ultrasound, and Medical Imaging****Summary****Questions****Problems****General Problems****Chapter 17 Temperature, Thermal Expansion, and the Ideal Gas Law****17-1 Atomic Theory of Matter****17-2 Temperature and Thermometers****17-3 Thermal Equilibrium and the Zeroth Law of Thermodynamics****17-4 Thermal Expansion****17-5 Thermal Stresses****17-6 The Gas Laws and Absolute Temperature****17-7 The Ideal Gas Law****17-8 Problem Solving with the Ideal Gas Law****17-9 Ideal Gas Law in Terms of Molecules: Avogadro's Number****17-10 Ideal Gas Temperature Scale-a Standard****Summary****Questions****Problems****General Problems****Chapter 18 Kinetic Theory of Gases****18-1 The Ideal Gas Law and the Molecular Interpretation of Temperature****18-2 Distribution of Molecular Speeds****18-3 Real Gases and Changes of Phase****18-4 Vapor Pressure and Humidity****18-5 Van der Waals Equation of State****18-6 Mean Free Path****18-7 Diffusion****Summary****Questions****Problems****General Problems****Chapter 19 Heat and the First Law of Thermodynamics****19-1 Heat as Energy Transfer****19-2 Internal Energy****19-3 Specific Heat****19-4 Calorimetry-Solving Problems****19-5 Latent Heat****19-6 The First Law of Thermodynamics****19-7 Applying the First Law of Thermodynamics Calculating the Work****19-8 Molar Specific Heats for Gases, and the Equipartition of Energy****19-9 Adiabatic Expansion of a Gas****19-10 Heat Transfer: Conduction, Convection, Radiation****Summary****Questions****Problems****General Problems****Chapter 20 Second Law of Thermodynamics****20-1 The Second Law of Thermodynamics-Introduction****20-2 Heat Engines****20-3 Reversible and Irreversible Processes the Carnot Engine****20-4 Refrigerators, Air Conditioners, and Heat Pumps****20-5 Entropy****20-6 Entropy and the Second Law of Thermodynamics****20-7 Order to Disorder****20-8 Unavailability of Energy Heat Death****20-9 Statistical Interpretation of Entropy and the Second Law****20-10 Thermodynamic Temperature Scale Absolute Zero and the Third Law of Thermodynamics****20-11 Thermal Pollution, Global Warming, and Energy Resources****Summary****Questions****Problems****General Problems****Chapter 21 electric Charge and Electric Field****21-1 Static Electricity Electric Charge and Its Conservation****21-2 Electric Charge in the Atom****21-3 Insulators and Conductors****21-4 Induced Charge the Electroscope****21-5 Coulomb's Law****21-6 The Electric Field****21-7 Electric Field Calculations for Continuous Charge Distributions****21-8 Field Lines****21-9 Electric Fields and Conductors****21-10 Motion of a Charged Particle in an Electric Field****21-11 Electric Dipoles****21-12 Electric Forces in Molecular Biology DNA****21-13 Photocopy Machines and Computer Printers Use Electrostatics****Summary****Questions****Problems****General Problems****Chapter 22 Gauss's Law****22-1 Electric Flux****22-2 Gauss's Law****22-3 Applications of Gauss's Law****22-4 Experimental Basis of Gauss's and Coulomb's Law****Summary****Questions****Problems****General Problems****Chapter 23 Electric Potential****23-1 Electric Potential Energy and Potential Difference****23-2 Relation between Electric Potential and Electric Field****23-3 Electric Potential Due to Point Charges****23-4 Potential Due to Any Charge Distribution****23-5 Equipotential Surfaces****23-6 Electric Dipole Potential****23-7 E Determined from V****23-8 Electrostatic Potential Energy the Electron Volt****23-9 Cathode Ray Tube: TV and Computer Monitors, Oscilloscope****Summary****Questions****Problems****General Problems****Chapter 24 Capacitance, Dielectrics, Electric Energy Storage****24-1 Capacitors****24-2 Determination of Capacitance****24-3 Capacitors in Series and Parallel****24-4 Electric Energy Storage****24-5 Dielectrics****24-6 Molecular Description of Dielectrics****Summary****Questions****Problems****General Problems****Chapter 25 Electric Currents and Resistance****25-1 The Electric Battery****25-2 Electric Current****25-3 Ohm's Law: Resistance and Resistors****25-4 Resistivity****25-5 Electric Power****25-6 Power in Household Circuits****25-7 Alternating Current****25-8 Microscopic View of Electric Current: Current Density and Drift Velocity****25-9 Superconductivity****25-10 Electrical Conduction in the Nervous System****Summary****Questions****Problems****General Problems****Chapter 26 DC Circuits****26-1 EMF and Terminal Voltage****26-2 Resistors in Series and in Parallel****26-3 Kirchoff's Rules****26-4 EMFs in Series and in Parallel Charging a Battery****26-5 Circuits Containing Resistor and Capacitor (RC Circuits)****26-6 Electric Hazards****26-7 Ammeters and Voltmeters****Summary****Questions****Problems****General Problems****Chapter 27 Magnetism****27-1 Magnets and Magnetic Fields****27-2 Electric Currents Produce Magnetic Fields****27-3 Force on an Electric Current in a Magnetic Field Definition of****27-4 Force on an Electric Charge Moving in a Magnetic Field****27-5 Torque on a Current Loop Magnetic Dipole Moment****27-6 Applications: Galvanometers, Motors, Loudspeakers****27-7 Discover and Properties of the Electron****27-8 The Hall Effect****27-9 Mass Spectrometer****Summary****Questions****Problems****General Problems****Chapter 28 Sources of Magnetic Field****28-1 Magnetic Field Due to a Straight Wire****28-2 Force between Two Parallel Wires****28-3 Definitions of the Ampere and the Coulomb****28-4 Ampere's Law****28-5 Magnetic Field of a Solenoid and a Toroid****28-6 Biot-Savart Law****28-7 Magnetic materials-Ferromagnetism****28-8 Electromagnets and Solenoids-Applications****28-9 Magnetic Fields in Magnetic Materials Hysteresis****28-10 Paramagnetism and Diamagnetism****Summary****Questions****Problems****General Problems****Chapter 29 Electromagnetic Induction and Faraday's Law****29-1 Induced EMF****29-2 Faraday's Law of Induction Lenz's Law****29-3 EMF Induced in a Moving Conductor****29-4 Electric Generators****29-5 Back EMF and Counter Torque Eddy Currents****29-6 Transformers and Transmission of Power****29-7 A Changing Magnetic Flux Produces an Electric Field****29-8 Applications of Induction: Sound Systems, Computer Memory, Seismograph, GFCI****Summary****Questions****Problems****General Problems****Chapter 30 Inductance, Electromagnetic Oscillations, and AC Circuits****30-1 Mutual Inductance****30-2 Self-Inductance****30-3 Energy Stored in a Magnetic Field****30-4 LR Circuits****30-5 LC Circuits and Electromagnetic Oscillations****30-6 LC Oscillations with Resistance (LRC Circuit)****30-7 AC Circuits with AC Source****30-8 LRC Series AC Circuit****30-9 Resonance in AC Circuits****30-10 Impedance Matching****Summary****Questions****Problems****General Problems****Chapter 31 Maxwell's Equations and Electromagnetic Waves****31-1 Changing Electric Fields Produce Magnetic Fields Ampere's Law and Displacement Current****31-2 Gauss's Law for Magnetism****31-3 Maxwell's Equations****31-4 Production of Electromagnetic Waves****31-5 Electromagnetic Waves, and Their Speed, from Maxwell's Equations****31-6 Light as an Electromagnetic Wave and the Electromagnetic Spectrum****31-7 Measuring the Speed of Light****31-8 Energy in EM Waves the Poynting Vector****31-9 Radiation Pressure****31-10 Radio and Television Wireless Communication****Summary****Questions****Problems****General Problems****Chapter 32 Light: Reflection and Refraction****32-1 The Ray Model of Light****32-2 The Speed of Light and Index of Refraction****32-3 Reflection Image Formation by a Plane Mirror****32-4 Formation of Images by Spherical Mirrors****32-5 Refraction: Snell's Law****32-6 Visible Spectrum and Dispersion****32-7 Total Internal Reflection Fiber Optics****32-8 Refraction at a Spherical Surface****Summary****Questions****Problems****General Problems****Chapter 33 Lenses and Optical Instruments****33-1 Thin Lenses Ray Tracing****33-2 The Thin Lens Equation Magnification****33-3 Combinations of Lenses****33-4 Lensmaker's Equation****33-5 Cameras, Film and Digital****33-6 The Human Eye Corrective Lenses****33-7 Magnifying Glass****33-8 Telescopes****33-9 Compound Microscope****33-10 Aberrations of Lenses and Mirrors****Summary****Questions****Problems****General Problems****Chapter 34 The Wave Nature of Light Interference****34-1 Waves Versus Particles Huygens' Principle and Diffraction****34-2 Huygens' Principle and the Law of Refraction****34-3 Interference-Young's Double-Slit Experiment****34-4 Intensity in the Double-Slit Interference Pattern****34-5 Interference in Thin Films****34-6 Michelson Interferometer****34-7 Luminous Intensity****Summary****Questions****Problems****General Problems****Chapter 35 Diffraction and Polarization****35-1 Diffraction by a Single Slit or Disk****35-2 Intensity in Single-Slit Diffraction Pattern****35-3 Diffraction in the Double-Slit Experiment****35-4 Limits of Resolution Circular Apertures****35-5 Resolution of Telescopes and Microscopes the ¿ Limit****35-6 Resolution of the Human Eye and Useful Magnification****35-7 Diffraction Grating****35-8 The Spectrometer and Spectroscopy****35-9 Peak Widths and Resolving Power for a Diffraction Grating****35-10 X-Rays and X-Ray Diffraction****35-11 Polarization****35-12 Liquid Crystal Displays (LCD)****35-13 Scattering of Light by the Atmosphere****Summary****Questions****Problems****General Problems****Chapter 36 Special Theory of Relativity****36-1 Galilean-Newtonian Relativity****36-2 The Michelson-Morley Experiment****36-3 Postulates of the Special Theory of Relativity****36-4 Simultaneity****36-5 Time Dilation and the Twin Paradox****36-6 Length Contraction****36-7 Four-Dimensional Space-Time****36-8 Galilean and Lorentz Transformations****36-9 Relativistic Momentum and Mass****36-10 The Ultimate Speed****36-11 Energy and Mass E=mc 2****36-12 Doppler Shift for Light****36-13 The Impact of Special Relativity****Summary****Questions****Problems****General Problems****Chapter 37 Early Quantum Theory and Models Of The Atom****37-1 Planck's Quantum Hypothesis****37-2 Photon Theory of Light and the Photoelectric Effect****37-3 Photons and the Compton Effect****37-4 Photon Interactions Pair Production****37-5 Wave-Particle Duality the Principle of Complementarity****37-6 Wave Nature of Matter****37-7 Electron Microscopes****37-8 Early Models of the Atom****37-9 Atomic Spectra: Key to the Structure of the Atom****37-10 The Bohr Model****37-11 DeBroglie's Hypothesis Applied to Atoms****Summary****Questions****Problems****General Problems****Chapter 38 Quantum Mechanics****38-1 Quantum Mechanics-A New Theory****38-2 The Wave Function and Its Interpretation the Double-Slit Experiment****38-3 The Heisenberg Uncertainty Principle****38-4 Philosophic Implications Probability Versus Determinism****38-5 The Schrodinger Equation in One Dimension-Time-Independent Form****38-6 Time-Dependent Schrodinger Equation****38-7 Free Particles Plane Waves and Wave Packets****38-8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box)****38-9 Finite Potential Well****38-10 Tunneling through a Barrier****Summary****Questions****Problems****General Problems****Chapter 39 Quantum Mechanics of Atoms****39-1 Quantum-Mechanical View of Atoms****39-2 Hydrogen Atom: Schrodinger Equation and Quantum Numbers****39-3 Hydrogen Atom Wave Functions****39-4 Complex Atoms the Exclusion Principle****39-5 The Periodic Table of Elements****39-6 X-Ray Spectra and Atomic Number****39-7 Magnetic Dipole Moments Total Angular Momentum****39-8 Fluorescence and Phosphorescence****39-9 Lasers****39-10 Holography****Summary****Questions****Problems****General Problems****Chapter 40 Molecules and Solids****40-1 Bonding in Molecules****40-2 Potential-Energy Diagrams for Molecules****40-3 Weak (van der Waals) Bonds****40-4 Molecular Spectra****40-5 Bonding in Solids****40-6 Free-Electron Theory of Metals****40-7 Band Theory of Solids****40-8 Semiconductors and Doping****40-9 Semiconductor Diodes****40-10 Transistors and Integrated Circuits****Summary****Questions****Problems****General Problems****Chapter 41 Nuclear Physics and Radioactivity****41-1 Structure and Properties of the Nucleus****41-2 Binding Energy and Nuclear Forces****41-3 Radioactivity****41-4 Alpha Decay****41-5 Beta Decay****41-6 Gamma Decay****41-7 Conservation of Nucleon Number and Other Conservation Laws****41-8 Half-Life and Rate of Decay****41-9 Decay Series****41-10 Radioactive Dating****41-11 Detection of Radiation****Summary****Questions****Problems****General Problems****Chapter 42 Nuclear Energy: Efects and Uses of Radiation****42-1 Nuclear Reactions and the Transmutations of Elements****42-2 Cross Section****42-3 Nuclear Fission****Nuclear Reactors****42-4 Fusion****42-5 Passage of radiation through matter Radiation Damage****42-6 Measurement of Radiation-Dosimetry****42-7 Radiation Therapy****42-8 Tracers****42-9 Imaging by Tomography: CAT Scans, and Emission Tomography****42-10 Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI)****Summary****Questions****Problems****General Problems****Chapter 43 Elementary Particles****43-1 High-Energy Particles****43-2 Particle Accelerators and Detectors****43-3 Beginnings of Elementary Particle Physics-Particle Exchange****43-4 Particles and Antiparticles****43-5 Particle Interactions and Conservation Laws****43-6 Particle Classification****43-7 Particle Stability and Resonances****43-8 Strange Particles****43-9 Quarks****43-10 The "Standard Model": Quantum Chromodynamics (QCD) and the Electroweak Theory****43-11 Grand Unified Theories****Summary****Questions****Problems****General Problems****Chapter 44 Astrophysics and Cosmology****44-1 Stars and Galaxies****44-2 Stellar Evolution the Birth and Death of Stars****44-3 General Relativity: Gravity and the Curvature of Space****44-4 The Expanding Universe****44-5 The Big Bang and the Cosmic Microwave Background****44-6 The Standard Cosmological Model: Early History of the Universe****44-7 The Future of the Universe?****Summary****Questions****Problems****General Problems**

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