Physics for Scientists & Engineers, 5th edition
Your access includes:
 Search, highlight, notes, and more
 Easily create flashcards
 Use the app for access anywhere
 14day refund guarantee
$10.99per month
Minimum 4month term, pay monthly or pay $43.96 upfront
Learn more, spend less

Listen on the go
Learn how you like with full eTextbook audio

Find it fast
Quickly navigate your eTextbook with search

Stay organized
Access all your eTextbooks in one place

Easily continue access
Keep learning with autorenew
Overview
Physics for Scientists and Engineers combines outstanding pedagogy and a clear direct narrative with applications to draw you into the physics at hand. You'll gain an understanding of the basic concepts of physics from mechanics to modern physics. Each topic begins with concrete observations and experiences that you can relate to your everyday life and to future professions, and then you'll move to generalizations and aspects of physics that show why we believe what we believe.
The 5th Edition presents new applications and includes the physics of digital and added problemsolving techniques.
Published by Pearson (June 11th 2021)  Copyright © 2022
ISBN13: 9780137488179
Subject: Physics
Category: CalculusBased Physics
Overview
Physics for Scientists & Engineers is available in the following versions:
 Complete version contains 44 Chapters including 9 Chapters of modern physics
 Classic version contains 37 Chapters, 35 on classical physics, plus one each on relativity and quantum theory
3 Volume version: Available separately or packaged together.
 Volume 1: Chapters 120 on mechanics, including fluids, oscillations, waves, plus heat and thermodynamics.
 Volume 2: Chapters 2135 on electricity and magnetism, plus light and optics.
 Volume 3: Chapters 3644 on modern physics: relativity, quantum theory, atomic physics, condensed matter, nuclear physics, elementary particles, cosmology and astrophysics.
Sections marked with a star * may be considered optional.
 Introduction, Measurement, Estimating
 1.1 How Science Works
 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
 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
 Kinematics in Two or Three Dimensions; Vectors
 3.1 Vectors and Scalars
 3.2 Addition of VectorsGraphical 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
 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 Weightthe Force of Gravity; and the Normal Force
 4.7 Solving Problems with Newton's Laws: FreeBody Diagrams
 4.8 Problem SolvingA General Approach
 Using Newton's Laws: Friction, Circular Motion, Drag Forces
 5.1 Using Newton's Laws with Friction
 5.2 Uniform Circular MotionKinematics
 5.3 Dynamics of Uniform Circular Motion
 5.4 Highway Curves: Banked and Unbanked
 5.5 Nonuniform Circular Motion
 *5.6 VelocityDependent Forces: Drag and Terminal Velocity
 Gravitation and Newton's 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
 6.4 Satellites and "Weightlessness"
 6.5 Planets, Kepler's Laws, and Newton's Synthesis
 6.6 Moon Rises an Hour Later Each Day
 6.7 Types of Forces in Nature
 *6.8 Gravitational Field
 *6.9 Principle of Equivalence; Curvature of Space; Black Holes
 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 WorkEnergy Principle
 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
 *8.10 Gravitational Assist (Slingshot)
 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 2 or 3 Dimensions
 9.8 Center of Mass (cm)
 9.9 Center of Mass and Translational Motion
 *9.10 Systems of Variable Mass; Rocket Propulsion
 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?
 Angular Momentum; General Rotation
 11.1 Angular Momentum Objects 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
 Static Equilibrium; Elasticity and Fracture
 12.1 The Conditions for Equilibrium
 12.2 Solving Statics Problems
 *12.3 Applications to Muscles and Joints
 12.4 Stability and Balance
 12.5 Elasticity; Stress and Strain
 12.6 Fracture
 *12.7 Trusses and Bridges
 *12.8 Arches and Domes
 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, Blood Flow
 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
 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
 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
 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
 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 Scaleâ€Ša Standard
 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 Temperature of Water Decrease with Altitude
 18.6 Van der Waals Equation of State
 18.7 Mean Free Path
 18.8 Diffusion
 Heat and the First Law of Thermodynamics
 19.1 Heat as Energy Transfer
 19.2 Internal Energy
 19.3 Specific Heat
 19.4 CalorimetrySolving Problems
 19.5 Latent Heat
 19.6 The First Law of Thermodynamics
 19.7 Thermodynamic Processes and the First Law
 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
 Second Law of Thermodynamics
 20.1 The Second Law of ThermodynamicsIntroduction
 20.2 Heat Engines
 20.3 The Carnot Engine; Reversible and Irreversible Processes
 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; Third Law of Thermodynamics
 20.11 Thermal Pollution, Global Warming, and Energy Resources
 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 Structure and Replication
 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 Laws
 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 Lines and Surfaces
 23.6 Potential Due to Electric Dipole; Dipole Moment
 23.7 E→Determined from V
 23.8 Electrostatic Potential Energy; the Electron Volt
 23.9 Digital; Binary Numbers; Signal Voltage
 *23.10 TV and Computer Monitors
 *23.11 Electrocardiogram (ECG or EKG)
 Capacitance, Dielectrics, Electric Energy Storage
 24.1 Capacitors
 24.2 Determination of Capacitance
 24.3 Capacitors in Series and Parallel
 24.4 Storage of Electric Energy
 24.5 Dielectrics
 *24.6 Molecular Description of Dielectrics
 Electric Current 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
 *25.9 Superconductivity
 *25.10 Electrical Conduction in the Human Nervous System
 DC Circuits
 26.1 EMF and Terminal Voltage
 26.2 Resistors in Series and in Parallel
 26.3 Kirchhoff's Rules
 26.4 EMFs in Series and in Parallel; Charging a Battery
 26.5 RC Circuits Resistor and Capacitor in Series
 26.6 Electric Hazards and Safety
 26.7 Ammeters and VoltmetersMeasurement Affects Quantity Measured
 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 B→
 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: Motors, Loudspeakers, Galvanometers
 27.7 Discovery and Properties of the Electron
 27.8 The Hall Effect
 27.9 Mass Spectrometer
 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 Ampère's Law
 28.5 Magnetic Field of a Solenoid and a Toroid
 28.6 BiotSavart Law
 28.7 Magnetic Field Due to a Single Moving Charge
 28.8 Magnetic MaterialsFerromagnetism
 28.9 Electromagnets and SolenoidsApplications
 28.10 Magnetic Fields in Magnetic Materials; Hysteresis
 *28.11 Paramagnetism and Diamagnetism
 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 Information Storage: Magnetic and Semiconductor
 *29.9 Applications of Induction: Microphone, Seismograph, GFCI
 Inductance, Electromagnetic Oscillations, and AC Circuits
 30.1 Mutual Inductance
 30.2 SelfInductance; Inductors
 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 and Reactance
 30.8 LRC Series AC Circuit; Phasor Diagrams
 30.9 Resonance in AC Circuits
 30.10 Impedance Matching
 *30.11 ThreePhase AC
 Maxwell's Equations and Electromagnetic Waves
 31.1 Changing Electric Fields Produce Magnetic Fields; 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, Derived 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
 Light: Reflection and Refraction
 32.1 The Ray Model of Light
 32.2 Reflection; Image Formation by a Plane Mirror
 32.3 Formation of Images by Spherical Mirrors
 32.4 Seeing Yourself in a Magnifying Mirror (Concave)
 32.5 Convex (Rearview) Mirrors
 32.6 Index of Refraction
 32.7 Refraction: Snell's Law
 32.8 The Visible Spectrum and Dispersion
 32.9 Total Internal Reflection; Fiber Optics
 *32.10 Refraction at a Spherical Surface
 Lenses and Optical Instruments
 33.1 Thin Lenses; Ray Tracing and Focal Length
 33.2 The Thin Lens Equation
 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
 The Wave Nature of Light: Interference and Polarization
 34.1 Waves vs. Particles; Huygens' Principle and Diffraction
 34.2 Huygens' Principle and the Law of Refraction
 34.3 InterferenceYoung's DoubleSlit Experiment
 34.4 Intensity in the DoubleSlit Interference Pattern
 34.5 Interference in Thin Films
 34.6 Michelson Interferometer
 34.7 Polarization
 *34.8 Liquid Crystal Displays (LCD)
 *34.9 Scattering of Light by the Atmosphere
 34.10 Lumens, Luminous Flux, and Luminous Intensity
 *34.11 Efficiency of Lightbulbs
 Diffraction
 35.1 Diffraction by a Single Slit or Disk
 35.2 Intensity in SingleSlit Diffraction Pattern
 35.3 Diffraction in the DoubleSlit Experiment
 35.4 Interference vs. Diffraction
 35.5 Limits of Resolution; Circular Apertures
 35.6 Resolution of Telescopes and Microscopes; the Î» Limit
 35.7 Resolution of the Human Eye and Useful Magnification
 35.8 Diffraction Grating
 35.9 The Spectrometer and Spectroscopy
 *35.10 Peak Widths and Resolving Power for a Diffraction Grating
 35.11 XRays and XRay Diffraction
 *35.12 XRay Imaging and Computed Tomography (CT Scan)
 *35.13 Specialty Microscopes and Contrast
 The 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 FourDimensional Space.Time
 36.8 Galilean and Lorentz Transformations
 36.9 Relativistic Momentum
 36.10 The Ultimate Speed
 36.11 E = mc2; Mass and Energy
 36.12 Doppler Shift for Light
 36.13 The Impact of Special Relativity
 Early Quantum Theory and Models of the Atom
 37.1 Blackbody Radiation; Planck's Quantum Hypothesis
 37.2 Photon Theory of Light and the Photoelectric Effect
 37.3 Energy, Mass, and Momentum of a Photon
 37.4 Compton Effect
 37.5 Photon Interactions; Pair Production
 37.6 Wave.Particle Duality; the Principle of Complementarity
 37.7 Wave Nature of Matter
 37.8 Electron Microscopes
 37.9 Early Models of the Atom
 37.10 Atomic Spectra: Key to the Structure of the Atom
 37.11 The Bohr Model
 37.12 de Broglie's Hypothesis Applied to Atoms
 Quantum Mechanics
 38.1 Quantum MechanicsA New Theory
 38.2 The Wave Function and Its Interpretation; the DoubleSlit Experiment
 38.3 The Heisenberg Uncertainty Principle
 38.4 Philosophic Implications; Probability Versus Determinism
 38.5 The Schrödinger Equation in One Dimension TimeIndependent Form
 *38.6 TimeDependent Schrödinger 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
 Quantum Mechanics of Atoms
 39.1 QuantumMechanical View of Atoms
 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers
 39.3 Hydrogen Atom Wave Functions
 39.4 Multielectron Atoms; the Exclusion Principle
 39.5 Periodic Table of Elements
 39.6 XRay Spectra and Atomic Number
 *39.7 Magnetic Dipole Moment; Total Angular Momentum
 39.8 Fluorescence and Phosphorescence
 39.9 Lasers
 *39.10 Holography
 Molecules and Solids
 40.1 Bonding in Molecules
 40.2 PotentialEnergy Diagrams for Molecules
 40.3 Weak (van der Waals) Bonds
 40.4 Molecular Spectra
 40.5 Bonding in Solids
 40.6 FreeElectron Theory of Metals; Fermi Energy
 40.7 Band Theory of Solids
 40.8 Semiconductors and Doping
 40.9 Semiconductor Diodes, LEDs, OLEDs
 40.10 Transistors: Bipolar and MOSFETs
 40.11 Integrated Circuits, 14nm Technology
 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 HalfLife and Rate of Decay
 41.9 Decay Series
 41.10 Radioactive Dating
 41.11 Detection of Particles
 Nuclear Energy; Effects and Uses of Radiation
 42.1 Nuclear Reactions and the Transmutation of Elements
 42.2 Cross Section
 42.3 Nuclear Fission; Nuclear Reactors
 42.4 Nuclear Fusion
 42.5 Passage of Radiation Through Matter; Biological Damage
 42.6 Measurement of RadiationDosimetry
 *42.7 Radiation Therapy
 *42.8 Tracers in Research and Medicine
 *42.9 Emission Tomography: PET and SPECT
 *42.10 Nuclear Magnetic Resonance (NMR); Magnetic Resonance Imaging (MRI)
 Elementary Particles
 43.1 HighEnergy Particles and Accelerators
 43.2 Beginnings of Elementary Particle PhysicsParticle Exchange
 43.3 Particles and Antiparticles
 43.4 Particle Interactions and Conservation Laws
 43.5 Neutrinos
 43.6 Particle Classification
 43.7 Particle Stability and Resonances
 43.8 Strangeness? Charm? Towards a New Model
 43.9 Quarks
 43.10 The Standard Model: QCD and Electroweak Theory
 43.11 Grand Unified Theories
 43.12 Strings and Supersymmetry
 Astrophysics and Cosmology
 44.1 Stars and Galaxies
 44.2 Stellar Evolution: Birth and Death of Stars, Nucleosynthesis
 44.3 Distance Measurements
 44.4 General Relativity: Gravity and the Curvature of Space
 44.5 The Expanding Universe: Redshift and Hubble's Law
 44.6 The Big Bang and the Cosmic Microwave Background
 44.7 The Standard Cosmological Model: Early History of the Universe
 44.8I nflation: Explaining Flatness, Uniformity, and Structure
 44.9 Dark Matter and Dark Energy
 44.10 LargeScale Structure of the Universe
 44.11 Gravitational WavesLIGO
 44.12 Finally . . .
APPENDICES
 A. Mathematical Formulas
 B. Derivatives and Integrals
 C. Numerical Integration
 D. More on Dimensional Analysis
 E. Gravitational Force Due to a Spherical Mass Distribution
 F. Differential Form of Maxwell's Equations
 G. Selected Isotopes
Your questions answered
When you purchase an eTextbook subscription, it will last 4 months. You can renew your subscription by selecting Extend subscription on the Manage subscription page in My account before your initial term ends.
If you extend your subscription, we'll automatically charge you every month. If you made a one‑time payment for your initial 4‑month term, you'll now pay monthly. To make sure your learning is uninterrupted, please check your card details.
To avoid the next payment charge, select Cancel subscription on the Manage subscription page in My account before the renewal date. You can subscribe again in the future by purchasing another eTextbook subscription.
When you purchase a Channels subscription it will last 1 month, 3 months or 12 months, depending on the plan you chose. Your subscription will automatically renew at the end of your term unless you cancel it.
We use your credit card to renew your subscription automatically. To make sure your learning is uninterrupted, please check your card details.