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College Physics

28.6 Relativistic Energy

College Physics28.6 Relativistic Energy
  1. Preface
  2. 1 Introduction: The Nature of Science and Physics
    1. Introduction to Science and the Realm of Physics, Physical Quantities, and Units
    2. 1.1 Physics: An Introduction
    3. 1.2 Physical Quantities and Units
    4. 1.3 Accuracy, Precision, and Significant Figures
    5. 1.4 Approximation
    6. Glossary
    7. Section Summary
    8. Conceptual Questions
    9. Problems & Exercises
  3. 2 Kinematics
    1. Introduction to One-Dimensional Kinematics
    2. 2.1 Displacement
    3. 2.2 Vectors, Scalars, and Coordinate Systems
    4. 2.3 Time, Velocity, and Speed
    5. 2.4 Acceleration
    6. 2.5 Motion Equations for Constant Acceleration in One Dimension
    7. 2.6 Problem-Solving Basics for One-Dimensional Kinematics
    8. 2.7 Falling Objects
    9. 2.8 Graphical Analysis of One-Dimensional Motion
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
  4. 3 Two-Dimensional Kinematics
    1. Introduction to Two-Dimensional Kinematics
    2. 3.1 Kinematics in Two Dimensions: An Introduction
    3. 3.2 Vector Addition and Subtraction: Graphical Methods
    4. 3.3 Vector Addition and Subtraction: Analytical Methods
    5. 3.4 Projectile Motion
    6. 3.5 Addition of Velocities
    7. Glossary
    8. Section Summary
    9. Conceptual Questions
    10. Problems & Exercises
  5. 4 Dynamics: Force and Newton's Laws of Motion
    1. Introduction to Dynamics: Newton’s Laws of Motion
    2. 4.1 Development of Force Concept
    3. 4.2 Newton’s First Law of Motion: Inertia
    4. 4.3 Newton’s Second Law of Motion: Concept of a System
    5. 4.4 Newton’s Third Law of Motion: Symmetry in Forces
    6. 4.5 Normal, Tension, and Other Examples of Forces
    7. 4.6 Problem-Solving Strategies
    8. 4.7 Further Applications of Newton’s Laws of Motion
    9. 4.8 Extended Topic: The Four Basic Forces—An Introduction
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
  6. 5 Further Applications of Newton's Laws: Friction, Drag, and Elasticity
    1. Introduction: Further Applications of Newton’s Laws
    2. 5.1 Friction
    3. 5.2 Drag Forces
    4. 5.3 Elasticity: Stress and Strain
    5. Glossary
    6. Section Summary
    7. Conceptual Questions
    8. Problems & Exercises
  7. 6 Uniform Circular Motion and Gravitation
    1. Introduction to Uniform Circular Motion and Gravitation
    2. 6.1 Rotation Angle and Angular Velocity
    3. 6.2 Centripetal Acceleration
    4. 6.3 Centripetal Force
    5. 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force
    6. 6.5 Newton’s Universal Law of Gravitation
    7. 6.6 Satellites and Kepler’s Laws: An Argument for Simplicity
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  8. 7 Work, Energy, and Energy Resources
    1. Introduction to Work, Energy, and Energy Resources
    2. 7.1 Work: The Scientific Definition
    3. 7.2 Kinetic Energy and the Work-Energy Theorem
    4. 7.3 Gravitational Potential Energy
    5. 7.4 Conservative Forces and Potential Energy
    6. 7.5 Nonconservative Forces
    7. 7.6 Conservation of Energy
    8. 7.7 Power
    9. 7.8 Work, Energy, and Power in Humans
    10. 7.9 World Energy Use
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
  9. 8 Linear Momentum and Collisions
    1. Introduction to Linear Momentum and Collisions
    2. 8.1 Linear Momentum and Force
    3. 8.2 Impulse
    4. 8.3 Conservation of Momentum
    5. 8.4 Elastic Collisions in One Dimension
    6. 8.5 Inelastic Collisions in One Dimension
    7. 8.6 Collisions of Point Masses in Two Dimensions
    8. 8.7 Introduction to Rocket Propulsion
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  10. 9 Statics and Torque
    1. Introduction to Statics and Torque
    2. 9.1 The First Condition for Equilibrium
    3. 9.2 The Second Condition for Equilibrium
    4. 9.3 Stability
    5. 9.4 Applications of Statics, Including Problem-Solving Strategies
    6. 9.5 Simple Machines
    7. 9.6 Forces and Torques in Muscles and Joints
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  11. 10 Rotational Motion and Angular Momentum
    1. Introduction to Rotational Motion and Angular Momentum
    2. 10.1 Angular Acceleration
    3. 10.2 Kinematics of Rotational Motion
    4. 10.3 Dynamics of Rotational Motion: Rotational Inertia
    5. 10.4 Rotational Kinetic Energy: Work and Energy Revisited
    6. 10.5 Angular Momentum and Its Conservation
    7. 10.6 Collisions of Extended Bodies in Two Dimensions
    8. 10.7 Gyroscopic Effects: Vector Aspects of Angular Momentum
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  12. 11 Fluid Statics
    1. Introduction to Fluid Statics
    2. 11.1 What Is a Fluid?
    3. 11.2 Density
    4. 11.3 Pressure
    5. 11.4 Variation of Pressure with Depth in a Fluid
    6. 11.5 Pascal’s Principle
    7. 11.6 Gauge Pressure, Absolute Pressure, and Pressure Measurement
    8. 11.7 Archimedes’ Principle
    9. 11.8 Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
    10. 11.9 Pressures in the Body
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
  13. 12 Fluid Dynamics and Its Biological and Medical Applications
    1. Introduction to Fluid Dynamics and Its Biological and Medical Applications
    2. 12.1 Flow Rate and Its Relation to Velocity
    3. 12.2 Bernoulli’s Equation
    4. 12.3 The Most General Applications of Bernoulli’s Equation
    5. 12.4 Viscosity and Laminar Flow; Poiseuille’s Law
    6. 12.5 The Onset of Turbulence
    7. 12.6 Motion of an Object in a Viscous Fluid
    8. 12.7 Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  14. 13 Temperature, Kinetic Theory, and the Gas Laws
    1. Introduction to Temperature, Kinetic Theory, and the Gas Laws
    2. 13.1 Temperature
    3. 13.2 Thermal Expansion of Solids and Liquids
    4. 13.3 The Ideal Gas Law
    5. 13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature
    6. 13.5 Phase Changes
    7. 13.6 Humidity, Evaporation, and Boiling
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  15. 14 Heat and Heat Transfer Methods
    1. Introduction to Heat and Heat Transfer Methods
    2. 14.1 Heat
    3. 14.2 Temperature Change and Heat Capacity
    4. 14.3 Phase Change and Latent Heat
    5. 14.4 Heat Transfer Methods
    6. 14.5 Conduction
    7. 14.6 Convection
    8. 14.7 Radiation
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  16. 15 Thermodynamics
    1. Introduction to Thermodynamics
    2. 15.1 The First Law of Thermodynamics
    3. 15.2 The First Law of Thermodynamics and Some Simple Processes
    4. 15.3 Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
    5. 15.4 Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
    6. 15.5 Applications of Thermodynamics: Heat Pumps and Refrigerators
    7. 15.6 Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
    8. 15.7 Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  17. 16 Oscillatory Motion and Waves
    1. Introduction to Oscillatory Motion and Waves
    2. 16.1 Hooke’s Law: Stress and Strain Revisited
    3. 16.2 Period and Frequency in Oscillations
    4. 16.3 Simple Harmonic Motion: A Special Periodic Motion
    5. 16.4 The Simple Pendulum
    6. 16.5 Energy and the Simple Harmonic Oscillator
    7. 16.6 Uniform Circular Motion and Simple Harmonic Motion
    8. 16.7 Damped Harmonic Motion
    9. 16.8 Forced Oscillations and Resonance
    10. 16.9 Waves
    11. 16.10 Superposition and Interference
    12. 16.11 Energy in Waves: Intensity
    13. Glossary
    14. Section Summary
    15. Conceptual Questions
    16. Problems & Exercises
  18. 17 Physics of Hearing
    1. Introduction to the Physics of Hearing
    2. 17.1 Sound
    3. 17.2 Speed of Sound, Frequency, and Wavelength
    4. 17.3 Sound Intensity and Sound Level
    5. 17.4 Doppler Effect and Sonic Booms
    6. 17.5 Sound Interference and Resonance: Standing Waves in Air Columns
    7. 17.6 Hearing
    8. 17.7 Ultrasound
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  19. 18 Electric Charge and Electric Field
    1. Introduction to Electric Charge and Electric Field
    2. 18.1 Static Electricity and Charge: Conservation of Charge
    3. 18.2 Conductors and Insulators
    4. 18.3 Coulomb’s Law
    5. 18.4 Electric Field: Concept of a Field Revisited
    6. 18.5 Electric Field Lines: Multiple Charges
    7. 18.6 Electric Forces in Biology
    8. 18.7 Conductors and Electric Fields in Static Equilibrium
    9. 18.8 Applications of Electrostatics
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
  20. 19 Electric Potential and Electric Field
    1. Introduction to Electric Potential and Electric Energy
    2. 19.1 Electric Potential Energy: Potential Difference
    3. 19.2 Electric Potential in a Uniform Electric Field
    4. 19.3 Electrical Potential Due to a Point Charge
    5. 19.4 Equipotential Lines
    6. 19.5 Capacitors and Dielectrics
    7. 19.6 Capacitors in Series and Parallel
    8. 19.7 Energy Stored in Capacitors
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  21. 20 Electric Current, Resistance, and Ohm's Law
    1. Introduction to Electric Current, Resistance, and Ohm's Law
    2. 20.1 Current
    3. 20.2 Ohm’s Law: Resistance and Simple Circuits
    4. 20.3 Resistance and Resistivity
    5. 20.4 Electric Power and Energy
    6. 20.5 Alternating Current versus Direct Current
    7. 20.6 Electric Hazards and the Human Body
    8. 20.7 Nerve Conduction–Electrocardiograms
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  22. 21 Circuits and DC Instruments
    1. Introduction to Circuits and DC Instruments
    2. 21.1 Resistors in Series and Parallel
    3. 21.2 Electromotive Force: Terminal Voltage
    4. 21.3 Kirchhoff’s Rules
    5. 21.4 DC Voltmeters and Ammeters
    6. 21.5 Null Measurements
    7. 21.6 DC Circuits Containing Resistors and Capacitors
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  23. 22 Magnetism
    1. Introduction to Magnetism
    2. 22.1 Magnets
    3. 22.2 Ferromagnets and Electromagnets
    4. 22.3 Magnetic Fields and Magnetic Field Lines
    5. 22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
    6. 22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications
    7. 22.6 The Hall Effect
    8. 22.7 Magnetic Force on a Current-Carrying Conductor
    9. 22.8 Torque on a Current Loop: Motors and Meters
    10. 22.9 Magnetic Fields Produced by Currents: Ampere’s Law
    11. 22.10 Magnetic Force between Two Parallel Conductors
    12. 22.11 More Applications of Magnetism
    13. Glossary
    14. Section Summary
    15. Conceptual Questions
    16. Problems & Exercises
  24. 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
    1. Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
    2. 23.1 Induced Emf and Magnetic Flux
    3. 23.2 Faraday’s Law of Induction: Lenz’s Law
    4. 23.3 Motional Emf
    5. 23.4 Eddy Currents and Magnetic Damping
    6. 23.5 Electric Generators
    7. 23.6 Back Emf
    8. 23.7 Transformers
    9. 23.8 Electrical Safety: Systems and Devices
    10. 23.9 Inductance
    11. 23.10 RL Circuits
    12. 23.11 Reactance, Inductive and Capacitive
    13. 23.12 RLC Series AC Circuits
    14. Glossary
    15. Section Summary
    16. Conceptual Questions
    17. Problems & Exercises
  25. 24 Electromagnetic Waves
    1. Introduction to Electromagnetic Waves
    2. 24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed
    3. 24.2 Production of Electromagnetic Waves
    4. 24.3 The Electromagnetic Spectrum
    5. 24.4 Energy in Electromagnetic Waves
    6. Glossary
    7. Section Summary
    8. Conceptual Questions
    9. Problems & Exercises
  26. 25 Geometric Optics
    1. Introduction to Geometric Optics
    2. 25.1 The Ray Aspect of Light
    3. 25.2 The Law of Reflection
    4. 25.3 The Law of Refraction
    5. 25.4 Total Internal Reflection
    6. 25.5 Dispersion: The Rainbow and Prisms
    7. 25.6 Image Formation by Lenses
    8. 25.7 Image Formation by Mirrors
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  27. 26 Vision and Optical Instruments
    1. Introduction to Vision and Optical Instruments
    2. 26.1 Physics of the Eye
    3. 26.2 Vision Correction
    4. 26.3 Color and Color Vision
    5. 26.4 Microscopes
    6. 26.5 Telescopes
    7. 26.6 Aberrations
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  28. 27 Wave Optics
    1. Introduction to Wave Optics
    2. 27.1 The Wave Aspect of Light: Interference
    3. 27.2 Huygens's Principle: Diffraction
    4. 27.3 Young’s Double Slit Experiment
    5. 27.4 Multiple Slit Diffraction
    6. 27.5 Single Slit Diffraction
    7. 27.6 Limits of Resolution: The Rayleigh Criterion
    8. 27.7 Thin Film Interference
    9. 27.8 Polarization
    10. 27.9 *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
  29. 28 Special Relativity
    1. Introduction to Special Relativity
    2. 28.1 Einstein’s Postulates
    3. 28.2 Simultaneity And Time Dilation
    4. 28.3 Length Contraction
    5. 28.4 Relativistic Addition of Velocities
    6. 28.5 Relativistic Momentum
    7. 28.6 Relativistic Energy
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  30. 29 Introduction to Quantum Physics
    1. Introduction to Quantum Physics
    2. 29.1 Quantization of Energy
    3. 29.2 The Photoelectric Effect
    4. 29.3 Photon Energies and the Electromagnetic Spectrum
    5. 29.4 Photon Momentum
    6. 29.5 The Particle-Wave Duality
    7. 29.6 The Wave Nature of Matter
    8. 29.7 Probability: The Heisenberg Uncertainty Principle
    9. 29.8 The Particle-Wave Duality Reviewed
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
  31. 30 Atomic Physics
    1. Introduction to Atomic Physics
    2. 30.1 Discovery of the Atom
    3. 30.2 Discovery of the Parts of the Atom: Electrons and Nuclei
    4. 30.3 Bohr’s Theory of the Hydrogen Atom
    5. 30.4 X Rays: Atomic Origins and Applications
    6. 30.5 Applications of Atomic Excitations and De-Excitations
    7. 30.6 The Wave Nature of Matter Causes Quantization
    8. 30.7 Patterns in Spectra Reveal More Quantization
    9. 30.8 Quantum Numbers and Rules
    10. 30.9 The Pauli Exclusion Principle
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
  32. 31 Radioactivity and Nuclear Physics
    1. Introduction to Radioactivity and Nuclear Physics
    2. 31.1 Nuclear Radioactivity
    3. 31.2 Radiation Detection and Detectors
    4. 31.3 Substructure of the Nucleus
    5. 31.4 Nuclear Decay and Conservation Laws
    6. 31.5 Half-Life and Activity
    7. 31.6 Binding Energy
    8. 31.7 Tunneling
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  33. 32 Medical Applications of Nuclear Physics
    1. Introduction to Applications of Nuclear Physics
    2. 32.1 Medical Imaging and Diagnostics
    3. 32.2 Biological Effects of Ionizing Radiation
    4. 32.3 Therapeutic Uses of Ionizing Radiation
    5. 32.4 Food Irradiation
    6. 32.5 Fusion
    7. 32.6 Fission
    8. 32.7 Nuclear Weapons
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  34. 33 Particle Physics
    1. Introduction to Particle Physics
    2. 33.1 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited
    3. 33.2 The Four Basic Forces
    4. 33.3 Accelerators Create Matter from Energy
    5. 33.4 Particles, Patterns, and Conservation Laws
    6. 33.5 Quarks: Is That All There Is?
    7. 33.6 GUTs: The Unification of Forces
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  35. 34 Frontiers of Physics
    1. Introduction to Frontiers of Physics
    2. 34.1 Cosmology and Particle Physics
    3. 34.2 General Relativity and Quantum Gravity
    4. 34.3 Superstrings
    5. 34.4 Dark Matter and Closure
    6. 34.5 Complexity and Chaos
    7. 34.6 High-temperature Superconductors
    8. 34.7 Some Questions We Know to Ask
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  36. A | Atomic Masses
  37. B | Selected Radioactive Isotopes
  38. C | Useful Information
  39. D | Glossary of Key Symbols and Notation
  40. Index
This photo shows the outside of the fusion reactor of the National Spherical Torus Experiment at the Princeton Plasma Physics Laboratory. The reactor, which sits in a large room, is connected to numerous tubes and instruments.
Figure 28.20 The National Spherical Torus Experiment (NSTX) has a fusion reactor in which hydrogen isotopes undergo fusion to produce helium. In this process, a relatively small mass of fuel is converted into a large amount of energy. (credit: Princeton Plasma Physics Laboratory)

A tokamak is a form of experimental fusion reactor, which can change mass to energy. Accomplishing this requires an understanding of relativistic energy. Nuclear reactors are proof of the conservation of relativistic energy.

Conservation of energy is one of the most important laws in physics. Not only does energy have many important forms, but each form can be converted to any other. We know that classically the total amount of energy in a system remains constant. Relativistically, energy is still conserved, provided its definition is altered to include the possibility of mass changing to energy, as in the reactions that occur within a nuclear reactor. Relativistic energy is intentionally defined so that it will be conserved in all inertial frames, just as is the case for relativistic momentum. As a consequence, we learn that several fundamental quantities are related in ways not known in classical physics. All of these relationships are verified by experiment and have fundamental consequences. The altered definition of energy contains some of the most fundamental and spectacular new insights into nature found in recent history.

Total Energy and Rest Energy

The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy is valid relativistically, if we define energy to include a relativistic factor.

Total Energy

Total energy EE size 12{E} {} is defined to be

E=γmc2,E=γmc2, size 12{E= ital "γmc" rSup { size 8{2} } } {}
28.43

where mm size 12{m} {} is mass, cc size 12{c} {} is the speed of light, γ=11v2c2γ=11v2c2 size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {}, and vv size 12{v} {} is the velocity of the mass relative to an observer. There are many aspects of the total energy EE size 12{E} {} that we will discuss—among them are how kinetic and potential energies are included in EE size 12{E} {}, and how EE size 12{E} {} is related to relativistic momentum. But first, note that at rest, total energy is not zero. Rather, when v=0v=0 size 12{v=0} {}, we have γ=1γ=1 size 12{γ=1} {}, and an object has rest energy.

Rest Energy

Rest energy is

E0=mc2.E0=mc2.
28.44

This is the correct form of Einstein’s most famous equation, which for the first time showed that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. The implications of these first two equations regarding relativistic energy are so broad that they were not completely recognized for some years after Einstein published them in 1907, nor was the experimental proof that they are correct widely recognized at first. Einstein, it should be noted, did understand and describe the meanings and implications of his theory.

Example 28.6 Calculating Rest Energy: Rest Energy is Very Large

Calculate the rest energy of a 1.00-g mass.

Strategy

One gram is a small mass—less than half the mass of a penny. We can multiply this mass, in SI units, by the speed of light squared to find the equivalent rest energy.

Solution

  1. Identify the knowns. m=1.00×103kgm=1.00×103kg size 12{m=1 "." "00" times "10" rSup { size 8{ - 3} } `"kg"} {}; c=3.00×108m/sc=3.00×108m/s size 12{c=3 "." "00" times "10" rSup { size 8{8} } `"m/s"} {}
  2. Identify the unknown. E0E0 size 12{E rSub { size 8{0} } } {}
  3. Choose the appropriate equation. E0=mc2E0=mc2 size 12{E rSub { size 8{0} } = ital "mc" rSup { size 8{2} } } {}
  4. Plug the knowns into the equation.
    E 0 = mc 2 = ( 1.00 × 10 3 kg ) ( 3.00 × 10 8 m/s ) 2 = 9.00 × 10 13 kg m 2 /s 2 E 0 = mc 2 = ( 1.00 × 10 3 kg ) ( 3.00 × 10 8 m/s ) 2 = 9.00 × 10 13 kg m 2 /s 2
    28.45
  5. Convert units.

    Noting that 1kgm2/s2=1 J1kgm2/s2=1 J size 12{1`"kg" cdot m rSup { size 8{2} } "/s" rSup { size 8{2} } =1`J} {}, we see the rest mass energy is

    E0=9.00×1013J.E0=9.00×1013J. size 12{E rSub { size 8{0} } =9 "." "00" times "10" rSup { size 8{"13"} } `J} {}
    28.46

Discussion

This is an enormous amount of energy for a 1.00-g mass. We do not notice this energy, because it is generally not available. Rest energy is large because the speed of light cc size 12{c} {} is a large number and c2c2 size 12{c rSup { size 8{2} } } {} is a very large number, so that mc2mc2 size 12{ ital "mc" rSup { size 8{2} } } {} is huge for any macroscopic mass. The 9.00×1013J9.00×1013J size 12{9 "." "00" times "10" rSup { size 8{"13"} } `J} {} rest mass energy for 1.00 g is about twice the energy released by the Hiroshima atomic bomb and about 10,000 times the kinetic energy of a large aircraft carrier. If a way can be found to convert rest mass energy into some other form (and all forms of energy can be converted into one another), then huge amounts of energy can be obtained from the destruction of mass.

Today, the practical applications of the conversion of mass into another form of energy, such as in nuclear weapons and nuclear power plants, are well known. But examples also existed when Einstein first proposed the correct form of relativistic energy, and he did describe some of them. Nuclear radiation had been discovered in the previous decade, and it had been a mystery as to where its energy originated. The explanation was that, in certain nuclear processes, a small amount of mass is destroyed and energy is released and carried by nuclear radiation. But the amount of mass destroyed is so small that it is difficult to detect that any is missing. Although Einstein proposed this as the source of energy in the radioactive salts then being studied, it was many years before there was broad recognition that mass could be and, in fact, commonly is converted to energy. (See Figure 28.21.)

Part a of the figure shows a solar storm on the Sun. Part b of the figure shows the Susquehanna Steam Electric Station, which produces electricity by nuclear fission.
Figure 28.21 The Sun (a) and the Susquehanna Steam Electric Station (b) both convert mass into energy—the Sun via nuclear fusion, the electric station via nuclear fission. (credits: (a) NASA/Goddard Space Flight Center, Scientific Visualization Studio; (b) U.S. government)

Because of the relationship of rest energy to mass, we now consider mass to be a form of energy rather than something separate. There had not even been a hint of this prior to Einstein’s work. Such conversion is now known to be the source of the Sun’s energy, the energy of nuclear decay, and even the source of energy keeping Earth’s interior hot.

Stored Energy and Potential Energy

What happens to energy stored in an object at rest, such as the energy put into a battery by charging it, or the energy stored in a toy gun’s compressed spring? The energy input becomes part of the total energy of the object and, thus, increases its rest mass. All stored and potential energy becomes mass in a system. Why is it we don’t ordinarily notice this? In fact, conservation of mass (meaning total mass is constant) was one of the great laws verified by 19th-century science. Why was it not noticed to be incorrect? The following example helps answer these questions.

Example 28.7 Calculating Rest Mass: A Small Mass Increase due to Energy Input

A car battery is rated to be able to move 600 ampere-hours (A·h)(A·h) of charge at 12.0 V. (a) Calculate the increase in rest mass of such a battery when it is taken from being fully depleted to being fully charged. (b) What percent increase is this, given the battery’s mass is 20.0 kg?

Strategy

In part (a), we first must find the energy stored in the battery, which equals what the battery can supply in the form of electrical potential energy. Since PEelec=qVPEelec=qV size 12{"PE" rSub { size 8{"elec"} } = ital "qV"} {}, we have to calculate the charge qq size 12{q} {} in 600A·h600A·h, which is the product of the current II and the time tt size 12{t} {}. We then multiply the result by 12.0 V. We can then calculate the battery’s increase in mass using ΔE=PEelec=(Δm)c2ΔE=PEelec=(Δm)c2 size 12{ΔE="PE" rSub { size 8{"elec"} } = \( Δm \) c rSup { size 8{2} } } {}. Part (b) is a simple ratio converted to a percentage.

Solution for (a)

  1. Identify the knowns. It=600 AhIt=600 Ah; V=12.0VV=12.0V size 12{V="12" "." 0`V} {}; c=3.00×108m/sc=3.00×108m/s
  2. Identify the unknown. ΔmΔm size 12{m} {}
  3. Choose the appropriate equation. PEelec=(Δm)c2PEelec=(Δm)c2
  4. Rearrange the equation to solve for the unknown. Δm=PEelecc2Δm=PEelecc2 size 12{m= { {"PE" rSub { size 8{"elec"} } } over {c rSup { size 8{2} } } } } {}
  5. Plug the knowns into the equation.
    Δm = PE elec c 2 = qV c 2 = ( I t ) V c 2 = ( 600 A h ) ( 12.0 V ) ( 3.00 × 10 8 ) 2 . Δm = PE elec c 2 = qV c 2 = ( I t ) V c 2 = ( 600 A h ) ( 12.0 V ) ( 3.00 × 10 8 ) 2 .
    28.47

    Write amperes A as coulombs per second (C/s), and convert hours to seconds.

    Δm = ( 600 C/s h 3600 s 1 h ( 12.0 J/C ) ( 3.00 × 10 8 m/s ) 2 = ( 2.16 × 10 6 C ) ( 12.0 J/C ) ( 3.00 × 10 8 m/s ) 2 Δm = ( 600 C/s h 3600 s 1 h ( 12.0 J/C ) ( 3.00 × 10 8 m/s ) 2 = ( 2.16 × 10 6 C ) ( 12.0 J/C ) ( 3.00 × 10 8 m/s ) 2
    28.48

    Using the conversion 1kgm2/s2=1J1kgm2/s2=1J, we can write the mass as

    Δm = 2.88 × 10 10 kg . Δm = 2.88 × 10 10 kg .

Solution for (b)

  1. Identify the knowns. Δm=2.88×1010kgΔm=2.88×1010kg; m=20.0 kgm=20.0 kg
  2. Identify the unknown. % change
  3. Choose the appropriate equation. % increase=Δmm×100%% increase=Δmm×100% size 12{%" increase"= { {Δm} over {m} } times "100"%} {}
  4. Plug the knowns into the equation.
    % increase = Δm m × 100% = 2.88 × 10 10 kg 20.0 kg × 100% = 1.44 × 10 9 % . % increase = Δm m × 100% = 2.88 × 10 10 kg 20.0 kg × 100% = 1.44 × 10 9 % .
    28.49

Discussion

Both the actual increase in mass and the percent increase are very small, since energy is divided by c2c2 size 12{c rSup { size 8{2} } } {}, a very large number. We would have to be able to measure the mass of the battery to a precision of a billionth of a percent, or 1 part in 10111011, to notice this increase. It is no wonder that the mass variation is not readily observed. In fact, this change in mass is so small that we may question how you could verify it is real. The answer is found in nuclear processes in which the percentage of mass destroyed is large enough to be measured. The mass of the fuel of a nuclear reactor, for example, is measurably smaller when its energy has been used. In that case, stored energy has been released (converted mostly to heat and electricity) and the rest mass has decreased. This is also the case when you use the energy stored in a battery, except that the stored energy is much greater in nuclear processes, making the change in mass measurable in practice as well as in theory.

Kinetic Energy and the Ultimate Speed Limit

Kinetic energy is energy of motion. Classically, kinetic energy has the familiar expression 12mv212mv2 size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } } {}. The relativistic expression for kinetic energy is obtained from the work-energy theorem. This theorem states that the net work on a system goes into kinetic energy. If our system starts from rest, then the work-energy theorem is

Wnet=KE.Wnet=KE. size 12{W rSub { size 8{"net"} } ="KE"} {}
28.50

Relativistically, at rest we have rest energy E0=mc2E0=mc2. The work increases this to the total energy E=γmc2E=γmc2. Thus,

Wnet=EE0=γmc2mc2=γ1mc2.Wnet=EE0=γmc2mc2=γ1mc2.
28.51

Relativistically, we have Wnet=KErelWnet=KErel size 12{W="KE" rSub { size 8{"rel"} } } {}.

Relativistic Kinetic Energy

Relativistic kinetic energy is

KErel=γ1mc2.KErel=γ1mc2. size 12{"KE" rSub { size 8{"rel"} } = left (γ - 1 right ) ital "mc" rSup { size 8{2} } } {}
28.52

When motionless, we have v=0v=0 size 12{v=0} {} and

γ=11v2c2=1,γ=11v2c2=1, size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } =1} {}
28.53

so that KErel=0KErel=0 size 12{"KE" rSub { size 8{"rel"} } =0} {} at rest, as expected. But the expression for relativistic kinetic energy (such as total energy and rest energy) does not look much like the classical 12mv212mv2 size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } } {}. To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for γγ size 12{γ} {} at low velocities gives

γ=1+12v2c2.γ=1+12v2c2. size 12{γ=1+ { {1} over {2} } { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } {}
28.54

A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In some cases, as in the limit of small velocity here, most terms are very small. Thus the expression derived for γγ size 12{γ} {} here is not exact, but it is a very accurate approximation. Thus, at low velocities,

γ1=12v2c2.γ1=12v2c2. size 12{γ - 1= { {1} over {2} } { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } {}
28.55

Entering this into the expression for relativistic kinetic energy gives

KErel=12v2c2mc2=12mv2=KEclass.KErel=12v2c2mc2=12mv2=KEclass.
28.56

So, in fact, relativistic kinetic energy does become the same as classical kinetic energy when v<<cv<<c size 12{v"<<"c} {}.

It is even more interesting to investigate what happens to kinetic energy when the velocity of an object approaches the speed of light. We know that γγ size 12{γ} {} becomes infinite as vv size 12{v} {} approaches cc size 12{c} {}, so that KErel also becomes infinite as the velocity approaches the speed of light. (See Figure 28.22.) An infinite amount of work (and, hence, an infinite amount of energy input) is required to accelerate a mass to the speed of light.

The Speed of Light

No object with mass can attain the speed of light.

So the speed of light is the ultimate speed limit for any particle having mass. All of this is consistent with the fact that velocities less than cc size 12{c} {} always add to less than cc size 12{c} {}. Both the relativistic form for kinetic energy and the ultimate speed limit being cc size 12{c} {} have been confirmed in detail in numerous experiments. No matter how much energy is put into accelerating a mass, its velocity can only approach—not reach—the speed of light.

In this figure a graph is shown on a coordinate system of axes. The x-axis is labeled as speed v (m/s). On the x-axis, velocity of the object is shown in terms of the speed of light starting from zero at origin to c, where c is the speed of light. The y-axis is labeled as Kinetic Energy K E (J). On the y-axis, relativistic kinetic energy is shown starting from 0 at origin to 1.0. The graph K sub r e l of relativistic kinetic energy is concave up and moving upward along the vertical line at x equals c. This graph shows that relativistic kinetic energy approaches infinity as the velocity of an object approaches the speed of light. Also shown is that when the speed of the object is equal to the speed of light c the kinetic energy is known as classical kinetic energy, which is denoted as K E sub class.
Figure 28.22 This graph of KErelKErel size 12{"KE" rSub { size 8{"rel"} } } {} versus velocity shows how kinetic energy approaches infinity as velocity approaches the speed of light. It is thus not possible for an object having mass to reach the speed of light. Also shown is KEclassKEclass size 12{"KE" rSub { size 8{"class"} } } {}, the classical kinetic energy, which is similar to relativistic kinetic energy at low velocities. Note that much more energy is required to reach high velocities than predicted classically.

Example 28.8 Comparing Kinetic Energy: Relativistic Energy Versus Classical Kinetic Energy

An electron has a velocity v=0.990cv=0.990c size 12{v=0 "." "990"c} {}. (a) Calculate the kinetic energy in MeV of the electron. (b) Compare this with the classical value for kinetic energy at this velocity. (The mass of an electron is 9.11×1031 kg9.11×1031 kg size 12{9 "." "11" times "10" rSup { size 8{ - "31"} } " kg"} {}.)

Strategy

The expression for relativistic kinetic energy is always correct, but for (a) it must be used since the velocity is highly relativistic (close to cc size 12{c} {}). First, we will calculate the relativistic factor γγ size 12{γ} {}, and then use it to determine the relativistic kinetic energy. For (b), we will calculate the classical kinetic energy (which would be close to the relativistic value if vv size 12{v} {} were less than a few percent of cc size 12{c} {}) and see that it is not the same.

Solution for (a)

  1. Identify the knowns. v=0.990cv=0.990c size 12{v=0 "." "990"c} {}; m=9.11×1031kgm=9.11×1031kg size 12{m=9 "." "11" times "10" rSup { size 8{ - "31"} } `"kg"} {}
  2. Identify the unknown. KErelKErel size 12{"KE" rSub { size 8{"rel"} } } {}
  3. Choose the appropriate equation. KErel=γ1mc2KErel=γ1mc2 size 12{"KE" rSub { size 8{"rel"} } = left (γ - 1 right ) ital "mc" rSup { size 8{2} } } {}
  4. Plug the knowns into the equation.

    First calculate γγ size 12{γ} {}. We will carry extra digits because this is an intermediate calculation.

    γ = 1 1 v 2 c 2 = 1 1 ( 0 . 990 c ) 2 c 2 = 1 1 ( 0 . 990 ) 2 = 7 . 0888 γ = 1 1 v 2 c 2 = 1 1 ( 0 . 990 c ) 2 c 2 = 1 1 ( 0 . 990 ) 2 = 7 . 0888 alignl { stack { size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {} # = { {1} over { sqrt {1 - { { \( 0 "." "990" ital " c" \) rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } {} # - { {1} over { sqrt {1 - \( 0 "." "990" \) rSup { size 8{2} } } } } {} # =7 "." "0888" {} } } {}
    28.57

    Next, we use this value to calculate the kinetic energy.

    KE rel = ( γ 1 ) mc 2 = ( 7.0888 1 ) ( 9.11 × 10 31 kg ) ( 3.00 × 10 8 m/s ) 2 = 4.99 × 10 –13 J KE rel = ( γ 1 ) mc 2 = ( 7.0888 1 ) ( 9.11 × 10 31 kg ) ( 3.00 × 10 8 m/s ) 2 = 4.99 × 10 –13 J
    28.58
  5. Convert units.
    KE rel = ( 4.99 × 10 –13 J ) 1 MeV 1.60 × 10 13 J = 3.12 MeV KE rel = ( 4.99 × 10 –13 J ) 1 MeV 1.60 × 10 13 J = 3.12 MeV alignl { stack { size 12{"KE" rSub { size 8{"rel"} } = \( 4 "." "99" times "10" rSup { size 8{"-13"} } " J" \) left ( { {"1 MeV"} over {1 "." "60" times "10" rSup { size 8{ - "13"} } " J"} } right )} {} # =3 "." "12"" MeV" {} } } {}
    28.59

Solution for (b)

  1. List the knowns. v=0.990cv=0.990c size 12{v=0 "." "990"c} {}; m=9.11×1031kgm=9.11×1031kg
  2. List the unknown. KEclassKEclass
  3. Choose the appropriate equation. KEclass=12mv2KEclass=12mv2
  4. Plug the knowns into the equation.
    KE class = 1 2 mv 2 = 1 2 ( 9.00 × 10 31 kg ) ( 0.990 ) 2 ( 3.00 × 10 8 m/s ) 2 = 4.02 × 10 14 J KE class = 1 2 mv 2 = 1 2 ( 9.00 × 10 31 kg ) ( 0.990 ) 2 ( 3.00 × 10 8 m/s ) 2 = 4.02 × 10 14 J
    28.60
  5. Convert units.
    KE class = 4.02 × 10 14 J 1 MeV 1.60 × 10 13 J = 0.251 MeV KE class = 4.02 × 10 14 J 1 MeV 1.60 × 10 13 J = 0.251 MeV
    28.61

Discussion

As might be expected, since the velocity is 99.0% of the speed of light, the classical kinetic energy is significantly off from the correct relativistic value. Note also that the classical value is much smaller than the relativistic value. In fact, KErel/KEclass=12.4KErel/KEclass=12.4 size 12{"KE" rSub { size 8{"rel"} } "/KE" rSub { size 8{"class"} } ="12" "." 4} {} here. This is some indication of how difficult it is to get a mass moving close to the speed of light. Much more energy is required than predicted classically. Some people interpret this extra energy as going into increasing the mass of the system, but, as discussed in Relativistic Momentum, this cannot be verified unambiguously. What is certain is that ever-increasing amounts of energy are needed to get the velocity of a mass a little closer to that of light. An energy of 3 MeV is a very small amount for an electron, and it can be achieved with present-day particle accelerators. SLAC, for example, can accelerate electrons to over 50×109eV=50,000 MeV50×109eV=50,000 MeV size 12{"50" times "10" rSup { size 8{9} } "eV"="50,000"`"MeV"} {}.

Is there any point in getting vv size 12{v} {} a little closer to c than 99.0% or 99.9%? The answer is yes. We learn a great deal by doing this. The energy that goes into a high-velocity mass can be converted to any other form, including into entirely new masses. (See Figure 28.23.) Most of what we know about the substructure of matter and the collection of exotic short-lived particles in nature has been learned this way. Particles are accelerated to extremely relativistic energies and made to collide with other particles, producing totally new species of particles. Patterns in the characteristics of these previously unknown particles hint at a basic substructure for all matter. These particles and some of their characteristics will be covered in Particle Physics.

An aerial view of the Fermi National Accelerator Laboratory. The accelerator has two large, ring shaped structures. There are circular ponds near the rings.
Figure 28.23 The Fermi National Accelerator Laboratory, near Batavia, Illinois, was a subatomic particle collider that accelerated protons and antiprotons to attain energies up to 1 Tev (a trillion electronvolts). The circular ponds near the rings were built to dissipate waste heat. This accelerator was shut down in September 2011. (credit: Fermilab, Reidar Hahn)

Relativistic Energy and Momentum

We know classically that kinetic energy and momentum are related to each other, since

KEclass=p22m=(mv)22m=12mv2.KEclass=p22m=(mv)22m=12mv2.
28.62

Relativistically, we can obtain a relationship between energy and momentum by algebraically manipulating their definitions. This produces

E2=(pc)2+(mc2)2,E2=(pc)2+(mc2)2, size 12{E rSup { size 8{2} } = \( ital "pc" \) rSup { size 8{2} } + \( ital "mc" \) rSup { size 8{2} } } {}
28.63

where EE size 12{E} {} is the relativistic total energy and pp size 12{p} {} is the relativistic momentum. This relationship between relativistic energy and relativistic momentum is more complicated than the classical, but we can gain some interesting new insights by examining it. First, total energy is related to momentum and rest mass. At rest, momentum is zero, and the equation gives the total energy to be the rest energy mc2mc2 (so this equation is consistent with the discussion of rest energy above). However, as the mass is accelerated, its momentum pp increases, thus increasing the total energy. At sufficiently high velocities, the rest energy term (mc2)2(mc2)2 becomes negligible compared with the momentum term (pc)2(pc)2; thus, E=pcE=pc at extremely relativistic velocities.

If we consider momentum pp size 12{p} {} to be distinct from mass, we can determine the implications of the equation E2=(pc)2+(mc2)2,E2=(pc)2+(mc2)2, size 12{E rSup { size 8{2} } = \( ital "pc" \) rSup { size 8{2} } + \( ital "mc" \) rSup { size 8{2} } } {} for a particle that has no mass. If we take mm size 12{m} {} to be zero in this equation, then E=pcE=pc size 12{E= ital "pc"} {}, or p=E/cp=E/c size 12{p=E/c} {}. Massless particles have this momentum. There are several massless particles found in nature, including photons (these are quanta of electromagnetic radiation). Another implication is that a massless particle must travel at speed cc size 12{c} {} and only at speed cc size 12{c} {}. While it is beyond the scope of this text to examine the relationship in the equation E2=(pc)2+(mc2)2,E2=(pc)2+(mc2)2, size 12{E rSup { size 8{2} } = \( ital "pc" \) rSup { size 8{2} } + \( ital "mc" \) rSup { size 8{2} } } {} in detail, we can see that the relationship has important implications in special relativity.

Problem-Solving Strategies for Relativity

  1. Examine the situation to determine that it is necessary to use relativity. Relativistic effects are related to γ=11v2c2γ=11v2c2 size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {}, the quantitative relativistic factor. If γγ size 12{γ} {} is very close to 1, then relativistic effects are small and differ very little from the usually easier classical calculations.
  2. Identify exactly what needs to be determined in the problem (identify the unknowns).
  3. Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Look in particular for information on relative velocity vv size 12{v} {}.
  4. Make certain you understand the conceptual aspects of the problem before making any calculations. Decide, for example, which observer sees time dilated or length contracted before plugging into equations. If you have thought about who sees what, who is moving with the event being observed, who sees proper time, and so on, you will find it much easier to determine if your calculation is reasonable.
  5. Determine the primary type of calculation to be done to find the unknowns identified above. You will find the section summary helpful in determining whether a length contraction, relativistic kinetic energy, or some other concept is involved.
  6. Do not round off during the calculation. As noted in the text, you must often perform your calculations to many digits to see the desired effect. You may round off at the very end of the problem, but do not use a rounded number in a subsequent calculation.
  7. Check the answer to see if it is reasonable: Does it make sense? This may be more difficult for relativity, since we do not encounter it directly. But you can look for velocities greater than cc size 12{c} {} or relativistic effects that are in the wrong direction (such as a time contraction where a dilation was expected).
Check Your Understanding
Exercise 4

A photon decays into an electron-positron pair. What is the kinetic energy of the electron if its speed is 0.992c0.992c size 12{c} {}?

Answer
KE rel = ( γ 1 ) mc 2 = 1 1 v 2 c 2 1 mc 2 = 1 1 ( 0.992 c ) 2 c 2 1 ( 9.11 × 10 31 kg ) ( 3.00 × 10 8 m/s ) 2 = 5.67 × 10 13 J KE rel = ( γ 1 ) mc 2 = 1 1 v 2 c 2 1 mc 2 = 1 1 ( 0.992 c ) 2 c 2 1 ( 9.11 × 10 31 kg ) ( 3.00 × 10 8 m/s ) 2 = 5.67 × 10 13 J
28.64
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