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College Physics

Problems & Exercises

College PhysicsProblems & Exercises
  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

33.1 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited

1.

A virtual particle having an approximate mass of 1014GeV/c21014GeV/c2 size 12{"10" rSup { size 8{"14"} } `"GeV/"c rSup { size 8{2} } } {} may be associated with the unification of the strong and electroweak forces. For what length of time could this virtual particle exist (in temporary violation of the conservation of mass-energy as allowed by the Heisenberg uncertainty principle)?

2.

Calculate the mass in GeV/c2GeV/c2 size 12{"GeV/"c rSup { size 8{2} } } {} of a virtual carrier particle that has a range limited to 10301030 size 12{"10" rSup { size 8{ - "30"} } } {} m by the Heisenberg uncertainty principle. Such a particle might be involved in the unification of the strong and electroweak forces.

3.

Another component of the strong nuclear force is transmitted by the exchange of virtual K-mesons. Taking K-mesons to have an average mass of 495MeV/c2495MeV/c2 size 12{"495"`"MeV/"c rSup { size 8{2} } } {}, what is the approximate range of this component of the strong force?

33.2 The Four Basic Forces

4.

(a) Find the ratio of the strengths of the weak and electromagnetic forces under ordinary circumstances.

(b) What does that ratio become under circumstances in which the forces are unified?

5.

The ratio of the strong to the weak force and the ratio of the strong force to the electromagnetic force become 1 under circumstances where they are unified. What are the ratios of the strong force to those two forces under normal circumstances?

33.3 Accelerators Create Matter from Energy

6.

At full energy, protons in the 2.00-km-diameter Fermilab synchrotron travel at nearly the speed of light, since their energy is about 1000 times their rest mass energy.

(a) How long does it take for a proton to complete one trip around?

(b) How many times per second will it pass through the target area?

7.

Suppose a WW size 12{W rSup { size 8{ - {}} } } {} created in a bubble chamber lives for 5.00×1025s.5.00×1025s size 12{5 "." "00" times "10" rSup { size 8{ - "25"} } `s} {}. What distance does it move in this time if it is traveling at 0.900 c? Since this distance is too short to make a track, the presence of the WW size 12{W rSup { size 8{ - {}} } } {} must be inferred from its decay products. Note that the time is longer than the given WW size 12{W rSup { size 8{ - {}} } } {} lifetime, which can be due to the statistical nature of decay or time dilation.

8.

What length track does a π+π+ size 12{π rSup { size 8{+{}} } } {} traveling at 0.100 c leave in a bubble chamber if it is created there and lives for 2.60×108s2.60×108s size 12{2 "." "60" times "10" rSup { size 8{ - 8} } `s} {}? (Those moving faster or living longer may escape the detector before decaying.)

9.

The 3.20-km-long SLAC produces a beam of 50.0-GeV electrons. If there are 15,000 accelerating tubes, what average voltage must be across the gaps between them to achieve this energy?

10.

Because of energy loss due to synchrotron radiation in the LHC at CERN, only 5.00 MeV is added to the energy of each proton during each revolution around the main ring. How many revolutions are needed to produce 7.00-TeV (7000 GeV) protons, if they are injected with an initial energy of 8.00 GeV?

11.

A proton and an antiproton collide head-on, with each having a kinetic energy of 7.00 TeV (such as in the LHC at CERN). How much collision energy is available, taking into account the annihilation of the two masses? (Note that this is not significantly greater than the extremely relativistic kinetic energy.)

12.

When an electron and positron collide at the SLAC facility, they each have 50.0 GeV kinetic energies. What is the total collision energy available, taking into account the annihilation energy? Note that the annihilation energy is insignificant, because the electrons are highly relativistic.

33.4 Particles, Patterns, and Conservation Laws

13.

The π0π0 size 12{π rSup { size 8{0} } } {} is its own antiparticle and decays in the following manner: π0γ+γπ0γ+γ size 12{π rSup { size 8{0} } rightarrow γ+γ} {}. What is the energy of each γγ size 12{γ} {} ray if the π0π0 size 12{π rSup { size 8{0} } } {} is at rest when it decays?

14.

The primary decay mode for the negative pion is πμ+ν-μπμ+ν-μ size 12{π rSup { size 8{ - {}} } rightarrow μ rSup { size 8{ - {}} } + { bar {ν}} rSub { size 8{μ} } } {}. What is the energy release in MeV in this decay?

15.

The mass of a theoretical particle that may be associated with the unification of the electroweak and strong forces is 1014GeV/c21014GeV/c2 size 12{"10" rSup { size 8{"14"} } `"GeV/"c rSup { size 8{2} } } {}.

(a) How many proton masses is this?

(b) How many electron masses is this? (This indicates how extremely relativistic the accelerator would have to be in order to make the particle, and how large the relativistic quantity γγ size 12{γ} {} would have to be.)

16.

The decay mode of the negative muon is μe+ν-e+νμμe+ν-e+νμ size 12{μ rSup { size 8{ - {}} } rightarrow e rSup { size 8{ - {}} } + { bar {ν}} rSub { size 8{e} } +ν rSub { size 8{μ} } } {}.

(a) Find the energy released in MeV.

(b) Verify that charge and lepton family numbers are conserved.

17.

The decay mode of the positive tau is τ+μ++νμ+ν-ττ+μ++νμ+ν-τ size 12{τ rSup { size 8{+{}} } rightarrow μ rSup { size 8{+{}} } +ν rSub { size 8{μ} } + { bar {ν}} rSub { size 8{τ} } } {}.

(a) What energy is released?

(b) Verify that charge and lepton family numbers are conserved.

(c) The τ+τ+ size 12{τ rSup { size 8{+{}} } } {} is the antiparticle of the ττ size 12{τ rSup { size 8{ - {}} } } {}.Verify that all the decay products of the τ+τ+ size 12{τ rSup { size 8{+{}} } } {} are the antiparticles of those in the decay of the ττ size 12{τ rSup { size 8{ - {}} } } {} given in the text.

18.

The principal decay mode of the sigma zero is Σ0Λ0+γΣ0Λ0+γ size 12{Σ rSup { size 8{0} } rightarrow Λ rSup { size 8{0} } +γ} {}.

(a) What energy is released?

(b) Considering the quark structure of the two baryons, does it appear that the Σ0Σ0 size 12{Σ rSup { size 8{0} } } {} is an excited state of the Λ0Λ0 size 12{Λ rSup { size 8{0} } } {}?

(c) Verify that strangeness, charge, and baryon number are conserved in the decay.

(d) Considering the preceding and the short lifetime, can the weak force be responsible? State why or why not.

19.

(a) What is the uncertainty in the energy released in the decay of a π0π0 size 12{π rSup { size 8{0} } } {} due to its short lifetime?

(b) What fraction of the decay energy is this, noting that the decay mode is π0γ+γπ0γ+γ size 12{π rSup { size 8{0} } rightarrow γ+γ} {} (so that all the π0π0 size 12{π rSup { size 8{0} } } {} mass is destroyed)?

20.

(a) What is the uncertainty in the energy released in the decay of a ττ size 12{τ rSup { size 8{ - {}} } } {} due to its short lifetime?

(b) Is the uncertainty in this energy greater than or less than the uncertainty in the mass of the tau neutrino? Discuss the source of the uncertainty.

33.5 Quarks: Is That All There Is?

21.

(a) Verify from its quark composition that the Δ+Δ+ size 12{Δ rSup { size 8{+{}} } } {} particle could be an excited state of the proton.

(b) There is a spread of about 100 MeV in the decay energy of the Δ+Δ+ size 12{Δ rSup { size 8{+{}} } } {}, interpreted as uncertainty due to its short lifetime. What is its approximate lifetime?

(c) Does its decay proceed via the strong or weak force?

22.

Accelerators such as the Triangle Universities Meson Facility (TRIUMF) in British Columbia produce secondary beams of pions by having an intense primary proton beam strike a target. Such “meson factories” have been used for many years to study the interaction of pions with nuclei and, hence, the strong nuclear force. One reaction that occurs is π++pΔ++π++pπ++pΔ++π++p size 12{π rSup { size 8{+{}} } +p rightarrow Δ rSup { size 8{"++"} } rightarrow π rSup { size 8{+{}} } +p} {}, where the Δ++Δ++ size 12{Δ rSup { size 8{"++"} } } {} is a very short-lived particle. The graph in Figure 33.26 shows the probability of this reaction as a function of energy. The width of the bump is the uncertainty in energy due to the short lifetime of the Δ++Δ++ size 12{Δ rSup { size 8{"++"} } } {}.

(a) Find this lifetime.

(b) Verify from the quark composition of the particles that this reaction annihilates and then re-creates a d quark and a d-d- size 12{ { bar {d}}} {} antiquark by writing the reaction and decay in terms of quarks.

(c) Draw a Feynman diagram of the production and decay of the Δ++Δ++ size 12{Δ rSup { size 8{"++"} } } {} showing the individual quarks involved.

The figure shows a graph of number of interactions on the y axis versus kinetic energy of pion on the x axis. The number of interactions reaches a peak at two hundred mega electron volts where the short lived particle delta plus plus is generated. The width of this peak is approximately hundred mega electron volts.
Figure 33.26 This graph shows the probability of an interaction between a π+π+ size 12{π rSup { size 8{+{}} } } {} and a proton as a function of energy. The bump is interpreted as a very short lived particle called a Δ++Δ++ size 12{Δ rSup { size 8{"++"} } } {}. The approximately 100-MeV width of the bump is due to the short lifetime of the Δ++Δ++ size 12{Δ rSup { size 8{"++"} } } {}.
23.

The reaction π++pΔ++π++pΔ++ size 12{π rSup { size 8{+{}} } +p rightarrow Δ rSup { size 8{"++"} } } {} (described in the preceding problem) takes place via the strong force. (a) What is the baryon number of the Δ++Δ++ size 12{Δ rSup { size 8{"++"} } } {} particle?

(b) Draw a Feynman diagram of the reaction showing the individual quarks involved.

24.

One of the decay modes of the omega minus is ΩΞ0+πΩΞ0+π size 12{ %OMEGA rSup { size 8{ - {}} } rightarrow Ξ rSup { size 8{0} } +π rSup { size 8{ - {}} } } {}.

(a) What is the change in strangeness?

(b) Verify that baryon number and charge are conserved, while lepton numbers are unaffected.

(c) Write the equation in terms of the constituent quarks, indicating that the weak force is responsible.

25.

Repeat the previous problem for the decay mode ΩΛ0+K.ΩΛ0+K. size 12{ %OMEGA rSup { size 8{ - {}} } rightarrow Λ rSup { size 8{0} } +K rSup { size 8{ - {}} } } {}

26.

One decay mode for the eta-zero meson is η0γ+γ.η0γ+γ. size 12{η rSup { size 8{0} } rightarrow γ+γ} {}

(a) Find the energy released.

(b) What is the uncertainty in the energy due to the short lifetime?

(c) Write the decay in terms of the constituent quarks.

(d) Verify that baryon number, lepton numbers, and charge are conserved.

27.

One decay mode for the eta-zero meson is η0π0+π0η0π0+π0 size 12{η rSup { size 8{0} } rightarrow π rSup { size 8{0} } +π rSup { size 8{0} } } {}.

(a) Write the decay in terms of the quark constituents.

(b) How much energy is released?

(c) What is the ultimate release of energy, given the decay mode for the pi zero is π0γ+γπ0γ+γ size 12{π rSup { size 8{0} } rightarrow γ+γ} {}?

28.

Is the decay ne++ene++e size 12{n rightarrow e rSup { size 8{+{}} } +e rSup { size 8{ - {}} } } {} possible considering the appropriate conservation laws? State why or why not.

29.

Is the decay μe+νe+νμμe+νe+νμ size 12{μ rSup { size 8{ - {}} } rightarrow e rSup { size 8{ - {}} } +ν rSub { size 8{e} } +ν rSub { size 8{μ} } } {} possible considering the appropriate conservation laws? State why or why not.

30.

(a) Is the decay Λ0n+π0Λ0n+π0 size 12{Λ rSup { size 8{0} } rightarrow n+π rSup { size 8{0} } } {} possible considering the appropriate conservation laws? State why or why not.

(b) Write the decay in terms of the quark constituents of the particles.

31.

(a) Is the decay Σn+πΣn+π size 12{Σ rSup { size 8{ - {}} } rightarrow n+π rSup { size 8{ - {}} } } {} possible considering the appropriate conservation laws? State why or why not. (b) Write the decay in terms of the quark constituents of the particles.

32.

The only combination of quark colors that produces a white baryon is RGB. Identify all the color combinations that can produce a white meson.

33.

(a) Three quarks form a baryon. How many combinations of the six known quarks are there if all combinations are possible?

(b) This number is less than the number of known baryons. Explain why.

34.

(a) Show that the conjectured decay of the proton, pπ0+e+pπ0+e+ size 12{p rightarrow π rSup { size 8{0} } +e rSup { size 8{+{}} } } {}, violates conservation of baryon number and conservation of lepton number.

(b) What is the analogous decay process for the antiproton?

35.

Verify the quantum numbers given for the Ω+Ω+ size 12{ %OMEGA rSup { size 8{+{}} } } {} in Table 33.2 by adding the quantum numbers for its quark constituents as inferred from Table 33.4.

36.

Verify the quantum numbers given for the proton and neutron in Table 33.2 by adding the quantum numbers for their quark constituents as given in Table 33.4.

37.

(a) How much energy would be released if the proton did decay via the conjectured reaction pπ0+e+pπ0+e+ size 12{p rightarrow π rSup { size 8{0} } +e rSup { size 8{+{}} } } {}?

(b) Given that the π0π0 size 12{π rSup { size 8{0} } } {} decays to two γγ size 12{γ} {} s and that the e+e+ size 12{e rSup { size 8{+{}} } } {} will find an electron to annihilate, what total energy is ultimately produced in proton decay?

(c) Why is this energy greater than the proton’s total mass (converted to energy)?

38.

(a) Find the charge, baryon number, strangeness, charm, and bottomness of the J/ΨJ/Ψ size 12{J/Ψ} {} particle from its quark composition.

(b) Do the same for the ϒ ϒ particle.

39.

There are particles called D-mesons. One of them is the D+D+ size 12{D rSup { size 8{+{}} } } {} meson, which has a single positive charge and a baryon number of zero, also the value of its strangeness, topness, and bottomness. It has a charm of +1.+1. size 12{+1} {} What is its quark configuration?

40.

There are particles called bottom mesons or B-mesons. One of them is the BB size 12{B rSup { size 8{ - {}} } } {} meson, which has a single negative charge; its baryon number is zero, as are its strangeness, charm, and topness. It has a bottomness of 11 size 12{ - 1} {}. What is its quark configuration?

41.

(a) What particle has the quark composition u-u-d-u-u-d- size 12{ { bar {u}} { bar {u}} { bar {d}}} {}?

(b) What should its decay mode be?

42.

(a) Show that all combinations of three quarks produce integral charges. Thus baryons must have integral charge.

(b) Show that all combinations of a quark and an antiquark produce only integral charges. Thus mesons must have integral charge.

33.6 GUTs: The Unification of Forces

43.

Integrated Concepts

The intensity of cosmic ray radiation decreases rapidly with increasing energy, but there are occasionally extremely energetic cosmic rays that create a shower of radiation from all the particles they create by striking a nucleus in the atmosphere as seen in the figure given below. Suppose a cosmic ray particle having an energy of 1010GeV1010GeV converts its energy into particles with masses averaging 200MeV/c2200MeV/c2. (a) How many particles are created? (b) If the particles rain down on a 1.00-km21.00-km2 area, how many particles are there per square meter?

The figure shows an extremely energetic cosmic ray penetrating into the Earth’s atmosphere. High up in the atmosphere, the cosmic ray disintegrates into a shower of particles that start a chain reaction by themselves creating further particles. All these particles shower the surface of the Earth.
Figure 33.27 An extremely energetic cosmic ray creates a shower of particles on earth. The energy of these rare cosmic rays can approach a joule (about 1010GeV1010GeV size 12{"10" rSup { size 8{"10"} } `"GeV"} {}) and, after multiple collisions, huge numbers of particles are created from this energy. Cosmic ray showers have been observed to extend over many square kilometers.
44.

Integrated Concepts

Assuming conservation of momentum, what is the energy of each γγ size 12{γ} {} ray produced in the decay of a neutral at rest pion, in the reaction π0γ+γπ0γ+γ size 12{π rSup { size 8{0} } rightarrow γ+γ} {}?

45.

Integrated Concepts

What is the wavelength of a 50-GeV electron, which is produced at SLAC? This provides an idea of the limit to the detail it can probe.

46.

Integrated Concepts

(a) Calculate the relativistic quantity γ=11v2/c2γ=11v2/c2 size 12{γ= { {1} over { sqrt {1 - v rSup { size 8{2} } /c rSup { size 8{2} } } } } } {} for 1.00-TeV protons produced at Fermilab. (b) If such a proton created a π+π+ size 12{π rSup { size 8{+{}} } } {} having the same speed, how long would its life be in the laboratory? (c) How far could it travel in this time?

47.

Integrated Concepts

The primary decay mode for the negative pion is πμ+ν-μπμ+ν-μ size 12{π rSup { size 8{ - {}} } rightarrow μ rSup { size 8{ - {}} } + { bar {ν}} rSub { size 8{μ} } } {}. (a) What is the energy release in MeV in this decay? (b) Using conservation of momentum, how much energy does each of the decay products receive, given the ππ size 12{π rSup { size 8{ - {}} } } {} is at rest when it decays? You may assume the muon antineutrino is massless and has momentum p=E/cp=E/c size 12{p=E/c} {}, just like a photon.

48.

Integrated Concepts

Plans for an accelerator that produces a secondary beam of K-mesons to scatter from nuclei, for the purpose of studying the strong force, call for them to have a kinetic energy of 500 MeV. (a) What would the relativistic quantity γ=11v2/c2γ=11v2/c2 size 12{γ= { {1} over { sqrt {1 - v rSup { size 8{2} } /c rSup { size 8{2} } } } } } {} be for these particles? (b) How long would their average lifetime be in the laboratory? (c) How far could they travel in this time?

49.

Integrated Concepts

Suppose you are designing a proton decay experiment and you can detect 50 percent of the proton decays in a tank of water. (a) How many kilograms of water would you need to see one decay per month, assuming a lifetime of 1031y1031y? (b) How many cubic meters of water is this? (c) If the actual lifetime is 1033y1033y, how long would you have to wait on an average to see a single proton decay?

50.

Integrated Concepts

In supernovas, neutrinos are produced in huge amounts. They were detected from the 1987A supernova in the Magellanic Cloud, which is about 120,000 light years away from the Earth (relatively close to our Milky Way galaxy). If neutrinos have a mass, they cannot travel at the speed of light, but if their mass is small, they can get close. (a) Suppose a neutrino with a 7-eV/c27-eV/c2 size 12{7"-eV/"c rSup { size 8{2} } } {} mass has a kinetic energy of 700 keV. Find the relativistic quantity γ=11v2/c2γ=11v2/c2 size 12{γ= { {1} over { sqrt {1 - v rSup { size 8{2} } /c rSup { size 8{2} } } } } } {} for it. (b) If the neutrino leaves the 1987A supernova at the same time as a photon and both travel to Earth, how much sooner does the photon arrive? This is not a large time difference, given that it is impossible to know which neutrino left with which photon and the poor efficiency of the neutrino detectors. Thus, the fact that neutrinos were observed within hours of the brightening of the supernova only places an upper limit on the neutrino’s mass. (Hint: You may need to use a series expansion to find v for the neutrino, since its γγ size 12{γ} {} is so large.)

51.

Construct Your Own Problem 

Consider an ultrahigh-energy cosmic ray entering the Earth’s atmosphere (some have energies approaching a joule). Construct a problem in which you calculate the energy of the particle based on the number of particles in an observed cosmic ray shower. Among the things to consider are the average mass of the shower particles, the average number per square meter, and the extent (number of square meters covered) of the shower. Express the energy in eV and joules.

52.

Construct Your Own Problem 

Consider a detector needed to observe the proposed, but extremely rare, decay of an electron. Construct a problem in which you calculate the amount of matter needed in the detector to be able to observe the decay, assuming that it has a signature that is clearly identifiable. Among the things to consider are the estimated half life (long for rare events), and the number of decays per unit time that you wish to observe, as well as the number of electrons in the detector substance.

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