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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

31.2 Radiation Detection and Detectors

1.

The energy of 30.0 eVeV is required to ionize a molecule of the gas inside a Geiger tube, thereby producing an ion pair. Suppose a particle of ionizing radiation deposits 0.500 MeV of energy in this Geiger tube. What maximum number of ion pairs can it create?

2.

A particle of ionizing radiation creates 4000 ion pairs in the gas inside a Geiger tube as it passes through. What minimum energy was deposited, if 30.0 eVeV is required to create each ion pair?

3.

(a) Repeat Exercise 31.2, and convert the energy to joules or calories. (b) If all of this energy is converted to thermal energy in the gas, what is its temperature increase, assuming 50.0 cm350.0 cm3 of ideal gas at 0.250-atm pressure? (The small answer is consistent with the fact that the energy is large on a quantum mechanical scale but small on a macroscopic scale.)

4.

Suppose a particle of ionizing radiation deposits 1.0 MeV in the gas of a Geiger tube, all of which goes to creating ion pairs. Each ion pair requires 30.0 eV of energy. (a) The applied voltage sweeps the ions out of the gas in 1.00μs1.00μs. What is the current? (b) This current is smaller than the actual current since the applied voltage in the Geiger tube accelerates the separated ions, which then create other ion pairs in subsequent collisions. What is the current if this last effect multiplies the number of ion pairs by 900?

31.3 Substructure of the Nucleus

5.

Verify that a 2.3×1017kg2.3×1017kg size 12{2 "." 3 times "10" rSup { size 8{"17"} } "kg"} {} mass of water at normal density would make a cube 60 km on a side, as claimed in Example 31.1. (This mass at nuclear density would make a cube 1.0 m on a side.)

6.

Find the length of a side of a cube having a mass of 1.0 kg and the density of nuclear matter, taking this to be 2.3×1017 kg/m32.3×1017 kg/m3 size 12{2 "." 3´"10" rSup { size 8{"17"} } " kg/m" rSup { size 8{3} } } {}.

7.

What is the radius of an αα size 12{α} {} particle?

8.

Find the radius of a 238Pu238Pu size 12{"" lSup { size 8{"238"} } "Pu"} {} nucleus. 238Pu238Pu size 12{"" lSup { size 8{"238"} } "Pu"} {} is a manufactured nuclide that is used as a power source on some space probes.

9.

(a) Calculate the radius of 58Ni58Ni size 12{"" lSup { size 8{"58"} } "Ni"} {}, one of the most tightly bound stable nuclei.

(b) What is the ratio of the radius of 58Ni58Ni size 12{"" lSup { size 8{"58"} } "Ni"} {} to that of 258Ha258Ha size 12{"" lSup { size 8{"258"} } "Ha"} {}, one of the largest nuclei ever made? Note that the radius of the largest nucleus is still much smaller than the size of an atom.

10.

The unified atomic mass unit is defined to be 1 u=1.6605×10−27kg1 u=1.6605×10−27kg size 12{1" u"=1 "." "6605"×"10" rSup { size 8{-"27"} } "kg"} {}. Verify that this amount of mass converted to energy yields 931.5 MeV. Note that you must use four-digit or better values for cc size 12{c} {} and qeqe size 12{ lline q rSub { size 8{e} } rline } {}.

11.

What is the ratio of the velocity of a ββ size 12{β} {} particle to that of an αα size 12{α} {} particle, if they have the same nonrelativistic kinetic energy?

12.

If a 1.50-cm-thick piece of lead can absorb 90.0% of the γγ size 12{γ} {} rays from a radioactive source, how many centimeters of lead are needed to absorb all but 0.100% of the γγ size 12{γ} {} rays?

13.

The detail observable using a probe is limited by its wavelength. Calculate the energy of a γγ size 12{γ} {}-ray photon that has a wavelength of 1×1016m1×1016m size 12{1 times "10" rSup { size 8{ - "16"} } m} {}, small enough to detect details about one-tenth the size of a nucleon. Note that a photon having this energy is difficult to produce and interacts poorly with the nucleus, limiting the practicability of this probe.

14.

(a) Show that if you assume the average nucleus is spherical with a radius r=r0A1/3r=r0A1/3 size 12{r=r rSub { size 8{0} } A rSup { size 8{1/3} } } {}, and with a mass of AA size 12{A} {} u, then its density is independent of AA size 12{A} {}.

(b) Calculate that density in u/fm3u/fm3 size 12{"u/fm" rSup { size 8{3} } } {} and kg/m3kg/m3 size 12{"kg/m" rSup { size 8{3} } } {}, and compare your results with those found in Example 31.1 for 56Fe56Fe size 12{"" lSup { size 8{"56"} } "Fe"} {}.

15.

What is the ratio of the velocity of a 5.00-MeV ββ size 12{β} {} ray to that of an αα size 12{β} {} particle with the same kinetic energy? This should confirm that ββ size 12{β} {}s travel much faster than αα size 12{β} {}s even when relativity is taken into consideration. (See also Exercise 31.11.)

16.

(a) What is the kinetic energy in MeV of a ββ size 12{β} {} ray that is traveling at 0.998c0.998c? This gives some idea of how energetic a ββ size 12{β} {} ray must be to travel at nearly the same speed as a γγ ray. (b) What is the velocity of the γγ ray relative to the ββ size 12{β} {} ray?

31.4 Nuclear Decay and Conservation Laws

In the following eight problems, write the complete decay equation for the given nuclide in the complete ZAXNZAXN size 12{"" lSub { size 8{Z} } lSup { size 8{A} } X rSub { size 8{N} } } {} notation. Refer to the periodic table for values of ZZ size 12{Z} {}.

17.

ββ size 12{β rSup { size 8{ - {}} } } {} decay of 3H3H size 12{"" lSup { size 8{3} } H} {} (tritium), a manufactured isotope of hydrogen used in some digital watch displays, and manufactured primarily for use in hydrogen bombs.

18.

ββ size 12{β rSup { size 8{ - {}} } } {} decay of 40K40K size 12{"" lSup { size 8{"40"} } K} {}, a naturally occurring rare isotope of potassium responsible for some of our exposure to background radiation.

19.

β+β+ size 12{β rSup { size 8{+{}} } } {} decay of 50Mn50Mn size 12{"" lSup { size 8{"50"} } "Mn"} {}.

20.

β+β+ size 12{β rSup { size 8{+{}} } } {} decay of 52Fe52Fe size 12{"" lSup { size 8{"52"} } "Fe"} {}.

21.

Electron capture by 7Be7Be size 12{"" lSup { size 8{7} } "Be"} {}.

22.

Electron capture by 106In106In size 12{"" lSup { size 8{"106"} } "In"} {}.

23.

αα size 12{α} {} decay of 210Po210Po size 12{"" lSup { size 8{"210"} } "Po"} {}, the isotope of polonium in the decay series of 238U238U size 12{"" lSup { size 8{"238"} } U} {} that was discovered by the Curies. A favorite isotope in physics labs, since it has a short half-life and decays to a stable nuclide.

24.

αα size 12{α} {} decay of 226Ra226Ra size 12{"" lSup { size 8{"226"} } "Ra"} {}, another isotope in the decay series of 238U238U size 12{"" lSup { size 8{"238"} } U} {}, first recognized as a new element by the Curies. Poses special problems because its daughter is a radioactive noble gas.

In the following four problems, identify the parent nuclide and write the complete decay equation in the ZAXNZAXN size 12{"" lSub { size 8{Z} } lSup { size 8{A} } X rSub { size 8{N} } } {} notation. Refer to the periodic table for values of ZZ size 12{Z} {}.

25.

ββ size 12{β rSup { size 8{ - {}} } } {} decay producing 137Ba137Ba size 12{"" lSup { size 8{"137"} } "Ba"} {}. The parent nuclide is a major waste product of reactors and has chemistry similar to potassium and sodium, resulting in its concentration in your cells if ingested.

26.

ββ size 12{β rSup { size 8{ - {}} } } {} decay producing 90Y90Y size 12{"" lSup { size 8{"90"} } Y} {}. The parent nuclide is a major waste product of reactors and has chemistry similar to calcium, so that it is concentrated in bones if ingested (90Y90Y size 12{"" lSup { size 8{"90"} } Y} {} is also radioactive.)

27.

αα size 12{α} {} decay producing 228Ra228Ra size 12{"" lSup { size 8{"228"} } "Ra"} {}. The parent nuclide is nearly 100% of the natural element and is found in gas lantern mantles and in metal alloys used in jets (228Ra228Ra size 12{"" lSup { size 8{"228"} } "Ra"} {} is also radioactive).

28.

αα size 12{α} {} decay producing 208Pb208Pb size 12{"" lSup { size 8{"208"} } "Pb"} {}. The parent nuclide is in the decay series produced by 232Th232Th size 12{"" lSup { size 8{"232"} } "Th"} {}, the only naturally occurring isotope of thorium.

29.

When an electron and positron annihilate, both their masses are destroyed, creating two equal energy photons to preserve momentum. (a) Confirm that the annihilation equation e++eγ+γe++eγ+γ size 12{e rSup { size 8{+{}} } +e rSup { size 8{ - {}} } rightarrow γ+γ} {} conserves charge, electron family number, and total number of nucleons. To do this, identify the values of each before and after the annihilation. (b) Find the energy of each γγ size 12{γ} {} ray, assuming the electron and positron are initially nearly at rest. (c) Explain why the two γγ size 12{γ} {} rays travel in exactly opposite directions if the center of mass of the electron-positron system is initially at rest.

30.

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for αα decay given in the equation ZAXN Z2A4 YN2 + 24 He2 ZAXN Z2A4 YN2 + 24 He2 . To do this, identify the values of each before and after the decay.

31.

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for ββ decay given in the equation ZA XN Z+1A Y N1 + β + ν¯eZA XN Z+1A Y N1 + β + ν¯e size 12{"" lSub { size 8{Z} } lSup { size 8{A} } X rSub { size 8{N} } rightarrow "" lSub { size 8{Z−1} } lSup { size 8{A} } Y rSub { size 8{N - 1} } +β rSup { size 8{ - {}} } + {overline {v rSub { size 8{e} } }} } {}. To do this, identify the values of each before and after the decay.

32.

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for ββ size 12{β rSup { size 8{ - {}} } } {} decay given in the equation ZA XN Z1A Y N1 + β + νeZA XN Z1A Y N1 + β + νe. To do this, identify the values of each before and after the decay.

33.

Confirm that charge, electron family number, and the total number of nucleons are all conserved by the rule for electron capture given in the equation ZAXN+eZ1AYN+1+νeZAXN+eZ1AYN+1+νe size 12{"" lSub { size 8{Z} } lSup { size 8{A} } X rSub { size 8{N} } +e rSup { size 8{ - {}} } rightarrow "" lSub { size 8{Z - 1} } lSup { size 8{A} } Y rSub { size 8{N+1} } +v rSub { size 8{e} } } {}. To do this, identify the values of each before and after the capture.

34.

A rare decay mode has been observed in which 222Ra222Ra emits a 14C14C nucleus. (a) The decay equation is 222RaAX+14C222RaAX+14C size 12{ {} rSup { size 8{"222"} } "Ra" rightarrow rSup { size 8{A} } "X+" rSup { size 8{"14"} } C} {}. Identify the nuclide AXAX. (b) Find the energy emitted in the decay. The mass of 222Ra222Ra size 12{"" lSup { size 8{"222"} } "Ra"} {} is 222.015353 u.

35.

(a) Write the complete αα size 12{α} {} decay equation for 226Ra226Ra size 12{"" lSup { size 8{"226"} } "Ra"} {}.

(b) Find the energy released in the decay.

36.

(a) Write the complete αα size 12{α} {} decay equation for 249Cf249Cf size 12{"" lSup { size 8{"249"} } "Cf"} {}.

(b) Find the energy released in the decay.

37.

(a) Write the complete ββ size 12{β rSup { size 8{ - {}} } } {} decay equation for the neutron. (b) Find the energy released in the decay.

38.

(a) Write the complete ββ size 12{β rSup { size 8{ - {}} } } {} decay equation for 90Sr90Sr size 12{"" lSup { size 8{"90"} } "Sr"} {}, a major waste product of nuclear reactors. (b) Find the energy released in the decay.

39.

Calculate the energy released in the β+β+ size 12{β rSup { size 8{+{}} } } {} decay of 22Na22Na, the equation for which is given in the text. The masses of 22Na22Na and 22Ne22Ne size 12{"" lSup { size 8{"22"} } "Ne"} {} are 21.994434 and 21.991383 u, respectively.

40.

(a) Write the complete β+β+ size 12{β rSup { size 8{+{}} } } {} decay equation for 11C11C size 12{"" lSup { size 8{"11"} } C} {}.

(b) Calculate the energy released in the decay. The masses of 11C11C size 12{"" lSup { size 8{"11"} } C} {} and 11B11B size 12{"" lSup { size 8{"11"} } B} {} are 11.011433 and 11.009305 u, respectively.

41.

(a) Calculate the energy released in the αα size 12{α} {} decay of 238U238U size 12{"" lSup { size 8{"238"} } U} {}.

(b) What fraction of the mass of a single 238U238U size 12{"" lSup { size 8{"238"} } U} {} is destroyed in the decay? The mass of 234Th234Th size 12{"" lSup { size 8{"234"} } "Th"} {} is 234.043593 u.

(c) Although the fractional mass loss is large for a single nucleus, it is difficult to observe for an entire macroscopic sample of uranium. Why is this?

42.

(a) Write the complete reaction equation for electron capture by 7Be.7Be. size 12{"" lSup { size 8{7} } "Be"} {}

(b) Calculate the energy released.

43.

(a) Write the complete reaction equation for electron capture by 15O15O size 12{"" lSup { size 8{"15"} } O} {}.

(b) Calculate the energy released.

31.5 Half-Life and Activity

Data from the appendices and the periodic table may be needed for these problems.

44.

An old campfire is uncovered during an archaeological dig. Its charcoal is found to contain less than 1/1000 the normal amount of 14C14C size 12{"" lSup { size 8{"14"} } C} {}. Estimate the minimum age of the charcoal, noting that 210=1024210=1024 size 12{2 rSup { size 8{"10"} } ="1024"} {}.

45.

A 60Co60Co size 12{"" lSup { size 8{"60"} } "Co"} {} source is labeled 4.00 mCi, but its present activity is found to be 1.85×1071.85×107 size 12{1 "." "85" times "10" rSup { size 8{7} } } {} Bq. (a) What is the present activity in mCi? (b) How long ago did it actually have a 4.00-mCi activity?

46.

(a) Calculate the activity RR size 12{R} {} in curies of 1.00 g of 226Ra226Ra size 12{"" lSup { size 8{"226"} } "Ra"} {}. (b) Discuss why your answer is not exactly 1.00 Ci, given that the curie was originally supposed to be exactly the activity of a gram of radium.

47.

Show that the activity of the 14C14C size 12{"" lSup { size 8{"14"} } C} {} in 1.00 g of 12C12C size 12{"" lSup { size 8{"12"} } C} {} found in living tissue is 0.250 Bq.

48.

Mantles for gas lanterns contain thorium, because it forms an oxide that can survive being heated to incandescence for long periods of time. Natural thorium is almost 100% 232Th232Th size 12{"" lSup { size 8{"232"} } "Th"} {}, with a half-life of 1.405×1010y1.405×1010y size 12{1 "." "405" times "10" rSup { size 8{"10"} } } {}. If an average lantern mantle contains 300 mg of thorium, what is its activity?

49.

Cow’s milk produced near nuclear reactors can be tested for as little as 1.00 pCi of 131I131I size 12{"" lSup { size 8{"131"} } I} {} per liter, to check for possible reactor leakage. What mass of 131I131I size 12{"" lSup { size 8{"131"} } I} {} has this activity?

50.

(a) Natural potassium contains 40K40K, which has a half-life of 1.277×1091.277×109 size 12{1 "." "277" times "10" rSup { size 8{9} } } {} y. What mass of 40K40K size 12{"" lSup { size 8{"40"} } K} {} in a person would have a decay rate of 4140 Bq? (b) What is the fraction of 40K40K size 12{"" lSup { size 8{"40"} } K} {} in natural potassium, given that the person has 140 g in his body? (These numbers are typical for a 70-kg adult.)

51.

There is more than one isotope of natural uranium. If a researcher isolates 1.00 mg of the relatively scarce 235U235U size 12{"" lSup { size 8{"235"} } U} {} and finds this mass to have an activity of 80.0 Bq, what is its half-life in years?

52.

50V50V has one of the longest known radioactive half-lives. In a difficult experiment, a researcher found that the activity of 1.00 kg of 50V50V is 1.75 Bq. What is the half-life in years?

53.

You can sometimes find deep red crystal vases in antique stores, called uranium glass because their color was produced by doping the glass with uranium. Look up the natural isotopes of uranium and their half-lives, and calculate the activity of such a vase assuming it has 2.00 g of uranium in it. Neglect the activity of any daughter nuclides.

54.

A tree falls in a forest. How many years must pass before the 14C14C size 12{"" lSup { size 8{"14"} } C} {} activity in 1.00 g of the tree’s carbon drops to 1.00 decay per hour?

55.

What fraction of the 40K40K size 12{"" lSup { size 8{"40"} } K} {} that was on Earth when it formed 4.5×1094.5×109 years ago is left today?

56.

A 5000-Ci 60Co60Co source used for cancer therapy is considered too weak to be useful when its activity falls to 3500 Ci. How long after its manufacture does this happen?

57.

Natural uranium is 0.7200% 235U235U and 99.27% 238U238U. What were the percentages of 235U235U and 238U238U in natural uranium when Earth formed 4.5×1094.5×109 years ago?

58.

The ββ size 12{β rSup { size 8{ - {}} } } {} particles emitted in the decay of 3H3H (tritium) interact with matter to create light in a glow-in-the-dark exit sign. At the time of manufacture, such a sign contains 15.0 Ci of 3H3H size 12{"" lSup { size 8{3} } H} {}. (a) What is the mass of the tritium? (b) What is its activity 5.00 y after manufacture?

59.

World War II aircraft had instruments with glowing radium-painted dials (see Figure 31.2). The activity of one such instrument was 1.0×1051.0×105 Bq when new. (a) What mass of 226Ra226Ra was present? (b) After some years, the phosphors on the dials deteriorated chemically, but the radium did not escape. What is the activity of this instrument 57.0 years after it was made?

60.

(a) The 210Po210Po source used in a physics laboratory is labeled as having an activity of 1.0μCi1.0μCi size 12{1 "." 0 m"Ci"} {} on the date it was prepared. A student measures the radioactivity of this source with a Geiger counter and observes 1500 counts per minute. She notices that the source was prepared 120 days before her lab. What fraction of the decays is she observing with her apparatus? (b) Identify some of the reasons that only a fraction of the αα size 12{α} {} s emitted are observed by the detector.

61.

Armor-piercing shells with depleted uranium cores are fired by aircraft at tanks. (The high density of the uranium makes them effective.) The uranium is called depleted because it has had its 235U235U removed for reactor use and is nearly pure 238U238U. Depleted uranium has been erroneously called non-radioactive. To demonstrate that this is wrong: (a) Calculate the activity of 60.0 g of pure 238U238U size 12{"" lSup { size 8{"238"} } U} {}. (b) Calculate the activity of 60.0 g of natural uranium, neglecting the 234U234U and all daughter nuclides.

62.

The ceramic glaze on a red-orange Fiestaware plate is U 2 O 3 U 2 O 3 and contains 50.0 grams of 238 U 238 U , but very little 235 U 235 U. (a) What is the activity of the plate? (b) Calculate the total energy that will be released by the 238 U 238 U decay. (c) If energy is worth 12.0 cents per kW h kWh, what is the monetary value of the energy emitted? (These plates went out of production some 30 years ago, but are still available as collectibles.)

63.

Large amounts of depleted uranium ( 238 U 238 U) are available as a by-product of uranium processing for reactor fuel and weapons. Uranium is very dense and makes good counter weights for aircraft. Suppose you have a 4000-kg block of 238 U 238 U. (a) Find its activity. (b) How many calories per day are generated by thermalization of the decay energy? (c) Do you think you could detect this as heat? Explain.

64.

The Galileo space probe was launched on its long journey past several planets in 1989, with an ultimate goal of Jupiter. Its power source is 11.0 kg of 238 Pu 238 Pu, a by-product of nuclear weapons plutonium production. Electrical energy is generated thermoelectrically from the heat produced when the 5.59-MeV α α particles emitted in each decay crash to a halt inside the plutonium and its shielding. The half-life of 238 Pu 238 Pu is 87.7 years. (a) What was the original activity of the 238 Pu 238 Pu in becquerel? (b) What power was emitted in kilowatts? (c) What power was emitted 12.0 y after launch? You may neglect any extra energy from daughter nuclides and any losses from escaping γ γ rays.

65.

Construct Your Own Problem

Consider the generation of electricity by a radioactive isotope in a space probe, such as described in Exercise 31.64. Construct a problem in which you calculate the mass of a radioactive isotope you need in order to supply power for a long space flight. Among the things to consider are the isotope chosen, its half-life and decay energy, the power needs of the probe and the length of the flight.

66.

Unreasonable Results

A nuclear physicist finds 1.0 μg 1.0μg of 236 U 236 U in a piece of uranium ore and assumes it is primordial since its half-life is 2.3 × 10 7 y 2.3× 10 7 y. (a) Calculate the amount of 236 U 236 Uthat would had to have been on Earth when it formed 4.5 × 10 9 y 4.5× 10 9 y ago for 1.0 μg 1.0μg to be left today. (b) What is unreasonable about this result? (c) What assumption is responsible?

67.

Unreasonable Results

(a) Repeat Exercise 31.57 but include the 0.0055% natural abundance of 234 U 234 U with its 2.45 × 10 5 y 2.45× 10 5 y half-life. (b) What is unreasonable about this result? (c) What assumption is responsible? (d) Where does the 234 U 234 U come from if it is not primordial?

68.

Unreasonable Results

The manufacturer of a smoke alarm decides that the smallest current of α α radiation he can detect is 1.00 μA 1.00μA. (a) Find the activity in curies of an α α emitter that produces a 1.00 μA 1.00μA current of α α particles. (b) What is unreasonable about this result? (c) What assumption is responsible?

31.6 Binding Energy

69.

2H2H is a loosely bound isotope of hydrogen. Called deuterium or heavy hydrogen, it is stable but relatively rare—it is 0.015% of natural hydrogen. Note that deuterium has Z=NZ=N size 12{Z=N} {}, which should tend to make it more tightly bound, but both are odd numbers. Calculate BE/ABE/A, the binding energy per nucleon, for 2H2H and compare it with the approximate value obtained from the graph in Figure 31.26.

70.

56Fe56Fe is among the most tightly bound of all nuclides. It is more than 90% of natural iron. Note that 56Fe56Fe has even numbers of both protons and neutrons. Calculate BE/ABE/A, the binding energy per nucleon, for 56Fe56Fe and compare it with the approximate value obtained from the graph in Figure 31.26.

71.

209Bi209Bi is the heaviest stable nuclide, and its BE/ABE/A is low compared with medium-mass nuclides. Calculate BE/ABE/A, the binding energy per nucleon, for 209Bi209Bi and compare it with the approximate value obtained from the graph in Figure 31.26.

72.

(a) Calculate BE/ABE/A for 235U235U, the rarer of the two most common uranium isotopes. (b) Calculate BE/ABE/A for 238U238U. (Most of uranium is 238U238U.) Note that 238U238U has even numbers of both protons and neutrons. Is the BE/ABE/A of 238U238U significantly different from that of 235U235U ?

73.

(a) Calculate BE/ABE/A for 12C12C. Stable and relatively tightly bound, this nuclide is most of natural carbon. (b) Calculate BE/ABE/A for 14C14C. Is the difference in BE/ABE/A between 12C12C and 14C14C significant? One is stable and common, and the other is unstable and rare.

74.

The fact that BE/ABE/A is greatest for AA near 60 implies that the range of the nuclear force is about the diameter of such nuclides. (a) Calculate the diameter of an A=60A=60 nucleus. (b) Compare BE/ABE/A for 58Ni58Ni and 90Sr90Sr. The first is one of the most tightly bound nuclides, while the second is larger and less tightly bound.

75.

The purpose of this problem is to show in three ways that the binding energy of the electron in a hydrogen atom is negligible compared with the masses of the proton and electron. (a) Calculate the mass equivalent in u of the 13.6-eV binding energy of an electron in a hydrogen atom, and compare this with the mass of the hydrogen atom obtained from Appendix A. (b) Subtract the mass of the proton given in Table 31.2 from the mass of the hydrogen atom given in Appendix A. You will find the difference is equal to the electron’s mass to three digits, implying the binding energy is small in comparison. (c) Take the ratio of the binding energy of the electron (13.6 eV) to the energy equivalent of the electron’s mass (0.511 MeV). (d) Discuss how your answers confirm the stated purpose of this problem.

76.

Unreasonable Results

A particle physicist discovers a neutral particle with a mass of 2.02733 u that he assumes is two neutrons bound together. (a) Find the binding energy. (b) What is unreasonable about this result? (c) What assumptions are unreasonable or inconsistent?

31.7 Tunneling

77.

Derive an approximate relationship between the energy of αα decay and half-life using the following data. It may be useful to graph the log of t1/2t1/2 against EαEα to find some straight-line relationship.

Nuclide E α (MeV) E α (MeV) t 1/2 t 1/2
216 Ra 216 Ra 9.5 9.5 0.18 μs 0.18 μs
194 Po 194 Po 7.0 7.0 0.7 s 0.7 s
240 Cm 240 Cm 6.4 6.4 27 d 27 d
226 Ra 226 Ra 4.91 4.91 1600 y 1600 y
232 Th 232 Th 4.1 4.1 1.4 × 10 10 y 1.4 × 10 10 y
Table 31.3 Energy and Half-Life for α α size 12{α} {} Decay
78.

Integrated Concepts

A 2.00-T magnetic field is applied perpendicular to the path of charged particles in a bubble chamber. What is the radius of curvature of the path of a 10 MeV proton in this field? Neglect any slowing along its path.

79.

(a) Write the decay equation for the αα decay of 235 U 235 U . (b) What energy is released in this decay? The mass of the daughter nuclide is 231.036298 u. (c) Assuming the residual nucleus is formed in its ground state, how much energy goes to the αα particle?

80.

Unreasonable Results

The relatively scarce naturally occurring calcium isotope 48 Ca 48 Ca has a half-life of about 2×1016y2×1016y. (a) A small sample of this isotope is labeled as having an activity of 1.0 Ci. What is the mass of the 48 Ca 48 Ca in the sample? (b) What is unreasonable about this result? (c) What assumption is responsible?

81.

Unreasonable Results

A physicist scatters γγ rays from a substance and sees evidence of a nucleus 7.5×10–13m7.5×10–13m in radius. (a) Find the atomic mass of such a nucleus. (b) What is unreasonable about this result? (c) What is unreasonable about the assumption?

82.

Unreasonable Results

A frazzled theoretical physicist reckons that all conservation laws are obeyed in the decay of a proton into a neutron, positron, and neutrino (as in β+β+ decay of a nucleus) and sends a paper to a journal to announce the reaction as a possible end of the universe due to the spontaneous decay of protons. (a) What energy is released in this decay? (b) What is unreasonable about this result? (c) What assumption is responsible?

83.

Construct Your Own Problem

Consider the decay of radioactive substances in the Earth’s interior. The energy emitted is converted to thermal energy that reaches the earth’s surface and is radiated away into cold dark space. Construct a problem in which you estimate the activity in a cubic meter of earth rock? And then calculate the power generated. Calculate how much power must cross each square meter of the Earth’s surface if the power is dissipated at the same rate as it is generated. Among the things to consider are the activity per cubic meter, the energy per decay, and the size of the Earth.

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