Physics for Scientists and Engineers (PDF)

 Physics for Scientists and Engineers (with PhysicsNOW and InfoTrac) Raymond A. Serway - Emeritus, James Madison University John W. Jewett - California State Polytechnic University, Pomona...

 Physics for Scientists and Engineers (with PhysicsNOW and InfoTrac) Raymond A. Serway - Emeritus, James Madison University John W. Jewett - California State Polytechnic University, Pomona ISBN 0534408427 1296 pages Case Bound 8 1/2 x 10 7/8 Thomson Brooks/Cole © 2004; 6th Edition This best-selling, calculus-based text is recognized for its carefully crafted, logical presentation of the basic concepts and principles of physics. PHYSICS FOR SCIENTISTS AND ENGINEERS, Sixth Edition, maintains the Serway traditions of concise writing for the students, carefully thought-out problem sets and worked examples, and evolving educational pedagogy. This edition introduces a new co-author, Dr. John Jewett, at Cal Poly – Pomona, known best for his teaching awards and his role in the recently published PRINCIPLES OF PHYSICS, Third Edition, also written with Ray Serway. Providing students with the tools they need to succeed in introductory physics, the Sixth Edition of this authoritative text features unparalleled media integration and a newly enhanced supplemental package for instructors and students! Features A GENERAL PROBLEM-SOLVING STRATEGY is outlined early in the text. This strategy provides a series of steps similar to those taken by professional physicists in solving problems. This problem solving strategy is integrated into the Coached Problems (within PhysicsNow) to reinforce this key skill. A large number of authoritative and highly realistic WORKED EXAMPLES promote interactivity and reinforce student understanding of problem-solving techniques. In many cases, these examples serve as models for solving end-of-chapter problems. The examples are set off from the text for ease of location and are given titles to describe their content. Many examples include specific references to the GENERAL PROBLEM-SOLVING STRATEGY to illustrate the underlying concepts and methodology used in arriving at a correct solution. This will help students understand the logic behind the solution and the advantage of using a particular approach to solve the problem. About one-third of the WORKED EXAMPLES include new WHAT IF? extensions. CONCEPTUAL EXAMPLES include detailed reasoning statements to help students learn how to think through physical situations. A concerted effort was made to place more emphasis on critical thinking and teaching physical concepts in this new edition. Both PROBLEM-SOLVING STRATEGIES and HINTS help students approach homework assignments with greater confidence. General strategies and suggestions are included for solving the types of problems featured in the worked examples, end-of-chapter problems, and PhysicsNow. This feature helps students identify the essential steps in solving problems and increases their skills as problem solvers. END-OF-CHAPTER PROBLEMS – An extensive set of problems is included at the end of each chapter. Answers to odd-numbered problems are given at the end of the book. For the convenience of both the student and instructor, about two thirds of the problems are keyed to specific sections of the chapter. All problems have been carefully worded and have been checked for clarity and accuracy. Solutions to approximately 20 percent of the end-of-chapter problems are included in the Student Solutions Manual and Study Guide. These problems are identified with a box around the problem number. Serway and Jewett have a clear, relaxed writing style in which they carefully define new terms and avoid jargon whenever possible. The presentation is accurate and precise. The International System of units (SI) is used throughout the book. The U.S. customary system of units is used only to a limited extent in the problem sets of the early chapters on mechanics.  Table of Contents Part I: MECHANICS 1 1. Physics and Measurement. 2 Standards of Length, Mass, and Time. Matter and Model Building. Density and Atomic Mass. Dimensional Analysis. Conversion of Units. Estimates and Order-of-Magnitude Calculations. Significant Figures. 2. Motion in One Dimension 23 Position, Velocity, and Speed. Instantaneous Velocity and Speed. Acceleration. Motion Diagrams. One-Dimensional Motion with Constant Acceleration. Freely Falling Objects. Kinematic Equations Derived from Calculus. General Problem-Solving Strategy. 3. Vectors. 58 Coordinate Systems. Vector and Scalar Quantities. Some Properties of Vectors. Components of a Vector and Unit Vectors. 4. Motion in Two Dimensions 77 The Position, Velocity, and Acceleration Vectors. Two-Dimensional Motion with Constant Acceleration. Projectile Motion. Uniform Circular Motion. Tangential and Radial Acceleration. Relative Velocity and Relative Acceleration. 5. The Laws of Motion 111 The Concept of Force. Newton's First Law and Inertial Frames. Mass. Newton's Second Law. The Gravitational Force and Weight. Newton's Third Law. Some Applications of Newton's Laws. Forces of Friction. 6. Circular Motion and Other Applications of Newton's Laws 150 Newton's Second Law Applied to Uniform Circular Motion. Nonuniform Circular Motion. Motion in Accelerated Frames. Motion in the Presence of Resistive Forces. Numerical Modeling in Particle Dynamics. 7. Energy and Energy Transfer 181 Systems and Environments. Work Done by a Constant Force. The Scalar Product of Two Vectors. Work Done by a Varying Force. Kinetic Energy and the Work--Kinetic Energy Theorem. The Non-Isolated System--Conservation of Energy. Situations Involving Kinetic Friction. Power. Energy and the Automobile. 8. Potential Energy 217 Potential Energy of a System. The Isolated System--Conservation of Mechanical Energy. Conservative and Nonconservative Forces. Changes in Mechanical Energy for Nonconservative Forces. Relationship Between Conservative Forces and Potential Energy. Energy Diagrams and Equilibrium of a System. 9. Linear Momentum and Collisions 251 Linear Momentum and Its Conservation. Impulse and Momentum. Collisions in One Dimension. Two-Dimensional Collisions. The Center of Mass. Motion of a System of Particles. Rocket Propulsion. 10. Rotation of a Rigid Object about a Fixed Axis 292 Angular Position, Velocity, and Acceleration. Rotational Kinematics: Rotational Motion with Constant Angular Acceleration. Angular and Linear Quantities. Rotational Kinetic Energy. Calculation of Moments of Inertia. Torque. Relationship Between Torque and Angular Acceleration. Work, Power, and Energy in Rotational Motion. Rolling Motion of a Rigid Object. 11. Angular Momentum 336 The Vector Product and Torque. Angular Momentum. Angular Momentum of a Rotating Rigid Object. Conservation of Angular Momentum. The Motion of Gyroscopes and Tops. Angular Momentum as a Fundamental Quantity. 12. Static Equilibrium and Elasticity 362 The Conditions for Equilibrium. More on the Center of Gravity. Examples of Rigid Objects in Static Equilibrium. Elastic Properties of Solids. 13. Universal Gravitation 389 Newton's Law of Universal Gravitation. Measuring the Gravitational Constant. Free-Fall Acceleration and the Gravitational Force. Kepler's Laws and the Motion of Planets. The Gravitational Field. Gravitational Potential Energy. Energy Considerations in Planetary and Satellite Motion. 14. Fluid Mechanics 420 Pressure. Variation of Pressure with Depth. Pressure Measurements. Buoyant Forces and Archimedes's Principle. Fluid Dynamics. Bernoulli's Equation. Other Applications of Fluid Dynamics. Part II: OSCILLATIONS AND MECHANICAL WAVES 451 15. Oscillatory Motion 452 Motion of an Object Attached to a Spring. Mathematical Representation of Simple Harmonic Motion. Energy of the Simple Harmonic Oscillator. Comparing Simple Harmonic Motion with Uniform Circular Motion. The Pendulum. Damped Oscillations/ Forced Oscillations. 16. Wave Motion 486 Propagation of a Disturbance. Sinusoidal Waves. The Speed of Waves on Strings. Reflection and Transmission. Rate of Energy Transfer by Sinusoidal Waves on Strings. The Linear Wave Equation. 17. Sound Waves 512 Speed of Sound Waves. Periodic Sound Waves. Intensity of Periodic Sound Waves. The Doppler Effect. Digital Sound Recording. Motion Picture Sound. 18. Superposition and Standing Waves 543 Superposition and Interference. Standing Waves. Standing Waves in a String Fixed at Both Ends. Resonance. Standing Waves in Air Columns. Standing Waves in Rods and Membranes. Beats: Interference in Time. Nonsinusoidal Wave Patterns.  Part III: THERMODYNAMICS 579 19. Temperature 580 Temperature and the Zeroth Law of Thermodynamics. Thermometers and the Celsius Temperature Scale. The Constant-Volume Gas Thermometer and the Absolute Temperature Scale. Thermal Expansion of Solids and Liquids. Macroscopic Description of an Ideal Gas. 20. Heat and the First Law of Thermodynamics 604 Heat and Internal Energy. Specific Heat and Calorimetry. Latent Heat. Work and Heat in Thermodynamic Processes. The First Law of Thermodynamics. Some Applications of the First Law of Thermodynamics. Energy Transfer Mechanisms. 21. The Kinetic Theory of Gases 640 Molecular Model of an Ideal Gas. Molar Specific Heat of an Ideal Gas. Adiabatic Processes for an Ideal Gas. The Equipartition of Energy. The Boltzmann Distribution Law. Distribution of Molecular Speeds/ Mean Free Path. 22. Heat Engines, Entropy, and the Second Law of Thermodynamics 667 Heat Engines and the Second Law of Thermodynamics. Heat Pumps and Refrigerators. Reversible and Irreversible Processes. The Carnot Engine. Gasoline and Diesel Engines. Entropy. Entropy Changes in Irreversible Processes. Entropy on a Microscopic Scale. Part IV: ELECTRICITY AND MAGNETISM 705 23. Electric Fields 706 Properties of Electric Charges. Charging Objects by Induction. Coulomb's Law. The Electric Field. Electric Field of a Continuous Charge Distribution. Electric Field Lines. Motion of Charged Particles in a Uniform Electric Field. 24. Gauss's Law 739 Electric Flux. Gauss's Law. Application of Gauss's Law to Various Charge Distributions. Conductors in Electrostatic Equilibrium. Formal Derivation of Gauss's Law. 25. Electric Potential 762 Potential Difference and Electric Potential. Potential Differences in a Uniform Electric Field. Electric Potential and Potential Energy Due to Point Charges. Obtaining the Value of the Electric Field from the Electric Potential. Electric Potential Due to Continuous Charge Distributions. Electric Potential Due to a Charged Conductor. The Millikan Oil-Drop Experiment. Applications of Electrostatics. 26. Capacitance and Dielectrics 795 Definition of Capacitance. Calculating Capacitance. Combinations of Capacitors. Energy Stored in a Charged Capacitor. Capacitors with Dielectrics. Electric Dipole in an Electric Field. An Atomic Description of Dielectrics. 27. Current and Resistance 831 Electric Current. Resistance. A Model for Electrical Conduction. Resistance and Temperature. Superconductors. Electrical Power. 28. Direct Current Circuits 858 Electromotive Force Resistors in Series and Parallel. Kirchhoff's Rules. RC Circuits. Electrical Meters. Household Wiring and Electrical Safety. 29. Magnetic Fields 894 Magnetic Field and Forces. Magnetic Force Acting on a Current-Carrying Conductor. Torque on a Current Loop in a Uniform Magnetic Field. Motion of a Charged Particle in a Uniform Magnetic Field. Applications Involving Charged Particles Moving in a Magnetic Field. The Hall Effect. 30. Sources of Magnetic Field 926 The Biot-Savart Law. The Magnetic Force Between Two Parallel Conductors. Ampere's Law. The Magnetic Field of a Solenoid. Magnetic Flux. Gauss's Law in Magnetism. Displacement Current and the General Form of Ampere's Law. Magnetism in Matter. The Magnetic Field of the Earth. 31. Faraday's Law 967 Faraday's Law of Induction. Motional emf. Lenz's Law. Induced emf and Electric Fields. Generators and Motors/ Eddy Currents. Maxwell's Equations. 32. Inductance 1003 Self-Inductance. RL Circuits. Energy in a Magnetic Field. Mutual Inductance. Oscillations in an LC Circuit. The RLC Circuit. 33. Alternating Current Circuits 1033 AC Sources. Resistors in an AC Circuit. Inductors in an AC Circuit. Capacitors in an AC Circuit. The RLC Series Circuit. Power in an AC Circuit. Resonance in a Series RLC Circuit. The Transformer and Power Transmission. Rectifiers and Filters. 34. Electromagnetic Waves 1066 Maxwell's Equations and Hertz's Discoveries. Plane Electromagnetic Waves. Energy Carried by Electromagnetic Waves. Momentum and Radiation Pressure. Production of Electromagnetic Waves by an Antenna. Part V: LIGHT AND OPTICS 1093 35. The Nature of Light and the Laws of Geometric Optics 1094 The Nature of Light. Measurements of the Speed of Light. The Ray Approximation in Geometric Optics. Reflection. Refraction. Huygens's Principle. Dispersion and Prisms. Total Internal Reflection. Fermat's Principle. 36. Image Formation 1126 Images Formed by Flat Mirrors. Images Formed by Spherical Mirrors. Images Formed by Refraction. Thin Lenses. Lens Aberrations. The Camera. The Eye. The Simple Magnifier. The Compound Microscope. The Telescope. 37. Interference of Light Waves 1176 Conditions for Interference. Young's Double-Slit Experiment. Intensity Distribution of the Double-Slit Interference Pattern. Phasor Addition of Waves. Change of Phase Due to Reflection. Interference in Thin Films. The Michelson Interferometer. 38. Diffraction Patterns and Polarization 1205 Introduction to Diffraction Patterns. Diffraction Patterns from Narrow Slits. Resolution of Single-Slit and Circular Apertures. The Diffraction Grating. Diffraction of X-rays by Crystals. Polarization of Light Waves. Part VI: MODERN PHYSICS 1243 39. Relativity 1244 The Principle of Galilean Relativity. The Michelson-Morley Experiment. Einstein's Principle of Relativity. Consequences of the Special Theory of Relativity. The Lorentz Transformation Equations. The Lorentz Velocity Transformation Equations Relativistic Linear Momentum and the Relativistic Form of Newton's Laws. Relativistic Energy. Mass and Energy. The General Theory of Relativity. APPENDIXES: A.1 A. Tables A.1 Conversion Factors. Symbols, Dimensions, and Units of Physical Quantities. Table of Atomic Masses. B. Mathematics Review A.14 Scientific Notation. Algebra. Geometry. Trigonometry. Series Expansions. Differential Calculus. Integral Calculus. Propagation of Uncertainty. C. Periodic Table of the Elements A.30 D. SI Units A.32 E. Nobel Prize Winners A.33 Answers to Odd-Numbered Problems A.37 Index I.1 Mechanics 1 PA R T hysics, the most fundamental physical science, is concerned with the basic P principles of the Universe. It is the foundation upon which the other sciences— astronomy, biology, chemistry, and geology—are based. The beauty of physics lies in the simplicity of the fundamental physical theories and in the manner in which just a small number of fundamental concepts, equations, and assumptions can alter and expand our view of the world around us. The study of physics can be divided into six main areas: 1. classical mechanics, which is concerned with the motion of objects that are large relative to atoms and move at speeds much slower than the speed of light; 2. relativity, which is a theory describing objects moving at any speed, even speeds approaching the speed of light; 3. thermodynamics, which deals with heat, work, temperature, and the statistical be- havior of systems with large numbers of particles; 4. electromagnetism, which is concerned with electricity, magnetism, and electro- magnetic fields; 5. optics, which is the study of the behavior of light and its interaction with materials; 6. quantum mechanics, a collection of theories connecting the behavior of matter at the submicroscopic level to macroscopic observations. The disciplines of mechanics and electromagnetism are basic to all other branches of classical physics (developed before 1900) and modern physics (c. 1900–present). The first part of this textbook deals with classical mechanics, sometimes referred to as Newtonian mechanics or simply mechanics. This is an ap- propriate place to begin an introductory text because many of the basic principles used to understand mechanical systems can later be used to describe such natural phenomena as waves and the transfer of energy by heat. Furthermore, the laws of conservation of energy and momentum introduced in mechanics retain their impor- tance in the fundamental theories of other areas of physics. Today, classical mechanics is of vital importance to students from all disciplines. It is highly successful in describing the motions of different objects, such as planets, rockets, and baseballs. In the first part of the text, we shall describe the laws of clas- sical mechanics and examine a wide range of phenomena that can be understood with these fundamental ideas. ! Liftoff of the space shuttle Columbia. The tragic accident of February 1, 2003 that took the lives of all seven astronauts aboard happened just before Volume 1 of this book went to press. The launch and operation of a space shuttle involves many fundamental principles of classical mechanics, thermodynamics, and electromagnetism. We study the principles of classical mechanics in Part 1 of this text, and apply these principles to rocket propulsion in Chapter 9. (NASA) 1  Chapter 1 Physics and Measurement CHAPTE R OUTLI N E 1.1 Standards of Length, Mass, and Time 1.2 Matter and Model Building 1.3 Density and Atomic Mass 1.4 Dimensional Analysis 1.5 Conversion of Units 1.6 Estimates and Order-of- Magnitude Calculations 1.7 Significant Figures ▲ The workings of a mechanical clock. Complicated timepieces have been built for cen- turies in an effort to measure time accurately. Time is one of the basic quantities that we use in studying the motion of objects. (elektraVision/Index Stock Imagery) 2 Like all other sciences, physics is based on experimental observations and quantitative measurements. The main objective of physics is to find the limited number of funda- mental laws that govern natural phenomena and to use them to develop theories that can predict the results of future experiments. The fundamental laws used in develop- ing theories are expressed in the language of mathematics, the tool that provides a bridge between theory and experiment. When a discrepancy between theory and experiment arises, new theories must be formulated to remove the discrepancy. Many times a theory is satisfactory only under limited conditions; a more general theory might be satisfactory without such limita- tions. For example, the laws of motion discovered by Isaac Newton (1642–1727) in the 17th century accurately describe the motion of objects moving at normal speeds but do not apply to objects moving at speeds comparable with the speed of light. In contrast, the special theory of relativity developed by Albert Einstein (1879–1955) in the early 1900s gives the same results as Newton’s laws at low speeds but also correctly describes motion at speeds approaching the speed of light. Hence, Einstein’s special theory of relativity is a more general theory of motion. Classical physics includes the theories, concepts, laws, and experiments in classical mechanics, thermodynamics, optics, and electromagnetism developed before 1900. Im- portant contributions to classical physics were provided by Newton, who developed classical mechanics as a systematic theory and was one of the originators of calculus as a mathematical tool. Major developments in mechanics continued in the 18th century, but the fields of thermodynamics and electricity and magnetism were not developed until the latter part of the 19th century, principally because before that time the appa- ratus for controlled experiments was either too crude or unavailable. A major revolution in physics, usually referred to as modern physics, began near the end of the 19th century. Modern physics developed mainly because of the discovery that many physical phenomena could not be explained by classical physics. The two most im- portant developments in this modern era were the theories of relativity and quantum mechanics. Einstein’s theory of relativity not only correctly described the motion of ob- jects moving at speeds comparable to the speed of light but also completely revolution- ized the traditional concepts of space, time, and energy. The theory of relativity also shows that the speed of light is the upper limit of the speed of an object and that mass and energy are related. Quantum mechanics was formulated by a number of distin- guished scientists to provide descriptions of physical phenomena at the atomic level. Scientists continually work at improving our understanding of fundamental laws, and new discoveries are made every day. In many research areas there is a great deal of overlap among physics, chemistry, and biology. Evidence for this overlap is seen in the names of some subspecialties in science—biophysics, biochemistry, chemical physics, biotechnology, and so on. Numerous technological advances in recent times are the re- sult of the efforts of many scientists, engineers, and technicians. Some of the most no- table developments in the latter half of the 20th century were (1) unmanned planetary explorations and manned moon landings, (2) microcircuitry and high-speed comput- ers, (3) sophisticated imaging techniques used in scientific research and medicine, and 3 4 C H A P T E R 1 Physics and Measurement (4) several remarkable results in genetic engineering. The impacts of such develop- ments and discoveries on our society have indeed been great, and it is very likely that future discoveries and developments will be exciting, challenging, and of great benefit to humanity. 1.1 Standards of Length, Mass, and Time The laws of physics are expressed as mathematical relationships among physical quanti- ties that we will introduce and discuss throughout the book. Most of these quantities are derived quantities, in that they can be expressed as combinations of a small number of basic quantities. In mechanics, the three basic quantities are length, mass, and time. All other quantities in mechanics can be expressed in terms of these three. If we are to report the results of a measurement to someone who wishes to repro- duce this measurement, a standard must be defined. It would be meaningless if a visitor from another planet were to talk to us about a length of 8 “glitches” if we do not know the meaning of the unit glitch. On the other hand, if someone familiar with our system of measurement reports that a wall is 2 meters high and our unit of length is defined to be 1 meter, we know that the height of the wall is twice our basic length unit. Like- wise, if we are told that a person has a mass of 75 kilograms and our unit of mass is de- fined to be 1 kilogram, then that person is 75 times as massive as our basic unit.1 What- ever is chosen as a standard must be readily accessible and possess some property that can be measured reliably. Measurements taken by different people in different places must yield the same result. In 1960, an international committee established a set of standards for the fundamen- tal quantities of science. It is called the SI (Système International), and its units of length, mass, and time are the meter, kilogram, and second, respectively. Other SI standards es- tablished by the committee are those for temperature (the kelvin), electric current (the ampere), luminous intensity (the candela), and the amount of substance (the mole). Length In A.D. 1120 the king of England decreed that the standard of length in his country would be named the yard and would be precisely equal to the distance from the tip of his nose to the end of his outstretched arm. Similarly, the original standard for the foot adopted by the French was the length of the royal foot of King Louis XIV. This stan- dard prevailed until 1799, when the legal standard of length in France became the me- ter, defined as one ten-millionth the distance from the equator to the North Pole along one particular longitudinal line that passes through Paris. Many other systems for measuring length have been developed over the years, but the advantages of the French system have caused it to prevail in almost all coun- tries and in scientific circles everywhere. As recently as 1960, the length of the meter was defined as the distance between two lines on a specific platinum–iridium bar stored under controlled conditions in France. This standard was abandoned for sev- eral reasons, a principal one being that the limited accuracy with which the separa- tion between the lines on the bar can be determined does not meet the current requirements of science and technology. In the 1960s and 1970s, the meter was de- fined as 1 650 763.73 wavelengths of orange-red light emitted from a krypton-86 lamp. However, in October 1983, the meter (m) was redefined as the distance traveled by light in vacuum during a time of 1/299 792 458 second. In effect, this 1 The need for assigning numerical values to various measured physical quantities was expressed by Lord Kelvin (William Thomson) as follows: “I often say that when you can measure what you are speaking about, and express it in numbers, you should know something about it, but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind. It may be the beginning of knowledge but you have scarcely in your thoughts advanced to the state of science.”  S E C T I O N 1. 1 Standards of Length, Mass, and Time 5 Table 1.1 Approximate Values of Some Measured Lengths ▲ PITFALL PREVENTION 1.1 No Commas in Length (m) Numbers with Many Distance from the Earth to the most remote known quasar 1.4 ! 1026 Digits Distance from the Earth to the most remote normal galaxies 9 ! 1025 We will use the standard scientific Distance from the Earth to the nearest large galaxy 2 ! 1022 notation for numbers with more (M 31, the Andromeda galaxy) than three digits, in which Distance from the Sun to the nearest star (Proxima Centauri) 4 ! 1016 groups of three digits are sepa- One lightyear 9.46 ! 1015 rated by spaces rather than commas. Thus, 10 000 is the Mean orbit radius of the Earth about the Sun 1.50 ! 1011 same as the common American Mean distance from the Earth to the Moon 3.84 ! 108 notation of 10,000. Similarly, Distance from the equator to the North Pole 1.00 ! 107 # $ 3.14159265 is written as Mean radius of the Earth 6.37 ! 106 3.141 592 65. Typical altitude (above the surface) of a 2 ! 105 satellite orbiting the Earth Length of a football field 9.1 ! 101 Length of a housefly 5 ! 10"3 Size of smallest dust particles # 10"4 Size of cells of most living organisms # 10"5 Table 1.2 Diameter of a hydrogen atom # 10"10 Diameter of an atomic nucleus # 10"14 Masses of Various Objects (Approximate Values) Diameter of a proton # 10"15 Mass (kg) latest definition establishes that the speed of light in vacuum is precisely 299 792 458 Observable # 1052 meters per second. Universe Table 1.1 lists approximate values of some measured lengths. You should study this Milky Way # 1042 galaxy table as well as the next two tables and begin to generate an intuition for what is meant Sun 1.99 ! 1030 by a length of 20 centimeters, for example, or a mass of 100 kilograms or a time inter- Earth 5.98 ! 1024 val of 3.2 ! 107 seconds. Moon 7.36 ! 1022 Shark # 103 Mass Human # 102 Frog # 10"1 The SI unit of mass, the kilogram (kg), is defined as the mass of a specific Mosquito # 10"5 platinum–iridium alloy cylinder kept at the International Bureau of Weights Bacterium # 1 ! 10"15 and Measures at Sèvres, France. This mass standard was established in 1887 and has Hydrogen 1.67 ! 10"27 not been changed since that time because platinum–iridium is an unusually stable al- atom loy. A duplicate of the Sèvres cylinder is kept at the National Institute of Standards and Electron 9.11 ! 10"31 Technology (NIST) in Gaithersburg, Maryland (Fig. 1.1a). Table 1.2 lists approximate values of the masses of various objects. Time Before 1960, the standard of time was defined in terms of the mean solar day for the year 1900. (A solar day is the time interval between successive appearances of the Sun ▲ PITFALL PREVENTION at the highest point it reaches in the sky each day.) The second was defined as 1.2 Reasonable Values ! "! "! " 1 60 1 60 1 of a mean solar day. The rotation of the Earth is now known to vary 24 Generating intuition about typi- slightly with time, however, and therefore this motion is not a good one to use for cal values of quantities is impor- defining a time standard. tant because when solving prob- lems you must think about your In 1967, the second was redefined to take advantage of the high precision attainable end result and determine if it in a device known as an atomic clock (Fig. 1.1b), which uses the characteristic frequency seems reasonable. If you are cal- of the cesium-133 atom as the “reference clock.” The second (s) is now defined as culating the mass of a housefly 9 192 631 770 times the period of vibration of radiation from the cesium atom.2 and arrive at a value of 100 kg, this is unreasonable—there is an 2 Period is defined as the time interval needed for one complete vibration. error somewhere. 6 C H A P T E R 1 Physics and Measurement (Courtesy of National Institute of Standards and Technology, U.S. Department of Commerce) (a) (b) Figure 1.1 (a) The National Standard Kilogram No. 20, an accurate copy of the International Standard Kilogram kept at Sèvres, France, is housed under a double bell jar in a vault at the National Institute of Standards and Technology. (b) The nation’s primary time standard is a cesium fountain atomic clock developed at the National Institute of Standards and Technology laboratories in Boulder, Colorado. The clock will neither gain nor lose a second in 20 million years. To keep these atomic clocks—and therefore all common clocks and watches that are set to them—synchronized, it has sometimes been necessary to add leap seconds to our clocks. Since Einstein’s discovery of the linkage between space and time, precise measure- ment of time intervals requires that we know both the state of motion of the clock used to measure the interval and, in some cases, the location of the clock as well. Otherwise, for example, global positioning system satellites might be unable to pinpoint your loca- tion with sufficient accuracy, should you need to be rescued. Approximate values of time intervals are presented in Table 1.3. Table 1.3 Approximate Values of Some Time Intervals Time Interval (s) Age of the Universe 5 ! 1017 Age of the Earth 1.3 ! 1017 Average age of a college student 6.3 ! 108 One year 3.2 ! 107 One day (time interval for one revolution of the Earth about its axis) 8.6 ! 104 One class period 3.0 ! 103 Time interval between normal heartbeats 8 ! 10"1 Period of audible sound waves # 10"3 Period of typical radio waves # 10"6 Period of vibration of an atom in a solid # 10"13 Period of visible light waves # 10"15 Duration of a nuclear collision # 10"22 Time interval for light to cross a proton # 10"24  S E C T I O N 1. 2 Matter and Model Building 7 Table 1.4 Prefixes for Powers of Ten Power Prefix Abbreviation 10"24 yocto y 10"21 zepto z 10"18 atto a 10"15 femto f 10"12 pico p 10"9 nano n 10"6 micro % 10"3 milli m 10"2 centi c 10"1 deci d 103 kilo k 106 mega M 109 giga G 1012 tera T 1015 peta P 1018 exa E 1021 zetta Z 1024 yotta Y In addition to SI, another system of units, the U.S. customary system, is still used in the United States despite acceptance of SI by the rest of the world. In this system, the units of length, mass, and time are the foot (ft), slug, and second, respectively. In this text we shall use SI units because they are almost universally accepted in science and industry. We shall make some limited use of U.S. customary units in the study of classical mechanics. In addition to the basic SI units of meter, kilogram, and second, we can also use other units, such as millimeters and nanoseconds, where the prefixes milli- and nano- denote multipliers of the basic units based on various powers of ten. Prefixes for the various powers of ten and their abbreviations are listed in Table 1.4. For example, 10" 3 m is equivalent to 1 millimeter (mm), and 103 m corresponds to 1 kilometer (km). Likewise, 1 kilogram (kg) is 103 grams (g), and 1 megavolt (MV) is 106 volts (V). 1.2 Matter and Model Building If physicists cannot interact with some phenomenon directly, they often imagine a model for a physical system that is related to the phenomenon. In this context, a model is a system of physical components, such as electrons and protons in an atom. Once we have identified the physical components, we make predictions about the behavior of the system, based on the interactions among the components of the sys- tem and/or the interaction between the system and the environment outside the system. As an example, consider the behavior of matter. A 1-kg cube of solid gold, such as that at the left of Figure 1.2, has a length of 3.73 cm on a side. Is this cube nothing but wall-to-wall gold, with no empty space? If the cube is cut in half, the two pieces still re- tain their chemical identity as solid gold. But what if the pieces are cut again and again, indefinitely? Will the smaller and smaller pieces always be gold? Questions such as these can be traced back to early Greek philosophers. Two of them—Leucippus and his student Democritus—could not accept the idea that such cuttings could go on for- ever. They speculated that the process ultimately must end when it produces a particle 8 C H A P T E R 1 Physics and Measurement Quark composition of a proton u u d Neutron Gold nucleus Nucleus Proton Gold cube Gold atoms Figure 1.2 Levels of organization in matter. Ordinary matter consists of atoms, and at the center of each atom is a compact nucleus consisting of protons and neutrons. Protons and neutrons are composed of quarks. The quark composition of a proton is shown. that can no longer be cut. In Greek, atomos means “not sliceable.” From this comes our English word atom. Let us review briefly a number of historical models of the structure of matter. The Greek model of the structure of matter was that all ordinary matter consists of atoms, as suggested to the lower right of the cube in Figure 1.2. Beyond that, no ad- ditional structure was specified in the model— atoms acted as small particles that in- teracted with each other, but internal structure of the atom was not a part of the model. In 1897, J. J. Thomson identified the electron as a charged particle and as a con- stituent of the atom. This led to the first model of the atom that contained internal structure. We shall discuss this model in Chapter 42. Following the discovery of the nucleus in 1911, a model was developed in which each atom is made up of electrons surrounding a central nucleus. A nucleus is shown in Figure 1.2. This model leads, however, to a new question—does the nucleus have structure? That is, is the nucleus a single particle or a collection of particles? The exact composition of the nucleus is not known completely even today, but by the early 1930s a model evolved that helped us understand how the nucleus behaves. Specifically, sci- entists determined that occupying the nucleus are two basic entities, protons and neu- trons. The proton carries a positive electric charge, and a specific chemical element is identified by the number of protons in its nucleus. This number is called the atomic number of the element. For instance, the nucleus of a hydrogen atom contains one proton (and so the atomic number of hydrogen is 1), the nucleus of a helium atom contains two protons (atomic number 2), and the nucleus of a uranium atom contains 92 protons (atomic number 92). In addition to atomic number, there is a second num- ber characterizing atoms—mass number, defined as the number of protons plus neu- trons in a nucleus. The atomic number of an element never varies (i.e., the number of protons does not vary) but the mass number can vary (i.e., the number of neutrons varies). The existence of neutrons was verified conclusively in 1932. A neutron has no charge and a mass that is about equal to that of a proton. One of its primary purposes  S E C T I O N 1. 3 Density and Atomic Mass 9 is to act as a “glue” that holds the nucleus together. If neutrons were not present in the nucleus, the repulsive force between the positively charged particles would cause the nucleus to come apart. But is this where the process of breaking down stops? Protons, neutrons, and a host of other exotic particles are now known to be composed of six different varieties of particles called quarks, which have been given the names of up, down, strange, charmed, 2 bottom, and top. The up, charmed, and top quarks have electric charges of ' 3 that of the proton, whereas the down, strange, and bottom quarks have charges of " 1 that 3 of the proton. The proton consists of two up quarks and one down quark, as shown at the top in Figure 1.2. You can easily show that this structure predicts the correct charge for the proton. Likewise, the neutron consists of two down quarks and one up quark, giving a net charge of zero. This process of building models is one that you should develop as you study physics. You will be challenged with many mathematical problems to solve in this study. One of the most important techniques is to build a model for the prob- lem—identify a system of physical components for the problem, and make predic- tions of the behavior of the system based on the interactions among the compo- nents of the system and/or the interaction between the system and its surrounding environment. 1.3 Density and Atomic Mass In Section 1.1, we explored three basic quantities in mechanics. Let us look now at an A table of the letters in the example of a derived quantity—density. The density & (Greek letter rho) of any sub- Greek alphabet is provided on stance is defined as its mass per unit volume: the back endsheet of the textbook. m &$ (1.1) V For example, aluminum has a density of 2.70 g/cm3, and lead has a density of 11.3 g/cm3. Therefore, a piece of aluminum of volume 10.0 cm3 has a mass of 27.0 g, whereas an equivalent volume of lead has a mass of 113 g. A list of densities for various substances is given in Table 1.5. The numbers of protons and neutrons in the nucleus of an atom of an element are re- lated to the atomic mass of the element, which is defined as the mass of a single atom of the element measured in atomic mass units (u) where 1 u $ 1.660 538 7 ! 10"27 kg. Table 1.5 Densities of Various Substances Substance Density ! (103 kg/m3) Platinum 21.45 Gold 19.3 Uranium 18.7 Lead 11.3 Copper 8.92 Iron 7.86 Aluminum 2.70 Magnesium 1.75 Water 1.00 Air at atmospheric pressure 0.0012 10 C H A P T E R 1 Physics and Measurement The atomic mass of lead is 207 u and that of aluminum is 27.0 u. However, the ratio of atomic masses, 207 u/27.0 u $ 7.67, does not correspond to the ratio of densities, (11.3 ! 103 kg/m3)/(2.70 ! 103 kg/m3) $ 4.19. This discrepancy is due to the differ- ence in atomic spacings and atomic arrangements in the crystal structures of the two elements. Quick Quiz 1.1 In a machine shop, two cams are produced, one of alu- minum and one of iron. Both cams have the same mass. Which cam is larger? (a) the aluminum cam (b) the iron cam (c) Both cams have the same size. Example 1.1 How Many Atoms in the Cube? A solid cube of aluminum (density 2.70 g/cm3) has a vol- write this relationship twice, once for the actual sample of ume of 0.200 cm3. It is known that 27.0 g of aluminum con- aluminum in the problem and once for a 27.0-g sample, and tains 6.02 ! 1023 atoms. How many aluminum atoms are then we divide the first equation by the second: contained in the cube? m sample $ kN sample m sample N : $ sample Solution Because density equals mass per unit volume, the m 27.0 g $ kN27.0 g m 27.0 g N27.0 g mass of the cube is Notice that the unknown proportionality constant k cancels, m $ &V $ (2.70 g/cm3)(0.200 cm3) $ 0.540 g so we do not need to know its value. We now substitute the values: To solve this problem, we will set up a ratio based on the fact that the mass of a sample of material is proportional to the 0.540 g N sample $ number of atoms contained in the sample. This technique 27.0 g 6.02 ! 1023 atoms of solving by ratios is very powerful and should be studied (0.540 g)(6.02 ! 1023 atoms) and understood so that it can be applied in future problem Nsample $ 27.0 g solving. Let us express our proportionality as m $ kN, where m is the mass of the sample, N is the number of atoms in the sample, and k is an unknown proportionality constant. We $ 1.20 ! 1022 atoms ▲ PITFALL PREVENTION 1.4 Dimensional Analysis 1.3 Setting Up Ratios The word dimension has a special meaning in physics. It denotes the physical nature of When using ratios to solve a a quantity. Whether a distance is measured in units of feet or meters or fathoms, it is problem, keep in mind that ratios still a distance. We say its dimension is length. come from equations. If you start The symbols we use in this book to specify the dimensions of length, mass, and from equations known to be cor- time are L, M, and T, respectively.3 We shall often use brackets [ ] to denote the dimen- rect and can divide one equation sions of a physical quantity. For example, the symbol we use for speed in this book is v, by the other as in Example 1.1 to and in our notation the dimensions of speed are written [v] $ L/T. As another exam- obtain a useful ratio, you will ple, the dimensions of area A are [A] $ L2. The dimensions and units of area, volume, avoid reasoning errors. So write speed, and acceleration are listed in Table 1.6. The dimensions of other quantities, the known equations first! such as force and energy, will be described as they are introduced in the text. In many situations, you may have to derive or check a specific equation. A useful and powerful procedure called dimensional analysis can be used to assist in the deriva- tion or to check your final expression. Dimensional analysis makes use of the fact that 3 The dimensions of a quantity will be symbolized by a capitalized, non-italic letter, such as L. The symbol for the quantity itself will be italicized, such as L for the length of an object, or t for time.  S E C T I O N 1. 4 Dimensional Analysis 11 Table 1.6 Units of Area, Volume, Velocity, Speed, and Acceleration ▲ PITFALL PREVENTION 1.4 Symbols for Area Volume Speed Acceleration Quantities System (L2) (L3) (L/T) (L/T 2) Some quantities have a small SI m2 m3 m/s m/s2 number of symbols that repre- U.S. customary ft2 ft3 ft/s ft/s2 sent them. For example, the sym- bol for time is almost always t. Others quantities might have var- ious symbols depending on the dimensions can be treated as algebraic quantities. For example, quantities can be usage. Length may be described added or subtracted only if they have the same dimensions. Furthermore, the terms on with symbols such as x, y, and z both sides of an equation must have the same dimensions. By following these simple (for position), r (for radius), a, b, and c (for the legs of a right tri- rules, you can use dimensional analysis to help determine whether an expression has angle), ! (for the length of an the correct form. The relationship can be correct only if the dimensions on both sides object), d (for a distance), h (for of the equation are the same. a height), etc. To illustrate this procedure, suppose you wish to derive an equation for the posi- tion x of a car at a time t if the car starts from rest and moves with constant accelera- tion a. In Chapter 2, we shall find that the correct expression is x $ 12 at 2. Let us use dimensional analysis to check the validity of this expression. The quantity x on the left side has the dimension of length. For the equation to be dimensionally correct, the quantity on the right side must also have the dimension of length. We can per- form a dimensional check by substituting the dimensions for acceleration, L/T2 (Table 1.6), and time, T, into the equation. That is, the dimensional form of the equation x $ 12 at 2 is L L$ ) T2 $ L T2 The dimensions of time cancel as shown, leaving the dimension of length on the right- hand side. A more general procedure using dimensional analysis is to set up an expression of the form x ( ant m where n and m are exponents that must be determined and the symbol ( indicates a proportionality. This relationship is correct only if the dimensions of both sides are the same. Because the dimension of the left side is length, the dimension of the right side must also be length. That is, [antm] $ L $ L1T0 Because the dimensions of acceleration are L/T2 and the dimension of time is T, we have (L/T2)n Tm $ L1T0 (Ln Tm "2n) $ L1T0 The exponents of L and T must be the same on both sides of the equation. From the exponents of L, we see immediately that n $ 1. From the exponents of T, we see that m " 2n $ 0, which, once we substitute for n, gives us m $ 2. Returning to our original expression x ( antm, we conclude that x ( at 2. This result differs by a factor of 12 from the correct expression, which is x $ 12 at 2. Quick Quiz 1.2 True or False: Dimensional analysis can give you the numeri- cal value of constants of proportionality that may appear in an algebraic expression. 12 C H A P T E R 1 Physics and Measurement Example 1.2 Analysis of an Equation Show that the expression v $ at is dimensionally correct, The same table gives us L/T2 for the dimensions of accelera- where v represents speed, a acceleration, and t an instant of tion, and so the dimensions of at are time. L L [at] $ 2 T$ T T Solution For the speed term, we have from Table 1.6 Therefore, the expression is dimensionally correct. (If the L expression were given as v $ at 2 it would be dimensionally [v] $ T incorrect. Try it and see!) Example 1.3 Analysis of a Power Law Suppose we are told that the acceleration a of a particle This dimensional equation is balanced under the conditions moving with uniform speed v in a circle of radius r is pro- portional to some power of r, say r n, and some power of v, say vm. Determine the values of n and m and write the sim- n ' m$ 1 and m$ 2 plest form of an equation for the acceleration. Therefore n $ " 1, and we can write the acceleration ex- Solution Let us take a to be pression as a $ kr nvm v2 a $ kr "1v 2 $ k where k is a dimensionless constant of proportionality. r Knowing the dimensions of a, r, and v, we see that the di- mensional equation must be When we discuss uniform circular motion later, we shall see that k $ 1 if a consistent set of units is used. The constant k ! " L L m Ln'm $ Ln $ would not equal 1 if, for example, v were in km/h and you T2 T Tm wanted a in m/s2. ▲ PITFALL PREVENTION 1.5 Conversion of Units 1.5 Always Include Units Sometimes it is necessary to convert units from one measurement system to another, or When performing calculations, to convert within a system, for example, from kilometers to meters. Equalities between include the units for every quan- SI and U.S. customary units of length are as follows: tity and carry the units through 1 mile $ 1 609 m $ 1.609 km 1 ft $ 0.304 8 m $ 30.48 cm the entire calculation. Avoid the temptation to drop the units 1 m $ 39.37 in. $ 3.281 ft 1 in. $ 0.025 4 m $ 2.54 cm (exactly) early and then attach the ex- pected units once you have an A more complete list of conversion factors can be found in Appendix A. answer. By including the units in Units can be treated as algebraic quantities that can cancel each other. For exam- every step, you can detect errors ple, suppose we wish to convert 15.0 in. to centimeters. Because 1 in. is defined as ex- if the units for the answer turn actly 2.54 cm, we find that out to be incorrect. 2.54 cm 15.0 in. $ (15.0 in.)! " $ 38.1 cm 1 in. where the ratio in parentheses is equal to 1. Notice that we choose to put the unit of an inch in the denominator and it cancels with the unit in the original quantity. The re- maining unit is the centimeter, which is our desired result. Quick Quiz 1.3 The distance between two cities is 100 mi. The number of kilo- meters between the two cities is (a) smaller than 100 (b) larger than 100 (c) equal to 100.  S E C T I O N 1. 6 Estimates and Order-of-Magnitude Calculations 13 Example 1.4 Is He Speeding? On an interstate highway in a rural region of Wyoming, a Figure 1.3 shows the speedometer of an automobile, with car is traveling at a speed of 38.0 m/s. Is this car exceeding speeds in both mi/h and km/h. Can you check the conver- the speed limit of 75.0 mi/h? sion we just performed using this photograph? Solution We first convert meters to miles: (38.0 m/s) ! 1 1609mim " $ 2.36 ! 10 "2 mi/s Now we convert seconds to hours: (2.36 ! 10 "2 mi/s) ! 160mins " ! 601min h " $ 85.0 mi/h Thus, the car is exceeding the speed limit and should slow down. What If? What if the driver is from outside the U.S. and is Phil Boorman/Getty Images familiar with speeds measured in km/h? What is the speed of the car in km/h? Answer We can convert our final answer to the appropriate units: Figure 1.3 The speedometer of a vehicle that (85.0 mi/h) ! 1.609 km 1 mi " $ 137 km/h shows speeds in both miles per hour and kilome- ters per hour. 1.6 Estimates and Order-of-Magnitude Calculations It is often useful to compute an approximate answer to a given physical problem even when little information is available. This answer can then be used to determine whether or not a more precise calculation is necessary. Such an approximation is usu- ally based on certain assumptions, which must be modified if greater precision is needed. We will sometimes refer to an order of magnitude of a certain quantity as the power of ten of the number that describes that quantity. Usually, when an order-of- magnitude calculation is made, the results are reliable to within about a factor of 10. If a quantity increases in value by three orders of magnitude, this means that its value in- creases by a factor of about 103 $ 1 000. We use the symbol # for “is on the order of.” Thus, 0.008 6 # 10"2 0.002 1 # 10"3 720 # 103 The spirit of order-of-magnitude calculations, sometimes referred to as “guessti- mates” or “ball-park figures,” is given in the following quotation: “Make an estimate before every calculation, try a simple physical argument... before every derivation, guess the answer to every puzzle.”4 Inaccuracies caused by guessing too low for one number are often canceled out by other guesses that are too high. You will find that with practice your guesstimates become better and better. Estimation problems can be fun to work as you freely drop digits, venture reasonable approximations for 4 E. Taylor and J. A. Wheeler, Spacetime Physics: Introduction to Special Relativity, 2nd ed., San Francisco, W. H. Freeman & Company, Publishers, 1992, p. 20. 14 C H A P T E R 1 Physics and Measurement unknown numbers, make simplifying assumptions, and turn the question around into something you can answer in your head or with minimal mathematical manipu- lation on paper. Because of the simplicity of these types of calculations, they can be performed on a small piece of paper, so these estimates are often called “back-of-the- envelope calculations.” Example 1.5 Breaths in a Lifetime Estimate the number of breaths taken during an average life in a year and the number of hours in a day are close span. enough for our purposes. Thus, in 70 years there will be (70 yr)(6 ! 105 min/yr) $ 4 ! 107 min. At a rate of 10 Solution We start by guessing that the typical life span is about 70 years. The only other estimate we must make in this breaths/min, an individual would take 4 ! 108 breaths example is the average number of breaths that a person in a lifetime, or on the order of 109 breaths. takes in 1 min. This number varies, depending on whether the person is exercising, sleeping, angry, serene, and so What If? What if the average life span were estimated as forth. To the nearest order of magnitude, we shall choose 10 80 years instead of 70? Would this change our final estimate? breaths per minute as our estimate of the average. (This is certainly closer to the true value than 1 breath per minute or Answer We could claim that (80 yr)(6 ! 105 min/yr) $ 100 breaths per minute.) The number of minutes in a year is 5 ! 107 min, so that our final estimate should be 5 ! 108 approximately breaths. This is still on the order of 109 breaths, so an order- of-magnitude estimate would be unchanged. Furthermore, 1 yr ! 4001 yrdays " ! 125dayh " ! 601min h " 5 $ 6 ! 10 min 80 years is 14% larger than 70 years, but we have overesti- mated the total time interval by using 400 days in a year in- stead of 365 and 25 hours in a day instead of 24. These two Notice how much simpler it is in the expression above to numbers together result in an overestimate of 14%, which multiply 400 ! 25 than it is to work with the more accurate cancels the effect of the increased life span! 365 ! 24. These approximate values for the number of days Example 1.6 It’s a Long Way to San Jose Estimate the number of steps a person would take walking Now we switch to scientific notation so that we can do the from New York to Los Angeles. calculation mentally: Solution Without looking up the distance between these (3 ! 103 mi)(2.5 ! 103 steps/mi) two cities, you might remember from a geography class that they are about 3 000 mi apart. The next approximation we $ 7.5 ! 106 steps # 107 steps must make is the length of one step. Of course, this length depends on the person doing the walking, but we can esti- So if we intend to walk across the United States, it will take mate that each step covers about 2 ft. With our estimated us on the order of ten million steps. This estimate is almost step size, we can determine the number of steps in 1 mi. Be- certainly too small because we have not accounted for curv- cause this is a rough calculation, we round 5 280 ft/mi to ing roads and going up and down hills and mountains. 5 000 ft/mi. (What percentage error does this introduce?) Nonetheless, it is probably within an order of magnitude of This conversion factor gives us the correct answer. 5 000 ft/mi $ 2 500 steps/mi 2 ft/step Example 1.7 How Much Gas Do We Use? Estimate the number of gallons of gasoline used each year distance each car travels per year is 10 000 mi. If we assume by all the cars in the United States. a gasoline consumption of 20 mi/gal or 0.05 gal/mi, then Solution Because there are about 280 million people in each car uses about 500 gal/yr. Multiplying this by the total the United States, an estimate of the number of cars in the number of cars in the United States gives an estimated total country is 100 million (guessing that there are between two consumption of 5 ! 1010 gal # 1011 gal. and three people per car). We also estimate that the average  S E C T I O N 1. 7 Significant Figures 15 1.7 Significant Figures When certain quantities are measured, the measured values are known only to within the limits of the experimental uncertainty. The value of this uncertainty can depend on various factors, such as the quality of the apparatus, the skill of the experimenter, and the number of measurements performed. The number of significant figures in a measurement can be used to express something about the uncertainty. As an example of significant figures, suppose that we are asked in a laboratory ex- periment to measure the area of a computer disk label using a meter stick as a measur- ing instrument. Let us assume that the accuracy to which we can measure the length of the label is * 0.1 cm. If the length is measured to be 5.5 cm, we can claim only that its length lies somewhere between 5.4 cm and 5.6 cm. In this case, we say that the mea- sured value has two significant figures. Note that the significant figures include the first estimated digit. Likewise, if the label’s width is measured to be 6.4 cm, the actual value lies between 6.3 cm and 6.5 cm. Thus we could write the measured values as (5.5 * 0.1) cm and (6.4 * 0.1) cm. Now suppose we want to find the area of the label by multiplying the two measured values. If we were to claim the area is (5.5 cm)(6.4 cm) $ 35.2 cm2, our answer would be unjustifiable because it contains three significant figures, which is greater than the number of significant figures in either of the measured quantities. A good rule of thumb to use in determining the number of significant figures that can be claimed in a multiplication or a division is as follows: When multiplying several quantities, the number of significant figures in the final answer is the same as the number of significant figures in the quantity having the lowest number of significant figures. The same rule applies to division. Applying this rule to the previous multiplication example, we see that the answer for the area can have only two significant figures because our measured quantities have only two significant figures. Thus, all we can claim is that the area is 35 cm2, realizing that the value can range between (5.4 cm)(6.3 cm) $ 34 cm2 and (5.6 cm)(6.5 cm) $ 36 cm2. Zeros may or may not be significant figures. Those used to position the decimal point in such numbers as 0.03 and 0.007 5 are not significant. Thus, there are one and two significant figures, respectively, in these two values. When the zeros come af- ter other digits, however, there is the possibility of misinterpretation. For example, suppose the mass of an object is given as 1 500 g. This value is ambiguous because we do not know whether the last two zeros are being used to locate the decimal point or whether they represent significant figures in the measurement. To remove this ambi- guity, it is common to use scientific notation to indicate the number of significant fig- ures. In this case, we would express the mass as 1.5 ! 103 g if there are two signifi- cant figures in the measured value, 1.50 ! 103 g if there are three significant figures, and 1.500 ! 103 g if there are four. The same rule holds for numbers less than 1, so that 2.3 ! 10"4 has two significant figures (and so could be written 0.000 23) and 2.30 ! 10"4 has three significant figures (also written 0.000 230). In general, a sig- ▲ PITFALL PREVENTION nificant figure in a measurement is a reliably known digit (other than a zero 1.6 Read Carefully used to locate the decimal point) or the first estimated digit. Notice that the rule for addition For addition and subtraction, you must consider the number of decimal places and subtraction is different from when you are determining how many significant figures to report: that for multiplication and divi- sion. For addition and subtrac- tion, the important consideration When numbers are added or subtracted, the number of decimal places in the result is the number of decimal places, should equal the smallest number of decimal places of any term in the sum. not the number of significant figures. 16 C H A P T E R 1 Physics and Measurement For example, if we wish to compute 123 ' 5.35, the answer is 128 and not 128.35. If we compute the sum 1.000 1 ' 0.000 3 $ 1.000 4, the result has five significant figures, even though one of the terms in the sum, 0.000 3, has only one significant figure. Like- wise, if we perform the subtraction 1.002 " 0.998 $ 0.004, the result has only one sig- nificant figure even though one term has four significant figures and the other has three. In this book, most of the numerical examples and end-of-chapter problems will yield answers having three significant figures. When carrying out estimates we shall typically work with a single significant figure. If the number of significant figures in the result of an addition or subtraction must be reduced, there is a general rule for rounding off numbers, which states that the last digit retained is to be increased by 1 if the last digit dropped is greater than 5. If the last digit dropped is less than 5, the last digit retained remains as it is. If the last digit dropped is equal to 5, the remaining digit should be rounded to the near- est even number. (This helps avoid accumulation of errors in long arithmetic processes.) A technique for avoiding error accumulation is to delay rounding of numbers in a long calculation until you have the final result. Wait until you are ready to copy the fi- nal answer from your calculator before rounding to the correct number of significant figures. Quick Quiz 1.4 Suppose you measure the position of a chair with a meter stick and record that the center of the seat is 1.043 860 564 2 m from a wall. What would a reader conclude from this recorded measurement? Example 1.8 Installing a Carpet A carpet is to be installed in a room whose length is mea- numbers should you claim? Our rule of thumb for multiplica- sured to be 12.71 m and whose width is measured to be tion tells us that you can claim only the number of significant 3.46 m. Find the area of the room. figures in your answer as are present in the measured quan- tity having the lowest number of significant figures. In this ex- ample, the lowest number of significant figures is three in Solution If you multiply 12.71 m by 3.46 m on your calcula- tor, you will see an answer of 43.976 6 m2. How many of these 3.46 m, so we should express our final answer as 44.0 m2. S U M MARY The three fundamental physical quantities of mechanics are length, mass, and time, Take a practice test for which in the SI system have the units meters (m), kilograms (kg), and seconds (s), re- this chapter by clicking on the Practice Test link at spectively. Prefixes indicating various powers of ten are used with these three basic http://www.pse6.com. units. The density of a substance is defined as its mass per unit volume. Different sub- stances have different densities mainly because of differences in their atomic masses and atomic arrangements. The method of dimensional analysis is very powerful in solving physics problems. Dimensions can be treated as algebraic quantities. By making estimates and perform- ing order-of-magnitude calculations, you should be able to approximate the answer to a problem when there is not enough information available to completely specify an ex- act solution. When you compute a result from several measured numbers, each of which has a certain accuracy, you should give the result with the correct number of significant fig- ures. When multiplying several quantities, the number of significant figures in the  Problems 17 final answer is the same as the number of significant figures in the quantity having the lowest number of significant figures. The same rule applies to division. When numbers are added or subtracted, the number of decimal places in the result should equal the smallest number of decimal places of any term in the sum. QU ESTIONS 1. What types of natural phenomena could serve as time stan- 6. If an equation is dimensionally correct, does this mean dards? that the equation must be true? If an equation is not di- 2. Suppose that the three fundamental standards of the mensionally correct, does this mean that the equation can- metric system were length, density, and time rather than not be true? length, mass, and time. The standard of density in this 7. Do an order-of-magnitude calculation for an everyday situ- system is to be defined as that of water. What considera- ation you encounter. For example, how far do you walk or tions about water would you need to address to make drive each day? sure that the standard of density is as accurate as 8. Find the order of magnitude of your age in seconds. possible? 9. What level of precision is implied in an order-of-magnitude 3. The height of a horse is sometimes given in units of calculation? “hands.” Why is this a poor standard of length? 10. Estimate the mass of this textbook in kilograms. If a scale is 4. Express the following quantities using the prefixes given in available, check your estimate. Table 1.4: (a) 3 ! 10"4 m (b) 5 ! 10"5 s (c) 72 ! 102 g. 11. In reply to a student’s question, a guard in a natural his- 5. Suppose that two quantities A and B have different dimen- tory museum says of the fossils near his station, “When I sions. Determine which of the following arithmetic opera- started work here twenty-four years ago, they were eighty tions could be physically meaningful: (a) A ' B (b) A/B million years old, so you can add it up.” What should the (c) B " A (d) AB. student conclude about the age of the fossils? PROBLEMS 1, 2, 3 = straightforward, intermediate, challenging = full solution available in the Student Solutions Manual and Study Guide = coached solution with hints available at http://www.pse6.com = computer useful in solving problem = paired numerical and symbolic problems Section 1.2 Matter and Model Building Note: Consult the endpapers, appendices, and tables in the text whenever necessary in solving problems. For this chapter, Appendix B.3 may be particularly useful. Answers to odd-numbered problems appear in the back of the L d book. (a) 1. A crystalline solid consists of atoms stacked up in a repeat- ing lattice structure. Consider a crystal as shown in Figure P1.1a. The atoms reside at the corners of cubes of side L $ 0.200 nm. One piece of evidence for the regular arrangement of atoms comes from the flat surfaces along which a crystal separates, or cleaves, when it is broken. Suppose this crystal cleaves along a face diagonal, as shown in Figure P1.1b. Calculate the spacing d between (b) two adjacent atomic planes that separate when the crystal Figure P1.1 cleaves. 18 C H A P T E R 1 Physics and Measurement Section 1.3 Density and Atomic Mass mass of a section 1.50 m long? (b) Assume that the atoms 2. Use information on the endpapers of this book to calcu- are predominantly iron, with atomic mass 55.9 u. How late the average density of the Earth. Where does the many atoms are in this section? value fit among those listed in Tables 1.5 and 14.1? Look up the density of a typical surface rock like granite in an- other source and compare the density of the Earth to it. 3. The standard kilogram is a platinum–iridium cylinder 15.0 cm 39.0 mm in height and 39.0 mm in diameter. What is the density of the material? 4. A major motor company displays a die-cast model of its 1.00 cm first automobile, made from 9.35 kg of iron. To celebrate 36.0 cm its hundredth year in business, a worker will recast the model in gold from the original dies. What mass of gold is needed to make the new model? 1.00 cm 5. What mass of a material with density & is required to make a hollow spherical shell having inner radius r 1 and outer Figure P1.11 radius r 2? 6. Two spheres are cut from a certain uniform rock. One has radius 4.50 cm. The mass of the other is five times greater. Find its radius. 12. A child at the beach digs a hole in the sand and uses a pail to fill it with water having a mass of 1.20 kg. The mass of 7. Calculate the mass of an atom of (a) helium, one molecule of water is 18.0 u. (a) Find the number of (b) iron, and (c) lead. Give your answers in grams. The water molecules in this pail of water. (b) Suppose the atomic masses of these atoms