physics for scientists and engineers 6th edition pdf

1 2 Units of Area, Volume, Velocity, Speed, and Acceleration 1.4 Symbols for Quantities Some quantities have a smallnumber of symbols that repre-sent them.. The average velocity v–x of a

6th Edition

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;

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

Physics for Scientists and Engineers (with PhysicsNOW and InfoTrac)

Table of Contents

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.

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.

Coordinate Systems.

Vector and Scalar Quantities.

Some Properties of Vectors.

Components of a Vector and Unit Vectors.

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.

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.

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.

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.

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.

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.

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.

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.

Pressure.

Variation of Pressure with Depth.

Pressure Measurements.

Buoyant Forces and Archimedes's Principle.

Fluid Dynamics Bernoulli's Equation.

Other Applications of Fluid Dynamics.

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.

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.

Speed of Sound Waves.

Periodic Sound Waves.

Intensity of Periodic Sound Waves.

The Doppler Effect.

Digital Sound Recording.

Motion Picture Sound.

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

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.

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.

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.

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.

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.

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.

Energy Stored in a Charged Capacitor.

Capacitors with Dielectrics.

Electric Dipole in an Electric Field.

An Atomic Description of Dielectrics.

Electric Current.

Resistance.

A Model for Electrical Conduction.

Resistance and Temperature.

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.

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.

Resonance in a Series RLC Circuit.

The Transformer and Power Transmission.

Rectifiers and Filters.

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.

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.

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.

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.

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.

Conversion Factors Symbols, Dimensions, and Units of Physical Quantities Table of Atomic Masses.

Scientific Notation Algebra Geometry Trigonometry Series Expansions Differential Calculus.

Integral Calculus Propagation of Uncertainty.

Mechanics P A R T

1

! 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)

hysics, the most fundamental physical science, is concerned with the basic

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 ■

P

1.2 Matter and Model Building

1.3 Density and Atomic Mass

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

(4) several remarkable results in genetic engineering The impacts of such ments and discoveries on our society have indeed been great, and it is very likely thatfuture discoveries and developments will be exciting, challenging, and of great benefit

develop-to humanity

1.1 Standards of Length, Mass, and Time

The laws of physics are expressed as mathematical relationships among physical ties that we will introduce and discuss throughout the book Most of these quantities

quanti-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 knowthe 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 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.1What-ever is chosen as a standard must be readily accessible and possess some property thatcan be measured reliably Measurements taken by different people in different placesmust yield the same result

Like-In 1960, an international committee established a set of standards for the tal quantities of science It is called the SI (Système International), and its units of length,

fundamen-mass, and time are the meter, kilogram, and second, respectively Other SI standards tablished by the committee are those for temperature (the kelvin), electric current (the

es-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 footadopted 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 meterwas defined as the distance between two lines on a specific platinum–iridium barstored 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 currentrequirements 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-86lamp However, in October 1983, the meter (m) was redefined as the distancetraveled by light in vacuum during a time of 1/299 792 458 second In effect, this

4 C H A P T E R 1 • Physics and Measurement

1 The need for assigning numerical values to various measured physical quantities was expressed byLord Kelvin (William Thomson) as follows: “I often say that when you can measure what you arespeaking about, and express it in numbers, you should know something about it, but when you cannotexpress 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.”

latest definition establishes that the speed of light in vacuum is precisely 299 792 458

meters per second

Table 1.1 lists approximate values of some measured lengths You should study this

table as well as the next two tables and begin to generate an intuition for what is meant

by a length of 20 centimeters, for example, or a mass of 100 kilograms or a time

inter-val of 3.2 ! 107seconds

Mass

The SI unit of mass, the kilogram (kg), is defined as the mass of a specific

platinum–iridium alloy cylinder kept at the International Bureau of Weights

and Measures at Sèvres, France This mass standard was established in 1887 and has

not been changed since that time because platinum–iridium is an unusually stable

al-loy A duplicate of the Sèvres cylinder is kept at the National Institute of Standards and

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

at the highest point it reaches in the sky each day.) The second was defined as

of a mean solar day The rotation of the Earth is now known to vary

slightly with time, however, and therefore this motion is not a good one to use for

defining a time standard

In 1967, the second was redefined to take advantage of the high precision attainable

in a device known as an atomic clock (Fig 1.1b), which uses the characteristic frequency

of the cesium-133 atom as the “reference clock.” The second (s) is now defined as

9 192 631 770 times the period of vibration of radiation from the cesium atom.2

!1

60"!1

60"!1

24"

S E C T I O N 1 1 • Standards of Length, Mass, and Time 5

2 Period is defined as the time interval needed for one complete vibration.

Length (m)

Distance from the Earth to the most remote known quasar 1.4 ! 1026

Distance from the Earth to the most remote normal galaxies 9 ! 1025

Distance from the Earth to the nearest large galaxy 2 ! 1022

(M 31, the Andromeda galaxy)

Distance from the Sun to the nearest star (Proxima Centauri) 4 ! 1016

Mean orbit radius of the Earth about the Sun 1.50 ! 1011

Mean distance from the Earth to the Moon 3.84 ! 108

Distance from the equator to the North Pole 1.00 ! 107

Typical altitude (above the surface) of a 2 ! 105

satellite orbiting the Earth

Size of smallest dust particles # 10"4

Size of cells of most living organisms # 10"5

typi-this is unreasonable—there is an

error somewhere

1.1 No Commas in Numbers with Many Digits

We will use the standard scientificnotation for numbers with morethan three digits, in whichgroups of three digits are sepa-rated by spaces rather thancommas Thus, 10 000 is thesame as the common Americannotation of 10,000 Similarly,

# $3.14159265 is written as3.141 592 65

Mass (kg)

Observable # 1052Universe

Milky Way # 1042galaxy

Sun 1.99 ! 1030Earth 5.98 ! 1024Moon 7.36 ! 1022

Frog # 10"1Mosquito # 10"5Bacterium # 1 ! 10"15Hydrogen 1.67 ! 10"27atom

Electron 9.11 ! 10"31

Table 1.2

Masses of Various Objects (Approximate Values)

To keep these atomic clocks—and therefore all common clocks and watches that areset to them—synchronized, it has sometimes been necessary to add leap seconds to ourclocks.

Since Einstein’s discovery of the linkage between space and time, precise ment of time intervals requires that we know both the state of motion of the clock used

measure-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

6 C H A P T E R 1 • Physics and Measurement

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 timestandard is a cesium fountain atomic clock developed at the National Institute of Standardsand Technology laboratories in Boulder, Colorado The clock will neither gain nor lose asecond in 20 million years

Time Interval (s)

One day (time interval for one revolution of the Earth about its axis) 8.6 ! 104

Time interval between normal heartbeats 8 ! 10"1

Period of vibration of an atom in a solid # 10"13

Time interval for light to cross a proton # 10"24

Approximate Values of Some Time Intervals

Table 1.3

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" 3m is equivalent to 1 millimeter (mm), and 103m corresponds to 1 kilometer

(km) Likewise, 1 kilogram (kg) is 103grams (g), and 1 megavolt (MV) is 106volts (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 syssys-tem 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

S E C T I O N 1 2 • Matter and Model Building 7

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 ofatoms, 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 themodel

In 1897, J J Thomson identified the electron as a charged particle and as a stituent of the atom This led to the first model of the atom that contained internalstructure We shall discuss this model in Chapter 42

con-Following the discovery of the nucleus in 1911, a model was developed in whicheach 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 havestructure? That is, is the nucleus a single particle or a collection of particles? The exactcomposition 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, 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 isidentified by the number of protons in its nucleus This number is called the atomicnumber of the element For instance, the nucleus of a hydrogen atom contains oneproton (and so the atomic number of hydrogen is 1), the nucleus of a helium atomcontains two protons (atomic number 2), and the nucleus of a uranium atom contains

sci-92 protons (atomic number sci-92) In addition to atomic number, there is a second 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 ofprotons does not vary) but the mass number can vary (i.e., the number of neutronsvaries)

num-The existence of neutrons was verified conclusively in 1932 A neutron has nocharge and a mass that is about equal to that of a proton One of its primary purposes

8 C H A P T E R 1 • Physics and Measurement

Gold atomsNucleus

Quark composition of a proton

u

d

Gold cube

Goldnucleus

Proton

Neutronu

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 andneutrons are composed of quarks The quark composition of a proton is shown

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,

the proton, whereas the down, strange, and bottom quarks have charges of that

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 interacpredic-tions 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

example of a derived quantity—density The density & (Greek letter rho) of any

sub-stance is defined as its mass per unit volume:

(1.1)

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 cm3has 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"27kg

& $ m

V

"13'23

S E C T I O N 1 3 • Density and Atomic Mass 9

A table of the letters in the Greek alphabet is provided on the back endsheet of the textbook.

Air at atmospheric pressure 0.0012

Densities of Various Substances

Table 1.5

The atomic mass of lead is 207 u and that of aluminum is 27.0 u However, the ratio ofatomic masses, 207 u/27.0 u $ 7.67, does not correspond to the ratio of densities,(11.3 ! 103kg/m3)/(2.70 ! 103kg/m3) $ 4.19 This discrepancy is due to the differ-ence in atomic spacings and atomic arrangements in the crystal structures of the twoelements.

1.4 Dimensional Analysis

The word dimension has a special meaning in physics It denotes the physical nature of

a quantity Whether a distance is measured in units of feet or meters or fathoms, it is

still a distance We say its dimension is length.

The symbols we use in this book to specify the dimensions of length, mass, andtime are L, M, and T, respectively.3We shall often use brackets [ ] to denote the dimen-

sions of a physical quantity For example, the symbol we use for speed in this book is v, and in our notation the dimensions of speed are written [v] $ L/T As another exam- ple, the dimensions of area A are [A] $ L2 The dimensions and units of area, volume,speed, and acceleration are listed in Table 1.6 The dimensions of other quantities,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

10 C H A P T E R 1 • Physics and Measurement

Quick Quiz 1.1 In a machine shop, two cams are produced, one of minum and one of iron Both cams have the same mass Which cam is larger? (a) thealuminum cam (b) the iron cam (c) Both cams have the same size

alu-Example 1.1 How Many Atoms in the Cube?

When using ratios to solve a

problem, keep in mind that ratios

come from equations If you start

from equations known to be

cor-rect and can divide one equation

by the other as in Example 1.1 to

obtain a useful ratio, you will

avoid reasoning errors So write

the known equations first!

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.

write this relationship twice, once for the actual sample ofaluminum in the problem and once for a 27.0-g sample, andthen we divide the first equation by the second:

Notice that the unknown proportionality constant k cancels,

so we do not need to know its value We now substitute thevalues:

$ 1.20 ! 1022 atoms

Nsample$ (0.540 g)(6.02 ! 1023 atoms)

27.0 g

0.540 g27.0 g $

Nsample

6.02 ! 1023 atoms

m27.0 g $kN27.0 g

msample $kNsample

A solid cube of aluminum (density 2.70 g/cm3) has a

vol-ume of 0.200 cm3 It is known that 27.0 g of aluminum

con-tains 6.02 ! 1023atoms How many aluminum atoms are

contained in the cube?

Solution Because density equals mass per unit volume, the

mass of the cube is

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

number of atoms contained in the sample This technique

of solving by ratios is very powerful and should be studied

and understood so that it can be applied in future problem

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

m $ &V $ (2.70 g/cm3)(0.200 cm3) $ 0.540 g

dimensions can be treated as algebraic quantities For example, quantities can be

added or subtracted only if they have the same dimensions Furthermore, the terms on

both sides of an equation must have the same dimensions By following these simple

rules, you can use dimensional analysis to help determine whether an expression has

the correct form The relationship can be correct only if the dimensions on both sides

of the equation are the same

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 $ at2 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

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,

[a n t m] $ L $ L1T0

Because the dimensions of acceleration are L/T2and the dimension of time is T, we have

(L/T2)nTm$L1T0

(LnTm "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 ( a n t m , we conclude that x ( at2 This result differs by a factor of from

the correct expression, which is x $12 at2

1 2

Units of Area, Volume, Velocity, Speed, and Acceleration

1.4 Symbols for Quantities

Some quantities have a smallnumber of symbols that repre-sent them For example, the sym-

bol for time is almost always t.

Others quantities might have ious symbols depending on theusage Length may be described

var-with symbols such as x, y, and z (for position), r (for radius), a, b, and c (for the legs of a right tri-

angle), ! (for the length of an

object), d (for a distance), h (for

a height), etc

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

1.5 Conversion of Units

Sometimes it is necessary to convert units from one measurement system to another, or

to convert within a system, for example, from kilometers to meters Equalities between

SI and U.S customary units of length are as follows:

1 mile $ 1 609 m $ 1.609 km 1 ft $ 0.304 8 m $ 30.48 cm

1 m $ 39.37 in $ 3.281 ft 1 in $ 0.025 4 m $ 2.54 cm (exactly)

A more complete list of conversion factors can be found in Appendix A

Units can be treated as algebraic quantities that can cancel each other For ple, suppose we wish to convert 15.0 in to centimeters Because 1 in is defined as ex-actly 2.54 cm, we find that

exam-where the ratio in parentheses is equal to 1 Notice that we choose to put the unit of aninch 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

15.0 in $ (15.0 in.)!2.54 cm

1 in "$38.1 cm

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,

where v represents speed, a acceleration, and t an instant of

time

Solution For the speed term, we have from Table 1.6

[v] $ L

T

The same table gives us L/T2for the dimensions of

accelera-tion, and so the dimensions of at are

Therefore, the expression is dimensionally correct (If the

expression were given as v $ at2it would be dimensionally

incorrect Try it and see!)

[at] $ TL2 T $ LT

Example 1.3 Analysis of a Power Law

Suppose we are told that the acceleration a of a particle

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 v m Determine the values of n and m and write the

sim-plest form of an equation for the acceleration

Solution Let us take a to be

where k is a dimensionless constant of proportionality.

Knowing the dimensions of a, r, and v, we see that the

di-mensional equation must be

1.5 Always Include Units

When performing calculations,

include the units for every

quan-tity and carry the units through

the entire calculation Avoid the

temptation to drop the units

early and then attach the

ex-pected units once you have an

answer By including the units in

every step, you can detect errors

if the units for the answer turn

When we discuss uniform circular motion later, we shall see

that k $ 1 if a consistent set of units is used The constant k would not equal 1 if, for example, v were in km/h and you wanted a in m/s2

k v2r

a $ kr"1v2$

21

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.”4Inaccuracies 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

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

car is traveling at a speed of 38.0 m/s Is this car exceeding

the speed limit of 75.0 mi/h?

Solution We first convert meters to miles:

Now we convert seconds to hours:

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

familiar with speeds measured in km/h? What is the speed

Figure 1.3 The speedometer of a vehicle that

shows speeds in both miles per hour and ters per hour

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

Example 1.5 Breaths in a Lifetime

Estimate the number of breaths taken during an average life

span

Solution We start by guessing that the typical life span is

about 70 years The only other estimate we must make in this

example is the average number of breaths that a person

takes in 1 min This number varies, depending on whether

the person is exercising, sleeping, angry, serene, and so

forth To the nearest order of magnitude, we shall choose 10

breaths per minute as our estimate of the average (This is

certainly closer to the true value than 1 breath per minute or

100 breaths per minute.) The number of minutes in a year is

approximately

Notice how much simpler it is in the expression above to

multiply 400 ! 25 than it is to work with the more accurate

365 ! 24 These approximate values for the number of days

in a lifetime, or on the order of 109breaths

What If? What if the average life span were estimated as

80 years instead of 70? Would this change our final estimate?

5 ! 107 min, so that our final estimate should be 5 ! 108

breaths This is still on the order of 109breaths, so an of-magnitude estimate would be unchanged Furthermore,

order-80 years is 14% larger than 70 years, but we have 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 twonumbers together result in an overestimate of 14%, whichcancels the effect of the increased life span!

overesti-4 ! 108 breaths

Example 1.6 It’s a Long Way to San Jose

Estimate the number of steps a person would take walking

from New York to Los Angeles

Solution Without looking up the distance between these

two cities, you might remember from a geography class that

they are about 3 000 mi apart The next approximation we

must make is the length of one step Of course, this length

depends on the person doing the walking, but we can

esti-mate that each step covers about 2 ft With our estiesti-mated

step size, we can determine the number of steps in 1 mi

Be-cause this is a rough calculation, we round 5 280 ft/mi to

5 000 ft/mi (What percentage error does this introduce?)

This conversion factor gives us

So if we intend to walk across the United States, it will take

us on the order of ten million steps This estimate is almostcertainly too small because we have not accounted for curv-ing roads and going up and down hills and mountains.Nonetheless, it is probably within an order of magnitude ofthe correct answer

7.5 ! 106 steps# 107 steps(3 ! 103 mi)(2.5 ! 103 steps/mi)

Example 1.7 How Much Gas Do We Use?

Estimate the number of gallons of gasoline used each year

by all the cars in the United States

Solution Because there are about 280 million people in

the United States, an estimate of the number of cars in the

country is 100 million (guessing that there are between two

and three people per car) We also estimate that the average

distance each car travels per year is 10 000 mi If we assume

a gasoline consumption of 20 mi/gal or 0.05 gal/mi, theneach car uses about 500 gal/yr Multiplying this by the totalnumber of cars in the United States gives an estimated total consumption of 5 ! 1010 gal# 1011 gal

unknown numbers, make simplifying assumptions, and turn the question aroundinto 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.”

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:

S E C T I O N 1 7 • Significant Figures 15

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 misinaf-terpretation 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 ! 103g if there are two

signifi-cant figures in the measured value, 1.50 ! 103g if there are three significant figures,

and 1.500 ! 103g if there are four The same rule holds for numbers less than 1, so

that 2.3 ! 10"4has two significant figures (and so could be written 0.000 23) and

2.30 ! 10"4has three significant figures (also written 0.000 230) In general,a

sig-nificant figure in a measurement is a reliably known digit (other than a zero

used to locate the decimal point) or the first estimated digit

For addition and subtraction, you must consider the number of decimal places

when you are determining how many significant figures to report:

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

1.6 Read Carefully

Notice that the rule for additionand subtraction is different fromthat for multiplication and divi-sion For addition and subtrac-tion, the important consideration

is the number of decimal places, not the number of significant figures.

For example, if we wish to compute 123 ' 5.35, the answer is 128 and not 128.35 If wecompute 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 hasthree In this book, most of the numerical examples and end-of-chapter problemswill yield answers having three significant figures When carrying out estimates weshall typically work with a single significant figure.

If the number of significant figures in the result of an addition or subtractionmust be reduced, there is a general rule for rounding off numbers, which states thatthe 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 thelast 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 arithmeticprocesses.)

A technique for avoiding error accumulation is to delay rounding of numbers in along 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 significantfigures

16 C H A P T E R 1 • Physics and Measurement

Quick Quiz 1.4 Suppose you measure the position of a chair with a meterstick and record that the center of the seat is 1.043 860 564 2 m from a wall Whatwould 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-sured to be 12.71 m and whose width is meamea-sured to be

3.46 m Find the area of the room

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

numbers should you claim? Our rule of thumb for tion tells us that you can claim only the number of significantfigures 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 3.46 m, so we should express our final answer as 44.0 m2

multiplica-The three fundamental physical quantities of mechanics are length, mass, and time,which in the SI system have the units meters (m), kilograms (kg), and seconds (s), re-spectively Prefixes indicating various powers of ten are used with these three basicunits

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 massesand 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 act solution

ex-When you compute a result from several measured numbers, each of which has acertain 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

S U M M A R Y

Take a practice test for

this chapter by clicking on

the Practice Test link at

http://www.pse6.com.

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

1 What types of natural phenomena could serve as time

stan-dards?

2 Suppose that the three fundamental standards of the

metric system were length, density, and time rather than

length, mass, and time The standard of density in this

system is to be defined as that of water What

considera-tions about water would you need to address to make

sure that the standard of density is as accurate as

possible?

3 The height of a horse is sometimes given in units of

“hands.” Why is this a poor standard of length?

4 Express the following quantities using the prefixes given in

Table 1.4: (a) 3 ! 10"4m (b) 5 ! 10"5s (c) 72 ! 102g

5 Suppose that two quantities A and B have different

dimen-sions Determine which of the following arithmetic

opera-tions could be physically meaningful: (a) A ' B (b) A/B

(c) B " A (d) AB

6 If an equation is dimensionally correct, does this mean

that the equation must be true? If an equation is not mensionally correct, does this mean that the equation can-not be true?

di-7 Do an order-of-magnitude calculation for an everyday

situ-ation you encounter For example, how far do you walk ordrive each day?

8 Find the order of magnitude of your age in seconds.

9 What level of precision is implied in an order-of-magnitude

calculation?

10 Estimate the mass of this textbook in kilograms If a scale is

available, check your estimate

11 In reply to a student’s question, a guard in a natural

his-tory museum says of the fossils near his station, “When Istarted work here twenty-four years ago, they were eightymillion years old, so you can add it up.” What should thestudent conclude about the age of the fossils?

Section 1.2 Matter and Model Building

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

two adjacent atomic planes that separate when the crystal

cleaves

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

book

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

P R O B L E M S

r1

r2

Figure P1.14

18 C H A P T E R 1 • Physics and Measurement

Section 1.3 Density and Atomic Mass

2 Use information on the endpapers of this book to

calcu-late the average density of the Earth Where does the

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

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

first automobile, made from 9.35 kg of iron To celebrate

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?

5.What mass of a material with density & is required to make

a hollow spherical shell having inner radius r1and outer

radius r2?

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

7. Calculate the mass of an atom of (a) helium,

(b) iron, and (c) lead Give your answers in grams The

atomic masses of these atoms are 4.00 u, 55.9 u, and 207 u,

respectively

8.The paragraph preceding Example 1.1 in the text

mentions that the atomic mass of aluminum is

27.0 u $ 27.0 ! 1.66 ! 10"27kg Example 1.1 says that

27.0 g of aluminum contains 6.02 ! 1023atoms (a) Prove

that each one of these two statements implies the other

(b) What If ? What if it’s not aluminum? Let M represent

the numerical value of the mass of one atom of any

chemi-cal element in atomic mass units Prove that M grams of the

substance contains a particular number of atoms, the same

number for all elements Calculate this number precisely

from the value for u quoted in the text The number of

atoms in M grams of an element is called Avogadro’s number

NA The idea can be extended: Avogadro’s number of

mol-ecules of a chemical compound has a mass of M grams,

where M atomic mass units is the mass of one molecule.

Avogadro’s number of atoms or molecules is called one

mole, symbolized as 1 mol A periodic table of the elements,

as in Appendix C, and the chemical formula for a

com-pound contain enough information to find the molar mass

of the compound (c) Calculate the mass of one mole of

water, H2O (d) Find the molar mass of CO2

9 On your wedding day your lover gives you a gold ring of

mass 3.80 g Fifty years later its mass is 3.35 g On the

aver-age, how many atoms were abraded from the ring during

each second of your marriage? The atomic mass of gold is

197 u

10 A small cube of iron is observed under a microscope The

edge of the cube is 5.00 ! 10"6cm long Find (a) the

mass of the cube and (b) the number of iron atoms in the

cube The atomic mass of iron is 55.9 u, and its density is

7.86 g/cm3

11.A structural I beam is made of steel A view of its

cross-section and its dimensions are shown in Figure P1.11 The

density of the steel is 7.56 ! 103kg/m3 (a) What is the

mass of a section 1.50 m long? (b) Assume that the atomsare predominantly iron, with atomic mass 55.9 u Howmany atoms are in this section?

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 ofone molecule of water is 18.0 u (a) Find the number ofwater molecules in this pail of water (b) Suppose thequantity of water on Earth is constant at 1.32 ! 1021kg.How many of the water molecules in this pail of water arelikely to have been in an equal quantity of water that oncefilled one particular claw print left by a Tyrannosaur hunt-ing on a similar beach?

Section 1.4 Dimensional Analysis

The position of a particle moving under uniform tion is some function of time and the acceleration Suppose

accelera-we write this position s $ ka m t n , where k is a dimensionless

constant Show by dimensional analysis that this expression

is satisfied if m $ 1 and n $ 2 Can this analysis give the value of k?

14 Figure P1.14 shows a frustrum of a cone Of the following

mensuration (geometrical) expressions, which describes(a) the total circumference of the flat circularfaces (b) the volume (c) the area of the curved sur-

face? (i) #(r1'r2)[h2'(r1"r2)2]1/2 (ii) 2#(r1'r2)

(iii) #h(r1 'r1r2'r2 )

13.

16 (a) A fundamental law of motion states that the acceleration

of an object is directly proportional to the resultant force

ex-erted on the object and inversely proportional to its mass If

the proportionality constant is defined to have no

dimen-sions, determine the dimensions of force (b) The newton is

the SI unit of force According to the results for (a), how can

you express a force having units of newtons using the

funda-mental units of mass, length, and time?

17 Newton’s law of universal gravitation is represented by

Here F is the magnitude of the gravitational force exerted by

one small object on another, M and m are the masses of the

objects, and r is a distance Force has the SI units kg · m/s2

What are the SI units of the proportionality constant G ?

Section 1.5 Conversion of Units

18 A worker is to paint the walls of a square room 8.00 ft high

and 12.0 ft along each side What surface area in square

meters must she cover?

19 Suppose your hair grows at the rate 1/32 in per day Find

the rate at which it grows in nanometers per second

Be-cause the distance between atoms in a molecule is on the

order of 0.1 nm, your answer suggests how rapidly layers of

atoms are assembled in this protein synthesis

using the definition 1 in $ 2.54 cm

A rectangular building lot is 100 ft by 150 ft Determine the

area of this lot in m2

22.An auditorium measures 40.0 m ! 20.0 m ! 12.0 m The

density of air is 1.20 kg/m3 What are (a) the volume of

the room in cubic feet and (b) the weight of air in the

room in pounds?

23 Assume that it takes 7.00 minutes to fill a 30.0-gal gasoline

tank (a) Calculate the rate at which the tank is filled in

gallons per second (b) Calculate the rate at which the

tank is filled in cubic meters per second (c) Determine

the time interval, in hours, required to fill a 1-m3volume

at the same rate (1 U.S gal $ 231 in.3)

24 Find the height or length of these natural wonders in

kilo-meters, meters and centimeters (a) The longest cave system

in the world is the Mammoth Cave system in central

Ken-tucky It has a mapped length of 348 mi (b) In the United

States, the waterfall with the greatest single drop is Ribbon

Falls, which falls 1 612 ft (c) Mount McKinley in Denali

Na-tional Park, Alaska, is America’s highest mountain at a

height of 20 320 ft (d) The deepest canyon in the United

States is King’s Canyon in California with a depth of 8 200 ft

A solid piece of lead has a mass of 23.94 g and a volume of

2.10 cm3 From these data, calculate the density of lead in

640 acres Determine the number of square meters in

1 acre

27 An ore loader moves 1 200 tons/h from a mine to the

sur-face Convert this rate to lb/s, using 1 ton $ 2 000 lb

28 (a) Find a conversion factor to convert from miles per

hour to kilometers per hour (b) In the past, a federal lawmandated that highway speed limits would be 55 mi/h.Use the conversion factor of part (a) to find this speed inkilometers per hour (c) The maximum highway speed isnow 65 mi/h in some places In kilometers per hour, howmuch increase is this over the 55 mi/h limit?

At the time of this book’s printing, the U.S national debt

is about $6 trillion (a) If payments were made at the rate

of $1 000 per second, how many years would it take to payoff the debt, assuming no interest were charged? (b) A dollar bill is about 15.5 cm long If six trillion dollar billswere laid end to end around the Earth’s equator, howmany times would they encircle the planet? Take the ra-

dius of the Earth at the equator to be 6 378 km (Note:

Be-fore doing any of these calculations, try to guess at the swers You may be very surprised.)

atom of hydrogen, of which the Sun is mostly composed, is1.67 ! 10"27kg How many atoms are in the Sun?

One gallon of paint (volume $ 3.78 ! 10"3m3) covers

an area of 25.0 m2 What is the thickness of the paint on the wall?

32 A pyramid has a height of 481 ft and its base covers an area

of 13.0 acres (Fig P1.32) If the volume of a pyramid is

given by the expression V $ Bh, where B is the area of the base and h is the height, find the volume of this pyra-

mid in cubic meters (1 acre $ 43 560 ft2)

1 3

31.

29.

Figure P1.32 Problems 32 and 33.

33 The pyramid described in Problem 32 contains

approxi-mately 2 million stone blocks that average 2.50 tons each.Find the weight of this pyramid in pounds

34 Assuming that 70% of the Earth’s surface is covered with

water at an average depth of 2.3 mi, estimate the mass ofthe water on the Earth in kilograms

35.A hydrogen atom has a diameter of approximately1.06 ! 10"10m, as defined by the diameter of the spheri-cal electron cloud around the nucleus The hydrogen nu-cleus has a diameter of approximately 2.40 ! 10"15m.(a) For a scale model, represent the diameter of the hy-drogen atom by the length of an American football field

(100 yd $ 300 ft), and determine the diameter of the

nucleus in millimeters (b) The atom is how many times

larger in volume than its nucleus?

36 The nearest stars to the Sun are in the Alpha Centauri

multiple-star system, about 4.0 ! 1013km away If the Sun,

with a diameter of 1.4 ! 109m, and Alpha Centauri A are

both represented by cherry pits 7.0 mm in diameter, how

far apart should the pits be placed to represent the Sun

and its neighbor to scale?

The diameter of our disk-shaped galaxy, the Milky Way, is

about 1.0 ! 105lightyears (ly) The distance to Messier 31,

which is Andromeda, the spiral galaxy nearest to the Milky

Way, is about 2.0 million ly If a scale model represents the

Milky Way and Andromeda galaxies as dinner plates 25 cm

in diameter, determine the distance between the two plates

the Moon is 1.74 ! 108 cm From these data calculate

(a) the ratio of the Earth’s surface area to that of the

Moon and (b) the ratio of the Earth’s volume to that of

the Moon Recall that the surface area of a sphere is 4#r2

and the volume of a sphere is

One cubic meter (1.00 m3) of aluminum has a mass

of 2.70 ! 103kg, and 1.00 m3 of iron has a mass of

7.86 ! 103kg Find the radius of a solid aluminum sphere

that will balance a solid iron sphere of radius 2.00 cm on

an equal-arm balance

40.Let &Alrepresent the density of aluminum and &Fethat of

iron Find the radius of a solid aluminum sphere that

bal-ances a solid iron sphere of radius rFeon an equal-arm

balance

Section 1.6 Estimates and Order-of-Magnitude

Calculations

Estimate the number of Ping-Pong balls that would fit

into a typical-size room (without being crushed) In your

solution state the quantities you measure or estimate and

the values you take for them

42.An automobile tire is rated to last for 50 000 miles To an

order of magnitude, through how many revolutions will it

turn? In your solution state the quantities you measure or

estimate and the values you take for them

43 Grass grows densely everywhere on a quarter-acre plot of

land What is the order of magnitude of the number of

blades of grass on this plot? Explain your reasoning Note

that 1 acre $ 43 560 ft2

44.Approximately how many raindrops fall on a one-acre lot

during a one-inch rainfall? Explain your reasoning

45 Compute the order of magnitude of the mass of a bathtub

half full of water Compute the order of magnitude of the

mass of a bathtub half full of pennies In your solution list

the quantities you take as data and the value you measure

or estimate for each

46.Soft drinks are commonly sold in aluminum containers To

an order of magnitude, how many such containers are

thrown away or recycled each year by U.S consumers?

To an order of magnitude, how many piano tuners are inNew York City? The physicist Enrico Fermi was famous forasking questions like this on oral Ph.D qualifying exami-nations His own facility in making order-of-magnitude cal-culations is exemplified in Problem 45.48

Section 1.7 Significant Figures

48 A rectangular plate has a length of (21.3 * 0.2) cm and a

width of (9.8 * 0.1) cm Calculate the area of the plate, cluding its uncertainty

in-49 The radius of a circle is measured to be (10.5 * 0.2) m.

Calculate the (a) area and (b) circumference of the circleand give the uncertainty in each value

50 How many significant figures are in the following

num-bers? (a) 78.9 * 0.2 (b) 3.788 ! 109 (c) 2.46 ! 10"6

(d) 0.005 3

51.The radius of a solid sphere is measured to be(6.50 * 0.20) cm, and its mass is measured to be(1.85 * 0.02) kg Determine the density of the sphere inkilograms per cubic meter and the uncertainty in the density

52 Carry out the following arithmetic operations: (a) the sum

of the measured values 756, 37.2, 0.83, and 2.5; (b) theproduct 0.003 2 ! 356.3; (c) the product 5.620 ! #

53 The tropical year, the time from vernal equinox to the next

vernal equinox, is the basis for our calendar It contains365.242 199 days Find the number of seconds in a tropicalyear

54 A farmer measures the distance around a rectangular field.

The length of the long sides of the rectangle is found to

be 38.44 m, and the length of the short sides is found to

be 19.5 m What is the total distance around the field?

55.A sidewalk is to be constructed around a swimming poolthat measures (10.0 * 0.1) m by (17.0 * 0.1) m If the side-walk is to measure (1.00 * 0.01) m wide by (9.0 * 0.1) cmthick, what volume of concrete is needed, and what is theapproximate uncertainty of this volume?

Additional Problems

56 In a situation where data are known to three significant

digits, we write 6.379 m $ 6.38 m and 6.374 m $ 6.37 m.When a number ends in 5, we arbitrarily choose to write6.375 m $ 6.38 m We could equally well write 6.375 m $6.37 m, “rounding down” instead of “rounding up,” be-cause we would change the number 6.375 by equal incre-ments in both cases Now consider an order-of-magnitude

Note: Appendix B.8 on propagation of uncertainty may be

useful in solving some problems in this section

47.

20 C H A P T E R 1 • Physics and Measurement

Problems 21

55.0˚

Figure P1.61

estimate, in which we consider factors rather than

incre-ments We write 500 m# 103m because 500 differs from

100 by a factor of 5 while it differs from 1 000 by only a

fac-tor of 2 We write 437 m# 103 m and 305 m# 102 m

What distance differs from 100 m and from 1 000 m

by equal factors, so that we could equally well choose to

represent its order of magnitude either as# 102 m or as

# 103m?

57.For many electronic applications, such as in computer

chips, it is desirable to make components as small as

possi-ble to keep the temperature of the components low and to

increase the speed of the device Thin metallic coatings

(films) can be used instead of wires to make electrical

con-nections Gold is especially useful because it does not

oxi-dize readily Its atomic mass is 197 u A gold film can be

no thinner than the size of a gold atom Calculate the

minimum coating thickness, assuming that a gold atom

oc-cupies a cubical volume in the film that is equal to the

vol-ume it occupies in a large piece of metal This geometric

model yields a result of the correct order of magnitude

58.The basic function of the carburetor of an automobile is to

“atomize” the gasoline and mix it with air to promote

rapid combustion As an example, assume that 30.0 cm3of

gasoline is atomized into N spherical droplets, each with a

radius of 2.00 ! 10"5m What is the total surface area of

these N spherical droplets?

The consumption of natural gas by a company

satis-fies the empirical equation V $ 1.50t ' 0.008 00t2, where

V is the volume in millions of cubic feet and t the time in

months Express this equation in units of cubic feet and

seconds Assign proper units to the coefficients Assume a

month is equal to 30.0 days

60. In physics it is important to use mathematical

approxi-mations Demonstrate that for small angles (+ 20°)

tan ,% sin , % , $ #,-/180°

where , is in radians and ,- is in degrees Use a calculator

to find the largest angle for which tan , may be

approxi-mated by sin , if the error is to be less than 10.0%

A high fountain of water is located at the center of a

circu-lar pool as in Figure P1.61 Not wishing to get his feet wet,

ele-62 Collectible coins are sometimes plated with gold to

en-hance their beauty and value Consider a commemorativequarter-dollar advertised for sale at $4.98 It has a diame-ter of 24.1 mm, a thickness of 1.78 mm, and is completelycovered with a layer of pure gold 0.180 %m thick The vol-ume of the plating is equal to the thickness of the layertimes the area to which it is applied The patterns on thefaces of the coin and the grooves on its edge have a negli-gible effect on its area Assume that the price of gold is

$10.0 per gram Find the cost of the gold added to thecoin Does the cost of the gold significantly enhance thevalue of the coin?

There are nearly # ! 107s in one year Find the age error in this approximation, where “percentage error’’

percent-is defined as

64 Assume that an object covers an area A and has a uniform

height h If its cross-sectional area is uniform over its height, then its volume is given by V $ Ah (a) Show that

V $ Ah is dimensionally correct (b) Show that the

vol-umes of a cylinder and of a rectangular box can be written

in the form V $ Ah, identifying A in each case (Note that

A, sometimes called the “footprint” of the object, can have

any shape and the height can be replaced by averagethickness in general.)

65.A child loves to watch as you fill a transparent plastic tle with shampoo Every horizontal cross-section is a cir-cle, but the diameters of the circles have different values,

bot-so that the bottle is much wider in bot-some places than ers You pour in bright green shampoo with constant vol-ume flow rate 16.5 cm3/s At what rate is its level in thebottle rising (a) at a point where the diameter of the bot-tle is 6.30 cm and (b) at a point where the diameter is1.35 cm?

(a) Determine the mass of 1.00 m3of water (b) Biologicalsubstances are 98% water Assume that they have the samedensity as water to estimate the masses of a cell that has a di-ameter of 1.0 %m, a human kidney, and a fly Model the kid-ney as a sphere with a radius of 4.0 cm and the fly as a cylin-der 4.0 mm long and 2.0 mm in diameter

Assume there are 100 million passenger cars in the UnitedStates and that the average fuel consumption is 20 mi/gal ofgasoline If the average distance traveled by each car is

10 000 mi/yr, how much gasoline would be saved per year ifaverage fuel consumption could be increased to 25 mi/gal?

68 A creature moves at a speed of 5.00 furlongs per fortnight

(not a very common unit of speed) Given that

1 furlong $ 220 yards and 1 fortnight $ 14 days, mine the speed of the creature in m/s What kind of crea-ture do you think it might be?

deter-67.

Percentage error $ &assumed value " true value&

true value !100%

63.

22 C H A P T E R 1 • Physics and Measurement

69 The distance from the Sun to the nearest star is about

4 ! 1016m The Milky Way galaxy is roughly a disk of

di-ameter# 1021m and thickness# 1019m Find the order

of magnitude of the number of stars in the Milky Way

Assume the distance between the Sun and our nearest

neighbor is typical

70 The data in the following table represent measurements

of the masses and dimensions of solid cylinders of

alu-minum, copper, brass, tin, and iron Use these data to

calculate the densities of these substances Compare your

results for aluminum, copper, and iron with those given

71.(a) How many seconds are in a year? (b) If one

microme-teorite (a sphere with a diameter of 1.00 ! 10" 6m)

strikes each square meter of the Moon each second, how

many years will it take to cover the Moon to a depth of

1.00 m? To solve this problem, you can consider a cubic

box on the Moon 1.00 m on each edge, and find how long

it will take to fill the box

Answers to Quick Quizzes

1.1 (a) Because the density of aluminum is smaller than that

of iron, a larger volume of aluminum is required for agiven mass than iron

1.2 False Dimensional analysis gives the units of the

propor-tionality constant but provides no information about itsnumerical value To determine its numerical value re-quires either experimental data or geometrical reason-ing For example, in the generation of the equation

, because the factor is dimensionless, there is

no way of determining it using dimensional analysis

1.3 (b) Because kilometers are shorter than miles, a larger

number of kilometers is required for a given distance thanmiles

1.4 Reporting all these digits implies you have determined the

location of the center of the chair’s seat to the est * 0.000 000 000 1 m This roughly corresponds to be-ing able to count the atoms in your meter stick becauseeach of them is about that size! It would be better torecord the measurement as 1.044 m: this indicates thatyou know the position to the nearest millimeter, assumingthe meter stick has millimeter markings on its scale

near-1 2

Motion in One Dimension

C H A P T E R O U T L I N E 2.1 Position, Velocity, and Speed 2.2 Instantaneous Velocity and Speed

2.3 Acceleration 2.4 Motion Diagrams 2.5 One-Dimensional Motion with Constant Acceleration 2.6 Freely Falling Objects 2.7 Kinematic Equations Derived from Calculus

▲ One of the physical quantities we will study in this chapter is the velocity of an object

moving in a straight line Downhill skiers can reach velocities with a magnitude greater than

100 km/h (Jean Y Ruszniewski/Getty Images)

Chapter 2

General Problem-Solving Strategy

As a first step in studying classical mechanics, we describe motion in terms of spaceand time while ignoring the agents that caused that motion This portion of classical

mechanics is called kinematics (The word kinematics has the same root as cinema Can

you see why?) In this chapter we consider only motion in one dimension, that is, tion along a straight line We first define position, displacement, velocity, and accelera-tion Then, using these concepts, we study the motion of objects traveling in one di-mension with a constant acceleration

mo-From everyday experience we recognize that motion represents a continuouschange in the position of an object In physics we can categorize motion into threetypes: translational, rotational, and vibrational A car moving down a highway is anexample of translational motion, the Earth’s spin on its axis is an example of rota-tional motion, and the back-and-forth movement of a pendulum is an example of vi-brational motion In this and the next few chapters, we are concerned only withtranslational motion (Later in the book we shall discuss rotational and vibrationalmotions.)

In our study of translational motion, we use what is called the particle model—

we describe the moving object as a particle regardless of its size In general, a particle

is a point-like object—that is, an object with mass but having infinitesimalsize For example, if we wish to describe the motion of the Earth around the Sun, wecan treat the Earth as a particle and obtain reasonably accurate data about its orbit.This approximation is justified because the radius of the Earth’s orbit is large com-pared with the dimensions of the Earth and the Sun As an example on a muchsmaller scale, it is possible to explain the pressure exerted by a gas on the walls of acontainer by treating the gas molecules as particles, without regard for the internalstructure of the molecules

2.1 Position, Velocity, and Speed

The motion of a particle is completely known if the particle’s position in space isknown at all times A particle’s position is the location of the particle with respect to achosen reference point that we can consider to be the origin of a coordinate system

Consider a car moving back and forth along the x axis as in Figure 2.1a When we

begin collecting position data, the car is 30 m to the right of a road sign, which we will

use to identify the reference position x ! 0 (Let us assume that all data in this

exam-ple are known to two significant figures To convey this information, we should reportthe initial position as 3.0 " 101m We have written this value in the simpler form 30 m

to make the discussion easier to follow.) We will use the particle model by identifyingsome point on the car, perhaps the front door handle, as a particle representing theentire car

We start our clock and once every 10 s note the car’s position relative to the sign at

x ! 0 As you can see from Table 2.1, the car moves to the right (which we have

S E C T I O N 2 1 • Position, Velocity, and Speed 25

Active Figure 2.1 (a) A car moves back and

forth along a straight line taken to be the x

axis Because we are interested only in thecar’s translational motion, we can model it as

a particle (b) Position–time graph for themotion of the “particle.”

defined as the positive direction) during the first 10 s of motion, from position ! to

position " After ", the position values begin to decrease, suggesting that the car is

backing up from position " through position & In fact, at $, 30 s after we start

mea-suring, the car is alongside the road sign (see Figure 2.1a) that we are using to mark

our origin of coordinates It continues moving to the left and is more than 50 m to the

left of the sign when we stop recording information after our sixth data point A

graph-ical representation of this information is presented in Figure 2.1b Such a plot is called

a position–time graph.

Given the data in Table 2.1, we can easily determine the change in position of the

car for various time intervals The displacement of a particle is defined as its change

in position in some time interval As it moves from an initial position x ito a final

posi-tion x f , the displacement of the particle is given by x f#x i We use the Greek letter

delta ($) to denote the change in a quantity Therefore, we write the displacement, or

change in position, of the particle as

26 C H A P T E R 2 • Motion in One Dimension

From this definition we see that $x is positive if x f is greater than x i and negative if x fis

less than x i

It is very important to recognize the difference between displacement and distancetraveled Distance is the length of a path followed by a particle Consider, for example,the basketball players in Figure 2.2 If a player runs from his own basket down the

court to the other team’s basket and then returns to his own basket, the displacement of

the player during this time interval is zero, because he ended up at the same point as

he started During this time interval, however, he covered a distance of twice the length

of the basketball court

Displacement is an example of a vector quantity Many other physical quantities, cluding position, velocity, and acceleration, also are vectors In general, a vector quan-tity requires the specification of both direction and magnitude By contrast, ascalar quantity has a numerical value and no direction In this chapter, we use pos-itive (%) and negative (#) signs to indicate vector direction We can do this becausethe chapter deals with one-dimensional motion only; this means that any object westudy can be moving only along a straight line For example, for horizontal motion let

in-us arbitrarily specify to the right as being the positive direction It follows that any

object always moving to the right undergoes a positive displacement $x & 0, and any object moving to the left undergoes a negative displacement, so that $x ' 0 We

shall treat vector quantities in greater detail in Chapter 3

For our basketball player in Figure 2.2, if the trip from his own basket to the ing basket is described by a displacement of % 28 m, the trip in the reverse directionrepresents a displacement of # 28 m Each trip, however, represents a distance of

oppos-28 m, because distance is a scalar quantity The total distance for the trip down thecourt and back is 56 m Distance, therefore, is always represented as a positive number,while displacement can be either positive or negative

There is one very important point that has not yet been mentioned Note that thedata in Table 2.1 results only in the six data points in the graph in Figure 2.1b The

smooth curve drawn through the six points in the graph is only a possibility of the actual

motion of the car We only have information about six instants of time—we have no

idea what happened in between the data points The smooth curve is a guess as to what happened, but keep in mind that it is only a guess

If the smooth curve does represent the actual motion of the car, the graph tains information about the entire 50-s interval during which we watch the car move

con-It is much easier to see changes in position from the graph than from a verbal scription or even a table of numbers For example, it is clear that the car was cover-ing more ground during the middle of the 50-s interval than at the end Between po-sitions # and $, the car traveled almost 40 m, but during the last 10 s, betweenpositions % and &, it moved less than half that far A common way of comparing

de-these different motions is to divide the displacement $x that occurs between two clock readings by the length of that particular time interval $t This turns out to be a

very useful ratio, one that we shall use many times This ratio has been given a special

name—average velocity The average velocity v–x of a particle is defined as the

Figure 2.2 On this basketball court,

players run back and forth for the entiregame The distance that the players runover the duration of the game is non-zero The displacement of the playersover the duration of the game isapproximately zero because they keepreturning to the same point over andover again

Average speed

S E C T I O N 2 1 • Position, Velocity, and Speed 27

particle’s displacement ∆x divided by the time interval ∆t during which that

displacement occurs:

(2.2)

where the subscript x indicates motion along the x axis From this definition we see

that average velocity has dimensions of length divided by time (L/T)—meters per

sec-ond in SI units

The average velocity of a particle moving in one dimension can be positive or

nega-tive, depending on the sign of the displacement (The time interval $t is always

posi-tive.) If the coordinate of the particle increases in time (that is, if x f&x i ), then $x is

positive and is positive This case corresponds to a particle moving in the

positive x direction, that is, toward larger values of x If the coordinate decreases in

time (that is, if x f'x i ) then $x is negative and hence is negative This case

corre-sponds to a particle moving in the negative x direction.

We can interpret average velocity geometrically by drawing a straight line between

any two points on the position–time graph in Figure 2.1b This line forms the

hy-potenuse of a right triangle of height $x and base $t The slope of this line is the ratio

the line between positions ! and " in Figure 2.1b has a slope equal to the average

ve-locity of the car between those two times, (52 m # 30 m)/(10 s # 0) ! 2.2 m/s

In everyday usage, the terms speed and velocity are interchangeable In physics,

how-ever, there is a clear distinction between these two quantities Consider a marathon

runner who runs more than 40 km, yet ends up at his starting point His total

displace-ment is zero, so his average velocity is zero! Nonetheless, we need to be able to quantify

how fast he was running A slightly different ratio accomplishes this for us The

aver-age speed of a particle, a scalar quantity, is defined as the total distance traveled

di-vided by the total time interval required to travel that distance:

(2.3)

The SI unit of average speed is the same as the unit of average velocity: meters per

sec-ond However, unlike average velocity, average speed has no direction and hence

car-ries no algebraic sign Notice the distinction between average velocity and average

speed—average velocity (Eq 2.2) is the displacement divided by the time interval, while

average speed (Eq 2.3) is the distance divided by the time interval.

Knowledge of the average velocity or average speed of a particle does not provide

in-formation about the details of the trip For example, suppose it takes you 45.0 s to travel

100 m down a long straight hallway toward your departure gate at an airport At the 100-m

mark, you realize you missed the rest room, and you return back 25.0 m along the

same hallway, taking 10.0 s to make the return trip The magnitude of the average

velocity for your trip is % 75.0 m/55.0 s ! % 1.36 m/s The average speed for your trip is

125 m/55.0 s ! 2.27 m/s You may have traveled at various speeds during the walk

Nei-ther average velocity nor average speed provides information about these details

Average speed ! total distance

The magnitude of the average

ve-locity is not the average speed.

For example, consider themarathon runner discussed here.The magnitude of the average ve-locity is zero, but the averagespeed is clearly not zero

Quick Quiz 2.1 Under which of the following conditions is the magnitude of

the average velocity of a particle moving in one dimension smaller than the average

speed over some time interval? (a) A particle moves in the%x direction without

revers-ing (b) A particle moves in the # x direction without reversrevers-ing (c) A particle moves in

the%x direction and then reverses the direction of its motion (d) There are no

con-ditions for which this is true

Average velocity

Example 2.1 Calculating the Average Velocity and Speed

28 C H A P T E R 2 • Motion in One Dimension

2.2 Instantaneous Velocity and Speed

Often we need to know the velocity of a particle at a particular instant in time, ratherthan the average velocity over a finite time interval For example, even though youmight want to calculate your average velocity during a long automobile trip, you would

be especially interested in knowing your velocity at the instant you noticed the police

car parked alongside the road ahead of you In other words, you would like to be able

to specify your velocity just as precisely as you can specify your position by noting what

is happening at a specific clock reading—that is, at some specific instant It may not beimmediately obvious how to do this What does it mean to talk about how fast some-thing is moving if we “freeze time” and talk only about an individual instant? This is asubtle point not thoroughly understood until the late 1600s At that time, with the in-vention of calculus, scientists began to understand how to describe an object’s motion

at any moment in time

To see how this is done, consider Figure 2.3a, which is a reproduction of the graph

in Figure 2.1b We have already discussed the average velocity for the interval duringwhich the car moved from position ! to position " (given by the slope of the darkblue line) and for the interval during which it moved from ! to & (represented bythe slope of the light blue line and calculated in Example 2.1) Which of these twolines do you think is a closer approximation of the initial velocity of the car? The carstarts out by moving to the right, which we defined to be the positive direction There-fore, being positive, the value of the average velocity during the ! to " interval ismore representative of the initial value than is the value of the average velocity duringthe ! to & interval, which we determined to be negative in Example 2.1 Now let usfocus on the dark blue line and slide point " to the left along the curve, toward point

!, as in Figure 2.3b The line between the points becomes steeper and steeper, and asthe two points become extremely close together, the line becomes a tangent line tothe curve, indicated by the green line in Figure 2.3b The slope of this tangent line

2.2 Slopes of Graphs

In any graph of physical data, the

slope represents the ratio of the

change in the quantity

repre-sented on the vertical axis to the

change in the quantity

repre-sented on the horizontal axis

Re-member that a slope has units

(un-less both axes have the same

units) The units of slope in

Figure 2.1b and Figure 2.3 are

m/s, the units of velocity

Find the displacement, average velocity, and average speed

of the car in Figure 2.1a between positions ! and &

Solution From the position–time graph given in Figure

2.1b, note that xA!30 m at tA!0 s and that xF! #53 m

at tF!50 s Using these values along with the definition of

displacement, Equation 2.1, we find that

This result means that the car ends up 83 m in the

nega-tive direction (to the left, in this case) from where it

started This number has the correct units and is of the

same order of magnitude as the supplied data A

quick look at Figure 2.1a indicates that this is the correct

answer

It is difficult to estimate the average velocity without

completing the calculation, but we expect the units to be

meters per second Because the car ends up to the left of

where we started taking data, we know the average velocity

must be negative From Equation 2.2,

#83 m

$x ! xF#xA! #53 m # 30 m ! We cannot unambiguously find the average speed of the

car from the data in Table 2.1, because we do not have mation about the positions of the car between the datapoints If we adopt the assumption that the details of thecar’s position are described by the curve in Figure 2.1b, thenthe distance traveled is 22 m (from ! to ") plus 105 m(from " to &) for a total of 127 m We find the car’s averagespeed for this trip by dividing the distance by the total time(Eq 2.3):

infor-2.5 m/sAverage speed ! 127 m

S E C T I O N 2 2 • Instantaneous Velocity and Speed 29

x(m)

t(s)

(a)

5040

3020

represents the velocity of the car at the moment we started taking data, at point !

What we have done is determine the instantaneous velocity at that moment In other

words, the instantaneous velocity vxequals the limiting value of the ratio !x(!t

as !t approaches zero:1

(2.4)

In calculus notation, this limit is called the derivative of x with respect to t, written dx/dt:

(2.5)

The instantaneous velocity can be positive, negative, or zero When the slope of the

position–time graph is positive, such as at any time during the first 10 s in Figure 2.3,

v x is positive—the car is moving toward larger values of x After point ", v xis

nega-tive because the slope is neganega-tive—the car is moving toward smaller values of x At

point ", the slope and the instantaneous velocity are zero—the car is momentarily at

rest

From here on, we use the word velocity to designate instantaneous velocity When it

is average velocity we are interested in, we shall always use the adjective average.

The instantaneous speed of a particle is defined as the magnitude of its

instan-taneous velocity As with average speed, instaninstan-taneous speed has no direction

associated with it and hence carries no algebraic sign For example, if one particle

has an instantaneous velocity of % 25 m/s along a given line and another particle

has an instantaneous velocity of # 25 m/s along the same line, both have a speed2

v x! lim

$t : 0

$x

$t

Active Figure 2.3 (a) Graph representing the motion of the car in Figure 2.1 (b) An

enlargement of the upper-left-hand corner of the graph shows how the blue line

between positions ! and " approaches the green tangent line as point " is moved

closer to point !

At the Active Figures link at http://www.pse6.com, you can move point "

as suggested in (b) and observe the blue line approaching the green tangent

line.

Instantaneous velocity

1 Note that the displacement $x also approaches zero as $t approaches zero, so that the ratio

looks like 0/0 As $x and $t become smaller and smaller, the ratio $x/$t approaches a value

equal to the slope of the line tangent to the x-versus-t curve.

2 As with velocity, we drop the adjective for instantaneous speed: “Speed” means instantaneous

speed

2.3 Instantaneous Speed and Instantaneous Velocity

In Pitfall Prevention 2.1, we gued that the magnitude of theaverage velocity is not the averagespeed Notice the differencewhen discussing instantaneousvalues The magnitude of the in-

ar-stantaneous velocity is the

instan-taneous speed In an infinitesimaltime interval, the magnitude ofthe displacement is equal to thedistance traveled by the particle

A particle moves along the x axis Its position varies with

time according to the expression x ! # 4t % 2t2where x is

in meters and t is in seconds.3The position–time graph for

this motion is shown in Figure 2.4 Note that the particle

moves in the negative x direction for the first second of

mo-tion, is momentarily at rest at the moment t ! 1 s, and

moves in the positive x direction at times t & 1 s.

(A) Determine the displacement of the particle in the time

intervals t ! 0 to t ! 1 s and t ! 1 s to t ! 3 s.

Solution During the first time interval, the slope is

nega-tive and hence the average velocity is neganega-tive Thus, we

know that the displacement between ! and " must be a

negative number having units of meters Similarly, we expect

the displacement between " and $ to be positive

In the first time interval, we set t i!tA!0 and

tf! tB! 1 s Using Equation 2.1, with x ! # 4t % 2t2, we

obtain for the displacement between t ! 0 and t ! 1 s,

To calculate the displacement during the second time

inter-val (t ! 1 s to t ! 3 s), we set t i !tB!1 s and t f !tD!3 s:

These displacements can also be read directly from the

posi-tion–time graph

(B) Calculate the average velocity during these two time

in-tervals

tB#tA! 1 s Therefore, using Equation 2.2 and the

dis-placement calculated in (a), we find that

! [#4(1) % 2(1)2] # [#4(0) % 2(0)2]

$xA : B ! x f#xi!xB#xA

30 C H A P T E R 2 • Motion in One Dimension

Conceptual Example 2.2 The Velocity of Different Objects

Example 2.3 Average and Instantaneous Velocity

Consider the following one-dimensional motions: (A)A ball

thrown directly upward rises to a highest point and falls

back into the thrower’s hand (B)A race car starts from rest

and speeds up to 100 m/s (C)A spacecraft drifts through

space at constant velocity Are there any points in the

mo-tion of these objects at which the instantaneous velocity has

the same value as the average velocity over the entire

mo-tion? If so, identify the point(s)

Solution (A) The average velocity for the thrown ball is

zero because the ball returns to the starting point; thus its

displacement is zero (Remember that average velocity is

defined as $x/$t.) There is one point at which the

instanta-neous velocity is zero—at the top of the motion

(B) The car’s average velocity cannot be evaluated biguously with the information given, but it must be somevalue between 0 and 100 m/s Because the car will haveevery instantaneous velocity between 0 and 100 m/s atsome time during the interval, there must be some instant

unam-at which the instantaneous velocity is equal to the averagevelocity

(C) Because the spacecraft’s instantaneous velocity is

con-stant, its instantaneous velocity at any time and its average velocity over any time interval are the same.

1086420–2–4

Figure 2.4 (Example 2.3) Position–time graph for a particle

having an x coordinate that varies in time according to the expression x ! # 4t % 2t2

In the second time interval, $t ! 2 s; therefore,

These values are the same as the slopes of the lines joiningthese points in Figure 2.4

(C) Find the instantaneous velocity of the particle at t ! 2.5 s.

Solution We can guess that this instantaneous velocity must

be of the same order of magnitude as our previous results,that is, a few meters per second By measuring the slope of

the green line at t ! 2.5 s in Figure 2.4, we find that

mensional consistency When we start our clocks at t ! 0, we usually

do not mean to limit the precision to a single digit Consider anyzero value in this book to have as many significant figures as youneed

Figure 2.5 (a) A car, modeled as a particle, moving along the x

axis from ! to " has velocity v xi at t ! t i and velocity v xf at t ! t f.(b) Velocity–time graph (rust) for the particle moving in astraight line The slope of the blue straight line connecting !and " is the average acceleration in the time interval

$t ! t f#t i

The average acceleration a– x of the particle is defined as the change in velocity

$v x divided by the time interval $t during which that change occurs:

2.3 Acceleration

In the last example, we worked with a situation in which the velocity of a particle

changes while the particle is moving This is an extremely common occurrence (How

constant is your velocity as you ride a city bus or drive on city streets?) It is possible to

quantify changes in velocity as a function of time similarly to the way in which we

quan-tify changes in position as a function of time When the velocity of a particle changes

with time, the particle is said to be accelerating For example, the magnitude of the

velocity of a car increases when you step on the gas and decreases when you apply the

brakes Let us see how to quantify acceleration

Suppose an object that can be modeled as a particle moving along the x axis has an

initial velocity v xi at time t i and a final velocity v xf at time t f, as in Figure 2.5a

Average acceleration

(2.6)

As with velocity, when the motion being analyzed is one-dimensional, we can use

positive and negative signs to indicate the direction of the acceleration Because the

di-mensions of velocity are L/T and the dimension of time is T, acceleration has

dimen-sions of length divided by time squared, or L/T2 The SI unit of acceleration is meters

per second squared (m/s2) It might be easier to interpret these units if you think of

them as meters per second per second For example, suppose an object has an

acceler-ation of % 2 m/s2 You should form a mental image of the object having a velocity that

is along a straight line and is increasing by 2 m/s during every interval of 1 s If the

ob-ject starts from rest, you should be able to picture it moving at a velocity of % 2 m/s

af-ter 1 s, at % 4 m/s afaf-ter 2 s, and so on

In some situations, the value of the average acceleration may be different over

different time intervals It is therefore useful to define the instantaneous acceleration

as the limit of the average acceleration as $t approaches zero This concept is

analo-gous to the definition of instantaneous velocity discussed in the previous section If

we imagine that point ! is brought closer and closer to point " in Figure 2.5a and

we take the limit of $v x /$t as $t approaches zero, we obtain the instantaneous