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Cuốn sách có nội dung phong phú và hình ảnh đẹp, được tác giả Serway và JewettChương I mô tả về các đại lượng Vật lý và các phép toán Vecto được sử dụng trong vật lý. Cuốn sách mô tả đầy đủ về các đại lượng vật lý, và có phần bài tập phù hợp. Cuốn sách phù hợp cho giáo viên dạy hsg và dạy các trường THPT Chuyên hoặc dạy học vật lý bằng tiếng Anh thì đây là tài liệu hết sức hữu dụng.
Licensed to: CengageBrain User This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. 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For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be e-mailed to permissionrequest@cengage.com International Edition: ISBN-13: 978-1-133-11000-2 ISBN-10: 1-133-11000-2 Cengage Learning International Offi ces Asia Australia/New Zealand www.cengageasia.com www.cengage.com.au tel: (65) 6410 1200 tel: (61) 3 9685 4111 India Brazil www.cengage.co.in www.cengage.com.br tel: (91) 11 4364 1111 tel: (55) 11 3665 9900 Latin America UK/Europe/Middle East/Africa www.cengage.com.mx www.cengage.co.uk tel: (52) 55 1500 6000 tel: (44) 0 1264 332 424 Represented in Canada by Nelson Education, Ltd. www.nelson.com tel: (416) 752 9100 / (800) 668 0671 Cengage Learning is a leading provider of customized learning solutions with offi ce locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local offi ce at: www.cengage.com/global For product information and free companion resources: www.cengage.com/international Visit your local offi ce: www.cengage.com/global Principles of Physics: A Calculus-Based Text, Fifth Edition, International Edition Raymond A. Serway John W. Jewett, Jr. Publisher, Physical Sciences: Mary Finch Publisher, Astronomy and Physics: Charles Hartford Development Editor: Ed Dodd Assistant Editor: Brandi Kirksey Editorial Assistant: Brendan Killion Media Editor: Rebecca Berardy Schwartz Marketing Manager: Jack Cooney Marketing Communications Manager: Darlene Macanan Content Project Manager: Cathy Brooks Art Director: Cate Rickard Barr Manufacturing Planner: Doug Bertke Rights Acquisition Specialist: Shalice Shah-Caldwell Production Service: MPS Limited Text Designer: Brian Salisbury Cover Image: André Leopold Compositor: MPS Limited Printed in China 1 2 3 4 5 6 7 16 15 14 13 12 We dedicate this book to our wives Elizabeth and Lisa and all our children and grandchildren for their loving understanding when we spent time on writing instead of being with them. 10002_FM.indd iv10002_FM.indd iv 2/7/12 12:20 PM2/7/12 12:20 PM Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Licensed to: CengageBrain User 4 Introduction and Vectors 1 Chapter Chapter Outline 1.1 Standards of Length, Mass, andTime 1.2 Dimensional Analysis 1.3 Conversion of Units 1.4 Order-of-Magnitude Calculations 1.5 Signifi cant Figures 1.6 Coordinate Systems 1.7 Vectors and Scalars 1.8 Some Properties of Vectors 1.9 Components of a Vector and Unit Vectors 1.10 Modeling, Alternative Representations, and Problem-Solving Strategy SUMMARY T he goal of physics is to provide a quantitative understanding of cer- tain basic phenomena that occur in our Universe. Physics is a science based on experimental observations and mathematical analyses. The main objectives behind suchexperiments and analyses are to develop theories that explain the phenomenon being studied and to relate those theories to other established theories. Fortunately, it is possible to explain the behavior of various physical systems using relatively few fundamental laws. Analytical procedures require the expression of those laws in the language of mathematics, the tool that provides a bridge between theory and experiment. In this chapter, we shall discuss a few math- ematical concepts and techniques that will be used throughout the text. In addition, we will outline an effective problem-solving strategy that should be adopted and used in your problem-solving activities throughout the text. 1.1 | Standards of Length, Mass, and Time To describe natural phenomena, we must make measurements associated with physical quantities, such as the length of an object. The laws of physics can be expressed as mathematical relationships among physical quantities that will be Raymond A. Serway A signpost in Saint Petersburg, Florida, shows the distance and direction to several cities. Quantities that are defi ned by both a magnitude and a direction are called vector quantities. Interactive content from this and other chapters may be assigned online in Enhanced WebAssign. 10293_ch01_001-034.indd 410293_ch01_001-034.indd 4 1/6/12 2:35 PM1/6/12 2:35 PM Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Licensed to: CengageBrain User 1.1 | Standards of Length, Mass, and Time5 introduced and discussed throughout the book. In mechanics, the three fundamen- tal quantities are length, mass, and time. All other quantities in mechanics can be expressed in terms of these three. If we measure a certain quantity and wish to describe it to someone, a unit for the quantity must be speci ed and de ned. For example, it would be meaningless for a visitor from another planet to talk to us about a length of 8.0 “glitches” if we did not know the meaning of the unit glitch. On the other hand, if someone familiar with our system of measurement reports that a wall is 2.0 meters high and our unit of length is de ned to be 1.0 meter, we then know that the height of the wall is twice our fundamental unit of length. An international committee has agreed on a system of de nitions and standards to describe fundamental physical quantities. It is called the SI system (Système International) of units. Its units of length, mass, and time are the meter, kilogram, and second, respectively. Length In .. 1120, King Henry I of England decreed that the standard of length in his country would be the yard and that the yard would be precisely equal to the distance from the tip of his nose to the end of his outstretched arm. Similarly, the original standard for the foot adopted by the French was the length of the royal foot of King Louis XIV. This standard prevailed until 1799, when the legal standard of length in France became the meter, de ned as one ten-millionth of the distance from the equator to the North Pole. Many other systems have been developed in addition to those just discussed, but the advantages of the French system have caused it to prevail in most countries and in scienti c circles everywhere. Until 1960, the length of the meter was de ned as the distance between two lines on a speci c bar of platinum–iridium alloy stored under controlled conditions. This standard was abandoned for several reasons, a principal one being that the limited accuracy with which the separation be tweenthe lines can be determined does not meet the current requirements of science and technology. The de nition of the meter was modi ed to be equal to 1 650 763.73 wavelengths of orange–red light emitted from a krypton-86 lamp. In October 1983, the meter was rede ned to be the distance traveled by light in a vacuum during a time interval of 1y299 792 458 second. This value arises from the establishment of the speed of light in a vacuum as exactly 299 792 458 meters per second. We will use the standard scienti c notation for numbers with more than three digits in which groups of three digits are separated by spaces rather than commas. Therefore, 1 650 763.73 and 299 792 458 in this paragraph are the same as the more popular American cultural notations of 1,650,763.73 and 299,792,458. Similarly, 5 3.14159265 is written as 3.141 592 65. Mass Mass represents a measure of the resistance of an object to changes in its motion. The SI unit of mass, the kilogram, is de ned as the mass of a speci c platinum–iridium alloy cylinder kept at the International Bureau of Weights and Measures at Sèvres, France. At this point, we should add a word of caution. Many beginning students of physics tend to confuse the physical quantities called weight and mass. For the pres- ent we shall not discuss the distinction between them; they will be clearly de ned in later chapters. For now you should note that they are distinctly different quantities. Time Before 1967, the standard of time was de ned in terms of the average length of a mean solar day. (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 basic unit of time, the sec- ond, was de ned to be (1/60)(1/60)(1/24) 5 1/86 400 of a mean solar day. In 1967, the second was rede ned to take advantage of the great precision obtainable with a device known as an atomic clock (Fig. 1.1), which uses the characteristic frequency of the c De nition of the meter c De nition of the kilogram Figure 1.1 A cesium fountain atomic clock. The clock will neither gain nor lose a second in 20 million years. © 2005 Geoffrey Wheeler Photography 10293_ch01_001-034.indd 510293_ch01_001-034.indd 5 1/6/12 2:35 PM1/6/12 2:35 PM Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Licensed to: CengageBrain User 6CHAPTER 1 | Introduction and Vectors cesium-133 atom as the “reference clock.” The second is now de ned as 9 192 631 770 times the period of oscillation of radiation from the cesium atom. It is possible today to purchase clocks and watches that receive radio signals from an atomic clock in Colorado, which the clock or watch uses to continuously reset itself to the correct time. Approximate Values for Length, Mass, and Time Approximate values of various lengths, masses, and time intervals are presented in Tables 1.1, 1.2, and 1.3, respectively. Note the wide range of values for these quantities. 1 You should study the tables and begin to generate an intuition for what is meant by a mass of 100 kilograms, for example, or by a time interval of 3.2 3 10 7 seconds. Systems of units commonly used in science, commerce, manufacturing, and everyday life are (1) the SI system, in which the units of length, mass, and time are the meter (m), kilogram (kg), and second (s), respectively; and (2) the U.S. customary system, in which the units of length, mass, and time are the foot (ft), slug, and second, respectively. Throughout most of this text we shall use SI units because they are almost universally accepted in science and industry. We will make limited use of U.S. customary units in the study of classical mechanics. Some of the most frequently used pre xes for the powers of ten and their abbreviations are listed in Table 1.4. For example, 10 23 m is equivalent to 1 millimeter (mm), and 10 3 m is 1 kilometer (km). Likewise, 1 kg is 10 3 grams (g), and 1 megavolt (MV) is 10 6 volts (V). The variables length, time, and mass are examples of fundamental quantities. A much larger list of variables contains derived quantities, or quantities that can be expressed as a mathematical combination of fundamental quantities. Common examples are area, which is a product of two lengths, and speed, which is a ratio of a length to a time interval. Another example of a derived quantity is density. The density (Greek letter rho; a table of the letters in the Greek alphabet is provided at the back of the book) of any substance is de ned as its mass per unit volume: ; m V 1.1 b c De nition of density Pitfall Prevention | 1.1 Reasonable Values Generating intuition about typical values of quantities when solving problems is important because you must think about your end result and determine if it seems reasonable. For example, if you are calculating the mass of a house y and arrive at a value of 100 kg, this answer is unreasonable and there is an error somewhere. 1 If you are unfamiliar with the use of powers of ten (scienti c notation), you should review Appendix B.1. TABLE 1.1 | Approximate Values of Some Measured Lengths Length (m) Distance from the Earth to the most remote quasar 1.4 3 10 26 Distance from the Earth to the most remote normal galaxies 9 3 10 25 Distance from the Earth to the nearest large galaxy (M 31, the Andromeda galaxy) 2 3 10 22 Distance from the Sun to the nearest star (Proxima Centauri) 4 3 10 16 One light-year 9.46 3 10 15 Mean orbit radius of the Earth 1.50 3 10 11 Mean distance from the Earth to the Moon 3.84 3 10 8 Distance from the equator to the North Pole 1.00 3 10 7 Mean radius of the Earth 6.37 3 10 6 Typical altitude (above the surface) of a satellite orbiting the Earth 2 3 10 5 Length of a football eld 9.1 3 10 1 Length of this textbook 2.8 3 10 21 Length of a house y 5 3 10 23 Size of smallest visible dust particles , 10 24 Size of cells of most living organisms , 10 25 Diameter of a hydrogen atom , 10 210 Diameter of a uranium nucleus , 10 214 Diameter of a proton , 10 215 TABLE 1.2 | Masses of Various Objects (Approximate Values) Mass (kg) Visible , 10 52 Universe Milky Way , 10 42 galaxy Sun 1.99 3 10 30 Earth 5.98 3 10 24 Moon 7.36 3 10 22 Shark , 10 3 Human , 10 2 Frog , 10 21 Mosquito , 10 25 Bacterium , 10 215 Hydrogen 1.67 3 10 227 atom Electron 9.11 3 10 231 c De nition of the second 10293_ch01_001-034.indd 610293_ch01_001-034.indd 6 1/6/12 2:35 PM1/6/12 2:35 PM Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Licensed to: CengageBrain User 1.2 | Dimensional Analysis7 which is a ratio of mass to a product of three lengths. For example, aluminum has a density of 2.70 3 10 3 kg/m 3 , and lead has a density of 11.3 3 10 3 kg/m 3 . An extreme difference in density can be imagined by thinking about holding a 10-centimeter (cm) cube of Styrofoam in one hand and a 10-cm cube of lead in the other. 1.2 | Dimensional Analysis In physics, the word dimension denotes the physical nature of a quantity. The dis- tance between two points, for example, can be measured in feet, meters, or furlongs, which are all different ways of expressing the dimension of length. The symbols used in this book to specify the dimensions 2 of length, mass, and time are L, M, and T, respectively. We shall often use square brackets [ ] to denote the dimensions of a physical quantity. For example, in this notation the dimensions of speed v are written [v] 5 L/T, and the dimensions of area A are [A] 5 L 2 . The di- mensions of area, volume, speed, and acceleration are listed in Table 1.5, along with their units in the two common systems. 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 be faced with having to derive or check a speci c equation. Although you may have forgotten the details of the derivation, a useful and powerful procedure called dimensional analysis can be used as a consistency check, to assist in the derivation, or to check your nal expression. Dimensional analysis makes use of the fact that dimensions can be treated as algebraic quantities. For example, quantities can be added or subtracted only if they have the same dimensions. Fur- thermore, 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 TABLE 1.4 | Some Prefi xes for Powers of Ten Power Pre x Abbreviation 10 224 yocto y 10 221 zepto z 10 218 atto a 10 215 femto f 10 212 pico p 10 29 nano n 10 26 micro 10 23 milli m 10 22 centi c 10 21 deci d 10 3 kilo k 10 6 mega M 10 9 giga G 10 12 tera T 10 15 peta P 10 18 exa E 10 21 zetta Z 10 24 yotta Y TABLE 1.3 | Approximate Values of Some Time Intervals Time Interval (s) Age of the Universe 4 3 10 17 Age of the Earth 1.3 3 10 17 Time interval since the fall of the Roman empire 5 3 10 12 Average age of a college student 6.3 3 10 8 One year 3.2 3 10 7 One day (time interval for one revolution of the Earth about its axis) 8.6 3 10 4 One class period 3.0 3 10 3 Time interval between normal heartbeats 8 3 10 21 Period of audible sound waves , 10 23 Period of typical radio waves , 10 26 Period of vibration of an atom in a solid , 10 213 Period of visible light waves , 10 215 Duration of a nuclear collision , 10 222 Time interval for light to cross a proton , 10 224 Pitfall Prevention | 1.2 Symbols for Quantities Some quantities have a small number of symbols that represent them. For example, the symbol for time is almost always t. Other quantities might have various symbols depending on the usage. Length may be described with symbols such as x, y, and z (for position); r (for radius); a, b, and c (for the legs of a right triangle); ℓ (for the length of an object); d (for a distance); h (for a height); and so forth. 2 The dimensions of a variable will be symbolized by a capitalized, nonitalic letter, such as, in the case of length, L. The symbol for the variable itself will be italicized, such as L for the length of an object or t for time. TABLE 1.5 | Dimensions and Units of Four Derived Quantities Quantity Area (A) Volume (V ) Speed (v) Acceleration (a) Dimensions L 2 L 3 L/T L/T 2 SI units m 2 m 3 m/s m/s 2 U.S. customary units ft 2 ft 3 ft/s ft/s 2 10293_ch01_001-034.indd 710293_ch01_001-034.indd 7 1/6/12 2:35 PM1/6/12 2:35 PM Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Licensed to: CengageBrain User 8CHAPTER 1 | Introduction and Vectors whether an expression has the correct form because the relationship can be correct only if the dimensions on the two sides of the equation are the same. To illustrate this procedure, suppose you wish to derive an expression for the posi- tion x of a car at a time t if the car starts from rest at t 5 0 and moves with constant acceleration a. In Chapter 2, we shall nd that the correct expression for this special case is x 5 1 2 at 2 . Let us check the validity of this expression from a dimensional analy- sis approach. 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 perform a dimensional check by substituting the basic dimensions for acceleration, L/T 2 (Table 1.5), and time, T, into the equation x 5 1 2 at 2 . That is, the dimensional form of the equation x 5 1 2 at 2 can be written as [x] 5 L T 2 T 2 5 L The dimensions of time cancel as shown, leaving the dimension of length, which is the correct dimension for the position x. Notice that the number 1 2 in the equation has no units, so it does not enter into the dimensional analysis. QUICK QUIZ 1.1 True or False: Dimensional analysis can give you the numerical value of constants of proportionality that may appear in an algebraic expression. 1.3 | Conversion of Units Sometimes it is necessary to convert units from one 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 mi l e ( mi ) 5 1 609 m 5 1 .609 k m 1 f t 5 0.30 4 8 m 5 30. 4 8 c m 1 m 5 39.37 in. 5 3.281 f t 1 inch (in.) 5 0.02 5 4 m 5 2. 5 4 cm A more complete list of equalities can be found in Appendix A. Units can be treated as algebraic quantities that can cancel each other. To per- form a conversion, a quantity can be multiplied by a conversion factor, which is a frac- tion equal to 1, with numerator and denominator having different units, to provide the desired units in the nal result. For example, suppose we wish to convert 15.0 in. to centimeters. Because 1 in. 5 2.54 cm, we multiply by a conversion factor that is the appropriate ratio of these equal quantities and nd that 15.0 in. 5 (15.0 in.) 1 2.54 cm 1 in. 2 5 38.1 cm Pitfall Prevention | 1.3 Always Include Units When performing calculations, make it a habit to include the units with every quantity and carry the units through the entire calculation. Avoid the temptation to drop the units during the calculation steps and then apply the expected unit to the number that results for an answer. By including the units in every step, you can detect errors if the units for the answer are incorrect. Example 1.1 | Analysis of an Equation Show that the expression v 5 at, where v represents speed, a acceleration, and t an instant of time, is dimensionally correct. SOLUTION Identify the dimensions of v from Table 1.5: [v] 5 L T Identify the dimensions of a from Table 1.5 and multiply [at] 5 L T 2 T 5 L T by the dimensions of t: Therefore, v 5 at is dimensionally correct because we have the same dimensions on both sides. (If the expression were given as v 5 at 2 , it would be dimensionally incorrect. Try it and see!) 10293_ch01_001-034.indd 810293_ch01_001-034.indd 8 1/6/12 2:35 PM1/6/12 2:35 PM Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Licensed to: CengageBrain User 1.4 | Order-of-Magnitude Calculations9 1.4 | Order-of-Magnitude Calculations Suppose someone asks you the number of bits of data on a typical musical compact disc. In response, it is not generally expected that you would provide the exact num- ber but rather an estimate, which may be expressed in scienti c notation. The esti- mate may be made even more approximate by expressing it as an order of magnitude, which is a power of ten determined as follows: 1. Express the number in scienti c notation, with the multiplier of the power of ten between 1 and 10 and a unit. 2. If the multiplier is less than 3.162 (the square root of ten), the order of magni- tude of the number is the power of ten in the scienti c notation. If the multi- plier is greater than 3.162, the order of magnitude is one larger than the power of ten in the scienti c notation. We use the symbol , for “is on the order of.” Use the procedure above to verify the orders of magnitude for the following lengths: 0.008 6 m , 10 22 m 0.002 1 m , 10 23 m 720 m , 10 3 m Usually, when an order-of-magnitude estimate is made, the results are reliable to within about a factor of ten. If a quantity increases in value by three orders of magni- tude, its value increases by a factor of about 10 3 5 1 000. where the ratio in parentheses is equal to 1. Notice that we express 1 as 2.54 cm/1 in. (rather than 1 in./2.54 cm) so that the inch cancels with the unit in the original quantity. The remaining unit is the centimeter, which is our desired result. QUICK QUIZ 1.2 The distance between two cities is 100 mi. What is the number of kilometers between the two cities? (a) smaller than 100 (b) larger than 100 (c) equal to 100 Example 1.2 | 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 the driver exceeding the speed limit of 75.0 mi/h? SOLUTION Convert meters in the (38.0 m/s) 1 1 mi 1 609 m 2 5 2.36 3 10 22 mi/s speed to miles: Convert seconds to hours: (2.36 3 10 22 mi/s) 1 60 s 1 min 2 1 60 min 1 h 2 5 85.0 mi/h The driver is indeed exceeding the speed limit and should slow down. What If? What if the driver were from outside the United States and is familiar with speeds measured in kilometers per hour? What is the speed of the car in km/h? Answer We can convert our nal answer to the appropriate units: (85.0 mi /h) 1 1.609 km 1 mi 2 5 137 km/h Figure 1.2 shows an automobile speedometer displaying speeds in both mi/h and km/h. Can you check the conversion we just performed using this photograph? Figure 1.2 (Example 1.2) The speedometer of a vehicle that shows speeds in both miles per hour and kilometers per hour. © Cengage Learning/Ed Dodd 10293_ch01_001-034.indd 910293_ch01_001-034.indd 9 1/6/12 2:35 PM1/6/12 2:35 PM Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Licensed to: CengageBrain User 10CHAPTER 1 | Introduction and Vectors Example 1.3 | The Number of Atoms in a Solid Estimate the number of atoms in 1 cm 3 of a solid. SOLUTION From Table 1.1 we note that the diameter d of an atom is about 10 210 m. Let us assume that the atoms in the solid are spheres of this diameter. Then the volume of each sphere is about 10 230 m 3 (more precisely, volume 5 4 r 3 /3 5 d 3 /6, where r 5 d/2). Therefore, because 1 cm 3 5 10 26 m 3 , the number of atoms in the solid is on the order of 10 26 /10 230 5 10 24 atoms. A more precise calculation would require additional knowledge that we could nd in tables. Our estimate, however, agrees with the more precise calculation to within a factor of 10. Example 1.4 | Breaths in a Lifetime Estimate the number of breaths taken during an average human lifetime. SOLUTION We start by guessing that the typical human lifetime is about 70 years. Think about 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. (This estimate is certainly closer to the true average value than an estimate of 1 breath per minute or 100 breaths per minute.) Find the approximate number of minutes in a year: 1 yr 1 400 days 1 y r 21 25 h 1 da y 21 60 min 1 h 2 5 6 3 10 5 min Find the approximate number of minutes in a 70-year number of minutes 5 (70 yr)(6 3 10 5 min/yr) lifetime: 5 4 3 10 7 min Find the approximate number of breaths in a lifetime: number of breaths 5 (10 breaths/min)(4 3 10 7 min) 5 4 3 10 8 breaths Therefore, a person takes on the order of 10 9 breaths in a lifetime. Notice how much simpler it is in the rst calculation above to multiply 400 3 25 than it is to work with the more accurate 365 3 24. What If? What if the average lifetime were estimated as 80 years instead of 70? Would that change our nal estimate? Answer We could claim that (80 yr)(6 3 10 5 min/yr) 5 5 3 10 7 min, so our nal estimate should be 5 3 10 8 breaths. This answer is still on the order of 10 9 breaths, so an order-of-magnitude estimate would be unchanged. 1.5 | Signifi cant 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 signi cant gures in a measurement can be used to express something about the uncertainty. The number of signi cant gures is related to the number of numerical digits used to express the measurement, as we discuss below. As an example of signi cant gures, suppose we are asked to measure the radius of a compact disc using a meterstick as a measuring instrument. Let us assume the accuracy to which we can measure the radius of the disc is 60.1 cm. Because of the uncertainty of 60.1 cm, if the radius is measured to be 6.0 cm, we can claim only that its radius lies somewhere between 5.9 cm and 6.1 cm. In this case, we say that the 10293_ch01_001-034.indd 1010293_ch01_001-034.indd 10 1/6/12 2:35 PM1/6/12 2:35 PM Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Licensed to: CengageBrain User [...]... claimed in a multiplication or a division is as follows: When multiplying several quantities, the number of significant figures in the final answer is the same as the number of significant figures in the quantity having the smallest number of significant figures The same rule applies to division Let’s apply this rule to find the area of the compact disc whose radius we measured above Using the equation... electric wire as a long, straight cylinder The particle model is an example of the second category of models, which we will call simplification models In a simplification model, details that are not significant in determining the outcome of the problem are ignored When we study rotation in Chapter 10, objects will be modeled as rigid objects All the molecules in a rigid object maintain their exact positions... representation It is often useful to redraw the pictorial representation without complicating details by applying a simplification model This process is similar to the discussion of the particle model described earlier In a pictorial representation of the Earth in orbit around the Sun, you might draw the Earth and the Sun as spheres, with possibly some attempt to draw continents to identify which sphere is... axis from position xi to position xf , as in Figure 1.7, its displacement is given by xf 2 xi (The indices i and f refer to the initial and final values.) We use the Greek letter delta (D) to denote the change in a quantity Therefore, we define the change in the position of the particle (the displacement) as Dx ; xf 2 xi 1.6b From this definition we see that Dx is positive if xf is greater than xi... situation and can predict what changes will occur in the situation This step is critical in approaching every problem • Pictorial representation Drawing a picture of the situation described in the word problem can be of great assistance in understanding the problem In Example 1.9, the pictorial representation in Figure 1.19 allows us to identify the triangle as a geometric model of the problem In architecture,... blueprint is a pictorial representation of a proposed building Generally, a pictorial representation describes what you would see if you were observing the situation in the problem For example, Figure 1.20 shows a pictorial representation of a baseball player hitting a short pop foul Any coordinate axes included in your pictorial representation will be in two dimensions: x and y axes • Simplified pictorial... problem Scientists must be able to communicate complex ideas to individuals without scientific backgrounds The best representation to use in conveying the information successfully will vary from one individual to the next Some will be convinced by a well-drawn graph, and others will require a picture Physicists are often persuaded to agree with a point of view by examining an equation, but nonphysicists... Energy is one of the physical concepts that we will investigate in this Context A fuel such as gasoline contains energy due to its chemical composition and its ability to undergo a combustion process The battery in an electric car also contains energy, again related to chemical composition, but in this case it is associated with an ability to produce an electric current One difficult social aspect of developing... be written 0.000 23) and 2.30 3 1024 has three significant figures (also written as 0.000 230) In problem solving, we often combine quantities mathematically through multiplication, division, addition, subtraction, and so forth When doing so, you must make sure that the result has the appropriate number of significant figures A good rule of thumb to use in determining the number of significant figures... them determine this force In a simplified model, the Earth is imagined to be a particle, an object with mass but zero size This replacement of an extended object by a particle is called the particle model, which is used extensively in physics By analyzing the motion of a particle with the mass of the Earth in orbit around the Sun, we find that the predictions of the particle’s motion are in excellent