44337_01_p1-22 10/13/04 2:13 PM Page 1 Thousands of years ago, people in southern England built Stonehenge, which was used as a calendar. The position of the sun and stars relative to the stones determined seasons for planting or harvesting. CHAPTER 1 O U T L I N E 1.1 Standards of Length, Introduction Mass, and Time 1.2 The Building Blocks of Matter The goal of physics is to provide an understanding of the physical world by developing theo- 1.3 Dimensional Analysis ries based on experiments. A physical theory is essentially a guess, usually expressed mathe- matically, about how a given physical system works. The theory makes certain predictions 1.4 Uncertainty in about the physical system which can then be checked by observations and experiments. If the Measurement and predictions turn out to correspond closely to what is actually observed, then the theory Significant Figures stands, although it remains provisional. No theory to date has given a complete description of 1.5 Conversion of Units all physical phenomena, even within a given subdiscipline of physics. Every theory is a work in 1.6 Estimates and progress. Order-of-Magnitude The basic laws of physics involve such physical quantities as force, velocity, volume, and Calculations acceleration, all of which can be described in terms of more fundamental quantities. In me- 1.7 Coordinate Systems chanics, the three most fundamental quantities are length (L), mass (M), and time (T); all 1.8 Trigonometry other physical quantities can be constructed from these three. 1.9 Problem-Solving Strategy 1.1 STANDARDS OF LENGTH, MASS, AND TIME To communicate the result of a measurement of a certain physical quantity, a unit for the quantity must be defined. For example, if our fundamental unit of length is defined to be 1.0 meter, and someone familiar with our system of measurement re- ports that a wall is 2.0 meters high, we know that the height of the wall is twice the fundamental unit of length. Likewise, if our fundamental unit of mass is defined as Throughout 1.0 kilogram, and we are told that a person has a mass of 75 kilograms, then that the text, the PhysicsNow icon indi- person has a mass 75 times as great as the fundamental unit of mass. cates an opportunity for you to test In 1960, an international committee agreed on a standard system of units for yourself on key concepts and to explore animations and interactions the fundamental quantities of science, called SI (Système International). Its units on the PhysicsNow website at of length, mass, and time are the meter, kilogram, and second, respectively. www.cp7e.com. 1 Stone/Getty Images 44337_01_p1-22 10/13/04 2:13 PM Page 2 2 Chapter 1 Introduction Length In 1799, the legal standard of length in France became the meter, defined as one ten- millionth of the distance from the equator to the North Pole. Until 1960, the official length of the meter was the distance between two lines on a specific bar of platinum- iridium alloy stored under controlled conditions. This standard was abandoned for several reasons, the principal one being that measurements of the separation be- tween the lines are not precise enough. In 1960, the meter was defined as 1 650 763.73 wavelengths of orange-red light emitted from a krypton-86 lamp. In October Definition of the meter 䊳 1983, this definition was abandoned also, and the meter was redefined as the distance traveled by light in vacuum during a time interval of 1/299 792 458 second. This lat- est definition establishes the speed of light at 299 792 458 meters per second. Mass Definition of the kilogram 䊳 The SI unit of mass, the kilogram, is defined as the mass of a specific platinum- iridium alloy cylinder kept at the International Bureau of Weights and Measures at Sèvres, France (similar to that shown in Figure 1.1a). As we’ll see in Chapter 4, mass is a quantity used to measure the resistance to a change in the motion of an object. It’s more difficult to cause a change in the motion of an object with a large mass than an object with a small mass. Time Before 1960, the time standard was defined in terms of the average length of a so- lar day in the year 1900. (A solar day is the time between successive appearances of the Sun at the highest point it reaches in the sky each day.) The basic unit of time, the second, was defined to be (1/60)(1/60)(1/24) ⫽ 1/86 400 of the average so- lar day. In 1967, the second was redefined to take advantage of the high precision attainable with an atomic clock, which uses the characteristic frequency of the Definition of the second 䊳 light emitted from the cesium-133 atom as its “reference clock.” The second is now defined as 9 192 631 700 times the period of oscillation of radiation from the cesium atom. The newest type of cesium atomic clock is shown in Figure 1.1b. (a) (b) Figure 1.1 (a) The National Standard Kilogram No. 20, an accurate copy of the International Stan- dard Kilogram kept at Sèvres, France, is housed under a double bell jar in a vault at the National Insti- tute of Standards and Technology. (b) The nation’s primary time standard is a cesium fountain atomic clock developed at the National Institute of Standards and Technology laboratories in Boulder, Colorado. This clock will neither gain nor lose a second in 20 million years. Courtesy of National Institute of Standards and Technology, U.S. Dept. of Commerce 44337_01_p1-22 10/13/04 2:13 PM Page 3 1.1 Standards of Length, Mass, and Time 3 TABLE 1.1 Approximate Values of Some Measured Lengths Length (m) 26 Distance from Earth to most remote known quasar 1 ⫻ 10 25 Distance from Earth to most remote known normal galaxies 4 ⫻ 10 22 Distance from Earth to nearest large galaxy (M31, the Andromeda galaxy) 2 ⫻ 10 16 Distance from Earth to nearest star (Proxima Centauri) 4 ⫻ 10 15 One light year 9 ⫻ 10 11 Mean orbit radius of Earth about Sun 2 ⫻ 10 8 Mean distance from Earth to Moon 4 ⫻ 10 6 Mean radius of Earth 6 ⫻ 10 5 Typical altitude of satellite orbiting Earth 2 ⫻ 10 1 Length of football field 9 ⫻ 10 ⫺3 Length of housefly 5 ⫻ 10 ⫺4 Size of smallest dust particles 1 ⫻ 10 ⫺5 Size of cells in most living organisms 1 ⫻ 10 ⫺10 Diameter of hydrogen atom 1 ⫻ 10 ⫺14 Diameter of atomic nucleus 1 ⫻ 10 ⫺15 Diameter of proton 1 ⫻ 10 TABLE 1.2 Approximate Values for Length, Mass, and Time Intervals Approximate Values of Some Approximate values of some lengths, masses, and time intervals are presented in Masses Tables 1.1, 1.2, and 1.3, respectively. Note the wide ranges of values. Study these Mass (kg) tables to get a feel for a kilogram of mass (this book has a mass of about 52 Observable 1 ⫻ 10 10 9 2 kilograms), a time interval of 10 seconds (one century is about 3 ⫻ 10 seconds), Universe or two meters of length (the approximate height of a forward on a basketball 41 Milky Way galaxy 7 ⫻ 10 team). Appendix A reviews the notation for powers of 10, such as the expression of 30 4 Sun 2 ⫻ 10 the number 50 000 in the form 5 ⫻ 10 . 24 Earth 6 ⫻ 10 Systems of units commonly used in physics are the Système International, in 22 Moon 7 ⫻ 10 which the units of length, mass, and time are the meter (m), kilogram (kg), and sec- 2 Shark 1 ⫻ 10 ond (s); the cgs, or Gaussian, system, in which the units of length, mass, and time 1 are the centimeter (cm), gram (g), and second; and the U.S. customary system, in Human 7 ⫻ 10 ⫺1 which the units of length, mass, and time are the foot (ft), slug, and second. SI units Frog 1 ⫻ 10 ⫺5 are almost universally accepted in science and industry, and will be used throughout Mosquito 1 ⫻ 10 the book. Limited use will be made of Gaussian and U.S. customary units. Bacterium 1 ⫻ 10⫺15 ⫺27 Hydrogen atom 2 ⫻ 10 ⫺31 Electron 9 ⫻ 10 TABLE 1.3 Approximate Values of Some Time Intervals Time Interval (s) 17 Age of Universe 5 ⫻ 10 17 Age of Earth 1 ⫻ 10 8 Average age of college student 6 ⫻ 10 7 One year 3 ⫻ 10 4 One day (time required for one revolution of Earth about its axis) 9 ⫻ 10 ⫺1 Time between normal heartbeats 8 ⫻ 10 a ⫺3 Period of audible sound waves 1 ⫻ 10 a ⫺6 Period of typical radio waves 1 ⫻ 10 a ⫺13 Period of vibration of atom in solid 1 ⫻ 10 a ⫺15 Period of visible light waves 2 ⫻ 10 ⫺22 Duration of nuclear collision 1 ⫻ 10 ⫺24 Time required for light to travel across a proton 3 ⫻ 10 a A period is defined as the time required for one complete vibration. 44337_01_p1-22 10/13/04 2:13 PM Page 4 4 Chapter 1 Introduction TABLE 1.4 Some of the most frequently used “metric” (SI and cgs) prefixes representing ⫺3 powers of 10 and their abbreviations are listed in Table 1.4. For example, 10 m is Some Prefixes for Powers of 3 equivalent to 1 millimeter (mm), and 10 m is 1 kilometer (km). Likewise, 1 kg is Ten Used with “Metric” 3 6 equal to 10 g, and 1 megavolt (MV) is 10 volts (V). (SI and cgs) Units Power Prefix Abbreviation ⫺18 10 atto- a 10⫺15 femto- f 1.2 THE BUILDING BLOCKS OF MATTER ⫺12 10 pico- p A 1-kg (⬇ 2-lb) cube of solid gold has a length of about 3.73 cm (⬇ 1.5 in.) on a ⫺9 10 nano- n side. Is this cube nothing but wall-to-wall gold, with no empty space? If the cube is 10⫺6 micro- ␮ cut in half, the two resulting pieces retain their chemical identity as solid gold. But 10⫺3 milli- m what if the pieces of the cube are cut again and again, indefinitely? Will the 10⫺2 centi- c smaller and smaller pieces always be the same substance, gold? Questions such as ⫺1 these can be traced back to early Greek philosophers. Two of them—Leucippus 10 deci- d 1 and Democritus—couldn’t accept the idea that such cutting could go on forever. 10 deka- da 3 They speculated that the process ultimately would end when it produced a particle 10 kilo- k that could no longer be cut. In Greek, atomos means “not sliceable.” From this 6 10 mega- M term comes our English word atom, once believed to be the smallest, ultimate par- 9 10 giga- G ticle of matter, but since found to be a composite of more elementary particles. 12 10 tera- T The atom can be visualized as a miniature Solar System, with a dense, positively 15 10 peta- P charged nucleus occupying the position of the Sun, with negatively charged elec- 18 10 exa- E trons orbiting like planets. This model of the atom, first developed by the great Danish physicist Niels Bohr nearly a century ago, led to the understanding of cer- tain properties of the simpler atoms such as hydrogen, but failed to explain many fine details of atomic structure. Notice the size of a hydrogen atom, listed in Table 1.1, and the size of a proton— the nucleus of a hydrogen atom—one hundred thousand times smaller. If the pro- ton were the size of a Ping Pong ball, the electron would be a tiny speck about the size of a bacterium, orbiting the proton a kilometer away! Other atoms are similarly constructed. So there is a surprising amount of empty space in ordinary matter. After the discovery of the nucleus in the early 1900s, questions arose concerning its structure. Is the nucleus a single particle or a collection of particles? The exact composition of the nucleus hasn’t been defined completely even today, but by the early 1930s a model evolved that helped us understand how the nucleus behaves. Scientists determined that two basic entities—protons and neutrons—occupy the nucleus. The proton is nature’s fundamental carrier of positive charge (equal in magnitude but opposite in sign to the charge on the electron), and the number of protons in a nucleus determines what the element is. For instance, a nucleus con- taining only one proton is the nucleus of an atom of hydrogen, regardless of how many neutrons may be present. Extra neutrons correspond to different isotopes of hydrogen—deuterium and tritium—which react chemically in exactly the same way as hydrogen, but are more massive. An atom having two protons in its nucleus, simi- larly, is always helium, although again, differing numbers of neutrons are possible. The existence of neutrons was verified conclusively in 1932. A neutron has no charge and has a mass about equal to that of a proton. One of its primary pur- poses is to act as a “glue” to hold the nucleus together. If neutrons were not pres- ent, the repulsive electrical force between the positively charged protons would cause the nucleus to fly apart. The division doesn’t stop here; it turns out that protons, neutrons, and a zoo of other exotic particles are now thought to be composed of six particles called quarks (rhymes with “forks,” though some rhyme it with “sharks”). These particles have been given the names up, down, strange, charm, bottom, and top. The up, charm, 2 and top quarks each carry a charge equal to ⫹ that of the proton, whereas the 3 1 down, strange, and bottom quarks each carry a charge equal to ⫺ the proton 3 charge. The proton consists of two up quarks and one down quark (see Fig. 1.2), giving the correct charge for the proton, ⫹ 1. The neutron is composed of two down quarks and one up quark and has a net charge of zero. The up and down quarks are sufficient to describe all normal matter, so the existence of the other four quarks indirectly observed in high-energy experiments, 44337_01_p1-22 10/13/04 2:13 PM Page 5 1.3 Dimensional Analysis 5 Quark composition of a proton Figure 1.2 Levels of organization in matter. Ordinary matter consists of u u atoms, and at the center of each atom Neutron is a compact nucleus consisting of protons and neutrons. Protons and neutrons are composed of quarks. d The quark composition of a proton is Gold shown. nucleus Nucleus Proton Gold cube Gold atoms is something of a mystery. It’s also possible that quarks themselves have internal structure. Many physicists believe that the most fundamental particles may be tiny loops of vibrating string. 1.3 DIMENSIONAL ANALYSIS In physics, the word dimension denotes the physical nature of a quantity. The distance between two points, for example, can be measured in feet, meters, or furlongs, which are different ways of expressing the dimension of length. The symbols that we use in this section to specify the dimensions of length, mass, and time are L, M, and T, respectively. Brackets [ ] will often be used to denote the dimensions of a physical quantity. For example, in this notation the dimensions of 2 velocity v are written [v] ⫽ L/T, and the dimensions of area A are [A] ⫽ L . The dimensions of area, volume, velocity, and acceleration are listed in Table 1.5, along with their units in the three common systems. The dimensions of other quantities, such as force and energy, will be described later as they are introduced. In physics, it’s often necessary either to derive a mathematical expression or equation or to check its correctness. A useful procedure for doing this is called di- mensional analysis, which makes use of the fact that dimensions can be treated as algebraic quantities. Such quantities can be added or subtracted only if they have the same dimensions. It follows that the terms on the opposite sides of an equation must have the same dimensions. If they don’t, the equation is wrong. If they do, the equation is probably correct, except for a possible constant factor. To illustrate this procedure, suppose we wish to derive a formula for the distance x traveled by a car in a time t if the car starts from rest and moves with constant ac- celeration a. The quantity x has the dimension length: [x] ⫽ L. Time t, of course, has dimension [t] ⫽ T. Acceleration is the change in velocity v with time. Since v has dimensions of length per unit time, or [v] ⫽ L/T, acceleration must have dimen- 2 sions [a] ⫽ L/T . We organize this information in the form of an equation: [v] L/T L [x] [a] ⫽ ⫽ ⫽ ⫽ 2 2 [t] T T [t] TABLE 1.5 Dimensions and Some Units of Area, Volume, Velocity, and Acceleration 2 3 2 System Area (L ) Volume (L ) Velocity (L/T) Acceleration (L/T ) 2 3 2 SI m m m/s m/s 2 3 2 cgs cm cm cm/s cm/s 2 3 2 U.S. customary ft ft ft/s ft/s 44337_01_p1-22 10/13/04 2:13 PM Page 6 6 Chapter 1 Introduction Looking at the left- and right-hand sides of this equation, we might now guess that a ⫽ tx2 : x ⫽ at 2 This is not quite correct, however, because there’s a constant of proportionality— a simple numerical factor—that can’t be determined solely through dimensional analysis. As will be seen in Chapter 2, it turns out that the correction expression is x ⫽ 12at 2. When we work algebraically with physical quantities, dimensional analysis allows us to check for errors in calculation, which often show up as discrepancies in units. If, for example, the left-hand side of an equation is in meters and the right-hand side is in meters per second, we know immediately that we’ve made an error. EXAMPLE 1.1 Analysis of an Equation Goal Check an equation using dimensional analysis. Problem Show that the expression v ⫽ v0 ⫹ at, is dimensionally correct, where v and v0 represent velocities, a is acceleration, and t is a time interval. Strategy Analyze each term, finding its dimensions, and then check to see if all the terms agree with each other. Solution L Find dimensions for v and v0. [v] ⫽ [v0] ⫽ T L L Find the dimensions of at. [at] ⫽ T 2 (T) ⫽ T Remarks All the terms agree, so the equation is dimensionally correct. Exercise 1.1 Determine whether the equation x ⫽ vt2 is dimensionally correct. If not, provide a correct expression, up to an over- all constant of proportionality. Answer Incorrect. The expression x ⫽ vt is dimensionally correct. EXAMPLE 1.2 Find an Equation Goal Derive an equation by using dimensional analysis. Problem Find a relationship between a constant acceleration a, speed v, and distance r from the origin for a parti- cle traveling in a circle. Strategy Start with the term having the most dimensionality, a. Find its dimensions, and then rewrite those dimen- sions in terms of the dimensions of v and r. The dimensions of time will have to be eliminated with v, since that’s the only quantity in which the dimension of time appears. Solution L Write down the dimensions of a: [a] ⫽ T2 L L Solve the dimensions of speed for T: [v] ⫽ : T ⫽ T [v] L L [v]2 Substitute the expression for T into the equation for [a] ⫽ T2 ⫽ (L/[v])2 ⫽ L [a]: 44337_01_p1-22 10/13/04 2:13 PM Page 7 1.4 Uncertainty In Measurement and Significant Figures 7 [v]2 v2 Substitute L ⫽ [r], and guess at the equation: [a] ⫽ : a ⫽ [r] r Remarks This is the correct equation for centripetal acceleration—acceleration towards the center of motion—to be discussed in Chapter 7. There isn’t any need in this case, to introduce a numerical factor. Such a factor is often displayed explicitly as a constant k in front of the right-hand side—for example, a ⫽ kv2/r. As it turns out, k ⫽ 1 gives the correct expression. Exercise 1.2 In physics, energy E carries dimensions of mass times length squared, divided by time squared. Use dimensional analysis to derive a relationship for energy in terms of mass m and speed v, up to a constant of proportionality. Set the speed equal to c, the speed of light, and the constant of proportionality equal to 1 to get the most famous equa- tion in physics. Answer E ⫽ kmv2 : E ⫽ mc2 when k ⫽ 1 and v ⫽ c. 1.4 UNCERTAINTY IN MEASUREMENT AND SIGNIFICANT FIGURES Physics is a science in which mathematical laws are tested by experiment. No physi- cal quantity can be determined with complete accuracy, because our senses are physically limited, even when extended with microscopes, cyclotrons, and other gadgets. Knowing the experimental uncertainties in any measurement is very important. Without this information, little can be said about the final measurement. Using a crude scale, for example, we might find that a gold nugget has a mass of 3 kilo- grams. A prospective client interested in purchasing the nugget would naturally want to know about the accuracy of the measurement, to ensure paying a fair price. He wouldn’t be happy to find that the measurement was good only to within a kilogram, because he might pay for three kilograms and get only two. Of course, he might get four kilograms for the price of three, but most people would be hesi- tant to gamble that an error would turn out in their favor. Accuracy of measurement depends on the sensitivity of the apparatus, the skill of the person carrying out the measurement, and the number of times the mea- surement is repeated. There are many ways of handling uncertainties, and here we’ll develop a basic and reliable method of keeping track of them in the measure- ment itself and in subsequent calculations. Suppose that in a laboratory experiment we measure the area of a rectangular plate with a meter stick. Let’s assume that the accuracy to which we can measure a particular dimension of the plate is ⫾ 0.1 cm. If the length of the plate is mea- sured to be 16.3 cm, we can claim only that it lies somewhere between 16.2 cm and 16.4 cm. In this case, we say that the measured value has three significant fig- ures. Likewise, if the plate’s width is measured to be 4.5 cm, the actual value lies between 4.4 cm and 4.6 cm. This measured value has only two significant figures. We could write the measured values as 16.3 ⫾ 0.1 cm and 4.5 ⫾ 0.1 cm. In gen- eral, a significant figure is a reliably known digit (other than a zero used to locate a decimal point). Suppose we would like to find the area of the plate by multiplying the two mea- sured values together. The final value can range between (16.3 ⫺ 0.1 cm)(4.5 ⫺ 0.1 cm) ⫽ (16.2 cm)(4.4 cm) ⫽ 71.28 cm2 and (16.3 ⫹ 0.1 cm)(4.5 ⫹ 0.1 cm) ⫽ (16.4 cm)(4.6 cm) ⫽ 75.44 cm2. Claiming to know anything about the hundredths place, or even the tenths place, doesn’t make any sense, because it’s clear we can’t even be certain of the units place, whether it’s the 1 in 71, the 5 in 75, or somewhere in between. The tenths and the hundredths places are clearly not significant. We have some information about the units place, so that number is 44337_01_p1-22 10/13/04 2:13 PM Page 8 8 Chapter 1 Introduction significant. Multiplying the numbers at the middle of the uncertainty ranges gives 2 (16.3 cm)(4.5 cm) ⫽ 73.35 cm , which is also in the middle of the area’s uncer- tainty range. Since the hundredths and tenths are not significant, we drop them 2 2 and take the answer to be 73 cm , with an uncertainty of ⫾ 2 cm . Note that the answer has two significant figures, the same number of figures as the least accu- rately known quantity being multiplied, the 4.5-cm width. There are two useful rules of thumb for determining the number of significant figures. The first, concerning multiplication and division, is as follows: In multiply- ing (dividing) two or more quantities, the number of significant figures in the final product (quotient) is the same as the number of significant figures in the least accurate of the factors being combined, where least accurate means having the lowest number of significant figures. To get the final number of significant figures, it’s usually necessary to do some rounding. If the last digit dropped is less than 5, simply drop the digit. If the last digit dropped is greater than or equal to 5, raise the last retained digit by one. EXAMPLE 1.3 Installing a Carpet Goal Apply the multiplication rule for significant figures. Problem A carpet is to be installed in a room of length 12.71 m and width 3.46 m. Find the area of the room, retaining the proper number of significant figures. Strategy Count the significant figures in each number. The smaller result is the number of significant figures in the answer. Solution Count significant figures: 12.71 m : 4 significant figures 3.46 m : 3 significant figures 2 2 Multiply the numbers, keeping only three digits: 12.71 m ⫻ 3.46 m ⫽ 43.9766 m : 44.0 m Remarks In reducing 43.976 6 to three significant figures, we used our rounding rule, adding 1 to the 9, which made 10 and resulted in carrying 1 to the unit’s place. Exercise 1.3 Repeat this problem, but with a room measuring 9.72 m long by 5.3 m wide. 2 Answer 52 m Zeros may or may not be significant figures. Zeros used to position the decimal TIP 1.1 Using Calculators point in such numbers as 0.03 and 0.007 5 are not significant (but are useful in Calculators were designed by engi- avoiding errors). Hence, 0.03 has one significant figure, and 0.007 5 has two. neers to yield as many digits as the When zeros are placed after other digits in a whole number, there is a possibility memory of the calculator chip per- of misinterpretation. For example, suppose the mass of an object is given as 1 500 g. mitted, so be sure to round the final answer down to the correct number This value is ambiguous, because we don’t know whether the last two zeros are be- of significant figures. ing used to locate the decimal point or whether they represent significant figures in the measurement. Using scientific notation to indicate the number of significant figures removes 3 this ambiguity. In this case, we express the mass as 1.5 ⫻ 10 g if there are two sig- 3 nificant figures in the measured value, 1.50 ⫻ 10 g if there are three significant 3 figures, and 1.500 ⫻ 10 g if there are four. Likewise, 0.000 15 is expressed in sci- ⫺4 ⫺4 entific notation as 1.5 ⫻ 10 if it has two significant figures or as 1.50 ⫻ 10 if it has three significant figures. The three zeros between the decimal point and the 44337_01_p1-22 10/13/04 2:13 PM Page 9 1.5 Conversion of Units 9 digit 1 in the number 0.000 15 are not counted as significant figures because they only locate the decimal point. In this book, most of the numerical ex- amples and end-of-chapter problems will yield answers having two or three signifi- cant figures. For addition and subtraction, it’s best to focus on the number of decimal places in the quantities involved rather than on the number of significant figures. When TIP 1.2 No Commas in numbers are added (subtracted), the number of decimal places in the result Numbers with Many Digits should equal the smallest number of decimal places of any term in the sum In science, numbers with more than three digits are written in groups of (difference). For example, if we wish to compute 123 (zero decimal places) three digits separated by spaces ⫹ 5.35 (two decimal places), the answer is 128 (zero decimal places) and not rather than commas; so that 10 000 128.35. If we compute the sum 1.000 1 (four decimal places) ⫹ 0.000 3 (four deci- is the same as the common American notation 10,000. Similarly, mal places) ⫽ 1.000 4, the result has the correct number of decimal places, ␲ ⫽ 3.14159265 is written as namely four. Observe that the rules for multiplying significant figures don’t work 3.141 592 65. here because the answer has five significant figures even though one of the terms in the sum, 0.000 3, has only one significant figure. Likewise, if we perform the subtraction 1.002 ⫺ 0.998 ⫽ 0.004, the result has three decimal places because each term in the subtraction has three decimal places. To show why this rule should hold, we return to the first example in which we added 123 and 5.35, and rewrite these numbers as 123.xxx and 5.35x. Digits writ- ten with an x are completely unknown and can be any digit from 0 to 9. Now we line up 123.xxx and 5.35x relative to the decimal point and perform the addition, using the rule that an unknown digit added to a known or unknown digit yields an unknown: 123.xxx ⫹ 5.35x 128.xxx The answer of 128.xxx means that we are justified only in keeping the number 128 because everything after the decimal point in the sum is actually unknown. The example shows that the controlling uncertainty is introduced into an addition or subtraction by the term with the smallest number of decimal places. In performing any calculation, especially one involving a number of steps, there will always be slight discrepancies introduced by both the rounding process and the algebraic order in which steps are carried out. For example, consider 2.35 ⫻ 5.89/1.57. This computation can be performed in three different orders. First, we have 2.35 ⫻ 5.89 ⫽ 13.842, which rounds to 13.8, followed by 13.8/1.57 ⫽ 8.789 8, rounding to 8.79. Second, 5.89/1.57 ⫽ 3.751 6, which rounds to 3.75, re- sulting in 2.35 ⫻ 3.75 ⫽ 8.812 5, rounding to 8.81. Finally, 2.35/1.57 ⫽ 1.496 8 rounds to 1.50, and 1.50 ⫻ 5.89 ⫽ 8.835 rounds to 8.84. So three different alge- braic orders, following the rules of rounding, lead to answers of 8.79, 8.81, and 8.84, respectively. Such minor discrepancies are to be expected, because the last significant digit is only one representative from a range of possible values, depend- ing on experimental uncertainty. The discrepancies can be reduced by carrying one or more extra digits during the calculation. In our examples, however, inter- mediate results will be rounded off to the proper number of significant figures, and only those digits will be carried forward. In experimental work, more sophisti- cated techniques are used to determine the accuracy of an experimental result. 1.5 CONVERSION OF UNITS Sometimes it’s necessary to convert units from one system to another. Conversion factors between the SI and U.S. customary systems for 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 A more extensive list of conversion factors can be found on the inside front cover of this book.