物理学基础_-_Fundamentals_of_Physics_by_Halliday,_Resnick,_Walker_1_Physics_and_Measurement

2 c h a p t e r Physics and Measurement For thousands of years the spinning Earth provided a natural standard for our measurements of time. However, since 1972 we have added more than 20 “leap seconds” to our clocks to keep them synchronized to the Earth. Why are such adjustments needed? What does it take to be a good standard? (Don Mason/The Stock Market and NASA) 1.1 Standards of Length, Mass, and Time 1.2 The Building Blocks of Matter 1.3 Density 1.4 Dimensional Analysis 1.5 Conversion of Units 1.6 Estimates and Order-of-Magnitude Calculations 1.7 Significant Figures C h a p t e r O u t l i n e P U Z Z L E RP U Z Z L E R 3 ike all other sciences, physics is based on experimental observations and quan- titative measurements. The main objective of physics is to find the limited num- ber of fundamental laws that govern natural phenomena and to use them to develop theories that can predict the results of future experiments. The funda- mental laws used in developing theories are expressed in the language of mathe- matics, 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 limitations. For example, the laws of motion discovered by Isaac Newton (1642–1727) in the 17th century accurately describe the motion of bodies at nor- mal 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 Ein- stein (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 is a more general theory of motion. Classical physics, which means all of the physics developed before 1900, in- cludes the theories, concepts, laws, and experiments in classical mechanics, ther- modynamics, and electromagnetism. Important contributions to classical physics were provided by Newton, who de- veloped 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 apparatus for controlled experiments was either too crude or unavailable. A new era 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 important developments in modern physics were the theories of relativity and quantum mechanics. Einstein’s theory of relativity revolutionized the tradi- tional concepts of space, time, and energy; quantum mechanics, which applies to both the microscopic and macroscopic worlds, was originally formulated by a num- ber of distinguished scientists to provide descriptions of physical phenomena at the atomic level. Scientists constantly work at improving our understanding of phenomena and fundamental laws, and new discoveries are made every day. In many research areas, a great deal of overlap exists between physics, chemistry, geology, and biology, as well as engineering. Some of the most notable developments are (1) numerous space missions and the landing of astronauts on the Moon, (2) microcircuitry and high-speed computers, and (3) sophisticated imaging tech- niques used in scientific research and medicine. The impact such developments and discoveries have had on our society has indeed been great, and it is very likely that future discoveries and developments will be just as exciting and challenging and of great benefit to humanity. STANDARDS OF LENGTH, MASS, AND TIME The laws of physics are expressed in terms of basic quantities that require a clear def- inition. In mechanics, the three basic quantities are length (L), mass (M), and time (T). All other quantities in mechanics can be expressed in terms of these three. 1.1 L 4 C H A P T E R 1 Physics and Measurements If we are to report the results of a measurement to someone who wishes to re- produce this measurement, a standard must be defined. It would be meaningless if a visitor from another planet were to talk to us about a length of 8 “glitches” if we do not know the meaning of the unit glitch. On the other hand, if someone famil- iar 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. Likewise, if we are told that a person has a mass of 75 kilo- grams and our unit of mass is defined to be 1 kilogram, then that person is 75 times as massive as our basic unit.1 Whatever is chosen as a standard must be read- ily accessible and possess some property that can be measured reliably—measure- ments taken by different people in different places must yield the same result. In 1960, an international committee established a set of standards for length, mass, and other basic quantities. The system established is an adaptation of the metric system, and it is called the SI system of units. (The abbreviation SI comes from the system’s French name “Système International.”) In this system, the units of length, mass, and time are the meter, kilogram, and second, respectively. Other SI standards established by the committee are those for temperature (the kelvin), electric current (the ampere), luminous intensity (the candela), and the amount of substance (the mole). In our study of mechanics we shall be concerned only with the units of length, mass, and time. Length In A.D. 1120 the king of England decreed that the standard of length in his coun- try would be named the yard and would be precisely equal to the distance from the tip of his nose to the end of his outstretched arm. Similarly, the original standard for the foot adopted by the French was the length of the royal foot of King Louis XIV. This standard prevailed until 1799, when the legal standard of length in France became the meter, defined as one ten-millionth the distance from the equa- tor 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 countries and in scientific circles everywhere. As recently as 1960, the length of the meter was defined as the distance between two lines on a specific platinum– iridium bar stored under controlled conditions in France. This standard was aban- doned for several reasons, a principal one being that the limited accuracy with which the separation between the lines on the bar can be determined does not meet the current requirements of science and technology. In the 1960s and 1970s, the meter was defined as 1 650 763.73 wavelengths of orange-red light emitted from a krypton-86 lamp. However, in October 1983, the meter (m) was redefined as the distance traveled by light in vacuum during a time of 1/299 792 458 second. In effect, this latest definition establishes that the speed of light in vac- uum is precisely 299 792 458 m per second. Table 1.1 lists approximate values of some measured lengths. 1 The need for assigning numerical values to various measured physical quantities was expressed by Lord Kelvin (William Thomson) as follows: “I often say that when you can measure what you are speak- ing about, and express it in numbers, you should know something about it, but when you cannot ex- press it in numbers, your knowledge is of a meagre and unsatisfactory kind. It may be the beginning of knowledge but you have scarcely in your thoughts advanced to the state of science.” 1.1 Standards of Length, Mass, and Time 5 Mass The basic SI unit of mass, the kilogram (kg), is defined as the mass of a spe- cific 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 alloy (Fig. 1.1a). A duplicate of the Sèvres cylinder is kept at the National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland. 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 1900.2 The mean solar second was originally defined as of a mean solar day. The rotation of the Earth is now known to vary slightly with time, how- ever, and therefore this motion is not a good one to use for defining a standard. In 1967, consequently, the second was redefined to take advantage of the high precision obtainable in a device known as an atomic clock (Fig. 1.1b). In this device, the frequencies associated with certain atomic transitions can be measured to a precision of one part in 1012. This is equivalent to an uncertainty of less than one second every 30 000 years. Thus, in 1967 the SI unit of time, the second, was rede- fined using the characteristic frequency of a particular kind of cesium atom as the “reference clock.” The basic SI unit of time, the second (s), is defined as 9 192 631 770 times the period of vibration of radiation from the cesium-133 atom.3 To keep these atomic clocks—and therefore all common clocks and ( 160)( 1 60)( 1 24) TABLE 1.1 Approximate Values of Some Measured Lengths Length (m) Distance from the Earth to most remote known quasar 1.4 � 1026 Distance from the Earth to most remote known normal galaxies 9 � 1025 Distance from the Earth to nearest large galaxy (M 31, the Andromeda galaxy) 2 � 1022 Distance from the Sun to nearest star (Proxima Centauri) 4 � 1016 One lightyear 9.46 � 1015 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 Mean radius of the Earth 6.37 � 106 Typical altitude (above the surface) of a satellite orbiting the Earth 2 � 105 Length of a football field 9.1 � 101 Length of a housefly 5 � 10�3 Size of smallest dust particles �10�4 Size of cells of most living organisms �10�5 Diameter of a hydrogen atom �10�10 Diameter of an atomic nucleus �10�14 Diameter of a proton �10�15 web Visit the Bureau at www.bipm.fr or the National Institute of Standards at www.NIST.gov 2 One solar day is the time interval between successive appearances of the Sun at the highest point it reaches in the sky each day. 3 Period is defined as the time interval needed for one complete vibration. TABLE 1.2 Masses of Various Bodies (Approximate Values) Body Mass (kg) Visible �1052 Universe Milky Way 7 � 1041 galaxy Sun 1.99 � 1030 Earth 5.98 � 1024 Moon 7.36 � 1022 Horse �103 Human �102 Frog �10�1 Mosquito �10�5 Bacterium �10�15 Hydrogen 1.67 � 10�27 atom Electron 9.11 � 10�31 6 C H A P T E R 1 Physics and Measurements watches that are set to them—synchronized, it has sometimes been necessary to add leap seconds to our clocks. This is not a new idea. In 46 B.C. Julius Caesar be- gan the practice of adding extra days to the calendar during leap years so that the seasons occurred at about the same date each year. Since Einstein’s discovery of the linkage between space and time, precise mea- surement of time intervals requires that we know both the state of motion of the clock used to measure the interval and, in some cases, the location of the clock as well. Otherwise, for example, global positioning system satellites might be unable to pinpoint your location with sufficient accuracy, should you need rescuing. Approximate values of time intervals are presented in Table 1.3. In addition to SI, another system of units, the British engineering system (some- times called the conventional system), is still used in the United States despite accep- tance of SI by the rest of the world. In this system, the units of length, mass, and Figure 1.1 (Top) The National Standard Kilogram No. 20, an accurate copy of the International Standard Kilo- gram kept at Sèvres, France, is housed under a double bell jar in a vault at the National Institute of Standards and Technology (NIST). (Bottom) The primary frequency stan- dard (an atomic clock) at the NIST. This device keeps time with an accuracy of about 3 millionths of a second per year. (Courtesy of National Institute of Standards and Technology, U.S. Department of Commerce) 1.1 Standards of Length, Mass, and Time 7 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 British engineering 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 various powers of ten. Some of the most frequently used prefixes for the various powers of ten and their abbreviations are listed in Table 1.4. For TABLE 1.3 Approximate Values of Some Time Intervals Interval (s) Age of the Universe 5 � 1017 Age of the Earth 1.3 � 1017 Average age of a college student 6.3 � 108 One year 3.16 � 107 One day (time for one rotation of the Earth about its axis) 8.64 � 104 Time between normal heartbeats 8 � 10�1 Period of audible sound waves �10�3 Period of typical radio waves �10�6 Period of vibration of an atom in a solid �10�13 Period of visible light waves �10�15 Duration of a nuclear collision �10�22 Time for light to cross a proton �10�24 TABLE 1.4 Prefixes for SI Units Power Prefix Abbreviation 10�24 yocto y 10�21 zepto z 10�18 atto a 10�15 femto f 10�12 pico p 10�9 nano n 10�6 micro � 10�3 milli m 10�2 centi c 10�1 deci d 101 deka da 103 kilo k 106 mega M 109 giga G 1012 tera T 1015 peta P 1018 exa E 1021 zetta Z 1024 yotta Y 8 C H A P T E R 1 Physics and Measurements example, 10�3 m is equivalent to 1 millimeter (mm), and 103 m corresponds to 1 kilometer (km). Likewise, 1 kg is 103 grams (g), and 1 megavolt (MV) is 106 volts (V). THE BUILDING BLOCKS OF MATTER A 1-kg cube of solid gold 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 retain 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? Ques- tions 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 cut- tings could go on forever. They speculated that the process ultimately must end when it produces a particle that can no longer be cut. In Greek, atomos means “not sliceable.” From this comes our English word atom. Let us review briefly what is known about the structure of matter. All ordinary matter consists of atoms, and each atom is made up of electrons surrounding a central nucleus. Following the discovery of the nucleus in 1911, the question arose: Does it have structure? That is, is the nucleus a single particle or a collection of particles? The exact composition of the nucleus is not known completely even today, but by the early 1930s a model evolved that helped us understand how the nucleus behaves. Specifically, scientists determined that occupying the nucleus are two basic entities, protons and neutrons. The proton carries a positive charge, and a specific element is identified by the number of protons in its nucleus. This num- ber is called the atomic number of the element. For instance, the nucleus of a hy- drogen atom contains one proton (and so the atomic number of hydrogen is 1), the nucleus of a helium atom contains two protons (atomic number 2), and the nucleus of a uranium atom contains 92 protons (atomic number 92). In addition to atomic number, there is a second number characterizing atoms—mass num- ber, defined as the number of protons plus neutrons in a nucleus. As we shall see, the atomic number of an element never varies (i.e., the number of protons does not vary) but the mass number can vary (i.e., the number of neutrons varies). Two or more atoms of the same element having different mass numbers are isotopes of one another. The existence of neutrons was verified conclusively in 1932. A neutron has no charge and a mass that is about equal to that of a proton. One of its primary pur- poses 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 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, charm, bottom, and top. The up, charm, and top quarks have charges of � that of 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 (Fig. 1.2), which you can easily show leads to the correct charge for the proton. Likewise, the neutron consists of two down quarks and one up quark, giving a net charge of zero. 1 3 2 3 1.2 Quark composition of a proton u u d Gold nucleus Gold atoms Gold cube Proton Neutron Nucleus Figure 1.2 Levels of organization in matter. Ordinary matter consists of atoms, and at the center of each atom is a compact nucleus consist- ing of protons and neutrons. Pro- tons and neutrons are composed of quarks. The quark composition of a proton is shown. 1.3 Density 9 DENSITY A property of any substance is its density � (Greek letter rho), defined as the amount of mass contained in a unit volume, which we usually express as 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 cm3 has a mass of 27.0 g, whereas an equivalent volume of lead has a mass of 113 g. A list of densities for various substances is given Table 1.5. The difference in density between aluminum and lead is due, in part, to their different atomic masses. The atomic mass of an element is the average mass of one atom in a sample of the element that contains all the element’s isotopes, where the relative amounts of isotopes are the same as the relative amounts found in nature. The unit for atomic mass is the atomic mass unit (u), where 1 u � 1.660 540 2 � 10�27 kg. The atomic mass of lead is 207 u, and that of aluminum is 27.0 u. How- ever, the ratio of atomic masses, 207 u/27.0 u � 7.67, does not correspond to the ratio of densities, (11.3 g/cm3)/(2.70 g/cm3) � 4.19. The discrepancy is due to the difference in atomic separations and atomic arrangements in the crystal struc- ture of these two substances. The mass of a nucleus is measured relative to the mass of the nucleus of the carbon-12 isotope, often written as 12C. (This isotope of carbon has six protons and six neutrons. Other carbon isotopes have six protons but different numbers of neutrons.) Practically all of the mass of an atom is contained within the nucleus. Because the atomic mass of 12C is defined to be exactly 12 u, the proton and neu- tron each have a mass of about 1 u. One mole (mol) of a substance is that amount of the substance that con- tains as many particles (atoms, molecules, or other particles) as there are atoms in 12 g of the carbon-12 isotope. One mole of substance A contains the same num