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P U Z Z L E R 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) c h a p t e r Physics and Measurement Chapter Outline 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 CuuDuongThanCong.com https://fb.com/tailieudientucntt L ike all other sciences, physics is based on experimental observations and quantitative measurements The main objective of physics is to find the limited number of fundamental 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 developing 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 limitations For example, the laws of motion discovered by Isaac Newton (1642 – 1727) in the 17th century accurately describe the motion of bodies at normal speeds but 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 is a more general theory of motion Classical physics, which means all of the physics developed before 1900, includes the theories, concepts, laws, and experiments in classical mechanics, thermodynamics, and electromagnetism Important 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 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 traditional concepts of space, time, and energy; quantum mechanics, which applies to both the microscopic and macroscopic worlds, was originally formulated by a number 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 techniques 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 1.1 STANDARDS OF LENGTH, MASS, AND TIME The laws of physics are expressed in terms of basic quantities that require a clear definition 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 CuuDuongThanCong.com https://fb.com/tailieudientucntt CHAPTER Physics and Measurements If we are to report the results of a measurement to someone who wishes to reproduce 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 “glitches” if we 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 meters high and our unit of length is defined to be 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 kilograms and our unit of mass is defined to be kilogram, then that person is 75 times as massive as our basic unit.1 Whatever is chosen as a standard must be readily accessible and possess some property that can be measured reliably — measurements taken by different people in different places must yield the same result In 1960, an international committee established a set of standards for 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 country would be named the yard and would be precisely equal to the distance from the tip of his nose to the end of his outstretched arm Similarly, the original standard for the foot adopted by the French was the length of the royal foot of King Louis XIV This standard prevailed until 1799, when the legal standard of length in France became the meter, 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 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 abandoned 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 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 vacuum is precisely 299 792 458 m per second Table 1.1 lists approximate values of some measured lengths The need for assigning numerical values to various measured physical quantities was expressed by Lord Kelvin (William Thomson) as follows: “I often say that when you can measure what you are speaking about, and express it in numbers, you should know something about it, but when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind It may be the beginning of knowledge but you have scarcely in your thoughts advanced to the state of science.” CuuDuongThanCong.com https://fb.com/tailieudientucntt 1.1 Standards of Length, Mass, and Time TABLE 1.1 Approximate Values of Some Measured Lengths Length (m) Distance from the Earth to most remote known quasar Distance from the Earth to most remote known normal galaxies Distance from the Earth to nearest large galaxy (M 31, the Andromeda galaxy) Distance from the Sun to nearest star (Proxima Centauri) One lightyear Mean orbit radius of the Earth about the Sun Mean distance from the Earth to the Moon Distance from the equator to the North Pole Mean radius of the Earth Typical altitude (above the surface) of a satellite orbiting the Earth Length of a football field Length of a housefly Size of smallest dust particles Size of cells of most living organisms Diameter of a hydrogen atom Diameter of an atomic nucleus Diameter of a proton 1.4 ϫ 1026 ϫ 1025 ϫ 1022 ϫ 1016 9.46 ϫ 1015 1.50 ϫ 1011 3.84 ϫ 108 1.00 ϫ 107 6.37 ϫ 106 ϫ 105 9.1 ϫ 101 ϫ 10Ϫ3 ϳ 10Ϫ4 ϳ 10Ϫ5 ϳ 10Ϫ10 ϳ 10Ϫ14 ϳ 10Ϫ15 Mass The basic 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 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 web Visit the Bureau at www.bipm.fr or the National Institute of Standards at www.NIST.gov TABLE 1.2 Time Before 1960, the standard of time was defined in terms of the mean solar day for the 1 )(60 )(24 ) of a mean year 1900.2 The mean solar second was originally defined as (60 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 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 redefined 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 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 One solar day is the time interval between successive appearances of the Sun at the highest point it reaches in the sky each day Masses of Various Bodies (Approximate Values) Body Visible Universe Milky Way galaxy Sun Earth Moon Horse Human Frog Mosquito Bacterium Hydrogen atom Electron Period is defined as the time interval needed for one complete vibration CuuDuongThanCong.com https://fb.com/tailieudientucntt Mass (kg) ϳ 1052 ϫ 1041 1.99 ϫ 1030 5.98 ϫ 1024 7.36 ϫ 1022 ϳ 103 ϳ 102 ϳ 10Ϫ1 ϳ 10Ϫ5 ϳ 10Ϫ15 1.67 ϫ 10Ϫ27 9.11 ϫ 10Ϫ31 CHAPTER Physics and Measurements Figure 1.1 (Top) 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 (NIST) (Bottom) The primary frequency standard (an atomic clock) at the NIST This device keeps time with an accuracy of about millionths of a second per year (Courtesy of National Institute of Standards and Technology, U.S Department of Commerce) 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 began 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 measurement 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 (sometimes called the conventional 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 CuuDuongThanCong.com https://fb.com/tailieudientucntt 1.1 Standards of Length, Mass, and Time TABLE 1.3 Approximate Values of Some Time Intervals Interval (s) Age of the Universe Age of the Earth Average age of a college student One year One day (time for one rotation of the Earth about its axis) Time between normal heartbeats Period of audible sound waves Period of typical radio waves Period of vibration of an atom in a solid Period of visible light waves Duration of a nuclear collision Time for light to cross a proton ϫ 1017 1.3 ϫ 1017 6.3 ϫ 108 3.16 ϫ 107 8.64 ϫ 104 ϫ 10Ϫ1 ϳ 10Ϫ3 ϳ 10Ϫ6 ϳ 10Ϫ13 ϳ 10Ϫ15 ϳ 10Ϫ22 ϳ 10Ϫ24 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 1.4 Prefixes for SI Units Power Prefix Abbreviation 10Ϫ24 10Ϫ21 10Ϫ18 10Ϫ15 10Ϫ12 10Ϫ9 10Ϫ6 10Ϫ3 10Ϫ2 10Ϫ1 101 103 106 109 1012 1015 1018 1021 1024 yocto zepto atto femto pico nano micro milli centi deci deka kilo mega giga tera peta exa zetta yotta y z a f p n m c d da k M G T P E Z Y CuuDuongThanCong.com https://fb.com/tailieudientucntt CHAPTER Physics and Measurements example, 10Ϫ3 m is equivalent to millimeter (mm), and 103 m corresponds to kilometer (km) Likewise, kg is 103 grams (g), and megavolt (MV) is 106 volts (V) u u 1.2 d Quark composition of a proton Proton Neutron Gold nucleus Nucleus Gold atoms Gold cube Figure 1.2 Levels of organization in matter Ordinary matter consists of atoms, and at the center of each atom is a compact nucleus consisting of protons and neutrons Protons and neutrons are composed of quarks The quark composition of a proton is shown CuuDuongThanCong.com 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? 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 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 number is called the atomic number of the element For instance, the nucleus of a hydrogen atom contains one proton (and so the atomic number of hydrogen is 1), the nucleus of a helium atom contains two protons (atomic number 2), and the nucleus of a uranium atom contains 92 protons (atomic number 92) In addition to atomic number, there is a second number characterizing atoms — mass number, 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 purposes 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 ϩ 32 that of the proton, whereas the down, strange, and bottom quarks have charges of Ϫ 13 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 https://fb.com/tailieudientucntt 1.3 1.3 Density 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: ϵ m V A table of the letters in the Greek alphabet is provided on the back endsheet of this textbook (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 u ϭ 1.660 540 ϫ 10Ϫ27 kg The atomic mass of lead is 207 u, and that of aluminum is 27.0 u However, the ratio of atomic masses, 207 u/27.0 u ϭ 7.67, does not correspond to the ratio of densities, (11.3 g/cm3)/(2.70 g/cm3) ϭ 4.19 The discrepancy is due to the difference in atomic separations and atomic arrangements in the crystal structure 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 neutron each have a mass of about u One mole (mol) of a substance is that amount of the substance that contains 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 number of particles as there are in mol of any other substance B For example, mol of aluminum contains the same number of atoms as mol of lead TABLE 1.5 Densities of Various Substances Substance Gold Uranium Lead Copper Iron Aluminum Magnesium Water Air Density (103 kg/m3) 19.3 18.7 11.3 8.92 7.86 2.70 1.75 1.00 0.0012 CuuDuongThanCong.com https://fb.com/tailieudientucntt 10 CHAPTER Physics and Measurements Experiments have shown that this number, known as Avogadro’s number, NA , is NA ϭ 6.022 137 ϫ 10 23 particles/mol Avogadro’s number is defined so that mol of carbon-12 atoms has a mass of exactly 12 g In general, the mass in mol of any element is the element’s atomic mass expressed in grams For example, mol of iron (atomic mass ϭ 55.85 u) has a mass of 55.85 g (we say its molar mass is 55.85 g/mol), and mol of lead (atomic mass ϭ 207 u) has a mass of 207 g (its molar mass is 207 g/mol) Because there are 6.02 ϫ 1023 particles in mol of any element, the mass per atom for a given element is m atom ϭ molar mass NA (1.2) For example, the mass of an iron atom is m Fe ϭ EXAMPLE 1.1 55.85 g/mol ϭ 9.28 ϫ 10 Ϫ23 g/atom 6.02 ϫ 10 23 atoms/mol How Many Atoms in the Cube? A solid cube of aluminum (density 2.7 g/cm3) has a volume of 0.20 cm3 How many aluminum atoms are contained in the cube? minum (27 g) contains 6.02 ϫ 1023 atoms: NA N ϭ 27 g 0.54 g Solution Since density equals mass per unit volume, the mass m of the cube is m ϭ V ϭ (2.7 g/cm3)(0.20 cm3) ϭ 0.54 g To find the number of atoms N in this mass of aluminum, we can set up a proportion using the fact that one mole of alu- 1.4 6.02 ϫ 10 23 atoms N ϭ 27 g 0.54 g Nϭ (0.54 g)(6.02 ϫ 10 23 atoms) 27 g ϭ 1.2 ϫ 10 22 atoms DIMENSIONAL ANALYSIS The word dimension has a special meaning in physics It usually denotes the physical nature of a quantity Whether a distance is measured in the length unit feet or the length unit meters, it is still a distance We say the dimension — the physical nature — of distance is length The symbols we use in this book to specify length, mass, and time are L, M, and T, respectively We shall often use brackets [ ] to denote the dimensions 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] ϭ L/T As another example, the dimensions of area, for which we use the symbol A, are [A] ϭ L2 The dimensions of area, volume, speed, and acceleration are listed in Table 1.6 In solving problems in physics, there is a useful and powerful procedure called dimensional analysis This procedure, which should always be used, will help minimize the need for rote memorization of equations Dimensional analysis makes use of the fact that dimensions can be treated as algebraic quantities That is, 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 CuuDuongThanCong.com https://fb.com/tailieudientucntt 1.4 Dimensional Analysis TABLE 1.6 Dimensions and Common Units of Area, Volume, Speed, and Acceleration System SI British engineering Area (L2) Volume (L3) Speed (L/T) Acceleration (L/T 2) m2 ft2 m3 ft3 m/s ft/s m/s2 ft/s2 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 are the same on both sides of the equation To illustrate this procedure, suppose you 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 acceleration a In Chapter 2, we shall find that the correct expression is x ϭ 12at 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 perform a dimensional check by substituting the dimensions for acceleration, L/T 2, and time, T, into the equation That is, the dimensional form of the equation x ϭ 12at is L иT ϭ L T2 Lϭ The units of time squared cancel as shown, leaving the unit of length A more general procedure using dimensional analysis is to set up an expression of the form x ϰ a nt m where n and m are exponents that must be determined and the symbol ϰ 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 nt m] ϭ L ϭ LT Because the dimensions of acceleration are L/T and the dimension of time is T, we have TL T n m ϭ L1 Ln T mϪ2n ϭ L1 Because the exponents of L and T must be the same on both sides, the dimensional equation is balanced under the conditions m Ϫ 2n ϭ 0, n ϭ 1, and m ϭ Returning to our original expression x ϰ a nt mwe result , conclude that x ϰ at 2This differs by a factor of from the correct expression, which is x ϭ 12at Because the factor 12 is dimensionless, there is no way of determining it using dimensional analysis CuuDuongThanCong.com https://fb.com/tailieudientucntt 11 A.30 APPENDIX B Table B.5 lists some useful indefinite integrals Table B.6 gives Gauss’s probability integral and other definite integrals A more complete list can be found in various handbooks, such as The Handbook of Chemistry and Physics, CRC Press TABLE B.5 Some Indefinite Integrals (An arbitrary constant should be added to each of these integrals.) ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵! ͵! ͵! ͵! ͵! ͵! ͵! ͵ ! ͵ x n dx ϭ x nϩ1 nϩ1 dx ϭ x x Ϫ1 dx ϭ ln x (provided n Ϫ1) dx ϭ ln(a ϩ bx) a ϩ bx b x a x dx ϭ Ϫ ln(a ϩ bx) a ϩ bx b b xϩa dx ϭ Ϫ ln x(x ϩ a) a x dx ϭϪ (a ϩ bx)2 b(a ϩ bx) dx x ϭ tanϪ1 a2 ϩ x a a dx aϩx ϭ ln Ϫx 2a aϪx (a Ϫ x Ͼ 0) dx xϪa ϭ ln x2 Ϫ a 2a xϩa (x Ϫ a Ͼ 0) a2 x dx ϭ Ϯ 12 ln(a Ϯ x 2) a2 Ϯ x dx Ϫ dx x2 Ϯ x dx a2 a2 x2 a2 Ϫ x x dx x2 Ϯ a2 ϭ sinϪ1 x x ϭ ϪcosϪ1 a a (a Ϫ x Ͼ 0) ϭ ln(x ϩ !x Ϯ a 2) ϭ Ϫ !a Ϫ x ϭ !x Ϯ a a Ϫ x dx ϭ 12 x !a Ϫ x ϩ a sinϪ1 x a x a Ϫ x dx ϭ Ϫ 13 (a Ϫ x 2)3/2 x Ϯ a dx ϭ 12 [x !x Ϯ a Ϯ a ln(x ϩ !x Ϯ a 2)] x( x Ϯ a 2) dx ϭ 13 (x Ϯ a 2)3/2 e ax dx ϭ ax e a CuuDuongThanCong.com ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ͵ ln ax dx ϭ (x ln ax) Ϫ x xe ax dx ϭ e ax (ax Ϫ 1) a2 dx x ϭ Ϫ ln(a ϩ be cx) a ϩ be cx a ac sin ax dx ϭ Ϫ cos ax a cos ax dx ϭ sin ax a tan ax dx ϭ 1 ln(cos ax) ϭ ln(sec ax) a a cot ax dx ϭ ln(sin ax) a sec ax dx ϭ csc ax dx ϭ ΄ ax ln(csc ax Ϫ cot ax) ϭ ln tan a a 1 ax ln(sec ax ϩ tan ax) ϭ ln tan ϩ a a sin2 ax dx ϭ sin ax x Ϫ 4a cos2 ax dx ϭ x sin ax ϩ 4a dx ϭ Ϫ cot ax sin2 ax a dx ϭ tan ax cos2 ax a tan2 ax dx ϭ (tan ax) Ϫ x a cot ax dx ϭ Ϫ (cot ax) Ϫ x a sinϪ1 ax dx ϭ x(sinϪ1 ax) ϩ cosϪ1 ax dx ϭ x(cosϪ1 ax) Ϫ !1 Ϫ a 2x a !1 Ϫ a 2x dx x ϭ (x ϩ a 2)3/2 a 2!x ϩ a x dx ϭϪ (x ϩ a 2)3/2 !x ϩ a https://fb.com/tailieudientucntt a ΅ B.7 Intergral Calculus TABLE B.6 Gauss’s Probability Integral and Other Definite Integrals ͵ ϱ x n e Ϫax dx ϭ I0 ϭ I1 ϭ I2 ϭ I3 ϭ I4 ϭ I5 ϭ и и и ͵ ͵ ͵ ͵ ͵ ͵ ϱ ϱ ϱ ϱ ϱ ϱ n! a nϩ1 e Ϫax dx ϭ xe Ϫax dx ϭ ! a (Gauss’s probability integral) 2a ! x 2e Ϫax dx ϭ Ϫ dI ϭ da x 3e Ϫax dx ϭ Ϫ dI 1 ϭ da 2a 2 x 4e Ϫax dx ϭ d 2I ϭ da x 5e Ϫax dx ϭ d 2I 1 ϭ da a 2 I 2n ϭ (Ϫ1)n dn I da n I 2nϩ1 ϭ (Ϫ1)n dn I da n CuuDuongThanCong.com ! a3 a5 https://fb.com/tailieudientucntt A.31 CuuDuongThanCong.com https://fb.com/tailieudientucntt CuuDuongThanCong.com https://fb.com/tailieudientucntt APPENDIX C • Periodic Table of the Elements Group I H Group II Transition elements 1.008 1s1 Li Be 6.94 9.012 2s1 2s Na 11 Mg 22.99 24.31 3s1 3s K 19 39.102 Ca 4s 4s 37 12 20 Sc 21 3d 4s Y 87.62 88.906 Cs 55 132.91 6s Fr Ba 56 3d 4s 57-71* 3d 4s 40 Nb Hf 41 Mo Ta W 180.95 183.85 Ra 88 89-103** Rf 5d 6s 104 Db (223) (226) (261) (262) 7s1 7s 6d 7s 6d 37s *Lanthanide series **Actinide series La 57 5d 6s 105 Ce 4d 5s Sg 5d 6s 106 (263) 58 Pr 44 75 Os Bh 5d 6s 107 Nd 76 Ir Hs 5d 76s 108 Pm Mt 61 Sm 140.12 140.91 144.24 (147) 150.4 5d 16s 5d 14f 16s 4f 36s 4f 46s 4f 56s 4f 66s 89 Th 90 Pa 91 U 92 Np 93 Pu 62 94 (227) (232) (231) 6d 17s 6d 27s 5f 26d 17s 5f 36d 17s 5f 46d 17s 5f 66d 07s (238) (239) (239) Atomic mass values given are averaged over isotopes in the percentages in which they exist in nature † For an unstable element, mass number of the most stable known isotope is given in parentheses †† Elements 110, 111, 112, and 114 have not yet been named ††† For a description of the atomic data, visit physics.nist.gov/atomic A.32 CuuDuongThanCong.com 109 (266) 138.91 Ac 77 192.2 (265) 60 45 4d 85s1 190.2 Rh 102.91 4d 5s (262) 59 Ru 27 3d 74s 2 101.1 Co 58.93 3d 4s 43 Re 26 186.2 Fe 55.85 Tc 74 2 25 (99) 178.49 5d 6s 3d 4s 42 2 4d 5s 73 Mn 54.94 95.94 4d 5s 72 24 3d 4s 92.91 Cr 51.996 91.22 4d 5s 23 50.94 Zr V Electron configuration 137.34 6s 87 4d 5s 22 39 5s Ti 47.90 85.47 5s Atomic number 4s 44.96 38 20 40.08 Atomic mass † Sr Ca Symbol 40.08 Rb https://fb.com/tailieudientucntt A.33 Periodic Table of the Elements Group III Group IV Group V Group VI Group VII H B 10.81 2p Al 13 Ni 28 58.71 3d 4s 29 63.54 Pd 3d 4s 46 10 Pt 30 10 Ag 3d 4s 47 112.40 10 Au 4d 5s 79 4p 48 10 Ga Hg In F 18.998 2p Si 14 P 3p 32 50 S 3p As 4p 2p 16 32.06 33 74.92 Sn 15 30.97 Ge 2p Se 34 78.96 4p Sb 51 Cl 3p 17 Te 52 35 I 53 121.75 127.60 126.90 Tl 81 Pb 82 5p Bi 83 5p Po 84 Xe 5p At 85 Rn 196.97 200.59 204.37 207.2 208.98 (210) (218) (222) 5d 96s1 5d 106s1 5d 106s 6p1 6p 6p 6p 6p 6p (269) Eu 111†† (272) 63 Gd 112†† Tb 86 114†† (277) 64 54 131.30 195.09 110†† 36 4p 118.69 5p Kr 83.80 5p 18 3p Br 4p Ar 39.948 79.91 10 20.18 35.453 Ne 2p 114.82 5p 80 1s 15.999 4p 49 O 1s 72.59 4.002 14.007 3p 31 N He 1.008 28.09 69.72 Cd 107.87 4d 5s 78 Zn 65.37 10 106.4 4d Cu 12.011 2p 26.98 3p C Group (289) 65 66 Ho 67 Er 68 Tm 69 Yb 70 Lu 71 152.0 157.25 162.50 164.93 167.26 168.93 173.04 174.97 4f 76s 5d 14f 76s 5d 14f 86s 4f 106s 4f 116s 4f 126s 4f 136s 4f 146s 5d 14f 146s Fm Md No Lr Am (243) 95 Cm (245) 158.92 Dy 96 Bk (247) 97 Cf (249) 98 Es (254) 99 (253) 100 (255) 101 102 (255) 5f 76d 07s 5f 76d 17s 5f 86d 17s 5f 106d 07s 5f 116d 07s 5f 126d 07s 5f 136d 07s 6d 07s CuuDuongThanCong.com 103 (257) 6d 17s https://fb.com/tailieudientucntt CuuDuongThanCong.com https://fb.com/tailieudientucntt APPENDIX D • SI Units TABLE D.1 SI Units SI Base Unit Base Quantity Name Symbol Length Mass Time Electric current Temperature Amount of substance Luminous intensity Meter Kilogram Second Ampere Kelvin Mole Candela m kg s A K mol cd TABLE D.2 Some Derived SI Units Quantity Name Symbol Expression in Terms of Base Units Plane angle Frequency Force Pressure Energy; work Power Electric charge Electric potential Capacitance Electric resistance Magnetic flux Magnetic field intensity Inductance radian hertz newton pascal joule watt coulomb volt farad ohm weber tesla henry rad Hz N Pa J W C V F ⍀ Wb T H m/m sϪ1 kg и m/s2 kg/m и s2 kg и m2/s kg и m2/s Aиs kg и m2/A и s A2 и s 4/kg и m2 kg и m2/A2 и s kg и m2/A и s kg/A и s2 kg и m2/A2 и s A.34 CuuDuongThanCong.com https://fb.com/tailieudientucntt Expression in Terms of Other SI Units J/m N/m2 Nиm J/s W/A C/V V/A Vиs T и m2/A CuuDuongThanCong.com https://fb.com/tailieudientucntt APPENDIX E • Nobel Prizes All Nobel Prizes in physics are listed (and marked with a P), as well as relevant Nobel Prizes in Chemistry (C) The key dates for some of the scientific work are supplied; they often antedate the prize considerably 1901 (P) Wilhelm Roentgen for discovering x-rays (1895) 1902 (P) Hendrik A Lorentz for predicting the Zeeman effect and Pieter Zeeman for discovering the Zeeman effect, the splitting of spectral lines in magnetic fields 1903 (P) Antoine-Henri Becquerel for discovering radioactivity (1896) and Pierre and Marie Curie for studying radioactivity 1904 (P) Lord Rayleigh for studying the density of gases and discovering argon (C) William Ramsay for discovering the inert gas elements helium, neon, xenon, and krypton, and placing them in the periodic table 1905 (P) Philipp Lenard for studying cathode rays, electrons (1898 – 1899) 1906 (P) J J Thomson for studying electrical discharge through gases and discovering the electron (1897) 1907 (P) Albert A Michelson for inventing optical instruments and measuring the speed of light (1880s) 1908 (P) Gabriel Lippmann for making the first color photographic plate, using interference methods (1891) (C) Ernest Rutherford for discovering that atoms can be broken apart by alpha rays and for studying radioactivity 1909 (P) Guglielmo Marconi and Carl Ferdinand Braun for developing wireless telegraphy 1910 (P) Johannes D van der Waals for studying the equation of state for gases and liquids (1881) 1911 (P) Wilhelm Wien for discovering Wien’s law giving the peak of a blackbody spectrum (1893) (C) Marie Curie for discovering radium and polonium (1898) and isolating radium 1912 (P) Nils Dalén for inventing automatic gas regulators for lighthouses 1913 (P) Heike Kamerlingh Onnes for the discovery of superconductivity and liquefying helium (1908) 1914 (P) Max T F von Laue for studying x-rays from their diffraction by crystals, showing that x-rays are electromagnetic waves (1912) (C) Theodore W Richards for determining the atomic weights of sixty elements, indicating the existence of isotopes 1915 (P) William Henry Bragg and William Lawrence Bragg, his son, for studying the diffraction of x-rays in crystals 1917 (P) Charles Barkla for studying atoms by x-ray scattering (1906) 1918 (P) Max Planck for discovering energy quanta (1900) 1919 (P) Johannes Stark, for discovering the Stark effect, the splitting of spectral lines in electric fields (1913) A.35 CuuDuongThanCong.com https://fb.com/tailieudientucntt A.36 APPENDIX E Nobel Prizes 1920 (P) Charles-Édouard Guillaume for discovering invar, a nickel-steel alloy with low coefficient of expansion (C) Walther Nernst for studying heat changes in chemical reactions and formulating the third law of thermodynamics (1918) 1921 (P) Albert Einstein for explaining the photoelectric effect and for his services to theoretical physics (1905) (C) Frederick Soddy for studying the chemistry of radioactive substances and discovering isotopes (1912) 1922 (P) Niels Bohr for his model of the atom and its radiation (1913) (C) Francis W Aston for using the mass spectrograph to study atomic weights, thus discovering 212 of the 287 naturally occurring isotopes 1923 (P) Robert A Millikan for measuring the charge on an electron (1911) and for studying the photoelectric effect experimentally (1914) 1924 (P) Karl M G Siegbahn for his work in x-ray spectroscopy 1925 (P) James Franck and Gustav Hertz for discovering the Franck-Hertz effect in electron-atom collisions 1926 (P) Jean-Baptiste Perrin for studying Brownian motion to validate the discontinuous structure of matter and measure the size of atoms 1927 (P) Arthur Holly Compton for discovering the Compton effect on x-rays, their change in wavelength when they collide with matter (1922), and Charles T R Wilson for inventing the cloud chamber, used to study charged particles (1906) 1928 (P) Owen W Richardson for studying the thermionic effect and electrons emitted by hot metals (1911) 1929 (P) Louis Victor de Broglie for discovering the wave nature of electrons (1923) 1930 (P) Chandrasekhara Venkata Raman for studying Raman scattering, the scattering of light by atoms and molecules with a change in wavelength (1928) 1932 (P) Werner Heisenberg for creating quantum mechanics (1925) 1933 (P) Erwin Schrödinger and Paul A M Dirac for developing wave mechanics (1925) and relativistic quantum mechanics (1927) (C) Harold Urey for discovering heavy hydrogen, deuterium (1931) 1935 (P) James Chadwick for discovering the neutron (1932) (C) Irène and Frédéric Joliot-Curie for synthesizing new radioactive elements 1936 (P) Carl D Anderson for discovering the positron in particular and antimatter in general (1932) and Victor F Hess for discovering cosmic rays (C) Peter J W Debye for studying dipole moments and diffraction of x-rays and electrons in gases 1937 (P) Clinton Davisson and George Thomson for discovering the diffraction of electrons by crystals, confirming de Broglie’s hypothesis (1927) 1938 (P) Enrico Fermi for producing the transuranic radioactive elements by neutron irradiation (1934 – 1937) 1939 (P) Ernest O Lawrence for inventing the cyclotron 1943 (P) Otto Stern for developing molecular-beam studies (1923), and using them to discover the magnetic moment of the proton (1933) 1944 (P) Isidor I Rabi for discovering nuclear magnetic resonance in atomic and molecular beams (C) Otto Hahn for discovering nuclear fission (1938) 1945 (P) Wolfgang Pauli for discovering the exclusion principle (1924) 1946 (P) Percy W Bridgman for studying physics at high pressures 1947 (P) Edward V Appleton for studying the ionosphere CuuDuongThanCong.com https://fb.com/tailieudientucntt Appendix E Nobel Prizes 1948 (P) Patrick M S Blackett for studying nuclear physics with cloud-chamber photographs of cosmic-ray interactions 1949 (P) Hideki Yukawa for predicting the existence of mesons (1935) 1950 (P) Cecil F Powell for developing the method of studying cosmic rays with photographic emulsions and discovering new mesons 1951 (P) John D Cockcroft and Ernest T S Walton for transmuting nuclei in an accelerator (1932) (C) Edwin M McMillan for producing neptunium (1940) and Glenn T Seaborg for producing plutonium (1941) and further transuranic elements 1952 (P) Felix Bloch and Edward Mills Purcell for discovering nuclear magnetic resonance in liquids and gases (1946) 1953 (P) Frits Zernike for inventing the phase-contrast microscope, which uses interference to provide high contrast 1954 (P) Max Born for interpreting the wave function as a probability (1926) and other quantum-mechanical discoveries and Walther Bothe for developing the coincidence method to study subatomic particles (1930 – 1931), producing, in particular, the particle interpreted by Chadwick as the neutron 1955 (P) Willis E Lamb, Jr., for discovering the Lamb shift in the hydrogen spectrum (1947) and Polykarp Kusch for determining the magnetic moment of the electron (1947) 1956 (P) John Bardeen, Walter H Brattain, and William Shockley for inventing the transistor (1956) 1957 (P) T.-D Lee and C.-N Yang for predicting that parity is not conserved in beta decay (1956) ˇ ˇ 1958 (P) Pavel A Cerenkov for discovering Cerenkov radiation (1935) and Ilya M Frank and Igor Tamm for interpreting it (1937) 1959 (P) Emilio G Segrè and Owen Chamberlain for discovering the antiproton (1955) 1960 (P) Donald A Glaser for inventing the bubble chamber to study elementary particles (1952) (C) Willard Libby for developing radiocarbon dating (1947) 1961 (P) Robert Hofstadter for discovering internal structure in protons and neutrons and Rudolf L Mössbauer for discovering the Mössbauer effect of recoilless gamma-ray emission (1957) 1962 (P) Lev Davidovich Landau for studying liquid helium and other condensed matter theoretically 1963 (P) Eugene P Wigner for applying symmetry principles to elementary-particle theory and Maria Goeppert Mayer and J Hans D Jensen for studying the shell model of nuclei (1947) 1964 (P) Charles H Townes, Nikolai G Basov, and Alexandr M Prokhorov for developing masers (1951 – 1952) and lasers 1965 (P) Sin-itiro Tomonaga, Julian S Schwinger, and Richard P Feynman for developing quantum electrodynamics (1948) 1966 (P) Alfred Kastler for his optical methods of studying atomic energy levels 1967 (P) Hans Albrecht Bethe for discovering the routes of energy production in stars (1939) 1968 (P) Luis W Alvarez for discovering resonance states of elementary particles 1969 (P) Murray Gell-Mann for classifying elementary particles (1963) 1970 (P) Hannes Alfvén for developing magnetohydrodynamic theory and Louis Eugène Félix Néel for discovering antiferromagnetism and ferrimagnetism (1930s) CuuDuongThanCong.com https://fb.com/tailieudientucntt A.37 A.38 APPENDIX E Nobel Prizes 1971 (P) Dennis Gabor for developing holography (1947) (C) Gerhard Herzberg for studying the structure of molecules spectroscopically 1972 (P) John Bardeen, Leon N Cooper, and John Robert Schrieffer for explaining superconductivity (1957) 1973 (P) Leo Esaki for discovering tunneling in semiconductors, Ivar Giaever for discovering tunneling in superconductors, and Brian D Josephson for predicting the Josephson effect, which involves tunneling of paired electrons (1958 – 1962) 1974 (P) Anthony Hewish for discovering pulsars and Martin Ryle for developing radio interferometry 1975 (P) Aage N Bohr, Ben R Mottelson, and James Rainwater for discovering why some nuclei take asymmetric shapes 1976 (P) Burton Richter and Samuel C C Ting for discovering the J/psi particle, the first charmed particle (1974) 1977 (P) John H Van Vleck, Nevill F Mott, and Philip W Anderson for studying solids quantum-mechanically (C) Ilya Prigogine for extending thermodynamics to show how life could arise in the face of the second law 1978 (P) Arno A Penzias and Robert W Wilson for discovering the cosmic background radiation (1965) and Pyotr Kapitsa for his studies of liquid helium 1979 (P) Sheldon L Glashow, Abdus Salam, and Steven Weinberg for developing the theory that unified the weak and electromagnetic forces (1958 – 1971) 1980 (P) Val Fitch and James W Cronin for discovering CP (charge-parity) violation (1964), which possibly explains the cosmological dominance of matter over antimatter 1981 (P) Nicolaas Bloembergen and Arthur L Schawlow for developing laser spectroscopy and Kai M Siegbahn for developing high-resolution electron spectroscopy (1958) 1982 (P) Kenneth G Wilson for developing a method of constructing theories of phase transitions to analyze critical phenomena 1983 (P) William A Fowler for theoretical studies of astrophysical nucleosynthesis and Subramanyan Chandrasekhar for studying physical processes of importance to stellar structure and evolution, including the prediction of white dwarf stars (1930) 1984 (P) Carlo Rubbia for discovering the W and Z particles, verifying the electroweak unification, and Simon van der Meer, for developing the method of stochastic cooling of the CERN beam that allowed the discovery (1982 – 1983) 1985 (P) Klaus von Klitzing for the quantized Hall effect, relating to conductivity in the presence of a magnetic field (1980) 1986 (P) Ernst Ruska for inventing the electron microscope (1931), and Gerd Binnig and Heinrich Rohrer for inventing the scanning-tunneling electron microscope (1981) 1987 (P) J Georg Bednorz and Karl Alex Müller for the discovery of high temperature superconductivity (1986) 1988 (P) Leon M Lederman, Melvin Schwartz, and Jack Steinberger for a collaborative experiment that led to the development of a new tool for studying the weak nuclear force, which affects the radioactive decay of atoms 1989 (P) Norman Ramsay (U.S.) for various techniques in atomic physics; and Hans Dehmelt (U.S.) and Wolfgang Paul (Germany) for the development of techniques for trapping single charge particles CuuDuongThanCong.com https://fb.com/tailieudientucntt Appendix E Nobel Prizes 1990 (P) Jerome Friedman, Henry Kendall (both U.S.), and Richard Taylor (Canada) for experiments important to the development of the quark model 1991 (P) Pierre-Gilles de Gennes for discovering that methods developed for studying order phenomena in simple systems can be generalized to more complex forms of matter, in particular to liquid crystals and polymers 1992 (P) George Charpak for developing detectors that trace the paths of evanescent subatomic particles produced in particle accelerators 1993 (P) Russell Hulse and Joseph Taylor for discovering evidence of gravitational waves 1994 (P) Bertram N Brockhouse and Clifford G Shull for pioneering work in neutron scattering 1995 (P) Martin L Perl and Frederick Reines for discovering the tau particle and the neutrino, respectively 1996 (P) David M Lee, Douglas C Osheroff, and Robert C Richardson for developing a superfluid using helium-3 1997 (P) Steven Chu, Claude Cohen-Tannoudji, and William D Phillips for developing methods to cool and trap atoms with laser light 1998 (P) Robert B Laughlin, Horst L Störmer, and Daniel C Tsui for discovering a new form of quantum fluid with fractionally charged excitations CuuDuongThanCong.com https://fb.com/tailieudientucntt A.39 CuuDuongThanCong.com https://fb.com/tailieudientucntt ... Length of a football field Length of a housefly Size of smallest dust particles Size of cells of most living organisms Diameter of a hydrogen atom Diameter of an atomic nucleus Diameter of a proton... 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. .. quarter-acre plot of land What is the order of magnitude of the number of blades of grass on this plot of land? Explain your reasoning (1 acre ϭ 43 560 ft2.) 46 Suppose that someone offers to give
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