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APPENDIX II Physical Topics
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _-'- APPENDIX II
Physical Topics A.
Units and Dimensions
Various internally consistent systems of units have been used in scientific work. The International System of Units (Le Systeme International-SI) is now widely accepted and is used in this book; other units in common use are referred to where particularly relevant. t The SI system utilizes the following base units Name
Quantity
meter kilogram second ampere kelvin mole candela
length mass time electric current thermodynamic temperature amount of substance luminous intensity
Symbol m kg A K
mole cd
The meter was originally defined as 10- 7 part of the arc length between the north pole and the equator. It is now defined as 1650763.73 times the wavelength in vacuum of the orange-red line of krypton 86. The kilogram was originally defined as 1000 times the mass of a cubic centimeter of distilled water at the temperature of its greatest density, but since 1889 it has been defined by the standard kilogram, a platinum-iridium cylinder which is carefully preserved in its original state. The second, formerly defined by the rotation rate of the earth, was defined by the 12th General Conference on Weights and Measures in 1967 as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of cesium 133. The quantities of length, mass, time are referred to as the dimensions of mechanics and are designated by L, M, and t, respectively. Other mechanical quantities, such as force, momentum, energy, etc., have dimensions which are made up of the three fundamental dimensions. The dimensions of all mechanical quantities may be determined from the definition of the quantity
t The SI system is comprehensively summarized in The Sf Metric Handbook, John L. Firrer, Scribner, New York, Chas. A. Bennett Co., Peoria, 1977.
412
APPENDIX II.
PHYSICAL TOPICS
or from a physical relation between quantities. Thus the dimensions of momentum and force are, respectively, mass x velocity
=
M Lt - 1
mass x acceleration = M Lt - 2 Electric current (A), thermodynamic temperature (K), and amount of substance (mole) also are regarded as dimensions in the SI system. B.
Significant Figures]
It is important in calculation to distinguish between mathematical numbers which are known to any accuracy required and physical numbers whose accuracy is limited by errors of measurement. It follows that the last digit of a physical number is uncertain. The number of digits in such a number, after any zeros to the left of the first number different from zero, is called the number of significant figures. Thus, if we can read a distance as 98 km, we say that there are two significant figures. We are not permitted to say that this is 98,000 m, for this erroneously gives the impression of a measurement accurate to the order of meters. Instead, we should say that the distance is 98 x 103 m. Mathematical numbers, which carry any number of significant figures, are treated as though they have at least as many significant figures as the largest number of significant figures in the problem. Often one more figure may be used than is significant in order to prevent round-off error from influencing the result. The mean of more than 10 and less than 1000 numbers may contain one more significant figure than the individual observations. This may sometimes be used to advantage, as Chapman demonstrated in finding the lunar tide in high latitudes from barometric observations.j In adding or subtracting, all numbers should have the same number of significant decimal places. In order to achieve this, digits beyond the last significant decimal place should be dropped. In multiplying and dividing the number of significant figures in the result is equal to the number of significant figures in that number which has the smallest number of significant figures. Special rules may be developed for handling the number of significant t An excellent book on the subject is Yardley Beers, Introduction of the Theory ofError, 2nd ed., Addison-Wesley, Cambridge, Massachusetts, 1957. t S. Chapman, Quart. J. Roy. Meteoro/. Soc. 44, 271 (1918).
D.
413
TABLE OF PHYSICAL CONSTANTS
figures of functions. The basic idea, however, is that the uncertainty of a function y = f(x) is expressed by dy
df
= dxdx
where dx represents the uncertainty of the error of measurement of x.
C. Quantity
Symbol
Force Charge Current Potential Resistance Capacitance Electrical field strength Magnetic induction" Work Power
F
Electromagnetic Conversion Tablea Sl system
Rationalized CGSe system
C E
I newton (I N) I coulomb (I C) I ampere (I A) I volt (I joule C - I) I ohm (I JA -2 s-I) I farad (I CV- I) INC-I
= 105dyn = 2.997925 X 109 Fr = 2.997925 x 109Frs- 1 = 3.335640 X 10- 3 erg Fr- [ = 1.112649 x 10- 12erg s Fr - 2 = 8.987556 X 1011 Fr 2 erg-I = 3.335640 x IO- SdynFr- 1
B
I weber m "? (I NA -I m- I)
= 3.335640 x 10- 7 dynsFr- 1 cm- 1
W P
I joule (I J) I watt (I W = I J s - I)
= 107 erg = 107ergs- 1
Q
I V R
Permeability (vacuum) flo =4n x 10- 7 weber A -1 m-I =4n x 1.112649 x 10- 21 dyn s? Fr- 2 Permittivity (vacuum) Eo = 107 (4nc6) -I farad m - I = in Fr 2 dyn - I ern - 2 Calculated from Nederlandse Natuurkundige Vereniging, Jaarboek, p. 131,1974-5. The magnetic induction is also expressed in the CGS electromagnetic unit, called the gauss, which is 10- 4 weber m- 2 . a b
D.
Table of Physical Constants
Universal Constants Velocity oflight in vacuum" Boltzmann constant" Planck constant" Avogadro's number" Mass of proton" Mass of electron" Charge of electron a Stefan-Boltzmann constant" Standard molar volume of gas" Molar gas constant" Gravitation constant" Wien's displacement constant"
(co)
(k)
(h) (No) (m+) (m_)
(e) (0) (Vm) (R) (G) (IX)
(2.99792458 ± 0.000000012) x 108 m S-1 (1.380662 ±0.000044) x 1O- 23JK- I (6.626176 ± 0.000036) x 10- 34 J s (6.022045 ± 0.000031) x 1023 mole- [ (1.6726485 ± 0.0000086) x 10- 27 kg (9.109534 ± 0.000047) x 10- 3 1 kg (1.6021892 ± 0.0000046) x 10- 19 C (5.67032 ± 0.00071) x 10- 8 Wm- 2 K- 4 (2.241383 ± 0.000070) x 10- 2 rn' mole- 1 (8.31441 ± 0.00026)1 mole-I K- I (6.6720 ± 0.0041) x 10- 11 N m 2 kg- 2 (2897.82 ± 0.013) Ilm K (cant.)
414
APPENDIX II.
D.
Table of Physical Constants (Continued) 12 (exact) 1.007822 ± 0.000003 15.994915 4.18580 J
Atomic mass, carbon-12 nucleus" Atomic mass, hydrogen-I atom" Atomic mass, oxygen-16 nucleus" Calorie' Sun Solar radius' Solar mass' Solar constant" Earth Earth mass' Mean solar day" Length of year" Acceleration of gravity at sea level" 45° 32' 33" Effective earth radius" Angular frequency of earth's rotation Air Standard sea level pressure" Standard sea level temperature' Average molecular mass at sea level' Thermal conductivity at sea level' 15°C Viscosity at sea level, 15°C' Diffusivity of water vapor in air at sea level, IYCf Water Molecular mass" Ice point'
PHYSICAL TOPICS
(£,)
(6.960 ± 0.001) x 108 m (1.991 ± 0.002) x 10 3 0 kg (1370 ± I) W m "?
(go)
(5.977 ± 0.004) x 102 4 kg 1.00273791 sidereal days 365.24219878 days 9.80665 m S-2
(Q)
6356.766 km 7.29212 x 10- 5
1.013250 288.15 K 28.9644
X
S-1
102 kPa
W m- 1 K - [
(A)
2.5326 x 10-
(Il) (D)
I. 7894 x 10- 5 kg m - 1 2.49 X 10- 4 m 2 S-1
3
S- 1
18.0160 (273.155 ± 0.015) K
International Council of Scientific Unions, CODATA Bulletin No. 11, Dec. 1973. Nederlandse Natuurkundige Vereniging, Jaarboek, 1962. 'c. W. Allen, Astrophysical Quantities, Univ. London, p. II, 1955. dR. C. Wilson, J. R. Hickey, The Solar Output and its Variation, O. R. White, ed., Univ. Colorado, 1977. e U.S. Standard Atmosphere, 1976, NOAA, NASA, USAF, Washington, D.C., 1976. f R. B. Montgomery, 1. Meteor., 4,193, 1947. 9 W. E. Forsythe, Smithsonian Physical Tables, 9th rev. ed., Smithsonian Institution, Washington, D.C., 1959. a
b
E.
415
STANDARD ATMOSPHERES
E. U.S. Standard Atmosphere 1976 and COSPAR International G
Reference Atmosphere (CIRA)b Height" Temp." (km) (K) 0 1 2 3 4 5 6 8 10 12 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120
288.15 281.65 275.15 268.66 262.17 255.68 249.19 236.22 223.25 216.65 216.65 216.65 221.55 226.51 236.51 250.35 264.16 270.65 260.77 247.02 233.29 219.59 208.40 198.64 188.89 186.87 188.42 195.08 208.84 240.00 300.00 360.00
Pressure" (kPa) 101.330 89.880 79.500 70.120 61.660 54.050 47.220 35.650 26.500 19.400 12.110 5.529 2.549 1.197 5.746 2.871 1.491 7.978 4.253 2.196 1.093 5.221 2.388 1.052 4.957 1.836 7.597 3.201 1.448 7.104 4.010 2.538
x x x x x x x x x x x x x x x x x x
10- 1 10- 1 10- 1 10- 2 10- 2 10- 2 10- 2 10- 3 10- 3 10- 3 10- 4 10- 4 10- 5 10- 5 10- 5 10- 6 10- 6 10- 6
Density" (kg m - 3) 1.225 1.112 1.007 9.093 8.194 7.364 6.601 5.258 4.135 3.119 1.948 8.891 4.008 1.841 8.463 3.996 1.966 1.027 5.681 3.097 1.632 8.283 3.992 1.846 8~220 3.416 1.393 5.604 2.325 9.708 4.289 2.222
x x x x x x x x x x x X X
x X X X X
x x x X X X X X
x x X
10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 2 10- 2 10- 2 10- 3 10- 3 10- 3 10- 3 10- 4 10- 4 10- 4 10- 5 10- 5 10- 5 10- 6 10- 6 10- 6 10- 7 10- 7 10- 8 10- 8 10- 8
Temp." (K)
221.7 230.7 241.5 255.3 267.7 271.6 263.9 249.3 232.7 216.2 205.0 195.0 185.1 183.8 190.3 203.5 228.0 265.5 317.1 380.6
Pressure" (kPa)
2.483 1.175 5.741 2.911 1.525 8.241 4.406 2.296 1.146 5.445 2.460 1.067 4.426 1.795 7.345 3.090 1.422 7.362 4.236 2.667
x x x x x x x x x x x x x x x x x x
10- 1 10- 1 10- 1 10- 2 10- 2 10- 2 10- 2 10- 3 10- 3 10- 3 10- 4 10- 4 10- 5 10- 5 10- 5 10- 6 10- 6 10- 6
Density" (kg m ":')
3.899 x 10- 2 1.774 x 10- 2 8.279 X 10- 3 3.972 X 10- 3 1.995 x 10- 3 1.057 X 10- 3 5.821 X 10- 3 3.206 X 10- 3 1.718 X 10- 3 8.770 X 10- 5 4.178 X 10- 5 1.905 X 10- 5 8.337 X 10- 6 3.396 X 10- 6 1.343 X 10- 6 5.297 X 10- 7 2.173 X 10- 7 9.661 X 10- 8 4.645 X 10- 8 2.438 X 10- 8
"U.S. Standard Atmosphere, 1976, NOAA, NASA, USAF, Washington, D.C., 1976. b COSPAR International Reference Atmosphere, 1972, Akademie Verlag, Berlin, 1972.
Physical Topics A.
Units and Dimensions
Various internally consistent systems of units have been used in scientific work. The International System of Units (Le Systeme International-SI) is now widely accepted and is used in this book; other units in common use are referred to where particularly relevant. t The SI system utilizes the following base units Name
Quantity
meter kilogram second ampere kelvin mole candela
length mass time electric current thermodynamic temperature amount of substance luminous intensity
Symbol m kg A K
mole cd
The meter was originally defined as 10- 7 part of the arc length between the north pole and the equator. It is now defined as 1650763.73 times the wavelength in vacuum of the orange-red line of krypton 86. The kilogram was originally defined as 1000 times the mass of a cubic centimeter of distilled water at the temperature of its greatest density, but since 1889 it has been defined by the standard kilogram, a platinum-iridium cylinder which is carefully preserved in its original state. The second, formerly defined by the rotation rate of the earth, was defined by the 12th General Conference on Weights and Measures in 1967 as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of cesium 133. The quantities of length, mass, time are referred to as the dimensions of mechanics and are designated by L, M, and t, respectively. Other mechanical quantities, such as force, momentum, energy, etc., have dimensions which are made up of the three fundamental dimensions. The dimensions of all mechanical quantities may be determined from the definition of the quantity
t The SI system is comprehensively summarized in The Sf Metric Handbook, John L. Firrer, Scribner, New York, Chas. A. Bennett Co., Peoria, 1977.
412
APPENDIX II.
PHYSICAL TOPICS
or from a physical relation between quantities. Thus the dimensions of momentum and force are, respectively, mass x velocity
=
M Lt - 1
mass x acceleration = M Lt - 2 Electric current (A), thermodynamic temperature (K), and amount of substance (mole) also are regarded as dimensions in the SI system. B.
Significant Figures]
It is important in calculation to distinguish between mathematical numbers which are known to any accuracy required and physical numbers whose accuracy is limited by errors of measurement. It follows that the last digit of a physical number is uncertain. The number of digits in such a number, after any zeros to the left of the first number different from zero, is called the number of significant figures. Thus, if we can read a distance as 98 km, we say that there are two significant figures. We are not permitted to say that this is 98,000 m, for this erroneously gives the impression of a measurement accurate to the order of meters. Instead, we should say that the distance is 98 x 103 m. Mathematical numbers, which carry any number of significant figures, are treated as though they have at least as many significant figures as the largest number of significant figures in the problem. Often one more figure may be used than is significant in order to prevent round-off error from influencing the result. The mean of more than 10 and less than 1000 numbers may contain one more significant figure than the individual observations. This may sometimes be used to advantage, as Chapman demonstrated in finding the lunar tide in high latitudes from barometric observations.j In adding or subtracting, all numbers should have the same number of significant decimal places. In order to achieve this, digits beyond the last significant decimal place should be dropped. In multiplying and dividing the number of significant figures in the result is equal to the number of significant figures in that number which has the smallest number of significant figures. Special rules may be developed for handling the number of significant t An excellent book on the subject is Yardley Beers, Introduction of the Theory ofError, 2nd ed., Addison-Wesley, Cambridge, Massachusetts, 1957. t S. Chapman, Quart. J. Roy. Meteoro/. Soc. 44, 271 (1918).
D.
413
TABLE OF PHYSICAL CONSTANTS
figures of functions. The basic idea, however, is that the uncertainty of a function y = f(x) is expressed by dy
df
= dxdx
where dx represents the uncertainty of the error of measurement of x.
C. Quantity
Symbol
Force Charge Current Potential Resistance Capacitance Electrical field strength Magnetic induction" Work Power
F
Electromagnetic Conversion Tablea Sl system
Rationalized CGSe system
C E
I newton (I N) I coulomb (I C) I ampere (I A) I volt (I joule C - I) I ohm (I JA -2 s-I) I farad (I CV- I) INC-I
= 105dyn = 2.997925 X 109 Fr = 2.997925 x 109Frs- 1 = 3.335640 X 10- 3 erg Fr- [ = 1.112649 x 10- 12erg s Fr - 2 = 8.987556 X 1011 Fr 2 erg-I = 3.335640 x IO- SdynFr- 1
B
I weber m "? (I NA -I m- I)
= 3.335640 x 10- 7 dynsFr- 1 cm- 1
W P
I joule (I J) I watt (I W = I J s - I)
= 107 erg = 107ergs- 1
Q
I V R
Permeability (vacuum) flo =4n x 10- 7 weber A -1 m-I =4n x 1.112649 x 10- 21 dyn s? Fr- 2 Permittivity (vacuum) Eo = 107 (4nc6) -I farad m - I = in Fr 2 dyn - I ern - 2 Calculated from Nederlandse Natuurkundige Vereniging, Jaarboek, p. 131,1974-5. The magnetic induction is also expressed in the CGS electromagnetic unit, called the gauss, which is 10- 4 weber m- 2 . a b
D.
Table of Physical Constants
Universal Constants Velocity oflight in vacuum" Boltzmann constant" Planck constant" Avogadro's number" Mass of proton" Mass of electron" Charge of electron a Stefan-Boltzmann constant" Standard molar volume of gas" Molar gas constant" Gravitation constant" Wien's displacement constant"
(co)
(k)
(h) (No) (m+) (m_)
(e) (0) (Vm) (R) (G) (IX)
(2.99792458 ± 0.000000012) x 108 m S-1 (1.380662 ±0.000044) x 1O- 23JK- I (6.626176 ± 0.000036) x 10- 34 J s (6.022045 ± 0.000031) x 1023 mole- [ (1.6726485 ± 0.0000086) x 10- 27 kg (9.109534 ± 0.000047) x 10- 3 1 kg (1.6021892 ± 0.0000046) x 10- 19 C (5.67032 ± 0.00071) x 10- 8 Wm- 2 K- 4 (2.241383 ± 0.000070) x 10- 2 rn' mole- 1 (8.31441 ± 0.00026)1 mole-I K- I (6.6720 ± 0.0041) x 10- 11 N m 2 kg- 2 (2897.82 ± 0.013) Ilm K (cant.)
414
APPENDIX II.
D.
Table of Physical Constants (Continued) 12 (exact) 1.007822 ± 0.000003 15.994915 4.18580 J
Atomic mass, carbon-12 nucleus" Atomic mass, hydrogen-I atom" Atomic mass, oxygen-16 nucleus" Calorie' Sun Solar radius' Solar mass' Solar constant" Earth Earth mass' Mean solar day" Length of year" Acceleration of gravity at sea level" 45° 32' 33" Effective earth radius" Angular frequency of earth's rotation Air Standard sea level pressure" Standard sea level temperature' Average molecular mass at sea level' Thermal conductivity at sea level' 15°C Viscosity at sea level, 15°C' Diffusivity of water vapor in air at sea level, IYCf Water Molecular mass" Ice point'
PHYSICAL TOPICS
(£,)
(6.960 ± 0.001) x 108 m (1.991 ± 0.002) x 10 3 0 kg (1370 ± I) W m "?
(go)
(5.977 ± 0.004) x 102 4 kg 1.00273791 sidereal days 365.24219878 days 9.80665 m S-2
(Q)
6356.766 km 7.29212 x 10- 5
1.013250 288.15 K 28.9644
X
S-1
102 kPa
W m- 1 K - [
(A)
2.5326 x 10-
(Il) (D)
I. 7894 x 10- 5 kg m - 1 2.49 X 10- 4 m 2 S-1
3
S- 1
18.0160 (273.155 ± 0.015) K
International Council of Scientific Unions, CODATA Bulletin No. 11, Dec. 1973. Nederlandse Natuurkundige Vereniging, Jaarboek, 1962. 'c. W. Allen, Astrophysical Quantities, Univ. London, p. II, 1955. dR. C. Wilson, J. R. Hickey, The Solar Output and its Variation, O. R. White, ed., Univ. Colorado, 1977. e U.S. Standard Atmosphere, 1976, NOAA, NASA, USAF, Washington, D.C., 1976. f R. B. Montgomery, 1. Meteor., 4,193, 1947. 9 W. E. Forsythe, Smithsonian Physical Tables, 9th rev. ed., Smithsonian Institution, Washington, D.C., 1959. a
b
E.
415
STANDARD ATMOSPHERES
E. U.S. Standard Atmosphere 1976 and COSPAR International G
Reference Atmosphere (CIRA)b Height" Temp." (km) (K) 0 1 2 3 4 5 6 8 10 12 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120
288.15 281.65 275.15 268.66 262.17 255.68 249.19 236.22 223.25 216.65 216.65 216.65 221.55 226.51 236.51 250.35 264.16 270.65 260.77 247.02 233.29 219.59 208.40 198.64 188.89 186.87 188.42 195.08 208.84 240.00 300.00 360.00
Pressure" (kPa) 101.330 89.880 79.500 70.120 61.660 54.050 47.220 35.650 26.500 19.400 12.110 5.529 2.549 1.197 5.746 2.871 1.491 7.978 4.253 2.196 1.093 5.221 2.388 1.052 4.957 1.836 7.597 3.201 1.448 7.104 4.010 2.538
x x x x x x x x x x x x x x x x x x
10- 1 10- 1 10- 1 10- 2 10- 2 10- 2 10- 2 10- 3 10- 3 10- 3 10- 4 10- 4 10- 5 10- 5 10- 5 10- 6 10- 6 10- 6
Density" (kg m - 3) 1.225 1.112 1.007 9.093 8.194 7.364 6.601 5.258 4.135 3.119 1.948 8.891 4.008 1.841 8.463 3.996 1.966 1.027 5.681 3.097 1.632 8.283 3.992 1.846 8~220 3.416 1.393 5.604 2.325 9.708 4.289 2.222
x x x x x x x x x x x X X
x X X X X
x x x X X X X X
x x X
10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 1 10- 2 10- 2 10- 2 10- 3 10- 3 10- 3 10- 3 10- 4 10- 4 10- 4 10- 5 10- 5 10- 5 10- 6 10- 6 10- 6 10- 7 10- 7 10- 8 10- 8 10- 8
Temp." (K)
221.7 230.7 241.5 255.3 267.7 271.6 263.9 249.3 232.7 216.2 205.0 195.0 185.1 183.8 190.3 203.5 228.0 265.5 317.1 380.6
Pressure" (kPa)
2.483 1.175 5.741 2.911 1.525 8.241 4.406 2.296 1.146 5.445 2.460 1.067 4.426 1.795 7.345 3.090 1.422 7.362 4.236 2.667
x x x x x x x x x x x x x x x x x x
10- 1 10- 1 10- 1 10- 2 10- 2 10- 2 10- 2 10- 3 10- 3 10- 3 10- 4 10- 4 10- 5 10- 5 10- 5 10- 6 10- 6 10- 6
Density" (kg m ":')
3.899 x 10- 2 1.774 x 10- 2 8.279 X 10- 3 3.972 X 10- 3 1.995 x 10- 3 1.057 X 10- 3 5.821 X 10- 3 3.206 X 10- 3 1.718 X 10- 3 8.770 X 10- 5 4.178 X 10- 5 1.905 X 10- 5 8.337 X 10- 6 3.396 X 10- 6 1.343 X 10- 6 5.297 X 10- 7 2.173 X 10- 7 9.661 X 10- 8 4.645 X 10- 8 2.438 X 10- 8
"U.S. Standard Atmosphere, 1976, NOAA, NASA, USAF, Washington, D.C., 1976. b COSPAR International Reference Atmosphere, 1972, Akademie Verlag, Berlin, 1972.