Conversion Factors A Table A.1 Length mcmkmin.ftmi 1 meter 1 102 10−3 39.37 3.281 6.214 × 10−4 1 centimeter 10−2 110−5 0.3937 3.281 × 10−2 6.214 × 10−6 1 kilometer 103 105 13.937 × 104 3.281 × 103 0.621 4 1inch 2.540 × 10−2 2.540 2.540 × 10−5 18.333 × 10−2 1.578 × 10−5 1 foot 0.3048 30.48 3.048 × 10−4 12 1 1.894 × 10−4 1 mile 1 609 1.609 × 105 1.609 6.336 × 104 5280 1 Table A.2 Time s min h day year 1 second 1 1.667 × 10−2 2.778 × 10−4 1.157 × 10−5 3.169 × 10−8 1 minute 60 1 1.667 × 10−2 6.994 × 10−4 1.901 × 10−6 1 hour 3600 60 1 4.167 × 10−2 1.141 × 10−4 1day 8.640 × 104 1440 24 1 2.738 × 10−5 1 year 3.156 × 107 5.259 × 105 8.766 × 103 365.2 1 Table A.3 Area m2 cm2 ft2 in.2 1 square meter 1 104 10.76 1550 1 square centimeter 10−4 11.076 × 10−3 0.1550 1 square foot 9.290 × 10−2 929.0 1 144 1 square inch 6.452 × 10−4 6.452 6.944 × 10−3 1 Note 1 square kilometer = 247.108 acres H. A. Radi and J. O. Rasmussen, Principles of Physics, 999 Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-642-23026-4, © Springer-Verlag Berlin Heidelberg 2013 1000 A Conversion Factors Table A.4 Volume m3 cm3 Lft3 in.3 1 cubic meter 1 106 1000 35.51 6.102 × 104 1 cubic centimeter 10−6 11.000 × 10−3 3.531 × 10−5 6.102 × 10−2 1liter 1.000 × 10−3 1000 1 3.531 × 10−2 61.02 1 cubic foot 2.832 × 10−4 1 28.32 1 1728 1 cubic inch 1.639 × 10−4 16.39 1.639 × 10−2 5.787 × 10−4 1 Note 1 U.S. fluid gallon = 3.786L Table A.5 Speed m/s cm/s ft/s mi/h km/h 1 meter per second 1 102 3.281 2.237 3.6 1 centimeter per second 10−2 13.281 × 10−2 2.237 × 10−2 3.6 × 10−2 1 foot per second 0.3048 30.48 1 0.6818 1.097 1 mile per hour 0.4470 44.70 1.467 1 1.609 1 kilometer per hour 0.2778 27.78 0.9113 0.6214 1 Table A.6 Mass kg g slug u 1 kilogram 1 103 6.852 × 10−2 6.024 × 1026 1gram 10−3 16.852 × 10−5 6.024 × 1023 1 slug 14.59 1.459 × 104 18.789 × 1027 1 atomic mass unit 1.660 × 10−27 1.660 × 10−24 1.137 × 10−28 1 Note 1 metric ton = 1000kg Table A.7 Force Nlb 1 newton 1 0.2248 1 pound 4.448 1 Table A.8 Work, energy, and heat Jft.lb eV 1 joule 1 0.737 6 6.242 × 1018 1 foot-pound 1.356 1 8.464 × 1018 1electronvolt 1.602 × 10−19 1.182 × 10−19 1 1 calorie 4.186 3.087 2.613 × 1019 1 British thermal unit 1.055 × 103 7.779 × 102 6.585 × 1021 1 kilowatt hour 3.600 × 106 2.655 × 106 2.247 × 1025 A Conversion Factors 1001 Table A.8 Continued cal Btu kWh 1 joule 0.2389 9.481 × 10−4 2.778 × 10−7 1 foot-pound 0.3239 1.285 × 10−3 3.766 × 10−7 1electronvolt 3.827 × 10−20 1.519 × 10−22 4.450 × 10−26 1 calorie 1 3.968 × 10−3 1.163 × 10−6 1 British thermal unit 2.520 × 102 12.930 × 10−4 1 kilowatt hour 8.601 × 105 3.413 × 102 1 Table A.9 Pressure Pa atm cm Hg lb/in.2 lb/ft2 1 pascal 1 9.869 × 10−6 7.501 × 10−4 1.450 × 10−4 2.089 × 10−2 1 atmosphere 1.013 × 105 1 76 14.70 2.116 × 103 1 centimeter mercurya 1.333 × 103 1.316 × 10−2 1 0.194 3 27.85 1 pound per square inch 6.895 × 103 6.805 × 10−2 5.171 1 144 1 pound per square foot 47.88 4.725 × 10−4 3.591 × 10−2 6.944 × 10−3 1 aAt 0◦C and at a location where the free-fall acceleration has its “standard” value, 9.80665m/s2 Basic Rules and Formulas B Scientific Notation When numbers in powers of 10 are expressed in scientific notation are being multi- plied or divided, the following rules are very useful: + 10m × 10n = 10m n m (B.1) 10 − = 10m n 10n When powers of a given quantity x are multiplied or divided, the following rules hold: xm × xn = xm+n xm (B.2) = xm−n xn The Distance Between Two Points In Fig. B.1, P(x1, y1) and Q(x2, y2) are two different points in the (x, y) plane. As we move from point P to point Q, the coordinates x and y change by amounts that we denote by x and y (read “delta x” and “delta y”). Thus: The change in x = x = x − x 2 1 (B.3) The change in y = y = y2 − y1 One can calculate the distance between the two points P and Q from the theorem of Pythagoras in geometry such that: H. A. Radi and J. O. Rasmussen, Principles of Physics, 1003 Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-642-23026-4, © Springer-Verlag Berlin Heidelberg 2013 1004 B Basic Rules and Formulas 2 2 2 2 The distance PQ = (x) + (y) = (x2 − x1) + (y2 − y1) (B.4) Fig. B.1 y-axis Q(x2,y2) y P(x1,y1) x-axis (0,0) o Slope and the Equation of a Straight Line The slope of a line (usually given the symbol m) on which two points P and Q lie, is defined as the ratio y/x, see Fig. B.2. Thus: y slope ≡ m = (B.5) x Fig. B.2 y-axis Q y P x (0,b) x-axis o Using this basic geometric property, we can find the equation of a straight line in terms of a general point (x, y), and the y intercept b of the line with the y-axis and the slope m of the line, as follows: y = mx + b (B.6) B Basic Rules and Formulas 1005 Exponential and Logarithmic Functions An exponential function with base a has the following forms: y = ax (a > 0, a = 1) (B.7) where x is a variable and a is a constant, i.e., the exponential function is a constant raised to a variable power. Exponential functions are continuous on the interval (−∞, ∞) with a range [0, ∞] and have one of the basic two shapes shown in Fig. B.3. Fig. B.3 y x y ax y a (0 < a < 1) (a> 1) x 0 Moreover, some algebraic properties of exponential functions are: 1. ax × ay = ax+y 2.(ab)x = ax × bx 3.(ax) y = a xy ax (B.8) 4. = a x−y ay √ √ 5. ax/q = q ax = ( q a)x,(q integer and q > 0) 6. a0 = 1, (for every positive real number a) The logarithmic function to the base a of x is introduced as the inverse of the = y. = exponential function x a That is, y loga x is the power (or exponent) to which a must be raised to produce x, so that: = ( ) = y y loga x is equivalent to x a (B.9) 1006 B Basic Rules and Formulas Additionally, some algebraic properties of logarithmic functions for any base a are as follows: . ( ) = ( ) + ( ) 1 loga xy loga x loga y Product property 2. log (x/y) = log (x) − log (y) Quotient property a a a (B.10) . ( r) = ( ) 3 loga x r loga x Power property . ( / ) =− ( ) 4 loga 1 x loga x Reciprocal property Historically, the first logarithmic base was 10, called the common logarithm.For such logarithms it is usual to suppress explicit reference to the base and write log x . rather than log10 x However, the most widely used logarithms in applications are the natural logarithms, which have an irrational base denoted by the letter e in honor of L. Euler, who first suggested its application to logarithms. This constant’s value to six decimal places is: e ≈ 2.718282 (B.11) This number arises as the horizontal asymptote of the graph of the equation y = (1 + 1/x)x. Therefore, as x →±∞this allows us to express e as a limit and ex as an infinite sum such that: x 1 1 e = lim 1 + = lim (1 + x) x (B.12) x→±∞ x x→0 x2 x3 ∞ xn ex = + x + + + ... = 1 ! ! ! (B.13) 2 3 n = 0 n where the symbol n! is read as “n factorial” and by definition 1!=1, 0!=1, and n! are given by: n!=n × (n − 1) × (n − 2)...× 3 × 2 × 1 (B.14) Both expressions (B.11) and (B.12) are sometimes taken to be the definition of the . , number e Thus, loge x is the natural logarithm to the base e of x and it is usually denoted by ln x, so that: ≡ ln x loge x (B.15) B Basic Rules and Formulas 1007 and thus: y = ex (is equivalent to) ln y = x (B.16) The exponential function f (x) = ex is called the natural exponential function.Tosim- plify the typography, this function is sometimes written as exp(x), that is exp(x) ≡ ex.