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Marks'

Standard Handbook for Mechanical Engineers

Section

1

Mathematical Tables and Measuring Units

BY

GEORGE F. BAUMEISTER President, EMC Process Co., Newport, DE JOHN T. BAUMEISTER Manager, Product Compliance Test Center, Unisys Corp.

1.1 MATHEMATICAL TABLES by George F. Baumeister Segments of Circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 Regular Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-4 Compound Interest and Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5 Statistical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-9 Decimal Equivalents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15 1.2 MEASURING UNITS by John T. Baumeister U.S. Customary System (USCS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16 Metric System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17

The International System of Units (SI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-17 Systems of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-24 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Terrestrial Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Mohs Scale of Hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Density and Relative Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-26 Conversion and Equivalency Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-27

1.1

MATHEMATICAL TABLES

by George F. Baumeister

"A Short Table of Integrals," Ginn. "Mathematical Tables from Handbook of Chemistry and Physics," Chemical Rubber Co. "Handbook of Mathematical Functions," NBS.

REFERENCES FOR MATHEMATICAL TABLES: Dwight, "Mathematical Tables of Elementary and Some Higher Mathematical Functions," McGraw-Hill. Dwight, "Tables of Integrals and Other Mathematical Data," Macmillan. Jahnke and Emde, "Tables of Functions," B. G. Teubner, Leipzig, or Dover. Pierce-Foster,

1-1

1-2

MATHEMATICAL TABLES

Table 1.1.1 Segments of Circles, Given h/c Given: h height; c chord. To find the diameter of the circle, the length of arc, or the area of the segment, form the ratio h/c, and find from the table the value of (diam/c), (arc/c); then, by a simple multiplication, diam arc area The table gives also the angle subtended at the center, and the ratio of h to D. h c .00 1 2 3 4 .05 6 7 8 9 .10 1 2 3 4 .15 6 7 8 9 .20 1 2 3 4 .25 6 7 8 9 .30 1 2 3 4 .35 6 7 8 9 .40 1 2 3 4 .45 6 7 8 9 .50 Diam c 25.010 12.520 8.363 6.290 5.050 4.227 3.641 3.205 2.868 2.600 2.383 2.203 2.053 1.926 1.817 1.723 1.641 1.569 1.506 1.450 1.400 1.356 1.317 1.282 1.250 1.222 1.196 1.173 1.152 1.133 1.116 1.101 1.088 1.075 1.064 1.054 1.046 1.038 1.031 1.025 1.020 1.015 1.011 1.008 1.006 1.003 1.002 1.001 1.000 1.000 Diff Arc c 1.000 1.000 1.001 1.002 1.004 1.007 1.010 1.013 1.017 1.021 1.026 1.032 1.038 1.044 1.051 1.059 1.067 1.075 1.084 1.094 1.103 1.114 1.124 1.136 1.147 1.159 1.171 1.184 1.197 1.211 1.225 1.239 1.254 1.269 1.284 1.300 1.316 1.332 1.349 1.366 1.383 1.401 1.419 1.437 1.455 1.474 1.493 1.512 1.531 1.551 1.571 Diff 0 1 1 2 3 3 3 4 4 5 6 6 6 7 8 8 8 9 10 9 11 10 12 11 12 12 13 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 18 19 19 19 19 20 20 Area h3c .6667 .6667 .6669 .6671 .6675 .6680 .6686 .6693 .6701 .6710 .6720 .6731 .6743 .6756 .6770 .6785 .6801 .6818 .6836 .6855 .6875 .6896 .6918 .6941 .6965 .6989 .7014 .7041 .7068 .7096 .7125 .7154 .7185 .7216 .7248 .7280 .7314 .7348 .7383 .7419 .7455 .7492 .7530 .7568 .7607 .7647 .7687 .7728 .7769 .7811 .7854 Diff 0 2 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 24 25 27 27 28 29 29 31 31 32 32 34 34 35 36 36 37 38 38 39 40 40 41 41 42 43 Central angle, v 0.008 4.58 9.16 13.73 18.30 22.848 27.37 31.88 36.36 40.82 45.248 49.63 53.98 58.30 62.57 66.808 70.98 75.11 79.20 83.23 87.218 91.13 95.00 98.81 102.56 106.268 109.90 113.48 117.00 120.45 123.868 127.20 130.48 133.70 136.86 139.978 143.02 146.01 148.94 151.82 154.648 157.41 160.12 162.78 165.39 167.958 170.46 172.91 175.32 177.69 180.008 Diff 458 458 457 457 454 453 451 448 446 442 439 435 432 427 423 418 413 409 403 399 392 387 381 375 370 364 358 352 345 341 334 328 322 316 311 305 299 293 288 282 277 271 266 261 256 251 245 241 237 231 h Diam .0000 .0004 .0016 .0036 .0064 .0099 .0142 .0192 .0250 .0314 .0385 .0462 .0545 .0633 .0727 .0826 .0929 .1036 .1147 .1263 .1379 .1499 .1622 .1746 .1873 .2000 .2128 .2258 .2387 .2517 .2647 .2777 .2906 .3034 .3162 .3289 .3414 .3538 .3661 .3783 .3902 .4021 .4137 .4252 .4364 .4475 .4584 .4691 .4796 .4899 .5000 Diff 4 12 20 28 35 43 50 58 64 71 77 83 88 94 99 103 107 111 116 116 120 123 124 127 127 128 130 129 130 130 130 129 128 128 127 125 124 123 122 119 119 116 115 112 111 109 107 105 103 101 c c h (diam/c) (arc/c) c (area/h

c)

12490 *4157 *2073 *1240 *823 *586 *436 *337 *268 *217 *180 *150 *127 *109 *94 *82 *72 *63 56 50 44 39 35 32 28 26 23 21 19 17 15 13 13 11 10 8 8 7 6 5 5 4 3 2 3 1 1 1 0

* Interpolation may be inaccurate at these points.

MATHEMATICAL TABLES Table 1.1.2 Segments of Circles, Given h/D Given: h height; D diameter of circle. To find the chord, the length of arc, or the area of the segment, form the ratio h/D, and find from the table the value of (chord/D), (arc/D), or (area/D2); then by a simple multiplication, chord arc area D (chord/D) D (arc/D) D 2 (area/D 2)

1-3

This table gives also the angle subtended at the center, the ratio of the arc of the segment of the whole circumference, and the ratio of the area of the segment to the area of the whole circle. h D .00 1 2 3 4 .05 6 7 8 9 .10 1 2 3 4 .15 6 7 8 9 .20 1 2 3 4 .25 6 7 8 9 .30 1 2 3 4 .35 6 7 8 9 .40 1 2 3 4 .45 6 7 8 9 .50 Arc D 0.000 .2003 .2838 .3482 .4027 .4510 .4949 .5355 .5735 .6094 .6435 .6761 .7075 .7377 .7670 .7954 .8230 .8500 .8763 .9021 0.9273 0.9521 0.9764 1.0004 1.0239 1.0472 1.0701 1.0928 1.1152 1.1374 1.1593 1.1810 1.2025 1.2239 1.2451 1.2661 1.2870 1.3078 1.3284 1.3490 1.3694 1.3898 1.4101 1.4303 1.4505 1.4706 1.4907 1.5108 1.5308 1.5508 1.5708 Diff 2003 *835 *644 *545 *483 *439 *406 *380 *359 *341 *326 *314 *302 *293 *284 276 270 263 258 252 248 243 240 235 233 229 227 224 222 219 217 215 214 212 210 209 208 206 206 204 204 203 202 202 201 201 201 200 200 200 Area D2 .0000 .0013 .0037 .0069 .0105 .0147 .0192 .0242 .0294 .0350 .0409 .0470 .0534 .0600 .0668 .0739 .0811 .0885 .0961 .1039 .1118 .1199 .1281 .1365 .1449 .1535 .1623 .1711 .1800 .1890 .1982 .2074 .2167 .2260 .2355 .2450 .2546 .2642 .2739 .2836 .2934 .3032 .3130 .3229 .3328 .3428 .3527 .3627 .3727 .3827 .3927 Diff 13 24 32 36 42 45 50 52 56 59 61 64 66 68 71 72 74 76 78 79 81 82 84 84 86 88 88 89 90 92 92 93 93 95 95 96 96 97 97 98 98 98 99 99 100 99 100 100 100 100 Central angle, v 0.008 22.96 32.52 39.90 46.15 51.688 56.72 61.37 65.72 69.83 73.748 77.48 81.07 84.54 87.89 91.158 94.31 97.40 100.42 103.37 106.268 109.10 111.89 114.63 117.34 120.008 122.63 125.23 127.79 130.33 132.848 135.33 137.80 140.25 142.67 145.088 147.48 149.86 152.23 154.58 156.938 159.26 161.59 163.90 166.22 168.528 170.82 173.12 175.41 177.71 180.008 Diff 2296 * 956 * 738 * 625 * 553

* *

Chord D .0000 .1990 .2800 .3412 .3919 .4359 .4750 .5103 .5426 .5724 .6000 .6258 .6499 .6726 .6940 .7141 .7332 .7513 .7684 .7846 .8000 .8146 .8285 .8417 .8542 .8660 .8773 .8879 .8980 .9075 .9165 .9250 .9330 .9404 .9474 .9539 .9600 .9656 .9708 .9755 .9798 .9837 .9871 .9902 .9928 .9950 .9968 .9982 .9992 .9998 1.0000

Diff *1990 *810 *612 *507 *440 *391 *353 *323 *298 *276 *258 *241 *227 *214 *201 *191 *181 *171 162 154 146 139 132 125 118 113 106 101 95 90 85 80 74 70 65 61 56 52 47 43 39 34 31 26 22 18 14 10 6 2

Arc Circum .0000 .0638 .0903 .1108 .1282 .1436 .1575 .1705 .1826 .1940 .2048 .2152 .2252 .2348 .2441 .2532 .2620 .2706 .2789 .2871 .2952 .3031 .3108 .3184 .3259 .3333 .3406 .3478 .3550 .3620 .3690 .3759 .3828 .3896 .3963 .4030 .4097 .4163 .4229 .4294 .4359 .4424 .4489 .4553 .4617 .4681 .4745 .4809 .4873 .4936 .5000

Diff *638 *265 *205 *174 *154 *139 *130 121 114 108 104 100 96 93 91 88 86 83 82 81 79 77 76 75 74 73 72 72 70 70 69 69 68 67 67 67 66 66 65 65 65 65 64 64 64 64 64 64 63 64

Area Circle .0000 .0017 .0048 .0087 .0134 .0187 .0245 .0308 .0375 .0446 .0520 .0598 .0680 .0764 .0851 .0941 .1033 .1127 .1224 .1323 .1424 .1527 .1631 .1738 .1846 .1955 .2066 .2178 .2292 .2407 .2523 .2640 .2759 .2878 .2998 .3119 .3241 .3364 .3487 .3611 .3735 .3860 .3986 .4112 .4238 .4364 .4491 .4618 .4745 .4873 .5000

Diff 17 31 39 47 53 58 63 67 71 74 78 82 84 87 90 92 94 97 99 101 103 104 107 108 109 111 112 114 115 116 117 119 119 120 121 122 123 123 124 124 125 126 126 126 126 127 127 127 128 127

504 465 * 435 * 411 * 391

* *

374 359 * 347 * 335 * 326 316 309 302 295 289 284 279 274 271 266 263 260 256 254 251 249 247 245 242 241 240 238 237 235 235 233 233 231 232 230 230 230 229 230 229

* Interpolation may be inaccurate at these points.

1-4

MATHEMATICAL TABLES

Table 1.1.3 Regular Polygons n number of sides v 3608/n angle subtended at the center by one side v v a length of one side 5 R a2 sin b 5 r a2 tan b 2 2 v v R radius of circumscribed circle 5 a a1/2 csc b 5 r asec b 2 2 v v r radius of inscribed circle 5 R acos b 5 a a1/2 cot b 2 2 v v Area a 2 a1/4 n cot b 5 R2 s 1/2 n sin vd 5 r 2 an tan b 2 2 n 3 4 5 6 7 8 9 10 12 15 16 20 24 32 48 64 v 1208 908 728 608 518.43 458 408 368 308 248 228.50 188 158 118.25 78.50 58.625 Area a2 0.4330 1.000 1.721 2.598 3.634 4.828 6.182 7.694 11.20 17.64 20.11 31.57 45.58 81.23 183.1 325.7 Area R2 1.299 2.000 2.378 2.598 2.736 2.828 2.893 2.939 3.000 3.051 3.062 3.090 3.106 3.121 3.133 3.137 Area r2 5.196 4.000 3.633 3.464 3.371 3.314 3.276 3.249 3.215 3.188 3.183 3.168 3.160 3.152 3.146 3.144 R a 0.5774 0.7071 0.8507 1.0000 1.152 1.307 1.462 1.618 1.932 2.405 2.563 3.196 3.831 5.101 7.645 10.19 R r 2.000 1.414 1.236 1.155 1.110 1.082 1.064 1.052 1.035 1.022 1.020 1.013 1.009 1.005 1.002 1.001 a R 1.732 1.414 1.176 1.000 0.8678 0.7654 0.6840 0.6180 0.5176 0.4158 0.3902 0.3129 0.2611 0.1960 0.1308 0.0981 a r 3.464 2.000 1.453 1.155 0.9631 0.8284 0.7279 0.6498 0.5359 0.4251 0.3978 0.3168 0.2633 0.1970 0.1311 0.0983 r R 0.5000 0.7071 0.8090 0.8660 0.9010 0.9239 0.9397 0.9511 0.9659 0.9781 0.9808 0.9877 0.9914 0.9952 0.9979 0.9968 r a 0.2887 0.5000 0.6882 0.8660 1.038 1.207 1.374 1.539 1.866 2.352 2.514 3.157 3.798 5.077 7.629 10.18

Table 1.1.4 (n)0 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1

Binomial Coefficients nsn 2 1d nsn 2 1dsn 2 2d nsn 2 1dsn 2 2d c[n 2 sr 2 1d] n snd3 5 . Other notations: nCr 5 a b 5 sndr (n)I n snd2 5 etc. in general sndr 5 132 13233 1 3 2 3 3 3 c3 r r (n)1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

14, (n)14

(n)0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(n)2

(n)3 1 4 10 20 35 56 84 120 165 220 286 364 455

(n)4 1 5 15 35 70 126 210 330 495 715 1001 1365

1.

(n)5 1 6 21 56 126 252 462 792 1287 2002 3003

(n)6 1 7 28 84 210 462 924 1716 3003 5005

(n)7 1 8 36 120 330 792 1716 3432 6435

(n)8 1 9 45 165 495 1287 3003 6435

(n)9 1 10 55 220 715 2002 5005

(n)10 1 11 66 286 1001 3003

(n)11 1 12 78 364 1365

(n)12 1 13 91 455

(n)13 1 14 105

1 3 6 10 15 21 28 36 45 55 66 78 91 105

1; for n 15, (n)14

NOTE: For n

15, and (n)15

MATHEMATICAL TABLES Table 1.1.5 Compound Interest. Amount of a Given Principal The amount A at the end of n years of a given principal P placed at compound interest today is A P x or A P percent per annum) is compounded annually, or continuously; the factor x or y being taken from the following tables. Values of x (interest compounded annually: A Years 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 60 r 2 3 1.0300 1.0609 1.0927 1.1255 1.1593 1.1941 1.2299 1.2668 1.3048 1.3439 1.3842 1.4258 1.4685 1.5126 1.5580 1.6047 1.6528 1.7024 1.7535 1.8061 2.0938 2.4273 3.2620 4.3839 5.8916 4 1.0400 1.0816 1.1249 1.1699 1.2167 1.2653 1.3159 1.3686 1.4233 1.4802 1.5395 1.6010 1.6651 1.7317 1.8009 1.8730 1.9479 2.0258 2.1068 2.1911 2.6658 3.2434 4.8010 7.1067 10.520

[1 (r/100)]n.

1-5

y, according as the interest (at the rate of r

P 7

x) 8 1.0800 1.1664 1.2597 1.3605 1.4693 1.5869 1.7138 1.8509 1.9990 2.1589 2.3316 2.5182 2.7196 2.9372 3.1722 3.4259 3.7000 3.9960 4.3157 4.6610 6.8485 10.063 21.725 46.902 101.26 10 1.1000 1.2100 1.3310 1.4641 1.6105 1.7716 1.9487 2.1436 2.3579 2.5937 2.8531 3.1384 3.4523 3.7975 4.1772 4.5950 5.0545 5.5599 6.1159 6.7275 10.835 17.449 45.259 117.39 304.48 12 1.1200 1.2544 1.4049 1.5735 1.7623 1.9738 2.2107 2.4760 2.7731 3.1058 3.4785 3.8960 4.3635 4.8871 5.4736 6.1304 6.8660 7.6900 8.6128 9.6463 17.000 29.960 93.051 289.00 897.60

5 1.0500 1.1025 1.1576 1.2155 1.2763 1.3401 1.4071 1.4775 1.5513 1.6289 1.7103 1.7959 1.8856 1.9799 2.0789 2.1829 2.2920 2.4066 2.5270 2.6533 3.3864 4.3219 7.0400 11.467 18.679

6 1.0600 1.1236 1.1910 1.2625 1.3382 1.4185 1.5036 1.5938 1.6895 1.7908 1.8983 2.0122 2.1329 2.2609 2.3966 2.5404 2.6928 2.8543 3.0256 3.2071 4.2919 5.7435 10.286 18.420 32.988

1.0200 1.0404 1.0612 1.0824 1.1041 1.1262 1.1487 1.1717 1.1951 1.2190 1.2434 1.2682 1.2936 1.3195 1.3459 1.3728 1.4002 1.4282 1.4568 1.4859 1.6406 1.8114 2.2080 2.6916 3.2810

1.0700 1.1449 1.2250 1.3108 1.4026 1.5007 1.6058 1.7182 1.8385 1.9672 2.1049 2.2522 2.4098 2.5785 2.7590 2.9522 3.1588 3.3799 3.6165 3.8697 5.4274 7.6123 14.974 29.457 57.946

NOTE: This table is computed from the formula x

Values of y (interest compounded continuously: A Years 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 60

FORMULA: y

P 7

y) 8 1.0833 1.1735 1.2712 1.3771 1.4918 1.6161 1.7507 1.8965 2.0544 2.2255 2.4109 2.6117 2.8292 3.0649 3.3201 3.5966 3.8962 4.2207 4.5722 4.9530 7.3891 11.023 24.533 54.598 121.51 10 1.1052 1.2214 1.3499 1.4918 1.6487 1.8221 2.0138 2.2255 2.4596 2.7183 3.0042 3.3201 3.6693 4.0552 4.4817 4.9530 5.4739 6.0496 6.6859 7.3891 12.182 20.086 54.598 148.41 403.43 12 1.1275 1.2712 1.4333 1.6161 1.8221 2.0544 2.3164 2.6117 2.9447 3.3201 3.7434 4.2207 4.7588 5.3656 6.0496 6.8210 7.6906 8.6711 9.7767 11.023 20.086 36.598 121.51 403.43 1339.4

r

2

3 1.0305 1.0618 1.0942 1.1275 1.1618 1.1972 1.2337 1.2712 1.3100 1.3499 1.3910 1.4333 1.4770 1.5220 1.5683 1.6161 1.6653 1.7160 1.7683 1.8221 2.1170 2.4596 3.3201 4.4817 6.0496

n

4 1.0408 1.0833 1.1275 1.1735 1.2214 1.2712 1.3231 1.3771 1.4333 1.4918 1.5527 1.6161 1.6820 1.7507 1.8221 1.8965 1.9739 2.0544 2.1383 2.2255 2.7183 3.3201 4.9530 7.3891 11.023

5 1.0513 1.1052 1.1618 1.2214 1.2840 1.3499 1.4191 1.4918 1.5683 1.6487 1.7333 1.8221 1.9155 2.0138 2.1170 2.2255 2.3396 2.4596 2.5857 2.7183 3.4903 4.4817 7.3891 12.182 20.086

6 1.0618 1.1275 1.1972 1.2712 1.3499 1.4333 1.5220 1.6161 1.7160 1.8221 1.9348 2.0544 2.1815 2.3164 2.4596 2.6117 2.7732 2.9447 3.1268 3.3201 4.4817 6.0496 11.023 20.086 36.598

1.0202 1.0408 1.0618 1.0833 1.1052 1.1275 1.1503 1.1735 1.1972 1.2214 1.2461 1.2712 1.2969 1.3231 1.3499 1.3771 1.4049 1.4333 1.4623 1.4918 1.6487 1.8221 2.2255 2.7183 3.3201

e(r/100) .

1.0725 1.1503 1.2337 1.3231 1.4191 1.5220 1.6323 1.7507 1.8776 2.0138 2.1598 2.3164 2.4843 2.6645 2.8577 3.0649 3.2871 3.5254 3.7810 4.0552 5.7546 8.1662 16.445 33.115 66.686

1-6

MATHEMATICAL TABLES Table 1.1.6 Principal Which Will Amount to a Given Sum The principal P, which, if placed at compound interest today, will amount to a given sum A at the end of n years P A xr or P A yr, according as the interest (at the rate of r percent per annum) is compounded annually, or continuously; the factor xr or yr being taken from the following tables. Values of xr (interest compounded annually: P Years 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 60 r 2 3 .97087 .94260 .91514 .88849 .86261 .83748 .81309 .78941 .76642 .74409 .72242 .70138 .68095 .66112 .64186 .62317 .60502 .58739 .57029 .55368 .47761 .41199 .30656 .22811 .16973

(r/100)]

n

A 7

xr) 8 .92593 .85734 .79383 .73503 .68058 .63017 .58349 .54027 .50025 .46319 .42888 .39711 .36770 .34046 .31524 .29189 .27027 .25025 .23171 .21455 .14602 .09938 .04603 .02132 .00988 10 .90909 .82645 .75131 .68301 .62092 .56447 .51316 .46651 .42410 .38554 .35049 .31863 .28966 .26333 .23939 .21763 .19784 .17986 .16351 .14864 .09230 .05731 .02209 .00852 .00328 12 .89286 .79719 .71178 .63552 .56743 .50663 .45235 .40388 .36061 .32197 .28748 .25668 .22917 .20462 .18270 .16312 .14564 .13004 .11611 .10367 .05882 .03338 .01075 .00346 .00111

4 .96154 .92456 .88900 .85480 .82193 .79031 .75992 .73069 .70259 .67556 .64958 .62460 .60057 .57748 .55526 .53391 .51337 .49363 .47464 .45639 .37512 .30832 .20829 .14071 .09506

5 .95238 .90703 .86384 .82270 .78353 .74622 .71068 .67684 .64461 .61391 .58468 .55684 .53032 .50507 .48102 .45811 .43630 .41552 .39573 .37689 .29530 .23138 .14205 .08720 .05354

6 .94340 .89000 .83962 .79209 .74726 .70496 .66506 .62741 .59190 .55839 .52679 .49697 .46884 .44230 .41727 .39365 .37136 .35034 .33051 .31180 .23300 .17411 .09722 .05429 .03031

.98039 .96117 .94232 .92385 .90573 .88797 .87056 .85349 .83676 .82035 .80426 .78849 .77303 .75788 .74301 .72845 .71416 .70016 .68643 .67297 .60953 .55207 .45289 .37153 .30478

[1

.93458 .87344 .81630 .76290 .71299 .66634 .62275 .58201 .54393 .50835 .47509 .44401 .41496 .38782 .36245 .33873 .31657 .29586 .27651 .25842 .18425 .13137 .06678 .03395 .01726

FORMULA: xr

1/x.

Values of yr (interest compounded continuously: P Years 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 60 r 2 3 .97045 .94176 .91393 .88692 .86071 .83527 .81058 .78663 .76338 .74082 .71892 .69768 .67706 .65705 .63763 .61878 .60050 .58275 .56553 .54881 .47237 .40657 .30119 .22313 .16530

1/y.

A

yr) 8 .92312 .85214 .78663 .72615 .67032 .61878 .57121 .52729 .48675 .44933 .41478 .38289 .35345. .32628 .30119 .27804 .25666 .23693 .21871 .20190 .13534 .09072 .04076 .01832 .00823 10 .90484 .81873 .74082 .67032 .60653 .54881 .49659 .44933 .40657 .36788 .33287 .30119 .27253 .24660 .22313 .20190 .18268 .16530 .14957 .13534 .08208 .04979 .01832 .00674 .00248 12 .88692 .78663 .69768 .61878 .54881 .48675 .43171 .38289 .33960 .30119 .26714 .23693 .21014 .18637 .16530 .14661 .13003 .11533 .10228 .09072 .04979 .02732 .00823 .00248 .00075

4 .96079 .92312 .88692 .85214 .81873 .78663 .75578 .72615 .69768 .67032 .64404 .61878 .59452 .57121 .54881 .52729 .50662 .48675 .46767 .44933 .36788 .30119 .20190 .13534 .09072

5 .95123 .90484 .86071 .81873 .77880 .74082 .70469 .67032 .63763 .60653 .57695 .54881 .52205 .49659 .47237 .44933 .42741 .40657 .38674 .36788 .28650 .22313 .13534 .08208 .04979

6 .94176 .88692 .83527 .78663 .74082 .69768 .65705 .61878 .58275 .54881 .51685 .48675 .45841 .43171 .40657 .38289 .36059 .33960 .31982 .30119 .22313 .16530 .09072 .04979 .02732

7 .93239 .86936 .81058 .75578 .70469 .65705 .61263 .57121 .53259 .49659 .46301 .43171 .40252 .37531 .34994 .32628 .30422 .28365 .26448 .24660 .17377 .12246 .06081 .03020 .01500

.98020 .96079 .94176 .92312 .90484 .88692 .86936 .85214 .83527 .81873 .80252 .78663 .77105 .75578 .74082 .72615 .71177 .69768 .68386 .67032 .60653 .54881 .44933 .36788 .30119

e

(r/100) n

FORMULA: yr

MATHEMATICAL TABLES Table 1.1.7 Amount of an Annuity The amount S accumulated at the end of n years by a given annual payment Y set aside at the end of each year is S following table (interest at r percent per annum, compounded annually). Values of v Years 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 60 r 2 3 1.0000 2.0300 3.0909 4.1836 5.3091 6.4684 7.6625 8.8923 10.159 11.464 12.808 14.192 15.618 17.086 18.599 20.157 21.762 23.414 25.117 26.870 36.459 47.575 75.401 112.80 163.05

1} (r/100) (x

1-7

Y

v, where the factor v is to be taken from the

4 1.0000 2.0400 3.1216 4.2465 5.4163 6.6330 7.8983 9.2142 10.583 12.006 13.486 15.026 16.627 18.292 20.024 21.825 23.698 25.645 27.671 29.778 41.646 56.085 95.026 152.67 237.99

1) (r/100).

5 1.0000 2.0500 3.1525 4.3101 5.5256 6.8019 8.1420 9.5491 11.027 12.578 14.207 15.917 17.713 19.599 21.579 23.657 25.840 28.132 30.539 33.066 47.727 66.439 120.80 209.35 353.58

6 1.0000 2.0600 3.1836 4.3746 5.6371 6.9753 8.3938 9.8975 11.491 13.181 14.972 16.870 18.882 21.015 23.276 25.673 28.213 30.906 33.760 36.786 54.865 79.058 154.76 290.34 533.13

7 1.0000 2.0700 3.2149 4.4399 5.7507 7.1533 8.6540 10.260 11.978 13.816 15.784 17.888 20.141 22.550 25.129 27.888 30.840 33.999 37.379 40.995 63.249 94.461 199.64 406.53 813.52

8 1.0000 2.0800 3.2464 4.5061 5.8666 7.3359 8.9228 10.637 12.488 14.487 16.645 18.977 21.495 24.215 27.152 30.324 33.750 37.450 41.446 45.762 73.106 113.28 259.06 573.77 1253.2

10 1.0000 2.1000 3.3100 4.6410 6.1051 7.7156 9.4872 11.436 13.579 15.937 18.531 21.384 24.523 27.975 31.772 35.950 40.545 45.599 51.159 57.275 98.347 164.49 442.59 1163.9 3034.8

12 1.0000 2.1200 3.3744 4.7793 6.3528 8.1152 10.089 12.300 14.776 17.549 20.655 24.133 28.029 32.393 37.280 42.753 48.884 55.750 63.440 72.052 133.33 241.33 767.09 2400.0 7471.6

1.0000 2.0200 3.0604 4.1216 5.2040 6.3081 7.4343 8.5830 9.7546 10.950 12.169 13.412 14.680 15.974 17.293 18.639 20.012 21.412 22.841 24.297 32.030 40.568 60.402 84.579 114.05

(r/100)]n

FORMULA: v {[1

Table 1.1.8 Annuity Which Will Amount to a Given Sum (Sinking Fund) The annual payment Y which, if set aside at the end of each year, will amount with accumulated interest to a given sum S at the end of n years is Y the factor vr is given below (interest at r percent per annum, compounded annually). Values of vr Years 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 60

FORMULA: v

S

vr, where

r

2

3 1.0000 .49261 .32353 .23903 .18835 .15460 .13051 .11246 .09843 .08723 .07808 .07046 .06403 .05853 .05377 .04961 .04595 .04271 .03981 .03722 .02743 .02102 .01326 .00887 .00613

{[1 (r/100)]n 1} 1/v.

4 1.0000 .49020 .32035 .23549 .18463 .15076 .12661 .10853 .09449 .08329 .07415 .06655 .06014 .05467 .04994 .04582 .04220 .03899 .03614 .03358 .02401 .01783 .01052 .00655 .00420

5 1.0000 .48780 .31721 .23201 .18097 .14702 .12282 .10472 .09069 .07950 .07039 .06283 .05646 .05102 .04634 .04227 .03870 .03555 .03275 .03024 .02095 .01505 .00828 .00478 .00283

6 1.0000 .48544 .31411 .22859 .17740 .14336 .11914 .10104 .08702 .07587 .06679 .05928 .05296 .04758 .04296 .03895 .03544 .03236 .02962 .02718 .01823 .01265 .00646 .00344 .00188

7 1.0000 .48309 .31105 .22523 .17389 .13980 .11555 .09747 .08349 .07238 .06336 .05590 .04965 .04434 .03979 .03586 .03243 .02941 .02675 .02439 .01581 .01059 .00501 .00246 .00123

8 1.0000 .48077 .30803 .22192 .17046 .13632 .11207 .09401 .08008 .06903 .06008 .05270 .04652 .04130 .03683 .03298 .02963 .02670 .02413 .02185 .01368 .00883 .00386 .00174 .00080

10 1.0000 .47619 .30211 .21547 .16380 .12961 .10541 .08744 .07364 .06275 .05396 .04676 .04078 .03575 .03147 .02782 .02466 .02193 .01955 .01746 .01017 .00608 .00226 .00086 .00033

12 1.0000 .47170 .29635 .20923 .15741 .12323 .09912 .08130 .06768 .05698 .04842 .04144 .03568 .03087 .02682 .02339 .02046 .01794 .01576 .01388 .00750 .00414 .00130 .00042 .00013

1.0000 .49505 .32675 .24262 .19216 .15853 .13451 .11651 .10252 .09133 .08218 .07456 .06812 .06260 .05783 .05365 .04997 .04670 .04378 .04116 .03122 .02465 .01656 .01182 .00877

(r/100)

1-8

MATHEMATICAL TABLES

Table 1.1.9 Present Worth of an Annuity The capital C which, if placed at interest today, will provide for a given annual payment Y for a term of n years before it is exhausted is C w is given below (interest at r percent per annum, compounded annually). Values of w Years 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 60

FORMULA: w

Y

w, where the factor

r

2

3 .97087 1.9135 2.8286 3.7171 4.5797 5.4172 6.2303 7.0197 7.7861 8.5302 9.2526 9.9540 10.635 11.296 11.938 12.561 13.166 13.754 14.324 14.877 17.413 19.600 23.115 25.730 27.676

(r/100)] n} [r/100] v/x.

4 .96154 1.8861 2.7751 3.6299 4.4518 5.2421 6.0021 6.7327 7.4353 8.1109 8.7605 9.3851 9.9856 10.563 11.118 11.652 12.166 12.659 13.134 13.590 15.622 17.292 19.793 21.482 22.623

5 .95238 1.8594 2.7232 3.5460 4.3295 5.0757 5.7864 6.4632 7.1078 7.7217 8.3064 8.8633 9.3936 9.8986 10.380 10.838 11.274 11.690 12.085 12.462 14.094 15.372 17.159 18.256 18.929

6 .94340 1.8334 2.6730 3.4651 4.2124 4.9173 5.5824 6.2098 6.8017 7.3601 7.8869 8.3838 8.8527 9.2950 9.7122 10.106 10.477 10.828 11.158 11.470 12.783 13.765 15.046 15.762 16.161

7 .93458 1.8080 2.6243 3.3872 4.1002 4.7665 5.3893 5.9713 6.5152 7.0236 7.4987 7.9427 8.3577 8.7455 9.1079 9.4466 9.7632 10.059 10.336 10.594 11.654 12.409 13.332 13.801 14.039

8 .92593 1.7833 2.5771 3.3121 3.9927 4.6229 5.2064 5.7466 6.2469 6.7101 7.1390 7.5361 7.9038 8.2442 8.5595 8.8514 9.1216 9.3719 9.6036 9.8181 10.675 11.258 11.925 12.233 12.377

10 .90909 1.7355 2.4869 3.1699 3.7908 4.3553 4.8684 5.3349 5.7590 6.1446 6.4951 6.8137 7.1034 7.3667 7.6061 7.8237 8.0216 8.2014 8.3649 8.5136 9.0770 9.4269 9.7791 9.9148 9.9672

12 .89286 1.6901 2.4018 3.0373 3.6048 4.1114 4.5638 4.9676 5.3282 5.6502 5.9377 6.1944 6.4235 6.6282 6.8109 6.9740 7.1196 7.2497 7.3658 7.4694 7.8431 8.0552 8.2438 8.3045 8.3240

.98039 1.9416 2.8839 3.8077 4.7135 5.6014 6.4720 7.3255 8.1622 8.9826 9.7868 10.575 11.348 12.106 12.849 13.578 14.292 14.992 15.678 16.351 19.523 22.396 27.355 31.424 34.761

{1 [1

Table 1.1.10 Annuity Provided for by a Given Capital The annual payment Y provided for a term of n years by a given capital C placed at interest today is Y the fund supposed to be exhausted at the end of the term). Values of wr Years 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 19 20 25 30 40 50 60

FORMULA: wr

C

wr (interest at r percent per annum, compounded annually;

r

2

3 1.0300 .52261 .35353 .26903 .21835 .18460 .16051 .14246 .12843 .11723 .10808 .10046 .09403 .08853 .08377 .07961 .07595 .07271 .06981 .06722 .05743 .05102 .04326 .03887 .03613

{1 [1 (r/100)] n} 1/w

4 1.0400 .53020 .36035 .27549 .22463 .19076 .16661 .14853 .13449 .12329 .11415 .10655 .10014 .09467 .08994 .08582 .08220 .07899 .07614 .07358 .06401 .05783 .05052 .04655 .04420

v (r/100).

5 1.0500 .53780 .36721 .28201 .23097 .19702 .17282 .15472 .14069 .12950 .12039 .11283 .10646 .10102 .09634 .09227 .08870 .08555 .08275 .08024 .07095 .06505 .05828 .05478 .05283

6 1.0600 .54544 .37411 .28859 .23740 .20336 .17914 .16104 .14702 .13587 .12679 .11928 .11296 .10758 .10296 .09895 .09544 .09236 .08962 .08718 .07823 .07265 .06646 .06344 .06188

7 1.0700 .55309 .38105 .29523 .24389 .20980 .18555 .16747 .15349 .14238 .13336 .12590 .11965 .11434 .10979 .10586 .10243 .09941 .09675 .09439 .08581 .08059 .07501 .07246 .07123

8 1.0800 .56077 .38803 .30192 .25046 .21632 .19207 .17401 .16008 .14903 .14008 .13270 .12652 .12130 .11683 .11298 .10963 .10670 .10413 .10185 .09368 .08883 .08386 .08174 .08080

10 1.1000 .57619 .40211 .31547 .26380 .22961 .20541 .18744 .17364 .16275 .15396 .14676 .14078 .13575 .13147 .12782 .12466 .12193 .11955 .11746 .11017 .10608 .10226 .10086 .10033

12 1.1200 .59170 .41635 .32923 .27741 .24323 .21912 .20130 .18768 .17698 .16842 .16144 .15568 .15087 .14682 .14339 .14046 .13794 .13576 .13388 .12750 .12414 .12130 .12042 .12013

1.0200 .51505 .34675 .26262 .21216 .17853 .15451 .13651 .12252 .11133 .10218 .09456 .08812 .08260 .07783 .07365 .06997 .06670 .06378 .06116 .05122 .04465 .03656 .03182 .02877

[r/100]

MATHEMATICAL TABLES Table 1.1.11 Ordinates of the Normal Density Function 1 2 fsxd 5 e2x >2 !2p x .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 .00 .3989 .3970 .3910 .3814 .3683 .3521 .3332 .3123 .2897 .2661 .2420 .2179 .1942 .1714 .1497 .1295 .1109 .0940 .0790 .0656 .0540 .0440 .0355 .0283 .0224 .0175 .0136 .0104 .0079 .0060 .0044 .0033 .0024 .0017 .0012 .0009 .0006 .0004 .0003 .0002 .01 .3989 .3965 .3902 .3802 .3668 .3503 .3312 .3101 .2874 .2637 .2396 .2155 .1919 .1691 .1476 .1276 .1092 .0925 .0775 .0644 .0529 .0431 .0347 .0277 .0219 .0171 .0132 .0101 .0077 .0058 .0043 .0032 .0023 .0017 .0012 .0008 .0006 .0004 .0003 .0002 .02 .3989 .3961 .3894 .3790 .3653 .3485 .3292 .3079 .2850 .2613 .2371 .2131 .1895 .1669 .1456 .1257 .1074 .0909 .0761 .0632 .0519 .0422 .0339 .0270 .0213 .0167 .0129 .0099 .0075 .0056 .0042 .0031 .0022 .0016 .0012 .0008 .0006 .0004 .0003 .0002 .03 .3988 .3956 .3885 .3778 .3637 .3467 .3271 .3056 .2827 .2589 .2347 .2107 .1872 .1647 .1435 .1238 .1057 .0893 .0748 .0620 .0508 .0413 .0332 .0264 .0208 .0163 .0126 .0096 .0073 .0055 .0040 .0030 .0022 .0016 .0011 .0008 .0005 .0004 .0003 .0002 .04 .3986 .3951 .3876 .3765 .3621 .3448 .3251 .3034 .2803 .2565 .2323 .2083 .1849 .1626 .1415 .1219 .1040 .0878 .0734 .0608 .0498 .0404 .0325 .0258 .0203 .0158 .0122 .0093 .0071 .0053 .0039 .0029 .0021 .0015 .0011 .0008 .0005 .0004 .0003 .0002

0.0404.

1-9

.05 .3984 .3945 .3867 .3752 .3605 .3429 .3230 .3011 .2780 .2541 .2299 .2059 .1826 .1604 .1394 .1200 .1023 .0863 .0721 .0596 .0488 .0396 .0317 .0252 .0198 .0154 .0119 .0091 .0069 .0051 .0038 .0028 .0020 .0015 .0010 .0007 .0005 .0004 .0002 .0002

.06 .3982 .3939 .3857 .3739 .3589 .3410 .3209 .2989 .2756 .2516 .2275 .2036 .1804 .1582 .1374 .1182 .1006 .0848 .0707 .0584 .0478 .0387 .0310 .0246 .0194 .0151 .0116 .0088 .0067 .0050 .0037 .0027 .0020 .0014 .0010 .0007 .0005 .0003 .0002 .0002

.07 .3980 .3932 .3847 .3725 .3572 .3391 .3187 .2966 .2732 .2492 .2251 .2012 .1781 .1561 .1354 .1163 .0989 .0833 .0694 .0573 .0468 .0379 .0303 .0241 .0189 .0147 .0113 .0086 .0065 .0048 .0036 .0026 .0019 .0014 .0010 .0007 .0005 .0003 .0002 .0002

.08 .3977 .3925 .3836 .3712 .3555 .3372 .3166 .2943 .2709 .2468 .2227 .1989 .1758 .1539 .1334 .1154 .0973 .0818 .0681 .0562 .0459 .0371 .0297 .0235 .0184 .0143 .0110 .0084 .0063 .0047 .0035 .0025 .0018 .0013 .0009 .0007 .0005 .0003 .0002 .0001

.09 .3973 .3918 .3825 .3697 .3538 .3352 .3144 .2920 .2685 .2444 .2203 .1965 .1736 .1518 .1315 .1127 .0957 .0804 .0669 .0551 .0449 .0363 .0290 .0229 .0180 .0139 .0107 .0081 .0061 .0046 .0034 .0025 .0018 .0013 .0009 .0006 .0004 .0003 .0002 .0001

NOTE: x is the value in left-hand column the value in top row. f(x) is the value in the body of the table. Example: x 2.14; f (x)

1-10

MATHEMATICAL TABLES Table 1.1.12 Fsxd 5 3 x .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4

x

Cumulative Normal Distribution 1 2 e2t / 2 dt 2` !2p .00 .5000 .5398 .5793 .6179 .6554 .6915 .7257 .7580 .7881 .8159 .8413 .8643 .8849 .9032 .9192 .9332 .9452 .9554 .9641 .9713 .9772 .9812 .9861 .9893 .9918 .9938 .9953 .9965 .9974 .9981 .9986 .9990 .9993 .9995 .9997 .01 .5040 .5438 .5832 .6217 .6591 .6950 .7291 .7611 .7910 .8186 .8438 .8665 .8869 .9049 .9207 .9345 .9463 .9564 .9649 .9719 .9778 .9826 .9864 .9896 .9920 .9940 .9955 .9966 .9975 .9982 .9987 .9991 .9993 .9995 .9997 .02 .5080 .5478 .5871 .6255 .6628 .6985 .7324 .7642 .7939 .8212 .8461 .8686 .8888 .9066 .9222 .9357 .9474 .9573 .9656 .9726 .9783 .9830 .9868 .9898 .9922 .9941 .9956 .9967 .9976 .9982 .9987 .9991 .9994 .9995 .9997 .03 .5120 .5517 .5910 .6293 .6664 .7019 .7357 .7673 .7967 .8238 .8485 .8708 .8906 .9082 .9236 .9370 .9484 .9582 .9664 .9732 .9788 .9834 .9871 .9901 .9925 .9943 .9957 .9968 .9977 .9983 .9988 .9991 .9994 .9996 .9997 .04 .5160 .5557 .5948 .6331 .6700 .7054 .7389 .7703 .7995 .8264 .8508 .8729 .8925 .9099 .9251 .9382 .9495 .9591 .9671 .9738 .9793 .9838 .9875 .9904 .9927 .9945 .9959 .9969 .9977 .9984 .9988 .9992 .9994 .9996 .9997 .05 .5199 .5596 .5987 .6368 .6736 .7088 .7422 .7734 .8023 .8289 .8531 .8749 .8943 .9115 .9265 .9394 .9505 .9599 .9678 .9744 .9798 .9842 .9878 .9906 .9929 .9946 .9960 .9970 .9978 .9984 .9989 .9992 .9994 .9996 .9997 .06 .5239 .5636 .6026 .6406 .6772 .7123 .7454 .7764 .8051 .8315 .8554 .8770 .8962 .9131 .9279 .9406 .9515 .9608 .9686 .9750 .9803 .9846 .9881 .9909 .9931 .9948 .9961 .9971 .9979 .9985 .9989 .9992 .9994 .9996 .9997 .07 .5279 .5675 .6064 .6443 .6808 .7157 .7486 .7793 .8078 .8340 .8577 .8790 .8980 .9147 .9292 .9418 .9525 .9616 .9693 .9756 .9808 .9850 .9884 .9911 .9932 .9949 .9962 .9972 .9979 .9985 .9989 .9992 .9995 .9996 .9997 .08 .5319 .5714 .6103 .6480 .6844 .7190 .7517 .7823 .8106 .8365 .8599 .8810 .8997 .9162 .9306 .9429 .9535 .9625 .9699 .9761 .9812 .9854 .9887 .9913 .9934 .9951 .9963 .9973 .9980 .9986 .9990 .9993 .9995 .9996 .9997 .09 .5359 .5735 .6141 .6517 .6879 .7224 .7549 .7852 .8133 .8389 .8621 .8830 .9015 .9177 .9319 .9441 .9545 .9633 .9706 .9767 .9817 .9857 .9890 .9916 .9936 .9952 .9964 .9974 .9981 .9986 .9990 .9993 .9995 .9997 .9998

NOTE: x (a m)/s where a is the observed value, m is the mean, and s is the standard deviation. x is the value in the left-hand column the value in the top row. F(x) is the probability that a point will be less than or equal to x. F(x) is the value in the body of the table. Example: The probability that an observation will be less than or equal to 1.04 is .8508. NOTE: F( x) 1 F(x).

MATHEMATICAL TABLES Table 1.1.13 Cumulative Chi-Square Distribution t x sn22d/2e2x/2 dx Fstd 5 3 n/2 0 2 [sn 2 2d/2]! F n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 .005 .000039 .0100 .0717 .207 .412 .676 .989 1.34 1.73 2.16 2.60 3.07 3.57 4.07 4.60 5.14 5.70 6.26 6.84 7.43 8.03 8.64 9.26 9.89 10.5 11.2 11.8 12.5 13.1 13.8 .010 .00016 .0201 .155 .297 .554 .872 1.24 1.65 2.09 2.56 3.05 3.57 4.11 4.66 5.23 5.81 6.41 7.01 7.63 8.26 8.90 9.54 10.2 10.9 11.5 12.2 12.9 13.6 14.3 15.0 .025 .00098 .0506 .216 .484 .831 1.24 1.69 2.18 2.70 3.25 3.82 4.40 5.01 5.63 6.26 6.91 7.56 8.23 8.91 9.59 10.3 11.0 11.7 12.4 13.1 13.8 14.6 15.3 16.0 16.8 .050 .0039 .103 .352 .711 1.15 1.64 2.17 2.73 3.33 3.94 4.57 5.23 5.89 6.57 7.26 7.96 8.67 9.39 10.1 10.9 11.6 12.3 13.1 13.8 14.6 15.4 16.2 16.9 17.7 18.5 .100 .0158 .211 .584 1.06 1.61 2.20 2.83 3.49 4.17 4.87 5.58 6.30 7.04 7.79 8.55 9.31 10.1 10.9 11.7 12.4 13.2 14.0 14.8 15.7 16.5 17.3 18.1 18.9 19.8 20.6 .250 .101 .575 1.21 1.92 2.67 3.45 4.25 5.07 5.90 6.74 7.58 8.44 9.30 10.2 11.0 11.9 12.8 13.7 14.6 15.5 16.3 17.2 18.1 19.0 19.9 20.8 21.7 22.7 23.6 24.5 .500 .455 1.39 2.37 3.36 4.35 5.35 6.35 7.34 8.34 9.34 10.3 11.3 12.3 13.3 14.3 15.3 16.3 17.3 18.3 19.3 20.3 21.3 22.3 23.3 24.3 25.3 26.3 27.3 28.3 29.3 .750 1.32 2.77 4.11 5.39 6.63 7.84 9.04 10.2 11.4 12.5 13.7 14.8 16.0 17.1 18.2 19.4 20.5 21.6 22.7 23.8 24.9 26.0 27.1 28.2 29.3 30.4 31.5 32.6 33.7 34.8

23.5 is .900.

1-11

.900 2.70 4.61 6.25 7.78 9.24 10.6 12.0 13.4 14.7 16.0 17.3 18.5 19.8 21.1 22.3 23.5 24.8 26.0 27.2 28.4 29.6 30.8 32.0 33.2 34.4 35.6 36.7 37.9 39.1 40.3

.950 3.84 5.99 7.81 9.49 11.1 12.6 14.1 15.5 16.9 18.3 19.7 21.0 22.4 23.7 25.0 26.3 27.6 28.9 30.1 31.4 32.7 33.9 35.2 36.4 37.7 38.9 40.1 41.3 42.6 43.8

.975 5.02 7.38 9.35 11.1 12.8 14.4 16.0 17.5 19.0 20.5 21.9 23.3 24.7 26.1 27.5 28.8 30.2 31.5 32.9 34.2 35.5 36.8 38.1 39.4 40.6 41.9 43.2 44.5 45.7 47.0

.990 6.62 9.21 11.3 13.3 15.1 16.8 18.5 20.1 21.7 23.2 24.7 26.2 27.7 29.1 30.6 32.0 33.4 34.8 36.2 37.6 38.9 40.3 41.6 43.0 44.3 45.6 47.0 48.3 49.6 50.9

.995 7.86 10.6 12.8 14.9 16.7 18.5 20.3 22.0 23.6 25.2 26.8 28.3 29.8 31.3 32.8 34.3 35.7 37.2 38.6 40.0 41.4 42.8 44.2 45.6 46.9 48.3 49.6 51.0 52.3 53.7

NOTE: n is the number of degrees of freedom. Values for t are in the body of the table. Example: The probability that, with 16 degrees of freedom, a point will be

1-12

MATHEMATICAL TABLES Table 1.1.14 Fstd 5 3

t

2`

Cumulative "Student's" Distribution n21 b! a 2 dx n22 x 2 sn11d/2 a b ! 2pn a1 1 b 2 n .75 .90 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.296 1.289 .95 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.671 1.658 .975 12.70 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000 1.980 .99 31.82 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.390 2.385 .995 63.66 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660 2.617 .9995 636.3 31.60 12.92 8.610 6.859 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.768 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.460 3.373

2.921 is

F n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 1.000 .816 .765 .741 .727 .718 .711 .706 .703 .700 .697 .695 .694 .692 .691 .690 .689 .688 .688 .687 .686 .686 .685 .685 .684 .684 .684 .683 .683 .683 .681 .679 .677

NOTE: n is the number of degrees of freedom. Values for t are in the body of the table. Example: The probability that, with 16 degrees of freedom, a point will be .995. NOTE: F( t) 1 F(t).

MATHEMATICAL TABLES Table 1.1.15 Cumulative F Distribution m degrees of freedom in numerator; n in denominator F [sm 1 n 2 2d/2]!m m/2nn/2x sm22d/2 sn 1 mxd2sm1nd/2 dx GsFd 5 3 [sm 2 2d/2]![sn 2 2d/2]! 0 Upper 5% points (F.95) Degrees of freedom for numerator 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 60 120 161 18.5 10.1 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.32 4.30 4.28 4.26 4.24 4.17 4.08 4.00 3.92 3.84

1-13

2

200 19.0 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.47 3.44 3.42 3.40 3.39 3.32 3.23 3.15 3.07 3.00

3

216 19.2 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 3.07 3.05 3.03 3.01 2.99 2.92 2.84 2.76 2.68 2.60

4

225 19.2 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.69 2.61 2.53 2.45 2.37

5

230 19.3 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.53 2.45 2.37 2.29 2.21

6

234 19.3 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.42 2.34 2.25 2.18 2.10

7

237 19.4 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.49 2.46 2.44 2.42 2.40 2.33 2.25 2.17 2.09 2.01

8

239 19.4 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.27 2.18 2.10 2.02 1.94

9

241 19.4 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.21 2.12 2.04 1.96 1.88

10

242 19.4 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.16 2.08 1.99 1.91 1.83

12

244 19.4 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.09 2.00 1.92 1.83 1.75

15

246 19.4 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.18 2.15 2.13 2.11 2.09 2.01 1.92 1.84 1.75 1.67

20

248 19.4 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.10 2.07 2.05 2.03 2.01 1.93 1.84 1.75 1.66 1.57

24

249 19.5 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.51 2.42 2.35 2.29 2.24 2.19 2.15 2.11 2.08 2.05 2.03 2.01 1.98 1.96 1.89 1.79 1.70 1.61 1.52

30

250 19.5 8.62 5.75 4.50 3.81 3.38 3.08 2.86 2.70 2.57 2.47 2.38 2.31 2.25 2.19 2.15 2.11 2.07 2.04 2.01 1.98 1.96 1.94 1.92 1.84 1.74 1.65 1.55 1.46

40

251 19.5 8.59 5.72 4.46 3.77 3.34 3.04 2.83 2.66 2.53 2.43 2.34 2.27 2.20 2.15 2.10 2.06 2.03 1.99 1.96 1.94 1.91 1.89 1.87 1.79 1.69 1.59 1.50 1.39

60

252 19.5 8.57 5.69 4.43 3.74 3.30 3.01 2.79 2.62 2.49 2.38 2.30 2.22 2.16 2.11 2.06 2.02 1.98 1.95 1.92 1.89 1.86 1.84 1.82 1.74 1.64 1.53 1.43 1.32

120

253 19.5 8.55 5.66 4.40 3.70 3.27 2.97 2.75 2.58 2.45 2.34 2.25 2.18 2.11 2.06 2.01 1.97 1.93 1.90 1.87 1.84 1.81 1.79 1.77 1.68 1.58 1.47 1.35 1.22 254 19.5 8.53 5.63 4.37 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71 1.62 1.51 1.39 1.25 1.00

Degrees of freedom for denominator

Upper 1% points (F.99) Degrees of freedom for numerator 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 60 120 4052 98.5 34.1 21.2 16.3 13.7 12.2 11.3 10.6 10.0 9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.19 8.10 8.02 7.95 7.88 7.82 7.77 7.56 7.31 7.08 6.85 6.63

2

5000 99.0 30.8 18.0 13.3 !0.9 9.55 8.65 8.02 7.56 7.21 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.78 5.72 5.66 5.61 5.57 5.39 5.18 4.98 4.79 4.61

3

5403 99.2 29.5 16.7 12.1 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.19 5.09 5.01 4.94 4.87 4.82 4.76 4.72 4.68 4.51 4.31 4.13 3.95 3.78

4

5625 99.2 28.7 16.0 11.4 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.37 4.31 4.26 4.22 4.18 4.02 3.83 3.65 3.48 3.32

5

5764 99.3 28.2 15.5 11.0 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.70 4.56 4.44 4.34 4.25 4.17 4.10 4.04 3.99 3.94 3.90 3.86 3.70 3.51 3.34 3.17 3.02

6

5859 99.3 27.9 15.2 10.7 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.81 3.76 3.71 3.67 3.63 3.47 3.29 3.12 2.96 2.80

7

5928 99.4 27.7 15.0 10.5 8.26 6.99 6.18 5.61 5.20 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.77 3.70 3.64 3.59 3.54 3.50 3.46 3.30 3.12 2.95 2.79 2.64

8

5982 99.4 27.5 14.8 10.3 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 3.71 3.63 3.56 3.51 3.45 3.41 3.36 3.32 3.17 2.99 2.82 2.66 2.51

9

6023 99.4 27.3 14.7 10.2 7.98 6.72 5.91 5.35 4.94 4.63 4.39 4.19 4.03 3.89 3.78 3.68 3.60 3.52 3.46 3.40 3.35 3.30 3.26 3.22 3.07 2.89 2.72 2.56 2.41

10

6056 99.4 27.2 14.5 10.1 7.87 6.62 5.81 5.26 4.85 4.54 4.30 4.10 3.94 3.80 3.69 3.59 3.51 3.43 3.37 3.31 3.26 3.21 3.17 3.13 2.98 2.80 2.63 2.47 2.32

12

6106 99.4 27.1 14.4 9.89 7.72 6.47 5.67 5.11 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.17 3.12 3.07 3.03 2.99 2.84 2.66 2.50 2.34 2.18

15

6157 99.4 26.9 14.2 9.72 7.56 6.31 5.52 4.96 4.56 4.25 4.01 3.82 3.66 3.52 3.41 3.31 3.23 3.15 3.09 3.03 2.98 2.93 2.89 2.85 2.70 2.52 2.35 2.19 2.04

20

6209 99.4 26.7 14.0 9.55 7.40 6.16 5.36 4.81 4.41 4.10 3.86 3.66 3.51 3.37 3.26 3.16 3.08 3.00 2.94 2.88 2.83 2.78 2.74 2.70 2.55 2.37 2.20 2.03 1.88

24

6235 99.5 26.6 13.9 9.47 7.31 6.07 5.28 4.73 4.33 4.02 3.78 3.59 3.43 3.29 3.18 3.08 3.00 2.92 2.86 2.80 2.75 2.70 2.66 2.62 2.47 2.29 2.12 1.95 1.79

30

6261 99.5 26.5 13.8 9.38 7.23 5.99 5.20 4.65 4.25 3.94 3.70 3.51 3.35 3.21 3.10 3.00 2.92 2.84 2.78 2.72 2.67 2.62 2.58 2.53 2.39 2.20 2.03 1.86 1.70

40

6287 99.5 26.4 13.7 9.29 7.14 5.91 5.12 4.57 4.17 3.86 3.62 3.43 3.27 3.13 3.02 2.92 2.84 2.76 2.69 2.64 2.58 2.54 2.49 2.45 2.30 2.11 1.94 1.76 1.59

60

6313 99.5 26.3 13.7 9.20 7.06 5.82 5.03 4.48 4.08 3.78 3.54 3.34 3.18 3.05 2.93 2.83 2.75 2.67 2.61 2.55 2.50 2.45 2.40 2.36 2.21 2.02 1.84 1.66 1.47

120

6339 99.5 26.2 13.6 9.11 6.97 5.74 4.95 4.40 4.40 3.69 3.45 3.25 3.09 2.96 2.84 2.75 2.66 2.58 2.52 2.46 2.40 2.35 2.31 2.27 2.11 1.92 1.73 1.53 1.32 6366 99.5 26.1 13.5 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.17 3.00 2.87 2.75 2.65 2.57 2.49 2.42 2.36 2.31 2.26 2.21 2.17 2.01 1.80 1.60 1.38 1.00

NOTE: m is the number of degrees of freedom in the numerator of F; n is the number of degrees of freedom in the denominator of F. Values for F are in the body of the table. G is the probability that a point, with m and n degrees of freedom will be F. Example: With 2 and 5 degrees of freedom, the probability that a point will be 13.3 is .99. SOURCE: "Chemical Engineers' Handbook," 5th edition, by R. H. Perry and C. H. Chilton, McGraw-Hill, 1973. Used with permission.

Degrees of freedom for denominator

1-14

MATHEMATICAL TABLES Table 1.1.16 a y r n a r 0.0 1 2 3 4 0.5 6 7 8 9 1.0 1 2 3 4 1.5 6 7 8 9 2.0 1 2 3 4 Standard Distribution of Residuals a

any positive quantity the number of residuals which are numerically the probable error of a single observation number of observations y n .000 .054 .107 .160 .213 .264 .314 .363 .411 .456 .500 .542 .582 .619 .655 .688 .719 .748 .775 .800 .823 .843 .862 .879 .895 Diff 54 53 53 53 51 50 49 48 45 44 42 40 37 36 33 31 29 27 25 23 20 19 17 16 13 a r 2.5 6 7 8 9 3.0 1 2 3 4 3.5 6 7 8 9 4.0 5.0

y n .908 .921 .931 .941 .950 .957 .963 .969 .974 .978 .982 .985 .987 .990 .991 .993 .999

Diff 13 10 10 9 7 6 6 5 4 4 3 2 3 1 2 6

Table 1.1.17 n

Factors for Computing Probable Error Bessel Peters 0.8453 !nsn 2 1d .5978 .3451 .2440 .1890 .1543 .1304 .1130 .0996 .0891 .0806 .0736 .0677 .0627 .0583 .0546 .0513 .0483 .0457 .0434 .0412 .0393 .0376 .0360 .0345 .0332 .0319 .0307 .0297 0.8453 n!n 2 1 .4227 .1993 .1220 .0845 .0630 .0493 .0399 .0332 .0282 .0243 .0212 .0188 .0167 .0151 .0136 .0124 .0114 .0105 .0097 .0090 .0084 .0078 .0073 .0069 .0065 .0061 .0058 .0055 30 31 32 33 34 35 36 37 38 39 40 45 50 55 60 65 70 75 80 85 90 95 100 n 0.6745 !sn 2 1d .1252 .1231 .1211 .1192 .1174 .1157 .1140 .1124 .1109 .1094 .1080 .1017 .0964 .0918 .0878 .0843 .0812 .0784 .0759 .0736 .0715 .0696 .0678 Bessel 0.6745 !nsn 2 1d .0229 .0221 .0214 .0208 .0201 .0196 .0190 .0185 .0180 .0175 .0171 .0152 .0136 .0124 .0113 .0105 .0097 .0091 .0085 .0080 .0075 .0071 .0068 0.8453 !nsn 2 1d .0287 .0277 .0268 .0260 .0252 .0245 .0238 .0232 .0225 .0220 .0214 .0190 .0171 .0155 .0142 .0131 .0122 .0113 .0106 .0100 .0094 .0089 .0085 Peters 0.8453 n!n 2 1 .0052 .0050 .0047 .0045 .0043 .0041 .0040 .0038 .0037 .0035 .0034 .0028 .0024 .0021 .0018 .0016 .0015 .0013 .0012 .0011 .0010 .0009 .0008

0.6745 !sn 2 1d 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 .6745 .4769 .3894 .3372 .3016 .2754 .2549 .2385 .2248 .2133 .2034 .1947 .1871 .1803 .1742 .1686 .1636 .1590 .1547 .1508 .1472 .1438 .1406 .1377 .1349 .1323 .1298 .1275

0.6745 !nsn 2 1d .4769 .2754 .1947 .1508 .1231 .1041 .0901 .0795 .0711 .0643 .0587 .0540 .0500 .0465 .0435 .0409 .0386 .0365 .0346 .0329 .0314 .0300 .0287 .0275 .0265 .0255 .0245 .0237

MATHEMATICAL TABLES Table 1.1.18 Decimal Equivalents Common fractions From minutes and seconds into decimal parts of a degree 0r 1 2 3 4 5r 6 7 8 9 10r 1 2 3 4 15r 6 7 8 9 20r 1 2 3 4 25r 6 7 8 9 30r 1 2 3 4 35r 6 7 8 9 40r 1 2 3 4 45r 6 7 8 9 50r 1 2 3 4 55r 6 7 8 9 60r 08.0000 .0167 .0333 .05 .0667 .0833 .10 .1167 .1333 .15 08.1667 .1833 .20 .2167 .2333 .25 .2667 .2833 .30 .3167 08.3333 .35 .3667 .3833 .40 .4167 .4333 .45 .4667 .4833 08.50 .5167 .5333 .55 .5667 .5833 .60 .6167 .6333 .65 08.6667 .6833 .70 .7167 .7333 .75 .7667 .7833 .80 .8167 08.8333 .85 .8667 .8833 .90 .9167 .9333 .95 .9667 .9833 1.00 0s 1 2 3 4 5s 6 7 8 9 10s 1 2 3 4 15s 6 7 8 9 20s 1 2 3 4 25s 6 7 8 9 30s 1 2 3 4 35s 6 7 8 9 40s 1 2 3 4 45s 6 7 8 9 50s 1 2 3 4 55s 6 7 8 9 60s 08.0000 .0003 .0006 .0008 .0011 .0014 .0017 .0019 .0022 .0025 08.0028 .0031 .0033 .0036 .0039 .0042 .0044 .0047 .005 .0053 08.0056 .0058 .0061 .0064 .0067 .0069 .0072 .0075 .0078 .0081 08.0083 .0086 .0089 .0092 .0094 .0097 .01 .0103 .0106 .0108 08.0111 .0114 .0117 .0119 .0122 .0125 .0128 .0131 .0133 .0136 08.0139 .0142 .0144 .0147 .015 .0153 .0156 .0158 .0161 .0164 08.0167 08.00 1 2 3 4 08.05 6 7 8 9 08.10 1 2 3 4 08.15 6 7 8 9 08.20 1 2 3 4 08.25 6 7 8 9 08 .30 1 2 3 4 08.35 6 7 8 9 08.40 1 2 3 4 08.45 6 7 8 9 08.50 From decimal parts of a degree into minutes and seconds (exact values) 0r 0r 36s 1r 12s 1r 48s 2r 24s 3r 3r 36s 4r 12s 4r 48s 5r 24s 6r 6r 36s 7r 12s 7r 48s 8r 24s 9r 9r 36s 10r 12s 10r 48s 11r 24s 12r 12r 36s 13r 12s 13r 48s 14r 24s 15r 15r 36s 16r 12s 16r 48s 17r 24s 18r 18r 36s 19r 12s 19r 48s 20r 24s 21r 21r 36s 22r 12s 22r 48s 23r 24s 24r 24r 36s 25r 12s 25r 48s 26r 24s 27r 27r 36s 28r 12s 28r 48s 29r 24s 30r 08.000 1 2 3 4 08.005 6 7 8 9 08.010 08.50 1 2 3 4 08.55 6 7 8 9 08.60 1 2 3 4 08.65 6 7 8 9 08.70 1 2 3 4 08.75 6 7 8 9 08.80 1 2 3 4 08.85 6 7 8 9 08.90 1 2 3 4 08.95 6 7 8 9 18.00 0s.0 3s.6 7s.2 10s.8 14s.4 18s 21s.6 25s.2 28s.8 32s.4 36s 30r 30r 36s 31r 12s 31r 48s 32r 24s 33r 33r 36s 34r 12s 34r 48s 35r 24s 36r 36r 36s 37r 12s 37r 48s 38r 24s 39r 39r 36s 40r 12s 40r 48s 41r 24s 42r 42r 36s 43r 12s 43r 48s 44r 24s 45r 45r 36s 46r 12s 46r 48s 47r 24s 48r 48r 36s 49r 12s 49r 48s 50r 24s 51r 51r 36s 52r 12s 52r 48s 53r 24s 54r 54r 36s 55r 12s 55r 48s 56r 24s 57r 57r 36s 58r 12s 58r 48s 59r 24s 60r 8 ths 16 ths 32 nds 1 1 2 3 1 2 4 5 3 6 7 2 4 8 9 5 10 11 3 6 12 13 7 14 15 4 8 16 17 9 18 19 5 10 20 21 11 22 23 6 12 24 25 13 26 27 7 14 28 29 15 30 31 64 ths 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

1-15

Exact decimal values .01 5625 .03 125 .04 6875 .06 25 .07 8125 .09 375 .10 9375 .12 5 .14 0625 .15 625 .17 1875 .18 75 .20 3125 .21 875 .23 4375 .25 .26 5625 .28 125 .29 6875 .31 25 .32 8125 .34 375 .35 9375 .37 5 .39 0625 .40 625 .42 1875 .43 75 .45 3125 .46 875 .48 4375 .50 .51 5625 .53 125 .54 6875 .56 25 .57 8125 .59 375 .60 9375 .62 5 .64 0625 .65 625 .67 1875 .68 75 .70 3125 .71 875 .73 4375 .75 .76 5625 .78 125 .79 6875 .81 25 .82 8125 .84 375 .85 9375 .87 5 .89 0625 .90 625 .92 1875 .93 75 .95 3125 .96 875 .98 4375

1.2

MEASURING UNITS

by John T. Baumeister

160 square rods 10 square chains 43,560 square feet 5,645 sq varas (Texas) 640 acres 1 square mile

REFERENCES: "International Critical Tables," McGraw-Hill. "Smithsonian Physical Tables," Smithsonian Institution. "Landolt-Börnstein: Zahlenwerte und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik," Springer. "Handbook of Chemistry and Physics," Chemical Rubber Co. "Units and Systems of Weights and Measures; Their Origin, Development, and Present Status," NBS LC 1035 (1976). "Weights and Measures Standards of the United States, a Brief History," NBS Spec. Pub. 447 (1976). "Standard Time," Code of Federal Regulations, Title 49. "Fluid Meters, Their Theory and Application," 6th ed., chaps. 1­2, ASME, 1971. H.E. Huntley, "Dimensional Analysis," Richard & Co., New York, 1951. "U.S. Standard Atmosphere, 1962," Government Printing Office. Public Law 89-387, "Uniform Time Act of 1966." Public Law 94-168, "Metric Conversion Act of 1975." ASTM E380-91a, "Use of the International Standards of Units (SI) (the Modernized Metric System)." The International System of Units," NIST Spec. Pub. 330. "Guide for the Use of the International System of Units (SI)," NIST Spec. Pub. 811. "Guidelines for Use of the Modernized Metric System," NBS LC 1120. "NBS Time and Frequency Dissemination Services," NBS Spec. Pub. 432. "Factors for High Precision Conversion," NBS LC 1071. American Society of Mechanical Engineers SI Series, ASME SI 1 9. Jespersen and FitzRandolph, "From Sundials to Atomic Clocks: Understanding Time and Frequency," NBS, Monograph 155. ANSI/IEEE Std 268-1992, "American National Standard for Metric Practice."

J

h j

1 acre 1 "section" of U.S. government-surveyed land 0.7854 sq in 1.2732 circular inches area of circle 0.001 in in diam 1 circular inch Units of volume

1 circular inch area of circle 1 inch in diameter 1 square inch 1 circular mil 1,000,000 cir mils

1,728 cubic inches 231 cubic inches 27 cubic feet 1 cord of wood 1 perch of masonry

1 cubic foot 1 gallon 1 cubic yard 128 cubic feet 161/2 to 25 cu ft

U.S. CUSTOMARY SYSTEM (USCS)

The USCS, often called the "inch-pound system," is the system of units most commonly used for measures of weight and length (Table 1.2.1). The units are identical for practical purposes with the corresponding English units, but the capacity measures differ from those used in the British Commonwealth, the U.S. gallon being defined as 231 cu in and the bushel as 2,150.42 cu in, whereas the corresponding British Imperial units are, respectively, 277.42 cu in and 2,219.36 cu in (1 Imp gal 1.2 U.S. gal, approx; 1 Imp bu 1.03 U.S. bu, approx).

Liquid or fluid measurements 4 gills 1 pint 2 pints 1 quart 4 quarts 1 gallon 7.4805 gallons 1 cubic foot (There is no standard liquid barrel; by trade custom, 1 bbl of petroleum oil, unrefined 42 gal. The capacity of the common steel barrel used for refined petroleum products and other liquids is 55 gal.) 60 minims 8 drams 16 ounces Apothecaries' liquid measurements 1 liquid dram or drachm 1 liquid ounce 1 pint

Table 1.2.1

U.S. Customary Units Units of length

12 inches 3 feet 51/2 yards 161/2 feet 40 poles 220 yards 8 furlongs 1,760 yards 5,280 feet 3 miles 4 inches 9 inches 6,076.11549 feet 6 feet 120 fathoms 1 nautical mile per hr 7.92 inches 100 links 66 ft 80 chains 331/3 inches

j

1 foot 1 yard 1 rod, pole, or perch 1 furlong 1 mile 1 league 1 hand 1 span Nautical units 1 international nautical mile 1 fathom 1 cable length 1 knot

Water measurements The miner's inch is a unit of water volume flow no longer used by the Bureau of Reclamation. It is used within particular water districts where its value is defined by statute. Specifically, within many of the states of the West the miner's inch is 1/50 cubic foot per second. In others it is equal to 1/40 cubic foot per second, while in the state of Colorado, 38.4 miner's inch is equal to 1 cubic-foot per second. In 10 6 m3/s, and .427 SI units, these correspond to .32 10 6 m3/s, .409 10 6 m3/s, respectively. Dry measures 1 quart 1 peck 1 bushel 7,056 cu in or 105 dry qt, struck measure

2 pints 8 quarts 4 pecks 1 std bbl for fruits and vegetables 1 Register ton 1 U.S. shipping ton 1 British shipping ton

Shipping measures 100 cu ft 40 cu ft 32.14 U.S. bu or 31.14 Imp bu 42 cu ft 32.70 Imp bu or 33.75 U.S. bu

Surveyor's or Gunter's units 1 link 4 rods 1 chain 1 mile 1 vara (Texas) Units of area

Board measurements (Based on nominal not actual dimensions; see Table 12.2.8) 144 cu in volume of board 1 board foot 1 ft sq and 1 in thick

h

The international log rule, based upon 1/4 in kerf, is expressed by the formula X 0.904762(0.22 D2 0.71 D)

144 square inches 9 square feet 301/4 square yards

1-16

1 square foot 1 square yard 1 square rod, pole, or perch

where X is the number of board feet in a 4-ft section of a log and D is the top diam in in. In computing the number of board feet in a log, the taper is taken at 1/2 in per 4 ft linear, and separate computation is made for each 4-ft section.

THE INTERNATIONAL SYSTEM OF UNITS (SI) Weights (The grain is the same in all systems.) 16 drams 437.5 grains 16 ounces 7,000 grains 100 pounds 2,000 pounds 2,240 pounds 1 std lime bbl, small 1 std lime bbl, large Also (in Great Britain): 14 pounds 2 stone 28 pounds 4 quarters 112 pounds 20 hundredweight Avoirdupois weights 1 ounce 1 pound 1 cental 1 short ton 1 long ton 180 lb net 280 lb net 1 stone 1 quarter 1 hundredweight (cwt) 1 long ton

1-17

25.4 mm exactly; and 1 lb (nearly).

0.453 592 37 kg, or 1 lb

453.59 g

THE INTERNATIONAL SYSTEM OF UNITS (SI)

In October 1960, the Eleventh General (International) Conference on Weights and Measures redefined some of the original metric units and expanded the system to include other physical and engineering units. This expanded system is called, in French, Le Système International d'Unités (abbreviated SI), and in English, The International System of

Units.

Troy weights 24 grains 1 pennyweight (dwt) 20 pennyweights 480 grains 1 ounce 12 ounces 5,760 grains 1 pound 1 assay ton 29,167 milligrams, or as many milligrams as there are troy ounces in a ton of 2,000 lb avoirdupois. Consequently, the number of milligrams of precious metal yielded by an assay ton of ore gives directly the number of troy ounces that would be obtained from a ton of 2,000 lb avoirdupois. 20 grains 3 scruples 60 grains 8 drams 12 ounces 5,760 grains Apothecaries' weights 1 scruple 1 dram 1 ounce 1 pound

Weight for precious stones 1 carat 200 milligrams (Used by almost all important nations) 60 seconds 60 minutes 90 degrees 360 degrees 57.2957795 degrees ( 57 17r44.806s) Circular measures 1 minute 1 degree 1 quadrant circumference 1 radian (or angle having arc of length equal to radius)

METRIC SYSTEM

In the United States the name "metric system" of length and mass units is commonly taken to refer to a system that was developed in France about 1800. The unit of length was equal to 1/10,000,000 of a quarter meridian (north pole to equator) and named the metre. A cube 1/10th metre on a side was the litre, the unit of volume. The mass of water filling this cube was the kilogram, or standard of mass; i.e., 1 litre of water 1 kilogram of mass. Metal bars and weights were constructed conforming to these prescriptions for the metre and kilogram. One bar and one weight were selected to be the primary representations. The kilogram and the metre are now defined independently, and the litre, although for many years defined as the volume of a kilogram of water at the temperature of its maximum density, 48C, and under a pressure of 76 cm of mercury, is now equal to 1 cubic decimeter. In 1866, the U.S. Congress formally recognized metric units as a legal system, thereby making their use permissible in the United States. In 1893, the Office of Weights and Measures (now the National Bureau of Standards), by executive order, fixed the values of the U.S. yard and pound in terms of the meter and kilogram, respectively, as 1 yard 3,600/3,937 m; and 1 lb 0.453 592 4277 kg. By agreement in 1959 among the national standards laboratories of the English-speaking nations, the relations in use now are: 1 yd 0.9144 m, whence 1 in

The Metric Conversion Act of 1975 codifies the voluntary conversion of the U.S. to the SI system. It is expected that in time all units in the United States will be in SI form. For this reason, additional tables of units, prefixes, equivalents, and conversion factors are included below (Tables 1.2.2 and 1.2.3). SI consists of seven base units, two supplementary units, a series of derived units consistent with the base and supplementary units, and a series of approved prefixes for the formation of multiples and submultiples of the various units (see Tables 1.2.2 and 1.2.3). Multiple and submultiple prefixes in steps of 1,000 are recommended. (See ASTM E380-91a for further details.) Base and supplementary units are defined [NIST Spec. Pub. 330 (2001)] as: Metre The metre is defined as the length of path traveled by light in a vacuum during a time interval 1/299 792 458 of a second. Kilogram The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. Second The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. Ampere The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible cross section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 10 7 newton per metre of length. Kelvin The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. Mole The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. (When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.) Candela The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. Radian The unit of measure of a plane angle with its vertex at the center of a circle and subtended by an arc equal in length to the radius. Steradian The unit of measure of a solid angle with its vertex at the center of a sphere and enclosing an area of the spherical surface equal to that of a square with sides equal in length to the radius. SI conversion factors are listed in Table 1.2.4 alphabetically (adapted from ASTM E380-91a, "Standard Practice for Use of the International System of Units (SI) (the Modernized Metric System)." Conversion factors are written as a number greater than one and less than ten with six or fewer decimal places. This number is followed by the letter E (for exponent), a plus or minus symbol, and two digits which indicate the power of 10 by which the number must be multiplied to obtain the correct value. For example: 3.523 907 E 02 is 3.523 907 10

2

or 0.035 239 07

An asterisk (*) after the sixth decimal place indicates that the conversion factor is exact and that all subsequent digits are zero. All other conversion factors have been rounded off.

1-18

MEASURING UNITS Table 1.2.2 SI Units Unit Base units* Length Mass Time Electric current Thermodynamic temperature Amount of substance Luminous intensity metre kilogram second ampere kelvin mole candela Supplementary units* Plane angle Solid angle radian steradian Derived units* Acceleration Activity (of a radioactive source) Angular acceleration Angular velocity Area Density Electric capacitance Electrical conductance Electric field strength Electric inductance Electric potential difference Electric resistance Electromotive force Energy Entropy Force Frequency Illuminance Luminance Luminous flux Magnetic field strength Magnetic flux Magnetic flux density Magnetic potential difference Power Pressure Quantity of electricity Quantity of heat Radiant intensity Specific heat capacity Stress Thermal conductivity Velocity Viscosity, dynamic Viscosity, kinematic Voltage Volume Wave number Work metre per second squared disintegration per second radian per second squared radian per second square metre kilogram per cubic metre farad siemens volt per metre henry volt ohm volt joule joule per kelvin newton hertz lux candela per square metre lumen ampere per metre weber tesla ampere watt pascal coulomb joule watt per steradian joule per kilogram-kelvin pascal watt per metre-kelvin metre per second pascal-second square metre per second volt cubic metre reciprocal metre joule Units in use with the SI Time minute hour day degree minute second litre metric ton unified atomic mass unit§ electronvolt§ min h d 8 r s L t u eV 1 min 60 s 1 h 60 min 3,600 s 1 d 24 h 86,400 s 18 p/180 rad 1r (1/60)8 (p/10,800) rad 1s (1/60)r (p/648,000) rad 1 L 1 dm3 10 3 m3 1 t 103 kg 1 u 1.660 54 10 27 kg 1 eV 1.602 18 10 19 J m/s2 (disintegration)/s rad/s2 rad/s m2 kg/m3 A s/V A/V V/m V s/A W/A V/A W/A N m J/K kg m/s2 1/s lm/m2 cd/m2 cd sr A/m V s Wb/m2 J/s N/m2 A s N m W/sr J/(kg K) N/m2 W/(m K) m/s Pa s m2/s W/A m3 1/m N m rad sr m kg s A K mol cd SI symbol Formula

Quantity

F S H V V J N Hz lx lm Wb T A W Pa C J

Pa

V

J

Plane angle

Volume Mass Energy

* ASTM E380-91a. These units are not part of SI, but their use is both so widespread and important that the International Committee for Weights and Measures in 1969 recognized their continued use with the SI (see NIST Spec. Pub. 330). Use discouraged, except for special fields such as cartography. § Values in SI units obtained experimentally. These units are to be used in specialized fields only.

THE INTERNATIONAL SYSTEM OF UNITS (SI) Table 1.2.3 SI Prefixes* Prefix 1024 1021 1018 1015 1012 109 106 103 102 101 10 1 10 2 10 3 10 6 10 9 10 12 10 15 10 18 10 21 10 24 yotta zetta exa peta tera giga mega kilo hecto deka deci centi milli micro nano pico femto atto zepto yocto SI symbol Y Z E P T G M k h da d c m m n P f a z y

1-19

Multiplication factors 1 000 000 000 000 000 000 000 000 1 000 000 000 000 000 000 000 1 000 000 000 000 000 000 1 000 000 000 000 000 1 000 000 000 000 1 000 000 000 1 000 000 1 000 100 10 0.1 0.01 0.001 0.000 001 0.000 000 001 0.000 000 000 001 0.000 000 000 000 001 0.000 000 000 000 000 001 0.000 000 000 000 000 000 001 0.000 000 000 000 000 000 000 001

* ANSI/IEEE Std 268-1992. To be avoided where practical.

Table 1.2.4

SI Conversion Factors to ampere (A) coulomb (C) farad (F) henry (H) siemens (S) ohm ( ) volt (V) metre3 (m3) metre2 (m2) ampere (A) ampere (A) coulomb (C) metre (m) metre2 (m2) metre (m) pascal (Pa) pascal (Pa) pascal (Pa) metre2 (m2) metre3 (m3) metre3 (m3) joule (J) joule (J) joule (J) joule (J) joule (J) joule (J) watt/metre2 (W/m2) watt/metre2 (W/m2) watt/metre2 (W/m2) watt/metre2 (W/m2) watt/metre-kelvin (W/m K) watt/metre-kelvin (W/m K) watt/metre-kelvin (W/m K) watt/metre-kelvin (W/m K) joule/metre (J/m ) joule/metre2 (J/m2) watt/metre2-kelvin (W/m2 K) watt/metre2-kelvin (W/m2 K) joule/kilogram (J/kg)

2 2

To convert from abampere abcoulomb abfarad abhenry abmho abohm abvolt acre-foot (U.S. survey)a acre (U.S. survey)a ampere, international U.S. (AINT US)b ampere, U.S. legal 1948 (AUS 48) ampere-hour angstrom are astronomical unit atmosphere (normal) atmosphere (technical 1 kgf/cm2) bar barn barrel (for crude petroleum, 42 gal) board foot British thermal unit (International Table)c British thermal unit (mean) British thermal unit (thermochemical) British thermal unit (398F) British thermal unit (598F) British thermal unit (608F) Btu (thermochemical)/foot2-second Btu (thermochemical)/foot2-minute Btu (thermochemical)/foot2-hour Btu (thermochemical)/inch2-second Btu (thermochemical) in/s ft2 8F (k, thermal conductivity) Btu (International Table) in/s ft2 8F (k, thermal conductivity) Btu (thermochemical) in/h ft2 8F (k, thermal conductivity) Btu (International Table) in/h ft2 8F (k, thermal conductivity) Btu (International Table)/ft2 Btu (thermochemical)/ft2 Btu (International Table)/h ft2 8F (C, thermal conductance) Btu (thermochemical)/h ft2 8F (C, thermal conductance) Btu (International Table)/pound-mass

Multiply by 1.000 000*E 1.000 000*E 1.000 000*E 1.000 000*E 1.000 000*E 1.000 000*E 1.000 000*E 1.233 489 E 4.046 873 E 9.998 43 E 1.000 008 E 3.600 000*E 1.000 000*E 1.000 000*E 1.495 98 E 1.013 25 E 9.806 650*E 1.000 000*E 1.000 000*E 1.589 873 E 2.359 737 E 1.055 056 E 1.055 87 E 1.054 350 E 1.059 67 E 1.054 80 E 1.054 68 E 1.134 893 E 1.891 489 E 3.152 481 E 1.634 246 E 5.188 732 E 01 01 09 09 09 09 08 03 03 01 00 03 10 02 11 05 04 05 28 01 03 03 03 03 03 03 03 04 02 00 06 02

5.192 204 E 02 1.441 314 E 01 1.442 279 E 01 1.135 653 E 04 1.134 893 E 04 5.678 263 E 00 5.674 466 E 00 2.326 000*E 03

1-20

MEASURING UNITS Table 1.2.4 SI Conversion Factors (Continued ) to joule/kilogram (J/kg) joule/kilogram-kelvin (J/kg K) joule/kilogram-kelvin (J/kg K) watt/metre -kelvin (W/m K) watt/metre2-kelvin (W/m2 K) watt (W) watt (W) watt (W) watt (W) metre3 (m3) joule (J) joule (J) joule (J) joule (J) joule (J) joule (J) joule (J) joule (J) watt/metre2 (W/m2) joule/metre2 (J/m2) watt/metre2 (W/m2) watt/metre-kelvin (W/m K) joule/kilogram (J/kg) joule/kilogram-kelvin (J/kg K) joule/kilogram (J/kg) joule/kilogram-kelvin (J/kg K) watt (W) watt (W) kilogram (kg) pascal (Pa) pascal (Pa) pascal-second (Pa s) metre2/second (m2/s) meter (m) meter (m) metre2 (m2) metre3 (m3) coulomb (C) coulomb (C) metre3 (m3) becquerel (Bq) second (s) second (s) radian (rad) kelvin (K) kelvin (K) degree Celsius kelvin (K) kelvin-metre2/watt (K m2/W) kelvin-metre2/watt (K m2/ W) kelvin (K) kilogram (kg) kilogram (kg) kilogram (kg) newton (N) newton-metre (N m) pascal (Pa) joule (J) farad (F) ampere (A) volt (V) henry (H) ohm ( ) farad (F) ampere (A) volt (V) henry (H)

2 2

To convert from Btu (thermochemical)/pound-mass Btu (International Table)/lbm 8F (c, heat capacity) Btu (thermochemical)/lbm 8F (c, heat capacity) Btu (International Table)/s ft2 8F Btu (thermochemical)/s ft2 8F Btu (International Table)/hour Btu (thermochemical)/second Btu (thermochemical)/minute Btu (thermochemical)/hour bushel (U.S.) calorie (International Table) calorie (mean) calorie (thermochemical) calorie (158C) calorie (208C) calorie (kilogram, International Table) calorie (kilogram, mean) calorie (kilogram, thermochemical) calorie (thermochemical)/centimetre2minute cal (thermochemical)/cm2 cal (thermochemical)/cm2 s cal (thermochemical)/cm s 8C cal (International Table)/g cal (International Table)/g 8C cal (thermochemical)/g cal (thermochemical)/g 8C calorie (thermochemical)/second calorie (thermochemical)/minute carat (metric) centimetre of mercury (08C) centimetre of water (48C) centipoise centistokes chain (engineer or ramden) chain (surveyor or gunter) circular mil cord coulomb, international U.S. (CINT US)b coulomb, U.S. legal 1948 (CUS 48) cup curie day (mean solar) day (sidereal) degree (angle) degree Celsius degree centigrade degree Fahrenheit degree Fahrenheit deg F h ft2/Btu (thermochemical) (R, thermal resistance) deg F h ft2/Btu (International Table) (R, thermal resistance) degree Rankine dram (avoirdupois) dram (troy or apothecary) dram (U.S. fluid) dyne dyne-centimetre dyne-centimetre2 electron volt EMU of capacitance EMU of current EMU of electric potential EMU of inductance EMU of resistance ESU of capacitance ESU of current ESU of electric potential ESU of inductance

Multiply by 2.324 444 E 03 4.186 800*E 03 4.184 000*E 03 2.044 175 E 2.042 808 E 2.930 711 E 1.054 350 E 1.757 250 E 2.928 751 E 3.523 907 E 4.186 800*E 4.190 02 E 4.184 000*E 4.185 80 E 4.181 90 E 4.186 800*E 4.190 02 E 4.184 000*E 6.973 333 E 4.184 000*E 4.184 000*E 4.184 000*E 4.186 800*E 4.186 800*E 4.184 000*E 4.184 000*E 4.184 000*E 6.973 333 E 2.000 000*E 1.333 22 E 9.806 38 E 1.000 000*E 1.000 000*E 3.048* E 2.011 684 E 5.067 075 E 3.624 556 E 9.998 43 E 04 04 01 03 01 01 02 00 00 00 00 00 03 03 03 02 04 04 02 03 03 03 03 00 02 04 03 01 03 06 01 01 10 00 01

1.000 008 E 00 2.365 882 E 04 3.700 000*E 10 8.640 000 E 04 8.616 409 E 04 1.745 329 E 02 tK t8C 273.15 tK t8C 273.15 t8C (t8F 32)/1.8 tK (t8F 459.67)/1.8 1.762 280 E 01 1.761 102 E 01 tK t8R/1.8 1.771 845 E 3.887 934 E 3.696 691 E 1.000 000*E 1.000 000*E 1.000 000*E 1.602 18 E 1.000 000*E 1.000 000*E 1.000 000*E 1.000 000*E 1.000 000*E 1.112 650 E 3.335 6 E 2.997 9 E 8.987 552 E

03 03 06 05 07 01 19 09 01 08 09 09 12 10 02 11

THE INTERNATIONAL SYSTEM OF UNITS (SI) Table 1.2.4 SI Conversion Factors (Continued ) to ohm ( ) joule (J) watt/metre2 (W/m2) watt (W) farad (F) coulomb (C) coulomb (C) coulomb (C) metre (m) metre (m) metre3 (m3) metre (m) metre (m) metre3/second (m3/s) metre3/second (m3/s) metre3 (m3) metre2 (m2) metre4 (m4) metre/second (m/s) metre/second (m/s) metre/second (m/s) metre2/second (m2/s) pascal (Pa) lumen/metre2 (lm/m2) lux (lx) candela/metre2 (cd/m2) joule (J) watt (W) watt (W) watt (W) joule (J) metre2/second (m2/s) metre/second2 (m/s2) metre/second2 (m/s2) metre (m) metre/second2 (m/s2) metre3 (m3) metre3 (m3) metre3 (m3) metre3 (m3) metre3/second (m3/s) metre3/second (m3/s) tesla (T) tesla (T) ampere-turn metre3 (m3) metre3 (m3) degree (angular) radian (rad) kilogram (kg) kilogram (kg) kilogram/metre3 (kg/m3) pascal (Pa) metre2 (m2) henry (H) metre3 (m3) watt (W) watt (W) watt (W) watt (W) watt (W) watt (W) second (s) second (s) kilogram (kg) kilogram (kg) metre (m) metre2 (m2) metre3 (m3) metre3/second (m3/s) metre4 (m4) metre/second (m/s) pascal (Pa) Multiply by 8.987 552 E 1.000 000*E 1.000 000*E 1.000 000*E 9.995 05 E 9.648 531 E 9.649 57 E 9.652 19 E 1.828 804 E 1.000 000*E 2.957 353 E 3.048 000*E 3.048 006 E 4.719 474 E 2.831 685 E 2.831 685 E 9.290 304*E 8.630 975 E 8.466 667 E 5.080 000*E 3.048 000*E 9.290 304*E 2.988 98 E 1.076 391 E 1.076 391 E 3.426 259 E 1.355 818 E 3.766 161 E 2.259 697 E 1.355 818 E 4.214 011 E 2.580 640*E 3.048 000*E 9.806 650*E 2.011 68 *E 1.000 000*E 4.546 090 E 4.546 092 E 4.404 884 E 3.785 412 E 4.381 264 E 6.309 020 E 1.000 000*E 1.000 000*E 7.957 747 E 1.420 653 E 1.182 941 E 9.000 000*E 1.570 796 E 6.479 891*E 1.000 000*E 1.000 000*E 9.806 650*E 1.000 000*E 1.000 495 E 2.384 809 E 7.456 999 E 9.809 50 E 7.460 000*E 7.354 99 E 7.460 43 E 7.457 0 E 3.600 000*E 3.590 170 E 5.080 235 E 4.535 924 E 2.540 000*E 6.451 600*E 1.638 706 E 2.731 177 E 4.162 314 E 2.540 000*E 3.386 38 E 11 07 03 07 01 04 04 04 00 15 05 01 01 04 02 02 02 03 05 03 01 02 03 01 01 00 00 04 02 00 02 05 01 00 02 02 03 03 03 03 08 05 09 04 01 04 04 01 02 05 03 03 01 04 00 01 02 03 02 02 02 02 03 03 01 01 02 04 05 07 07 02 03

1-21

To convert from ESU of resistance erg erg/centimetre2-second erg/second farad, international U.S. (FINT US) faraday (based on carbon 12) faraday (chemical) faraday (physical) fathom (U.S. survey)a fermi (femtometer) fluid ounce (U.S.) foot foot (U.S. survey)a foot3/minute foot3/second foot3 (volume and section modulus) foot2 foot4 (moment of section)d foot/hour foot/minute foot/second foot2/second foot of water (39.28F) footcandle footcandle footlambert foot-pound-force foot-pound-force/hour foot-pound-force/minute foot-pound-force/second foot-poundal ft2/h (thermal diffusivity) foot/second2 free fall, standard furlong gal gallon (Canadian liquid) gallon (U.K. liquid) gallon (U.S. dry) gallon (U.S. liquid) gallon (U.S. liquid)/day gallon (U.S. liquid)/minute gamma gauss gilbert gill (U.K.) gill (U.S.) grade grade grain (1/7,000 lbm avoirdupois) gram gram/centimetre3 gram-force/centimetre2 hectare henry, international U.S. (HINT US) hogshead (U.S.) horsepower (550 ft lbf/s) horsepower (boiler) horsepower (electric) horsepower (metric) horsepower (water) horsepower (U.K.) hour (mean solar) hour (sidereal) hundredweight (long) hundredweight (short) inch inch2 inch3 (volume and section modulus) inch3/minute inch4 (moment of section)d inch/second inch of mercury (328F)

1-22

MEASURING UNITS Table 1.2.4 SI Conversion Factors (Continued ) to pascal (Pa) pascal (Pa) pascal (Pa) metre/second2 (m/s2) joule (J) joule (J) 1/metre (1/m) degree Celsius watt (W) watt (W) newton (N) newton-metre (N m) kilogram (kg) pascal (Pa) pascal (Pa) pascal (Pa) kilogram (kg) metre/second (m/s) newton (N) joule (J) joule (J) joule (J) newton (N) pascal (Pa) metre/second (m/s) candela/metre2 (cd/m2) joule/metre2 (J/m2) metre (m) metre (m) metre (m) metre (m) metre (m) metre (m) metre3 (m3) lumen/metre2 (lm/m2) weber (Wb) siemens (S) metre (m) metre (m) metre (m) metre (m) metre (m) metre (m) metre (m) metre2 (m2) metre2 (m2) metre/second (m/s) kilometre/hour pascal (Pa) radian (rad) second (s) second (s) second (s) ampere/metre (A/m) ohm ( ) ohm-metre ( m) newton (N) newton-metre (N m) kilogram (kg) kilogram (kg) kilogram/metre2 (kg/m2) kilogram/metre3 (kg/m3) metre3 (m3) metre3 (m3) metre (m) metre3 (m3) kilogram (kg) kilogram/pascal-secondmetre2 (kg/Pa s m2) kilogram/pascal-secondmetre2 (kg/Pa s m2) Multiply by 3.376 85 E 03 2.490 82 E 02 2.488 4 E 02 2.540 000*E 02 1.000 182 E 00 1.000 017 E 00 1.000 000*E 02 tC tK 273.15 6.973 333 E 01 4.184 000*E 03 9.806 650*E 00 9.806 650*E 00 9.806 650*E 00 9.806 650*E 04 9.806 650*E 00 9.806 650*E 06 1.000 000*E 00 2.777 778 E 01 9.806 650*E 00 3.600 000*E 06 3.600 655 E 06 3.600 061 E 06 4.448 222 E 6.894 757 E 5.144 444 E 3.183 099 E 4.184 000*E 5.556 000*E 4.828 041 E 5.559 552*E 9.460 54 E 3.048* E 2.011 68* E 1.000 000*E 1.000 000*E 1.000 000*E 1.000 000*E 2.540 000*E 1.000 000*E 2.540 000*E 1.852 000*E 1.853 184*E 1.609 344*E 1.609 347 E 2.589 988 E 2.589 998 E 4.470 400*E 1.609 344*E 1.333 224 E 2.908 882 E 6.000 000 E 5.983 617 E 2.268 000 E 7.957 747 E 1.000 495 E 1.000 000*E 2.780 139 E 7.061 552 E 2.834 952 E 3.110 348 E 3.390 575 E 1.729 994 E 2.841 306 E 2.957 353 E 3.085 678 E 8.809 768 E 1.555 174 E 5.721 35 E 03 06 01 03 04 03 03 03 15 01 01 03 00 08 00 08 06 05 03 03 03 03 06 06 01 00 02 04 01 01 06 01 00 02 01 03 02 02 02 03 05 05 16 03 03 11

To convert from inch of mercury (608F) inch of water (39.28F) inch of water (608F) inch/second2 joule, international U.S. (JINT US)b joule, U.S. legal 1948 (JUS 48) kayser kelvin kilocalorie (thermochemical)/minute kilocalorie (thermochemical)/second kilogram-force (kgf ) kilogram-force-metre kilogram-force-second2/metre (mass) kilogram-force/centimetre2 kilogram-force/metre3 kilogram-force/millimetre2 kilogram-mass kilometre/hour kilopond kilowatt hour kilowatt hour, international U.S. (kWhINT US)b kilowatt hour, U.S. legal 1948 (kWhUS 48) kip (1,000 lbf ) kip/inch2 (ksi) knot (international) lambert langley league, nautical (international and U.S.) league (U.S. survey)a league, nautical (U.K.) light year (365.2425 days) link (engineer or ramden) link (surveyor or gunter) litree lux maxwell mho microinch micron (micrometre) mil mile, nautical (international and U.S.) mile, nautical (U.K.) mile (international) mile (U.S. survey)a mile2 (international) mile2 (U.S. survey)a mile/hour (international) mile/hour (international) millimetre of mercury (08C) minute (angle) minute (mean solar) minute (sidereal) month (mean calendar) oersted ohm, international U.S. ( INT­US) ohm-centimetre ounce-force (avoirdupois) ounce-force-inch ounce-mass (avoirdupois) ounce-mass (troy or apothecary) ounce-mass/yard2 ounce (avoirdupois)(mass)/inch3 ounce (U.K. fluid) ounce (U.S. fluid) parsec peck (U.S.) pennyweight perm (08C) perm (23 8C)

5.745 25 E 11

THE INTERNATIONAL SYSTEM OF UNITS (SI) Table 1.2.4 SI Conversion Factors (Continued ) to kilogram/pascal-secondmetre (kg/Pa s m) kilogram/pascal-secondmetre (kg/Pa s m) lumen/metre2 (lm/m2) metre (m) metre3 (m3) metre3 (m3) metre pascal-second (Pa s) newton (N) pascal (Pa) pascal-second (Pa s) newton (N) newton-metre (N m) newton-metre (N m) newton-metre/metre (N m/m) newton-metre/metre (N m/m) newton/metre (N/m) newton/metre (N/m) pascal (Pa) pascal (Pa) pascal-second (Pa s) kilogram (kg) kilogram (kg) kilogram-metre2 (kg m2) kilogram-metre2 (kg m2) kilogram/metre2 (kg/m2) kilogram/second (kg/s) kilogram/second (kg/s) kilogram/metre3 (kg/m3) kilogram/metre3 (kg/m3) kilogram/metre3 (kg/m3) kilogram/metre3 (kg/m3) pascal-second (Pa s) metre3 (m3) metre3 (m3) gray (Gy) sievert (Sv) metre2/newton-second (m2/N s) metre (m) coulomb/kilogram (C/kg) radian (rad) second (s) metre2 (m2) second (s) kilogram (kg) kilogram/metre3 (kg/m3) pascal-second (Pa s) ampere (A) coulomb (C) farad (F) henry (H) siemens (S) ohm ( ) volt (V) metre3 (m3) candela/metre2 (cd/m2) metre2/second (m2/s) metre3 (m3) metre3 (m3) kilogram (kg) kilogram (kg) kilogram (kg) joule (J) metre3 (m3) kilogram (kg) kilogram/second (kg/s) kilogram/metre3 (kg/m3) kilogram (kg) pascal (Pa) metre2 (m2) weber (Wb) Multiply by 1.453 22 E 12 1.459 29 E 12 1.000 000*E 4.217 518 E 5.506 105 E 4.731 765 E 3.514 598 E 1.000 000*E 1.382 550 E 1.488 164 E 1.488 164 E 4.448 222 E 1.129 848 E 1.355 818 E 5.337 866 E 4.448 222 E 1.751 268 E 1.459 390 E 4.788 026 E 6.894 757 E 4.788 026 E 4.535 924 E 3.732 417 E 4.214 011 E 2.926 397 E 4.882 428 E 4.535 924 E 7.559 873 E 1.601 846 E 2.767 990 E 9.977 637 E 1.198 264 E 1.488 164 E 1.101 221 E 9.463 529 E 1.000 000*E 1.000 000*E 1.000 000*E 5.029 210 E 2.580 000*E 4.848 137 E 9.972 696 E 2.589 998 E 1.000 000*E 1.459 390 E 5.153 788 E 4.788 026 E 3.335 641 E 3.335 641 E 1.112 650 E 8.987 552 E 1.112 650 E 8.987 552 E 2.997 925 E 1.000 000*E 1.000 000*E 1.000 000*E 1.478 676 E 4.928 922 E 2.916 667 E 1.016 047 E 1.000 000*E 4.184 000*E 2.831 685 E 9.071 847 E 2.519 958 E 1.328 939 E 1.000 000*E 1.333 22 E 9.323 994 E 1.256 637 E 04 03 04 04 04 01 01 00 00 00 01 00 01 00 02 01 01 03 01 01 01 02 04 00 01 03 01 04 01 02 00 03 04 02 02 01 00 04 06 01 06 08 01 02 01 10 10 12 11 12 11 02 00 04 04 05 06 02 03 03 09 00 02 01 03 03 02 07 07

1-23

To convert from perm-inch (08C) perm-inch (238C) phot pica (printer's) pint (U.S. dry) pint (U.S. liquid) point (printer's) poise (absolute viscosity) poundal poundal/foot2 poundal-second/foot2 pound-force (lbf avoirdupois) pound-force-inch pound-force-foot pound-force-foot/inch pound-force-inch/inch pound-force/inch pound-force/foot pound-force/foot2 pound-force/inch2 (psi) pound-force-second/foot2 pound-mass (lbm avoirdupois) pound-mass (troy or apothecary) pound-mass-foot2 (moment of inertia) pound-mass-inch2 (moment of inertia) pound-mass/foot2 pound-mass/second pound-mass/minute pound-mass/foot3 pound-mass/inch3 pound-mass/gallon (U.K. liquid) pound-mass/gallon (U.S. liquid) pound-mass/foot-second quart (U.S. dry) quart (U.S. liquid) rad (radiation dose absorbed) rem (dose equivalent) rhe rod (U.S. survey)a roentgen second (angle) second (sidereal) section (U.S. survey)a shake slug slug/foot3 slug/foot-second statampere statcoulomb statfarad stathenry statmho statohm statvolt stere stilb stokes (kinematic viscosity) tablespoon teaspoon ton (assay) ton (long, 2,240 lbm) ton (metric) ton (nuclear equivalent of TNT) ton (register) ton (short, 2,000 lbm) ton (short, mass)/hour ton (long, mass)/yard3 tonne torr (mm Hg, 08C) township (U.S. survey)a unit pole

1-24

MEASURING UNITS Table 1.2.4 SI Conversion Factors (Continued ) to volt (V) volt (V) watt (W) watt (W) watt/metre2 (W/m2) joule (J) joule (J) metre (m) metre2 (m2) metre3 (m3) metre3/second (m3/s) second (s) second (s) second (s) Multiply by 1.000 338 E 1.000 008 E 1.000 182 E 1.000 017 E 1.000 000*E 3.600 000*E 1.000 000*E 9.144 000*E 8.361 274 E 7.645 549 E 1.274 258 E 3.153 600*E 3.155 815 E 3.155 693 E 00 00 00 00 04 03 00 01 01 01 02 07 07 07

To convert from volt, international U.S. (VINT US)b volt, U.S. legal 1948 (VUS 48) watt, international U.S. (WINT US)b watt, U.S. legal 1948 (WUS 48) watt/centimetre2 watt-hour watt-second yard yard2 yard3 yard3/minute year (calendar) year (sidereal) year (tropical)

a Based on the U.S. survey foot (1 ft 1,200/3,937 m). b In 1948 a new international agreement was reached on absolute electrical units, which changed the value of the volt used in this country by about 300 parts per million. Again in 1969 a new base of reference was internationally adopted making a further change of 8.4 parts per million. These changes (and also changes in ampere, joule, watt, coulomb) require careful terminology and conversion factors for exact use of old information. Terms used in this guide are: Volt as used prior to January 1948--volt, international U.S. (VINT US) Volt as used between January 1948 and January 1969--volt, U.S. legal 1948 (VINT 48) Volt as used since January 1969--volt (V) Identical treatment is given the ampere, coulomb, watt, and joule. c This value was adopted in 1956. Some of the older International Tables use the value 1.055 04 E 03. The exact conversion factor is 1.055 055 852 62*E 03. d Moment of inertia of a plane section about a specified axis. e In 1964, the General Conference on Weights and Measures adopted the name "litre" as a special name for the cubic decimetre. Prior to this decision the litre differed slightly (previous value, 1.000028 dm3), and in expression of precision, volume measurement, this fact must be kept in mind.

SYSTEMS OF UNITS

The principal units of interest to mechanical engineers can be derived from three base units which are considered to be dimensionally independent of each other. The British "gravitational system," in common use in the United States, uses units of length, force, and time as base units and is also called the "foot-pound-second system." The metric system, on the other hand, is based on the meter, kilogram, and second, units of length, mass, and time, and is often designated as the "MKS system." During the nineteenth century a metric "gravitational system," based on a kilogram-force (also called a "kilopond") came into general use. With the development of the International System of Units (SI), based as it is on the original metric system for mechanical units, and the general requirements by members of the European Community that only SI units be used, it is anticipated that the kilogram-force will fall into disuse to be replaced by the newton, the SI unit of force. Table 1.2.5 gives the base units of four systems with the corresponding derived unit given in parentheses. In the definitions given below, the "standard kilogram body" refers to the international kilogram prototype, a platinum-iridium cylinder kept in the International Bureau of Weights and Measures in Sèvres, just outside Paris. The "standard pound body" is related to the kilogram by a precise numerical factor: 1 lb 0.453 592 37 kg. This new "unified" pound has replaced the somewhat smaller Imperial pound of the United Kingdom and the slightly larger pound of the United States (see NBS Spec. Pub. 447). The "standard locality" means sea level, 458 latitude,

or more strictly any locality in which the acceleration due to gravity has 32.1740 ft/s2, which may be called the the value 9.80 665 m/s2 standard acceleration (Table 1.2.6). The pound force is the force required to support the standard pound body against gravity, in vacuo, in the standard locality; or, it is the force which, if applied to the standard pound body, supposed free to move, would give that body the "standard acceleration." The word pound is used for the unit of both force and mass and consequently is ambiguous. To avoid uncertainty, it is desirable to call the units "pound force" and "pound mass," respectively. The slug has been defined as that mass which will accelerate at 1 ft/s2 when acted upon by a one pound force. It is therefore equal to 32.1740 pound-mass. The kilogram force is the force required to support the standard kilogram against gravity, in vacuo, in the standard locality; or, it is the force which, if applied to the standard kilogram body, supposed free to move, would give that body the "standard acceleration." The word kilogram is used for the unit of both force and mass and consequently is ambiguous. It is for this reason that the General Conference on Weights and Measures declared (in 1901) that the kilogram was the unit of mass, a concept incorporated into SI when it was formally approved in 1960. The dyne is the force which, if applied to the standard gram body, would give that body an acceleration of 1 cm/s2; i.e., 1 dyne 1/980.665 of a gram force. The newton is that force which will impart to a 1-kilogram mass an acceleration of 1 m/s2.

Table 1.2.5

Systems of Units Dimensions of units in terms of L/M/F/T L M F T British "gravitational system" 1 ft (1 slug) 1 lb 1s Metric "gravitational system" 1m 1 kg 1s CGS system 1 cm 1g (1 dyne) 1s SI system 1m 1 kg (1 N) 1s

Quantity Length Mass Force Time

TIME Table 1.2.6 Latitude, deg 0 10 20 30 40 Acceleration of Gravity g m/s2 9.780 9.782 9.786 9.793 9.802 ft/s2 32.088 32.093 32.108 32.130 32.158 g/g0 0.9973 0.9975 0.9979 0.9986 0.9995 Latitude, deg 50 60 70 80 90 g m/s2 9.811 9.819 9.826 9.831 9.832 ft/s2 32.187 32.215 32.238 32.253 32.258 g/g0 1.0004 1.0013 1.0020 1.0024 1.0026

1-25

NOTE: Correction for altitude above sea level: 3 mm/s2 for each 1,000 m; SOURCE: U.S. Coast and Geodetic Survey, 1912.

0.003 ft/s2 for each 1,000 ft.

TEMPERATURE

The SI unit for thermodynamic temperature is the kelvin, K, which is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. Thus 273.16 K is the fixed (base) point on the kelvin scale. Another unit used for the measurement of temperature is degrees Celsius (formerly centigrade), 8C. The relation between a thermodynamic temperature T and a Celsius temperature t is t 5 T 2 273.15 K (the ice point of water) Thus the unit Celsius degree is equal to the unit kelvin, and a difference of temperature would be the same on either scale. In the USCS temperature is measured in degrees Fahrenheit, F. The relation between the Celsius and the Fahrenheit scales is t8C 5 st8F 2 32d/1.8 (For temperature-conversion tables, see Sec. 4.)

TERRESTRIAL GRAVITY Standard acceleration of gravity is g 0 9.80665 m per sec per sec, or 32.1740 ft per sec per sec. This value g0 is assumed to be the value of g at sea level and latitude 458. MOHS SCALE OF HARDNESS

This scale is an arbitrary one which is used to describe the hardness of several mineral substances on a scale of 1 through 10 (Table 1.2.7). The given number indicates a higher relative hardness compared with that of substances below it; and a lower relative hardness than those above it. For example, an unknown substance is scratched by quartz, but it, in turn, scratches feldspar. The unknown has a hardness of between 6 and 7 on the Mohs scale.

Table 1.2.7 1. 2. 3. 4. TIME Kinds of Time Three kinds of time are recognized by astronomers: sidereal, apparent solar, and mean solar time. The sidereal day is the interval between two consecutive transits of some fixed celestial object across any given meridian, or it is the interval required by the earth to make one complete revolution on its axis. The interval is constant, but it is inconvenient as a time unit because the noon of the sidereal day occurs at all hours of the day and night. The apparent solar day is the interval between two consecutive transits of the sun across any given meridian. On account of the variable distance between the sun and earth, the variable speed of the earth in its orbit, the effect of the moon, etc., this interval is not constant and consequently cannot be kept by any simple mechanisms, such as clocks or watches. To overcome the objection noted above, the mean solar day was devised. The mean solar day is Talc Gypsum Calc-spar Fluorspar Mohs Scale of Hardness 5. Apatite 6. Feldspar 7. Quartz 8. Topaz 9. Sapphire 10. Diamond

the length of the average apparent solar day. Like the sidereal day it is constant, and like the apparent solar day its noon always occurs at approximately the same time of day. By international agreement, beginning Jan. 1, 1925, the astronomical day, like the civil day, is from midnight to midnight. The hours of the astronomical day run from 0 to 24, and the hours of the civil day usually run from 0 to 12 A.M. and 0 to 12 P.M. In some countries the hours of the civil day also run from 0 to 24. The Year Three different kinds of year are used: the sidereal, the tropical, and the anomalistic. The sidereal year is the time taken by the earth to complete one revolution around the sun from a given star to the same star again. Its length is 365 days, 6 hours, 9 minutes, and 9 seconds. The tropical year is the time included between two successive passages of the vernal equinox by the sun, and since the equinox moves westward 50.2 seconds of arc a year, the tropical year is shorter by 20 minutes 23 seconds in time than the sidereal year. As the seasons depend upon the earth's position with respect to the equinox, the tropical year is the year of civil reckoning. The anomalistic year is the interval between two successive passages of the perihelion, viz., the time of the earth's nearest approach to the sun. The anomalistic year is used only in special calculations in astronomy. The Second Although the second is ordinarily defined as 1/86,400 of the mean solar day, this is not sufficiently precise for many scientific purposes. Scientists have adopted more precise definitions for specific purposes: in 1956, one in terms of the length of the tropical year 1900 and, more recently, in 1967, one in terms of a specific atomic frequency. Frequency is the reciprocal of time for 1 cycle; the unit of frequency is the hertz (Hz), defined as 1 cycle/s. The Calendar The Gregorian calendar, now used in most of the civilized world, was adopted in Catholic countries of Europe in 1582 and in Great Britain and her colonies Jan. 1, 1752. The average length of the Gregorian calendar year is 365 1/4 2 3/400 days, or 365.2425 days. This is equivalent to 365 days, 5 hours, 49 minutes, 12 seconds. The length of the tropical year is 365.2422 days, or 365 days, 5 hours, 48 minutes, 46 seconds. Thus the Gregorian calendar year is longer than the tropical year by 0.0003 day, or 26 seconds. This difference amounts to 1 day in slightly more than 3,300 years and can properly be neglected. Standard Time Prior to 1883, each city of the United States had its own time, which was determined by the time of passage of the sun across the local meridian. A system of standard time had been used since its first adoption by the railroads in 1883 but was first legalized on Mar. 19, 1918, when Congress directed the Interstate Commerce Commission to establish limits of the standard time zones. Congress took no further steps until the Uniform Time Act of 1966 was enacted, followed with an amendment in 1972. This legislation, referred to as "the Act," transferred the regulation and enforcement of the law to the Department of Transportation. By the legislation of 1918, with some modifications by the Act, the contiguous United States is divided into four time zones, each of which, theoretically, was to span 15 degrees of longitude. The first, the Eastern zone, extends from the Atlantic westward to include most of Michigan and Indiana, the eastern parts of Kentucky and Tennessee, Georgia, and Florida, except the west half of the panhandle. Eastern standard time is

1-26

MEASURING UNITS

based upon the mean solar time of the 75th meridian west of Greenwich, and is 5 hours slower than Greenwich Mean Time (GMT). (See also discussion of UTC below.) The second or Central zone extends westward to include most of North Dakota, about half of South Dakota and Nebraska, most of Kansas, Oklahoma, and all but the two most westerly counties of Texas. Central standard time is based upon the mean solar time of the 90th meridian west of Greenwich, and is 6 hours slower than GMT. The third or Mountain zone extends westward to include Montana, most of Idaho, one county of Oregon, Utah, and Arizona. Mountain standard time is based upon the mean solar time of the 105th meridian west of Greenwich, and is 7 hours slower than GMT. The fourth or Pacific zone includes all of the remaining 48 contiguous states. Pacific standard time is based on the mean solar time of the 120th meridian west of Greenwich, and is 8 hours slower than GMT. Exact locations of boundaries may be obtained from the Department of Transportation. In addition to the above four zones there are four others that apply to the noncontiguous states and islands. The most easterly is the Atlantic zone, which includes Puerto Rico and the Virgin Islands, where the time is 4 hours slower than GMT. Eastern standard time is used in the Panama Canal strip. To the west of the Pacific time zone there are the Yukon, the Alaska-Hawaii, and Bering zones where the times are, respectively, 9, 10, and 11 hours slower than GMT. The system of standard time has been adopted in all civilized countries and is used by ships on the high seas. The Act directs that from the first Sunday in April to the last Sunday in October, the time in each zone is to be advanced one hour for advanced time or daylight saving time (DST). However, any state-bystate enactment may exempt the entire state from using advanced time. By this provision Arizona and Hawaii do not observe advanced time (as of 1973). By the 1972 amendment to the Act, a state split by a timezone boundary may exempt from using advanced time all that part which is in one zone without affecting the rest of the state. By this amendment, 80 counties of Indiana in the Eastern zone are exempt from using advanced time, while 6 counties in the northwest corner and 6 counties in the southwest, which are in Central zone, do observe advanced time. Pursuant to its assignment of carrying out the Act, the Department of Transportation has stipulated that municipalities located on the boundary between the Eastern and Central zones are in the Central zone; those on the boundary between the Central and Mountain zones are in the Mountain zone (except that Murdo, SD, is in the Central zone); those on the boundary between Mountain and Pacific time zones are in the Mountain zone. In such places, when the time is given, it should be specified as Central, Mountain, etc. Standard Time Signals The National Institute of Standards and Technology broadcasts time signals from station WWV, Ft. Collins, CO, and from station WWVH, near Kekaha, Kaui, HI. The broadcasts by WWV are on radio carrier frequencies of 2.5, 5, 10, 15, and 20 MHz, while those by WWVH are on radio carrier frequencies of 2.5, 5, 10, and 15 MHz. Effective Jan. 1, 1975, time announcements by both WWV and WWVH are referred to as Coordinated Universal Time, UTC, the international coordinated time scale used around the world for most timekeeping purposes. UTC is generated by reference to International Atomic Time (TAI), which is determined by the Bureau International de l'Heure on the basis of atomic clocks operating in various establishments in accordance with the definition of the second. Since the difference between UTC and TAI is defined to be a whole number of seconds, a "leap second" is periodically added to or subtracted from UTC to take into account variations in the rotation of the earth. Time (i.e., clock time) is given in terms of 0 to 24 hours a day, starting with 0000 at midnight at Greenwich zero longitude. The beginning of each 0.8-second-long audio tone marks the end of an announced time interval. For example, at 2:15 P.M., UTC, the voice announcement would be: "At the tone fourteen hours fifteen minutes Coordinated Universal Time," given during the last 7.5 seconds of each minute. The tone markers from both stations are given simultaneously, but owing to propagation interferences may not be received simultaneously.

Beginning 1 minute after the hour, a 600-Hz signal is broadcast for about 45 s. At 2 min after the hour, the standard musical pitch of 440 Hz is broadcast for about 45 s. For the remaining 57 min of the hour, alternating tones of 600 and 500 Hz are broadcast for the first 45 s of each minute (see NIST Spec. Pub. 432). The time signal can also be received via long-distance telephone service from Ft. Collins. In addition to providing the musical pitch, these tone signals may be of use as markers for automated recorders and other such devices.

DENSITY AND RELATIVE DENSITY Density of a body is its mass per unit volume. With SI units densities

are in kilograms per cubic meter. However, giving densities in grams per cubic centimeter has been common. With the USCS, densities are given in pounds per mass cubic foot.

Table 1.2.8 Relative Densities at 608/608F Corresponding to Degrees API and Weights per U.S. Gallon at 608F 141.5 ¢Calculated from the formula, relative density 5 131.5 1 deg API Degrees API 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 Relative density 1.0000 0.9930 0.9861 0.9792 0.9725 0.9659 0.9593 0.9529 0.9465 0.9402 0.9340 0.9279 0.9218 0.9159 0.9100 0.9042 0.8984 0.8927 0.8871 0.8816 0.8762 0.8708 0.8654 0.8602 0.8550 0.8498 0.8448 0.8398 0.8348 0.8299 0.8251 0.8203 0.8155 0.8109 0.8063 0.8017 0.7972 0.7927 0.7883 0.7839 0.7796 0.7753 0.7711 0.7669 0.7628 0.7587 Lb per U.S. gallon 8.328 8.270 8.212 8.155 8.099 8.044 7.989 7.935 7.882 7.830 7.778 7.727 7.676 7.627 7.578 7.529 7.481 7.434 7.387 7.341 7.296 7.251 7.206 7.163 7.119 7.076 7.034 6.993 6.951 6.910 6.870 6.830 6.790 6.752 6.713 6.675 6.637 6.600 6.563 6.526 6.490 6.455 6.420 6.385 6.350 6.316 Degrees API 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Relative density 0.7547 0.7507 0.7467 0.7428 0.7389 0.7351 0.7313 0.7275 0.7238 0.7201 0.7165 0.7128 0.7093 0.7057 0.7022 0.6988 0.6953 0.6919 0.6886 0.6852 0.6819 0.6787 0.6754 0.6722 0.6690 0.6659 0.6628 0.6597 0.6566 0.6536 0.6506 0.6476 0.6446 0.6417 0.6388 0.6360 0.6331 0.6303 0.6275 0.6247 0.6220 0.6193 0.6166 0.6139 0.6112

Lb per U.S. gallon 6.283 6.249 6.216 6.184 6.151 6.119 6.087 6.056 6.025 5.994 5.964 5.934 5.904 5.874 5.845 5.817 5.788 5.759 5.731 5.703 5.676 5.649 5.622 5.595 5.568 5.542 5.516 5.491 5.465 5.440 5.415 5.390 5.365 5.341 5.316 5.293 5.269 5.246 5.222 5.199 5.176 5.154 5.131 5.109 5.086

NOTE: The weights in this table are weights in air at 608F with humidity 50 percent and pressure 760 mm.

CONVERSION AND EQUIVALENCY TABLES Table 1.2.9 Relative Densities at 608/608F Corresponding to Degrees Baumé for Liquids Lighter than Water and Weights per U.S. Gallon at 608F 608 140 ¢Calculated from the formula, relative density F5 608 130 1 deg Baumé Degrees Baumé 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 43.0 44.0 45.0 46.0 47.0 48.0 49.0 50.0 51.0 52.0 53.0 54.0 55.0 Relative density 1.0000 0.9929 0.9859 0.9790 0.9722 0.9655 0.9589 0.9524 0.9459 0.9396 0.9333 0.9272 0.9211 0.9150 0.9091 0.9032 0.8974 0.8917 0.8861 0.8805 0.8750 0.8696 0.8642 0.8589 0.8537 0.8485 0.8434 0.8383 0.8333 0.8284 0.8235 0.8187 0.8140 0.8092 0.8046 0.8000 0.7955 0.7910 0.7865 0.7821 0.7778 0.7735 0.7692 0.7650 0.7609 0.7568 Lb per gallon 8.328 8.269 8.211 8.153 8.096 8.041 7.986 7.931 7.877 7.825 7.772 7.721 7.670 7.620 7.570 7.522 7.473 7.425 7.378 7.332 7.286 7.241 7.196 7.152 7.108 7.065 7.022 6.980 6.939 6.898 6.857 6.817 6.777 6.738 6.699 6.661 6.623 6.586 6.548 6.511 6.476 6.440 6.404 6.369 6.334 6.300 Degrees Baumé 56.0 57.0 58.0 59.0 60.0 61.0 62.0 63.0 64.0 65.0 66.0 67.0 68.0 69.0 70.0 71.0 72.0 73.0 74.0 75.0 76.0 77.0 78.0 79.0 80.0 81.0 82.0 83.0 84.0 85.0 86.0 87.0 88.0 89.0 90.0 91.0 92.0 93.0 94.0 95.0 96.0 97.0 98.0 99.0 100.0 Relative density 0.7527 0.7487 0.7447 0.7407 0.7368 0.7330 0.7292 0.7254 0.7216 0.7179 0.7143 0.7107 0.7071 0.7035 0.7000 0.6965 0.6931 0.6897 0.6863 0.6829 0.6796 0.6763 0.6731 0.6699 0.6667 0.6635 0.6604 0.6573 0.6542 0.6512 0.6482 0.6452 0.6422 0.6393 0.6364 0.6335 0.6306 0.6278 0.6250 0.6222 0.6195 0.6167 0.6140 0.6114 0.6087

1-27

Lb per gallon 6.266 6.233 6.199 6.166 6.134 6.102 6.070 6.038 6.007 5.976 5.946 5.916 5.886 5.856 5.827 5.798 5.769 5.741 5.712 5.685 5.657 5.629 5.602 5.576 5.549 5.522 5.497 5.471 5.445 5.420 5.395 5.370 5.345 5.320 5.296 5.272 5.248 5.225 5.201 5.178 5.155 5.132 5.100 5.088 5.066

equal volumes, the ratio of the molecular weight of the gas to that of air may be used as the relative density of the gas. When this is done, the molecular weight of air may be taken as 28.9644. The relative density of liquids is usually measured by means of a hydrometer. In addition to a scale reading in relative density as defined above, other arbitrary scales for hydrometers are used in various trades and industries. The most common of these are the API and Baumé. The API (American Petroleum Institute) scale is approved by the American Petroleum Institute, the ASTM, the U.S. Bureau of Mines, and the National Bureau of Standards and is recommended for exclusive use in the U.S. petroleum industry, superseding the Baumé scale for liquids lighter than water. The relation between API degrees and relative density (see Table 1.2.8) is expressed by the following equation: Degrees API 5 141.5 2 131.5 rel dens 608/608F

The relative densities corresponding to the indications of the Baumé

hydrometer are given in Tables 1.2.9 and 1.2.10. Table 1.2.10 Relative Densities at 608/608F Corresponding to Degrees Baumé for Liquids Heavier than Water 608 145 ¢Calculated from the formula, relative density F5 608 145 2 deg Baumé Degrees Baumé 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Relative density 1.0000 1.0069 1.0140 1.0211 1.0284 1.0357 1.0432 1.0507 1.0584 1.0662 1.0741 1.0821 1.0902 1.0985 1.1069 1.1154 1.1240 1.1328 1.1417 1.1508 1.1600 1.1694 1.1789 1.1885 Degrees Baumé 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 Relative density 1.1983 1.2083 1.2185 1.2288 1.2393 1.2500 1.2609 1.2719 1.2832 1.2946 1.3063 1.3182 1.3303 1.3426 1.3551 1.3679 1.3810 1.3942 1.4078 1.4216 1.4356 1.4500 1.4646 1.4796 Degrees Baumé 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

Relative density 1.4948 1.5104 1.5263 1.5426 1.5591 1.5761 1.5934 1.6111 1.6292 1.6477 1.6667 1.6860 1.7059 1.7262 1.7470 1.7683 1.7901 1.8125 1.8354 1.8590 1.8831 1.9079 1.9333

CONVERSION AND EQUIVALENCY TABLES Note for Use of Conversion Tables (Tables 1.2.11 through 1.2.34)

Relative density is the ratio of the density of one substance to that of a

second (or reference) substance, both at some specified temperature. Use of the earlier term specific gravity for this quantity is discouraged. For solids and liquids water is almost universally used as the reference substance. Physicists use a reference temperature of 48C ( 39.28F); U.S. engineers commonly use 608F. With the introduction of SI units, it may be found desirable to use 598F, since 598F and 158C are equivalents. For gases, relative density is generally the ratio of the density of the gas to that of air, both at the same temperature, pressure, and dryness (as regards water vapor). Because equal numbers of moles of gases occupy

Subscripts after any figure, 0s, 9s, etc., mean that that figure is to be repeated the indicated number of times.

1-28

MEASURING UNITS Length Equivalents Inches 0.3937 1 12 36 39.37 792 39370 63360 Feet 0.03281 0.08333 1 3 3.281 66 3281 5280 Yards 0.01094 0.02778 0.3333 1 1.0936 22 1093.6 1760 Metres 0.01 0.0254 0.3048 0.9144 1 20.12 1000 1609 Chains 0.034971 0.001263 0.01515 0.04545 0.04971 1 49.71 80 Kilometres 10 5 0.04254 0.033048 0.039144 0.001 0.02012 1 1.609 Miles 0.056214 0.041578 0.031894 0.035682 0.036214 0.0125 0.6214 1

Table 1.2.11 Centimetres 1 2.540 30.48 91.44 100 2012 100000 160934

(As used by metrology laboratories for precise measurements, including measurements of surface texture)*

Angstrom units Å 1 254 2937.5 10,000 25,400 254,000 10,000,000 254,000,000

Surface texture (U.S.), microinch min 0.003937 1 11.566 39.37 100 1000 39,370 1,000,000

Light bands, monochromatic helium light count 0.0003404 0.086 1 3.404 8.646 86.46 3404 86,460

Surface texture foreign, mm 0.0001 0.0254 0.29375 1 2.54 25.4 1000 25,400

Precision measurements, § 0.0001 in 0.043937 0.01 0.11566 0.3937 1 10 393.7 10,000

Close-tolerance measurements, 0.001 in (mils) 0.053937 0.001 0.011566 0.03937 0.1 1 39.37 1000

Metric unit, mm 0.061 0.04254 0.0329375 0.001 0.00254 0.0254 1 25.4

USCS unit, in 0.083937 0.051 0.0411566 0.043937 0.0001 0.001 0.03937 1

* Computed by J. A. Broadston. One light band equals one-half corresponding wavelength. Visible-light wavelengths range from red at 6,500 Å to violet at 4,100 Å. One helium light band 0.000011661 in 2937.5 Å; one krypton 86 light band 0.0000119 in 3,022.5 Å; one mercury 198 light band § The designations "precision measurements," etc., are not necessarily used in all metrology laboratories.

0.00001075 in

2,730 Å.

CONVERSION AND EQUIVALENCY TABLES Table 1.2.12 Conversion of Lengths* Millimetres to inches 0.03937 0.07874 0.1181 0.1575 0.1969 0.2362 0.2756 0.3150 0.3543 Feet to metres 0.3048 0.6096 0.9144 1.219 1.524 1.829 2.134 2.438 2.743 Metres to feet 3.281 6.562 9.843 13.12 16.40 19.69 22.97 26.25 29.53 Yards to metres 0.9144 1.829 2.743 3.658 4.572 5.486 6.401 7.315 8.230 Metres to yards 1.094 2.187 3.281 4.374 5.468 6.562 7.655 8.749 9.843 Miles to kilometres 1.609 3.219 4.828 6.437 6.047 9.656 11.27 12.87 14.48

1-29

Inches to millimetres 1 2 3 4 5 6 7 8 9 25.40 50.80 76.20 101.60 127.00 152.40 177.80 203.20 228.60

* EXAMPLE: 1 in 25.40 mm.

Kilometres to miles 0.6214 1.243 1.864 2.485 3.107 3.728 4.350 4.971 5.592

Common fractions of an inch to millimetres (from 1/64 to 1 in) 64ths 1 2 3 4 5 6 7 8 9 10 11 12 Millimetres 0.397 0.794 1.191 1.588 1.984 2.381 2.778 3.175 3.572 3.969 4.366 4.762 64ths 13 14 15 16 17 18 19 20 21 22 23 24 Millimetres 5.159 5.556 5.953 6.350 6.747 7.144 7.541 7.938 8.334 8.731 9.128 9.525 64th 25 26 27 28 29 30 31 32 33 34 35 36 Millimetres 9.922 10.319 10.716 11.112 11.509 11.906 12.303 12.700 13.097 13.494 13.891 14.288 64ths 37 38 39 40 41 42 43 44 45 46 47 48 Millimetres 14.684 15.081 15.478 15.875 16.272 16.669 17.066 17.462 17.859 18.256 18.653 19.050 64ths 49 50 51 52 53 54 55 56 Millimetres 19.447 19.844 20.241 20.638 21.034. 21.431 21.828 22.225 64ths 57 58 59 60 61 62 63 64 Millimetres 22.622 23.019 23.416 23.812 24.209 24.606 25.003 25.400

Decimals of an inch to millimetres (0.01 to 0.99 in) 0 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 2.540 5.080 7.620 10.160 12.700 15.240 17.780 20.320 22.860 1 0.254 2.794 5.334 7.874 10.414 12.954 15.494 18.034 20.574 23.114 2 0.508 3.048 5.588 8.128 10.668 13.208 15.748 18.288 20.828 23.368 3 0.762 3.302 5.842 8.382 10.922 13.462 16.002 18.542 21.082 23.622 4 1.016 3.556 6.096 8.636 11.176 13.716 16.256 18.796 21.336 23.876 5 1.270 3.810 6.350 8.890 11.430 13.970 16.510 19.050 21.590 24.130 6 1.524 4.064 6.604 9.144 11.684 14.224 16.764 19.304 21.844 24.384 7 1.778 4.318 6.858 9.398 11.938 14.478 17.018 19.558 22.098 24.638 8 2.032 4.572 7.112 9.652 12.192 14.732 17.272 19.812 22.352 24.892 9 2.286 4.826 7.366 9.906 12.446 14.986 17.526 20.066 22.606 25.146

Millimetres to decimals of an inch (from 1 to 99 mm) 0. 0 1 2 3 4 5 6 7 8 9 0.3937 0.7874 1.1811 1.5748 1.9685 2.3622 2.7559 3.1496 3.5433 1. 0.0394 0.4331 0.8268 1.2205 1.6142 2.0079 2.4016 2.7953 3.1890 3.5827 2. 0.0787 0.4724 0.8661 1.2598 1.6535 2.0472 2.4409 2.8346 3.2283 3.6220 3. 0.1181 0.5118 0.9055 1.2992 1.6929 2.0866 2.4803 2.8740 3.2677 3.6614 4. 0.1575 0.5512 0.9449 1.3386 1.7323 2.1260 2.5197 2.9134 3.3071 3.7008 5. 0.1969 0.5906 0.9843 1.3780 1.7717 2.1654 2.5591 2.9528 3.3465 3.7402 6. 0.2362 0.6299 1.0236 1.4173 1.8110 2.2047 2.5984 2.9921 3.3858 3.7795 7. 0.2756 0.6693 1.0630 1.4567 1.8504 2.2441 2.6378 3.0315 3.4252 3.8189 8. 0.3150 0.7087 1.1024 1.4961 1.8898 2.2835 2.6772 3.0709 3.4646 3.8583 9. 0.3543 0.7480 1.1417 1.5354 1.9291 2.3228 2.7165 3.1102 3.5039 3.8976

1-30

MEASURING UNITS

Table 1.2.13 Area Equivalents (1 hectare 100 ares 10,000 centiares or square metres) Square metres 1 0.036452 0.09290 0.8361 25.29 404.7 1012 4047 2589988 Square inches 1550 1 144 1296 39204 627264 1568160 6272640 Square feet 10.76 0.006944 1 9 272.25 4356 10890 43560 27878400 Square yards 1.196 0.037716 0.1111 1 30.25 484 1210 4840 3097600 Square rods 0.0395 0.042551 0.003673 0.03306 1 16 40 160 102400 Square chains 0.002471 0.051594 0.032296 0.002066 0.0625 1 2.5 10 6400 Roods 0.039884 0.066377 0.049183 0.038264 0.02500 0.4 1 4 2560 Acres 0.032471 0.061594 0.042296 0.0002066 0.00625 0.1 0.25 1 640 Square miles or sections 0.063861 0.092491 0.073587 0.063228 0.059766 0.0001562 0.033906 0.001562 1

Table 1.2.14

Conversion of Areas* Sq cm to sq in 0.1550 0.3100 0.4650 0.6200 0.7750 0.9300 1.085 1.240 1.395

6.452 cm2.

Sq in to sq cm 1 2 3 4 5 6 7 8 9 6.452 12.90 19.35 25.81 32.26 38.71 45.16 51.61 58.06

* EXAMPLE: 1 in2

Sq ft to sq m 0.0929 0.1858 0.2787 0.3716 0.4645 0.5574 0.6503 0.7432 0.8361

Sq m to sq ft 10.76 21.53 32.29: 43.06 53.82 64.58 75.35 86.11 96.88

Sq yd to sq m 0.8361 1.672 2.508 3.345 4.181 5.017 5.853 6.689 7.525

Sq m to sq yd 1.196 2.392 3.588 4.784 5.980 7.176 8.372 9.568 10.764

Acres to hectares 0.4047 0.8094 1.214 1.619 2.023 2.428 2.833 3.237 3.642

Hectares to acres 2.471 4.942 7.413 9.884 12.355 14.826 17.297 19.768 22.239

Sq mi to sq km 2.590 5.180 7.770 10.360 12.950 15.540 18.130 20.720 23.310

Sq km to sq mi 0.3861 0.7722 1.158 1.544 1.931 2.317 2.703 3.089 3.475

Table 1.2.15 Cubic inches 1 1728 46656 1.805 57.75 67.20 231 2150 61.02

Volume and Capacity Equivalents Cubic feet 0.035787 1 27 0.001044 0.03342 0.03889 0.1337 1.244 0.03531 Cubic yards 0.042143 0.03704 1 0.043868 0.001238 0.001440 0.004951 0.04609 0.001308 U.S. Apothecary fluid ounces 0.5541 957.5 25853 1 32 37.24 128 1192 33.81 U.S. quarts Liquid 0.01732 29.92 807.9 0.03125 1 1.164 4 37.24 1.057 Dry 0.01488 25.71 694.3 0.02686 0.8594 1 3.437 32 0.9081 U.S. gallons 0.024329 7.481 202.2 0.007812 0.25 0.2909 1 9.309 0.2642 U.S. bushels 0.034650 0.8036 21.70 0.038392 0.02686 0.03125 0.1074 1 0.02838 Cubic decimetres or litres 0.01639 28.32 764.6 0.02957 0.9464 1.101 3.785 35.24 1

Table 1.2.16 Cu in to mL 1 2 3 4 5 6 7 8 9 16.39 32.77 49.16 65.55 81.94 98.32 114.7 131.1 147.5

Conversion of Volumes or Cubic Measure* mL to cu in 0.06102 0.1220 0.1831 0.2441 0.3051 0.3661 0.4272 0.4882 0.5492

16.39 mL.

Cu ft to cu m 0.02832 0.05663 0.08495 0.1133 0.1416 0.1699 0.1982 0.2265 0.2549

Cu m to cu ft 35.31 70.63 105.9 141.3 176.6 211.9 247.2 282.5 317.8

Cu yd to cu m 0.7646 1.529 2.294 3.058 3.823 4.587 5.352 6.116 6.881

Cu m to cu yd 1.308 2.616 3.924 5.232 6.540 7.848 9.156 10.46 11.77

Gallons to cu ft 0.1337 0.2674 0.4010 0.5347 0.6684 0.8021 0.9358 1.069 1.203

Cu ft to gallons 7.481 14.96 22.44 29.92 37.40 44.88 52.36 59.84 67.32

* EXAMPLE: 1 in3

CONVERSION AND EQUIVALENCY TABLES Table 1.2.17 Conversion of Volumes or Capacities* mL to fluid ounces 0.03381 0.06763 0.1014 0.1353 0.1691 0.2092 0.2367 0.2705 0.3043

29.57 mL.

1-31

Fluid ounces to mL 1 2 3 4 5 6 7 8 9 29.57 59.15 88.72 118.3 147.9 177.4 207.0 236.6 266.2

* EXAMPLE: 1 fluid oz

Liquid pints to litres 0.4732 0.9463 1.420 1.893 2.366 2.839 3.312 3.785 4.259

Litres to liquid pints 2.113 4.227 6.340 8.454 10.57 12.68 14.79 16.91 19.02

Liquid quarts to litres 0.9463 1.893 2.839 3.785 4.732 5.678 6.624 7.571 8.517

Litres to liquid quarts 1.057 2.113 3.170 4.227 5.284 6.340 7.397 8.454 9.510

Gallons to litres 3.785 7.571 11.36 15.14 18.93 22.71 26.50 30.28 34.07

Litres to gallons 0.2642 0.5284 0.7925 1.057 1.321 1.585 1.849 2.113 2.378

Bushels to hectolitres 0.3524 0.7048 1.057 1.410 1.762 2.114 2.467 2.819 3.171

Hectolitres to bushels 2.838 5.676 8.513 11.35 14.19 17.03 19.86 22.70 25.54

Table 1.2.18

Mass Equivalents Ounces Pounds Avoirdupois 35.27 0.022286 1.09714 1 13.17 16 3203 35840 35274 Troy and apoth 2.6792 0.031736 0.08333 0.07595 1 1.215 2431 2722 2679 Avoirdupois 2.205 0.031429 0.06857 0.0625 0.8229 1 2000 2240 2205 Short 0.021102 0.077143 0.043429 0.043125 0.034114 0.0005 1 1.12 1.102 Tons Long 0.039842 0.076378 0.043061 0.042790 0.033673 0.034464 0.8929 1 0.9842 Metric 0.001 0.076480 0.043110 0.042835 0.033732 0.034536 0.9072 1.016 1

Kilograms 1 0.046480 0.03110 0.02835 0.3732 0.4536 907.2 1016 1000

Grains 15432 1 480 437.5 5760 7000 1406 156804 15432356

Troy and apoth 32.15 0.022083 1 0.9115 12 14.58 29167 32667 32151

Table 1.2.19

Conversion of Masses* Short tons (2000 lb) to metric tons 0.907 1.814 2.722 3.629 4.536 5.443 6.350 7.257 8.165 Metric tons (1000 kg) to short tons 1.102 2.205 3.307 4.409 5.512 6.614 7.716 8.818 9.921 Long tons (2240 lb) to metric tons 1.016 2.032 3.048 4.064 5.080 6.096 7.112 8.128 9.144 Metric tons to long tons 0.984 1.968 2.953 3.937 4.921 5.905 6.889 7.874 8.858

Grains to grams 1 2 3 4 5 6 7 8 9 0.06480 0.1296 0.1944 0.2592 0.3240 0.3888 0.4536 0.5184 0.5832

* EXAMPLE: 1 grain

Grams to grains 15.43 30.86 46.30 61.73 77.16 92.59 108.03 123.46 138.89

0.06480 grams.

Ounces (avdp) to grams 28.35 56.70 85.05 113.40 141.75 170.10 198.45 226.80 255.15

Grams to ounces (avdp) 0.03527 0.07055 0.1058 0.1411 0.1764 0.2116 0.2469 0.2822 0.3175

Pounds (avdp) to kilograms 0.4536 0.9072 1.361 1.814 2.268 2.722 3.175 3.629 4.082

Kilograms to pounds (avdp) 2.205 4.409 6.614 8.818 11.02 13.23 15.43 17.64 19.84

Table 1.2.20

Pressure Equivalents Columns of mercury at temperature 08C and g 9.80665 m/s2 Atmospheres 0.00001 0.9869 0.06805 1 0.01316 0.03342 0.000967 0.002456 cm 0.00075 75.01 5.171 76.000 1 2.540 0.07349 0.1867 in 0.000295 29.53 2.036 29.92 0.3937 1 0.02893 0.07349 Columns of water at temperature 158C and g 9.80665 m/s2 cm 0.01021 1020.7 70.37 1034 13.61 34.56 1 2.540 in 0.00402 401.8 27.703 407.1 5.357 13.61 0.3937 1

Pascals N/m2 1 100000 6894.8 101326 1333 3386 97.98 248.9

Bars 105 N/m2 10 1 0.068948 1.0132 0.0133 0.03386 0.0009798 0.002489

5

Poundsf per in2 0.000145 14.504 1 14.696 0.1934 0.4912 0.01421 0.03609

1-32

MEASURING UNITS Table 1.2.21

2

Conversion of Pressures* Bars to lb/in2 14.504 29.008 43.511 58.015 72.519 87.023 101.53 116.03 130.53

0.06895 bar.

Lb/in to bars 1 2 3 4 5 6 7 8 9 0.06895 0.13790 0.20684 0.27579 0.34474 0.41368 0.48263 0.55158 0.62053

* EXAMPLE: 1 lb/in2

Lb/in2 to atmospheres 0.06805 0.13609 0.20414 0.27218 0.34823 0.40826 0.47632 0.54436 0.61241

Atmospheres to lb/in2 14.696 29.392 44.098 58.784 73.480 88.176 102.87 117.57 132.26

Bars to atmospheres 0.98692 1.9738 2.9607 3.9477 4.9346 5.9215 6.9085 7.8954 8.8823

Atmospheres to bars 1.01325 2.0265 3.0397 4.0530 5.0663 6.0795 7.0927 8.1060 9.1192

Table 1.2.22 cm/s 1 100 1.667 27.78 30.48 0.5080 44.70 51.44

Velocity Equivalents m/s 0.01 1 0.01667 0.2778 0.3048 0.005080 0.4470 0.5144 m/min 0.6 60 1 16.67 18.29 0.3048 26.82 30.87 km/h 0.036 3.6 0.06 1 1.097 0.01829 1.609 1.852 ft/s 0.03281 3.281 0.05468 0.9113 1 0.01667 1.467 1.688 ft/min 1.9685 196.85 3.281 54.68 60 1 88 101.3 mi/h 0.02237 2.237 0.03728 0.6214 0.6818 0.01136 1 1.151 Knots 0.01944 1.944 0.03240 0.53996 0.59248 0.00987 0.86898 1

Table 1.2.23

Conversion of Linear and Angular Velocities* ft/min to cm/s 0.508 1.016 1.524 2.032 2.540 3.048 3.556 4.064 4.572

1.97 ft/min.

cm/s to ft/min 1 2 3 4 5 6 7 8 9 1.97 3.94 5.91 7.87 9.84 11.81 13.78 15.75 17.72

* EXAMPLE: 1 cm/s

cm/s to mi/h 0.0224 0.0447 0.0671 0.0895 0.1118 0.1342 0.1566 0.1790 0.2013

mi/h to cm/s 44.70 89.41 134.1 178.8 223.5 268.2 312.9 357.6 402.3

ft/s to mi/h 0.682 1.364 2.045 2.727 3.409 4.091 4.773 5.455 6.136

mi/h to ft/s 1.47 2.93 4.40 5.87 7.33 8.80 10.27 11.73 13.20

rad/s to r/min 9.55 19.10 28.65 38.20 47.75 57.30 66.84 76.39 85.94

r/min to rad/s 0.1047 0.2094 0.3142 0.4189 0.5236 0.6283 0.7330 0.8378 0.9425

Table 1.2.24 cm/s

2

Acceleration Equivalents m/s2 0.01 1 0.0002778 0.2778 0.00008467 0.3048 0.00008467 0.4470 0.5144 m/(h s) 36.00 3600 1 1000 0.3048 1097 0.3048 1609 1852 km/(h s) 0.036 3.6 0.001 1 0.0003048 1.097 0.0003048 1.609 1.852 ft/(h s) 118.1 11811 3.281 3281 1 3600 1 5280 6076 ft/s2 0.03281 3.281 0.0009113 0.9113 0.0002778 1 0.0002778 1.467 1.688 ft/min2 118.1 11811 3.281 3281 1 3600 1 5280 6076 mi/(h s) 0.02237 2.237 0.0006214 0.6214 0.0001894 0.6818 0.0001894 1 1.151 knots/s 0.01944 1.944 0.0005400 0.5400 0.0001646 0.4572 0.0001646 0.8690 1

1 100 0.02778 27.78 0.008467 30.48 0.008467 44.70 51.44

CONVERSION AND EQUIVALENCY TABLES Table 1.2.25 cm/s to ft/min2 1 2 3 4 5 6 7 8 9 118.1 236.2 354.3 472.4 590.6 708.7 826.8 944.9 1063

* EXAMPLE: 1 cm/s2

2

1-33

Conversion of Accelerations* km/(h s) to mi/(h s) 0.6214 1.243 1.864 2.485 3.107 3.728 4.350 4.971 5.592

118.1 ft/min2.

km/(h s) to knots/s 0.5400 1.080 1.620 2.160 2.700 3.240 3.780 4.320 4.860

ft/s2 to mi/(h s) 0.6818 1.364 2.045 2.727 3.409 4.091 4.772 5.454 6.136

ft/s2 to knots/s 0.4572 0.9144 1.372 1.829 2.286 2.743 3.200 3.658 4.115

ft/min2 to cm/s2 0.008467 0.01693 0.02540 0.03387 0.04233 0.05080 0.05927 0.06774 0.07620

mi/(h s) to km/(h s) 1.609 3.219 4.828 6.437 8.046 9.656 11.27 12.87 14.48

mi/(h s) to knots/s 0.8690 1.738 2.607 3.476 4.345 5.214 6.083 6.952 7.821

knots/s to mi/(h s) 1.151 2.302 3.452 4.603 5.754 6.905 8.056 9.206 10.36

knots/s to km/(h s) 1.852 3.704 5.556 7.408 9.260 11.11 12.96 14.82 16.67

Table 1.2.26 Joules or Newton-metres 1 9.80665 1.356 3.600 106 2.648 106 2.6845 106 101.33 4186.8 1055

Energy or Work Equivalents Kilogramfmetres 0.10197 1 0.1383 3.671 105 270000 2.7375 105 10.333 426.9 107.6 Kilowatt hours 0.062778 0.052724 0.063766 1 0.7355 0.7457 0.042815 0.001163 0.032931 Metric horsepowerhours 0.063777 0.0537037 0.0651206 1.3596 1 1.0139 0.043827 0.001581 0.033985 Horsepowerhours 0.063725 0.053653 0.0650505 1.341 0.9863 1 0.043775 0.001560 0.033930 Litreatmospheres 0.009869 0.09678 0.01338 35528 26131 26493 1 41.32 10.41 British thermal units 0.039478 0.009295 0.001285 3412 2510 2544 0.09604 3.968 1

Foot-poundsf 0.7376 7.233 1 2.655 106 1.9529 106 1.98 106 74.74 3088 778.2

Kilocalories 0.032388 0.002342 0.033238 859.9 632.4 641.2 0.02420 1 0.25200

Table 1.2.27 Ft lbf to joules 1 2 3 4 5 6 7 8 9

Conversion of Energy, Work, Heat* Joules to ft lbf 0.7376 1.4751 2.2127 2.9503 3.6879 4.4254 5.1630 5.9006 6.6381

1.3558 J.

Ft lbf to Btu 0.001285 0.002570 0.003855 0.005140 0.006425 0.007710 0.008995 0.01028 0.01156

Btu to ft lbf 778.2 1,556 2,334 3,113 3,891 4,669 5,447 6,225 7,003

Kilogramfmetres to kilocalories 0.002342 0.004685 0.007027 0.009369 0.01172 0.01405 0.01640 0.01874 0.02108

Kilocalories to kilogramfmetres 426.9 853.9 1,281 1,708 2,135 2,562 2,989 3,415 3,842

Joules to calories 0.2388 0.4777 0.7165 0.9554 1.194 1.433 1.672 1.911 2.150

Calories to joules 4.187 8.374 12.56 16.75 20.93 25.12 29.31 33.49 37.68

1.3558 2.7116 4.0674 5.4232 6.7790 8.1348 9.4906 10.8464 12.2022

* EXAMPLE: 1 ft lbf

Table 1.2.28 Horsepower 1 1.341 0.9863 0.01315 0.00182 5.615 1.415

Power Equivalents Kilowatts 0.7457 1 0.7355 0.009807 0.001356 4.187 1.055 Metric horsepower 1.014 1.360 1 0.01333 0.00184 5.692 1.434 Kgf m per s 76.04 102.0 75 1 0.1383 426.9 107.6 Ft lbf per s 550 737.6 542.5 7.233 1 3088 778.2 Kilocalories per s 0.1781 0.2388 0.1757 0.002342 0.033238 1 0.2520 Btu per s 0.7068 0.9478 0.6971 0.009295 0.001285 3.968 1

1-34

MEASURING UNITS Table 1.2.29 Conversion of Power* Kilowatts to horsepower 1.341 2.682 4.023 5.364 6.705 8.046 9.387 10.73 12.07

0.7457 kW.

Horsepower to kilowatts 1 2 3 4 5 6 7 8 9 0.7457 1.491 2.237 2.983 3.729 4.474 5.220 5.966 6.711

* EXAMPLE: 1 hp

Metric horsepower to kilowatts 0.7355 1.471 2.206 2.942 3.677 4.412 5.147 5.883 6.618

Kilowatts to metric horsepower 1.360 2.719 4.079 5.438 6.798 8.158 9.520 10.88 12.24

Horsepower to metric horsepower 1.014 2.028 3.042 4.055 5.069 6.083 7.097 8.111 9.125

Metric horsepower to horsepower 0.9863 1.973 2.959 3.945 4.932 5.918 6.904 7.891 8.877

Table 1.2.30 Grams per mL 1 27.68 0.01602 1.187 0.1198

Density Equivalents* Lb per cu in 0.03613 1 0.035787 0.04287 0.004329 Lb per cu ft 62.43 1728 1 74.7 7.481 Short tons (2,000 lb) per cu yd 0.8428 23.33 0.0135 1 0.1010 Lb per U.S. gal 8.345 231 0.1337 9.902 1

Table 1.2.31

Conversion of Density Grams per mL to lb per cu ft Lb per cu ft to grams per mL 0.01602 0.03204 0.04805 0.06407 0.08009 0.09611 0.11213 0.12814 0.14416 0.16018 Grams per mL to short tons per cu yd 0.8428 1.6856 2.5283 3.3711 4.2139 5.0567 5.8995 6.7423 7.5850 8.4278 Short tons per cu yd to grams per mL 1.187 2.373 3.560 4.746 5.933 7.119 8.306 9.492 10.679 11.866

1 2 3 4 5 6 7 8 9 10

62.43 124.86 187.28 249.71 312.14 374.57 437.00 499.43 561.85 624.28

* EXAMPLE: 1 g per mL

62.43 lb per cu ft.

Table 1.2.32 Calories per cm s 8C 1 0.2388 0.0002778 0.004134 0.00001435

Thermal Conductivity Watts per cm 8C 4.1868 1 0.001163 0.01731 0.00006009 Calories per cm h 8C 3,600 860 1 14.88 0.05167 Btu ft per ft2 h 8F 241.9 57.79 0.0672 1 0.00347 Btu in per ft2 day 8F 69,670 16,641 19.35 288 1

Table 1.2.33 Calories per cm2 s 8C 1 0.2388 0.0002778 0.0001356 0.000005651

Thermal Conductance Watts per cm2 8C 4.1868 1 0.001163 0.0005678 0.00002366 Calories per cm2 h 8C 3,600 860 1 0.4882 0.02034 Btu per ft2 h 8F 7,373 1,761 2.048 1 0.04167 Btu per ft2 day 8F 176,962 42,267 49.16 24 1

Table 1.2.34 Calories per cm2 s 1 0.2388 0.0002778 0.00007535 0.000003139

Heat Flow Watts per cm2 4.1868 1 0.001163 0.0003154 0.00001314 Calories per cm2 h 3,600 860 1 0.2712 0.01130 Btu per ft2 h 13,272 3,170 3.687 1 0.04167 Btu per ft2 day 318,531 76,081 88.48 24 1

Marks'

Standard Handbook for Mechanical Engineers

Revised by a staff of specialists

EUGENE A. AVALLONE

Editor

Consulting Engineer; Professor of Mechanical Engineering, Emeritus The City College of the City University of New York

THEODORE BAUMEISTER III

Editor

Retired Consultant, Information Systems Department E. I. du Pont de Nemours & Co.

ALI M. SADEGH

Editor

Consulting Engineer; Professor of Mechanical Engineering The City College of the City University of New York

Eleventh Edition

New York Chicago San Francisco Lisbon London Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

Madrid

Library of Congress Cataloged The First Issue of this title as follows: Standard handbook for mechanical engineers. 1st-ed.; 1916­ New York, McGraw-Hill. v. Illus. 18­24 cm. Title varies: 1916­58; Mechanical engineers' handbook. Editors: 1916­51, L. S. Marks.--1958­ T. Baumeister. Includes bibliographies. 1. Mechanical engineering--Handbooks, manuals, etc. I. Marks, Lionel Simeon, 1871­ ed. II. Baumeister, Theodore, 1897­ ed. III. Title; Mechanical engineers' handbook. TJ151.S82 502'.4'621 16­12915 Library of Congress Catalog Card Number: 87-641192

MARKS' STANDARD HANDBOOK FOR MECHANICAL ENGINEERS Copyright © 2007, 1996, 1987, 1978 by The McGraw-Hill Companies, Inc. Copyright © 1967, renewed 1995, and 1958, renewed 1986, by Theodore Baumeister III. Copyright © 1951, renewed 1979, by Lionel P. Marks and Alison P. Marks. Copyright © 1941, renewed 1969, and 1930, renewed 1958, by Lionel Peabody Marks. Copyright © 1924, renewed 1952 by Lionel S. Marks. Copyright © 1916 by Lionel S. Marks. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 1 2 3 4 5 6 7 8 9 0 DOW/DOW 0 1 0 9 8 7 6 ISBN-13: 978-0-07-142867-5 ISBN-10: 0-07-142867-4

The sponsoring editor for this book was Larry S. Hager, the editing supervisor was David E. Fogarty, and the production supervisor was Richard C. Ruzycka. It was set in Times Roman by International Typesetting and Composition. The art director for the cover was Anthony Landi. Printed and bound by RR Donnelley. This book is printed on acid-free paper. The editors and the publisher will be grateful to readers who notify them of any inaccuracy or important omission in this book.

Information contained in this work has been obtained by The McGraw-Hill Companies, Inc. ("McGraw-Hill") from sources believed to be reliable. However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGrawHill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought.

Contents

For the detailed contents of any section consult the title page of that section.

Contributors ix The Editors xiii Preface to the Eleventh Edition xv Preface to the First Edition xvii Symbols and Abbreviations xix

1. Mathematical Tables and Measuring Units . . . . . . . . . . . . . .

1.1 Mathematical Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Measuring Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-1

1-1 1-16

2. Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-1

2-2 2-40

3. Mechanics of Solids and Fluids . . . . . . . . . . . . . . . . . . . . . .

3.1 3.2 3.3 3.4 Mechanics of Solids Friction . . . . . . . . . . . Mechanics of Fluids . Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-1

3-2 3-20 3-29 3-61

4. Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 4.2 4.3 4.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Substances . . . . . . . . Radiant Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . Transmission of Heat by Conduction and Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-1

4-2 4-32 4-63 4-79

5. Strength of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 5.2 5.3 5.4 5.5 5.6 Mechanical Properties of Materials . . . . . Mechanics of Materials . . . . . . . . . . . . . . Pipeline Flexure Stresses . . . . . . . . . . . . Nondestructive Testing . . . . . . . . . . . . . . Experimental Stress and Strain Analysis Mechanics of Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5-1

5-2 5-14 5-51 5-57 5-63 5-71

6. Materials of Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 General Properties of Materials . . . . . . . . . . . . . . . Iron and Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iron and Steel Castings . . . . . . . . . . . . . . . . . . . . . . Nonferrous Metals and Alloys; Metallic Specialties Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paints and Protective Coatings . . . . . . . . . . . . . . . . Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonmetallic Materials . . . . . . . . . . . . . . . . . . . . . . . Cement, Mortar, and Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6-1

6-3 6-12 6-34 6-46 6-92 6-111 6-115 6-131 6-162

v

vi

CONTENTS

6.10 6.11 6.12 6.13

Water . . . . . . . . . . . . . . . . . Lubricants and Lubrication Plastics . . . . . . . . . . . . . . . Fiber Composite Materials

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6-171 6-180 6-189 6-206

7. Fuels and Furnaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1 7.2 7.3 7.4 7.5 Fuels . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbonization of Coal and Gas Making Combustion Furnaces . . . . . . . . . . . . . . Municipal Waste Combustion . . . . . . . . Electric Furnaces and Ovens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7-1

7-2 7-30 7-43 7-48 7-54

8. Machine Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1 8.2 8.3 8.4 8.5 8.6 8.7 Mechanism . . . . . . . . . . . . . . . . Machine Elements . . . . . . . . . . Gearing . . . . . . . . . . . . . . . . . . Fluid-Film Bearings . . . . . . . . . Bearings with Rolling Contact Packings, Gaskets, and Seals . Pipe, Pipe Fittings, and Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8-1

8-3 8-10 8-83 8-111 8-127 8-133 8-138

9. Power Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Sources of Energy . . . . . . . . . Steam Boilers . . . . . . . . . . . . . Steam Engines . . . . . . . . . . . . Steam Turbines . . . . . . . . . . . Power-Plant Heat Exchangers Internal-Combustion Engines Gas Turbines . . . . . . . . . . . . . Nuclear Power . . . . . . . . . . . . Hydraulic Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9-1

9-3 9-29 9-56 9-58 9-78 9-93 9-127 9-138 9-154

10. Materials Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.1 10.2 10.3 10.4 10.5 10.6 10.7 Materials Holding, Feeding, and Metering Lifting, Hoisting, and Elevating . . . . . . . . . Dragging, Pulling, and Pushing . . . . . . . . Loading, Carrying, and Excavating . . . . . . Conveyor Moving and Handling . . . . . . . . Automatic Guided Vehicles and Robots . . Material Storage and Warehousing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-1

10-2 10-4 10-22 10-26 10-42 10-63 10-69

11. Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Automotive Engineering . . . . . . . . . . . Railway Engineering . . . . . . . . . . . . . . Marine Engineering . . . . . . . . . . . . . . . Aeronautics . . . . . . . . . . . . . . . . . . . . . Jet Propulsion and Aircraft Propellers Astronautics . . . . . . . . . . . . . . . . . . . . Pipeline Transmission . . . . . . . . . . . . . Containerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11-1

11-3 11-18 11-40 11-58 11-83 11-104 11-139 11-149

12. Building Construction and Equipment . . . . . . . . . . . . . . . . .

12.1 12.2 12.3 12.4 12.5 12.6 Industrial Plants . . . . . . . . . . . . . . . . . . . . . . . . Structural Design of Buildings . . . . . . . . . . . . . Reinforced Concrete Design and Construction Air Conditioning, Heating, and Ventilating . . . . Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound, Noise, and Ultrasonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12-1

12-2 12-19 12-37 12-49 12-88 12-96

13. Manufacturing Processes . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.1 Foundry Practice and Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13-1

13-3

CONTENTS

vii

13.2 13.3 13.4 13.5 13.6 13.7

Plastic Working of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Welding and Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Machining Processes and Machine Tools . . . . . . . . . . . . . . . . . Surface Texture Designation, Production, and Quality Control Woodcutting Tools and Machines . . . . . . . . . . . . . . . . . . . . . . . Precision Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . .

13-9 13-29 13-50 13-72 13-77 13-80

14. Fans, Pumps, and Compressors . . . . . . . . . . . . . . . . . . . . . .

14.1 14.2 14.3 14.4 14.5 Displacement Pumps Centrifugal Pumps . . . Compressors . . . . . . . High-Vacuum Pumps . Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14-1

14-2 14-15 14-26 14-39 14-46

15. Electrical and Electronics Engineering . . . . . . . . . . . . . . . .

15.1 Electrical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15-1

15-2 15-68

16. Instruments and Controls . . . . . . . . . . . . . . . . . . . . . . . . . . .

16.1 Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Automatic Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Surveying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16-1

16-2 16-21 16-52

17. Industrial Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17.1 17.2 17.3 17.4 17.5 17.6 17.7 Operations Management . . . . . . . . . . . . . . . Cost Accounting . . . . . . . . . . . . . . . . . . . . . Engineering Statistics and Quality Control Methods Engineering . . . . . . . . . . . . . . . . . Cost of Electric Power . . . . . . . . . . . . . . . . . Human Factors and Ergonomics . . . . . . . . Automatic Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17-1

17-3 17-11 17-18 17-25 17-31 17-39 17-43

18. The Regulatory Environment . . . . . . . . . . . . . . . . . . . . . . . .

18.1 18.2 18.3 18.4 Environmental Control . . . . . . . . . . . Occupational Safety and Health . . . . Fire Protection . . . . . . . . . . . . . . . . . . Patents, Trademarks, and Copyrights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18-1

18-2 18-18 18-22 18-27

19. Refrigeration, Cryogenics, and Optics . . . . . . . . . . . . . . . . .

19.1 Mechanical Refrigeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Cryogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19-1

19-2 19-26 19-41

20. Emerging Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12 20.13 An Introduction to Microelectromechanical Systems (MEMS) . . . . . . . . . Introduction to Nanotechnology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ferroelectrics/Piezoelectrics and Shape Memory Alloys . . . . . . . . . . . . . Introduction to the Finite-Element Method . . . . . . . . . . . . . . . . . . . . . . . . Computer-Aided Design, Computer-Aided Engineering, and Variational Design . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . Experimental Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Biomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Human Injury Tolerance and Anthropometric Test Devices . . . . . . . . . . . Air-Inflated Fabric Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robotics, Mechatronics, and Intelligent Automation . . . . . . . . . . . . . . . . Rapid Prototyping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20-1

20-3 20-13 20-20 20-28 20-44 20-51 20-63 20-79 20-104 20-108 20-118 20-132 20-135

Index follows Section 20

Contributors

Abraham Abramowitz* Consulting Engineer; Professor of Electrical Engineering,

Emeritus, The City College of The City University of New York (ILLUMINATION) Vincent M. Altamuro President, VMA Inc., Toms River, NJ (MATERIAL HOLDING, FEEDING, AND METERING. CONVEYOR MOVING AND HANDLING. AUTOMATED GUIDED VEHICLES AND ROBOTS. MATERIAL STORAGE AND WAREHOUSING. METHODS ENGINEERING. AUTOMATIC MANUFACTURING. INDUSTRIAL PLANTS) Charles A. Amann Principal Engineer, KAB Engineering (AUTOMOTIVE ENGINEERING) Farid M. Amirouche Professor of Mechanical and Industrial Engineering, University of Illinois at Chicago (INTRODUCTION TO THE FINITE-ELEMENT METHOD. COMPUTER-AIDED DESIGN, COMPUTER-AIDED ENGINEERING, AND VARIATIONAL DESIGN) Yiannis Andreopoulos Professor of Mechanical Engineering, The City College of the City University of New York (EXPERIMENTAL FLUID MECHANICS) William Antis* Technical Director, Maynard Research Council, Inc., Pittsburgh, PA (METHODS ENGINEERING) Glenn E. Asauskas Lubrication Engineer, Chevron Corp. (LUBRICANTS AND LUBRICATION) Dennis N. Assanis Professor of Mechanical Engineering, University of Michigan (INTERNAL COMBUSTION ENGINES) Eugene A. Avallone Consulting Engineer; Professor of Mechanical Engineering, Emeritus, The City College of The City University of New York (MECHANICAL PROPERTIES OF MATERIALS. GENERAL PROPERTIES OF MATERIALS. PIPE, PIPE FITTINGS, AND VALVES. SOURCES OF ENERGY. STEAM ENGINES. MISCELLANY) Klemens C. Baczewski Consulting Engineer (CARBONIZATION OF COAL AND GAS MAKING) Glenn W. Baggley* Former Manager, Regenerative Systems, Bloom Engineering Co., Inc. (COMBUSTION FURNACES) Frederick G. Baily Consulting Engineer; Steam Turbines, General Electric Co. (STEAM TURBINES) Robert D. Bartholomew Associate, Sheppard T. Powell Associates, LLC (CORROSION) George F. Baumeister President, EMC Process Corp., Newport, DE (MATHEMATICAL TABLES) John T. Baumeister Manager, Product Compliance Test Center, Unisys Corp. (MEASURING UNITS) E. R. Behnke* Product Manager, CM Chain Division, Columbus, McKinnon Corp. (CHAINS) John T. Benedict* Retired Standards Engineer and Consultant, Society of Automotive Engineers (AUTOMOTIVE ENGINEERING) Bernadette M. Bennett, Esq. Associate; Carter, DeLuca, Farrell and Schmidt, LLP Melville, NY (PATENTS, TRADEMARKS, AND COPYRIGHTS) Louis Bialy Director, Codes & Product Safety, Otis Elevator Company (ELEVATORS, DUMBWAITERS, AND ESCALATORS) Malcolm Blair Technical and Research Director, Steel Founders Society of America (IRON AND STEEL CASTINGS) Omer W. Blodgett Senior Design Consultant, Lincoln Electric Co. (WELDING AND CUTTING) B. Douglas Bode Engineering Supervisor, Product Customization and Vehicle Enhancement, Construction and Forestry Div., John Deere (OFF-HIGHWAY VEHICLES AND EARTHMOVING EQUIPMENT) Donald E. Bolt* Engineering Manager, Heat Transfer Products Dept., Foster Wheeler Energy Corp. (POWER PLANT HEAT EXCHANGERS) G. David Bounds Senior Engineer, Duke Energy Corp. (PIPELINE TRANSMISSION) William J. Bow* Director, Retired, Heat Transfer Products Department, Foster Wheeler Energy Corp. (POWER PLANT HEAT EXCHANGERS)

James L. Bowman* Senior Engineering Consultant, Rotary-Reciprocating Compressor

Division, Ingersoll-Rand Co. (COMPRESSORS)

Walter H. Boyes, Jr. Editor-in-Chief/Publisher, Control Magazine (INSTRUMENTS) Richard L. Brazill Technology Specialist, ALCOA Technical Center, ALCOA (ALUMINUM

AND ITS ALLOYS)

Frederic W. Buse* Chief Engineer, Standard Pump Division, Ingersoll-Rand Co.

(DISPLACEMENT PUMPS)

Charles P. Butterfield

*Contributions by authors whose names are marked with an asterisk were made for the previous edition and have been revised or rewritten by others for this edition. The stated professional position in these cases is that held by the author at the time of his or her contribution.

Chief Engineer, National Wind Technology Center, National Renewable Energy Laboratory (WIND POWER) C. L. Carlson* Late Fellow Engineer, Research Labs., Westinghouse Electric Corp. (NONFERROUS METALS AND ALLOYS. METALS AND ALLOYS FOR NUCLEAR ENERGY APPLICATIONS) Scott W. Case Professor of Engineering Science & Mechanics, Virginia Polytechnic Institute and State University (MECHANICS OF COMPOSITE MATERIALS) Vittorio (Rino) Castelli Senior Research Fellow, Retired, Xerox Corp.; Engineering Consultant (FRICTION. FLUID FILM BEARINGS) Paul V. Cavallaro Senior Mechanical Research Engineer, Naval Underwater Warfare Center (AIR-INFLATED FABRIC STRUCTURES) Eric L. Christiansen Johnson Space Center, NASA (METEOROID/ORBITAL DEBRIS SHIELDING) Robin O. Cleveland Associate Professor of Aerospace and Mechanical Engineering, Boston University (SOUND, NOISE, AND ULTRASONICS) Gary L. Cloud Professor, Department of Mechanical Engineering, Michigan State University (EXPERIMENTAL STRESS AND STRAIN ANALYSIS) Ashley C. Cockerill Vice President and Event Coordinator, nanoTech Business, Inc. (ENGINEERING STATISTICS AND QUALITY CONTROL) Timothy M. Cockerill Senior Project Manager, University of Illinois (ELECTRONICS) Thomas J. Cockerill Advisory Engineer, International Business Machines Corp. (COMPUTERS) Aaron Cohen Retired Center Director, Lyndon B. Johnson Space Center, NASA; Zachry Professor, Texas A&M University (ASTRONAUTICS) Arthur Cohen Former Manager, Standards and Safety Engineering, Copper Development Assn. (COPPER AND COPPER ALLOYS) D. E. Cole Director, Office for Study of Automotive Transportation, Transportation Research Institute, University of Michigan (INTERNAL COMBUSTION ENGINES) James M. Connolly Section Head, Projects Department, Jacksonville Electric Authority (COST OF ELECTRIC POWER) Alexander Couzis Professor of Chemical Engineering, The City College of the City University of New York (INTRODUCTION TO NANOTECHNOLOGY) Terry L. Creasy Assistant Professor of Mechanical Engineering, Texas A&M University (STRUCTURAL COMPOSITES) M. R. M. Crespo da Silva* University of Cincinnati (ATTITUDE DYNAMICS, STABILIZATION, AND CONTROL OF SPACECRAFT) Richard A. Dahlin Vice President, Engineering, Walker Magnetics (LIFTING MAGNETS) Benjamin C. Davenny Acoustical Consultant, Acentech Inc., Cambridge, MA (SOUND, NOISE, AND ULTRASONICS) William H. Day President, Longview Energy Associates, LLC; formerly Founder and Board Chairman, The Gas Turbine Association (GAS TURBINES) Benjamin B. Dayton Consulting Physicist, East Flat Rock, NC (HIGH-VACUUM PUMPS) Horacio M. de la Fuente Senior Engineer, NASA Johnson Space Center (TRANSHAB) Donald D. Dodge Supervisor, Retired, Product Quality and Inspection Technology, Manufacturing Development, Ford Motor Co. (NONDESTRUCTIVE TESTING) Andrew M. Donaldson Project Director, Parsons E&C, Reading, PA (COST OF ELECTRIC POWER) Joseph S. Dorson Senior Engineer, Columbus McKinnon Corp. (CHAIN) James Drago Manager, Engineering, Garlock Sealing Technologies (PACKING, GASKETS, AND SEALS) Michael B. Duke Chief, Solar Systems Exploration, Johnson Space Center, NASA (DYNAMIC ENVIRONMENTS) F. J. Edeskuty Retired Associate, Las Alamos National Laboratory (CRYOGENICS)

ix

x

CONTRIBUTORS

University of Cincinnati (SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND

AND

O. Elnan*

Byron M. Jones* Retired Assistant Professor of Electrical Engineering, School of

Engineering, University of Tennessee at Chattanooga (ELECTRONICS) Professor, Department of Accounting & MIS, Alfred Lerner College of Business & Economics, University of Delaware (COST ACCOUNTING) Robert Jorgensen Engineering Consultant (FANS) Serope Kalpakjian Professor Emeritus of Mechanical and Materials Engineering, Illinois Institute of Technology (MACHINING PROCESSES AND MACHINE TOOLS) Igor J. Karassik* Late Senior Consulting Engineer, Ingersoll Dresser Pump Co. (CENTRIFUGAL AND AXIAL FLOW PUMPS) Jonathan D. Kemp Vibration Consultant, Acentech, Inc., Cambridge, MA (SOUND, NOISE, AND ULTRASONICS) J. Randolph Kissell President, The TGB Partnership (ALUMINUM AND ITS ALLOYS) John B. Kitto, Jr. Babcock & Wilcox Co. (STEAM BOILERS) Andrew C. Klein Professor of Nuclear Engineering, Oregon State University; Director of Training, Education and Research Partnership, Idaho National Laboratories (NUCLEAR POWER. ENVIRONMENTAL CONTROL. OCCUPATIONAL SAFETY AND HEALTH. FIRE PROTECTION) Doyle Knight Professor of Mechanical and Aerospace Engineering, Rutgers University (INTRODUCTION TO COMPUTATIONAL FLUID MECHANICS) Ronald M. Kohner President, Landmark Engineering Services, Ltd. (DERRICKS) Ezra S. Krendel Emeritus Professor of Operations Research and Statistics, University of Pennsylvania (HUMAN FACTORS AND ERGONOMICS. MUSCLE GENERATED POWER) A. G. Kromis* University of Cincinnati (SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND PERFORMANCE) Srirangam Kumaresan Biomechanics Institute, Santa Barbara, California (HUMAN INJURY TOLERANCE AND ANTHROPOMETRIC TEST DEVICES) L. D. Kunsman* Late Fellow Engineer, Research Labs, Westinghouse, Electric Corp. (NONFERROUS METALS AND ALLOYS. METALS AND ALLOYS FOR NUCLEAR ENERGY APPLICATIONS) Colin K. Larsen Vice President, Blue Giant U.S.A. Corp. (SURFACE HANDLING) Stan Lebow Research Forest Products Technologist, Forest Products Lab., USDA (WOOD) John H. Lewis Engineering Consultant; Formerly Engineering Staff, Pratt & Whitney Division, United Technologies Corp.; Adjunct Associate Professor, Hartford Graduate Center, Renssealear Polytechnic Institute (GAS TURBINES) Jackie Jie Li Professor of Mechanical Engineering, The City College of the City University of New York (FERROELECTRICS/PIEZOELECTRICS AND SHAPE MEMORY ALLOYS) Peter E. Liley Professor Emeritus of Mechanical Engineering, Purdue University (THERMODYNAMICS, THERMODYNAMIC PROPERTIES OF SUBSTANCES) James P. Locke Flight Surgeon, NASA Johnson Space Center (PORTABLE HYPERBARIC CHAMBER) Ernst K. H. Marburg Manager, Product Standards and Service, Columbus McKinnon Corp. (LIFTING, HOISTING, AND ELEVATING. DRAGGING, PULLING, AND PUSHING. LOADING, CARRYING, AND EXCAVATING) Larry D. Means President, Means Engineering and Consulting (WIRE ROPE) Leonard Meirovitch University Distinguished Professor Emeritus, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University (VIBRATION) George W. Michalec Late Consulting Engineer (GEARING) Duane K. Miller Manager, Engineering Services, Lincoln Electric Co. (WELDING AND CUTTING) Patrick C. Mock Principal Electron Optical Scientist, Science and Engineering Associates, Inc. (OPTICS) Thomas L. Moser Deputy Associate Administrator, Office of Space Flight, NASA Headquarters, NASA (SPACE-VEHICLE STRUCTURES) George J. Moshos* Professor Emeritus of Computer and Information Science, New Jersey Institute of Technology (COMPUTERS) Eduard Muljadi Senior Engineer, National Wind Technology Center, National Renewable Energy Laboratory (WIND POWER) Otto Muller-Girard* Consulting Engineer (INSTRUMENTS) James W. Murdock Late Consulting Engineer (MECHANICS OF FLUIDS) Gregory V. Murphy Professor, Department of Electrical and Computer Engineering, College of Engineering, University of Alabama (AUTOMATIC CONTROLS) Joseph F. Murphy Research General Engineer, Forest Products Lab., USDA (WOOD) John Nagy Retired Supervisory Physical Scientist, U.S. Department of Labor, Mine Safety and Health Administration (DUST EXPLOSIONS) B. W. Niebel* Professor Emeritus of Industrial Engineering, The Pennsylvania State University (INDUSTRIAL ECONOMICS AND MANAGEMENT) James J. Noble Formerly Research Associate Professor of Chemical and Biological Engineering, Tufts University (RADIANT HEAT TRANSFER) Charles Osborn Business Manager, Precision Cleaning Division, PTI Industries, Inc. (PRECISION CLEANING) D. J. Patterson Professor of Mechanical Engineering, Emeritus, University of Michigan (INTERNAL COMBUSTION ENGINES) Harold W. Paxton United States Steel Professor Emeritus, Carnegie Mellon University (IRON AND STEEL) Richard W. Perkins Professor Emeritus of Mechanical, Aerospace, and Manufacturing Engineering, Syracuse University (WOODCUTTING TOOLS AND MACHINES) W. R. Perry* University of Cincinnati (ORBITAL MECHANICS. SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND PERFORMANCE)

PERFORMANCE. ORBITAL MECHANICS)

Robert E. Eppich Vice President, Technology, American Foundry Society (IRON

STEEL CASTINGS)

Scott K. Jones

C. James Erickson*

Retired Principal Consultant, Engineering Department, E. I. du Pont de Nemours & Co. (ELECTRICAL ENGINEERING) George H. Ewing* Retired President and Chief Executive Officer, Texas Eastern Gas Pipeline Co. and Transwestern Pipeline Co. (PIPELINE TRANSMISSION) Heimir Fanner Chief Design Engineer, Ariel Corp. (COMPRESSORS) Erich A. Farber Distinguished Service Professor Emeritus, Director Emeritus of Solar Energy and Energy Conversion Lab., University of Florida [STIRLING (HOT AIR) ENGINES. SOLAR ENERGY. DIRECT ENERGY CONVERSION] Raymond E. Farrell, Esq. Partner; Carter, DeLuca, Farrell and Schmidt, LLP, Melville, NY (PATENTS, TRADEMARKS, AND COPYRIGHTS) D. W. Fellenz* University of Cincinnati (SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND PERFORMANCE. ATMOSPHERIC ENTRY) Chuck Fennell Program Manager, Dalton Foundries (FOUNDARY PRACTICE AND EQUIPMENT) Arthur J. Fiehn* Late Retired Vice President, Project Operations Division, Burns & Roe, Inc. (COST OF ELECTRIC POWER) Sanford Fleeter McAllister Distinguished Professor, School of Mechanical Engineering, Purdue University (JET PROPULSION AND AIRCRAFT PROPELLERS) Luc G. Fréchette Professor of Mechanical Engineering, Université de Sherbrooke, Canada [AN INTRODUCTION TO MICROELECTROMECHANICAL SYSTEMS (MEMS)] William L. Gamble Professor Emeritus of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign (CEMENT, MORTAR, AND CONCRETE. REINFORCED CONCRETE DESIGN AND CONSTRUCTION) Robert F. Gambon Power Plant Design and Development Consultant (COST OF ELECTRIC POWER) Burt Garofab Senior Engineer, Pittston Corp. (MINES, HOISTS, AND SKIPS. LOCOMOTIVE HAULAGE, COAL MINES) Siamak Ghofranian Senior Engineer, Rockwell Aerospace (DOCKING OF TWO FREEFLYING SPACECRAFT) Samuel V. Glorioso Section Chief, Metallic Materials, Johnson Space Center, NASA (STRESS CORROSION CRACKING) Norman Goldberg Consulting Engineer, Economides and Goldberg (AIR-CONDITIONING, HEATING, AND VENTILATING) Andrew Goldenberg Professor of Mechanical and Industrial Engineering, University of Toronto, Canada; President and CEO of Engineering Service Inc. (ESI) Toronto (ROBOTICS, MECHATRONICS, AND INTELLIGENT AUTOMATION) David T. Goldman* Late Deputy Manager, U.S. Department of Energy, Chicago Operations Office (MEASURING UNITS) Frank E. Goodwin Executive Vice President, ILZRO, Inc. (BEARING METALS. LOWMELTING-POINT METALS AND ALLOYS. ZINC AND ZINC ALLOYS) Don Graham Manager, Turning Products, Carboloy, Inc. (CEMENTED CARBIDES) David W. Green Supervisory Research General Engineer, Forest Products Lab., USDA (WOOD) Leonard M. Grillo Principal, Grillo Engineering Co. (MUNICIPAL WASTE COMBUSTION) Walter W. Guy Chief, Crew and Thermal Systems Division, Johnson Space Center, NASA (SPACECRAFT LIFE SUPPORT AND THERMAL MANAGEMENT) Marsbed Hablanian Retired Manager of Engineering and R&D, Varian Vacuum Technologies (HIGH-VACUUM PUMPS) Christopher P. Hansen Structures and Mechanism Engineer, NASA Johnson Space Center (PORTABLE HYPERBARIC CHAMBER) Harold V. Hawkins* Late Manager, Product Standards and Services, Columbus McKinnon Corp. (DRAGGING, PULLING, AND PUSHING. PIPELINE FLEXURE STRESSES) Keith L. Hawthorne Vice President--Technology, Transportation Technology Center, Inc. (RAILWAY ENGINEERING) V. Terrey Hawthorne Late Senior Engineer, LTK Engineering Services (RAILWAY ENGINEERING) J. Edmund Hay U.S. Department of the Interior (EXPLOSIVES) Terry L. Henshaw Consulting Engineer, Magnolia, TX (DISPLACEMENT PUMPS. CENTRIFUGAL PUMPS) Roland Hernandez Research Engineer, Forest Products Lab., USDA (WOOD) David T. Holmes Manager of Engineering, Lift-Tech International Div. of Columbus McKinnon Corp. (MONORAILS. OVERHEAD TRAVELING CRANES) Hoyt C. Hottel Late Professor Emeritus, Massachusetts Institute of Technology (RADIANT HEAT TRANSFER) Michael W. Hyer Professor of Engineering Science & Mechanics, Virginia Polytechnic Institute and State University (MECHANICS OF COMPOSITE MATERIALS) Timothy J. Jacobs Research Fellow, Department of Mechanical Engineering, University of Michigan (INTERNAL COMBUSTION ENGINES) Michael W. M. Jenkins Professor, Aerospace Design, Georgia Institute of Technology (AERONAUTICS) Peter K. Johnson Consultant (POWDERED METALS) Randolph T. Johnson Naval Surface Warfare Center (ROCKET FUELS) Robert L. Johnston Branch Chief, Materials, Johnson Space Center, NASA (METALLIC MATERIALS FOR AEROSPACE APPLICATIONS. MATERIALS FOR USE IN HIGH-PRESSURE OXYGEN SYSTEMS)

CONTRIBUTORS Kenneth A. Phair

POWER)

xi

Senior Project Engineer, Shaw Stone & Webster (GEOTHERMAL

Wiliam C. Schneider

Orvis E. Pigg Henry O. Pohl

Section Head, Structural Analysis, Johnson Space Center, NASA (SPACE Chief, Propulsion and Power Division, Johnson Space Center, NASA

VEHICLE STRUCTURES)

(SPACE PROPULSION)

Nicholas R. Rafferty

Retired Technical Associate, E. I. du Pont de Nemours & Co., Inc. (ELECTRICAL ENGINEERING) Rama Ramakumar PSO/Albrecht Naeter Professor and Director, Engineering Energy Laboratory, Oklahoma State University (WIND POWER) Pascal M. Rapier* Scientist III, Retired, Lawrence Berkeley Laboratory (ENVIRONMENTAL CONTROL. OCCUPATIONAL SAFETY AND HEALTH. FIRE PROTECTION) James D. Redmond President, Technical Marketing Services, Inc. (STAINLESS STEELS) Darrold E. Roen Late Manager, Sales & Special Engineering & Government Products, John Deere (OFF-HIGHWAY VEHICLES) Ivan L. Ross* International Manager, Chain Conveyor Division, ACCO (OVERHEAD CONVEYORS) Robert J. Ross Supervisory Research General Engineer, Forest Products Lab., USDA (WOOD) J. W. Russell* University of Cincinnati (SPACE-VEHICLE TRAJECTORIES, FLIGHT MECHANICS, AND PERFORMANCE. LUNAR- AND INTERPLANETARY FLIGHT MECHANICS) A. J. Rydzewski Consultant, DuPont Engineering, E. I. du Pont de Nemours and Company (MECHANICAL REFRIGERATION) Ali M. Sadegh Professor of Mechanical Engineering, The City College of The City University of New York (MECHANICS OF MATERIALS. NONMETALLIC MATERIALS. MECHANISM. MACHINE ELEMENTS. SURFACE TEXTURE DESIGNATION, PRODUCTION, AND QUALITY CONTROL. INTRODUCTION TO BIOMECHANICS. AIR-INFLATED FABRIC STRUCTURES. RAPID PROTOTYPING.) Anthony Sances, Jr. Biomechanics Institute, Santa Barbara, CA (HUMAN INJURY TOLERANCE AND ANTHROPOMETRIC TEST DEVICES) C. Edward Sandifer Professor, Western Connecticut State University, Danbury, CT (MATHEMATICS) Erwin M. Saniga Dana Johnson Professor of Information Technology and Professor of Operations Management, University of Delaware (OPERATIONS MANAGEMENT) Adel F. Sarofim Presidential Professor of Chemical Engineering, University of Utah (RADIANT HEAT TRANSFER) Martin D. Schlesinger Late Consultant, Wallingford Group, Ltd. (FUELS) John R. Schley Manager, Technical Marketing, RMI Titanium Co. (TITANIUM AND ZIRCONIUM) Matthew S. Schmidt Senior Engineer, Rockwell Aerospace (DOCKING OF TWO FREEFLYING SPACECRAFT)

Retired Assistant Director Engineering/Senior Engineer, NASA Johnson Space Center; Visiting Professor, Texas A&M University (ASTRONAUTICS) James D. Shearouse, III Late Senior Development Engineer, The Dow Chemical Co. (MAGNESIUM AND MAGNESIUM ALLOYS) David A. Shifler Metallurgist, MERA Metallurgical Services (CORROSION) Rajiv Shivpuri Professor of Industrial, Welding, and Systems Engineering, Ohio State University (PLASTIC WORKING OF METALS) James C. Simmons Senior Vice President, Business Development, Core Furnace Systems Corp. (ELECTRIC FURNACES AND OVENS) William T. Simpson Research Forest Products Technologist, Forest Products Lab., USDA (WOOD) Kenneth A. Smith Edward R. Gilliland Professor of Chemical Engineering, Massachusetts Institute of Technology (TRANSMISSION OF HEAT BY CONDUCTION AND CONVECTION) Lawrence H. Sobel* University of Cincinnati (VIBRATION OF STRUCTURES) James G. Speight Western Research Institute (FUELS) Robert D. Steele Project Manager, Voith Siemens Hydro Power Generation, Inc. (HYDRAULIC TURBINES) Stephen R. Swanson Professor of Mechanical Engineering, University of Utah (FIBER COMPOSITE MATERIALS) John Symonds* Fellow Engineer, Retired, Oceanic Division, Westinghouse Electric Corp. (MECHANICAL PROPERTIES OF MATERIALS) Peter L. Tea, Jr. Professor of Physics Emeritus, The City College of the City University of New York (MECHANICS OF SOLIDS) Anton TenWolde Supervisory Research Physicist, Forest Products Lab., USDA (WOOD) W. David Teter Retired Professor of Civil Engineering, University of Delaware (SURVEYING) Michael C. Tracy Rear Admiral, U.S. Navy (MARINE ENGINEERING) John H. Tundermann Former Vice President, Research and Technology, INCO Intl., Inc. (METALS AND ALLOYS FOR USE AT ELEVATED TEMPERATURES. NICKEL AND NICKEL ALLOYS) Charles O. Velzy Consultant (MUNICIPAL WASTE COMBUSTION) Harry C. Verakis Supervisory Physical Scientist, U.S. Department of Labor, Mine Safety and Health Administration (DUST EXPLOSIONS) Arnold S. Vernick Former Associate, Geraghty & Miller, Inc. (WATER) Robert J. Vondrasek* Assistant Vice President of Engineering, National Fire Protection Assoc. (COST OF ELECTRIC POWER) Michael W. Washo Senior Engineer, Retired, Eastman Kodak, Co. (BEARINGS WITH ROLLING CONTACT) Larry F. Wieserman Senior Technical Supervisor, ALCOA (ALUMINUM AND ITS ALLOYS) Robert H. White Supervisory Wood Scientist, Forest Products Lab., USDA (WOOD) John W. Wood, Jr. Manager, Technical Services, Garlock Klozure (PACKING, GASKETS, AND SEALS)

The Editors

EUGENE A. AVALLONE, Editor, is Professor of Mechanical Engineering, Emeritus, The City College of the City University of New York. He has been engaged for many years as a consultant to industry and to a number of local and national governmental agencies. THEODORE BAUMEISTER, III, Editor, is now retired from Du Pont where he was an internal consultant. His specialties are operations research, business decision making, and longrange planning. He has also taught financial modeling in the United States, South America, and the Far East. ALI M. SADEGH, Editor, is Professor of Mechanical Engineering, The City College of the City University of New York. He is also Director of the Center for Advanced Engineering Design and Development. He is actively engaged in research in the areas of machine design, manufacturing, and biomechanics.

xiii

Preface to the Eleventh Edition

The evolutionary trends underlying modern engineering practice are grounded not only on the tried and true principles and techniques of the past, but also on more recent and current advances. Thus, in the preparation of the eleventh edition of "Marks'," the Editors have considered the broad enterprise falling under the rubric of "Mechanical Engineering" and have added to and/or amended the contents to include subject areas that will be of maximum utility to the practicing engineer. That said, the Editors note that the publication of this eleventh edition has been accomplished through the combined and coordinated efforts of contributors, readers, and the Editors. First, we recognize, with pleasure, the input from our many contributors--past, continuing, and those newly engaged. Their contributions have been prepared with care, and are authoritative, informative, and concise. Second, our readers, who are practitioners in their own wise, have found that the global treatment of the subjects presented in the "Marks'" permits of great utility and serves as a convenient ready reference. The reading public has had access to "Marks'" since 1916, when Lionel S. Marks edited the first edition. This eleventh edition follows 90 years later. During the intervening years, "Marks'" and "Handbook for Mechanical Engineers" have become synonymous to a wide readership which includes mechanical engineers, engineers in the associated disciplines, and others. Our readership derives from a wide spectrum of interests, and it appears many find the "Marks'" useful as they pursue their professional endeavors. The Editors consider it a given that every successive edition must balance the requests to broaden the range or depth of subject matter printed, the incorporation of new material which will be useful to the widest possible audience, and the requirement to keep the size of the Handbook reasonable and manageable. We are aware that the current engineering practitioner learns quickly that the revolutionary developments of the recent past soon become standard practice. By the same token, it is prudent to realize that as a consequence of rapid developments, some cutting-edge technologies prove to have a short shelf life and soon are regarded as obsolescent--if not obsolete. The Editors are fortunate to have had, from time to time, input from readers and reviewers, who have proffered cogent commentary and suggestions; a number are included in this edition. Indeed, the synergy between Editors, contributors, and readers has been instrumental in the continuing usefulness of successive editions of "Marks' Standard Handbook for Mechanical Engineers." The reader will note that a considerable portion of the tabular data and running text continue to be presented in dual units; i.e., USCS and SI. The date for a projected full transition to SI units is not yet firm, and the "Marks'" reflects that. We look to the future in that regard. Society is in an era of information technology, as manifest by the practicing engineer's working tools. For example: the ubiquitous personal computer, its derivative use of software programs of a vast variety and number, printers, computer-aided design and drawing, universal access to the Internet, and so on. It is recognized, too, that the great leaps forward which

xv

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PREFACE TO THE ELEVENTH EDITION

are thereby enhanced still require the engineer to exercise sound and rational judgment as to the reliability of the solutions provided. Last, the Handbook is ultimately the responsibility of the Editors. The utmost care has been exercised to avoid errors, but if any inadvertently are included, the Publisher and Editors will appreciate being so informed. Corrections will be incorporated into subsequent printings. Ardsley, NY Newark, DE Franklin Lakes, NJ EUGENE A. AVALLONE THEODORE BAUMEISTER III ALI M. SADEGH

Preface to the First Edition*

This Handbook is intended to supply both the practicing engineer and the student with a reference work which is authoritative in character and which covers the field of mechanical engineering in a comprehensive manner. It is no longer possible for a single individual or a small group of individuals to have so intimate an acquaintance with any major division of engineering as is necessary if criticial judgment is to be exercised in the statement of current practice and the selection of engineering data. Only by the cooperation of a considerable number of specialists is it possible to obtain the desirable degree of reliability. This Handbook represents the work of fifty specialists. Each contributor is to be regarded as responsible for the accuracy of his section. The number of contributors required to ensure sufficiently specialized knowledge for all the topics treated is necessarily large. It was found desirable to enlist the services of thirteen specialists for an adequate handling of the "Properties of Engineering Materials." Such topics as "Automobiles," "Aeronautics," "Illumination," "Patent Law," "Cost Accounting," "Industrial Buildings," "Corrosion," "Air Conditioning, "Fire Protection," "Prevention of Accidents," etc., though occupying relatively small spaces in the book, demanded each a separate writer. A number of the contributions which deal with engineering practice, after examination by the Editor-in-Chief, were submitted by him to one or more specialists for criticism and suggestions. Their cooperation has proved of great value in securing greater accuracy and in ensuring that the subject matter does not embody solely the practice of one individual but is truly representative. An accuracy of four significant figures has been assumed as the desirable limit; figures in excess of this number have been deleted, except in special cases. In the mathematical tables only four significant figures have been kept. The Editor-in-Chief desires to express here his appreciation of the spirit of cooperation shown by the Contributors and of their patience in submitting to modifications of their sections. He wishes also to thank the Publishers for giving him complete freedom and hearty assistance in all matters relating to the book from the choice of contributors to the details of typography. Cambridge, Mass. April 23, 1916 LIONEL S. MARKS

*Excerpt.

xvii

Symbols and Abbreviations

For symbols of chemical elements, see Sec. 6; for abbreviations applying to metric weights and measures and SI units, Sec. 1; SI unit prefixes are listed on p. 1­19. Pairs of parentheses, brackets, etc., are frequently used in this work to indicate corresponding values. For example, the statement that "the cost per kW of a 30,000-kW plant is $86; of a 15,000-kW plant, $98; and of an 8,000-kW plant, $112," is condensed as follows: The cost per kW of a 30,000 (15,000) [8,000]-kW plant is $86 (98) [112]. In the citation of references readers should always attempt to consult the latest edition of referenced publications.

A or Å A AA AAA AAMA AAR AAS ABAI abs a.c. a-c, ac ACI ACM ACRMA ACS ACSR ACV A.D. AEC a-f, af AFBMA AFS AGA AGMA ahp AIChE AIEE AIME AIP AISC AISE AISI Al. Assn. a.m. a-m, am Am. Mach. AMA AMCA amu AN AN-FO ANC Angstrom unit 10 10 m; 3.937 10 11 in mass number N Z; ampere arithmetical average Am. Automobile Assoc. American Automobile Manufacturers' Assoc. Assoc. of Am. Railroads Am. Astronautical Soc. Am. Boiler & Affiliated Industries absolute aerodynamic center alternating current Am. Concrete Inst. Assoc. for Computing Machinery Air Conditioning and Refrigerating Manufacturers Assoc. Am. Chemical Soc. aluminum cable steel-reinforced air cushion vehicle anno Domini (in the year of our Lord) Atomic Energy Commission (U.S.) audio frequency Anti-friction Bearings Manufacturers' Assoc. Am. Foundrymen's Soc. Am. Gas Assoc. Am. Gear Manufacturers' Assoc. air horsepower Am. Inst. of Chemical Engineers Am. Inst. of Electrical Engineers (see IEEE) Am. Inst. of Mining Engineers Am. Inst. of Physics American Institute of Steel Construction, Inc. Am. Iron & Steel Engineers Am. Iron and Steel Inst. Aluminum Association ante meridiem (before noon) amplitude modulation Am. Machinist (New York) Acoustical Materials Assoc. Air Moving & Conditioning Assoc., Inc. atomic mass unit ammonium nitrate (explosive); Army-Navy Specification ammonium nitrate-fuel oil (explosive) Army-Navy Civil Aeronautics Committee ANS ANSI antilog API approx APWA AREA ARI ARS ASCE ASHRAE ASLE ASM ASME ASST ASTM ASTME atm Auto. Ind. avdp avg, ave AWG AWPA AWS AWWA b bar B&S bbl B.C. B.C.C. Bé B.G. bgd BHN bhp BLC B.M. bmep B of M, BuMines Am. Nuclear Soc. American National Standards Institute antilogarithm of Am. Petroleum Inst. approximately Am. Public Works Assoc. Am. Railroad Eng. Assoc. Air Conditioning and Refrigeration Inst. Am. Rocket Soc. Am. Soc. of Civil Engineers Am. Soc. of Heating, Refrigerating, and Air Conditioning Engineers Am. Soc. of Lubricating Engineers Am. Soc. of Metals Am. Soc. of Mechanical Engineers Am. Soc. of Steel Treating Am. Soc. for Testing and Materials Am. Soc. of Tool & Manufacturing Engineers atmosphere Automotive Industries (New York) avoirdupois average Am. Wire Gage Am. Wood Preservation Assoc. American Welding Soc. American Water Works Assoc. barns barometer Brown & Sharp (gage); Beams and Stringers barrels before Christ body centered cubic Baumé (degrees) Birmingham gage (hoop and sheet) billions of gallons per day Brinnell Hardness Number brake horsepower boundary layer control board measure; bench mark brake mean effective pressure Bureau of Mines

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xx

SYMBOLS AND ABBREVIATIONS biochemical oxygen demand boiling point bequerel brake specific fuel consumption British Standards Inst. British thermal units Btu per hr bushels Bulletin Bureau of Weapons, U.S. Navy Birmingham wire gage velocity of light degrees Celsius (centigrade) coulomb Civil Aeronautics Board Compressed Air & Gas Inst. calories chemical, biological & radiological (filters) Columbia Broadcasting System cubic centimetres critical compression ratio center to center candela centrifugal force confer (compare) cubic feet per hour cubic feet per minute Cooperative Fuel Research cubic feet per second center of gravity centimetre-gram-second Chemical Eng'g (New York) centrigrade heat unit cast iron circular circular mils centimetres Chartered Mech. Engr. (IMechE) cetane number coefficient U.S. Committee on Extension to the Standard Atmosphere column cologarithm of constant cosine of angle whose cosine is, inverse cosine of hyperbolic cosine of inverse hyperbolic cosine of cotangent of angle whose cotangent is (see cos 1) hyperbolic cotangent of inverse hyperbolic cotangent of coversed sine of circular pitch; center of pressure candle power coef of performance chemically pure close packed hexagonal cycles per minute cycles per second Canadian Standards Assoc. cosecant of angle whose cosecant is (see cos 1) hyperbolic cosecant of inverse hyperbolic cosecant of cubic cylinder db, dB d-c, dc def deg diam. (dia) DO D2O d.p. DP DPH DST d 2 tons DX e EAP EDR EEI eff e.g. ehp EHV El. Wld. elec elong emf Engg. Engr. ENT EP ERDA Eq. est etc. et seq. eV evap exp exsec ext F F FAA F.C. FCC F.C.C. ff. fhp Fig. F.I.T. f-m, fm F.O.B. FP FPC fpm, ft/min fps ft/s F.S. FSB fsp ft fc fL ft lb g g gal decibel direct current definition degrees diameter dissolved oxygen deuterium (heavy water) double pole Diametral pitch diamond pyramid hardness daylight saving time breaking strength, d chain wire diam. in. direct expansion base of Napierian logarithmic system ( 2.7182 ) equivalent air pressure equivalent direct radiation Edison Electric Inst. efficiency exempli gratia (for example) effective horsepower extra high voltage Electrical World (New York) electric elongation electromotive force Engineering (London) The Engineer (London) emergency negative thrust extreme pressure (lubricant) Energy Research & Development Administration (successor to AEC; see also NRC) equation estimated et cetera (and so forth) et sequens (and the following) electron volts evaporation exponential function of exterior secant of external degrees Fahrenheit farad Federal Aviation Agency fixed carbon, % Federal Communications Commission; Federal Constructive Council face-centered-cubic (alloys) following (pages) friction horsepower figure Federal income tax frequency modulation free on board (cars) fore perpendicular Federal Power Commission feet per minute foot-pound-second system feet per second Federal Specifications Federal Specifications Board fiber saturation point feet foot candles foot lamberts foot-pounds acceleration due to gravity grams gallons

BOD bp Bq bsfc BSI Btu Btub, Btu/h bu Bull. Buweaps BWG c C C CAB CAGI cal C-B-R CBS cc, cm3 CCR c to c cd c.f. cf. cfh, ft3/h cfm, ft3/min C.F.R. cfs, ft3/s cg cgs Chm. Eng. chu C.I. cir cir mil cm CME C.N. coef COESA col colog const cos cos 1 cosh cosh 1 cot cot 1 coth coth 1 covers c.p. cp cp CP CPH cpm cycles/min cps, cycles/s CSA csc csc 1 csch csch 1 cu cyl

SYMBOLS AND ABBREVIATIONS gc GCA g cal gd G.E. GEM GFI G.M. GMT GNP gpcd gpd gpm, gal/min gps, gal/s gpt H h h HEPA h-f, hf hhv horiz hp h-p HPAC hp hr hr, h HSS H.T. HTHW Hz IACS IAeS ibid. ICAO ICC ICE ICI I.C.T. I.D., ID i.e. IEC IEEE IES i-f, if IGT ihp IMechE imep Imp in., in in lb, in lb INA Ind. & Eng. Chem. int i-p, ip ipm, in/min ipr IPS IRE IRS ISO isoth ISTM IUPAC gigacycles per second ground-controlled approach gram-calories Gudermannian of General Electric Co. ground effect machine gullet feed index General Motors Co. Greenwich Mean Time gross national product gallons per capita day gallons per day, grams per denier gallons per minute gallons per second grams per tex henry Planck's constant 6.624 10 27 org-sec Planck's constant, h h/2 high efficiency particulate matter high frequency high heat value horizontal horsepower high-pressure Heating, Piping, & Air Conditioning (Chicago) horsepower-hour hours high speed steel heat-treated high temperature hot water hertz 1 cycle/s (cps) International Annealed Copper Standard Institute of Aerospace Sciences ibidem (in the same place) International Civil Aviation Organization Interstate Commerce Commission Inst. of Civil Engineers International Commission on Illumination International Critical Tables inside diameter id est (that is) International Electrotechnical Commission Inst. of Electrical & Electronics Engineers (successor to AIEE, q.v.) Illuminating Engineering Soc. intermediate frequency Inst. of Gas Technology indicated horsepower Inst. of Mechanical Engineers indicated mean effective pressure Imperial inches inch-pounds Inst. of Naval Architects Industrial & Eng'g Chemistry (Easton, PA) internal intermediate pressure inches per minute inches per revolution iron pipe size Inst. of Radio Engineers (see IEEE) Internal Revenue Service International Organization for Standardization isothermal International Soc. for Testing Materials International Union of Pure & Applied Chemistry J J&P Jour. JP k K K kB kc kcps kg kg cal kg m kip kips km kmc kmcps kpsi ksi kts kVA kW kWh L l, L £ lb L.B.P. lhv lim lin ln loc. cit. log LOX l-p, lp LPG lpw, lm/W lx L.W.L. lm m M mA Machy. max MBh mc m.c. Mcf mcps Mech. Eng. mep METO me V MF mhc mi MIL-STD min mip MKS MKSA mL ml, mL mlhc mm joule joists and planks Journal jet propulsion fuel isentropic exponent; conductivity degrees Kelvin (Celsius abs) Knudsen number kilo Btu (1000 Btu) kilocycles kilocycles per second kilograms kilogram-calories kilogram-metres 1000 lb or 1 kilo-pound thousands of pounds kilometres kilomegacycles per second kilomegacycles per second thousands of pounds per sq in one kip per sq in, 1000 psi (lb/in2) knots kilovolt-amperes kilowatts kilowatt-hours lamberts litres Laplace operational symbol pounds length between perpendiculars low heat value limit linear Napierian logarithm of loco citato (place already cited) common logarithm of liquid oxygen explosive low pressure liquified petroleum gas lumens per watt lux load water line lumen metres thousand; Mach number; moisture, % milliamperes Machinery (New York) maximum thousands of Btu per hr megacycles per second moisture content thousand cubic feet megacycles per second Mechanical Eng'g (ASME) mean effective pressure maximum, except during take-off million electron volts maintenance factor mean horizontal candles mile U.S. Military Standard minutes; minimum mean indicated pressure metre-kilogram-second system metre-kilogram-second-ampere system millilamberts millilitre 1.000027 cm3 mean lower hemispherical candles millimetres

xxi

xxii

SYMBOLS AND ABBREVIATIONS mineral matter free magnetomotive force mole melting point maximum permissible concentration miles per hour mean radiant temperature manuscript; milliseconds mean spherical candles Manufacturers Standardization Soc. of the Valve & Fittings Industry micron, micro megawatts megawatt day mean water temperature polytropic exponent number (in mathematical tables) number of neutrons; newton specific speed not available National Assoc. of Accountants National Advisory Committee on Aeronautics (see NASA) National Assoc. of Chain Manufacturers National Aeronautics and Space Administration natural National Broadcasting Company National Board of Fire Underwriters National Bureau of Standards (see NIST) nitrocarbonitrate (explosive) National District Hearing Assoc. National Electric Code® (National Electrical Code® and NEC® are registered trademarks of the National Fire Protection Association, Inc., Quincy, MA.) National Electrical Manufacturers Assoc. National Fire Protection Assoc. National Institute of Standards and Technology National Lubricating Grease Institute nautical miles number(s) net positive suction head Nuclear Regulator Commission (successor to AEC; see also ERDA) normal temperature and pressure outside diameter (pipes) open-hearth (steel) octane number opere citato (work already cited) Occupational Safety & Health Administration Office of Saline Water Office of Technical Services, U.S. Dept. of Commerce ounces page (pages) pascal propulsive coefficient polyethylene polyethylene glycol proportional elastic limit an explosive power factor Pipe Fabrication Inst. peak inverse voltage post meridiem (after noon) preventive maintenance performance number parts per billion plan position indicator parts per million pressure Proc. PSD psi, lb/in2 psia psig pt PVC Q qt q.v. r R R rad RBE R-C RCA R&D RDX rem rev r-f, rf RMA rms rpm, r/min rps, r/s RSHF ry. s s S SAE sat SBI scfm SCR sec sec­1 Sec. sech sech­1 segm SE No. SEI sfc sfm, sfpm shp SI sin sin­1 sinh sinh­1 SME SNAME SP sp specif sp gr sp ht spp SPS sq sr SSF SSU std Proceedings power spectral density, g2/cps lb per sq in lb per sq in. abs lb per sq in. gage point; pint polyvinyl chloride 1018 Btu quarts quod vide (which see) roentgens gas constant deg Rankine (Fahrenheit abs); Reynolds number radius; radiation absorbed dose; radian see rem resistor-capacitor Radio Corporation of America research and development cyclonite, a military explosive Roentgen equivalent man (formerly RBE) revolutions radio frequency Rubber Manufacturers Assoc. square root of mean square revolutions per minute revolutions per second room sensible heat factor railway entropy seconds sulfur, %; siemens Soc. of Automotive Engineers saturated steel Boiler Inst. standard cu ft per min silicon controlled rectifier secant of angle whose secant is (see cos­1) Section hyperbolic secant of inverse hyperbolic secant of segment steam emulsion number Structural Engineering Institute specific fuel consumption, lb per hphr surface feet per minute shaft horsepower International System of Units (Le Systéme International d'Unites) sine of angle whose sine is (see cos­1) hyperbolic sine of inverse hyperbolic sine of Society of Manufacturing Engineers (successor to ASTME) Soc. of Naval Architects and Marine Engineers static pressure specific specification specific gravity specific heat species unspecified (botanical) standard pipe size square steradian sec Saybolt Furol seconds Saybolt Universal (same as SUS) standard

mm-free mmf mol mp MPC mph, mi/h MRT ms msc MSS mu MW MW day MWT n N N Ns NA NAA NACA NACM NASA nat. NBC NBFU NBS NCN NDHA NEC®

NEMA NFPA NIST NLGI nm No. (Nos.) NPSH NRC NTP O.D., OD O.H. O.N. op. cit. OSHA OSW OTS oz p. (pp.) Pa P.C. PE PEG P.E.L. PETN pf PFI PIV p.m. PM P.N. ppb PPI ppm press

SYMBOLS AND ABBREVIATIONS SUS SWG T TAC tan tan 1 tanh tanh 1 TDH TEL temp THI thp TNT torr TP tph tpi TR Trans. T.S. tsi ttd UHF UKAEA UL ult UMS USAF USCG USCS USDA USFPL USGS USHEW USN USP Saybolt Universal seconds (same as SSU) Standard (British) wire gage tesla Technical Advisory Committee on Weather Design Conditions (ASHRAE) tangent of angle whose tangent is (see cos 1) hyperbolic tangent of inverse hyperbolic tangent of total dynamic head tetraethyl lead temperature temperature-humidity (discomfort) index thrust horsepower trinitrotoluol (explosive) 1 mm Hg 1.332 millibars (1/760) atm (1.013250/760) dynes per cm2 total pressure tons per hour turns per in transmitter-receiver Transactions tensile strength; tensile stress tons per sq in terminal temperature difference ultra high frequency United Kingdom Atomic Energy Authority Underwriters' Laboratory ultimate universal maintenance standards U.S. Air Force U.S. Coast Guard U.S. Commercial Standard; U.S. Customary System U.S. Dept. of Agriculture U.S. Forest Products Laboratory U.S. Geologic Survey U.S. Dept. of Health, Education & Welfare U.S. Navy U.S. pharmacopoeia USPHS USS USSG UTC V VCF VCI VDI vel vers vert VHF VI viz. V.M. vol VP vs. W Wb W&M w.g. WHO W.I. W.P.A. wt yd Y.P. yr Y.S. z Zeit. mc s, s m mm U.S. Public Health Service United States Standard U.S. Standard Gage Coordinated Universal Time volt visual comfort factor visual comfort index Verein Deutscher Ingenieure velocity versed sine of vertical very high frequency viscosity index videlicet (namely) volatile matter, % volume velocity pressure versus watt weber Washburn & Moen wire gage water gage World Health Organization wrought iron Western Pine Assoc. weight yards yield point year(s) yield strength; yield stress atomic number, figure of merit Zeitschrift mass defect microcurie Boltzmann constant micro ( 10 6 ), as in ms micrometre (micron) 10 6 m (10 ohm

xxiii

3

mm)

MATHEMATICAL SIGNS AND SYMBOLS plus (sign of addition) positive minus (sign of subtraction) Negative plus or minus (minus or plus) times, by (multiplication sign) multiplied by sign of division divided by ratio sign, divided by, is to equals, as (proportion) less than greater than much less than much greater than equals identical with similar to approximately equals approximately equals, congruent equal to or less than equal to or greater than 2 S8 ` 2 3 2 not equal to approaches varies as infinity square root of cube root of therefore parallel to parentheses, brackets and braces; quantities enclosed by them to be taken together in multiplying, dividing, etc. length of line from A to B pi ( 3.14159 ) degrees minutes seconds angle differential of x (delta) difference increment of x partial derivative of u with respect to x integral of

( )

/ :

y () [] {} AB p r rr l dx x 'u/'x e

V W ; , L >

xxiv 3

a

SYMBOLS AND ABBREVIATIONS integral of, between limits a and b line integral around a closed path (sigma) summation of functions of x [e 2.71828 (base of natural, or Napierian, logarithms)] del or nabla, vector differential operator Laplacian operator Laplace operational symbol 4! |x| # x $ x A factorial 4 4 3 2 1 absolute value of x first derivative of x with respect to time second derivative of x with respect to time vector product; magnitude of A times magnitude of B times sine of the angle from A to B; AB sin AB scalar product; magnitude of A times magnitude of B times cosine of the angle from A to B; AB cos AB

b

r o f (x), F(x) exp x ex = =2 £

B

A B

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