Thus the number 52 is 57.58 percent larger than the number 33. We also can say that 33 increased by 57.58 percent is equal to 52; that is, 0.5758 × 33 ϩ 33 ϭ 52. Now Thus the number 52 minus 36.54 percent of itself is 33. We also can say that 33 is 36.54 percent less than 52; that is, 0.3654 ϫ 52 = 19 and 52 Ϫ 19 ϭ 33. The number 33 is what percent of 52? That is, 33/52 ϭ 0.6346. Therefore, 33 is 63.46 percent of 52. 2.6 Decimal Equivalents and Millimeter Chart (see Fig. 2.79) 52 33 52 0 3654 − = . Mathematics for Machinists and Metalworkers 69 Figure 2.79 Decimal equivalents and millimeters. Walsh CH02 8/30/05 9:11 PM Page 69 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers 2.7 Degrees and Radians Chart (see Fig. 2.80) 2.8 Mathematical Signs and Symbols (see Table 2.1) 2.9 Greek Alphabet (see Table 2.2) 2.10 Sine Bar and Sine Plate Calculations Sine bar procedures. Referring to Fig. 2.81, find the sine bar–setting height for an angle of 34°25′ using a 5-in sine bar. sin 34°25′ ϭ x/5 (34°25′ ϭ 34.416667 decimal degrees) 70 Chapter Two Figure 2.80 Degrees to radians conversion chart. Walsh CH02 8/30/05 9:11 PM Page 70 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers TABLE 2.1 Mathematical Signs and Symbols + Plus, positive − Minus, negative × or и Times, multiplied by ÷ or / Divided by = Is equal to ϵ Is identical to Х Is congruent to or approximately equal to ∼ Is approximately equal to or is similar to < and Is less than, is not less than > and Is greater than, is not greater than Is not equal to ± Plus or minus, respectively ؏ Minus or plus, respectively ␣ Is proportional to → Approaches, e.g., as x → 0 ≤, Ϲ Less than or equal to ≥, м More than or equal to І Therefore : Is to, is proportional to Q.E.D. Which was to be proved, end of proof % Percent # Number @At Є or ѯ Angle Њ ′′′ Degrees, minutes, seconds ԽԽ,// Parallel to ֊ Perpendicular to e Base of natural logs, 2.71828 . . . π Pi, 3.14159 . . . ( ) Parentheses [ ] Brackets { } Braces ′ Prime, f ′(x) ′′ Double prime, f ″(x) √, ͙ n ෆෆ Square root, nth root 1/x or x −1 Reciprocal of x ! Factorial ° Infinity ∆ Delta, increment of ∂ Curly “d,” partial differentiation Σ Sigma, summation of terms Π The product of terms, product arc As in arcsine (the angle whose sine is) f Function, as f (x) rms Root mean square ⏐x⏐ Absolute value of x i For −1 j Operator, equal to −1 TABLE 2.2 The Greek Alphabet αΑalpha ιΙ iota ρΡ rho βΒbeta κκ kappa σΣ sigma γΓgamma λΛ lambda τΤtau δ∆delta µΜ mu υΥupsilon εΕepsilon νΝ nu φΦphi ζΖzeta ϕΞ xi χΧchi ηΗeta οΟ omicron ψΨpsi θΘtheta πΠ pi ωΩomega Walsh CH02 8/30/05 9:11 PM Page 71 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers sin 34.416667° ϭ x/5 x ϭ 5 ϫ 0.565207 x ϭ 2.826 in Set the sine bar height with Jo-blocks or precision blocks to 2.826 in. From this example it is apparent that the setting height can be found for any sine bar length simply by multiplying the length of the sine bar times the natural sine value of the required angle. The simplicity, speed, and accuracy possible for setting sine bars with the aid of the pocket calculator render sine bar tables obso- lete. No sine bar table will give you the required setting height for such an angle as 42°17′26′′, but by using the calculator proce- Figure 2.81 (a) Sine bar. (b) Sine bar. 72 72 Chapter Two Walsh CH02 8/30/05 9:11 PM Page 72 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers Mathematics for Machinists and Metalworkers 73 dure, this becomes a routine, simple process with less chance for errors. Method 1. Convert the required angle to decimal degrees. 2. Find the natural sine of the required angle. 3. Multiply the natural sine of the angle by the length of the sine bar to find the bar-setting height (see Fig. 2.81). Formulas for finding angles. Refer to Fig. 2.82 when angles ␣ and are known to find angles X, A, B, and C. tan X = tan ␣ cos sin C = cos ␣/cos X Angle B = 180° Ϫ (angle A ϩ angle C) D = true angle tan D = tan sin T tan sin tan C D = θ tan sin sin sin (sin cos ) A C C = − α φα Figure 2.82 Finding the unknown angles. Walsh CH02 8/30/05 9:11 PM Page 73 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers 74 Chapter Two cos A = cos E cos G cos A = sin sin T Formulas for finding true and apparent angles. See Fig. 2.83a, where ␣ϭapparent angle, ϭtrue angle, and ϭangle of rotation. Note: Apparent angle ␣ is OA triangle projected onto plane OB. See also Fig. 2.83b. tan ϭK/L tan ␣ϭK/(L cos ) tan ␣ cos ϭK/L K/L ϭ tan ϭcos tan ␣ or tan ϭcos tan ␣ and tan ␣ϭtan /cos The three-dimensional relationships shown for the angles and triangles in the preceding figures and formulas are of impor- tance and should be understood. This will help in the setting of compound sine plates when it is required to set a compound angle. Setting compound sine plates. For setting two known angles at 90° to each other, proceed as shown in Fig. 2.84. Example: First angle ϭ 22.45°. Second angle ϭ 38.58° (see Fig. 2.84). To find the amount the intermediate plate must be raised from the base plate (X dimension in Fig. 2.84b) to obtain the desired first angle, 1. Find the natural cosine of the second angle (38.58°), and multiply this times the natural tangent of the first angle (22.45°). 2. Find the arctangent of this product, and then find the natural sine of this angle. tan tan (tan )MT= ( ) +θ 2 2 Walsh CH02 8/30/05 9:11 PM Page 74 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers Mathematics for Machinists and Metalworkers 75 3. This natural sine is now multiplied by the length of the sine plate to find the X dimension in Fig. 2.84b to which the intermediate plate must be set. 4. Set up the Jo-blocks to equal the X dimension, and set them in position between the base plate and the intermediate plate. Figure 2.83 True and apparent angles. Walsh CH02 8/30/05 9:11 PM Page 75 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers 76 Chapter Two Figure 2.84 Setting angles on a sine plate. (a) (b) (c) Walsh CH02 8/30/05 9:11 PM Page 76 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers Mathematics for Machinists and Metalworkers 77 cos 38.58° ϭ 0.781738 tan 22.45° ϭ 0.413192 0.781738 ϫ 0.413192 ϭ 0.323008 arctan 0.323008 ϭ 17.900872° sin 17.900872° ϭ 0.307371 0.307371 ϫ 10 in (for 10-in sine plate) ϭ 3.0737 in Therefore, set X dimension to 3.074 in (to three decimal places). To find the amount the top plate must be raised (the Y dimension in Fig. 2.84c) above the intermediate plate to obtain the desired second angle, 1. Find the natural sine of the second angle, and multiply this by the length of the sine plate. 2. Set up the Jo-blocks to equal the Y dimension, and set them in position between the top plate and the intermediate plate. sin 38.58° ϭ 0.632607 0.632607 ϫ 10 in (for l0-in sine plate) ϭ 6.32607 Therefore, set the Y dimension to 6.326 in (to three decimal places). 2.11 Solutions to Problems in Machining and Metalworking The following sample problems will show in detail the importance of trigonometry and basic algebraic operations as they apply to machining and metalworking. By using the methods and proce- dures shown in this chapter of the Handbook, you will be able to solve many basic and complex machining and metalworking problems. Taper (Fig. 2.85). Solve for x if y is given; solve for y if x is given; solve for d. Use the tangent function: tan A ϭ y/x d ϭ D Ϫ 2y Walsh CH02 8/30/05 9:11 PM Page 77 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers 78 Chapter Two where A ϭ taper angle D ϭ outside diameter of rod d ϭ diameter at end of taper x ϭ length of taper y ϭ drop of taper Example: If the rod diameter ϭ 0.9375 diameter, taper length ϭ 0.875 ϭ x, and taper angle ϭ 20° ϭ angle A, find y and d from tan 20° ϭ y/x y ϭ x tan 20° ϭ 0.875(0.36397) = 0.318 d ϭ D – 2y ϭ 0.9375 – 2(0.318) ϭ 0.9375 – 0.636 ϭ 0.3015 Countersink depths (three methods for calculating) Method 1: To find the tool, travel y from the top surface of the part for a given countersink finished diameter at the part surface: y D A = / tan / 2 2 (Fig. 2.86) Figure 2.85 Taper. Walsh CH02 8/30/05 9:11 PM Page 78 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website. Mathematics for Machinists and Metalworkers [...]... 11XX 12XX 13XX 23 XX 25 XX 31XX 32XX 33XX 34XX 40XX 41XX 43XX 44XX 46XX 47XX 48XX 50XX 51XX 50XXX 51XXX UNS G10XX0 G11XX0 G12XX0 G13XX0 G23XX0 G25XX0 G31XX0 G32XX0 G33XX0 G34XX0 G40XX0 G41XX0 G43XX0 G44XX0 G46XX0 G47XX0 G48XX0 G50XX0 G51XX0 G50XX6 G51XX6 Alloy Steels 404 and 124 9 124 9 124 9 124 9 124 9 124 9 124 9 404 and 124 9 404 and 124 9 404 and 124 9 404 and 124 9 404 and 124 9 404 404 and 124 9 404 and 124 9... water Pounds of water per minute 1.094 1.667 3 .28 1 0.05468 196.8 3 .28 1 10−6 10−3 10−3 0.1 0.03937 1.54 723 2. 909 × 10−4 16 437.5 0.0 625 28 .349 527 0.9115 2. 790 × 10−5 2. 835 × 10−5 480 20 0.08333 31.103481 1.09714 1.805 0. 029 57 0.0 625 24 1.55517 0.05 4.1667 × 10−3 16 25 6 7000 453.5 924 1 .21 528 14.5833 5760 24 0 12 373 .24 177 0. 822 857 13.1657 0.016 02 27.68 0.1198 2. 670 × 10−4 93 Yards Centimeters per second... 0.061 02 1, 728 .0 0. 028 3 0.0370 7.481 28 . 32 29. 922 2 16.39 0.0005787 0.00001639 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 04 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Walsh CH03 8/30/05 9 :24 PM Page 87 U.S Customary and Metric (SI) Measures and Conversions U.S Customary and. .. in2 Pascals Kilograms per m2 Kilograms per m2 Kilograms per m2 Kilograms per m3 Pascals Pounds per ft2 Pounds per yd2 Pounds per ft3 Ounces per in3 Grams per cm3 3 0.1000 0.5780 14 .22 3 98,066.5 9.8066 0 .20 48 1.8433 0.0 624 3 1.7300 Pounds per ft Pounds per ft2 Pounds per ft2 3 Kilograms per cm Kilograms per m2 Pascals 16.019 4.8 824 47.880 Pounds per in2 Pounds per in2 Pounds per yd2 Kilograms per cm2... grams by 0.0648 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 04 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Walsh CH04 8/30/05 9: 32 PM Page 97 Source: McGraw-Hill Machining and Metalworking Handbook Chapter 4 Materials: Physical Properties, Characteristics, and Uses Materials,... liter Horsepower Horsepower Horsepower By: To obtain: 3600 0.01745 0.1667 0.0 027 78 0.0 625 1.771845 30.48 12 0.3048 1 ⁄3 0. 029 50 0.5080 0.01667 0.01 829 0.3048 18 .29 3785 0.1337 23 1 3.785 × 10−3 4.95 × 10−3 3.785 8 4 1 .20 095 0.8 326 7 8.3453 2. 228 × 10−3 0.06308 8. 020 8 980.7 15.43 10−3 103 0.03 527 0.0 321 5 2. 205 × 10−3 5.600 × 10−3 62. 43 Seconds Radians per second Revolutions per minute Revolutions per second... Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 04 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Walsh CH 02 8/30/05 9:11 PM Page 84 Mathematics for Machinists and Metalworkers Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 04... converted as described here The Kelvin and Celsius scales are related by the equation K ϭ 27 3.18 ϩ °C Thus 0°C ϭ 27 3.18 K Absolute zero is equal to 27 3.18°C Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 04 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Walsh CH03 8/30/05 9 :24 PM... diameter, and A ϭ 1 2 countersink angle, 41°, tan A ϭ x/y y ϭ x/tan A or x/(1 2 countersink angle) First, find x from D ϭ H ϩ 2x If D ϭ 0.875 and H ϭ 0.500, 0.875 ϭ 0.500 + 2x 2x ϭ 0.375 x ϭ 0.1875 Now solve for y, the tool advance: y ϭ x/tan A Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 04 The McGraw-Hill Companies All rights reserved Any... Angle α ϭ 35.0 826 52 Then solve triangle A′ B′ C′ , where y′ ϭ 0.9375 or 1 2 diameter of rod: Angle C ϭ 90° Ϫ 17.541 326 ° ϭ 72. 458674° Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 04 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Walsh CH 02 8/30/05 9:11 PM Page 82 Mathematics . plate. Figure 2. 83 True and apparent angles. Walsh CH 02 8/30/05 9:11 PM Page 75 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 20 04 The McGraw-Hill. website. Source: McGraw-Hill Machining and Metalworking Handbook To convert from: to: Multiply by: Rods Meters 5. 029 2 Yards Centimeters 91.44 Yards Feet 3.0 Yards Inches 36.0 Yards Meters 0.9144 Pressure Dynes. countersink finished diameter at the part surface: y D A = / tan / 2 2 (Fig. 2. 86) Figure 2. 85 Taper. Walsh CH 02 8/30/05 9:11 PM Page 78 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright