Part 36

ULTIMATE STRENGTH OE MATERIALS. ──────────────────┬────────────┬────────────┬────────────┬──────────── MATERIAL. │ Tension in │Compression │Shearing in │ Modulus of │lb. per sq. │ in lb. per │lb. per sq. │ Elasticity │ in. │ sq. in. │ in. │ in lb. per │ │ │ │ sq. in. ──────────────────┼────────────┼────────────┼────────────┼──────────── Cast Iron │ 11,000 to│ 50,000 to│ │ 14,000,000 │ 30,000│ 130,000│ │ to │ │ │ │ 23,000,000 „ aver.│ 16,000│ 95,000│ 11,000│ Wrought Iron │ 40,000 to│ │ │ 26,000,000 │ 70,000│ │ │ to │ │ │ │ 31,000,000 „ aver.│ 50,000│ 50,000│ 40,000│ Soft Steel │ 60,000 to│ │ │ 30,000,000 │ 100,000│ │ │ to │ │ │ │ 36,000,000 Soft Steel aver.│ 80,000│ 70,000│ 55,000│ Cast Steel aver.│ 120,000│ │ │ 15,000,000 │ │ │ │ to │ │ │ │ 17,000,000 Copper, Cast │ 19,000│ 58,000│ │ „ Wrought │ 34,000│ │ │ 16,000,000 Brass, Cast │ 18,000│ 10,500│ │ 9,170,000 Gun Metal │ 34,000│ │ │ 11,500,000 Phosphor Bronze │ 58,000│ │ 43,000│ 13,500,000 Wood, Ash │ 17,000│ 9,300│ 1,400│ „ Beech │ 16,000│ 8,500│ │ „ Pine │ 11,000│ 6,000│ 650│ 1,400,000 „ Oak │ 15,000│ 10,000│ 2,300│ 1,500,000 Leather │ 4,200│ │ │ 25,000 ──────────────────┴────────────┴────────────┴────────────┴────────────

POWERS, ROOTS, ETC., OF USEFUL FACTORS. n │(1)/(n)│ n^2 │ n^3 │ √̅n │ (1)/(√̅n) │ ∛̅n │ (1)/(∛̅n) ────────────────┼─────────┼───────┼──────────┼──────┬┴───────────┬─┴────┬──┴───────── π = 3·142 │ 0·318│ 9·870│ 31·006│ 1·772│ 0·564│ 1·465│ 0·683 2π= 6·283 │ 0·159│ 39·478│ 248·050│ 2·507│ 0·399│ 1·845│ 0·542 (π)/(2) = 1·571 │ 0·637│ 2·467│ 3·878│ 1·253│ 0·798│ 1·162│ 0·860 (π)/(3) = 1·047 │ 0·955│ 1·097│ 1·148│ 1·023│ 0·977│ 1·016│ 0·985 (4)/(3)π = 4·189│ 0·239│ 17·546│ 73·496│ 2·047│ 0·489│ 1·612│ 0·622 (π)/(4) = 0·785 │ 1·274│ 0·617│ 0·484│ 0·886│ 1·128│ 0·923│ 1·084 (π)/(6) = 0·524 │ 1·910│ 0·274│ 0·144│ 0·724│ 1·382│ 0·806│ 1·241 π^2 = 9·870 │ 0·101│ 97·409│ 961·390│ 3·142│ 0·318│ 2·145│ 0·466 π^3 = 31·006 │ 0·032│961·390│29,809·910│ 5·568│ 1·796│ 3·142│ 0·318 (π)/(32) = 0·098│ 10·186│ 0·0095│ 0·001│ 0·313│ 3·192│ 0·461│ 2·168 g = 32·2 │ 0·031│1036·84│ 33,386·24│ 5·674│ 0·176│ 3·181│ 0·314 2g = 64·4 │ 0·015│4147·36│ 267,090│ 8·025│ 0·125│ 4·007│ 0·249 ────────────────┴─────────┴───────┴──────────┴──────┴────────────┴──────┴────────────

HYDRAULIC EQUIVALENTS.

1 foot head = 0·434 lb. per square inch. 1 lb. per square inch = 2·31 ft. head. 1 imperial gallon = 277·274 cubic inches. 1 imperial gallon = 0·16045 cubic foot. 1 imperial gallon = 10 lb. 1 cubic foot of water = 62·32 lb. = 6·232 imperial gallons. 1 cubic foot of sea water = 64·00 lb. 1 cubic inch of water = 0·03616 lb. 1 cubic inch of sea water = 0·037037 lb. 1 cylindrical foot of water = 48·96 lb. 1 cylindrical inch of water = 0·0284 lb. A column of water 12 in. long 1 in. square = 0·434 lb. A column of water 12 in. long 1 in. diameter = 0·340 lb. Capacity of a 12 in. cube = 6·232 gallons. Capacity of a 1 in. square 1 ft. long = 0·0434 gallon. Capacity of a 1 ft. diameter 1 ft. long = 4·896 gallons. Capacity of a cylinder 1 in. diameter 1 ft. long = 0·034 gallon. Capacity of a cylindrical inch = 0·002832 gallon. Capacity of a cubic inch = 0·003606 gallon. Capacity of a sphere 12 in. diameter = 3·263 gallons. Capacity of a sphere 1 in. diameter = 0·00188 gallon. 1 imperial gallon = 1·2 United States gallon. 1 imperial gallon = 4·543 litres of water. 1 United States gallon = 231·0 cubic inches. 1 United States gallon = 0·83 imperial gallon. 1 United States gallon = 3·8 litres of water. 1 cubic foot of water = 7·476 United States gallons. 1 cubic foot of water = 28·375 litres of water. 1 litre of water = 0·22 imperial gallon. 1 litre of water = 0·264 United States gallon. 1 litre of water = 61·0 cubic inches. 1 litre of water = 0·0353 cubic foot.

───────────────────────────────────────────────────────────────────────── EQUIVALENTS OF POUNDS AVOIRDUPOIS. ─┬───────┬────────────┬────────────────┬────────────────┬──────────────── │ 10 │ 100 │ 1000 │ 10,000 │ 100,000 ─┼───────┼────────────┼────────────────┼────────────────┼──────────────── │qr. lb.│cwt. qr. lb.│ton cwt. qr. lb.│ton cwt. qr. lb.│ton cwt. qr. lb. 1│ 0 10│ 0 3 16│ 0 8 3 20│ 4 9 1 4│ 44 12 3 12 2│ 0 20│ 1 3 4│ 0 17 3 12│ 8 18 2 8│ 89 5 2 24 3│ 1 2│ 2 2 20│ 1 6 3 4│ 13 7 3 12│133 18 2 8 4│ 1 12│ 3 2 8│ 1 15 2 24│ 17 17 0 16│178 11 1 20 5│ 1 22│ 4 1 24│ 2 4 2 16│ 22 6 1 20│223 4 1 4 6│ 2 4│ 5 1 12│ 2 13 2 8│ 26 15 2 24│267 17 0 16 7│ 2 14│ 6 1 0│ 3 2 2 0│ 31 5 0 0│312 10 0 0 8│ 2 24│ 7 0 16│ 3 11 1 20│ 35 14 1 4│357 2 3 12 9│ 3 6│ 8 0 4│ 4 0 1 12│ 40 3 2 8│401 15 2 24 ─┴───────┴────────────┴────────────────┴────────────────┴────────────────

TRIGONOMETRICAL FUNCTIONS.

RIGHT-ANGLED TRIANGLES.

Sin. A = (a)/(b) Sec. A = (b)/(c) Tan. A = (a)/(c)

Cos. A = (c)/(b) Cosec. A = (b)/(a) Cotan. A = (c)/(a)

Versin. A = (b − c)/(b). Coversin. A = (b − a)/(b).

───────┬─────────┬───────────────────────────────────────────────────── Given. │Required.│ Formulæ. ───────┼─────────┼───────────────────────────────────────────────────── a,b│ A,C,c │Sin. A = (a)/(b) Cos. C = (a)/(b) c = │ │ √((b + a)(b − a)) │ │ a,c│ A,C,b │Tan. A = (a)/(c) Cotan. B = (a)/(c) b = │ │ √(a^2 + c^2) │ │ A,a │C,c,b│ C = 90° − A c = a × Cotan. A b = │ │ (a)/(Sin. A) │ │ A,b │C,a,c│C = 90° − A a = b × Sin. A c = b × Cos. │ │ A │ │ A,c │C,a,b│ C = 90° − A a = c × Tan. A b = │ │ (c)/(Cos. A) │ │ ───────┴─────────┴─────────────────────────────────────────────────────

OBLIQUE-ANGLED TRIANGLES.

s = ½(a + b + c)

───────────┬─────────┬───────────────────────────────────────────────── Given. │ │ Formulæ. ───────────┼─────────┼───────────────────────────────────────────────── A,B,C,a │ Area= │(a^2 × Sin. B × Sin. C) ÷ 2 Sin. A A,b,c │ „ │½(c × b × Sin. A) a,b,c│ „ │√(s(s − a)(s − b)(s − c)) ───────────┼─────────┼───────────────────────────────────────────────── Given. │Required.│ Formulæ. ───────────┼─────────┼───────────────────────────────────────────────── A,C,a │ c │ c = a(Sin. C)/(Sin. A) ───────────┼─────────┼───────────────────────────────────────────────── A,a,c │ C │ Sin. C = (c Sin. A)/(a) ───────────┼─────────┼───────────────────────────────────────────────── a,c,B │ A │ Tan. A = (a Sin. B)/(c − a Cos. B) ───────────┼─────────┼───────────────────────────────────────────────── a,b,c│ A │Sin. ½A = √(((s − b)(s − c))/(b × c)) „ │ „ │ Cos. ½A = √((s(s − a))/(b × c)); „ │ „ │ Tan. ½A = √(((s − b)(s − c))/(s(s − │ │ a))) ───────────┴─────────┴─────────────────────────────────────────────────

COMPOUND ANGLES.

Sin. (A + B) = Sin. A Cos. B + Cos. A Sin. B. Sin. (A − B) = Sin. A Cos. B − Cos. A Sin. B. Cos. (A + B) = Cos. A Cos. B − Sin. A Sin. B. Cos. (A − B) = Cos. A Cos. B + Sin. A Sin. B.

Tan. (A + B) = (Tan. A + Tan. B)/(1 − Tan. A Tan. B).

Tan. (A − B) = (Tan. A − Tan. B)/(1 + Tan. A Tan. B).

SLIDE RULE DATA SLIPS, COMPILED BY C. N. PICKWORTH, WH.SC.

(It is suggested that this page be removed by cutting through the above line, and selected portions of the Sectional Data Slips attached to the back of the Slide Rule.)

¹⁄₃₂ │0·03125 ¹⁄₁₆ │0·0625 ³⁄₃₂ │0·09375 ⅛ │0·125 ⁵⁄₃₂ │0·15625 ³⁄₁₆ │0·1875 ⁷⁄₃₂ │0·21875 ¼ │0·25 ⁹⁄₃₂ │0·28125 ⁵⁄₁₆ │0·3125 ¹¹⁄₃₂ │0·34375 ⅜ │0·375 ¹³⁄₃₂ │0·40625 ⁷⁄₁₆ │0·4375 ¹⁵⁄₃₂ │0·46875 ¹⁷⁄₃₂ │0·53125 ⁹⁄₁₆ │0·5625 ¹⁹⁄₃₂ │0·59375 ⅝ │0·625 ²¹⁄₃₂ │0·65625 ¹¹⁄₁₆ │0·6875 ²³⁄₃₂ │0·71875 ¾ │0·75 ²⁵⁄₃₂ │0·78125 ¹³⁄₁₆ │0·8125 ²⁷⁄₃₂ │0·84375 ⅞ │0·875 ²⁹⁄₃₂ │0·90625 ¹⁵⁄₁₆ │0·9375 ³¹⁄₃₂ │0·96875

Circ. of circle = 3·1416 d.

Area „ „ = 0·7854 d^2.

Sq. eq. area to cir., s = 0·886 d.

Circle eq. to sq., d = 1·128 s.

Sq. inscbd. in circ., s = 0·707 d.

Circsb. circ. of sq., d = 1·414 s.

Area of ellipse = 0.7854 a × b.