Ex. 7. Find the separate weights of a cast-iron ball, 4 in. in radius, and of a copper cylinder 3 ft. long, the diameter of whose base is 1 in. Determine also the diminution in the weight of the ball if a hole were cut through it which the cylinder would exactly fit, the axis of the cylinder passing through the centre of the sphere. Also, find the error that results from considering the part cut away a perfect cylinder.

Ex. 8. If a 10 in. shell were of cast iron, and were 2 in. thick, what would be its weight supposing it complete? If the weight of a 10 in. shell were 86 lbs. what would be its thickness supposing it complete?

Ans. (1) 107 lbs. (2) 1.41 in.

Ex. 9. A hammer consists of a rectangular mass of wrought iron, 6 in. long, and 3 in. by 2 in. in section; its handle is of oak, and is a cylinder 3 ft. 6 in long, on a base of 1 in. in radius. Determine its weight.

Ans. 12.83 lbs.

Ex. 10. A pendulum consists of a cylindrical rod of steel 40 in. long, on a base whose diameter measures 4 in.; to the end of this is screwed a steel cylinder in. thick, and 14 in. in radius, which fits accurately a hollow cylinder of glass, containing mercury 6 in. deep, the glass vessel weighing Ans. 360-8 oz.

3 oz.

Determine the weight of the pendulum.

Ex. 11. Determine the weight of a leaden cone whose height is 1 ft. and radius of base 6 in.; determine also the external radius of that hollow cast iron sphere which is 1 in. thick, and equals the cone in weight.

Ans. (1) 185-74 lbs. (2) 8.02 in.

Ex. 12. A rectangular mass of cast iron 6 ft. long, 6 in. wide, and 3 in. deep, has fitted square to its end a cube of the same materials whose edge is 1 ft. long; find its weight. Ans. 1858 lbs.

Ex. 13. It is reckoned that a foot length of iron pipe weighs 64.4 lbs. when the diameter of the bore is 4 in. and the thickness of the metal 14 in.: what does this assume to be the specific gravity of iron? Ans. 7.197.

Ex. 14. A cast-iron column 10 ft. high and 6 in. in diameter will safely support a weight of 171⁄2 tons, whether it be solid, or hollow and 1 in. thick; determine:-(1) the weight of a solid column; (2) the number of equally strong hollow columns that can be made out of 500 solid columns; (3) the price of 500 solid columns at 10s. per cwt. and of 500 hollow columns at 11s. 3d. per cwt.; (4) the cost of sending the 500 solid and the 500 hollow columns to a given place at the rate of 10s. 6d. per ton.

Ans. (1) 884.4 lbs. (2) 900. (3) 19741. 3s. solid. 1233l. 16s. hollow. (4) 103l. 13s. solid. 571. 12s. hollow.

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Ex. 15. Determine the weight of a hollow leaden cylinder whose length is 3 in., internal radius 14 in., and thickness 11⁄2 in.

Ans. 26-121 lbs.

Ex. 16. Determine the weight of a grindstone 4 ft. in diameter and 8 in. thick, fitted with a wrought-iron axis of which the part within the stone is 2 in. square, and the projecting parts each 4 in. long with a section 2 in. in diameter. Ans. 1135 lbs.

Ex. 17. Determine the weight of an oak door 7 ft. high, 3 ft. wide, and 11⁄2 in. thick. Ans. 1534 lbs. Ex. 18. There is a fly wheel of cast iron the external radius of whose rim is 5 ft. and internal radius 4ft. 6 in.; it is 4 in. thick and is connected with the centre by 8 spokes 4 in. wide and 1 in. thick, strengthened by a flange on each side 1 in. square (so that their section is a cross 4 in. long and 3 in. wide), each spoke is 4 ft. long; the centre to which they join the rim has the same thickness as the rim, is solid, and (of course) 6 in. in radius: determine the weight of the whole. Ans. 2959 lbs.

Ex. 19. There are 2 rooms each 100 ft. long and 30 ft. wide; the one is floored with oak planking 14 in. thick; the other with deal planking (Riga fir) 11⁄2 in. thick. Determine the weights of the floors and their cost, the price of deal being 3s. and oak 7s. per cubic foot.

Ans. Deal floor weighs 17,648 lbs. costs 561. 5s.

Oak

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18,242 lbs. Ex. 20.-A cubic foot of copper is drawn into wire meter; what length of wire is made ?

Ex. 21. It is said that gold can be drawn into wire one millionth part of an inch thick; what will be the length of such a wire that can be made from an ounce of pure gold? Ans. 1,793,448 miles.

1

Ex. 22. It is said that silver leaf can be made 데이이이 of an inch thick; how many ounces of silver would be required to make an acre of such silver leaf? Ans. 254.3 oz,

4. Brickwork. The measurement and determination of the weight of a mass of brickwork depend upon the following data :

(1) A rod of brickwork has a surface of 1 square rod (or 30+ square yards) and a thickness of a brick and a half, i. e. of 1 ft. 11⁄2 in., or it contains 306 cubic feet.

(2) A rod of brickwork contains about 4500 bricks in mortar, or 5000 bricks laid dry.

(3) A rod of brickwork requires 31⁄2 loads (i. e. 31⁄2 cubic yards) of sand and 18 bushels of stone lime.

(4) A brick measures 84 x 44 x 24 inches, i. e. a quarter of an inch each way less than 9 x 44 x 3 inches.

(5) A bricklayer's hod measures 16 x 9 x 9 inches, and can contain 20 bricks. Labourers, however, commonly put 10 or 12 bricks into it.*

Ex. 23. How many rods of brickwork are there in a square tower 117 ft. high and 28 ft. by 7 ft. at its base, externally, and 3 bricks thick? Determine the number of bricks required to build the tower and their price at 11. 10s. per thousand.

Ans. (1) 52-43 rods. (2) 236,000 bricks. (3) 3547.

Ex. 24. A tower the base of which measures externally 9 ft. square is 50 ft. high and 2 bricks thick; how many bricks are required to build it, and how many loads of sand and bushels of lime? Determine also the cost of the materials if the bricks cost 17. 10s. per thousand, sand 5s. 4d. per load, and lime 1s. 8d. per bushel.

Ans. (1) 7:35 rods. (2) 33,000 bricks, 25 loads of sand, 132 bushels of lime. (3) Cost 671. 8s. 2d.

Ex. 25. How many rods of brickwork are there in a reservoir of a rectangular form, the internal measurements of which are 20 ft. long, 6 ft. wide, and 12 ft. deep; the work being 2 bricks thick, viz. both walls and floor; and the reservoir being open at the top? Ans. 4.43.

Ex. 26. Find how many rods of brickwork there are in a wall 360ft. long, 17 ft. high, and 2 bricks thick; and determine the cost of the material from the data in Ex. 24. Ans. (1) 30 rods. (2) 275l. 10s.

Ex. 27. If the wall in the last example had an additional 2 ft. of foundation 3 bricks thick, and were supported by 20 square buttresses reaching to the top of the wall 2 bricks thick, on foundations 3 bricks thick, and measuring 2 ft. in a direction perpendicular to the face of the wall; determine the number of rods of brickwork in the foundations and buttresses. Ans. 10-2 rods.

Ex. 28. What would be the cost of the carriage of the bricks in the wall described in the last two examples at 5s. 6d. per thousand? Ans. 491. 15s.

Ex. 29. The following are the actual dimensions of the brickwork of the outer shell of the chimney of St. Rollox, Glasgow. Commencing from the top, there are five divisions; the tops of these divisions are respectively 435, 350, 2102, 1144, 542 ft., above the ground; the external diameters at the tops of the divisions are respectively 13 ft. 6 in., 16 ft. 9 in., 24 ft., 30 ft. 6 in., 35 ft. The diameter on the ground is 40 ft.; the thicknesses of

* Weale's Contractor's Price Book for 1859, p. 280.

the divisions are respectively 14, 2, 24, 3, and 3) bricks; below ground the brickwork reaches 14 ft., with a uniform external diameter of 40 ft.; the first 8 feet are 3 ft. thick; in the remaining 6 feet the thickness gradually increases to 12 ft. thick. Determine the number of rods of brickwork contained in the chimney; the number of thousand bricks employed, their cost at 11. 11s. 3d. per thousand; also, if the mortar were of sand and stone lime, determine the number of loads of sand and bushels of stone lime required, and their cost at 5s. 4d. per load, and 1s. 8d. per bushel, respectively.

[The surface of each division of the chimney may be considered as that of a conic frustum; the real volume of each division will be the difference between the volumes of two conic frustums. A sufficiently close approximation may be obtained by multiplying the mean surface by the thickness and considering the slant side equal to the height; the volume of the part below ground is to be determined accurately.]

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Ans. (1) 218 rods, or 981,000 bricks. (2) Cost of bricks, 15327. 16s. 3d. (3) 763 loads of sand, costing 203l. 9s. 4d. (4) 3924 bushels of lime, costing 3271.

5. Expansion and contraction by heat. It is found that all bodies experience a small change of volume on the application of heat. In general, the change is one of increase,* and with sufficient accuracy may be considered to obey the following law within moderate ranges of temperature. If a volume v be increased by k v for an addition of one degree of heat, it will be increased by n×kv for an addition of n degrees of heat, i. e. the increase of volume is proportional to the increase of temperature. The same rule holds for the expansions in length, which a body experiences from an increase of temperature. In order to fix the conception of a degree of heat (with sufficient accuracy for our present purpose), it will be proper to mention that when heat is applied to ice the water produced by melting retains a constant temperature until the whole of the ice is melted. This temperature serves as one fixed point, and is called the freezing point. Moreover, boiling water in free contact with the air also keeps at a constant temperature (at least when the barometer stands

* Water, near freezing point, is a conspicuous exception.

at a given height). This fact, therefore, supplies a second fixed point, and is called the boiling point, viz., when the barometer stands at 30 inches. These two points being fixed, the graduation is arbitrary. The scale of Fahrenheit's thermometer (which is commonly used in England) is constructed by dividing the space between the freezing and boiling points into 180 equal parts, termed degrees, and by commencing the graduation 32° below freezing point, so that the freezing point is marked 32°, and the boiling point 212°. In the centigrade thermometer (now commonly used in scientific investigation) the graduation begins at the freezing point, and the interval between the freezing and boiling points is divided into 100 equal parts called degrees.* It is easy to see that if at any temperature Fahrenheit's thermometer stood at Fo and the centigrade at co, we should have

Ex. 30. The density of water is greatest at 30.9 on the centigrade scale; what is the same temperature called on Fahrenheit's scale? Ans. 39°.02 F.

Ex. 31. The standard temperature not unfrequently referred to in English experiments is 60° F.; what would the same temperature be called on the centigrade scale? Ans. 15° 55 С.

Ex. 32. If the centigrade thermometer stood at 5o below zero, or at 5° C, what would the same temperature be marked on Fahrenheit's scale ? Ans. 23° F.

Ex. 33. What degree on the centigrade scale would be equivalent to Ans. -20° С.

-4° on Fahrenheit's scale?

The following Table gives the fractional part of the whole by which substances expand when heated:†

* In Reaumur's thermometer the freezing point is marked zero, and the boiling point 80°: consequently 180 80

F°-32-RO

† From Dr. Young's Natural Philosophy, vol. ii. p. 390.