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Ex. 34. The length of the base line of the Ordnance Survey on Hounslow Heath was found to be 27,404 ft.; this was measured first by glass tubes, and then by steel chains; if, in correcting the glass tubes for temperature a uniform error of 1o in excess had been committed, and in correcting the steel chain an error of 1o in defect had been committed, what would have been the difference between the apparent measurements ?

Ans. 3.51 in.

Ex. 35. If the wrought-iron rails on a railway are 10 miles long when at a temperature of 32° below freezing, by how much will they lengthen if their temperature is raised to 88° F.? Ans. 29-83ft.

Ex. 36.-Ramsden's brass yard exceeded Shuckburgh's by 0.002505 of an inch; what would be the difference of their temperatures when accurately the same length ? Ans. 6°-6 F.

Ex. 37. Two rods, respectively of iron and brass, A B and CD are fas

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cular to the bars, at 62° F.; in consequence of the unequal expansion or contraction of the bars the tongues will assume different positions, as shown by the dotted lines; it is required to determine the length of ce, that the point E may remain unmoved by the expansion or contraction of the bar. The length of a B is 10 ft. and the distance A c is 1.725 in.

Ans. CE=4.426 in.

Ex. 38. If the expansion in length of a substance is e times the length at a given temperature, show that the expansion in volume will be very nearly 3 e times the volume at that temperature.

Ex. 39. The volume of a mass of lead being a cubic foot at 60° F. what will be its volume at 0°F.? and what at 88° F.?

Ans. At 0° F. 0-997134 cubic ft.

At 88° F. 1.-00133728 cubic ft. Ex. 40. There is half a cubic inch of mercury in a thermometer at 32° F.; when the temperature is raised to 92° F. the mercury ascends 4 in.; what is the diameter of the bore of the glass tube? Ans. 0.0288 in.

6. Elongation produced by strain. -The principle on which this determination is made is the following:-Suppose the length of a beam or bar to be L feet, the area of its section to be K square inches, then if by the application of a strain of P lbs. its length becomes L+l, it appears from experiment that

P

1:L:::E

K

where E is a constant number depending on the nature of the material, and is called the Modulus of Elasticity.

It is found that all substances obey this law when the degree of extension does not exceed certain limits; the limits are different in different substances, and in many are very narrow. It appears also that within these limits (i. e. the limits of elasticity) a strain producing a certain degree of extension will, if applied in the opposite direction so as to become a thrust, produce an equal degree of compression.

P

K

It will be observed that is the strain or thrust per square inch on the section of the beam or bar. It is also plain that if were equal to E then would I be equal to L,

P

K

so that the modulus of elasticity is that strain per square inch of the section of a bar which would double its length if its elasticity continued perfect. It is, perhaps, unnecessary to remark that no solid substance has limits of elasticity any way approaching this in extent.

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Ex. 41. By how much would a bar of wrought iron of an inch square and 100 ft. long lengthen under a strain of 2 tons (neglecting the weight of the bar)? Ans. 0-247 ft.

Ex. 42. Determine the elongation of a steel bar 2 in. square and 40 ft. long when subjected to a strain of 40 tons. What would have been its elongation had it been of cast brass ? Ans. Steel 0.03 ft. Brass 0.1 ft. Ex. 43. A bar of wrought iron 2 in. square has its ends fixed between two immovable blocks when the temperature is 20° F.; what pressure will it exert against them if the temperature becomes 96° F.? Ans. 254 tons.

Ex. 44. A wall of brickwork 2 ft. thick and 12 ft. high is supported by columns of oak 6 inches in radius, 18 ft. high and 14 ft. apart from centre to centre; determine the thrust per square inch exerted on the section of the columns, and the amount of their compression.

Ans. (1) 332.7 lbs. (2) in. nearly.

Ex. 45. In the last example if the wall had been of Portland stone and 14 ft. thick, what would have been the pressure per square inch, and the degree of compression? Ans. (1) 248.9 lbs. (2) in.

Ex. 46. In the last example if the oak column were replaced by a wrought-iron bar 2 inches square, what would be the degree of compression? and at what temperature would the iron rod have the same length as it has when unpressed at 32° F.? Ans. (1) in. (2) 69.8° F.

Ex. 47. A bar of wrought iron a square inch in section is fixed firmly between two immovable blocks which are 50 ft. apart; if the temperature is raised 50° F. above that which the bar had when fixed, find the pressure produced against these blocks. Ans. 9309 lbs.

Ex. 48. In the last example, if only one of the blocks were immovable and the other were capable of revolving round a joint 12 ft. below the point at which it is met by the rod, determine the angle through which it will be turned by the expansion of the rod. Ans. 0° 4′ 36′′.

* Based on Mr. Moseley's Mech. Eng. p. 622, compared with Mr. Rankine's Applied Mechanics, p. 631.

Ex. 49. It is observed that two opposite walls of an ancient building are each 3o out of the vertical, the inclination being outward; to bring them into the perpendicular, the following means are employed; at certain intervals iron bars are placed across the building, their ends passing through the walls and projecting on the outside, on these ends strong plates or washers are screwed; the rods are then heated and expand, in this state the washers are screwed tightly against the outside of the walls and the rods allowed to cool, when they contract and draw the walls together; the process being continued until the walls become vertical.* If we suppose the rods to be 50 ft. long and 3 square inches in section, and to be fastened 15 ft. above the joint of the masonry, round which walls will be made to turn; and if the range of temperature is from 60° F. to 240° F.; determine the number of times the bars must be heated before the operation is complete, and the pressure which would tend to draw the walls together if they were entirely immovable. Ans. (1) 27 times. (2) 100,572 lbs.

7. Resistance to rupture by tearing or tenacity. When a strain which elongates a bar attains a certain magnitude, the bar will break. If we determine by experiment this force in lbs. per square inch, we obtain the tenacity of the substance. It is manifest that the strain which will tear a bar whose section is n square inches will ben times the tenacity.

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Ex. 50. How great a strain will a cylindrical bar of wrought iron bear which is of an inch in diameter? and by what fraction of its length would it lengthen under this strain if the elasticity continued perfect?

Ans. (1) 3298 lbs. (2) 0.0023.

Ex. 51. How many iron wires of an inch in diameter must be put together to sustain a strain of 3 tons? Ans. 13.

* The walls of Armagh Cathedral were restored to a vertical position by this process. Daniell's Chemistry, p. 103.

Ex. 52. What is the length of a bar of wrought iron which being suspended vertically would break by its own weight?

Ans. 19,880 ft.

Ex. 53. What strain will a bar of oak 14 in. square sustain ?

Ans. 38,925 lbs. Ex. 54. What strain will a cylindrical bar of larch 14 in. in diameter sustain? Ans. 17,671 lbs. Ex. 55. If a rope be made of wires whose diameter is d, show that the number of wires in each square inch of the section of the rope is very

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of an inch in diameter must be put together Ans. 115.

to form a rope a square inch in section?

Ex. 57. If the number of wires of an inch in diameter which must be put together to form a rope one square inch in section be determined by each of the formulæ in Ex. 55, what is the difference between the results? Ans. 4.8.

Ex. 58. Show that the number of lbs. weight in a foot length of a rope made of iron wire is given by the formula (circ. in inches) x 0.244 very nearly; the specific gravity of iron wire being assumed to be the same as that of wrought iron.

Ex. 59. Show that if a rope of hemp has the same strength as a rope of iron wire, the circumference of the latter is about of the circumference, and its weight about of the weight of the former.

8. Resistance to rupture by compression. There are as many as five forms which the results of crushing assume in different bodies. They are enumerated as follows by Mr. Rankine :*

(1) Crushing by splitting, when the substance divides in a direction nearly parallel to the direction of the pressure. This occurs in the case of hard homogeneous substances of a glassy texture.

(2) Crushing by shearing, when the substance divides along a plane inclined at a certain angle to the direction of the force, the upper part of the substance sliding upon the lower. This fact was ascertained, and its conditions investigated, by Mr. Hodgkinson. It takes place in the case of substances of a granular texture, such as cast iron,

* Applied Mechanics, p. 303. See also Mr. Moseley's Mechanics of Engineering, pp. 549, 579,

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