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where M and m are constants depending on the nature of the iron, D the external and d the internal diameters of the column in inches, and I the length in feet. The values of M and m vary considerably with different kinds of iron, but may be taken at 42 tons. The limits of variation in the values of m are 49.94 and 39.60.*

Ex. 68. Determine the breaking weight of a solid cast-iron column 20 ft. high and 6 in. in diameter. Ans. 168-3 tons,

Ex. 69. Determine the breaking weight of the column in the last example if it were hollow and 1 in. thick.

Ex. 70. Determine the thickness of a column 20 ft. external diameter, which is as strong as that in Ex. 68.

Ans. 127.6 tons. high and 7 in. in Ans. 0.774 in.

* Proceedings of the Royal Society, vol. viii. p. 318.

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CHAPTER II.

ON WORK; OR, THE EFFICIENCY OF AGENTS.

11. Definition of work. An agent is said to do work when it causes the point of application of the force it exerts to move through a certain space; thus a carpenter employed in planing wood works, since he causes the point of application of the force he exerts to move through a certain space, and the same is true of any agent that works in the sense here intended. For the sake of distinctness it may be observed that the union of force and motion is essential to the conception of work; thus when the expansive force of steam lifts the piston of a steam engine it does work. In the boiler, though it produces an enormous pressure, it does no work, since the pressure is unaccompanied by motion. The unit by which the work of different agents is expressed numerically is called the unit of work; according to the practice of English writers it is defined as follows:

Def. The work done when a force of 1 Ib. is exerted through a space of 1 ft. in the direction of the force is called a unit of work.

The following important principle is closely connected with this definition. When a force of P lbs. is exerted through a space of s ft., it does PS units of work, the force being exerted along the line in which its point of application is made to move. For since a unit of work is done when a force of 1 lb. is exerted through 1 ft., there must be 2 units of work done when a force of 2 lbs. is exerted through 1 ft., 3 units of work when a force of 3lbs. is exerted through

COMPARISON OF THE EFFICIENCY OF AGENTS. 19

1 ft., and generally p units of work when a force of P lbs. is exerted through 1 ft. Again, since P units of work are done when a force of P lbs. is exerted through 1 ft., there must be 2 p done when it is exerted through 2 ft., 3 p when it is exerted through 3 ft., and generally ps units must be done when the force of P lbs. is exerted through s ft.

Ex. 71. How many units of work are expended in raising 2 cwts. through 30 fathoms? Ans. 40,320.

Ex. 72. The mean pressure on the piston of a steam engine is 15 lbs. per sq. in., the length of the stroke is 6 ft.; if the area of the piston is 448 sq. in., how many units of work are done per stroke? Ans. 40,320.

12. Comparison of the efficiency of agents. If the above examples are compared, it will be seen that the work done during each stroke by the steam on the piston of the engine is equivalent to the work expended in raising 2 cwts. through a height of 30 fathoms; and whatever agent raises this weight must do as much work as that done by the steam. In these examples we have not considered the time in which the work is done; let us then suppose that the engine in Ex. 72 makes 10 strokes per minute; the expansive force of the steam will then do 403,200 units of work per minute. Now, if we suppose an agent, or a number of agents, to raise a weight of 1 ton through 30 fathoms in one minute, they will do exactly 2240 × 180 or 403,200 units of work per minute. It is plain that under these circumstances the comparison is complete between the efficiency of the expansive force of the steam and the efficiency of the other agents, and that they are reciprocally equivalent. Hence we infer the general principle

The number of units of work yielded by any agent in a given time is the true measure of its efficiency or working power.

Of course it follows from this principle that the working powers of two agents are in the ratio of the number of units of work done by them in the same time.

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The most familiar instance of this mode of measuring the power of an agent is furnished by the steam engine, whose efficiency is estimated in horse-power, as when we speak of an engine of twenty horse-power.' From some experiments, Mr. Watt concluded that a horse is capable of yielding 33,000 units of work per minute. The conclusion, as far as regards the efficiency of the animal, is not very correct; it has, however, fixed the meaning of the term horse-power when applied to a steam engine. Hence Def.-A steam engine works with one horse-power when it yields 33,000 units of work per minute.

Of course an engine of n horse-power yields n times 33,000 units of work per minute.

Ex. 73. The piston of a steam engine is 15 in. in diameter, in stroke is 21⁄2 ft. long; it makes 40 strokes per minute; the mean pressure of the steam on it is 15 lbs. per square inch; what number of units of work is done by the steam per minute, and what is the horse-power of the engine?

Ans. 265,072 units of work. 8.03 H.-P.

Ex. 74. A weight of 11⁄2 tons is to be raised from a depth of 50 fathoms in 1 minute; determine the horse-power of the engine capable of doing the work? Ans. 30 H.-P.

Ex. 75. The resistance to the motion of a certain weight is 440 lbs., how many units of work must be expended in making this weight move over 30 miles in 1 hour? What must be the horse-power of an engine that does the same number of units of work in the same time?

Ans. 69,696,000 units of work. 35 H.-P.

13. Application of the foregoing principles. A considerable number of practical questions can be answered by means of the principles already laid down, viz. such questions as the horse-power of the engine required to do a certain amount of work, the time in which an engine of a certain power will do a certain amount of work, &c. They are all done by following the same method, viz. First, from a consideration of the work to be done, obtain the number of units of work that must be expended in a certain time. Next, from a consideration of the power of the agent obtain the number of units yielded in the same time. One

....

of these expressions will contain an unknown quantity, but, since by the terms of the question they are equal, they will form an equation from which the unknown quantity can be readily determined.

Ex. 76. An engine is required to raise a weight of 13 cwts. from a depth of 140 fathoms in 3 minutes; determine its horse-power.

Let x be the required horse-power; then the units of work yielded in 3 minutes will equal 33,000 xxx 3; also the number of units of work required to raise 13 cwts. from a depth of 140 fath. equals 13 x 112 x 140 x 6. And since these two numbers are equal we have

33,000 x 3 x x = 13 x 112 x 140 x 6.
...x=12-35 H.-P.

Ex. 77.-In how many minutes would an engine working at 25 horsepower raise a load of 12 cwts. from a depth of 160 fathoms?

Ans. 1.564 min.

Ex. 78. A locomotive engine draws a gross load of 60 tons at the rate of 20 miles an hour; the resistances are at the rate of 8 lbs. per ton; what must be the horse-power of the engine?

[The reader must bear in mind that the work to be done is to overcome a resistance of 480 lbs. through 20 miles in one hour.] Ans. 25.6 H.-P.

Ex. 79. What must be the horse-power of an engine that raises 20 cubic feet of water per minute from a depth of 200 fathoms?

Ans. 45 H.-P.

Ex. 80. How many cubic feet of water would an engine working at 100 horse-power raise per minute from a depth of 25 fathoms ? Ans. 352. Ex. 81. How many cubic feet of water will an engine of 250 horse-power raise per minute from a depth of 200 fathoms? Ans. 110 cub. ft.

Ex. 82. It being required to raise 100 cubic feet of water per minute from a depth of 495 ft., what must be the horse-power of the engine?

Ans. 93 H.-P.

Ex. 83. There is a mine with three shafts which are respectively 300, 450, and 500 ft. deep: it is required to raise from the first 80, from the second 60, from the third 40 cubic feet of water per minute; what must be the horse-power of the engine ? Ans. 134 H.-P.

Ex. 84. At what rate per hour will a locomotive engine of 30 horsepower draw a train weighing 90 tons gross, the resistances being 8 lbs. per ton? Ans. 15.625 miles.

Ex. 85. What is the gross weight of a train which an engine of 25 horsepower will draw at the rate of 25 miles an hour, resistances being 8 lbs. per ton? Ans. 46-875 tons.

Ex. 86. A train whose gross weight is 80 tons travels at the rate of 20 miles an hour; if the resistance is 8 lbs. per ton what is the horse-power of the engine? Ans. 34 H.-P.

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