Ex. 87. An engine working with the same power as that in the last example draws a train at the rate of 30 miles an hour; the resistances being 7 lbs. per ton, what is the gross weight of the train? Ans. 60 tons. Ex. 88. What must be the length of the stroke of the piston of an engine, the surface of which is 1500 square inches, which makes 20 strokes per minute, so that with a mean pressure of 12 lbs. on each square inch of the piston, the engine may be of 80 horse-power? Ans. 7 ft. Ex. 89. The diameter of the piston of an engine is 80 in., the length of the stroke is 10 ft., it makes 11 strokes per minute, and the mean pressure of the steam on the piston is 12 lbs. per square inch: what is the horsepower? Ans. 201-06 H.-P. Ex. 90. Find the horse-power of an engine that will raise in one minute 100 cubic feet of water from a depth of 600 feet. Ans. 113 H.-P. Ex. 91. A train weighing 50 tons is drawn along a railway at the rate of 20 miles an hour; the resistances being 8 lbs. per ton, find the horsepower of the engine. Ans. 21 H.-P. Ex. 92. The cylinder of a steam engine has an internal diameter of 3 ft.; the length of the stroke is 6 ft.; it makes 6 strokes per minute; under what effective pressure per square inch would it have to work in order that 75 horse-power may be done on the piston? Ans. 67-54 lbs. Ex. 93. What must be the horse-power of a stationary engine that draws a weight of 150 tons along a horizontal road at the rate of 30 miles per hour; friction being 8 lbs. per ton? Ans. 96 H.-P. 14. Modulus of a machine.-An agent rarely, if ever, does a considerable amount of useful work directly, but nearly always through the intervention of a machine, by which the motive power of the agent is so applied as to overcome the resistance in the most convenient manner. For instance, when a steam engine raises water out of a shaft, the motive power is the expansive pressure of the steam on the piston, the resistance to be overcome is the weight of the water, the beam, crank, &c., of the engine are the means by which the motive power is applied so as to overcome the resistance. Now it will be remarked that each part of the machine offers more or less resistance to the motion, so that a certain part of the work done by the motive power must be expended in overcoming these resistances, i. e. in reference to the purpose of the ma chine, must be expended uselessly. The remainder of the work done by the motive power will be expended usefully in accomplishing that purpose. If the number of units of work done by the agent is represented by u, the number expended in overcoming prejudicial resistances by U, and the number expended usefully by u1, all in the same given time, then it admits of proof in the case of a machine moving uniformly, that U=Uo+U1 It also appears that in most machines u1 bears to u a constant ratio, so that U1=KU where the letter K denotes some proper fraction, depending on the nature of the machine; this fraction is called the modulus of the machine; the following table, taken from General Morin's Aide-Mémoire de Mécanique Pratique, gives the value of K for different classes of steam engines: High-pressure engines, working without ex pansion or condensation 20 30 0.60 0.48 دو دو 30 40 0.65 0.52 above 40 0.70 0.56 Ex. 94. The diameter of the piston of a steam engine is 60 in.; it makes 11 strokes per minute; the length of each stroke is 8 ft.; the mean pressure per square inch, 15 lbs. The modulus of the engine being 0.65, determine the number of cubic feet of water it will raise per hour from a depth of 50 fathoms. [The number of units of work done by steam on piston in one hour equals π× 302 × 8 × 15 ×11×60; this number multiplied by 0.65 will give the number of units usefully spent in raising water; hence the number of cubic feet of water is found.] Ans. 7763 cub. ft. Ex. 95. The diameter of the piston of an engine is 80 in., the mean pressure of the steam is 12 lbs. per square inch, the length of the stroke is 10 ft., the number of strokes made per minute is 11. How many cubic feet of water will it raise per minute from a depth of 250 fathoms, its modulus being 0.6? Ans. 42-46 cub. ft. Ex. 96. If the engine in the last example had raised 55 cubic feet of water per minute from a depth of 250 fathoms, what would have been its modulus? Ans. 07771. Ex. 97.-How many strokes per minute must the engine in Ex. 95 make in order to raise 15 cubic feet of water per minute from the given depth? Ans. 4. Ex. 98. What must be the length of the stroke of an engine whose modulus is 0.65, and whose other dimensions and conditions of working are the same as in Ex. 95, if they both do the same useful work? Ans. 9-23 ft. Ex. 99. The diameter of the cylinder of an engine is 80 inches, the piston makes per minute 8 strokes of 104 ft. under a mean pressure of 15 lbs. per square inch; the modulus of the engine is 0.55. How many cubic feet of water will it raise from a depth of 112 ft. in one minute ? Ans. 485-78 cub. ft. Ex. 100. If in the last example the engine raised a weigh of 66,433 lbs. through 90 ft. in one minute, what must be the mean pressure per square inch on the piston? Ans. 26-37 lbs. Ex. 101. If the diameter of the piston of the engine in Ex. 99 had been 85 in. what addition in horse-power would that make in the useful power of the engine? Ans. 13.28 H.-P. 15. Work of water-wheels.-Hitherto we have considered only one kind of motive power, viz. the pressure of steam. The same principles are applicable to machines worked by any other motive power, as by the muscular force of animal agents, the pressure of moving air, or of falling water. The last of these, viz. the power of falling water, is, next to steam, the most conspicuous example of work done on a large scale by an inanimate agent. We shall therefore consider somewhat particularly the application of this power by means of water-wheels. It is plain that 1 lb. of water, in descending through 1 foot, must accumulate as much work as would be required to raise it through 1 foot, and hence if P lbs. of water descend through h feet, they will accumulate Ph units of work; and if, moreover, we suppose this water to descend against an obstacle, such as the float boards of a waterwheel, the amount of work so accumulated will be done upon the wheel, and this work may then be applied to any useful purpose after a certain deduction has been made on account of prejudicial resistances. It must be borne in mind that the height of the fall is the difference between the levels of the surface of the water in the reservoir and in the exit canal; in the case of overshot wheels it is supposed that the extreme circumference of the wheel is just in contact with the surface of the water in the exit canal. The height is represented by A B in the accompanying figures; of which fig. 2 represents the ordinary undershot wheel with plane float boards; fig. 3 the breast wheel, in which the water acts upon the float boards considerably above the level of the exit canal. Fig. 4 represents the overshot wheel. FIG. 4. A B The following table exhibits the moduli of various kinds of water-wheels. It is founded on results given in General Morin's Aide-Mémoire. In the table a denotes the length of the line A B in figs. 2, 3, 4, and h denotes the length of Bcin fig. 3:— celet's construction) (3) Breast wheels with curved float boards (Pon for a greater than 65 feet 0.60 to 0.65 (4) Overshot wheels, when the velocity is small and the buckets half filled 0.70 to 0.75 |