WEIGHT ASD ELASTICITF.] 



APPLIED MECHANICS. 



837 



foot. The pressure on the valve is, therefore, the weight 

 of 4 cubic feet of water ; namely, 4 X 62| = 250 Ibs. 

 The effect of the weight to lift the valve by means of its 

 lever must be somewhat greater than this. If the valve 

 could be suddenly lifted so as to leave an opening to the 

 full extent of its area, 1 square foot, the water would 

 descend in C at the rate of 16 feet per second, the 

 velocity due to 4 feet bead ; but as the valve opens gra- 

 dually, and is only open for a short time, and that 

 not to its full extent, we may take the velocity of 

 the descending column at not more than 4th of this rate ; 

 that is to say, at 4 feet per second. When this move- 

 ment ensues, the valve is pressed on not only by the 

 weight of the column above it, as before, but also by the 

 weight of such a column as is due to a velocity of 4 feet 

 per second, which will be found by calculation to be ^th 

 of a foot high, adding 62i X J, about 15 Ibs. to the 

 load on the valve. This additional load overcomes the 

 leverage of the weight, and closes the valve; but the 

 column of water, 1 foot in area and 4 feet high, contained 

 in C, amounting to 4 cubic feet, is thus arrested whilst 

 moving at the rate of 4 feet per second ; and its mo- 

 mentum, which is equivalent to that of 4 X 4 = 16 

 cubic feet, moving at 1 foot per second, or 1 cubic foot 

 at 16 feet per second, must be given out as a force pro- 

 pelling the water along the small pipe at the side. If 

 we suppose that this pipe communicates with a cistern 

 36 feet high, the velocity due to that height is 48 feet 

 per second ; and the momentum of 1 cubic foot, moving 

 at lj feet per second, being equivalent to Jrd of a cubic 

 foot moving at 48 feet per second, we should expect that 

 this quantity Jrd of a cubic foot would be propelled 

 upwards to tin high cistern at each closing of the valve. 

 We must, however, recollect, that the momentum of the 

 larger volume has not only to balance that of the smaller 

 it must considerably exceed it, because it has to lift 

 the valve, to give the motion upwards to the column, to 

 overcome friction in the pipes and other impediments, 

 and, upon the whole, may be reckoned effective to the 

 extent of not more than j-th of its estimated force. We 

 may estimate, then, that -^Wth of a cubic foot is propelled 

 up to the high cistern by the descent of 4 cubic feet 

 through 4 feet from the lower cistern. Should the action 

 not last so long as a second, a smaller volume will 

 descend, and a proportionally smaller quantity will 

 be sent upwards, and conversely ; but the number 

 of times the descent and ascent take place ill a given 

 period will be greater or less accordingly. 



We have presented this calculation only as a rough 

 approximation to the practical results. We arc not 

 aware that any carefuuy-conducted experiments with 

 hydraulic rams have been given to the world, and we 

 therefore do not venture to offer any estimate as a guide 

 for practice ; but have merely discussed a case with the 

 view of opening the questions that have to be considered 

 in dealing with such apparatus. In many situations 

 where it may be desirable to raise water for the purpose 

 of ornamental fountains, or of domestic supply, the 

 hydraulic ram is applicable with great advantage. A 

 neighbouring stream may be dammed so as to provide a 

 fall of a few feet, if it have not sufficient local fall natu- 

 rally ; and the apparatus once fixed and properly ad- 

 justed, will continue effective for a long period. It is 

 exceedingly simple, entirely self-acting, and seldom liable 

 to derangement, if care be taken to fix gratings on the 

 pipe, so that dirt or extraneous matter of any kind may be 

 prevented from interfering with the action of the valves. 



3. WEIGHT AND ELASTICITY OF BODIES. 

 The gravitating attraction which the earth exerts on 

 bodies near its surface, and the force with which elastic 

 bodies tend to resume the condition from which they 

 have been withdrawn, are frequently employed forgiving 

 motion to machinery, chiefly* when the power required 

 is small, but exerted for considerable periods. Weight 

 and elasticity are not really sources of power ; they 

 rather afford the means of storing up efforts of short 

 duration for subsequent use during longer periods. 

 U'-f'ire a body can act by its weight, it must be lifted to ' 

 the height whence it has to descend; and, in the same way, [ 



before a body can act by its elasticity, its condition must 

 be changed to the extent through which it has afterwards 

 to return. Thus, in winding up a clock or watch, as 

 much power is exerted as is afterwards given out by the 

 weight or spring in moving the machinery with which 

 either is connected. The weight of the clock, and the 

 spring of the watch, are merely instruments for absorb- 

 ing, -at the moment, a certain amount of power, and 

 giving it out by degrees afterwards. The simplest, and, 

 it may be said, the only general mode of employing the 

 weight of a solid body to give impulse to machinery, is 

 to attach it by a chain or string to a cylindrical barrel 

 mounted on bearings at some height from the ground, 

 and connected by gearing with the machinery which it is 

 intended to move. The barrel being caused to revolve 

 by hand, or any other convenient power applied to it, 

 the string is wound round it, and the weight raised from 

 the ground. When Jeft to itself, the weight, attracted 

 again towards the earth, descends, unwinding the string 

 from the barrel, causing it to revolve, and thus giving 

 the necessary impulse to the machinery. Did the 

 machinery offer no resistance, the weight would always 

 descend with a speed accelerated at every moment of its 

 descent by the continued action of gravity, which exerts 

 as much influence on a body in motion as on one at 

 rest. In dropping a stone from a considerable height, 

 whatever be its weight, we find that, during the first 

 second of its descent, it acquires a velocity of 32 feet per 

 second. Its velocity at the commencement was nothing, 

 for it began to move from a state of rest ; at every one 

 of the instants into which we may conceive a second of 

 time divided, it acquired more and more velocity, until 

 it attained the final velocity of 32 feet per second. All 

 these acquisitions in speed are equal in equal times, 

 because the force of gravity is constant, and therefore 

 exerts equal influences in equal times. Had tha body 

 descended during the whole second at the final velocity 

 of 32 feet per second, it would of course have passed 

 through 32 foet of space ; had its velocity remained the 

 initial velocity, which was nothing, it would have 

 descended through feet ; but as the velocity began 

 with and ended with 32, its average throughout the 

 second was 16 feet per second ; and therefore the body 

 descends in the first second through 16 feet. During 

 the next second, the body, starting with a velocity of 32, 

 acquires an additional velocity of 32, and therefore ends 

 with a velocity of 64 feet per second ; the average being 

 48 feet per second, and therefore the descent being 48 

 feet of height. Adding this to the space descended 

 during the first second, 16 feet, we find that in the first 

 2 seconds the total descent is 64. Were we to pursue 

 the investigation farther, we should find the velocity at 

 the end of the third second 96 feet per second, and the 

 total descent 144 feet, and so on according to the follow- 

 ing law : The velocity (in feet per second) acquired by 

 a falling body is 32 times the time (in seconds). 



The space (in feet) passed through by a falling body 

 is 16 times the square of the time, or the square of 4 

 times the time. Henco it follows that the time (in 

 seconds) occupied by tha descent of a falling body 

 through a given height (in feet), is Jth of the square root 

 of the height ; that the velocity (in feet per second) is 8 

 times the square root of the height ; and that the height 

 is the square of Jth of the velocity. 



Example 1. Kequired the velocity acquired by a fall- 

 ing body in 5 seconds. 



32 X 6 = 160 feet per second. 



Example. 2, Eequired the height fallen by a body in 

 5 seconds. 



16 X 5 X 5=400 feet, or (4 X 5=) 20 X 20=400. 



Exam/pie 3. Required the time occupied by a body 

 falling through 400 feet. 



Sq. root of 400 is 20, and Jth of 20 is 5 seconds. 



Example 4. Required the velocity acquired in falling 

 400 feet. 



Sq. root of 400 is 20, and 8 times 20 is 160 feet per 

 second. 



Example 5. Required the distance fallen by a body 

 when it has acquired a velocity of 100 feet per second. 



