THE PULLEY. 



279 



treble instead of a double proportion, as will be evident by attending to trie 

 numbers which express the tensions in the figure. One rope would render 

 the weight three times the power ; two ropes nine times ; three ropes twenty- 

 seven times, and so on. An arrangement of pulleys is represented in fig. 17, 

 by which each rope, instead of being finally attached to a fixed point, as in 

 fig. 15, is attached to the weight. The weight is in this case supported by 

 three ropes : one stretched with a force equal to the power ; another with a 

 force equal to twice the power ; and a third with a force equal to four times 

 the power. The weight is, therefore, in this case, seven times the power. 



If the ropes, instead of being attached to the weight, pass through wheels, 

 as in fig. 18, and are finally attached to the pulleys above, the power of the 

 machine will be considerably increased. In the system here represented, the 

 weight is twenty-six times the power. 



In considering these several combinations of pulleys, we have omitted to 

 estimate the effects produced by the weights of the sheaves and blocks. With- 

 out entering into the details of this computation, it may be observed generally 

 that in the systems represented in figs. 15 and 16, the weight of the wheel and 

 blocks acts against the power ; but that in figs. 17 and 18, they assist the power 

 in supporting the weight. In the systems represented in fig. 12, the weight 

 of the pulleys, to a certain extent, neutralize each other. 



It will in all cases be found that that quantity by which the weight exceeds 

 the power, is supported by fixed points ; and therefore, although it be commonly 

 stated that a small power supports a great weight, yet, in the pulley, as in all 

 other machines, the power supports no more of the weight than is exactly equal 

 to its own amount. It will not be necessary to establish this in each of the 

 examples which have been given ; having explained it in one instance, the stu- 

 dent will find no difficulty in applying the same reasoning to others. In fig. 

 15, the fixed pulley sustains a force equal to twice the power, and by it the 

 power giving tension to the first rope sustains a part of the weight equal to 

 itself. The first hook sustains a portion of the weight equal to the tension of 

 the first string, or to the power. The second hook sustains a force equal to 

 twice the power ; and the third hook sustains a force equal to four times the 

 power. The three hooks therefore sustain a portion of the weight equal to 

 seven times the power ; and the weight itself being eight times the power, it 

 is evident that the part of the weight which remains to be supported by the 

 power, is equal to the power itself. 



When a weight is raised by any of the systems of pulleys which have been 

 last described, the proportion between the velocity of the weight and the ve- 

 locity of the power, so frequently noticed in other machines, will always be 

 observed. In the system of pulleys represented in fig. 15, the weight being 

 eight times the power, the velocity of the power will be eight times that of 

 the weight. If the power be moved through eight feet, that part of the rope 

 between the fixed pulley and the first moveable pulley will be shortened by 

 eight feet. And since the two parts which lie above the first moveable pulley 

 must be equally shortened, each will be diminished by four feet ; therefore the 

 first pulley will rise through four feet, while the power moves' through eight 

 feet. In the same way it may be shown that while the first pulley moves 

 through four feet, the second moves through two ; and while the second moves 

 through two, the third, to which the weight is attached, is raised through one 

 foot. While the power, therefore, is carried through eight feet, the weight is 

 moved through one foot. 



By reasoning similar to this it may be shown that the space through which 

 the power is moved in every case is as many times greater than the height 

 through which the weight is raised, as the weight is greater than the power. 



