396 



THE POPULAR EDUCATOR. 



MECHANICS. XIV. 



COMPOUND PULLEYS. 



WE are now in a position more clearly to understand the 

 remaining mechanical powers. We have explained the principle 

 of the simple pulley, and seen how to find out the advantage 

 gained by it, both when the cords are parallel and when they 

 are inclined at an angle. But there are various combinations 

 of fixed and movable pulleys which are caUed compound pulleys, 

 and are very frequently used in ships and in raising heavy 

 weights, or exerting powerful strains. We must examine the 

 principle of these, and see how to ascertain the advantages 

 gained by using them. 



They are usually classed in three systems. 

 In the first system, which is represented in Fig. 72, each 

 pulley hangs by a separate cord, and all 

 are movable, at least all that have any 

 effect, the runner, D, being introduced 

 merely for the sake of reversing the direc- 

 tion in which the power acts. 



Now let us try and see what is the 

 ratio the power bears to the weight when 

 the system is in equilibrium. We will 

 suppose, for the sake of simplicity, that 

 the power applied to P is 1 pound. The 

 strain or tension of the cord P o is the 

 same throughout its entire length (we 

 are not taking into consideration now 

 friction and the imperfect flexibility of 

 the ropes). The pulley c is therefore 

 kept at rest by the tensions of the three 

 cords, a c, DC, and B c, and as these are 

 parallel forces, the strain on B C is equal 

 to the sum of the other two, each of 

 which is 1 pound. The tension of each 

 part of the cord c F is therefore 2 pounds. 

 In the same way we see that the tension 

 of the next cord, B E, is double that of 

 c F, that is, 4 pounds. But the weight, 

 w, is supported by the two cords, B A, 

 A E, or rather by these two parts of 



' fco same cord, and as each has a strain of 4 pounds, the 

 total weight supported is 8 pounds. In this case then, there 

 is a gain of 8, a power of 1 pound balancing a weight 

 of 8 pounds. Similarly, if ar.other pulley were added, a power 

 of 1 pound would balance a weight of 16 pounds, each additional 

 pulley doubling the weight supported; and thus we have the fol- 

 lowing rule for determining the gain in the 

 first system of pulleys : 



Multiply 2 by itself as many times as there 

 are movable pulleys ; the result will show 

 the mechanical gain. 



Thus, if there are five pulleys, the gain is 

 2x2x2x2x2, that is, 32. You must be 

 careful, in calculating this, not to count the 

 fixed pulley, as that has no effect. 



In the second system of pulleys, instead of 

 each having a separate cord, the same one 

 passes round all, and they are arranged in 

 two blocks, one of which is fixed, and the 

 other (usually the lower) is movable. Fig. 

 73 represents this system. One end of the 

 cord is here fastened to the hook A on the 

 fixed block, and it then passes in succession 

 round the pulleys B, c, D, E, F, and G. In 

 this case, the weight is supported by six folds 

 of the same cord, and each bears an equal 

 part, the cord being equally strained through- 

 out. Each part, therefore, sustains a portion 

 of the weight equal to P, and w is therefore 

 six times as great as p. 



If we take away one pulley or sheave, as 

 it is called, from the lower block, leaving 

 two only, the weight will be divided be- 

 tween four folds of the cord, and thus only 

 four times the weight of P will be supported. 

 Similarly, were we to add another sheave to each block, we 

 should have a mechanical advantage of 8. We see then, that 



IV 72. 



Fig. 73. 



Fig. 74. 



in this system, the advantage is always ttrice as great as the 

 number of pulleys in the movable blocks. 



We have in this calculation supposed the cords to be 

 parallel. They are not, however, strictly so, still the dif- 

 ference is so slight that we need not notice it. 

 A trifling loss of power, however results from 

 it. 



Now there is one disadvantage about this sys- 

 tem when made as shown in our illustration, and 

 that is, that the weight must, on account of the 

 length of the blocks in which the pulleys are set, 

 be a long way below the point to which the 

 upper block is fixed. If we are using it, for 

 instance, to strain a telegraph wire or to tighten 

 a rope in the rigging of a ship, purposes for 

 which blocks are constantly employed, we should 

 fasten our rope to the hook from which w hangs, 

 and fasten the fixed block to the " dead eye" OH 

 the side of the ship ; but then we should not be 

 able to bring the rope within some considerable 

 distance of the eye. Another form of this sama 

 system has therefore been contrived which ob- 

 viates this difficulty. This is shown in Fig. 74. 



Here the required number of sheaves, three in 

 the present case, are fixed side by side in each 

 block, and the cord is fixed to a hook or staple A 



in the upper one, and then passes in succession 

 over each sheave. The weight is, of course, as 

 before, divided between the six ropes. In all 

 these cases it must be remembered that, as the 

 lower block is suspended from the cords, it 

 forms a part of the weight lifted, and the 

 weight W is therefore less by this amount than 

 our calculation makes it appear. 



There is a third plan for arranging this system 

 of pulleys which has the advantage of greatly 

 reducing the amount of friction, there being 

 only one sheave to turn on the axis instead of 

 several. This will bo understood from Fig. 75. 

 Each sheave is here a compound one. as if several 

 simple sheaves increasing in size were laid on 

 each other. A little attention will show that 

 for every inch the weight is raised 1 inch of cord 

 will pass over the smallest pulley, and as the 

 cord from this to No. 2 must also be shortened 

 an inch, 2 inches will pass over No. 2. In like 

 manner 3 inches must pass over No. 3. Their 

 sizes must therefore bo in the proportion of the numbers 1, 2, 3, 

 etc., or else the cord will grate on the pulleys. This is another 

 illustration of our fundamental law ; and we see further that, if 

 the weight is to be raised 1 inch, each of the cords supporting 

 it must be shortened by that amount, so that in this case P 

 must fall 6 inches to raise w 1 inch ; but a power of 1 

 pound will balance a weight of 6 pounds ; therefore, here also, 

 the power multiplied by the distance through which it moves 

 is equal to the weight multiplied by its distance. 



These two systems are those in most general 

 use. We must, however, just look at the third 

 system, which is represented in Fig. 76. B is a- 

 pulley fastened to the cord B D, which passes 

 over the runner A, and is made fast to the 

 weight at the other end ; c is likewise fixed to 

 a cord which passes over B to the weight ; 

 a third cord passes over c, and P acts at the 

 extremity of this. Each cord is thus fastened 

 to the weight. Now in this case it is rather 

 more difficult to find the advantage gained, as 

 the weight is not shared equally by the three 

 cords. The first cord is stretched by p, which, 

 as before, we will call 1 pound ; the part c F, 

 therefore, supports 1 pound of the weight. The 

 next cord is stretched by the tension of the 

 two parts of the first, and its strain is there- 

 fore 2 pounds, which is the portion of the 

 weight it sustains. Similarly the tension of 

 the cord which passes over the pulley A is 4 

 pounds, and therefore the entire weight sup- 

 ported by P is the sum of 4, 2, and 1, that is, 7 pounds. Were 



75. 



Fig. 76. 



