MECHANICS. 



33 



happen to be true, appears from this, 

 that if we suppose the wheels and blocks 

 abandoned, and the ropes merely to 

 pass through rings, but to be perfectly 

 flexible and to act without friction, all 

 the properties of the pulley may be 

 established. 



In our diagrams of pulleys, we shall 

 always represent the cords as acting 

 in the usual way over wheels ; but 

 our demonstrations will be founded only 

 on the supposed flexibility of the string, 

 and its perfect power of transmitting 

 force by its tension. 



(74.) The mechanical efficacy of every 

 system of pullies may be immediately 

 derived from this single principle, that 

 the same flexible string must always 

 suffer the same tension in every part of 

 its length. Thus, if the weight W (Jig. 

 56) be supported by the string AB, the 

 parts of the string 

 A and B will be 

 equally stretched, 

 and consequently 

 the tw r o hooks are 

 Fig. 56. equally engaged 

 in sustaining the 

 weight. Hence, 

 each part A and 

 B of the string 

 must sustain half 

 the weight. In 

 this case we sup- 

 pose the string to 

 be perfectly free in 

 passing through 

 the ring, and the parts A and B to be 

 parallel. 



(75.) Pulleys are fixed and moveable. 

 A fixed pulley has no mechanical ad- 

 vantage, since the power and weight are 

 equal. This apparatus is represented 

 mfig. 55. It is, however, very conve- 

 nient in accommodating the direction 

 of the power to that of the resistance. 

 Thus, by pulling downwards, we are 

 able to draw a weight upwards. It has 

 been already observed, that by means of 

 this simple machine, a power in any di- 

 rection whatever may be opposed to a 

 resistance in any other direction. 



(76.) The single moveable pulley, 

 sometimes called a runner, is repre- 

 sented in fig. 57. In this machine the 

 same rope extends from the power P 

 to the fixed point E, and has the same 

 tension throughout its whole length. 



It is evident that this tension is equal 

 to the power, for in that part P B of the 

 rope, between the power and the fixed 

 pulley, the power is supported by this 



tension. > The weight W is supported by 

 the parts C A and D E of the string, and 



Fig. 57. 



Fig. 58. 



must be equal to the sum of the ten- 

 sions, that is, to twice the tension of the 

 rope, or to twice the 

 power. In this machine, 

 therefore, a power is 

 capable of opposing a 

 resistance of twice its 

 own amount. 



We have not noticed 

 the effect of the weight 

 of the pulley A. If this 

 be taken into account, 

 it is only necessary to 

 add it to the weight. 

 The single moveable pul- 

 ley may also be so con- 

 structed that the weight 

 will be three times the 

 power. This is evidently 

 the case in the arrange- 

 ment in fig. 58. 



(77.) There are several systems of 

 pullies worked by a single rope. In all 

 these there is one moveable block, in 

 which wheels or sheaves are fixed, over 

 which the rope runs, and to which 

 the weight is attached. In estimating 

 the mechanical effect, the rooveable 

 block is to be considered a? a part of 

 the weight. Since the same rope is 

 successively passed over all the wheels, 

 it must have in every part the same 

 tension ; and since the part K sus- 

 tains the power, this tension must be 

 equal to the power. The weight (in- 

 cluding the weight of the block to which 

 it is attached) is supported equally by 

 each part of the rope, which passes 

 between it and the fixed block. In fig. 

 59, the weight is distributed equally 

 among four ropes, each of which is 

 stretched by the force of the power. 



