MECHANICS. 



the memory the general principles upon 

 which the mechanical agency of puUies 

 is to be investigated. 



Like all other machines, the pulley 

 obeys the principle of virtual vdocitUi ; 

 that is, the ascent of the weight is as 

 many times less than the simultaneous 

 descent of the power as the weight 

 is greater than the p 



Thus in the single moveable pulley 

 represented in fig. 57, if the power 

 descend through two feet, two feet of the 

 rope CADE will pass over the fixed 

 pulley B C. Hence, that part of the 

 rope will be shortened by two feet, and 

 therefore, each of the parts C A and D 

 E will he shortened by one foot. Thus 

 the weight ascends through one foot 

 while the power descends through two 

 feet, that is, the velocity of the power 

 is twice that of the weight. But by 

 (76) it appears that the weight is equal 

 to twice the power. 



In the same manner it may be proved 

 that in fig* 58, while the power descends 

 through three feet, the rope extending 

 from the fixed pulley to that end which 

 is attached to the pulley which supports 

 the weight is shortened by three feet, 

 and therefore, each of the three parts 

 engaged in supporting the weight is 

 shortened by one foot. Hence, it ap- 

 pears, that the velocity of the power i-> 

 three times that of the weight ; by (76) 

 it was proved that the weight is three 

 times the power. 



In general, in all systems of pullies 

 in which there is but one rope, the 

 space through which the power de- 

 scends is equal to the entire length by 

 which the rope extending from the pul- 



i-\t the power to its extrem: 

 shortened. But this length is m 

 buted equally between all the parts of 

 the rope which are engaged in support- 

 ing the weight. Hence each part will be 

 shortened by a quantity as many times 

 less than ihe descent "of the power as 

 there are parts of the rope engaged in 

 supporting the weight. But the number 

 of tl expresses the proportion 



of the weight to the power. 



This re;^ iu be easily applied 



to the systems represented injlp*. 

 60, 63, &c. We shall not pursue this 

 m to the other systems. By 

 adopting a similar method of reasoning, 

 the student will find no difficulty in ner- 

 ceiving that it is applicable to all of 

 them/ and that universally as we gain 

 great mechanical efficacy, that is, raise 

 a very great weight with a very small 



power, we invariably lose just as much 

 in velocity as we gain in force. 



(88.) We hav hitherto supposed that 

 the ropes by which the pulleys are sus- 

 tained arc 'all in the vertical direction. 

 \Vhen this is not the case, the several 

 results which we have obtained are not 

 applicable. ln/fg. 71 the power sus- 

 tains the weight by the tension of a 

 rope, in which the parts are not parallel 

 Let 1 F (fc.71.) be the vertical line 



\\- 



Fiy.fl. 



through which the centre of gravity of 

 the weight passes, and from F draw F G 

 and FH parallel to DC and A B. At the 

 point E three forces may be considered 

 as acting, which are in equilibrium, viz. 

 the tensions in the directions E H and 

 EG, and the weight W. Hence, by 

 Treatise I. (9), these forces must be 

 represented by the lines E H, E G, 

 and E F. But, since the tension of 

 every part of the rope is the same, and 

 equal to the power P, the sides E H 

 and E G of the parallelogram must be 

 equal, and therefore the diagonal E F 

 must divide the angle G E H into two 

 equal parts. Hence, it follows that the 

 weight will always settle itself into that 

 position in which the two parts A B, 

 D C of the rope will be equally inclined 

 to the vertical line, and it will have to 

 the power the same ratio as E F to E H. 



Those who are conversant with tri- 

 gonometry will perceive, that if the 

 angle A B D, at which the'parts of the 

 suspending rope are inclined, be called 

 E, we have 



E F : K H : : Sin. E : Sin. * E. 

 But Sin. E 2 Sin, * E Cos. * E. 

 Heiu-e 

 E F : E H : : 2 Sin. * E Cos. 4 E : 



Sin. $ E. 



F. H ' 2 Cos, 4 E : 1 

 . . W : P : : 2 Cos. E : 1 



W ~ 2 P Cos. 4 E ; 

 that is, twice the power, multiplied by 

 the cosine of half the angle under the 

 ropes, is equal to the weight. 



(89.) In the same way the effect of 

 the obliquity of the ropes may be de- 



