' 



I \Nir\l, PHILOSOPHY. -STATICS. 



[ri-LLKTH. 





ThU : :ll nvcGMBirily diminish the ten 



i, as the cosine will always bo I 



n less t! 



In thf ><- ',{ nytttm of jntlltys, the 

 t<> which ; .. i )>asscs 



.11 tho j.iilh-yK, as shown ! 



Tim pulloyi art) divided into 

 two groups, tin; appear working in a 

 ."ii l>lock winch U fixed, and 

 tho lower in a common movablo block. 

 The other extremity of the cord to 

 w liicli P is attached, after passing over 

 all the sheaves, U fixed to the upper 

 block. W is attached to tho lower 

 block. 



The weight or force P will commu- 

 nicate tho same tension <, throughout 

 tho rope. 



It is evident, therefore, that if we 

 lor the ropes as parallel, and 

 t the weight of the lower block 

 MM its sheaves when there is equili- 

 brium, we shall have 



W-6P 



and generally we slmll have Wi=nP 

 n is the number of the portions 

 into which the cord is divided by tho 

 two Mocks. 



\Vhon tlio weights of the sheaves 

 anil blocks are taken into considera- 

 tion, it Li evident that tho weight of 

 tho lower block and its sheaves must 

 be added to W. 



If the positions of tho cords be 

 not parallel, each portion must have its tension multi- 

 plied by tho cosine of the 

 illicit; to which it is inclined 

 to the direction of \V. as in 

 the first system of pulleys. 



In Fig. 131 we have repre- 

 sented a very powerful ar- 

 rangement, having the pul- 

 leys in each block arranged 

 as shown in the figure. The 

 sheaves of the upper row in 

 the upper block, as well as 

 those of the lower row of 

 the lower block, are all of 

 ' the same diameter ; those 

 of tho lower row of the 

 upper, and upper row of the 



Hf rv-f- rwi lower, are equal to each other 

 111 In, ]! IH. in diameter, though less than 

 tho former. The cord is 

 li W omitted for the sake of clear- 

 ness, but one extremity is 

 Mi Ml Mi fi**l to the hook A below 

 < YJ J, ; j I 'j (he lower block ; it thru 

 passes in succession under 



*^*> ***^J U^< IF^S. U"^ i * ,t i i 



each of the sheaves in the 

 lowor block, and over those 

 in tlio upper in tho order 

 shown by the numbers at- 

 tached to tho sheaves in 

 the figure : the power is ap- 

 plied to tho extremity of 

 So oord which passes over 

 e pulley marked w 20 . 

 As there arc twenty portions of the cord between the 

 two blocks, in this case W-20P. 



Thin arrangement permits the portions of tho cords to 

 be nearly parallel, and has tho additional advantage that 

 the ] * immediately over the weight W, so that 



the I'MMl'it-luin of the cords will not bo deranged, as it 

 would otherwise be liable to be ; it has, however, serious 

 I>ntrti<-.il defects. 



If the weight \V be raised by tho power P a given 



't, say for instance one foot, cnch portion of the 



oord Iwtwocu the two blocks must be raised the same 



Fig 132. 



which will necessarily cause P to d-r- 

 he arrangement of the cord, it is evident th;it one 

 foot of the cord will pass over the pulley w, , two foot 

 over w s , three feet over w,, &c., and 20 over w ao . 



Now since the sheaves w,, w., w s , &c., are of the 

 same diameter, and revolve each round its own axis, 

 and twice as much cord passes over w., as w, three time* 

 as much over w a as w,, and BO on ; it follows that w 2 

 will revolve twice as fast, w s three times as fast, as w, , 

 and so on for the other sheaves. 



This inequality of motion leads to great inequality in 

 the wear of the various parts of tho machine. If we 

 attempt to remedy this defect by fixing the sheaves in 

 the same row to a common axis, parts of the cord must 

 necessarily scrape or slide over the grooves of -the sheave* 

 instead of moving with them ; and this will lead to great 

 increase of friction. To remedy these dofects, Mr. Wliito 

 contrived the pulley known by his name. The sheave* 

 in each block were cut out of one piece of wood or metal, 

 being formed of a series of parallel and circular grooves, 

 each increasing in diameter. The diameters of the 

 grooves were made to bear to each other the same pro- 

 portions as the portions of rope which were to pass over 

 them those for the lower block being as the odd num- 

 bers 1, 3, 5, 7, <tc. , and those in the upper as tho even 

 numbers 2, 4, 6, <fcc. 



By this arrangement all 

 inequality of wear was 

 supposed to be obviated, 

 and the friction was re- 

 duced nearly to that of 

 the two axles of the blocks. 

 But this ingenious con- 

 trivance is found prac- 

 tically to fail, as the elas- 

 ticity of the cord, which 

 is supposed by tho ar- 

 rangement to be inelastic 

 or iuextensible, is subject 

 to continual change by 

 the moisture or dryness 

 of the atmosphere, and 

 by the friction of tho cord 

 upon the grooves. 



In the third system of 

 pulleys each cord passes 

 over a pulley, and has one 

 extremity attached to the 

 weight W, as shown in 

 Fig. 132, and the cords 

 are supposed to bo parallel 

 to each other. 



Let w,, w s , w 3 , w 4 , re- 

 present the weights of tho 

 pulleys, u t,, t s , t t , the 

 tension of tho cords pass- 

 ing over them. 



Then 



t 1 P. 



< t + w, = 2 P + w 

 t a = 2t 3 + w a - 2* P + 2w, 

 <t - M a + w a = 2P 3 + 2-w, + 



^+W 3 . 



Therefore W = , + < 2 -f t, + t t . 



- (1 + 2 + 2' + 2=-) P + (1 + 2 + 2=) 

 w, -h(l + 2)w A +w,. 



- (2* - 1) P + (2 - 1) w, + (2* - 1) 

 w, + (2 - 1) w,. 



The same reasoning may be extended to any number 

 of pulleys n, in which case 



W = (2" - l)P+(2- - 1 -l)w, +(2" - a -l)w 2 +<fec. 



When the weights of tho pulleys are neglected, w,, 

 w 2 , A-c., w are equal to zero, and W = (2" 1) P. 



Wln-n tho strings are not parallel, their respective 

 tensions must bo multiplied by tho cosino of the angle 

 they mako with the direction of W, as in the preceding 

 systems. 



