PDLLEYS. ] 



MECHANICAL PHILOSOPHY. STATICS. 



729 



Fijr. 126. 



circular wheel B C, Fig. 126, having a smooth groove cut 



in its circumference. This wheel revolves freely round 



an axis or pivot passing through its centre at right angles 



lo its surface, and having its extremities fixed to a 



wooden or metallic frame F G. This frame may be either 



fixed or movable. 



The wheel is called 



the sheaf, shiver, 



orrundle: the axis 



the gudgeon ; and 



the frame in which 



the axis is fixed, 



the block of the 



pulley. 



The force P is 

 applied to one ex- 

 tremity of a cord 

 passing freely over 

 the grooved cir- 

 cumference of the 

 sheaf, which innves 

 with it round its 



axis, thus dimin- ** 



ishing the friction of the cord ; and the weight W, wliicl 

 is to be overcome, is attached to the other extremity oi 

 the cord 



Several pulleys may be combined together, forming 

 what is called a system. The cords which pass over the 

 Kheaves are supposed to be perfectly flexible and inex- 

 tunsible ; the friction of the cords, as well as that of the 

 axis, and when not specially mentioned, the weights ol 

 the blocks and sheaves, are not considered in our 

 calculations. 



?V 127. The single pulley, 

 _~- ?----_..! with its block fixed, af- 

 fords no mechanical ad- 

 vantage of the power 

 over the weight, for the power P 

 (Fig. 126) exerted on the extremity 

 of the cord, communicates a tension 

 T proportional to P throughout its 

 length to its extremity D, at which 

 W acts. When there is equilibrium, 

 these forces P and W must be both 

 equal to the tension T of the cord 

 at the points A and D, or the ten- 

 sion of the cord will produce a 

 force equal and opposite to P at A, 

 and equal and opposite to W (it D. 



Hence P = W = T. 

 The single fixed pulley affords a 



convenient means for altering the direction of the appli- 

 cation of a force to a machine. Thus a weight which 

 i tically downwards, may be made to exert an 

 equal force in any direction we require, by a proper posi- 

 tion of the pulley and the support to which its block is 

 fixed. 



Single Movable PiMey. Let a weight P be attached 

 to one extremity of a cord passing over the fixed pulley 

 FE (Fig. 127) round the movable pulley C B, and having 

 its other extremity fastened to a fixed beam at A, the 

 block of the pulley F E being fixed to the beam at G in 

 such a manner that E C and A B shall be parallel. The 

 weight W is attached to the block D of the pulley C B. 

 Neglecting the weight of the pulley C B, P will commu- 

 nicate a tension throughout the cord, and this tension 

 will produce a force P acting in the direction C E at E, 

 and another force P acting in the direction B A at B 

 (Fig. 127). 



Hence when there is equilibrium, we shall have 



W = P + P = 2P. 



When the weight of the pulley is taken into account, 

 this weight nitut be added to W. When the cords are 

 not parallel, resolving the forces P acting at C, and P 

 acting at B, into vertical and horizontal forces, as shown 

 in Fig. 1L'H, and equating the horizontal and vertical 

 forces acting on 15 C, we have 



P sin. <f> = P sin. 0, hence = 

 VOL. i. 



Fig. 12S. 



and W = P cos. -f P cos 

 =P cos. + Pcos. =2P 

 cos. 0. 



Where equals half the 

 angle A B produced, makes 

 with E C produced. 



In the first system of pul- 

 leys, each pulley hangs by a 

 separate cord, as shown in 

 Fig. 129, so that the cords 

 may be parallel. 



Let Wu W 2 , W 3 , be 

 weights of the pulleys, t , , 

 *, t 3 the tension of tin 



, 3 

 cords. 



W + w, = 2t,, <, + vr t = 2t 2 , t z + w s = 

 , = P. 



Hence P = -* + *| = ^ + L 2 + sg 



2 3 , and 



W 



23 



If there had been four pulleys we should have had 



W 



4- - 4- 

 ' 2< ' 



+ 2 



Neglecting the weights of the pulleys, w,, W 2 , (fee., 

 or three pulleys we shall have P = -^, for four P _ 



W 



and generally for n pulleys P= 7^' 



When the cords are not parallel, W3 may see, from the 

 ase of the single pulley, that each tension, t,, t.,, t.,, 

 :c., must be multiplied by the cosine of the angle which 

 ts cord makes with a line drawn parallel to the direction 

 n which W acts. 



5 A 



