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WOKK 



two free ends of rope be called the weight end and the power end 

 respectively, and let us suppose that the arrangement is such that, 

 in order to move the weight end through 1 inch, the power end 

 must be moved through n inches. Let a weight W be attached to 

 the weight end, and let us suppose that it is found that a force P 

 must be applied to the power end to maintain equilibrium. 



We now have forces P and F in equilibrium. To find the rela- 

 tion between them, let us give the system a small displacement. 

 Let us move the weight W a distance ds, then, if the rope is not to 

 be stretched, we must suppose the power P moved through a dis- 

 tance nds. The work done by external force consists solely of 

 the work performed on the power end of the rope, namely P n ds, 

 and the work performed in moving the weight against gravity, 

 namely W ds. These are of opposite signs, if we raise the weight, 

 W ds must be taken positively and P nds negatively, and vice versa. 

 If the system was initially in equilibrium, the total work performed 

 by external forces in this small displacement must vanish, so that 

 the equation of equilibrium is seen to be 



K Wds-Pnds = Q, 



so that P = > 



n 



giving the relation between power and weight. 

 This investigation assumes that friction, 

 etc., may be neglected, and also neglects the 

 weight of the moving ropes and pulleys. 



As an instance of a system of pulleys, let us 

 consider the arrangement shown in fig. 91. 

 FlG 91 There are two blocks of pulleys, A and B. 



The former is fixed, while the latter is free to 



move, and has the weight W suspended from it. The rope, starting 

 from the power end, passes first round a pulley of block A, then round 

 one of block B, then round one of block A, and so on any number of 

 times, until finally its end is fastened to block B. To find the relation 

 between P and W, we need only find the number n. Let us suppose that 

 in addition to the free power end of the rope the number of vertical 

 ropes is s. Then, if we pull the power end until the weight end is raised 



