32 MECHANICS. 



in cases where this supposed flexibility 

 is not found, it will be necessary to 

 apply corrections to our results to ren- 

 der them practically applicable. 



(72.) A perfectly flexible rope, or 

 thread, is a machine which, indepen- 

 dently of the usual advantage attending 

 the use of machines, of enabling us, by 

 the aid of fixed points, to support 

 a considerable weight by a small power, 

 offers what may be considered great 

 mechanical convenience, even in cases 

 in which the power is equal to the 

 weight or resistance. 



A flexible rope may be used in trans- 

 mitting force from one point to another 



s\v 



Fig. 53. 



in the direction of its 

 length. Thus the 

 force of the weight 

 W (fig. 53.) is trans- 

 mitted by the string 

 a to the hook H, 

 and presses on the 

 hook in the same 

 manner as if it were 

 immediately suspend 

 ed from it without the 

 intervention of the 

 rope a. This power 

 of transmitting pres- 

 sure in the direction 

 of its length, is not owing to the flexi- 

 bility of the rope, but to its inextensi- 

 bility. This quality the rope enjoys in 

 common with an inflexible bar, which 

 would also transmit the force of the 

 weight to the hook in the same way. 

 An inflexible bar, however, has the ad- 

 vantage of the flexible rope in transmit- 

 ting force in the direction of its length ; 

 for, although a flexible rope will trans- 

 mit a force applied at one end to the 

 other end when that force is directed 

 from the end to which it is to be trans- 

 mitted, it will totally fail if the force be 

 directed towards the other, in which 

 case the inflexible bar will be effectual. 

 Thus, if the weight acted towards the 

 hook H, which would be the ease if it 

 were below W, the string a will fail to 

 transmit the force, but if it were an in- 

 flexible bar it would do so. 



(73.) One of the greatest conveniences 

 attending the use of a flexible rope, is 

 that by its means a force in any one di- 

 rection may be made to balance an equal 

 force in any other direction. Thus, if it 

 be necessary to sustain the weight W, 

 (fig. 54.) acting vertically downwards, by 

 a power which acts in the direction P H, 

 let a point P be assumed directly over 

 the weight, and iu the line P II, and let 



a flexible string be Fig. 54. 

 attached to the weight, 

 passed through a ring 

 at P, and connected 

 with the power at H, 

 and the object will be 

 evidently accomplish- 

 ed. For if the rope be 

 supposed to be per- 

 fectly flexible and 

 smooth, it will suffer 

 no resistance either 

 from rigidity or friction, in passing 

 through the ring at P, and the string 

 will be stretched by the same force in 

 its entire length, that tension being 

 equal to the weight W. 



But since it is impossible in practice 

 to obtain ropes which are perfectly 

 smooth and flexible, nor to construct 

 rings whose surfaces are free from all 

 asperities, it is usual, p- 55 



instead of passing 

 the rope through a 

 ring, to pass it over a 

 grooved wheel which 

 turns freely on an 

 axle, or on pivots, 

 as in Fi%. 55. 



The substitution of 

 the wheel for the 

 ring is attended in practice with two 

 obvious advantages: first, it removes 

 in a great degree the effects of the fric- 

 tion of the rope with the surface of the 

 ring, for instead of the surface of the 

 rope sliding on the surface of the wheel, 

 that surface turns with it. Secondly, it 

 diminishes very much the effects of the 

 imperfect flexibility of the rope, which 

 instead of being suddenly and sharply 

 bent, as over the ring, is gradually 

 deflected upon the curvature of the rim 

 of the wheel. 



The wheel, therefore, is not used to im- 

 part any mechanical advantage properly 

 speaking to the machine, nor is it at all 

 necessary to be taken into account in 

 the theory in which the perfect smooth- 

 ness and flexibility, the want of which 

 it is introduced to remedy, are pre-sup- 

 posed. 



Thewheel thus used is called npulley, 

 and hence that name has been given to 

 the machine itself, and its various modi- 

 fications. Some writers have even 

 ascribed to the wheel thus used, the 

 whole mechanical virtue of the machine, 

 and have established the conditions of 

 equilibrium by considering it as a lever. 

 That such investigations are founded on 

 wrong principles, although their results 



