FRICTION OF STRAPS AND ROPES.] 



APPLIED MECHANICS. 



903 



Speed of strap, 150x3| ft. X 3f=1650 feet per minute. 



Breadth, 



10X800 

 1650 ' 



5 inches nearly. 



When less than half the circumference of the pulley is 

 embraced, the strap must be proportionally wider ; and 

 when more than half the circumference is embraced, its 

 width may be less. 



The law according to which the friction of a strap in- 

 creases with an increased arc of contact, is of a peculiar 

 character, but may be readily understood by comparing 

 the friction on arcs of different lengths. If a pulley (of 

 any diameter whatever) were prevented from revolving, 

 and a strap passing over part of its circumference were 

 stretched by a certain weight at each end, additions 

 might be made to the weight at one end until the strap 

 began to slip over the pulley. The ratio which the 

 weight so increased might bear to the weight at the other 

 end, would measure the amount of friction. For ex- 

 ample, in experiments made to test a theoretical investi- 

 gation on this subject, a strap passing over a pulley in 

 contact with 60, or Jth of its circumference, was 

 stretched by a weight of 10 Ibs. at each end. One of the 

 weights was increased until it amounted to 16 Ibs , when 

 the strap began to slip. The ratio of 16 to 10, or }J = 

 16 was then the measure of the friction. When 20 Ibs. 

 at each end were used to stretch the strap, the one weight 

 was increased to 32 Ibs., giving the ratio of H= io = l'6, 

 tli -iame as before. And likewise when 5 Ibs. were used 

 for stretching, the weight at one end was increased to 

 8 Ibs., giving still the same ratio f = I'd, So far, then, 

 the friction was precisely proportional to the stretching 

 weight, as might have been expected from the ordinarily 

 received doctrine on the subject of friction. On extend- 

 ing the arc of contact to 120 = jths or Jrd of the cir- 

 cumference, the ratio was found to be 2'5C, or 1'6 mul- 

 tiplied by 1'6, or the square of 16. And again, on em- 

 bracing 180 = Jths or ^ of the circumference, the ratio 

 was found to be 4-1, very nearly the cube of 1'6. The 

 theoretical investigation (given shortly in the note be- 

 low*) brought out this result independently, and 

 the following law may therefore be taken as estab- 

 lished : 



If, for any given arc of contact, the one weight bears 

 to the other, at the point of slipping, a certain ratio 

 for double the arc, the ratio will be squared ; for triple 

 the arc it will be cubed ; for 4 times the arc it will be 

 raised to the 4th power ; and so on. 



In all cases, however, much depends on the tightness 

 of the strap, the limits to the force with which it is 



* If IP be a weieht straining one end of a strap, and W a 

 greater weight applied at the other end, just sufficient to cause 

 the slipping of the strap when it is in contact over an arc of the 



\V 



circumference ; then, while remains constant, the ratio 



w 



which may be called/, is constant, whatever the absolute weights 

 W and w may be, or the friction is proportional to the pressure ; 

 for W IP is the measure of the friction ; and while W and u> 

 maintain the same proportion, W w also maintains the same 

 proportion to either of them. But if the strap supporting W 

 be bent farther round the circumference, so as to embrace an 

 trc increased by ^0, W will have to be increased by a quantity 

 A W to produce slip, which must be proportional to W, the 

 friction on any arc being proportional to the pressure. Hence 

 ultimately 



d \V 



= n W, when a is tome constant depending on the nature 

 il it 



of the rubbing surfaces. Integrating, we have log. W =: a -(- C. 

 When = o, \V = w, because the friction is = o. Hence 



\V \V 



log. = a 9, or =/= t*. If when the arc is e, the 



_ 



friction be F = t'* and hence /= e =: ( e ) 8 F e ' 



or the ratio expressing the friction over any arc is, as the friction 

 over any other arc raised, to the power expressed by the ratio 

 of the one arc to the other 



strained being, first, the tensive strength of the strap 

 itself, and, secondly, the amount of pressure that it may 

 be convenient to throw upon the shaft and its bearings. 

 New straps become extended by use, and it is therefore 



Fig. 269. 



frequently necessary to tulce them up or shorten them. 

 Before use, they should be strained for some time by 

 weights suspended from them, so aa to leave less room 

 for extension while in use. Wherever straps are em- 

 ployed, they should be of the greatest breadth, and travel 

 at the greatest speed consistent with convenience, as it 

 is most important to have the requisite strength in the 

 form best suited for flexure, and the least possible strain 

 on the shafts and bearings. 



When ropes or chains are employed, as in cranes, cap- 

 stans, windlasses, or the like, for raising heavy weights 

 or resisting great strains, the requisite amount of friction 

 is obtained by coiling them more than once round the 

 barrel of the apparatus. It is found that one complete 

 coil of a rope, as in Fig. 269, produces a friction equiva- 

 lent to 9 times the tension on the rope, the barrel being 

 fixed ; that is to say, 1 Ib. or 1 cwt. of tension on the 

 end of a rope at A, can support 9 Ibs. or 9 cwt. of ten- 

 sion at B. Were the end B of the rope coiled again 

 round a barrel, it would support 9 times its tension ; that 

 is, 9 X 9, or 81 times the tension of A, and so on, coil 

 after coil increasing the friction in a very high ratio, as 

 may be calculated according to the law given above, 

 thus : 



Multiply 9 by itself as many times as there are coils, 

 and the product will be the number of times the tension 

 at one end that will be supported at the other. For ex- 

 ample, 1 cwt. at one end of a rope coiled three times 

 round a barrel, would support 9x9x9 = 729 cwt., or 

 36J tons at the other end of the rope. The diameter of 

 the barrel does not affect the result. Having regard to 

 these facts, we may readily understand the force witli 

 which a knot on a cord or rope resists the slip of the coils 

 of which it consists, for the several parts of the cord act 

 as small ba Tels, round which the other parts are coiled ; 

 and the yielding nature of the material of which the 

 barrels are composed, permits the coils to become im- 

 pressed into their substance on the application of force, 

 and prevents them from slipping more effectually than if 

 they were coiled on a hard and resisting barrel. 



MOTION OF FLUIDS. Before concluding this part 

 of the subject, we may briefly allude to suggestions that 

 have from time to time been made for communicating 

 power by means of the movement of fluids. Some of 

 these have been practically carried out with very good 

 ellect, and we believe that much may yet be done in this 

 direction. 



Manufactories, containing numerous machines, are 

 generally arranged in such a manner that the power of a 



