MECHANICS. 



17 



end of the motion the rope and weights 

 are in the position A'C'D'B', which 

 would, in fact, be the case if the rope 

 were perfectly rigid, and the friction 

 with the wheel sufficient to prevent it 

 from sliding in the groove. In this 

 position .the weight added to A, instead 

 of acting against B with an equal le- 

 verage CO, would act with the dimi- 

 nished leverage EO against B resisting 

 with the increased leverage FO, (the 

 lines A'E and B'F being drawn perpen- 

 dicular to CD and its production.) Thus 

 it appears that if the rope were perfectly 

 rigid, any power which would com- 

 mence to" turn the wheel would very 

 soon bring the apparatus into such a 

 position, the opposing weight or re- 

 sistance gaining leverage, while on the 

 other hand the moving power would be 

 losing its leverage, that the machine 

 would come to equilibrium, and no fur- 

 ther motion would ensue. 



Now let us suppose, what is generally 

 the case, that, the rope, without being 

 absolutely and perfectly rigid, has a 

 certain degree of stiffness. First, sup- 

 pose the apparatus to assume the posi- 

 tion represented in the last figure. The 

 weights A' and B', acting upon the par- 

 tially flexible ropes A'C'C and B D', 

 will evidently bend them into curves 

 such a. represented mfig. 6. From A' 

 6. 



and B', as before, draw the perpendicu- 

 lars A'E and B'F, and it is obvious that 

 the increase of weight given to A' works 

 with a diminished leverage EO, while 

 the unaltered weight B' receives an in- 

 creased leverage FO. If the weight 

 which is added to A' multiplied by EO be 

 not greater than the weight B' multiplied 

 by FO, no motion can ensue ; and thus, 

 owing to the effect of the rigidity of the 

 rope, a fixed pulley may be loaded with 



unequal weights, and yet continue in 

 equilibrium. 



If we consider the effec-ts which the 

 weights A' and B' produce upon the 

 rope as the wheel revolves, we shall find 

 them very different. The weight B 

 continually bends the rope B'D, so as 

 to give it at the point D' a curvature 

 equal to that of the groove of the wheel. 

 On the other hand, the weight A' is 

 employed in destroying the curvature 

 which' the rope had when resting in the 

 groove, and even in giving it a curva- 

 ture in the opposite direction. The effort 

 which the rope in this case makes to 

 retain its curvature at C tends to dimi- 

 nish the leverage by which A' acts, but 

 the effect of A' in giving curvature to the 

 rope in the', opposite direction below C' 

 counteracts this effect, and has a ten- 

 dency to increase the leverage of A' ; 

 the difference between these effects is 

 what produces the diminution of the le- 

 verage of A'. On the other hand, the 

 resistance which the rope B D offers to 

 flexure is opposed to the effect of B' ; 

 and this resistance, undiminished by any 

 other cause, is wholly effective in in- 

 creasing the leverage of B'. We may, 

 therefore, anticipate that the increase 

 DF of the leverage of B' is much greater 

 than the diminution CE of the leverage 

 of A'. This effect, which is found by 

 actual experiment to obtain, is of some 

 importance in simplifying the theory 

 of rigidity. For we find, in general, 

 that the effect of the weight A' upon the 

 rope is so considerable that the diminu- 

 tion CE of its leverage, owing to the 

 rigidity of the rope on the side C, is so 

 small that it may be entirely neglected, 

 and that in our investigations we may, 

 without sensible error, consider the 

 weight A' as acting with the leverage 

 OC, or the radius of the wheel. 



On the contraiy, for the reasons just 

 assigned, the increase DF of the lever- 

 age of B', owing to the rigidity of the 

 rope D B', is considerable, and forms an 

 important element in the determination 

 of the effects of rigidity. 



[Let x express the weight which 

 must be added to A', in order just to put 

 the wheel in motion, and we have by 

 the principles of the lever, 



where r expresses the radius of the pul- 

 ley and b = DF. Hence, 



but since A' = B', therefore A'r 2= B r. 

 Taking these equals from both, we find 



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