24 



MECHANICS. 



The manner in which the pressures of 

 ihe power and weight are divided be- 

 tween the pivots, may be determined by 

 the principles established in Chap. iii. 

 By these principles it appears, that the 

 pressures of the power and weight are 

 divided respectively between the pivots 

 in the inverse proportion of their dis- 

 tances from them ; that is, the part of the 

 \veight or power which presses on the 

 one pivot, is to the part which presses 

 on the other as the distance of the 

 weight or power from the latter pivot is 

 to its distance from the former. 



In this investigation it should be re- 

 membered, that we do not consider the 

 machine as in motion, but merely that 

 the power sustains the weight. 



(58.) In the simple wheel and axle, 

 whatever be its form or peculiarity, its 

 mechanical efficacy depends on the ratio 

 of the radius of the wheel to the radius 

 of the axle, or, to speak more gene- 

 rally, the length of the lever by which 

 the power acts, to the radius of the 

 cylinder on which the rope which raises 

 the weight is coiled. There are then 

 two, and only two, ways, in which this 

 efficacy may be increased ; viz. by in- 

 creasing the leverage of the power, or 

 by diminishing the radius of the cylin- 

 der which supports the weight. Now, 

 although, in a theoretical view, there is 

 no limit to our power of increasing this 

 efficacy, since we can conceive the le- 

 verage of the power increased without 

 limit, or the thickness of the cylinder 

 diminished without limit, yet there is a 

 practical limit to the increase of the 

 mechanical efficacy of this engine. 



If we attempt to increase this efficacy 

 by giving to the power a considerable 

 leverage, the machine will become un- 

 wieldy ; the power will work through a 

 most inconveniently great space, and 

 the practical disadvantages which will 

 arise, will more than counterbalance any 

 thing which can be gained by the in- 

 creased power. 



If, on the other hand, we attempt to 

 increase the efficacy by diminishing the 

 thickness of the axle, we diminish the 

 strength of that part of the machine 

 which must support the weight; and 

 the cases in which this great efficacy is 

 required are precisely those cases in 

 which the greatest weights are to be 

 raised, or resistances to be overcome, 

 and, therefore, where the greatest 

 strength in those parts of the machine 

 on which the weight or resistance acts 

 are indispensably necessary. 



In cases, therefore, where great re- 

 sistances are to-be overcome, it is a 

 problem of considerable importance to 

 assign a method, by w r hich, without 

 rendering the machine more complex, 

 sufficient strength may be preserved by 

 preserving the thickness of the axle, the 

 machine may not be rendered unwieldy 

 by increasing very much the leverage of 

 the power, and a high degree of power 

 may be gained. 



(59.) All these ends are attained by the 

 simple modification of the wheel and 

 axle in Jig. 36. The axle or cylinder 



Fig. 36. 



A B consists of two parts, the diameter 

 of one part being less than that of the 

 other. A wheel and block C is attached 

 to the weight, and round the wheel the 

 rope which elevates the weight is 

 passed, and is coiled in the same direc- 

 tion on the thicker and thinner parts of 

 the axle. The elevation of the weight 

 is thus effected : upon turning the axle 

 in such a direction that the rope shall 

 be coiled upon the thicker part, it will 

 necessarily be rolled off the thinner 

 part. Upon every revolution of the 

 wheel, therefore, a portion of the rope 

 equal to the circumference of the thicker 

 part will be drawn up ; but, at the same 

 time, a portion, equal to the circum- 

 ference of the thinner part, will be let 

 down. On the whole, therefore, the 

 effect of one revolution will be to 

 shorten the entire length of that part of 

 the rope, by which the weight is sus- 

 pended, by a length equal to the differ- 

 ence between the circumferences of the 

 thicker and thinner parts of the axle ; 

 consequently, half that portion of the 

 rope will be shortened by half the dif- 

 ference between these circumferences. 

 But half that part of the rope by which 

 the weight is suspended, is evidently 

 equal to the distance of the wheel to 

 which the weight is attached from the 

 cylinder. Hence, we perceive that every 

 revolution of the cylinder raises the 

 weight through a space equal to half 

 the difference between the circum- 



