252 



THE LEVER AND WHEELWOR.K. 



the weight as A C does to A B. In the same manner, considering B as a ful- 

 crum, and the pressure of the prop A as the power, it may be proved that the 

 pressure of A bears the same proportion to the weight as the line B C does to 

 A B. It therefore appears that the pressure on the prop A is greater than the 

 weight. 



When great power is required, and it is inconvenient to construct a long 

 lever, a combination of levers may be used. In fig. 9, such a system of levers 



Pig. 9. 



is represented, consisting of three levers of the first kind. The manner in 

 which the effect of the power is transmitted to the weight may be investigated 

 by considering the effect of each lever successively. The power at P produces 

 an upward force at P 7 , which bears to P the same proportion as P 7 F to P F. 

 Therefore the effect at P' is as many times the power as the line P F is of P 7 

 F. Thus, if P F be ten times P 7 F, the upward force at P 7 is ten times the 

 power. The arm, P 7 F 7 , of the second lever is pressed upward by a force 

 equal to ten times the power at P. In the same manner this may be shown to 

 produce an effect at P 7/ as many times greater than P 7 as P 7 F x is greater than 

 p// F/ Thus, if P' F 7 be twelve times P 7 ' F x , the effect at P 77 will be twelve 

 times that of P / . But this last was ten times the power, and therefore the P" 

 will be one hundred and twenty times the power. In the same manner it may 

 be shown that the weight is as many times greater than the effect at P 77 as P 77 

 F 77 is greater than W F 7/ . If P 77 F 77 be five times W F 77 , the weight will be 

 five times the effect at P 77 . But this effect is one hundred and twenty times 

 the power, and therefore the weight would be six hundred times the power. 



In the same manner the effect of any compound system of levers may be 

 ascertained by taking the proportion of the weight to the power in each lever 

 separately, and multiplying these numbers together. In the example given, 

 these proportions are 10, 12, and 5, which, multiplied together, give 600. In 

 fig. 9, the levers composing the system are of the first kind ; but the principles 

 of the calculation will not be altered if they be of the second or third kind, or 

 some of one kind and some of another. 



That number which expresses the proportion of the weight to the equilibra- 

 ting power in any machine we shall call the power of the machine. Thus, if, 

 in a lever, a power of one pound support a weight of ten pounds, the power of 

 the machine is ten. If a power of 2 Ibs. support a weight of 1 1 Ibs., the power 

 of the machine is 5 J, 2 being contained in 1 1 5^ times. 



As the distances of the power and weight from the fulcrum of a lever may 

 be varied at pleasure, and any assigned proportion given to them, a lever may 

 always be conceived having a power equal to that of any given machine. Such 

 a lever may be called, in relation to that machine, the equivalent lever. 



As every complex machine consists of a number of simple machines acting 

 one upon another, and as each simple machine may be represented by an equiv- 

 alent lever, the complex machine will be represented by a compound system 

 of equivalent levers. From what has been proved in fig. 9 it therefore fol- 

 lows that the power of a complex machine may be calculated by multiplying 

 together the powers of the several simple machines of which it is composed. 



