MKCMANICS. 



319 



for equilibrium, must pom* through tlie fulcrum, cannot Ue between 



thorn. M r. .\ r. aa it has been shown there that the distances 



of o from A au.i iff. 44, page 804) are inversely M 



roes, no here the distances P r and w r. mvuit be invenely 



I.OWIT ami resistance, or, what IB equivalent, the power 



multiplied by its arm f r ia equal to tho weight multiplied by 



r of 



levers, aa in the former, it .sin ml. 1 

 bo obaerved that thcro ia a 

 mechanical advantage gained 

 a larger weight at w ia overcome 

 by a IIS.SIT ut i', a result always 

 to be aeonred whore tho larger 

 arm con bo in von to tho power. 



: example of this lover, take the crowbar in tho illustration 

 18, used differently from that in Fig. 40. The workman 

 makes tho ground at the point of hi bar hia fulcrum, 

 throws the weight of the chest about tho middle, and, instead 

 of i>unhing downwards with his himd, lifts upwards. The 

 mechanical advantage is clearly on hia aide. The oar of a boat 

 ia also a lever of tho second order ; tho arms of the oarsman 

 furnish the powor ; but moat persona at first imagine that the 

 rowlock is tho fulcrum. This is natural, for it looks very like 

 one, but that it is not such is evident from the fact that tho 



boat is the thing he wants 

 to move. To spurt tho 

 water about with tho blade 

 is not hia object, but with 

 each stroke ho makes a 

 temporary fulcrum of the 

 water, by which he imparts 

 a smart blow to bis boat, 



and Bends it ahead. The fulcrum is then in the water at one 

 end, the resistance in the middle, and the power at the other 

 end. A nut-cracker furnishes another instance the fulcrum 

 t the joint, the resisting nut in the middle. 



Third Order. Here again the Resistance and Power, aa in 

 Pig. 49, are parallel forces acting in opposite directions, and the 

 condition of equilibrium is the same as in the last order, and for 





Fig. 50. 



a similar reason ; but the mechanical advantage is against the 

 power, which from being nearer the fulcrum must be greater 

 than the resistance. The best examples are found in the limbs 

 of animals. The leg of a horse is a pair of levers with a joint 

 in the middle, which he can make into one or use separately as 

 he likes by means of the muscles attached to them along their 

 lengths. The fulcrum is in the shoulder-joint or the knee-joint, 

 and the resistance is at the hoof when he puts forth his strength 

 to pull a load. 



If a man stretches his arm out straight, and so lifts a weight, 

 that weight is the resistance ; the shoulder Is the fulcrum, and 

 he must put forth a strength by his muscles in the middle 

 greater than the weight before he succeeds in lifting it. If he 

 moves only the lower joint, as in Fig. 50, his elbow ia the 

 fulcrum, and the power is midway. 



It may be asked, Why 

 ever use a lever in which f 



the power is at a mechanical 

 disadvantage ? The answer 

 to be given is, that to lift 

 a large weight by a small 

 foroe is not the only object aimed at in mechanism, natural 

 or artificial. It is as often desirable to give the end of a 

 lever a very rapid motion, and this can be done with most 

 advantage when it is of the third order. Tho amount of force 



Fig. 51. 



h in aooh oases u no consideration in comparison to 



of action, especially in animal mechanic* To strike 



a awif t and amort blow with the cloed hand, or with a word 



in the hand, aa it ia often ntomnry to do, a lever of the third 



order u the moat effective. 



Levers of the vorioua order* are often worked together, mo aa 

 to make compound levern, the resistance end of one working 

 into the powor end of the other. In thin way the effect of a 

 amall power ia often rery largely multiplied, and a very great 

 resistance easily overcome. Such a compound lever is that in 

 Fig. 51, where all are of the first order, three fulcrums at r, r,, r t , 

 a power at P overcoming a resistance at P,, and there multiplied 

 overcoming a second resistance at P,, and this eventually 

 tho Btill greater weight w. Tho power ia multiplied in the nr*t 

 lever inversely aa tho length of the arms, also in the second. 

 and so also in the third. Suppose, for example, the power at v 

 in one pound, and the short arm of each lever a third of the 

 long one, then the 1 pound at P produce* at the end of the long 

 arm of the second lever at P, a force of 3 pounds. This again 

 produces at P, in the third lever 3 times 3, or 9 pounds; and 

 thus 1 pound eventually balances a weight of 27 pounds at w, the 

 mechanical advantage gained by tho combination being 27 to 1. 



But suppose that the lengths of the arms were in the propor- 

 tion of any other numbers in the several levers say 9 to 4 in 

 tho first, 7 to 3 in the second, 5 to 2 in the third ; what weight 

 would 1 pound at P support at w ? It is not difficult to discover, 

 if you know something about multiplying fractions. Now, in 

 the first lever, by the principle of momenta, already explained, 9 

 times the 1 pound at P is equal to 4 times the power produced by 

 that pound in the second lever at P, ; that ia to say, this second 

 power is J of a pound. But this force, for the same reason, 

 is multiplied at P S in the proportion of 7 to 3, and therefore 

 becomes J of J of a pound, and this eventually balance* a 

 weight at w of of J of f of that unit, or, on making the 

 calculation, the I pound balances 13 pounds 2 ounces. And, 

 of course, what ii true of these numbers ia true of all others, 

 and the rule you arrive at is this 



Rule. Multiply together tho fractions which represent the 

 ratios of the Power arms to the Resistance arms, and the product 

 obtained is the number of pounds of the Resistance which each 

 pound of the Power balances. When the Power ia more than 

 1 pound, multiply this number into that of the pounds and 

 fractions of a pound in it. 



And this leads us to another result, which expresses the rela- 

 tion between the power and resistance without fractions. Since, 

 in the above example, we had the resistance equal to | of 

 I of J of the power, it is evident that the three denominators 

 multiplied into the resistance must be equal to the three nume- 

 rators into the power, and thus, extending the principle, we 

 may say that 



The Power multiplied by the several lengths of the Power 

 arms is equal to the Resistance multiplied by those of the Resist- 

 ance arms. 



And yon thus have a result not unlike that established above 

 for a single lever. And observe that this, though proved above 

 only for a combination of levers of the first order, holds equally 

 good of other combinations, mixed or unmixed, all of the second, 

 or all of the third, or of two kinds, or of all three together. 

 The principle of momenta is true for each kind, and therefore 

 for their combinations. For thia reason I have above avoided, 

 in the statement of the general principle, the terms " long arm " 

 and " short arm," but used instead " power arm " and " resist- 

 ance arm," indicating thereby the arms that work with the 

 power or \uith the resistance. 



The example of a combination of levers which ia most likely 

 to interest you, is the tsommon weighing machine, used for 

 weighing loaded market carts, or luggage at railway stations. 

 In Fig. 52 is a ground-plan of this piece of mechanism, 

 where at A, B, c, D, the four corners of the bottom of a 

 shallow box, are the fnlcrums of four levers of the second 

 order, which meet, two and two, on either side at r, and 

 are joined across by a stout steel pin, by which they are 

 also connected with tho lover of the second order, o, 

 which has its fulcrum at . The end. o, of this lever is 

 connected by a rod which ascends perpendicularly from the 

 ground, and ia attached above to the short arm of another 

 lever one of tho first order, generally a steelyard, to be 

 afterwards described to tho longer arm of which the weighing 



