284 



THE POPULAR EDUCATOR. 



Fig. 47. 

 Second Order. This is a no less important class ; 



forces, acting downwards. A poker put into a fire to raise 

 the coals is also an example, the bar of the grate being the 

 fulcrum ; the handle by which a pump is worked is another. A 

 pair of scissors is a double lever of this kind, of which the 

 connecting rivet is the fulcrum, the force of the finger 

 and thumb at one end being the power which overcomes the 

 resistance of the cloth to be out. A gardener at work with his 



spade is also a familiar il- 

 lustration. After he has 

 driven it into the ground 

 he forces the handle down- 

 wards, making a temporary 

 fulcrum of the harder earth 

 at its back. In all the 

 principle is the same. 



but the 



Power and Eesistance, not as in the former case, act in opposite 

 directions, as in Fig. 47 ; and this accounts for the fulcrum 

 having both these forces on one side of it, for, as I have shown 

 in the last lesson, the forces being opposite, the resultant, which, 

 for equilibrium, must pass through the fulcrum, cannot lie between 

 them. Moreover, as it has been shown there that the distances 

 of o from A and B (see Fig. 44, page 250) are inversely as 

 the forces, so here the distances P F and w p must be inversely 

 as the power and resistance, or, what is equivalent, the power 

 multiplied by its arm p p is equal to the weight multiplied by 

 its arm w p. In this order of 

 levers, as in the former, it should 

 be observed that there is a 

 mechanical advantage gained 

 a larger weight at w is overcome 

 by a lesser at p, a result always 

 to be secured where the larger 

 arm can be given to the power. 



Fig. 48. 



As an example of this lever, take the crowbar in the illustration 

 in Fig. 48, used differently from that in Fig. 46. The workman 

 makes the ground at the point of his bar his fulcrum, 

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

 of pushing downwards with his hand, lifts upwards. The 

 mechanical advantage is clearly on his side. The oar of a boat 

 is also a lever of the second order ; the arms of the oarsman 

 furnish the power ; but most persons at first imagine that the 

 rowlock is the 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 the 

 water about with the blade 

 is not his object, but with 

 each stroke he makes a 

 temporary fulcrum of the 



F . jg water, by which he imparts 



a smart blow to his boat, 



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

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

 end. A nut-cracker furnishes another instance the fulcrum 

 at the joint, the resisting nut in the middle. 



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

 Fig. 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 poll 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 tho middle 

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

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

 fulcrum, and the power is midway. 



It may be asked, Why 

 ever use a lever in which 

 the power is at a mechanical 

 disadvantage ? The answer 

 to be given is, that to lift 

 a large weight by a small 



Fig. 51. 



force 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. The amount of force 

 put forth in such cases is no consideration in comparison to 

 rapidity of action, especially in animal mechanics. To strike 

 a swift and smart blow with the closed hand, or with a sword 

 in the hand, as it is often necessary to do, a lever of the third 

 order is the most effective. 



Levers of the various orders are often worked togetlier, so as 

 to make compound levers, the resistance end of one working 

 into the power end of the other. In this way the effect of a 

 small power is often very 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 p, P t , P., 

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

 overcoming a second resistance at P 2 , and this eventually lifting 

 the still greater weight w. The power is multiplied in the first 

 lever inversely as the length of the arms, also in the second, 

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

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

 long one, then the 1 pound at p produces at the end of the long 

 arm of the second lever at P x 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 the 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 

 the 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 moments, 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 PJ ; that is to say, this second 

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

 is multiplied at P 2 in the proportion of 7 to 3, and therefore 

 becomes J of g of a pound, and this eventually balances a 

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

 calculation, the 1 pound balances 13 pounds 2 ounces. And, 

 of course, what 13 true of these numbers is true of all others, 

 and the rule you arrive at is this 



Rule. Multiply together the fractions which represent tho 

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

 obtained is the number of pounds of the Eesistance which each 

 pound of the Power balances. When the Power is 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 

 | of | 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 Eesistance multiplied by those of the Eesist- 

 ance arms. 



And you thus have a result not unlike that established above 

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

 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. 



