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THE POPULAR EDUCATOR. 



MECHANICS. VIII. 



INTRODUCTION TO THE MECHANICAL POWERS. 

 IN the account given of parallel forces in Lesson IV., such only 

 were considered as act in the same direction, pull or push 

 together, each adding to the effect of every other ; and of these 

 the subject of the centre of gravity in Lessons V. and VI. fur- 

 nished numerous exemplifications, the forces all pulling towards 



the earth's centre. 



A Q Now you have 



-^* ****- to consider two 



forces, unequal 

 and parallel, but 

 acting in opposite 

 directions. 



Suppose two 

 such applied to a 

 body, as in Fig. 43, 

 where A and B are 

 the points of ap- 

 plication, and the 

 arrows A P, B Q, 

 represent their 



magnitudes and directions. Let A p be 7 pounds and B Q 

 3 pounds ; how can we find their resultant ? From a very 

 simple consideration. Whatever it be, or at whatever point 

 it acts, it must be such that a force at that point, equal and 

 opposite to it, will balance it, and therefore make equilibrium 

 with its components A P, B Q. Now, that point cannot be 

 inside the line A B, for in that case the resultant of the two 

 which pull together could not be opposite to the third. The 

 point must, therefore, be outside A B, and on the side of the 

 greater force A p. Let the point, therefore, be o, and o R the 

 resultant, o s being the force equal and opposite to it, which 

 makes equilibrium with A p and B Q. 



Then, since there is equilibrium, the resultant of the two that 

 pull together, B Q and o s, must be equal and opposite to A p ; 

 and therefore, as proved in Lesson IV., A P is the sum of B Q 

 and o s. But A P being 7 pounds, and B Q 3 pounds, o a must 

 be 4 pounds, the difference ot these forces. The resultant in 

 magnitude, therefore, is the difference of the components. 



Now for the point of application. Since the resultant of 4 

 I founds at o and 3 pounds at B must cut B o at A inversely as the 

 forces, if I divide A B into four equal parts, three of them will 

 be in A o ; or, which is the same thing, seven parts in B o and 

 three parts in A o, showing that o is the point whose distances 

 from A and B are inversely as the forces. Putting all together, 

 wo learn that 1. The Eesultant of two Unequal Parallel 

 Forces which act at two points of a body in opposite directions 

 is equal in magnitude to their difference. 2. Its point of appli- 

 cation is outside of the greater force, at distances from the 

 points of application of the components, which are inversely as 

 these forces. 



The rule to be observed practically in finding this centre is. 

 to cut A B into as many equal parts as there are pounds, or 

 other units, or fractions of a unit, in the difference of the forces, 

 and then to measure outwards from A along the production of 

 A B as many of these parts as there are pounds or other units 

 in B Q ; the point o so obtained is the parallel centre required. 

 And you see that what is thus proved for the numbers 3 and 7 

 must hold equally for other numbers, whatever they be. 



There is one particular case of this principle, which I shall 

 jusfc notice. Suppose A p becomes equal to B Q ; what of their 

 resultant P how large is it, and where applied P In magnitude 

 it is nothing, being the difference of the forces ; and the point 

 of application is nowhere, at least within reach ; for on A B 

 produced no point o can be found such that A o be equal to 

 B o. Pairs of forces of this kind are termed " couples," and 

 they play an important part in Mechanics, in producing a ten- 

 dency to rotation ; but we shall not consider them here. 



One consequence more : How find the resultant of any num- 

 ber of parallel forces, some acting in one direction, others in the 

 opposite ? Evidently by compounding separately, and finding 

 the centres of, those which act in the opposite directions. Tou 

 thus get two single parallel and opposite forces the resultants 

 of the opposing sets, and their centres of application ; and there- 

 1'ore, by the aid of the principle above established, learn that 

 1. The Eesultant of a system of Parallel Forces, which act, 



some in one direction, others in the opposite, is in magnitude 

 the Difference of the Sums of the Opposing sets of Forces. 



2. Its Point of Application is had by finding the parallel 

 centre of each opposing set, and taking a point on the side of 

 the greater sum, on the production of the line joining these 

 centres whose distances from these points are inversely as the 

 sums of the opposing forces. 



For example : Suppose eight parallel forces are applied to the 

 eight corners of a box, five of 2, 4, 6, 7, and 9 pounds directed 

 to the east, and three of 10, 11, and 15 pounds to the west ; 

 the resultant will be 8 pounds, acting towards the west and at 

 a point on the line joining the parallel centres of the two sets, 

 and outside the greater, whose distances from these centres are 

 inversely as 36 to 28. 



These principles, with others previously established, we now 

 apply to the Lever ; first taking the cases in which the forces, 

 usually termed the "Power" and the "Resistance," or 

 " Weight," are parallel. The principle of leverage may be un- 

 derstood by the aid of Fig. 44. Two balls, say of iron, connected 

 by a thin bar, 

 are supported 

 by a cord at a 

 point o. How is 

 this point to be 

 selected so that 

 the balls may 

 equally balance 

 each other, the 

 weight of the rod 

 not being taken 

 into consideration? Again, having recourse to numbers, let tho 

 balls be 13 pounds and 4 pounds, and their centres the points A 

 and B ; how is o to be found ? Evidently by cutting A B so that 

 A o be to BO inversely as 13 to 4 ; or, on dividing that line 

 into seventeen equal parts, so that four of them be in A o and 

 thirteen in B o. If the bar be supported by the cord from 

 above, or by a prop from below, at this point there is equilibrium. 

 This is the principle of the Lever, of which the ball B may be 

 considered the Power, and the ball A the Resistance. We say, 

 therefore, that the support, or prop, commonly called the 

 "fulcrum," must be so placed that the arms A o, B o of the 

 lever on each side of it be to one another inversely as the 

 Power and Resistance. 



But, as inverse ratio puzzles some persons, I shall put the 

 matter in another light. You observe that at the end, A, 

 of this lever, there are only 4 equal parts in the arm, but 

 13 pounds in the resistance, while in the arm B o the parts 

 are 13, and the pounds only 4. Now, suppose the parts were 

 all inches, then if you at either end multiply the number of 

 inches in an arm by the number of pounds on that arm, you 

 get the same number namely, 52, for product. Choose any 

 other numbers different for 13 and 4, and the result is the 

 same ; the numbers at either end multiplied together give 

 the same product. Therefore, another way of stating the 

 Condition of Equilibrium in a lever is, that the product of 

 the Power and arm on one side should be equal to that of 

 the Resistance and arm on the other. 



But here be careful to be clear as to what is meant by " the 

 product of Power and arm, Resistance and arm." This puzzles 

 some persons extremely, from its never being clearly explained 

 to them. Strictly speaking, the product of a force and a line, 

 or of a resistance and an arm, is nonsense. Multiply a bag 

 of flour by the iron beam from the end of which it hangs, and 

 who can divine what the result of the operation is to be P 

 Neither flour nor iron, but something between ! Well, then, to 

 remove every possibility of confusion on this point, keep in 

 mind (as tho example above shows) that we multiply numbers 

 only, not the Power and its arm, or the Resistance and its arm, 

 but the NUMBER which denotes the units of FOKCE in one, ly the 

 NUMBER which denotes the units of LENGTH in the other. Then 

 you can make no mistake ; there will be no confusion ; and you 

 can still say, knowing the meaning of your words, that the 

 Power multiplied by its arm is equal to the Resistance multi- 

 plied by the other arm. This product is commonly termed the 

 " Moment" of the Power or Resistance, and the Condition of 

 Equilibrium is stated as follows : 



For Equilibrium in a Lever the Moments of the Power, with 

 reference to the fulcrum, and Resistance should be equal. 



