84 



THE POPULAR EDUCATOR. 



Fisr, 1. 



denote the putting together, compounding, or finding the resul- 

 tant of any number of forces, and by the latter the separating, 

 or resolving, of any given force into the two or more to which it 

 may be considered equivalent. The composition we first 

 Consider j but this requires a short digression on 



THE PARALLELOGRAM. 



The resultant of two forces is found by the aid of the " paral- 

 lelogram of forces ; " and, as some of you may not know what a 

 parallelogram precisely is, I shall explain the term, and tell you 

 a few things about it which, in Mechanics, it is desirable you 

 should know. 



A parallelogram is a four-sided figure whose pairs of opposite 

 sides and opposite angles are equal. In the adj oining figure, A B c D 



is a parallelogram, if the 

 side A B is equal to D c, 

 and also B c to A D. The 

 two cross lines, A c and 

 B D, are called the "dia- 

 gonals of the parallelo- 

 gram." Now, if you ex- 

 amine the two triangles, 

 A B C, A D C, which are 

 on opposite sides of the diagonal, A c, you will see reason for 

 believing that thpy must be equal to each other. They are, 

 in fact, the sawuv'triangle on opposite sides of that line ; for 

 they have A c for a common side, and the two other pairs of 

 sides are equal, namely, A B equal to D 0, and A r> to B c ; and you 

 cannot out of three straight lines make two different triangles. 

 This you can satisfy yourselves of by experiment, by putting three 

 rods of different lengths together so as to form a closed figure. 



Now, the point to which I am trying to lead you, and which 

 you will soon find of importance, is that, since these triangles 

 are equal in fact, one and the same triangle in two positions 

 their angles must be equal to each other. Hence we arrive at 

 the following important properties of a parallelogram : 



1. That the opposite angles, ABC and ADC, are equal, also 

 the opposite angles, BAD and BCD. 



2. That either diagonal makes equal angles with the pairs of 

 opposite sides, A B D equal to c D B, and A D B equal to c B D. 



It is on account of this latter property the figure is called 

 " parallelogram." The opposite sides are not only equal, but 

 parallel, on account of their making equal angles with either 

 diagonal. However, keep in mind that these angles are equal, 

 for this knowledge is necessary to your properly understanding 

 what we next come to, namely 



THE PARALLELOGRAM OF FORCES. 



The forces in our cuts and diagrams being represented, as 

 agreed on, by lines, and their directions by arrow-heads attached 

 to their remote ends, this principle may be stated as follows : 

 If two forces applied to a point are represented in magnitude 

 and direction by two straight lines, their resultant is represented 

 in magnitude and direction by the diagonal passing through 

 that point of the parallelogram of which these lines are two 

 adjoining sides. 



In Fig. 2 let o P, o Q be the two forces, and draw from p and Q 

 the two dotted lines parallel to them which meet in R, then the 

 dotted diagonal, o R, of the parallelogram thus formed is the 

 resultant, both in magnitude and direction, of o p and o Q. 



Now, I shall not here 

 give you the strict ma- 

 thematical proof of. this 

 proposition ; it is too 

 complicated, and involves 

 so much close reasoning, 

 that to force it on a 

 student in the begin- 

 ning of a treatise on me- 



Fig. 2. 



chanics would be tc throw an unnecessary difficulty in his way. 

 The best course is to defer it until you have become more 

 accustomed to mechanical reasoning, and then return to it. In 

 the meantime you can satisfy yourselves that it is true by a 

 reference to the two following experiments, one derived from 

 equilibrium, the other from motion. 



First Experiment. Let three weights, tr v w, be attached to 

 three cordfl, as in Fig. 3, which are knotted together at o ; and 

 let two of the cards, longer than the third, with their at- 



3. 



tached weights, be thrown over two pulleys, p Q, which move 



freely in the same plane round axles fastened into a wall or 



upright board. Arrange, then, the weights and cords until 



equilibrium is produced. It is evident, from the principle stated 



at the close of the last lesson, that the force, w, must bt- aqual 



and opposite to the resultant of u and v, acting over the 



pulleys at o. Now, take on p 



the cord o p, a length o A, 



equal in inches to the num- 



ber of pounds in u, and on 



o Q another, o B, equal to 



the pounds in v, and then 



draw the parallels, A R and 



B R, to O p and O Q, meeting 



in R ; O R will then be the 



resultant of u and v, if the 



principle of the parallelogram 



of forces be true. It should, 



therefore, be opposite in di- 



rection to the force w, and 



the number of inches in it 



should be equal to the num- 



ber of pounds in w. Now, 



on trial it is found that O R 



is opposite to w, that is to say, that it points vertically 



upwards in the plomb-line ; and it is also found that the number 



of inches in its length is that of the pounds in w. 



Second Experiment. Let us suppose that a parallelogram 

 O A R B is described anyhow on a perfectly smooth horizontal 

 table, and that at the point O, two springs are fitted so that one 

 of them, on being let go, would make the unit ivory ball move 

 over o A in the same time that the other would make it move over 

 o B. It is evident that the lines o A and o B would then represent 

 these forces. Furthermore, it should follow, if the principle of 

 the parallelogram of forces be true, that, when both springs are 

 let go together so as together to strike the ball, it should move 

 over the dia- 

 gonal o B of 

 the parallelo- 

 gram in the 

 same time as 

 the ball moved 

 over o A and 

 OBwhenstruck 

 s e parately. 

 Now, this is 

 what, on trial, 

 exactly hap- 



- 4 - 



pens. The ball does move over the diagonal, and moves over 

 it in the same time that it previously moved over the sides. 

 This it could not do if the resultant of two forces was not repre- 

 sented in magnitude and direction by the diagonal. Instruments 

 are fitted up for lecture-rooms by which the experiment can be 

 made, and the result always is as I have stated. 



Taking the principle, then, as established, let us observe its 

 consequences. You are given two forces, acting at a point, and 

 you want their resultant. Make, you will immediately say, a 

 parallelogram of the two forces, and the diagonal is the required 

 line. Not so fast ; you need not describe the whole of that 

 figure, a part will suffice. Now, if from the end A of o A, you 

 draw A R parallel and A n 



equal to o B, it is clear 

 you do not want to draw 

 B R at all. A R gives you 

 the far end of the result- 

 ant, and all you have to 

 do then is to join R with 

 o, and your object is o p. B 



gained. Thus your paral- 

 lelogram of forces suddenly becomes a triangle of forces ; an 1 

 you may lay this down as your rule in future for compounding 

 two forces. 



Draw from the extremity of one of the forces a line equal, 

 and parallel to, the other force; and the third side of the 

 triangle so formed by joining the end of this line with the point 

 of application is the resultant. 



There is great advantage in this substitution of the triangle 

 for the parallelogram, for it saves the drawing of unnecessary 



