MECHANICS. 



mean a point " in a body ; " aud that will save me always 

 ii'lding tin- I:ittr words when I use the former. Of 

 forces applied to "a inali'riul point" arc included in tho 

 : inn, and thoso you will find to be of groat importance. 



j.iint offoot of two or more forces so applied is termed 

 fln-ir ' " no wo name tho separate forcesof whii-h it in 



the effect its components. There are thus two operation*, tho 

 a of Forces, and tho Resolution of Forces, with which 

 i y bo concerned : by tho former wo denote the putting to- 

 gether, compounding, or finding tho resultant 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 bo considered 

 equivalent. Tho Composition wo first consider ; but this re- 

 quires a short digression on tho nature of a parallelogram. 



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

 wdes and opposite angleo are equal. In tho odj oining figure, A u < i > 



is a parallelogram, if tho 

 side A B is equal to D c, 

 and also B c to A D. Tho 

 two cross lines, A c and 

 B D, are called tho " dia- 

 gonals of the parallelo- 

 gram." Now, if you ex- 

 amine the two triangles, 

 A B c, A D c, which are 



Fig. 1. 



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

 lu'lii'ving that they must be equal to each other. They are, 

 in fact, the same 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 c, and A D to B c ; and you 

 cannot out of three straight lines make two different triangles. 

 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 bo equal to each other. Hence we arrive at 

 the following important properties of a parallelogram : 



1. That tho opposite angles, ABC aud ADC, are equal, also 

 the opposite angles, BAD aud 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 

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

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

 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 



Pi 2 <* student in the begin- 



ning of a treatise on me- 

 chanics would be to 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 tho two following experiments, one derived from 

 equilibrium, tho other from motion. 



Pint Experiment. Let three weights, u v w, be attached to 

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

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



Fig. 3. 



t*ohed weight*, be thrown over two pulley*, f Q, which more 

 frooly in tliu i-am> piano round axles fattened into a wall or 

 upright board. Arrange, then, tho weight* and oordi until 

 r.|ii:lii,riii!n in produced. It w evident, from the principle stated 

 at tho close of tho hut Won, that the force, w, mtut be equal 

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

 pulleys at o. Now, take on 

 tho cord o r, a length o A, 

 equal in im-in-- to the num- 

 ber of pounds in u, and on 

 o Q another, o B, equal to 

 tho 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 tho num- 

 ber of pounds in w. Now, 

 on trial it is found that o B 

 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 tho 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 tho lines o A and o B would then represent 

 those 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 R of 

 the parallelo- 

 gram in the 

 same time as 

 the ball moved 

 over o A and 

 OB whenstruck 

 s e paratoly. 

 Now, this is 

 what, on trial, 

 exactly hap- 



Fig. 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 tho two forces, and the diagonal is the required 

 line. Not so fast ; yon 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 

 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 . B 



gained. Thus your paral- gf ' 



lelogram of forces suddenly becomes a triangle of forces ; and 

 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 tho triangle 

 for the parallelogram, for it saves the drawing of unnecessary 



