MECHANICS. 



lines, which, as you will see, when many forces havo to bo com- 



i.-d, \vould i-:uiM> imirii H ynur figures. 



supply this pnii. -i]. If in.w to compound any number of 

 1 tin-re In- : ..1 illui*- 



. < rulo OB well u a thousand could. Suppose forces, o A, 



o i 



bo tin- juiiiit 



tint triangular rule, if I 

 draw A B equal and 

 parallel to o B, the line 

 joining <> with u iri the 



?1 ^ B3 resultant of the first two 

 forces. I shall not actu- 

 ally draw this line, o B ; 

 let us suppose it drawn. 



Fig. 6. Now, if I compound this 



resultant with o c, I havo 

 tho resultant of three of 

 the forces. But that, by 



the same rule, is got by drawing from B a lino R it, equal 

 and parallel to O C. The line o B is this resultant of three. 

 Again we shall not draw it. The resultant of this and o D, 

 for tho same reason, would be o B 2 , got by drawing R, K 2 

 parallel and equal to o D, and, lastly, tho resultant of this 

 and o E would be o B 3 , the lino, B a B 3 , being equal and 

 parallel to o E. We have thus exhausted all tho forces, 

 and evidently o B 3 is tho resultant of the whole five. There 

 was here no confusing ourselves with parallelograms ; all wo 

 had to do was to draw line after line, one attached to the 

 other, carefully observing to keep their magnitudes and difec- 

 tions aright. A kind of unfinished polygon was thus formed, and 

 the line o B 3 , which closes up tho polygon, joining tho last point 

 B S , with the point of application, is the resultant in magnitude 

 and in direction. Thus you have made another step in advance, 

 and arrived at the Polygon of Forces. You have learned how, 

 by the mere careful drawing of linos, to determine the resultant 

 of any number of forces. All you require is paper and pencil, 

 a rule, compasses, a scale, and a pair of parallel rulers. 



Now, there is one point about this polygon I wish you 

 carefully to note. You will observe that the arrows on its 

 ides, representing tho directions of the forces you have com- 

 pounded, all point from left to right, as you go round the figure, 

 turning it with you so as to bring each side in succession to tho 

 top. The resultant, however, points in the opposite direction, 

 from right to left, when that side is uppermost. This is as it 

 should be ; the direction of tho resultant, as you go round the 

 figure, must bo opposite to those of the components. The use of 

 this you will see in tho next lesson. 



Now, let us suppose that, in determining tho resultant after 

 this method, as we come to the end of the operation, the end, 

 BJ, of the lost line, K 2 B 3 , chanced to coincide with, or fall upon 

 the point of application, o. The polygon would close of itself 

 without any joining line ; what is the meaning of this ? It 

 means that there is no resultant ; the line, o B 3 , is nothing, and 

 therefore the resultant is nothing, and tho forces produce equili- 

 brium. What a valuable result we have arrived at ! A method 

 by which we can, l>y rule and compass, tell at once whether any 

 number of forces make equilibrium at a point or not. All we 

 have to do is to describe the polygon of forces, and if it closes' 

 up of itself, there is equilibrium ; if it does not, there cannot bo 

 equilibrium, and tho resultant is in magnitude the side which is 

 necessary to close the figure. 



Deferring tho further expansion of this subject to the next 

 lesson, I shall now turn back and apply these principles to a few 

 elementary examples. 



First Example. Three equal forces act at a point in different 

 directions what condition should they fulfil in order to be in 

 equilibrium ? Get your ruler and compass, and commence 

 constructing the figure by which their resultant may bo found. 

 From the end of one of tho forces you are to draw a line equal 

 and parallel to the second equal force, and from the end of that 

 another line, equal and parallel to tho third. You will thus 

 have three lines strung together, all equal to each other. But 

 if the forces are in equilibrium, the end of the last line must 

 fall on the point of application, that is to say, the polygon of 

 forces must close up, and form a triangle. Your construction 

 will then give you a triangle of three equal sides, commonly 

 called an equilateral triangle. But such a triangle must have 



all iU angle* equal ; also tho angle* between the sides of tho 

 triangle polygon of forces, are the angle* between the 



forces themselves, since they are parallel to these force*. Thin 

 in evident from the propertied, 1 and 2, of the parallelogram 

 referred to above ; therefore, in the CAM we are considering, the 

 three equal force* must act at equal angle*, as I showed othea 

 wise must be the case at the close of the hurt lesion. 



Second Example. Let a weight hang from tho ceiling by 

 means of two cords of unequal length, as in Fig. 7. 

 evidently at rest. Whatever bo the forces called into action, 

 thi-y i>rodu<-<: c<(tiilil<riiun. Ii th- : nothing further to ascertain? 

 There in ; it may bo desirable to know by how much each cord 

 is strained. Our assurance that tho cords will support the 

 weight depends on this knowledge. Let P and Q be tho two 

 points of support of tho strings which meet at o. Now, what- 

 ever be the (strains on tho cords, o P, o Q, they make equilibrium 

 witli \v, the weight. Therefore, if we suppose a length, o A, of 

 o P to represent the strain 

 on o r, and from A draw a 

 line, A B, parallel to o Q, 

 equal to the strain, o B, 

 on O Q, then, since the three 

 forces are in equilibrium, 

 tho lino, B o, closing uj tho 

 triangle must be equal to, 

 and bo in tho direction as, 

 tho third force, or weight, w. 

 Tlu's, then, tells us what to 

 do. Measure on o B upward 

 as many inches as there are 

 pounds in w ; and from B 

 then draw B A parallel to Fig. 



the cord o Q to meet the cord o A. The number of inches in 

 o A will represent in pounds the strain en o P, and those on 

 B A the strain on o Q. All, therefore, that we desire to know is 

 determined. 



Third Example. A horse pulls a roller up a smooth inclined 

 plane or slope ; what is the force he must exert when he just 

 keeps the roller at rest? And by how much does the roller 

 press on the plane ? 



Let tho horse pull in any direction, o A. Then there will be 

 three forces acting on tho roller ; namely, its own weight right 

 downwards, the horse's pull, and the resistance of the plane or 

 slope, perpendicular to itself. There must be this third force, 

 for the other two, B 



not being opposite 

 to each other, can- 

 not make equili- 

 brium. The roller 

 is somehow sup- 

 ported by the plane; 

 and that it cannot 

 be unless by its re- 

 sistance ; and a 

 plane cannot resist 

 except perpendicu- 

 larly to itself . This 

 third force, you 

 thus see, must be 



Fig. 8. 



taken perpendicular to the plane. It is represented in th* 

 figure by o B. Now apply the polygon of forces. Let o o 

 represent tho weight of tho roller, and from c suppose a 

 line, c B, drawn equal and parallel to o A, the horse's pull. 

 Then, sinco there is equilibrium, the polygon of forces should 

 close up and become a triangle that is, the line joining B 

 with o should bo the pressure, and therefore should be per- 

 perdicular to the plane. What then are we to do? Take 



c, equal in inches to the number of pounds in the roller, 

 draw then from c a line c B parallel to tho horse's pull, to 

 meet the line drawn from the centre o of th<> roller perpen- 

 dicularly to the plane ; c B so determined will in inches tell 

 the pounds in the horse's pull, and o B tho amount by which 

 the roller presses the plane. You can easily see from this that 

 as the slope increases the pull will increase and the pressure, 

 diminish. This is what naturally we should expect. Tho plane 



1 havo supposed to be smooth ; for, where there is friction 

 against the roller caused by roughness in itself or in the plane, ot 

 in both, the question is much altered, as in dun time yon will see. 



