164 



STEENGTH OF MATERIALS 



Since the position of the pole is entirely arbitrary, there is an 

 infinite number of equilibrium polygons corresponding to any jjivrn 

 set of forces. The position and magnitude of the resultant R, how- 

 ever, is independent of the choice of the pole, and will be the same, 

 no matter where is placed. 



Problem 157. The ends of a cord are fastened to supports and weights attached 

 at different points of its length. Show that the position assumed by the string is 

 the equilibrium polygon for the given system of loads. 



132. Application of equilibrium polygon to determining reactions. 



One of the principal applications of the equilibrium polygon is in 

 determining the unknown reactions of a beam or truss. To illustrate 

 its use for this purpose, consider a simple beam placed horizon- 

 tally and bearing a number of vertical loads P lt P 2 , etc. (Fig. \'1\ >. 

 To determine the reactions R^ and R z , the force diagram is first 



FIG. 121 



constructed by laying off the loads P lt P 2 , etc., to scale on a line AF, 

 choosing any convenient point as pole and drawing the rays OA> 

 OB, etc. The equilibrium polygon corresponding to this force diagram 

 is then constructed, starting from any point, say A', in R r 



Now the closing side A'G 1 of the equilibrium polygon determines 

 the line of action of the resultants P 1 and P" at A' and G f respectively. 

 For a simple beam, however, the reactions are vertical. Therefore, in 

 order to find these reactions each of the forces P r and P" must be 

 resolved into two components, one of which shall be vertical. To 

 accomplish this, suppose that a line Off is drawn from the pole in 



