35 



FORCE POLYGON. 



41 



The concurrent forces may be reduced to a single force R. 

 See Fig. 26 (c) and Art. 35. 



Any number of non-concurrent forces are, in general, equiv- 

 alent to a single force and a couple. These are again reducible 

 to a single force unless R = 0, in which case the resultant is a 

 couple. 



In the special case when R = 0, the resultant is a couple, 

 and in the special case when M = 0, the resultant is a single 

 force (30). 



FOR COMPLETE EQUILIBRIUM 



R = and M = 0. 



These conditions will now be shown graphically, remember- 

 ing that R refers to the concurrent forces and M to the couples. 



35. Force Polygon. The resultant R will now be found, 

 that is, the resultant of half the forces at in Fig. 26. This may 

 be done by means of the force triangle (32). In Fig. 26 (c) the 

 resultant of P a and P 2 is R r ; of /t t and P 3 is #2; of R 2 and P 4 is 

 R. The given forces are not in equilibrium since their resultant 

 is not zero. The equilibrant is ea. It is important to note that 

 the given forces must be drawn so as to follow each other in 

 direction in each triangle, and this results in having their direc- 

 tions the same around the polygon abcdea, when E is one of them. 

 This polygon is the FORCE POLYGON and it follows that, 



When a number of forces form a closed polygon, they are in 

 equilibrium if their directions around the polygon are the same 

 (clockiuise or anti-clockwise), and any one is the equilibrant of 

 all the others. If any one force has an opposite direction to that 

 of all the others, it is their resultant. 



It is evidently not necessary, in drawing a force polygon, to 

 draw the intermediate resultants, E^, R 2 , etc., but only to lay out, 

 in magnitude and direction, one force at the end of the preceding 

 one. If the polygon closes the forces are in equilibrium; if not, 

 the closing line will be the equilibrant or resultant. If the forces 

 had been numbered in a different order, the appearance of the 

 polygon would have been entirely different, but the equilibrant 

 would have been the same (Fig. 26c). 



It remains to find the value of M if the forces are non- 

 concurrent. If they are concurrent the point (Fig. 26) may 

 be taken at their common point of intersection, in which case the 

 arms a of the couples all become zero and M = 0. For concurrent 

 forces, therefore, the number of independent equations of equili- 

 brium is one less than for non-concurrent forces. 



