PROBLEMS 



PROBLEM 2. RESULTANT OF NON-CONCURRENT FORCES. 



(a) Problem. Given the following non-concurrent forces: 



720 270 350 



*>,-- (7-0", 6.3"), P, t ;(o.5", 9-3"), P~ ;- (7-o", 6.6"), 

 150 o 205 



210 



F 4 (0.8", 8.5"). 



Forces are given in pounds. Find resultant, R, by means of force 

 and equilibrium polygons. Check by calculating R by means of a new 

 force polygon and a new equilibrium polygon. Also check as described 

 below. Scale for force polygon, i" = 100 Ibs. 



(b) Methods. Start force polygon at (7.0", i.o"). Take pole 

 at (3.8", o.o"). Start equilibrium polygon at (7.0", 6.3"). Take new 

 pole at (2.2", o.o"), and draw new polygon starting at (7.0", 6.3"). 



(c) Results. The resultant is a force, R, and acts through the 

 intersection of the strings d, e and d' , e' , and is parallel to the closing 

 line in the force polygon. If corresponding strings in the equilibrium 

 are produced to an intersection, the points of intersection, i, 2, 3, 4, 5, 

 will lie in a straight line which will be parallel to the line O-O r joining 

 the poles of the force polygons. This relation is due to the reciprocal 

 nature of the force and equilibrium polygons, and may be proved as 

 follows : In the force polygon the force P 4 may be resolved into the 

 rays c and d, it may likewise be resolved into the rays c' and d'. In 

 like manner it may be seen that the force O-O' can be resolved into the 

 d and d', or into c and c'. Now if the strings d and d' are drawn 

 through the point 4, and the strings c and c' are drawn, they must in- 

 tersect in the point 3, and 4-3 must be parallel to O-O'. For the re- 

 sultant of d and d f is equal to O-O' and must act in a line parallel to 

 O-O'; likewise the resultant of c and c' is equal to O-O' and must 

 act parallel to O-O'; and in order to have equilibrium 3-4 must be 

 parallel to O-O'. 



In like manner it may be proved that I, 2, 3, 4, 5, are in a straight 

 line parallel to O-O'. 



From the above it will be seen that to have equilibrium in a sys- 

 tem of non-concurrent forces it is necessary that the force polygon and 

 its corresponding equilibrium polygons must close, or that two equi- 

 librium polygons must close. 



