18 RESISTANCE OF MATERIALS 



The total weight W of all the particles is 



W= tv + w + - - 4- w n = w ; 



that is, W is the resultant of the n parallel forces w^ w^ . ., w n . 

 The location of this resultant W may be determined by applying 

 the theorem of moments. Thus, let x^ # 2 , , x n denote the perpen- 

 dicular distances of w^ w^ , w n from 

 any fixed point (Fig. il). Then, if 



&r 3 x ^ Denotes th e perpendicular distance 



*IHf?rJJ?5a.- ____ __^)P 4 of the resultant W from 0, by the 



| theorem of moments 



W 



whence x = 



FIG. 11 



or, since W= 2) w -> this may also be written 

 (10) tf = 



This relation determines the line of action of W for the given 

 position of the system. If, now, the system is turned through any 

 angle in its plane, and the process repeated, a new line of action 

 for W will be determined. The point of intersection of two such 

 lines is called the center of gravity of the system. From the method 

 of determining this point it is evident that if the entire weight of 

 the system was concentrated at its center of gravity, this single 

 weight, or force, would be equivalent to the given system of forces, 

 no matter what the position of the system might be. 



If the particles do not all lie in the same plane, a reference plane 

 must be drawn through instead of a reference line. In this case 



^-\ 



the equation x = ^ determines the position of a plane in which 







the resultant force W must lie. The intersection of three such 

 planes corresponding to different positions of the system of particles 

 will then determine a point which is the required center of gravity. 



