FIRST AND SECOND MOMENTS 



17 



If the forces F 1 and F 2 are parallel, introduce two equal and 

 opposite forces If, - H, as shown in Fig. 10, and combine the TTs 

 with F 1 and F 2 into resultants F(, F 2 . Transferring these resultants 

 FV F 2 to their point of intersection 0, they may now be resolved 

 into their original components, giving two equal and opposite forces, 

 -h H and - H, which cancel, and a resultant F l + F 2 parallel to F l 

 and F 2 . 



Moreover, applying the theorem of moments proved above to the 

 concurrent forces F^ F 2 at 0, the sum of their moments about any 

 point is equal to the moment 

 of their resultant F 1 + F 2 

 about the same point. But 

 the moment of F[ is equal 

 to the sum of the moments 

 of F^ and H, and, simi- 

 larly, the moment of F 2 is 

 equal to the sum of the 

 moments of F 2 and + H. 

 Since the forces + H and 

 - H have the same line 

 of action, then* moments 

 about any point cancel, and therefore the theorem of moments is 

 also valid for parallel forces. 



This theorem may obviously be extended to any number of forces 

 by combining the moments of any two of them into a resultant 

 moment, combining this resultant moment with the moment of the 

 third force, etc. Hence, 



The sum of the moments of any number of forces lying in the same 

 plane with respect to a given point in this plane is equal to the moment 

 of their resultant with respect to this point. 



12. Center of gravity. An important application of the theorem 

 of moments arises in considering a system of particles lying in the 

 same plane and rigidly connected. The weights w^ w 2 , ,> of the 

 particles are forces directed toward the center of the earth. Since 

 this is relatively at an infinite distance as compared with the dis- 

 tances between the particles, their weights may be regarded as a 

 system of parallel forces. 



FIG. 10 



_v 



