16 FORCES IN THE SAME PLANE. [CHAI\ II. 



CHAPTEK II. 



FORCES IN THE SAME PLANE DIFFERENT POINTS OF APPLICATION.* 



16. Resultant of Two Forces in a Plane Different 

 Points of Application. Heretofore we have considered 

 forces having a common point of application, and have seen 

 that in any case the direction and intensity of the resultant is 

 easily found by closing the force polygon. 



But suppose we have two forces P t P 2 having different 

 points of application A x A 2 ; required the position and direc* 

 tiou of the resultant [PI. 3, Fig. 8]. 



Any force acting in a plane may be considered as acting at 

 any point in its line of direction. 



P! and P 2 may then be supposed to act at their common 

 point of intersection a, and through this point the resultant 

 should pass. The case reduces therefore to a common point 

 of application. The resultant is given' in intensity and direc- 

 tion as before by the force polygon (J), and its position is deter- 

 mined by the point of intersection a. At this point, or at any 

 point in the line through a, parallel to 2, the resultant may 

 be supposed to act. 



But the direction of the forces may not intersect within 

 reasonable limits, or the forces may be supposed parallel to 

 each other, so that they may not intersect at all. In any case 

 the force polygon will still give the intensity and direction of 

 action of the resultant, but its position in the plane of the 

 forces remains yet to be determined. Now we have seen [Art. 

 5] that we can decompose a force into two components in any 

 desired directions, by choosing a "pole " and drawing lines to 

 the beginning and end of the force in the force polygon. Let 

 us choose then a pole C [Fig. 8 (J)] and decompose the result- 

 ant thus into two forces given in intensity by the lines C 

 and 2 C. The forces P! P 2 being supposed to act at the 

 points A! AS in the common plane, at what point in the plane 

 and in what direction must the resultant 2 be applied to keep 



* This chapter follows the development of Bauschinger as given in his 

 ''Elemente der Graphischen Statik." 



