I39-] PLANE STATICS. 83 



V. Plane Statics. 



I. THE CONDITIONS OF EQUILIBRIUM. 



139. Suppose a rigid body to be acted upon by any number 

 of forces, all of which are situated in the same plane. To 

 reduce such a plane system of forces to its simplest form the 

 proposition of Art. 136 may be used. This proposition allows 

 us to transfer all the forces to a common origin, by introducing, 

 in addition to each force, a certain couple in the same plane. 

 The concurrent forces can then be combined into their result- 

 ant by geometric addition, or by forming their force polygon 

 (Art. 98) ; and the couples lying all in the same plane combine 

 by algebraic addition of their moments into a resultant couple 

 (Art. 134). 



Thus, let F (Fig. 36) be one of the forces of the given plane 

 system, P its point of application. Selecting any point O in 

 the plane as origin, apply at 

 O two equal and opposite 

 forces F, F, each equal 

 and parallel to the given 

 force F\ and let / be the 

 perpendicular distance of 

 the origin O from the line 



of action of the given force Fig. 36. 



'F. The force F at P is 



equivalent to the force F at O in connection with the couple 

 formed by .Fat P and Fat O ; the moment of this couple is 

 Fp, its vector is perpendicular to the plane of the system. 



Proceeding in the same way with every force of the given 

 system, all forces are transferred to the common origin O. 

 The whole system is therefore equivalent to their resultant R 

 passing through O, in connection with the resulting couple 



