200.] 



CONDITIONS OF EQUILIBRIUM. 



121 



VI. Solid Statics. 



I. THE CONDITIONS OF EQUILIBRIUM. 



200. The equilibrium of a rigid body in the most general 

 case, that is, when acted upon by any number of forces F in a 

 space of three dimensions, can be investigated in a manner 

 similar to that adopted for the plane system in Art. 139. 



Selecting as origin any point O rigidly connected with the 

 body, let two equal and opposite forces F, F be applied at O, 

 for every one of the given forces F (Fig. 65). The effect of the 

 given system of forces on 

 the body is not changed by 

 the introduction of these 

 forces at O. But we may 

 now regard the given force 

 F acting at its point of 

 application P as replaced by 

 the equal and parallel force 

 Fat O, in combination with 

 the couple formed by the original force F at P and the force F 

 at O. All the forces of the given system are thus transferred 

 to a common point of application O, and can therefore be com- 

 pounded into a single resultant R, passing through O and 

 represented in magnitude and direction by the geometric sum of 

 the forces. In addition to this resultant R, we obtain as many 

 couples (F t F) as there were forces given ; and their resultant 

 is found by geometrically adding the vectors of the couples 

 (Art. 134). 



Thus the given system of forces is seen to be equivalent to a 

 resultant R in combination with a couple whose vector we shall 

 call H\ in other words, it has been proved that any system of 

 forces acting on a rigid body can be reduced to a single resultant 

 force in combination isjith a single resultant couple. 



Fig. 65. 



