178 RESISTANCE OF MATERIALS 



to F^ acting at 0. Similarly, each of the other forces is equivalent 

 to a couple plus a single force (equal and parallel to the given 

 force) acting at 0. The given force system is therefore equivalent 



to a system of equal but concur- 

 rent forces acting at 0, and an 

 equal number of couples, the 

 moment of each couple being 

 equal to the moment of the cor- 

 responding given force with re- 

 spect to the point 0. 



/ ^\F This concurrent force system, 



however, may now be combined 

 into a single resultant force, and 

 the couples also combined into 



a single resultant couple, as just explained. Consequently, we 

 have the following general theorem: 



Any system of forces lying in the same plane is equivalent to a single 

 force acting at any assigned point in this plane plus a couple whose 

 moment is equal to the sum of the moments of the given forces with 

 respect to this point. 



108. Conditions of equilibrium of a system of coplanar forces. 

 When a body acted upon by two or more forces is at rest or in 

 uniform motion relative to any system of coordinate axes, it is 

 said to be in equilibrium, and the forces acting on it are said to 

 equilibrate. The conditions for equilibrium are, therefore, that the 

 resultant force acting on the body must be zero, and that the result- 

 ant moment or couple acting on it must also be zero. That is to 

 say, the algebraic sum of all the forces acting on the body must be 

 zero, and the algebraic sum of the moments of these forces with 

 respect to any point must also be zero. Expressed symbolically the 

 conditions of equilibrium are 



In general it is convenient in applying these conditions to resolve 

 each force F into rectangular components X, Y, and replace the single 

 condition F = by the two independent conditions X = 0, 



