FORCES IN SPACE 



107 



X 



84. THEOREM. Any system of forces acting on a rigid body can 

 be replaced by a force and a couple of which the axis is parallel to 

 the line of action of the force. 



By the theorem just proved, the system may first be replaced 

 by a force acting at any point 0, and a couple. Let the force be 

 of amount R, having OP as its line of action, and let the couple 

 be of moment 6r, having OQ for its axis. If the angle POQ is 

 denoted by 6, we may resolve the couple into two couples : 



(a) a couple of moment G cos 0, 

 having OP for axis; 



(b) a couple of moment G sin 0, 

 having its axis perpendicular to OP. 



The second of these couples may 

 be replaced by any two forces pro- 

 vided these are chosen so as to be 

 equivalent to the couple. Let us 

 choose the first force to be a force 

 R acting along OP, i.e. the force 

 which will exactly neutralize the 

 force R which we already have acting along OP. The second 

 force of the couple must then be a force R acting along a line 

 parallel to OP but at a distance from it equal to G sin 9/R. 



The system has now been replaced by 



(a) forces + R, R acting along OP; 



(b) a force R parallel to OP; 



(c) a couple G cos 6 of which the axis is parallel to OP. 



The two forces (a) neutralize and we are left with a force R and 

 a couple G cos of which the axis is parallel to the line of action 

 of the force. This proves the theorem. 



The line of action of the force, which is now also the axis of 

 the couple, is called the central axis of the system of forces. 

 A system of forces is most simply specified by a knowledge of the 

 magnitude of the force and couple, and of the position and direction 

 of the central axis. Such a system is called a " wrench." 



FIG. 59 



