59a] 



THEOREMS ON FORCE -COUPLES. 



207 



Fig. 51. 



Theorem I. A couple may be transported parallel to itself either 

 in its own or a parallel plane without changing its effect. 



Consider the forces P l and P 2 both equal to P, applied perpendi- 

 cularly at the ends of AB (Fig. 51). At the ends of an equal 

 and parallel line A'B' apply 

 four equal and opposite forces 

 P 3 ,P 4 ,P 5 ,P 6 , each equal to P, 

 which are in equilibrium. The 

 resultant of the equal parallel ^ 

 forces P 17 P 6 is a force 2P | 

 applied half - way between A 

 and B'. The resultant of P 2 

 and P 5 is a force 2P in the 

 opposite direction applied half- 

 way between A and B. Since 

 ABB' A' is a parallelogram 

 these two points of application 

 coincide and the two resultants neutralize each other. We have left 

 the couple P 3 P 4 equivalent in effect to the original couple. 



Theorem II. A couple may be turned in its plane about its 

 center of symmetry without changing its effect. 



Let A'B' be a line of the same length and with the same center 

 as AB, the arm of the couple, and in the plane of the couple 

 (Fig. 52). Apply at 

 A' and B' four equal 

 and opposite forces 

 in equilibrium, each 

 equal to P, and 

 perpendicular to 

 A'B' and in the 

 plane of the couple. 

 Consider P l and P 5 

 applied at (7, their 

 point of inter- 

 section, and by 

 symmetry their re- 

 sultant will be along 

 OC. Similarly the 

 resultant of P 2 and 

 P 6 is an equal force 

 along OD in the 

 opposite direction. 

 These two resultants neutralize each other, leaving the couple P 3 P 4 

 which has the same effect as the original couple. 



