CO-PLANAK FORCES 67 



moments about the point P, and the perpendicular from P to the 

 line of action of a force is spoken of as the arm of the moment 

 of this force. 



52. THEOREM. When three forces, acting in a plane, keep a body 

 or system of bodies in equilibrium, these three forces must meet in 

 a point. 



For let P, Q, R be the forces, and let P, Q intersect in the 

 point A. Then the sum of the moments of P, Q, and R about A 

 must vanish, and those of P and Q are already known to vanish. 

 Thus the moment of R about A must vanish, that is, R must 

 pass through the point A, or, what is the same thing, the three 

 forces must intersect in a single point. 



An application of this principle is often sufficient in itself for 

 the solution of statical problems in which the applied forces can 

 be reduced to three. 



ILLUSTRATIVE EXAMPLES 



1. The seesaw. Two persons of weights Wi, TF 2 stand on a plank which rests 

 on a rough support about which it is free to turn. Neglecting the weight of the 

 plank, find how the persons must place themselves in order that the plank may 

 balance. 



The forces may be supposed all to act in one plane, namely the vertical 

 plane through the central line of the 

 plank. The forces are 



(a) the weight Wi of the person at 

 one end ; r- 



(b) the weight Wz of the person at W\ /\ ^2 

 the other end ; / \ 



(c) the reaction between the plank 



K IG. OT: 



and its support. 



Let a, 6 be the distances of the persons from the support ; then, on taking 

 moments about the point of support, we have 



Thus the two persons should stand at distances from the support which are 

 inversely proportional to their weights. 



Notice that in this problem the system is acted on by three forces, which meet in 

 a point, the point being at infinity. 



