CONDITIONS FOE, EQUILIBRIUM 39 



30. Polygon of forces. THEOREM. If forces acting on a particle 

 are represented by straight lines, the particle will be in equilibrium 

 if the polygon formed by taking all these straight lines as edges is 

 a closed polygon, i.e. if after putting all the lines end to end we 

 come back to the starting point. 



Let AB, BC, CD, , MN represent in magnitude and direction 

 any number of forces which act simultaneously on a particle. 

 Since force is a vector the forces represented by AB and BC are 

 equivalent to a single force repre- 

 sented by AC, and may therefore 

 be replaced by this force. 



Thus the system of forces may 

 now be supposed to be forces repre- 

 sented by the lines A C, CD, , MN. 

 The first two of these may again 

 be replaced by a single force repre- 

 sented by AD, so that the system 

 is reduced to forces represented by 

 AD, DE, -, MN. We can proceed FlG< 16 



in this way until we are left with only a single force represented 

 by AN. This therefore represents the resultant of all the forces. 



If the polygon is a closed polygon, the points A and N coincide, 

 so that the resultant force represented by AN vanishes and the 

 particle is in equilibrium. Conversely, if the particle is in equilib- 

 rium, AN vanishes, so that the polygon is a closed polygon. Thus 

 the condition for equilibrium expressed by the theorem just proved 

 is necessary and sufficient, necessary because the condition must 

 be satisfied if the particle is to be in equilibrium, and sufficient 

 because equilibrium is insured as soon as the condition is satisfied. 



31. Triangle of forces. If there are only three forces, the theo- 

 rem reduces to a simpler theorem known as the triangle of forces. 

 This is as follows : 



THEOREM. If a particle is acted on by three forces represented 

 by straight lines, the particle will be in equilibrium if these three 

 straight lines placed end to end form the sides of a triangle. 



