CONDITION OF EQUILIBRIUM 103 



We can express the condition for equilibrium in a different form : 



A system of forces in a plane will be in equilibrium if the sums 

 of the moments about any three non-collinear points are each zero. 



For, if the moment about any one point is zero, the resultant 

 cannot be a couple. It must, therefore, be a single force. If the 

 moments about each of two points A, B vanish, the line of action 

 of this force must in general be AB, but if the moment about 

 some third point C, not in the line AB y also vanishes, then the 

 force itself must vanish. 



EXAMPLES 



1. Parallel forces of 6, 12, and 7 pounds act at the two ends and middle 

 point, respectively, of a line 2 feet in length. Find the magnitude and line of 

 action of their resultant. 



2. Find the resultant of the forces in the last question when their magnitudes 

 are respectively 5, 12, and 7 pounds. 



3. Find the resultant of three forces, each of amount P, acting along the 

 sides of an equilateral triangle, taken in order. 



4. Prove that a system of forces acting along and represented by the sides 

 of a plane polygon taken in order, is equivalent to a couple, whose moment is 

 represented by twice the area of the polygon. 



5. If the sums of the moments of any co-planar forces about three points 

 which are not in a straight line are equal, and not each zero, prove that the 

 system is equivalent to a couple. . 



6. A uniform rod is of length 3 feet and weight 24 pounds. Weights of 16 

 and 18 pounds are clamped to its two ends. Find at what point the rod must be 

 supported so as just to balance. 



7. A uniform beam weighing 20 pounds is suspended at its two ends, 

 and has a weight of 50 pounds suspended from a point distant 7 feet and 

 3 feet from the two ends. Find the pressures at the points of suspension of 

 the beam. 



8. A uniform rod of weight 60 pounds and length 18 feet is carried on the 

 shoulders of two men who walk at distances of 2 feet and 3 feet respectively 

 from the two ends. A weight of 60 pounds is suspended from the middle point 

 of the beam. Find the total weight carried by each man. 



9. A dumb-bell weighing 32 pounds is formed of two equal spheres, each of 

 radius 3 inches, connected by a bar of iron so that the centers of the spheres 

 are 16 inches apart. One of the spheres is now removed, and the remainder of 

 the dumb-bell is found to weigh 20 pounds. Find where this remaining part 

 must be supported in order that it may just balance. 



