56 STATICS. [102. 



102. As the projection on any line of any closed polygon, 

 even when its sides do not all lie in the same plane, is equal to 

 o, it follows that the proposition of Art. 89 holds for any num- 

 ber of concurrent forces. 



103. Exercises. 



(1) Show that three forces that are in equilibrium must lie in the 

 same plane and pass through the same point. 



(2) Six forces of i, 2, 3, 4, 5, 6 Ibs., respectively, act in the same 

 plane on the same point, making angles of 60 with each other. Find 

 their resultant in magnitude and direction : (a) graphically ; (b) analyti- 

 cally. 



(3) Let AB = c (Fig. 20) be the vertical post, AC = b the jib, of a 

 crane, the ends BC being connected by a chain of length a. If a 

 weight W be suspended from C, find the tension 

 T produced by it in the chain and the thrust P" 

 in AC. 



(4) Let AC be hinged at A (Fig. 20) so as to 

 turn freely in a vertical plane, and let the chain 

 pass over a pulley at C and carry the weight W. 

 In what position of A C will there be equilibrium? 



Fig. 20. 



(5) Find the resultant R of three concurrent 



forces A, B, C lying in the same plane and making angles a, (3, y with 

 each other. 



(6) Prove that the moment of the resultant of any number of 

 concurrent forces lying in the same plane about any point in this 

 plane is equal to the sum of the moments of the forces about the same 

 point. 



(7) By means of Ex. (6), express the conditions of equilibrium of 

 any number of concurrent forces in the same plane. 



(8) When three forces are in equilibrium, show that they are pro- 

 portional and parallel to the sides of a triangle. 



(9) When any number of concurrent forces are in equilibrium, show 

 that any one of them reversed is the resultant of all the others. '- 



