100 STATICS. [167. 



A _ ft _ cos a cos ft w A sm a cos P w & _ cos sin /? r^ 

 x ~~sin(+) y ~ sin ( + /?) *~~ sin (a + 0) 



As the horizontal thrusts at ^4 and ^ are equal, it makes no difference 

 whether the rods be hinged to the support at A and B, or whether the 

 thrust is taken up by lateral supports, or by a string connecting the ends 

 A, B of the rods. 



167. Two equal homogeneous rods AC, BC (Fig. 46) are hinged at 

 A, B, C so as to form a triangle whose height h is vertical and whose base 

 AB = 2 b is horizontal. The weight of each rod 

 being W,Jind the reactions at the joints. 



Owing to the symmetry of the figure, the reac- 

 tions at C must be equal and opposite and 

 horizontal. The rod AC is subject to three 

 forces only, viz. the horizontal reaction C, the 

 weight Wj and the reaction A the latter must 

 46 * therefore pass through the intersection D of 



C and W. 



If the direction of W intersect AB at E and the scale of forces be 

 taken so as to have W represented by DE = h, DEA will be the force 

 polygon ; hence EA represents C and AD represents A on the same 

 scale on which W is represented by h. 



Analytically, the reactions are found by resolving the forces horizon- 

 tally and vertically and taking moments about A : 



whence C=mW y A = 



where m == 



zh 



168. Two equal homogeneous rods AC, BC, each of weight W, are 

 hinged at C ; their ends A, B rest on a smooth horizontal plane ; a third 

 redDE is hinged to them, connecting their middle points (Fig. 47). 



The plane AB being smooth, the reaction at A is vertical ; the reac- 

 tion at C is horizontal owing to the symmetry ; that at D is likewise 

 horizontal if the weight of the rod DE be neglected, for then this rod is 

 subject only to the reactions at its ends. 



Resolving horizontally and vertically and taking moments about Z>, we 

 find in this case 





