217.] 



CONDITIONS OF EQUILIBRIUM. 



If G falls outside the triangle, one or two of the points A ly A 2 , A 3 

 will be subject to pressures vertically upwards. If G be the centroid of 

 the triangular area A^A^, we have p^/h^ p^/h^ =/ 3 /^ 3 = 1/3 ; hence 

 in this case the three reactions are equal. 



217. The axis of the hinges of a door is inclined at an angle 6 to the 

 horizon. The door is turned out of its position of equilibrium by an 

 angle <f>, and held in this position 

 by a force F perpendicular to the 

 plane of the door. Determine F 

 and the reaction of the hinges A, 

 B (Fig. 72). 



Let the axis of the hinges be 

 taken as the axis of x t the verti- 

 cal plane through it as the plane 

 zx, and the point midway be- 

 tween the hinges A, B as the 

 origin O. Regarding the door 

 as a homogeneous rectangular 

 plate whose dimensions are AB 

 = 2 a, OC= 2&, the co-ordinates 

 of its centroid G are o, b sin <, 

 b cos cf>. If the force F be ap- 

 plied at a point Pon the middle line OC at the distance OPp from 

 O, the co-ordinates of its point of application P are o, / sin <, p cos <. 



To proceed systematically, we may tabulate the components of the 

 forces, and the co-ordinates of their points of application, and then 

 form the component couples, as shown below. The components 

 of the unknown reactions A, B of the hinges are called A x , A y , A s , 

 *, B B s . 



Fig. 72. 



