A door has mass M, width W, and height H. It is attached to a door frame by only two hinges: one at 5H/6 (the top hinge) and one at H/6 (the bottom hinge). Calculate the force that the door frame exerts on each hinge.

Answer 1

Imagine the points: A and B are the bottom and top hinges, respectively, O is the center of the door.
Use Lami’s Theorem(If three forces acting on a particle keep it in equilibrium, each is proportional to the sine of the angle between the other two, and the lines of forces are intersected in the same point).
Examine two similar triangles: OAB and the force triangle with following sides: gravity force (directed downwards), and reactions of bottom and top hinges,R(A) and R(B) (along the lines AO and BO, respectively):
mg/AB =R(A)/OA= R(B)/OB
OA = OB (as the hinges are located symmetrically)
AB = 2H/3
OA = sqrt ((H/3)^2 +(W/2)^2).
Then R(A) = R(B) = OA• mg/AB = sqrt ((H/3)^2 +(W/2)^2)• mg/(2H/3)
R(A) directed along AO (from A)
R(B) directed along AO (to B)