Answer
An object is in equilibrium if the linear momentum of its center of mass is constant and if its angular momentum about its center of mass is constant:
P = constant
L = constant
An object is in static equilibrium if its linear momentum and angular momentum is equal to zero:
P = 0 kg m/s
L = 0 kg m2/s
Requirements for Equilibrium
If a body is in translational equilibrium then dP/dt = 0, or
If a body is in rotational equilibrium then dL/dt=0
Work Step by Step
Vertical force acting on rod by the hinge will be equal to the difference of two forces; the weight of rope and the vertical component of tension on the rope. Since vertical component of tensions in situations (1),(2) and (3) are same (because torques are equal and the distances are equal); the vertical force will be same in all 3 situations.
Therefore,
$F_{y}\left( 1\right) =F_{y}\left( 2\right) =F_{y}\left( 3\right) $
