What are our Givens?
L = 3m
H = 1.5-m
W = 205-N
(a) What must the tension in the wire be for the force on the upper hinge to have no horizontal component?
We can do this by realizing that the net torque is 0.
Set the reference point at the bottom hinge.
Tq = T * cos(45) * L/2 + T*sin(45)*H - W * L/2 = 0
Solve for T
T * (cos(45) * L / 2 + sin(45) * H ) = W * L / 2
T = ( W * L / 2 ) / (cos(45) * L / 2 + sin(45) * H )
T = (205-N * 3-m / 2) / (cos(45) * 3-m / 2 + sin(45) * 1.5-m)
T = 307.5 / (1.06 + 1.06)
T = 145.05-N
(b) What is the horizontal force on the lower hinge?
For this we will need to express the forces in the x direction.
H_b = bottom hinge horizontal force
Fx = H_b - T*cos(45) = 0
H_b = T*cos(45)
H_b = 145.05-N * cos(45)
H_b = 102.57-N
(c) What are the vertical forces on the hinges?
For this we will need to express the forces in the y direction.
V_b = bottom hinge vertical force
V_t = top hinge vertical force
Fy = V_b + V_t + T*sin(45) - W = 0
V_b + V_t = W - T*sin(45)
V_b + V_t = 205-N - 145.05-N * sin(45)
V_b + V_t = 102.43-N