Two boxes are hung from a ceiling and wall as shown in the diagram. Each box is supported by two cables that are interconnected through rings. What is the maximum tension in any rope other than the ropes directly connected to the boxes?

In this example, there are two different loads that interact through a system of cables. Since all ropes are connected to at least one ring, it would seem reasonable to focus on the ring objects and make sure they are in static equilibrium.

Free-body Diagrams of Ring E

Drawing a free-body diagram of ring E, identifies two rope forces, F_DE and F_EC, that are not known. The direction of the loads are known since the force must act in the direction of the rope.

The rope forces can be determined by applying force equilibrium in the x and y directions.

     ΣF_x = 0
     F_DE cos30 + F_EC cos55 = 0
     F_DE = 0.6623 F_EC

     ΣF_y = 0
     -10 + F_DE sin30 + F_EC sin 55 = 0
     0.6623 F_EC (0.5) + F_EC (0.8192) = 10

Solving the two equations give,
F_EC = 8.693 lb
F_DE = 5.757 lb

Now that the force in rope EC is know, ring C can be analyzed. Again, drawing a free-body diagram of ring C identifies two unknown forces, F_AC and F_CB. Applying the equilibrium equations gives,

     ΣF_x = 0
     F_CB cos45 - 8.693 cos55 = 0
     F_CB = 7.051 lb

     ΣF_y = 0
     F_AC + 7.051 sin45 - 8 - 8.693 sin 55 = 0
     F_AC = 10.14 lb

Maximum force occurs in rope AC,
     F_max = 10.14 lb