Problem 1: Equivalent Systems (25%)
Saruman has set his Uruk-hai to work building war machines for his assault on Helm's Deep. Realizing that he would rather have them fighting than building, Saruman decides to replace the efforts of the 4 Uruk-hai with one cave troll, each one exerting a force ($F_1, F_2, F_3, F_4$) on the machine. Determine the magnitude, direction, and location of the equivalent force that the cave troll must exert on the arm to achieve an equivalent loading on the machine.
Take:
$F_1 = 100 lb$
$F_2 = 150 lb$
$F_3 = 260 lb$
$F_4 = 80 lb$
$a = 6 ft$
$b = 2 ft$
$c = 4 ft$
$d = 3 ft$
$\theta = 30^\circ$
Find: