Question

A rope extend from pt B (2,0,4) ft. Through a metal loop attach to a wall at point A (6,7,0) ft. to point C (12,0,6) ft. The rope exerts forces Fab and Fac on the loop at A. If Fab = Fac = 200lbs, determine the resultant force exerted on the loop by the rope.

Answer 1

Answer on Question #61307-Physics-Mechanics | Relativity

A rope extend from pt B (2,0,4) ft. Through a metal loop attach to a wall at point A (6,7,0) ft. to point C (12,0,6) ft. The rope exerts forces Fab and Fac on the loop at A. If Fab = Fac = 200lbs, determine the resultant force exerted on the loop by the rope.

Solution

$$F_{ab} = F_{ac} = F$$ $$ab = (2 - 6, 0 - 7, 4 - 0) = (-4, -7, 4)ft$$ $$ac = (2 - 12, 0 - 0, 4 - 6) = (-10, 0, -2)ft$$ $$\cos \theta = \frac{ab \cdot ac}{|ab||ac|} = \frac{(-4)(-10) + (-7)(0) + (4)(-2)}{\sqrt{(-4)^2 + (-7)^2 + (4)^2} \sqrt{(-10)^2 + (0)^2 + (-2)^2}} = \frac{8\sqrt{26}}{117}$$

The resultant force exerted on the loop by the rope is

$$F_{res} = \sqrt{F^2 + F^2 - 2F^2 \cos \theta} = F \sqrt{2(1 - \cos^2 \theta)} = 200 \sqrt{2 \left(1 - \left(\frac{8\sqrt{26}}{117}\right)^2\right)} = 265 \text{ lbs}$$

Answer: 265 lbs.

http://www.AssignmentExpert.com