You are camping at the rim of the Grand Canyon, where bears are abundant. To pro

Question

You are camping at the rim of the Grand Canyon, where bears are abundant. To protect your food (and yourself) from them, you decide to abide by regulations and hang your food in a backpack from a bear sling, as shown in the picture. The backpack has a mass m, the distance between the two trees is 1 and the height above the ground of the attachments of the rope to the trees is h. Consider the rope massless and the pulleys ideal. Initially, as the backpack is on the ground, what is the force F you need to pull with (sec figure) to lift the backpack of the ground? Find a formula for the force F which describes the force for a situation, when the backpack is a distance d above the ground. Note: This formula will depend on m, g, h, d and I. For m= 10.0kg, h=5.0m, d=4.0m (i.e. the rope sags by h-d=1.0m) and l=8.0m, what is F. Use your formula from b (or a physical argument) to argue if you can pull the rope with sufficient force to have no sagging at all, or if this is impossible. How do your answers change if the backpack is not in the middle of the rope but more to the right?

Explanation / Answer

Tension in the rope, T*sin() + T*sin() = m*g
2*Tsin() = m*g
Where, sin() = (h-d)/sqrt((h-d)^2 + (l/2)^2)

2*T(h-d)/sqrt((h-d)^2 + (l/2)^2) = m*g

Force needed to pull,
F = T = [m*g * sqrt((h-d)^2 + (l/2)^2)]/2(h - d)

(a)
When the back pack is at grond, d = 0
F = [m*g*sqrt((h)^2 + (l/2)^2)]/2h

(b)
F = [m*g * sqrt((h-d)^2 + (l/2)^2)]/2(h - d)

(c)
Substituting Values,
m = 10 , h = 5.0 , d = 4.0 , l = 8.0
F =  [10*9.8*sqrt((1)^2 + (4)^2)]/2(5 - 4)
F = 202.2 N
(d)
If there has to be no sagging, h - d = 0
Which means Force needed will be much high.

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