How can I determine the curve and tension in a hammock with damaged cords?

[ShaddollDa9u]

Homework Statement

You want to rest on a hammock where the cords are damaged.
a) If you don't want to the cords to break during your nap, do you have to place the hammock at the horizontal or, on the contrary, put the knot of the first rope higher than the other ?

b) What is the curve made by the hammock and the expression of the tension in every point ? We will define the hammock as a wire of linear density λ.

Homework Equations

I couldn't find the curve made by the hammock and the tension in every point, how can I express them ?

The Attempt at a Solution

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a) It should be at the horizontal so that the tension between the two knots of the hammock have the same value. Making a knot higher than the other will increase the tension of that knot, which could break.

b) I have assumed that the system is in equilibrium, so ΣF = 0.

So P - (T1 + T2) = 0, where P is the Weight of the object on the hammock, and T1 and T2 the tension of each knot.

So T1 = (m / 2λ) sin (α) where α is the angle between the first knot and the floor.
T2 = (m / 2λ) sin(β) where β is the angle between the second knot and the floor.

Since P = m1g*, (m1 the mass of the object on the hammock ), we have:

m1*g - (m/2λ) (sin(α) + sin(β)) = 0.

But it looks messed up and I think it's wrong, also, I don't know how to get the tension on each point. Could you help me?

 

[Nidum]

Edit : A similar problem with a link to catenary curves was discussed here .