Tension in a rope hanging between two trees

[Buffu]

Homework Statement

A uniform rope of weight $W$ hangs between two trees. The ends of the rope are same height, and they each make an angle $\theta$ with the trees. Find :

a): The tension at the either end of the rope.

b): The tension in the middle of the rope.

Homework Equations

The Attempt at a Solution

[/B]
For the tension at the end, We split the tension in horizontal and vertical compoenents. $T \sin\theta$ and $T \cos \theta$ respectively.

Since the vertical forces balance the weight, $2T \cos\theta = W \iff T = \dfrac{W}{2\cos\theta}$,

I am stuck at second part.
I know the tension would be tangent to the curve. I thought I will integrate over the curve to find the total tension but the curve is not a semi circle instead it is a catenary which has a locus $y = \alpha \cosh(x/\alpha)$, nevertheless this locus looks a complete mess to.

Is there a way apart from this mess ?

 

[Orodruin]

If you draw a free body diagram of an arbitrary section of the rope, what is the total horizontal force component?

 

[Buffu]

If you draw a free body diagram of an arbitrary section of the rope, what is the total horizontal force component?

Ok let me try.

 

[Buffu]

So the total horizontal force would be $dT \cos(da/2) \cos a$ is this correct ? Now should I integrate from $0 \to \pi/2$ ?

 

[Orodruin]

It is much (much) simpler than that. You just need to draw the free-body diagram of the correct part of the rope. Hint: Try drawing the free-body diagram for half the rope. What forces act on that half in the different directions?

 

[Buffu]

It is much (much) simpler than that. You just need to draw the free-body diagram of the correct part of the rope. Hint: Try drawing the free-body diagram for half the rope. What forces act on that half in the different directions?

Ok I did that, the horizontal force toward right is $T\sin \theta$ and towards left is $T^\prime$, since rope is at rest, $T^\prime = T\sin \theta = W\tan\theta /2$.

Is this correct ?

 

[scottdave]

As you move along the rope, how would you expect the vertical component of tension to change? How would you expect horizontal component to change along the rope?

 

[Orodruin]

Ok I did that, the horizontal force toward right is $T\sin \theta$ and towards left is $T^\prime$, since rope is at rest, $T^\prime = T\sin \theta = W\tan\theta /2$.

Is this correct ?

Yes.