Tension, A Rope with Mass problem

[k-rod AP 2010]

Homework Statement

A clothsline with mass M is hung from two poles, and each end makes angle [theta] with the horizontal. The middle of the rope sags under its own weight. a) what is the tension in the ends of the rope? b)the tension at the sagging point in the rope?

Homework Equations

The Attempt at a Solution

I drew free body diagrams for the ends and middle of the rope and i figured that the rope's weight (m*g) is the only force and is balanced by the tension T at the ends of the rope. So the tension at each end would be 1/2(m*g) b/c each end is holding half the weight.

And the T at the middle of the rope is 0 b/c it is sagging and the (m*g) isn't being balanced by any other force.

Am I correct? i am honestly not too sure how tension works so any help would be appreciated.

 

[PhanthomJay]

No, this is not correct. A rope sagging under its own weight takes the shape of a catenary, but as long as the sag is not a large percentage of the span, you can approximate the curve as a parabola with very small error. So in your free body diagram that isolates the left end and middle point of the rope from the right half of the rope, identify all forces acting on the rope. There is the weight force acting down, and forces at each end of the rope. The tension in the rope at midpoint is NOT zero.
HINT: Tension forces always act tangent to the curve of the rope and pull away from it, that is, tension is a pulling force acting along the longitudiinal curved axis of the rope, and varies in magnitude and direction along the rope's length. Since the rope is in equilibrium, apply Newton's first law. You might want to calculate the vertical support reactions first, before drawing the FBD of the left half of the rope.

 

[k-rod AP 2010]

Ok that makes sense, ur explanation and my teacher's helped me get the correct answer, thanks.