Tension of a rope between two trees

AI Thread Summary
The discussion centers on calculating the tension in a rope of mass m hanging between two trees at the same height, forming an angle with the trees. The tension at the ends is expressed as T=(mg)/(2cos(theta)), but there is confusion about determining the tension in the middle of the rope without using integrals. Participants suggest using free body diagrams to analyze the forces acting on the rope, particularly focusing on the left half to derive the tension. One contributor notes that their calculations incorrectly indicate zero force in the middle, which is deemed incorrect. The conversation concludes with a consensus that while complex links are provided, simpler methods should suffice for solving the problem.
Elphaba123
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1. Suppose a rope of mass m hangs between two trees. The ends of the rope are at the same height and they make an angle ! with the trees.



I believe the tension at either end of the rope is T=(mg)/(2cos(theta)) but I don't know how to solve the for the middle. The only way I can think of doing it is by using integrals but my teacher told me we won't be using integrals till later in the year.
 
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How did you get the answer at the ends? The same working may help for the middle.

If you haven't already, try drawing a force diagram for just, say, the left half of the rope.
 
To find the rope at the end, I drew a free body diagram. From that I found the sum of the forces to be Tcos(theta)-(1/2)mg=0 because there is no movement. From this I found the tension. The problem is if I use this way I end up with the force in the middle being zero which I don't think is true.
 
Elphaba123 said:
To find the rope at the end, I drew a free body diagram. From that I found the sum of the forces to be Tcos(theta)-(1/2)mg=0 because there is no movement. From this I found the tension. The problem is if I use this way I end up with the force in the middle being zero which I don't think is true.
As Modulated suggests, draw a free body diagram of the left half of the rope, and sum forces in the x and y directions to solve for the unknown tension at the middle of the rope.
 

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