What is the Tension at the Ends of a Hanging Rope?

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In summary, the tension at each end of the uniform rope with weight W, hanging between two trees with the same height, and making an angle of theta with the trees is equal to half of the weight (W/2). This is found by summing the forces in the y-direction, where the sum of the vertical component of tension (Ty) is equal to half of the weight (W/2). The horizontal tension (Tx) is constant and can be found using the equation T = T*sin(theta).
  • #1
Patta1667
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Homework Statement



A uniform rope of weight W hangs between two trees. The ends of the rope are the same height, and each make angle [tex] \theta [/tex] with the trees (the angle is between the rope and the lower part of the tree). Find the tension at either end of the rope.

Homework Equations



The Attempt at a Solution



I'm having troubles understanding that the tension at the rope ends. The rope's weight is W, so to me it seems that there is a force of W/2 on each tree to keep the rope up. Then the tension at the rope end would be [tex] F = (W/2) \cos \theta [/tex], but the book answer says otherwise. Is the force at each end W or W/2?
 
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  • #2
The vertical component of the tension at each end is indeed (W/2)*cos(theta)

But that is not the total tension in each end.

Draw a Free body diagram and sum the forces in both x and y directions.

Edit: T_y=W/2 sorry, I answered too quickly :/
 
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  • #3
Do you mean that the total tension is W/2? I cannot see anything when it comes to ropes, but I have read in a book that the horizontal tension is constant and it makes sense that [tex] T_x = T \sin \theta [/tex]. Using this equation and [tex] T_y = (W/2) \cos \theta [/tex] I can find the tension at an end for a given [tex] \theta [/tex] (for theta = 45 degrees, the tension is [tex] W/\sqrt{2} [/tex]), but I cannot see the forces at work here.
 
  • #4
I'll see if I can help. Each tree holds up half the rope. By symmetry, each holds up half the weight. Each tree only knows about the end of the rope, not what the middle does. It knows 1) the direction the rope is pulling and 2) how much downward force there must be.
 
  • #5
Then how can I know the horizontal tension in the rope in order to find the total tension at an end?
 
  • #6
You do not need to. If T is the tension built up in the rope at some angle theta, then by applying Newtons 2nd Law in the y-direction you have that

[itex]\sum F_y=0=2*T\cos\theta - W[/itex]

so what is T?

EDIT: Also, I answered to hastily in post #2. The vertical component Ty is NOT (W/2)*cos(theta), it simply Ty=W/2

But Ty=T*cos(theta) which when solved for T will give you the same results as above.
 
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  • #7
Okay, thanks :smile:. Some sort of brain block there!
 

FAQ: What is the Tension at the Ends of a Hanging Rope?

1. What are the benefits of using a rope hung between two trees?

There are several benefits of using a rope hung between two trees. First, it allows for a stable and secure support for activities such as hammocking, zip lining, and rope climbing. It also provides a natural and eco-friendly alternative to using man-made structures for these activities. Additionally, it can be easily set up and taken down, making it a convenient option for outdoor adventures.

2. How do you properly hang a rope between two trees?

To properly hang a rope between two trees, you will need to find two sturdy trees that are at least 10 feet apart. Tie one end of the rope to the first tree using a secure knot, such as a bowline or figure-eight. Then, walk to the second tree and wrap the rope around it multiple times, making sure to evenly distribute the weight. Finally, tie the free end of the rope to the second tree, again using a secure knot. Test the rope to make sure it is stable before using it for any activities.

3. How much weight can a rope hung between two trees hold?

The weight capacity of a rope hung between two trees will depend on several factors, including the type and quality of the rope, the distance between the trees, and the knots used to secure the rope. It is important to always follow the manufacturer's recommendations and to regularly check the rope for any signs of wear and tear. As a general rule, it is recommended to not exceed 400-500 pounds on a rope hung between two trees.

4. Is it safe to hang a rope between two trees?

When done properly, hanging a rope between two trees can be a safe and enjoyable activity. However, it is important to always assess the trees and the surrounding area for any potential hazards, such as weak or diseased trees, sharp branches, or uneven terrain. It is also crucial to use appropriate safety gear, such as helmets and harnesses, when engaging in activities on the rope.

5. Can a rope hung between two trees be left up permanently?

While some ropes are designed for permanent installation, it is generally not recommended to leave a rope hung between two trees up permanently. Over time, the weight of the rope and any activities performed on it can cause damage to the trees, potentially harming their health and stability. It is best to set up the rope only when it will be used and to take it down afterwards to preserve the trees and the surrounding environment.

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