Mass of Hanging Rope: Calculate Easily

[Mark128]

Hi, I have a question about a hanging rope - how do you find it's mass? I've been searching a long time, stumbled across some advanced calculus involving catenary functions and equations, but couldn't quite figure it out.

 

[kuruman]

If all that's given is $h$ and $d$ as shown in your figure, you cannot find its mass. The catenary shape will be the same if you hang ropes of different linear mass densities but equal lengths from the two points. You will need additional information such as the tension at the point of suspension to find the mass.

 

[phinds]

It's been 50+ years since I did any catenary math but what you show is just a catenary. The shape is in no way dependent on the mass, assuming the mass is uniform throughout the rope (and if it isn't you probably wouldn't have a catenary anyway)

EDIT: I see kuruman beat me to it.

 

[Mark128]

If all that's given is h and d as shown in your figure, you cannot find its mass. The catenary shape will be the same if you hang ropes of different linear mass densities but equal lengths from the two points. You will need additional information such as the tension at the point of suspension to find the mass.

And if a tension at the lowest point is given? Then how to calculate its mass?

 

[Lnewqban]

And if a tension at the lowest point is given? Then how to calculate its mass?

That external force N makes it a weighted catenary.

Please, see:
https://en.wikipedia.org/wiki/Weighted_catenary

http://arch-re-review.blogspot.com/2010/09/catenary-and-parabola.html

 

[phinds]

Hi, I have a question about a hanging rope - how do you find it's mass?

@Mark128 do you have a practical reason for asking this or are you just playing around with math. If you are playing around with math, then learn the math. If you have a practical reason, then consult a structural engineer to help with your project (and be very specific about exactly what you are doing, WITH numbers)

 

[kuruman]

And if a tension at the lowest point is given? Then how to calculate its mass?

If you know the tension $T_0$ at the lowest point, then

1. Use the catenary equation to find the slope $\dfrac{dy}{dx}$ at an arbitrary point.
2. Find the length $s$ of a piece of rope from the lowest point to a point at an arbitrary value of $x$.
3. Draw a free body diagram of that piece. Note that if you let $w_0$ be the rope's weight per unit length, the weight of the piece of length $s$ is $w_0s$.
4. Use the FBD to find a relation $w_0 =\dots$ in terms of $T_0$ and $h$.
5. Find the total length of the rope in terms of $h$ and $d$ and multiply by your result for $w_0$ to find the weight of the rope.
6. Divide by $g$ to get the mass.

Spoiler

$m=\dfrac{2T_0}{g}\sinh\left(\dfrac{d}{2h}\right).$