Calculating parabola of drooping chain

[Molydood]

Hi,

I was giving a pub quiz type quesstion the other day, which I managed to solve without the use of formulae/mathematics, but it got me thinking about how to solve it mathematically too.

Imagine a chain stretched between two posts, that then droops in the middle, and forms a natural parabola. What are the variables involved and what is the relationship that determines the curve?

I guess that it would mean calculating for all positions on the chain (using calculus?), in order to give a final curve, but this would require a basic starting formulae stating the relationship.

So variable would be, for any specific point on the chain:
Chain tension
Chain weight
Chain angle (tangent to curve)

Any others?
Any idea on the relationship?

looking forward to hearing your comments

thanks,
Martin

 

[Integral]

You are on the right track. Trouble is the shape is not a parabola, it is a catenary.

I cannot help you off the top of my head, but this type of analysis can be found in books on Partial Differential equations.

 

[Chi Meson]

This site says it all:
http://teacher.sduhsd.k12.ca.us/abrown/Activities/Matching/answers/Catenary.htm

 

[Molydood]

thanks guys, that's great!

 

[Loren Booda]

Molydood

which I managed to solve without the use of formulae/mathematics

How so?

 

[Molydood]

Hi,

'Solve' is perhaps the wrong word. I'll give you the problem then post the answer a little later if required:
two posts of 4 metre height support a chain of length 6 metres. The chain hangs 1 metre from the floor at its lowest point. How far apart are the posts?

 

[Chi Meson]

Hi,

'Solve' is perhaps the wrong word. I'll give you the problem then post the answer a little later if required:
two posts of 4 metre height support a chain of length 6 metres. The chain hangs 1 metre from the floor at its lowest point. How far apart are the posts?

AHA! THis is a brain teaser that does not require any knowledge of the catenary! (I've heard this one before, and I never got the chance to discover it for myself, so I won't give it away)

 

[Chen]

As Chi Meson said this has nothing to do with physics... but if the said chain hung 2 meters from the floor you couldn't solve it.

 

[chroot]

Yeah that brain teaser's pretty dumb... hehe.

- Warren

 

[Integral]

I can't believe that I did not see the solution instantly, had to think about it for a few minutes.

 

[Meithan]

Nice brain teaser. I'll challenge my friends at school. I bet it'll drive them crazy!

 

[ShawnD]

So how is it solved without using the caternary formula?

 

[cookiemonster]

Here's a hint: Draw it to scale.

cookiemonster

 

[ShawnD]

It's a tad hard to draw something that fits this equation

$$y = \frac{e^{ax} + e^{-ax}}{2a}$$

Besides, it aparently can be done without using that formula.

 

[cookiemonster]

Heh, start with the posts. Then add in the minimum height off the ground. Calculate a distance or two, like how far the chain is drooping and how long the chain is. The rest comes pretty naturally.

cookiemonster

 

[Integral]

LOL, come on... Think man.

Consider the problem statement.

You are going to hate yourself when the solution pops into your head.

 

[ShawnD]

The only thing I can think of doing is breaking the length of the chain into little triangles then integrating it.

 

[Integral]

Let it all hang out!

 

[cookiemonster]

You're working way too hard here, Shawn.

cookiemonster

 

[ShawnD]

Well what's the answer? I've asked a 3rd year chem honors student as well as a 2nd year mechanical engineer and neither of them can figure it out. I'm stumped as well.

 

[cookiemonster]

Calculate a distance or two, like how far the chain is drooping and how long the chain is.

That there is probably too big a hint...

cookiemonster

 

[ShawnD]

The chain is drooping 3 and it's 6 long. Those are both given in the question.

 

[cookiemonster]

Yeah, now look at those two numbers for just one second.

cookiemonster

 

[Janitor]

Think degenerate solution.

 

[ShawnD]

Yes I've seen that relationship but it still has to satisfy that equation.

$$y = \frac{e^0 + \frac{1}{e^0}}{2a}$$

$$y = \frac{1 + \frac{1}{1}}{2a}$$

$$y = \frac{2}{2a}$$

$$y = \frac{1}{a}$$

*
Nevermind, the 0 probably screws up the equation. 0's tend to do that.

 

[Michael D. Sewell]

The only thing I can think of doing is breaking the length of the chain into little triangles then integrating it.

Draw it as one triangle.

 

[Molydood]

Shawn, did you ever get the answer? I'm interested to hear your reaction when you get it.
Martin

 

[Chen]

I'm more interested to see the look on his face when he gets it.

 

[NateTG]

The chain is drooping 3 and it's 6 long. Those are both given in the question.

Consider approximating the chain's path with a triangle.

 

[Jack Martinelli]

Consider approximating the chain's path with a triangle.

lol! Got it.

 

[Michael D. Sewell]

Consider approximating the chain's path with a triangle.

Correct!
-Mike

 

[turin]

This thread gave me a good laugh at the expense of ShawnD. I'm sorry Shawn, but, whenever you do get the answer, you'll have to admit that this was pretty funny. What amused me the most, though, was:

I've asked a 3rd year chem honors student as well as a 2nd year mechanical engineer and neither of them can figure it out.

I think this would only work on the educated people. Uneducated people would naturally start out with an oversimplified model and get the answer so easily. It's those of us that know it should be a catenary, for instance, that have a beet red face when they get to the answer. If it makes you feel any better, I would have gone down the same path as you if Chi Meson would not have said it's a brain teaser.

 

[Michael D. Sewell]

In a lot of cases, there is no substitute for a pencil and a piece of paper.

 

[turin]

In a lot of cases, there is no substitute for a pencil and a piece of paper.

Very true. But, if you're already thinking about catenaries, then you would probably stare at your scribbled napkin for an embarrasingly long time in the pub.

 

[Chi Meson]

If it makes you feel any better, I would have gone down the same path as you if Chi Meson would not have said it's a brain teaser.

To think I nearly gave it away at the start. I actually had to edit out the answer before I posted. What a "benny" I would have been.