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.