Questions about this question on Hooke's Law in Balloons

[cnd4747]

Hi, I would like to do an experiment for my physics class about which balloon has the highest stretch ratio and found the following page on this forum:

https://www.physicsforums.com/threads/hookes-law-for-a-balloon.670566/

First of all, can you please explain this function? σ=σ(λ). I'm assuming the first σ should be prime. Also, λ is the stretch ratio, so λ=l/L, where l is final length and L is the initial length. If I were to find the stretch ratio of a balloon, would I use the initial length of the uninflated balloon and then the length of how far I can stretch the balloon, or would I use the final length as the length of the balloon after I put in a certain amount of volume?

I also wanted to know how Mr. Miller got from where he was to the final equation.

Do you know the name of that equation I mentioned earlier? Thanks

 

[Simon Bridge]

First of all, can you please explain this function? σ=σ(λ). I'm assuming the first σ should be prime.

When you see $a=a(b)$, you should read that to mean that $a$ depends on $b$.
Mathematically it is just saying $a=a$ ... only the RHS has extra information. This is how you use maths notation as a language, complete with the nuances of implied context and inferences.

In this case $\sigma=\sigma(\lambda)$ is just saying that "the tensile stress within the sheet σ (force per unit area) will be a non-linear function of the stretch ratio λ" ... just as @Chestermiller says in post #4. There is no reason to assume that the tensile stress should be anything in particular. I don't know what you mean by "prime" in this context (the word does not seem to appear in the link.)

If I were to find the stretch ratio of a balloon, would I use the initial length of the uninflated balloon and then the length of how far I can stretch the balloon, or would I use the final length as the length of the balloon after I put in a certain amount of volume?

You should use stretch ratios for balloons the same way as you would for anything. L does not have to be taken off the unstretched balloon.
So L would be from whatever your initial state for the balloon is, and l would be for whatever the final state of the balloon is.

I also wanted to know how Mr. Miller got from where he was to the final equation.

He explains his derivation as he goes - where did he lose you?
(I've tagged him to this post so he can respond.)

Do you know the name of that equation I mentioned earlier?

It has no special name ... it is a general statement saying that something depends on something else: that is just how you write that sentence in maths.

 

[cnd4747]

Thanks for the answers and sorry it took me so long to get back to you. I figured out how he derived the rest of the equation on my own

 

[Simon Bridge]

Well done - if you post the answer to your question you will help others with a similar question.

 

[cnd4747]

Unfortunately it is part of a paper I am writing for my class and I think if I shared it online it would discredit me and make it look like I just copied it from online rather than actually figuring it out myself.

As a second question, though, would it make any difference in the stretch coefficient if I inflated it or just pulled a part of the balloon? When I presented my experimental idea to my physics teacher she said it would be easier to cut a test square out of the balloon and just stretch that to calculate the stretch factor. Would you get the same stretch factor either way (that is, using the above formula and inflating the balloon vs just pulling on part of it)? I think if the entire balloon was inflated that the stretching would go in different directions, stretching the balloon somewhat thinner and making the overall elastic coefficient lower than if a test square was used, since a test square would only pull from 2 edges instead of all sides.

Thanks