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:

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
Homework Helper
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.

Chestermiller and Delta2
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