• cnd4747
In summary, the conversation discusses an experiment for a physics class about the stretch ratio of balloons. The function σ=σ(λ) is explained as the tensile stress within the sheet being a non-linear function of the stretch ratio. The question is raised about which length to use for the stretch ratio of a balloon, and it is clarified that the initial length of the uninflated balloon should be used. The derivation of the final equation is explained, and it is mentioned that it has no special name. The person asking the questions figures out the rest of the equation on their own, but is unable to share it online due to concerns of it being seen as plagiarism. Another question is raised about whether the stretch coefficient would differ if the

#### 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:

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

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

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

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

## 1. What is Hooke's Law?

Hooke's Law states that the force exerted by a spring or similar object is directly proportional to the amount of stretch or compression of the object.

## 2. How does Hooke's Law apply to balloons?

Hooke's Law applies to balloons because they are made of an elastic material that can be stretched or compressed, similar to a spring.

## 3. What is the formula for Hooke's Law?

The formula for Hooke's Law is F = -kx, where F is the force applied, k is the spring constant, and x is the displacement from the equilibrium position.

## 4. How does the volume of a balloon affect Hooke's Law?

The volume of a balloon can affect Hooke's Law by changing the amount of displacement and therefore changing the force exerted on the balloon. A larger volume may require more force to stretch or compress the balloon compared to a smaller volume.

## 5. How is Hooke's Law used in the study of mechanics?

Hooke's Law is used in the study of mechanics to understand the behavior of elastic materials and how they respond to applied forces. It is also used in engineering to design and test the strength and durability of various structures and objects.