Confusion Calculating Young's Modulus

[AndrewsRT_16]

Homework Statement

c) One of the bestselling items at the stall are gummy worms, so you decide to
perfect the recipe. The ideal gummy will stretch a little as you bite into it, but
not too much. The mold you use produces worms with a length of 0.1m and a
diameter of 0.03m.

You narrow it down to two recipes. In the graph below are the two possible
options you produce when you hang weights from the bottom of your two
most likely gummy worms and measure the extension.

Calculate the Young Modulus of the gummy worms by converting one of the options
into a stress/ strain graph (to be shown with your answers) and the other by using
calculations. Choose which gummy worm you think would be better and explain why.

Relevant Equations

Stress = F/A
Strain = Change in length / Original Length
Youngs Modulus = Stress/Strainhttps://www.physicsforums.com/account/

'Gummy 1' = Black Line
'Gummy 2' = Orange Line

Approach:
Using the diameter given and the force measurements on the vertical axis of the graph, I've worked out the cross-sectional area of the mold to be 0.000706858, which I've rounded to 0.00071.

Using this data and the equation Stress = F/A, I've calculated the Stress for each point on the graph 0.2, 0.4, 0.6 0.8.

For strain I've used the initial length (0.1m) noted in the question divided by the extension at each point, I've maintained the measurement in metres for simplicity. the two tables of measurements i've produced for these are as follows:

So from this, I have to calculate Young's modulus for one gummy using a graph and one from just calculations.

This is the part I'm getting confused about, I understand that you use the equation Young's Modulus = Stress/Strain for some point within the linear region of the graph, but how do I choose where, and how do I know that is the correct point, of course, if I had some object where the measured constant of Young's modulus is known I could compare, but to my knowledge, I cant do that in this scenario, I just feel as if I'm guessing.

My calculations:

Gummy 1:

I used 0.2 again as it's my first value in my table, another point of confusion for, how do I know that this is within the linear region of the graph? It

Gummy 2 with Graph.

My conclusion:

"Choose which gummy worm you think would be better and explain why."

I believe the second choice would be better, as highlighted the ideal gummy should stretch a little but not too much, and the results of the second recipe appear to mirror this. The lower Young's Modulus of the second recipe indicates that the gummy worm will be more flexible and deform easier than the first recipe which has a considerably higher Young's Modulus. The first recipe maintains stress levels between 9000-10000 Pa while the gummy continues to stretch from 0.002m to 0.008m without showing any indication of breaking, this means that this recipe is likely too firm and stretches excessively without breaking adding to the fact that it's not suitable for the stall.

My overall issue is mostly with the calculation of Young's Modulus, I'm not confident in the answers I've provided, I get with other problems I've seen on this topic you typically get materials like steel, in which you can compare your result with a constant but that's not applicable in this case. It seems any point that I choose within the linear region gives me a different value, which just confuses me even more.

Any advice on where I may be going wrong would be much appreciated
Thanks

 

[Lnewqban]

...Using this data and the equation Stress = F/A, I've calculated the Stress for each point on the graph 0.2, 0.4, 0.6 0.8.
...
This is the part I'm getting confused about, I understand that you use the equation Young's Modulus = Stress/Strain for some point within the linear region of the graph, but how do I choose where, and how do I know that is the correct point

Welcome, @AndrewsRT_16 !

One thing is to estimate the Young's modulus of each recipe, and another is to create its stress–strain curve.

For the former, you should only use several points within the linear (purely elastic) region of the graphs or data.

Copied from
https://en.wikipedia.org/wiki/Young's_modulus

"Young's modulus is defined as the ratio of the stress (force per unit area) applied to the object and the resulting axial strain (displacement or deformation) in the linear elastic region of the material."

 

[haruspex]

I understand that you use the equation Young's Modulus = Stress/Strain for some point within the linear region of the graph

So why did you not do that? Linear region means where it is a straight line. For Gummy that is for extensions up to 0.1cm, but the shortest extension in your table is 0.2cm.

 

[AndrewsRT_16]

So why did you not do that? Linear region means where it is a straight line. For Gummy that is for extensions up to 0.1cm, but the shortest extension in your table is 0.2cm.

Yeah, I see what you're saying, it doesnt make sense to use 0.2cm onward. I did try to use stress/strain values at 0.1cm intervals but after coming up with the answers, I somehow convinced myself it wasn't right and decided to use 0.2 instead. Thanks for pointing that out.

Welcome, @AndrewsRT_16 !

One thing is to estimate the Young's modulus of each recipe, and another is to create its stress–strain curve.

For the former, you should only use several points within the linear (purely elastic) region of the graphs or data.

Copied from
https://en.wikipedia.org/wiki/Young's_modulus

"Young's modulus is defined as the ratio of the stress (force per unit area) applied to the object and the resulting axial strain (displacement or deformation) in the linear elastic region of the material."

I've redone the tables with values from 0.1 - 0.8 and so far this is what I've got.

So for the first Gummy, the only extension value within the Linear elastic region is 0.1cm (on the table)
The Youngs Modulus at this value should be:

7042.25 / 0.01
= 704225 Pa

Just to further confirm I can work out the value at 0.05cm of extension

Force = 2.5N
Extension = 0.0005 (M)

Stress = 2.5 / 0.00071
= 3521.13 Pa

Strain = 0.0005 / 0.1
= 0.005

So 3521.126 / 0.005
= 704225.2 Pa

As for Gummy 2, both 0.1 and 0.2 cm are within the linear elastic region, the new graph looks like this.

As I calculated previously the value at 0.2cm extension is:
5915.5/0.02 = 295775 Pa

Now with the value of 0.1cm:
2957.75/0.01 = 295775 Pa

That seems to confirm that's the correct value.

Looking back at my conclusion I still think it remains the same, the second option is more suitable for the stand. The first option appears too stiff and inflexible, requiring much more applied stress to deform a similar amount to the second option, it generally wouldn't be a good choice for a gummy.

 

[haruspex]

The first option appears too stiff and inflexible, requiring much more applied stress to deform a similar amount to the second option, it generally wouldn't be a good choice for a gummy.

How can you tell it is too stiff? Maybe option 2 is too soft.
The question seems to require some ergonomic or comparison data, such as the modulus of some common item you can relate to.

 

[KingRich]

I am also in the process of answering the same question and have concluded that Gummy 2 is the better choice. The problem is that in the traditional means, we would have to use a graph representing ductility, with values of 100mpa being considered ductile. However, we wouldn't refer to a gummy as ductile. So, because the young modulus values are small for gummy one and gummy two, I am not sure we could use the same ductile graph as a reference. What is certain is that gummy 1 has higher values of young modulus than gummy 2, suggesting stiffer and more brittle in the traditional terms.