How Does Young's Modulus Differ from Spring Constant in Shape and Geometry?

[Jimmy87]

Homework Statement

Use data from your experiment to support the idea that Young's Modulus relates the material and is independent of shape and geometry whilst the spring constant is a function of the shape and geometry. The experiment involved stretching identicle springs (starting off with one all the way to to 5) for series and parallel with a constant load of 1kg.

Homework Equations

YM = FL/Ax (YM - youngs modulus, F - force, L - original length, A is area and x is extension).
spring constant = F/x

The Attempt at a Solution

I was thinking that when they are in series as you add more springs the original length increases but in the same proportion to the extension. The area is and force are fixed so YM is constant. Is that correct? In parallel, as you add more springs the extension increases but the area decreases in proportional so that the product of Ax is constant. Force and original length are constant so again YM is constant. With the spring constant I guess F in this case refers to the force on each individual spring whereas F in the YM equation refers to the overall force otherwise I don't see how the spring constant can change (which it must do if it depends on shape/geometry). When relating to a spring what exactly is A? Is it the area of one coil or the cross sectional area?

Also, would I be right in saying that strain for the springs in series is constant but for parallel it isn't because the force is fixed in parallel but the area goes down as you add more springs? So is strain dependent on shape/geometry?

Thanks for any help

 

[haruspex]

In parallel, as you add more springs the extension increases but the area decreases

Is that what you meant to write?

When relating to a spring what exactly is A? Is it the area of one coil or the cross sectional area?

Good question. Think about how a small section of a coiled spring deforms as the spring is extended. Does the cross sectional area change?

 

[Jimmy87]

Is that what you meant to write?

No sorry I meant the extension decreases.

Thanks for your help.

Good question. Think about how a small section of a coiled spring deforms as the spring is extended. Does the cross sectional area change?

So the cross-sectional area represent 'A' in the Young's Modulus equation then? If the equation is YM = FL/Ax then F and L are fixed. The extension 'x' definitely decreases so A must increase to satisfy the condition that YM only depends on the material . We put a thin wooden pole through the springs in parallel and added a slotted mass on the centre of the wooden pole. Does each extra spring in parallel add an additional area equal to the cross-sectional area of the spring then?

 

[haruspex]

So the cross-sectional area represent 'A' in the Young's Modulus equation then? If the equation is YM = FL/Ax then F and L are fixed. The extension 'x' definitely decreases so A must increase to satisfy the condition that YM only depends on the material . We put a thin wooden pole through the springs in parallel and added a slotted mass on the centre of the wooden pole. Does each extra spring in parallel add an additional area equal to the cross-sectional area of the spring then?

Sorry, I couldn't get the picture of what you did from that.
Unfortunately the task as given is flawed. Coiled springs do not depend on elasticity in the sense of Young's modulus. The answer to my earlier question is that each part of the spring undergoes torsion. So the elasticity of the spring depends on the shear modulus, not Young's modulus.
(The situation is further confused by the existence of 'torsion springs'. Their operation depends on the bending modulus, not the shear modulus. They are called torsion springs because they provide a torque, but they do not themselves undergo torsion of the wire they are made of.)

 

[Jimmy87]

Sorry, I couldn't get the picture of what you did from that.
Unfortunately the task as given is flawed. Coiled springs do not depend on elasticity in the sense of Young's modulus. The answer to my earlier question is that each part of the spring undergoes torsion. So the elasticity of the spring depends on the shear modulus, not Young's modulus.
(The situation is further confused by the existence of 'torsion springs'. Their operation depends on the bending modulus, not the shear modulus. They are called torsion springs because they provide a torque, but they do not themselves undergo torsion of the wire they are made of.)

Sorry maybe a picture is better:

So we started off with one spring and then added an extra one and measured the new extension. The load from the slotted masses was kept constant. So can you not apply Young's Modulus to this? Please could you explain why? What is wrong with saying the following:

The force (F) is constant
The original length (L) is constant
Doubling the number of springs halves the extension (x)
Doubling the number of springs results in twice the cross sectional area being pulled (A)

Is it significantly flawed as it is a practical from the actual exam board that needs to be passed as part of the course. Answering the question isn't necessary to pass apparently but it is still on the sheet given to us.

 

[haruspex]

So we started off with one spring and then added an extra one and measured the new extension. The load from the slotted masses was kept constant. So can you not apply Young's Modulus to this? Please could you explain why? What is wrong with saying the following:

The force (F) is constant
The original length (L) is constant
Doubling the number of springs halves the extension (x)
Doubling the number of springs results in twice the cross sectional area being pulled (A)

Is it significantly flawed as it is a practical from the actual exam board that needs to be passed as part of the course. Answering the question isn't necessary to pass apparently but it is still on the sheet given to us.

That's all correct, except that it is not the cross-sectional area.
Consider a single spring, but varying the radius. If it were just the cross-sectional area that mattered (as for stretching a straight wire) then making the spring wire twice the radius would quadruple the spring constant. But torsion resistance rises as the fourth power, giving sixteen times the constant. See http://www.engineersedge.com/spring_comp_calc_k.htm and https://en.m.wikipedia.org/wiki/Shear_modulus.
So although doubling the number of springs does double the spring constant, it cannot be explained in terms of doubling the area with the same modulus.

It is rather a serious flaw because what you are asked to do, to conclude something about Young's modulus, is not possible. It should be asking what you can conclude about shear modulus, but I suspect the course material has never covered how coiled springs actually work.

Which exam board? Can you provide a link?

 

[Jimmy87]

That's all correct, except that it is not the cross-sectional area.
Consider a single spring, but varying the radius. If it were just the cross-sectional area that mattered (as for stretching a straight wire) then making the spring wire twice the radius would quadruple the spring constant. But torsion resistance rises as the fourth power, giving sixteen times the constant. See http://www.engineersedge.com/spring_comp_calc_k.htm and https://en.m.wikipedia.org/wiki/Shear_modulus.
So although doubling the number of springs does double the spring constant, it cannot be explained in terms of doubling the area with the same modulus.

It is rather a serious flaw because what you are asked to do, to conclude something about Young's modulus, is not possible. It should be asking what you can conclude about shear modulus, but I suspect the course material has never covered how coiled springs actually work.

Which exam board? Can you provide a link?

Thanks for the useful information. The exam board is OCR. I have scanned in the investigation sheet we were given from my folder and attached it to this post.
It might be misunderstanding the question. The one my post relates to is right at the end under 'extension opportunities'. It does clearly say to talk about Young's Modulus by using the two experiments we did which I interpret as they are implying Young's Modulus is somehow used to explain springs in series and parallel? Having looked at the links it does seem that the question is flawed. What do you think?

 

[haruspex]

Thanks for the useful information. The exam board is OCR. I have scanned in the investigation sheet we were given from my folder and attached it to this post.
It might be misunderstanding the question. The one my post relates to is right at the end under 'extension opportunities'. It does clearly say to talk about Young's Modulus by using the two experiments we did which I interpret as they are implying Young's Modulus is somehow used to explain springs in series and parallel? Having looked at the links it does seem that the question is flawed. What do you think?

I would like to contact OCR, but it would help to have a bit more detail. What course and level is this? Is there a specific name or link for the paper?

Edit: I came across a Wikipedia entry that has the same confusion. Ouch.

 

[Jimmy87]

I would like to contact OCR, but it would help to have a bit more detail. What course and level is this? Is there a specific name or link for the paper?

Edit: I came across a Wikipedia entry that has the same confusion. Ouch.

Sorry for the delayed reply I had to find out the information you needed in school today from my teacher. These practical assessments are called 'PAGS' by OCR and I have no idea what that stands for. This PAG was 2.2 - Connecting Springs in Series & Parallel so I think if you quote that to them they will know what you are talking about. My teacher did say that he thinks the PAGS are generally not very well written and he said he thinks they are contracted out by OCR (i.e. they don't put them together themselves).

Thanks for your help haruspex!

 

[L-x]

What OCR wants to get at is that if you have a wire of some dimensions, doubling the length halves k without affecting YM, and having two wires (effectively doubling the cross sectional surface area) doubles k without affecting the YM. This let's you nicely draw a distinction between the YM of a material and the spring constant of an object. Unfortunately it's difficult to accurately measure the extension of a wire (certainly if you only want to look at elastic deformation), so OCR has decided to use springs instead, hand wave, and say "these springs deform like wires to a good approximation, they just aren't as stiff".

 

[haruspex]

What OCR wants to get at is that if you have a wire of some dimensions, doubling the length halves k without affecting YM, and having two wires (effectively doubling the cross sectional surface area) doubles k without affecting the YM. This let's you nicely draw a distinction between the YM of a material and the spring constant of an object. Unfortunately it's difficult to accurately measure the extension of a wire (certainly if you only want to look at elastic deformation), so OCR has decided to use springs instead, hand wave, and say "these springs deform like wires to a good approximation, they just aren't as stiff".

You appear not to have read my posts, or did not understand them, or disagree with them.
Yes, doubling the number of springs in parallel will double the effective spring constant, but it is not to do with any surface area. If you were to double the cross-sectional area of the springs by increasing the radius of the wires they are made of the spring constant would quadruple.