# Thread: Length of inside and outside of a material when wound in a spiral

1. ## Length of inside and outside of a material when wound in a spiral

Hi, firstly, appologies if I'm posting in the wrong forums! Its been along time since I was at school and I have a feeling this may be a simple question for knowledgable people so I have placed it in the high school area and I'm not 100% where this questions lies in type of mathematics.

As for my question -

I'm trying to work out the length of a material of a known thickness whilst it is wound up into a tight sprial.

Imagine a carboard tube with an external diameter of say 300mm, and a material with a thickness of say 1mm which is 100000mm (100 Linear M) in length. The material is attached at one end to the tube and wound up tightly around the tube over and over against itself with no space between each wrap/layer until all 100000mm is wrapped up.

My thoughts are because the circumference of the material on the inside of each single wrap is smaller than on the outside (because the material is 1mm thick), the side of the material on the inside must be travelling less distance than the side of the material on the outside? Does that make sense??

This would lead me to believe that there is some compression of the material somewhere (on the inside) or extension (on the outside) to allow this to happen (the material must be extensible?).

What I want to calculate is the length of the inside of the material and the length of the outside of the material, if it where to be measured whilst wound up on this tube.

Thanks!
Batfink

2. Originally Posted by thebatfink
Hi, firstly, appologies if I'm posting in the wrong forums! Its been along time since I was at school and I have a feeling this may be a simple question for knowledgable people so I have placed it in the high school area and I'm not 100% where this questions lies in type of mathematics.

As for my question -

I'm trying to work out the length of a material of a known thickness whilst it is wound up into a tight sprial.

Imagine a carboard tube with an external diameter of say 300mm, and a material with a thickness of say 1mm which is 100000mm (100 Linear M) in length. The material is attached at one end to the tube and wound up tightly around the tube over and over against itself with no space between each wrap/layer until all 100000mm is wrapped up.

My thoughts are because the circumference of the material on the inside of each single wrap is smaller than on the outside (because the material is 1mm thick), the side of the material on the inside must be travelling less distance than the side of the material on the outside? Does that make sense??

This would lead me to believe that there is some compression of the material somewhere (on the inside) or extension (on the outside) to allow this to happen (the material must be extensible?).

What I want to calculate is the length of the inside of the material and the length of the outside of the material, if it where to be measured whilst wound up on this tube.

Thanks!
Batfink
see Wikipedia: HELIX
A circular helix has constant band curvature and constant torsion. The pitch of a helix is the width of one complete helix turn, measured along the helix axis.

The fractional amount to be added to the circumference caused by the pitch of the helix:
$f = \sqrt {(300 \pi)^2 + 1} - (300 \pi)$

The circumference of the inside of the material is
$C_I = 300mm \times \pi +f$

The circumference of the outside of the material is
$C_O = (300+1)mm \times \pi +f$

The total number of wraps will be $\dfrac{100000}{C_O} \, or \, \dfrac{100000}{C_I} \, or \, \dfrac{200000}{C_O + C_I}$

depending on whether you assume:
1) there is no inside compression and only the outside is stretched;
2) that the outside is NOT elongated but the inside is compressed;
3) equal stretching/compression of the outside/inside coils,

The OUTSIDE LENGTH = $C_O \times wraps$

The INSIDE LENGTH = $C_I \times wraps$

3. Hi! Thanks for the reply!!

Could you please explain if the fractional amount formula - in particular - 300*pi is my tube diameter or just "300".. I suspect its obviously the tube but I'd like to confirm.

Also, which of the wraps formula is for each type of extension / compression? are the 1 / 2 / 3 from left to right?

I've got this all down on a spreadsheet and its thrown one of my previous workings into doubt now (assuming this set of formula are correct for calculating number of wraps). I use this to work out my final roll diameter -

SQRT(((materiallengthLM*(4000*materialthickness))/PI)+tubediameter²)

Using your formulas I am taking the number of wraps, doubling them and multiplying by the material thickness and adding the tube diameter to coarsely work out the finished roll diameter when wound up and for a 300mm tub with 0.5mm material thickness and 100LM roll length its coming out 17mm smaller with my diameter formula.

Can I assume 100% that your formula is correct and I might not be working out final roll diameter correctly? Obviously when I started calculating roll diameters I wasn't thinking about compression or extension of material.

I'm questioning both because I'm getting differing answers now to roll diameter and also because I had expected there to be a larger difference in the lengths on the insides and outsides than the formula is giving me.

But then obviously, like I originally said.. Its been a LONG time since I was being taught math!

Thanks again for your help with this!

4. Originally Posted by thebatfink

Could you please explain if the fractional amount formula - in particular - 300*pi is my tube diameter or just "300".. I suspect its obviously the tube but I'd like to confirm.

Also, which of the wraps formula is for each type of extension / compression? are the 1 / 2 / 3 from left to right?

I've got this all down on a spreadsheet and its thrown one of my previous workings into doubt now (assuming this set of formula are correct for calculating number of wraps). I use this to work out my final roll diameter -

SQRT(((materiallengthLM*(4000*materialthickness))/PI)+tubediameter²)

Using your formulas I am taking the number of wraps, doubling them and multiplying by the material thickness and adding the tube diameter to coarsely work out the finished roll diameter when wound up and for a 300mm tub with 0.5mm material thickness and 100LM roll length its coming out 17mm smaller with my diameter formula.

Can I assume 100% that your formula is correct and I might not be working out final roll diameter correctly? Obviously when I started calculating roll diameters I wasn't thinking about compression or extension of material.

I'm questioning both because I'm getting differing answers now to roll diameter and also because I had expected there to be a larger difference in the lengths on the insides and outsides than the formula is giving me.

But then obviously, like I originally said.. Its been a LONG time since I was being taught math!

Thanks again for your help with this!
From where did this arrive?

FinalRollDiameter = $\sqrt{ \dfrac{ materiallengthLM \cdot 4000 \cdot materialthickness}{\pi} + tubediameter^2}$

$\dfrac{100m}{0.94248m}$ = 106.1 wraps total