# Wrap a wire around a cone [ ]

1. Dec 24, 2012

### Helpls

wrap a wire around a cone [urgent]

Hello everyone, i need help solving a problem i'm facing so i can continue my project. So the problem i'm facing is that i got a cone 60cm height and 45 cm base diametre, and i want to wrap a wire with (0,3±0,1)cm diametre around the cone, but i must do it in order to get the same length of wire in the top and in the bottom, so i will need to gradualy increase the gaps between the wires, i was thinking in start with 0,5cm of gap between the first and second winding. But i dont know how to do this right in order to get the same lenght of wire in the top and bottom. Please help me , i want this so bad but i simply dont know.

Something like this: http://img11.imageshack.us/img11/6627/conewire.jpg [Broken]

http://img11.imageshack.us/img11/6627/conewire.jpg [Broken]

Last edited by a moderator: May 6, 2017
2. Dec 24, 2012

### sophiecentaur

Re: wrap a wire around a cone [urgent]

Hi,
I assume this means that you want to be using the same length of wire per cm of cone height?
The pitch of the tapered helix is fairly small so I think you can treat the problem as putting circles around the cone, rather than turns on a helix. So the problem reduces to the similar, siosceles 'triangles' as in your diagram. (X ∏ for the actual length of wire for each turn to give you the length of wire actually needed)
So my 'obvious' (?) reaction is that the number of turns per unit length should be inversely proportional to their length. So the spacing of the turns needs to be proportional to the length of wire in each turn. The length of each turn will be 45/60 (=3/4) times the distance from the apex so the spacing (X) between turns at distance D needs to be
X = x D/d.
Where d is the distance of first turn from apex and x is the first spacing.

length of each turn will be 3∏ D /4

I think this is OK.

3. Dec 25, 2012

### Helpls

Re: wrap a wire around a cone [urgent]

Thanks alot , it really helped me to understand better the problem i was facing by simplifying it :D

4. Dec 25, 2012

### sophiecentaur

Re: wrap a wire around a cone [urgent]

Check through the detail before you commit but I think it' s a way into the prob. Merry Xmas

5. Dec 25, 2012

### Helpls

Re: wrap a wire around a cone [urgent]

I will, thanks. Merry Christmas and Happy New Year to you to. :D

6. Dec 26, 2012

### Helpls

Re: wrap a wire around a cone [urgent]

Dude im getting problem, i tryed to use the formule that u gave me for spacing between turns, until the 30cm of height is fine but for exemple when i do the X=0.1(60/1) gives me 6 cm and i already ended with 3.3cm at the base of the cone. the formule to the lenght of wire works just fine i already confirmed. i already tryed so many things but i cant find out why the spacing doesnt work. According with the formula with my 0.1 first space and 1cm from top, i should be increasing the space by 0.1 and thats what i did until the base of the cone where i get 3.3cm but then i tryed to confirm in the formula by doing X=0.1(60/1) and gives me 6cm :S

7. Dec 26, 2012

### sophiecentaur

Re: wrap a wire around a cone [urgent]

I have to go out now but I'll think about it some more. I actually can't make sense of what you are saying about when it goes wrong.
It has to be true to say that, for the same quantity of wire at each position, [Edit: number of turns per unit distance] times circumference must stay the same - i.e. if the loops are twice as long the spacing needs to be [Edit: double - this may be the problem!!?].

Last edited: Dec 26, 2012
8. Dec 26, 2012

### Helpls

Re: wrap a wire around a cone [urgent]

i will do a draw to try explain better

9. Dec 26, 2012

### Helpls

Re: wrap a wire around a cone [urgent]

Is right, i just didnt undestand it very well, but now im getting it. im trying to relate the spacing and the lenght of wire i will actualy need. i will get it! :D Now i completely undestand

tyvm for the help i will be able to continue and get SOME GREAT Results ;D

Last edited: Dec 26, 2012
10. Dec 26, 2012

### Studiot

Re: wrap a wire around a cone [urgent]

You have drawn your windings horizontal. Of course they must be sloping or the wiring could not wind continuously and progressively round the cone.

You need to find and expression for the curve length of a helix in terms of 3d cordinates and adjust to relate to your criteria for constant segmental length, which I did not fully grasp.

If you can explain again exactly what you want, bearing in mind what I said about the 'coils' perhaps we can help further.

11. Dec 26, 2012

### Helpls

Re: wrap a wire around a cone [urgent]

thanks but i already solve my problem. i just wasnt undestanding what sophiecentaur gaveme but now that i undestand is perfect and simple. so thanks agian for the help, im happy xd

12. Dec 26, 2012

### Studiot

Re: wrap a wire around a cone [urgent]

It's always better to solve it yourself. Well Done.

13. Dec 27, 2012

### Studiot

Re: wrap a wire around a cone [urgent]

You may like to compare your solution with my analysis.

A cone is a developable surface.

This means that you can open out the surface to form a flat sector of a circle.

I have drawn this in the attachment. The slope length (=radius of circle) and cone angle are functions of the cone size.

A curve that wraps round the minimum distance between two points on the surface will be a straight line on the development.

So if the sector is ABC and the pitch of your has n turns the pitch is the (slope length)/n

AC is he curve that when bent round to a cone forms the base, AD is the straight line on the development that becomes the first turn of your tapered helix. DC is the pitch of the helix. EF, GH etc are subsequent turns parall to AD

You can calculate each triangle such as ADB from the cosine rule.

It should be noted that a hoop or sting loop dropped onto a cone will lie slant ways, not horizontal as this is the shortest distance.

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14. Dec 27, 2012

### sophiecentaur

Re: wrap a wire around a cone [urgent]

Just got back from an Xmas jolly.
I know the drawing isn't accurate but, for many turns per cm, the error is irrelevant. I did say, initially, that it was an approximation. With a slope or not, we are still dealing with similar figures and a linear relationship so the simple formula is fine (turns per cm times length of turn is constant). It may not deliver an answer for exactly how much wire you need but I think that's really the only thing wrong. Imagine using a series of rings (my model) then split each ring and join it to the the adjacent rings. You would just need to add a short length to join them and that additional length would also be proportional to the distance from the apex. Not a bad fudge really.

15. Dec 27, 2012

### sophiecentaur

Re: wrap a wire around a cone [urgent]

Just got back from an Xmas jolly.
I know the drawing isn't accurate but, for many turns per cm, the error is irrelevant. I did say, initially, that it was an approximation. With a slope or not, we are still dealing with similar figures and a linear relationship so the simple formula is fine (turns per cm times length of turn is constant). It may not deliver an answer for exactly how much wire you need but I think that's really the only thing wrong. Imagine using a series of rings (my model) then split each ring and join it to the the adjacent rings. You would just need to add a short length to join them and that additional length would also be proportional to the distance from the apex. Not a bad fudge really.

btw, I like your version of the developable surface and the straight bits of wire. If you use that instead of my simple version you would get the right lengths of wire too, using my same formula (still similar figures but a different apex angle).

16. Dec 27, 2012

### rcgldr

Re: wrap a wire around a cone [urgent]

I have the impression that what the OP wanted is equal length segments as the wire wraps around. For opened up surface, this would mean a spiral with equal length segments every 360°. This would require a huge pitch at the start (nearly radial), and then nearly horizontal at the end. I'm not sure if it's even possible to get 2 complete wraps with "equal length" segments.

I'm not sure why the OP didn't want the spacing between wires to be constant, which would translate into an Archimedean spiral (r = a θ + b) on the opened surface.

Last edited: Dec 27, 2012
17. Dec 27, 2012

### Studiot

Re: wrap a wire around a cone [urgent]

I did post the 3D approach using the Frenet formulae, but couldn't complete because of this.

The developable surface properties are fortituitous and reduce the problem to 2 dimensions and we can work on the projections of quantities, however we still need a clear objective to be able to establish boundary conditions.

Don't forget my comment at the end. It is not possible to wind a fixed length of wire round a cone as a circle, without gluing it on, since it is not a geodesic. It will just slip down at an angle if you try it. The error on each coil may be small but they will certainly add up if there are many coils.

18. Dec 27, 2012

### sophiecentaur

Re: wrap a wire around a cone [urgent]

Yet again we suffer from not knowing exactly what was wanted. Should we devise some sort of pro-forma for practical enquiries (assuming this really was one), including a box for "what do you think the finished product will look like"?

I was assuming it was some electrical appliance which needed equal heat output per unit length for an element wound round a conical former. Yours sounds more like a question that a Mathematician would ask. I wonder which was the original intended question.

19. Dec 29, 2012

### Helpls

Re: wrap a wire around a cone [urgent]

sorry guys, just read this now i already wraped the wire in one of the cones, i m not a expert in maths geometry etc thats why took me so long to figure out the sophiecenta formula :D i will draw some things, to explain what my experiment is about, so u guys that are more smart than me, maybe if interested try out :D

it will take some time cuz i want to write and draw my ideias.

20. Dec 29, 2012

### rcgldr

Re: wrap a wire around a cone [urgent]

I should have stated this as the 3d cone eqivalent of a 2d Archimediean spiral, one with equal spacing between loops of wire. For a 3d cone in spherical coordinates, this tranlates into ρ = a θ , ϕ = constant.

An unrolled cone only forms a partial disk, so I'm not sure it's a good analogy to use.

Last edited: Dec 29, 2012