So say I have a cylinder confined in a rectangular cuboid of space, as I tilt the cylinder onto its edge first the rectangle increases in height, then, the rectangle decreases in height as I begin to lay the cylinder on its edge.
Now let's say, I wrap the cylinder around another cylinder.
Let's say this second cylinder is infinitely small, so we have a coil, effectively coiling around itself.
Its angle exists in the range (90,R].
It can't reach zero, even if it has zero thickness, and it can't reach 90 regardless of thickness; however as its thickness decreases its minimum angle inceases.
How does the angle of the coil, effect the ammount of length L1 the cylinder A needs to be to get to the end of cylinder B's length L2, given cylinder A has a fixed diameter D1.
Cylinder B has a Diameter D2 of 0.
However as we increase D2 of B, its circumference C, would logically modify the amount of L1 A needs to be to reach the end of L2 while wrapping around it at an arbitrary angle θ[R,90).
What I am asking for in practical physical terms is:
If I have a wire and I wrap it around a rod, how tightly can I pack that wire given its cross-sectional diameter, and how much longer does the wire need to be than the rod, for it's top side's circular edge to be tangent with the plane of the rod's top edge.
This is given the rod's thickness [0,inf)
And the wires thickness [1/inf,inf)
The angle of the coil of wire should start with the tightest possible coil given the thickness of the wire, and the length of the rod; and then converge on what the number nearest 90 degrees in the sig figs you choose to use.
Because 89.9 degrees is still a very loose coil, but 90 degrees is a straight line of wire, so would be equal to the rod in length.
I'm curious to know what ratio of D1 to D2 produces a minimum θ of 45.
R is determined by D1, but I'm not sure if it's as simple as diamter equals rise over run of some fixed number, or what.
I want to know the rules for coils because...
...As you know twisting something does not decrease its height, however coiling it does. For physical materials, like a ribbon of woven fabric for example, I wonder if, after a 360 degree twist, if the fibers become taught around the center-point of the ribbon, and then begin to skew at an angle determined by the thickness of the fiber and the widtg of the ribbon so as to begin to coil proportionally to how much they are additionally twisted.
Just food for thought, but if true, it would create a more realistic model for how much fabric deforms as it is stressed, because current understanding is that something can be twisted infinite times without loosing length, and this works for a stack of coins or stacks of any shapes, but when the shape in question is a whole object, not an object in slices, it seems impossible to dramatically twist it without also coiling it to some degree, as if the fibers inside the objects, are pushing up against eachother, and pulling on one-another with angular torque. -just my personal ponderance.
In my personal opinion a twisted ribbon does not behave the same way as a bunch of thin rectangular slices spiralling in a 3D graph, because otherwise it would be a lot harder to wring out a wet cloth than it proves to be.
Some degree of twisting a continuous 3D object, creates a point of stress where the fibers running the length of it, begin to collide and coil around one another messily, which gets amplified the higher above a single full twist you go.
So if if you imagine 2 parrallel strings of fiber, that are only a fixed distance apart at the ends, held at that distance by two rigid bars or paperclip segments.
As you twise the two strings, their endpoints remain rectangular, but the distance between their strings decreases as you approach the center of the twist, where the two strings begin to coil around one-another.
As you twist them more and more, the line becomes more taught, and the coiled portion spreads out from the center, slightly decreasing the length of the ribbon.
Triangles of empty space will form between each of the two bars and the two sides of the coiled portion.
As these triangles reduce in height, the stress on the materials will go up, until something breaks or stretches or bends.
If breakage stretching and bending are all prohibited, then there must be a point of singularity where the ability to twist it further is made impossible and it just becomes rigid and stops.
It would be nice if this point was when the triangles became equilateral, but I somehow doubt it is that simple.
Also most ribbons have "train-tracks" of flexible fibers between the two opposite lines that want to form the "X" shape.
But because they can slide around and be compressed, ie, since they can pull but can't push, I imagine that for the sake of washcloths and ribbons, some perhaps fractional ammount of coiling is still taking place when a purely twisting motion is applied to both ends, resulting in a slightly shorter cloth, and also a lot of internal pressure releasing some of any water trapped between the pores of the cloth.
Because as we saw with the two strings and two fixed ends, example of a twisting ribbon, the space between is creased, so if we had a soap bubble layer between the two strings and then twisted them, the bubble walls would thicken as the ammount of bubble surface area decreased and/or the excess bubble solution would spill off.
I just find this fascinating.
My many thanks to anywone whom helps me figure out this formula.
Specifically the wire and rod coil formula.
My other hypothesis about the ribbon and such is just my motivation well that and I want to know how much copper wire any future magnetism experiments I do, will need.
Given the radius of the rod $R$, and the diameter of the wire $d$, you want the wire to make 1 full rotation, while it advances a distance of $b$ along the axis the rod. This means that the slope of the wire with respect to the base of the rod is
$ \tan \theta = \dfrac{ b }{2 \pi R} $
i.e.
$ b = 2 \pi R \tan \theta $
The minimum distance $b$ is the wire diameter $d$.
The length of the wire in 1 revolution is
$ L_1 = 2 \pi R \sec \theta = 2 \pi R \sqrt{ 1 + \left(\dfrac{b}{2 \pi R}\right)^2 } = \sqrt{ (2 \pi R)^2 + b^2 } $
The total length of the wire $W$ over the length of the rod is the length of 1 revolution times $\dfrac{L}{b} $ where $L$ is the length of the rod.
$\begin{equation} \begin{split} W &= \bigg(\dfrac{L}{b}\bigg) L_1 \\ &= \bigg( \dfrac{L}{b} \bigg) \sqrt{ (2 \pi R)^2 + b^2 } \\ &= L \sqrt{ 1 + \left(\dfrac{2 \pi R}{b}\right)^2 } \\ &= L \sqrt{1 +\cot^2 \theta} \\&= L \csc \theta \\ &= \dfrac{L}{\sin \theta} \end{split}\end{equation}$
In summary,
$ W = \dfrac{L}{\sin \theta} $
Note that $ \theta \ge \theta_{MIN} $
where $\theta_{MIN} = \tan^{-1} \left( \dfrac{d}{2 \pi R} \right) $