In strip winding of a cylindrical surface like this
What is the length of one turn along the axis? Or what is the distance between two similar points on consecutive turns along the axis of cylinder?
This video https://www.youtube.com/watch?v=qowe3iim-B4 says that it is $2 \pi r$ but I can't understand how.
Can someone explain?
On
There should be side by side contact after a single rotation and after one rotation you should neither be able see the cylinder nor should the strips climb on top of previous strips with any overlap.
The repeating distance you ask is never calculated mathematically /numerically.It is fixed by choice, by machine setting what is often called a lead screw pitch on a lathe machine.
Radius of mandrel or cylinder used to wind the strips $r$, standard (single lead) pitch employed for the helix/screw $p$, strip or tape width $W$, then,
$$ \; \; \tan \alpha = \frac{p}{ 2 \pi r} ; \cos \alpha = \frac{W}{p}. $$
Denote by $r$ the radius of the cylinder, by $a$ the with of the tape, and by $\ell$ the length of tape needed for a full turn. From the following figure one deduces $$\sin\alpha={a\over 2\pi r},\quad \ell={2\pi r\over\cos\alpha}\ .$$ It follows that $$\ell={2\pi r\over\sqrt{1-\sin^2\alpha}}={2\pi r\over\sqrt{1-\left({a\over2\pi r}\right)^2}}\ .$$