What length is needed to coil around a tube?

Question

For a project I'm doing, I'm wrapping an led strip light around a tube. The tube is 19mm in diameter and 915mm tall. I'm going to coil the led strip around the tube from top to bottom and the strip is 8mm wide, so the coils will be 8mm apart. How long does the led strip need to be to fully cover the tube?

This reminds me of a popular question on Math SE about a toilet paper roll, but slightly different. I estimated this by measuring how many 8mm wide circles could fit around the tube, then multiplied by the circumference. However, I don't know how to calculate the exact length of the coil. Out of curiosity, how would you find the exact length of the coil wrapping around the tube with each coil being 8mm apart?

Answer 1

The tape make a helix around the tube. the lenth of one revolution is the leghth of the hypotenuse of a triangle, one leg of which is $\pi d$,where d is diameter of tube and the other is the lead of one revolution which is $8 mm$. There fore the exact value of the length is:

$l=\frac{915}{8}\sqrt{(\pi.19)^2 + 8^2}=6885$ $mm$

Note that the thickness of tape must also be considered, here we had a mathematical imagination for that and supposed is is zero which is not true. for more accurate result you have to add the thickness of tape to the diameter of tube.

Answer 2

You are measuring an helix.

Let the parametric equations be

$$\begin{cases}x=\dfrac d2\cos\theta,\\y=\dfrac d2\sin\theta,\\z=\dfrac s{2\pi}\theta.\end{cases}$$ where $d$ is the diameter (of the axis of the strip) and $s$ is the spacing (between axis) of two windings after a full turn.

Integrating the element of length,

$$L=\int_0^{2\pi H/s}\sqrt{\frac{d^2}4+\frac{s^2}{4\pi^2}}d\theta=H\sqrt{\pi^2\frac{d^2}{s^2}+1}$$ where $H$ is the height of the coil.

If the diameter of the tube is $d_t$ and the thickness of the strip is $d_s$, you take $d=d_t+d_s$.

If the width of the strip is $w_s$ and the windings are in contact, by trigonometry you have $s=\dfrac{\pi dw_s}{\sqrt{\pi^2d^2-w_s^2}}$.