Calculating the length of tape when it is wound up

Question

Goal: calculate the length of a cylindrical roll of tape given the thickness of the tape, the internal and external diameter of the roll.

Firstly, I considered the roll of tape to be a bunch of concentric "circles" rather than a spiral.

Let

Then,

$$L=\sum_{i=0}^{\alpha-1} \pi(d+2it)$$

$$=\sum_{i=0}^{\alpha-1} \pi d + \sum_{i=1}^{\alpha-1} 2\pi it$$

$$=\alpha \pi d + \alpha (\alpha -1) \pi t$$

$$=\alpha \pi [d + (\alpha -1)t]$$

I have tested this formula for a few different values of $\alpha$, $d$ and $t$ and it seems to give the correct answer.

However, a roll of tape is not a bunch of concentric circles.

Is there any way to calculate the exact length of tape given these parameters?

Furthermore, how effective would using the formula above be?

score 2 · Accepted Answer · answered Feb 15 '17 at 17:17

2

Find the area of rolled tape and that would be equal to the area of wounded up tape.

answered Feb 15 '17 at 17:17

Seyed

that is not what I am asking – Desmoz Feb 15 '17 at 17:21
@ConnorGaughan, But didn't you ask this: ...Is there any way to calculate the exact length of tape given these parameters? – Seyed Feb 15 '17 at 17:23
I probably shouldn't have said "using these parameters" as the parameters don't make sense when we have a spiral instead of concentric circles, my bad (inner and outer diameters don't exist right?) – Desmoz Feb 15 '17 at 17:31
1

@ConnorGaughan, Since the thickness of an usual tape is very thin then this method is quite accurate, but when it comes to a noticeable thick material then we need a different way of calculation. – Seyed Feb 15 '17 at 17:37
I have posted a similar question (about the thickness of a toilet paper) in my FB. I found your picture is much nicer than mine. Can I borrow it? – Mick Mar 03 '17 at 04:25
@Mick, Yes of course my friend, you can use it. – Seyed Mar 03 '17 at 09:53

1 Answers1