# What is the length of the roll?

sonhouse
Posers and Puzzles 31 Oct '14 09:44
1. sonhouse
Fast and Curious
31 Oct '14 09:44
So a new manufacturing process can make graphene in unlimited lengths and you start rolling the stuff up on a 1 cm diameter roller, 1 meter in length and you start rolling up this graphene as it comes off the machine making it. Now we take graphene as being a 1 dimensional object, but it is in fact about 3 angnstroms thick per layer.

So starting with that first layer, how long it is when it reaches 1 meter in diameter? Like a long roll of toilet paper but of course a lot thinner.
2. forkedknight
Defend the Universe
31 Oct '14 19:522 edits
I figured there had to be a pretty simple equation for the generalized problem of calculating the length of a roll based on thickness and diameter.

I was right:
http://www.handymath.com/cgi-bin/rollen.cgi

For this problem,
3 angstroms is 0.0000003 mm
center diameter of 1 cm
outer diameter of 1 m

Calculated Length:
2617732078.6037m

That's 2.6 million km. doesn't actually seem like that much
3. forkedknight
Defend the Universe
31 Oct '14 20:05
Funny story. That's about the same distance that this guy drove his car:
http://www.allpar.com/old/high-miles/vaillancourt.php
4. sonhouse
Fast and Curious
01 Nov '14 11:473 edits
Originally posted by forkedknight
Funny story. That's about the same distance that this guy drove his car:
http://www.allpar.com/old/high-miles/vaillancourt.php

2.6 million km would wrap around the moon and back three times!
I calculated the first cm on the roll and it came to about 2600 miles but was unsure of how to do the whole thing. I tried to go to your site but get an error message 'server error' or some such, tried three times. I wanted to see what this guy's equation would show for that first cm.

2.6 million km would wrap around the moon and back three times!
5. forkedknight
Defend the Universe
03 Nov '14 00:041 edit
Hmm, yes, I thought I tried my link before posting, but it doesn't seem to work.

Try
http://www.handymath.com/calculators.html

and select the "Calculator for Rolled Length of Roll of Material" link

The equation itself isn't actually listed, which is slightly frustrating from a curiosity standpoint.
6. forkedknight
Defend the Universe
03 Nov '14 20:37
I found another description of an equation to use:

The length of the material will be the product of the number
of layers and the average length of one layer in the roll:

Number of layers = (Do-Di)/(2t) [total thickness/one layer]

Average layer = pi(Do+Di)/2 [circumference at average diameter]

where Do and Di are the outer and inner diameters, and t is the
thickness of the material.

So the length is

L = pi(Do+Di)(Do-Di)/(4t)

= pi/4 (Do^2 - Di^2)/t

I ran that through a calculator, the answer is the same
7. forkedknight
Defend the Universe
04 Nov '14 17:27
Originally posted by sonhouse
I calculated the first cm on the roll and it came to about 2600 miles but was unsure of how to do the whole thing. I tried to go to your site but get an error message 'server error' or some such, tried three times. I wanted to see what this guy's equation would show for that first cm.

2.6 million km would wrap around the moon and back three times!
As for the first cm, that calculator gets ~262km
8. sonhouse
Fast and Curious
09 Nov '14 18:431 edit
Originally posted by forkedknight
As for the first cm, that calculator gets ~262km
Yeah, saw that a bit late.

That guy's Plymouth went through 6 engines but still, he got 278,000 miles out of each one on average. The piece says he changed the oil once a week! That must have been expensive, 52 oil changes per year. Oil must have been cheap in Canada back then.