The other day I started realized that toilet paper is pretty non-linear. (As I’m not the first person to point out.) The last part of the roll disappears a lot quicker than the start because each sheet makes up more and more of a warp around the roll.
Conceptually one can look at it as a series of concentric shells of radius Rn and length 2*pi*Rn. A roll with no center tube looks like:
L = pi*R^2/T + pi*R
where L is the length of paper, R is the radius, T is the paper thickness.
The Clackson Scroll Formula discounts the R term (and plugging in some numbers from my case confirms that it’s fairly negligible). And if one then subtracts the length of paper that would have been inside the cardboard tube (radius R0), the length is:
L = pi/T * (R^2-R0^2)
If one takes a constant rate of paper being unrolled, as is done elsewhere, the rate which the radius goes down over time, dR/dt, goes as 1/R.
But I’m not interested in day to day use, so I plotted it discretely, as one would use the toilet paper, both 1 sheet at a time or, as I think is probably more common, 3 sheets at a time. I also plotted diameter because I feel that is the most apparent visual dimension.
(One of the tricky things, I should note, is that with an easily compressible material like toilet paper, one basically cannot measure its thickness reliably. I used known variables to calculate the thickness, which turned out to be about 0.22mm.)
Three sheets at a time:
Three sheets at a time, for the last portion:
I’m a little disappointed that the non-linearity isn’t more pronounced, but you can see how the dots in the last one seem to stretch out. That’s when you say, “Geez, we just had a full roll here and now it’s gone!”
Here’s a different way to think about it, the % of the diameter that’s used up by diameter and by length. The length is completely linear; as you can see, the diameter is not.
Feel free to let me know if you see any mistakes.