What is the equation of adhesive tape?

Question

I know that equation of spiral is $r*e^\theta$ but it seems that it can't depict fragment of adhesive tape irrespective of the value of $r$. So, I was wondering what could be the equation of it. Surely, it isn't a bunch of circles brought together to form a disc.

What would have been the case if its curvature was changing at every point?

@Arthur Undoubtedly the OP was thinking about the logaritmic spiral $r=Ae^{k\theta}$ and just unwittingly overloaded symbols. Admittedly confusingly enough. Anyway, as Fabian's answer explains, we need an Archimedean spiral here. A similar spiral is seen as the groove on vinyl records. — Jyrki Lahtonen, Aug 24 '17 at 05:47
May be https://math.stackexchange.com/questions/1633704/the-length-of-toilet-roll could be of interest. — Claude Leibovici, Aug 24 '17 at 05:54

score 3 · Answer 1 · answered Aug 24 '17 at 05:37

You need to know the thickness $d$ of the tape. The equation is then determined from the fact that $$r(\theta+2\pi) = r(\theta) + d. \tag{1}$$

With the additional assumption that the increase of radius is constant, $$r'(\theta) = c, \tag{2}$$ which is fulfilled once you are far away from the initial part of the tape (where the function depends on a lot of details of the material of the tape).

Integrating (2) from $\theta$ to $\theta+2\pi$ and using (1), we obtain the relation $$ 2\pi c =d $$ which determines the constant $c$.

So the equation of the tape is given by $$ r(\theta) = \frac{d}{2\pi} \theta.$$

What is the equation of adhesive tape?

1 Answers1