Thursday, February 3, 2011

Roll of Tape

Let’s make a mathematical model for a roll of tape that will allow us to relate the thickness (τ), number of wraps (n), length (L), and the inside (r0) and outside (rn) radii of the role to each other.
We approximate the role as a series of concentric circles.  For simplicity, our diagram contains exactly two “wraps” of tape.  Note that it takes the first two circles just to obtain the first wrap since we are considering it’s thickness – sort of.  From thereon, the outside of the ith wrap is the inside of the (i+1)th wrap.  Although we are making some consideration for the thickness, we are only intending to apply our results to cases were the thickness is very small.  In particular, we assume that the place where the next wrap begins to overlap the previous wrap does not cause the shape of the wrap to significantly deviate from that of a circle.
concentricCircles
We observe that the number of wraps n is given by the difference of the inner and outer radii divided by the thickness of a single wrap of tape:
$n=\frac{r_n - r_0}{\tau}$
We further note that the radius of the ith wrap is given by ri = r0 + τi and so the circumference of the ith wrap is given by Ci = 2π(r0 + τi) = 2πr0 + 2πτi. So the total length of the tape (L) is given thus:

$L=\sum_{i=1}^n{C_i} = \sum_{i=1}^n{(2\pi r_0+2 \pi \tau i)}$
$ = \sum_{i=1}^n{2 \pi r_0} + \sum_{i=1}^n{2 \pi \tau i}$
$ = 2 \pi r_0 n + 2 \pi \tau \sum_{i=1}^n{i}$
$ = 2 \pi r_0 n + 2 \pi \tau n (n+1)/2$
$ = 2 \pi r_0 n + \pi \tau n (n+1)$
By substituting the equation for n into the equation for L we can relate the thickness, τ, to the length and inner and outer radius:
$L = 2 \pi n r_0 + \pi \tau n (n+1)$
$L = 2 \pi ({{r_n - r_0}\over{\tau}})r_0 + \pi \tau (\frac{r_n - r_0}{\tau})(\frac{r_n - r_0}{\tau}+1)$
$\tau L = 2 \pi (r_n - r_0)r_0 + \pi (r_n - r_0)(r_n - r_0 + \tau)$
$\tau L = 2 \pi (r_n - r_0)r_0 + \pi (r_n - r_0)(r_n - r_0) + \pi (r_n - r_0) \tau$
$\tau L - \pi (r_n - r_0) \tau = 2 \pi (r_n - r_0)r_0 + \pi (r_n - r_0)^2$
$(L - \pi (r_n - r_0)) \tau = 2 \pi (r_n - r_0)r_0 + \pi (r_n - r_0)^2$
$\tau = \frac{\pi (r_n - r_0)(2 r_0 + r_n - r_0)}{L - \pi (r_n - r_0)}$
$\tau = \frac{\pi (r_n - r_0)(r_n + r_0)}{L - \pi (r_n - r_0)}$
Now that we have τ in terms of known values, our equation for n can be used to determine the number of wraps. Clearly we will get only an approximate number unless our measurements were exact.  What is it good for?  I have no idea.  Maybe we can use some of our formulae to model other things that have no application that I’m aware of. 
We’ve worked this model backwards in this case; that is, we have determined things about tape that we have been given already rolled up.  Maybe we can use the formula for length to determine how long it will take to roll up a sheet given the rate of rolling, the length and thickness of the sheet, and the outside radius of the roll that we are putting the sheet on to.