Question

Hi, I'm doing a project for college and I've hit a stumbling block.

Essentially I'm designing a large 120ft tape measure. Tape width 50mm and height 1mm.

I need to find the reel size of this tape when fully retracted so that I can design the case.

Any help would be greatly appreciated!
Cheers

Answer 1

I'm doing a project for college and I've hit a stumbling block. Essentially I'm designing a large 120ft tape measure. Tape width 50mm and height 1mm. I need to find the reel size of this tape when fully retracted so that I can design the case.

Solution.

Case 1.

We have to count quantity of hanks of a tape on a reel. The reel size (radius) of this tape equals quantity of hanks (in mm). And the sum of lengths of all hanks has to equal 120 foots. Circle length is

$$S = 2 \pi R.$$

Then we have

$$2 \pi (1 + 2 + \dots + k) = 1 2 0 \cdot 0, 3 0 4 8 \cdot 1 0 0 0 (m m),$$ $$2 \pi \cdot \frac {1 + k}{2} \cdot k = 1 2 0 \cdot 0, 3 0 4 8 \cdot 1 0 0 0$$

where $k$ is quantity of hanks of a tape on reel. In the left side of last equation we used the formula for sum of an arithmetic progression. Then we have

$$k ^ {2} + k - 1 1 6 4 2. 5 0 = 0,$$ $$D = 1 ^ {2} - 4 \cdot 1 \cdot (- 1 1 6 4 2. 5 0) = 1 + 4 6 5 7 0 = 4 6 5 7 1,$$ $$k _ {1, 2} = \frac {- 1 \pm \sqrt {D}}{2}.$$

And finally

$$k = \frac {- 1 + \sqrt {4 6 5 7 1}}{2} \approx 1 0 8.$$

Then the radius of reel is $108\mathrm{mm}$.

Case 2.

Let $r$ (mm) is initial value of radius of a reel. Then we have

$$\begin{array}{l} 2 \pi ((1 + r) + (2 + r) + \dots + (k + r)) = 120 \cdot 0,3048 \cdot 1000 \, (\text{mm}), \\ 2 \pi \cdot \frac{1 + k}{2} \cdot k + 2 \pi k r = 120 \cdot 0,3048 \cdot 1000 \\ k^2 + (1 + 2r)k - 11642.50 = 0, \\ D = (1 + 2r)^2 - 4 \cdot 1 \cdot (-11642.50) = (1 + 2r)^2 + 46570, \\ k_{1,2} = \frac{-(1 + 2r) \pm \sqrt{D}}{2}. \end{array}$$

And finally

$$k = \frac{-(1 + 2r) + \sqrt{(1 + 2r)^2 + 46570}}{2}.$$

Then the radius of reel is $\frac{-(1 + 2r) + \sqrt{(1 + 2r)^2 + 46570}}{2}$ (mm).

Answer:

Case 1. The radius of reel is $108\mathrm{mm}$.

Case 2. The radius of reel is $\frac{-(1 + 2r) + \sqrt{(1 + 2r)^2 + 46570}}{2}$ (mm).