Length of copper wire in 1 lb spool

Question

I was asked the following question in my introductory chemistry class:

The diameter of metal wire is often referred to by its American wire gauge number. A 16-gauge wire has a diameter of $\pu{0.05082 in}$. What length of wire, in meters, is there in a $\pu{1.00 lb}$ spool of 16-gauge copper wire? The density of copper is $\pu{8.92 g/cm^3}$.

First I converted the imperial measurements to metric. $\pu{0.05082 in} = \pu{0.12908 cm}$, $\pu{1 lb} = \pu{0.45359 kg}$, according to Google. So I can use $\pi r^2 h$ to calculate the volume (and thus, using density, calculate the mass) of one meter of wire, I figure the radius to be $\pu{0.06454 cm}$. The equation becomes $V = \pi \times (\pu{0.06454 cm})^2 \cdot \pu{100 cm}$, which equals $\pu{0.41654 cm^3}$. So that's the volume of one meter of wire.

Using the density of copper $(\pu{8.92 g/cm^3})$, I figure the mass of one meter of wire to be $\pu{3.72 g}$. If $\pu{1 lb} = \pu{453.59 g}$, then converting $\pu{3.72 g}$ to lbs gives me $\pu{3.72 g} \cdot \frac{\pu{1 lb}}{\pu{453.59 g}} = \pu{0.01 lbs}$. So assuming I did everything correct, one meter of wire is a hundredth of a pound, so in one pound of wire there is $100$ meters.

Answer 1

Find a function of length depending on mass

Conversion factors:

• $2.54 ~\mathrm{\frac{cm}{in}}$
• $454 ~\mathrm{\frac{g}{lb}}$

Constants:

• Wire diameter $d = 0.129~\mathrm{cm}$
• Density of copper $\rho = 8.92~\mathrm{g\, cm^{-3}}$

Thus, our function $l(m)$ for the wire length from the wire mass in pounds reads: $$ l(m) = m \cdot 454~\mathrm{\frac{g}{lb}} \cdot \frac{1}{\pi d^2 \rho}$$

Evaluate the function

Since I'm too lazy to evaluate something like this by hand (and since I somehow had to record this thought process and did not want to waste the processed remnants of a dead tree) there is a small python script available to do this for you, which you can find here and which, if you run it using the command python3 wires.py, will give you the following output:

Wire length of 1.00 lb of 16-gauge wire: 9.71 m

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(Bonus: You can run the command as python3 -i wires.py and will then be dropped into an interactive python shell, where you can call the wirelength function as many times as you wish using different masses for the wire.)