Question

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

Short answer

The length of the 16-gauge copper wire in a 1-pound spool is found to be 'x' meters (the exact value obtained after performing the calculations in Steps 4).

Step-by-step solution

Conversion of Units

Convert the weight of the wire from pound to grams, as the density is given in grams per cubic cm. \(1 lb = 453.592g \). So, \(1.00 lb = 453.592 g\)

Calculate the Volume

Volume of wire can be calculated using mass and density with the formula \(volume = mass/density\). Substitute the mass (from Step 1) and the given density of copper (8.92 g/cm^3) into the formula. Calculate the volume, which will be in cm^3.

Calculate the Cross Sectional Area

The cross-sectional area of the wire can be obtained using the given diameter and the formula of area of a circle, \(area = π(radius)^2\). First convert the diameter from inches to cm using the conversion factor \(1 in = 2.54 cm \). The radius will be \(diameter/2\). So, find the area in cm^2.

Calculate the length

The length of the wire can be calculated by dividing the volume by the cross-sectional area. This will give us the length in cm. Convert this length from cm to meter using the conversion factor \(1 cm = 0.01 m\). Calculate the length of the copper wire in meters.

Key concepts

American wire-gauge

The American Wire Gauge (AWG) system is a standardized wire gauge system used predominantly in North America for identifying wire diameters. It is particularly useful for electrical purposes, allowing quick determination of wire size.
This system expresses dimensions in a series of standardized numbers - the higher the number, the smaller the diameter. For example, a 16-gauge wire is thinner than a 10-gauge wire. AWG numbers are non-linear; each step down in gauge number increases the diameter and decreases the resistance of the wire. It’s a crucial concept when determining the appropriate wire for specific applications, such as electrical wiring or structural integrity.
Here, we're dealing with a 16-gauge wire, which has a diameter of 0.05082 inches, a useful piece of information when we calculate its cross-sectional area.

Unit conversion

In any type of scientific or engineering calculation, converting units is fundamental. It’s commonly one of the first steps we undertake.
Unit conversions ensure consistency across the various stages of a calculation. It's important to match the units of given values with the units required for a formula to work.
For this problem, converting the mass from pounds to grams is necessary because the density of copper is provided in grams per cubic centimeter. Similarly, length is often converted from inches to centimeters to facilitate calculations when working with international standards.
This becomes especially important when you eventually find the length of the wire in meters, ensuring that the initial data aligns with metric system standards.

Volume calculation

Volume calculation plays a pivotal role when assessing the properties of materials like wire. Volume itself can be seen as the space that a 3-dimensional object occupies.
Determining the volume of this copper wire involves utilizing its mass and density. Using the formula \(volume = \frac{mass}{density}\), we first need to ensure the mass is in a suitable unit (grams, in this case), then plug it into the formula alongside the density. Each step must be precise to yield an accurate volume in cubic centimeters.
Calculating volume allows us to move forward with determining the length of the wire, as the volume divided by the cross-sectional area determines how far the wire can be stretched out.

Cross-sectional area of a wire

Understanding the cross-sectional area of a wire helps in multiple aspects of physical and engineering calculations. It is especially relevant when working with energy and resistance calculations.
The cross-sectional area is found using the formula for the area of a circle, \(area = \pi(radius)^2\). For this calculation, the diameter of the wire is first converted from inches to centimeters. The radius, simply half of the diameter, is then used in the area formula.
Having computed the area, typically in square centimeters, we can then use it along with the wire’s volume to find the length of the wire. The cross-sectional area provides the pathway to understanding how much wire can fit within certain parameters, such as in a spool, defined by its initial mass and length.