Question

A copper wire (density \(=8.96 \mathrm{g} / \mathrm{cm}^{3} )\) has a diameter of 0.25 \(\mathrm{mm}\) . If a sample of this copper wire has a mass of 22 \(\mathrm{g}\) , how long is the wire?

Short answer

The length of the copper wire is approximately \(2077.44 \textrm{ cm}\).

Step-by-step solution

Calculate the volume of the copper wire

To find the volume of the copper wire, we can use the formula: Volume \(= \frac{\text{mass}}{\text{density}}\) We are given the mass of the wire as 22 g and the density of copper as \(8.96 \textrm{ g/cm}^3\). Plugging in the values, we get: Volume \(= \frac{22 \textrm{ g}}{8.96 \, \textrm{g/cm}^3}\)

Calculate the cross-sectional area of the wire

The copper wire has a circular cross-section, and we are given its diameter as 0.25 mm. To calculate the cross-sectional area, we first need to find the wire's radius. The radius of the wire is half of the diameter: Radius \(= \frac{\text{diameter}}{2} = \frac{0.25 \textrm{mm}}{2} = 0.125 \textrm{mm}\) Now, to find the cross-sectional area of the wire, we use the formula for the area of a circle: Area \(= \pi (\text{radius})^2\) Plugging in the value of the radius, we get: Area \(= \pi (0.125 \textrm{mm})^2\) It is important to convert the radius to cm before proceeding: Radius \(= 0.0125 \textrm{ cm}\) Area \(= \pi (0.0125 \textrm{cm})^2\)

Calculate the length of the wire

Now that we have the volume and cross-sectional area of the wire, we can find its length using the formula: Length \(= \frac{\text{volume}}{\text{area}}\) Plugging in the values, we get: Length \(= \frac{\frac{22}{8.96}}{\pi (0.0125)^2}\) Now, solve for the length: Length \(= \frac{22}{8.96 \times \pi (0.0125)^2}\textrm{ cm}\) Length \(= 2077.44 \textrm{ cm}\) So, the length of the copper wire is approximately 2077.44 cm.

Key concepts

Volume Calculation

To solve for the volume of a copper wire, you need to use a very simple formula involving mass and density. The formula is:

• Volume = \( \frac{\text{mass}}{\text{density}} \)

In this particular problem, the mass of the copper wire is given as 22 grams, and its density is provided as \(8.96 \, \text{g/cm}^3\). By substituting the given values into the formula:

• Volume = \( \frac{22 \, \text{g}}{8.96 \, \text{g/cm}^3} \)

When you calculate, it results in:

• Volume ≈ 2.455 \( \text{cm}^3 \)

The volume signifies the amount of space the copper wire occupies. Calculating the volume is crucial because it allows us to find out other dimensions like the length when the cross-sectional area is known.

Cross-Sectional Area

Understanding the cross-sectional area of a copper wire involves some basic geometry. Since a wire is generally cylindrical, you can imagine its cross-section as a circle. The formula to calculate the circular area is:

• Area = \( \pi (\text{radius})^2 \)

First, obtain the radius of the wire. Given that the diameter is 0.25 mm, calculate the radius:

• Radius = \( \frac{0.25 \, \text{mm}}{2} = 0.125 \, \text{mm} \)

It's important to convert millimeters to centimeters for consistency in units:

• Radius = 0.0125 cm

Now, plug this value into the area formula:

• Area = \( \pi (0.0125 \, \text{cm})^2 \)
• Area ≈ 0.0004909 \( \text{cm}^2 \)

The cross-sectional area helps determine how much space the copper wire uses in a plane perpendicular to its length, which is necessary for calculating its length.

Wire Length Calculation

Now, with the volume and cross-sectional area known, the length of the copper wire can be found using another simple formula:

• Length = \( \frac{\text{volume}}{\text{area}} \)

Using the calculations from earlier:

• Volume = 2.455 \( \text{cm}^3 \)
• Area ≈ 0.0004909 \( \text{cm}^2 \)

You substitute these values into the formula to find length:

• Length = \( \frac{2.455}{0.0004909} \)

This results in:

• Length ≈ 5000 cm

The length calculation tells you how far the copper wire would stretch out if it were laid in a straight line. This step demonstrates the application of geometry and physics in solving practical problems related to physical dimensions.