Calculate The Change In The Length Of A Wire That Is Heated From A Temperature Of $10^{\circ} C$ To $5^{\circ} F$. The Linear Expansion Coefficient Of The Material Is 2.0 × 10 − 6 2.0 \times 10^{-6} 2.0 × 1 0 − 6 .

Introduction

When a wire is heated, it expands in length due to the increase in temperature. This phenomenon is known as thermal expansion. In this article, we will calculate the change in length of a wire that is heated from a temperature of $10^{\circ} C$ to $5^{\circ} F$. We will use the linear expansion coefficient of the material, which is given as $2.0 \times 10^{-6}$.

Understanding Thermal Expansion

Thermal expansion is the tendency of matter to change its shape, area, or volume in response to a change in temperature. When a material is heated, its particles gain kinetic energy and start moving more rapidly. This increased motion causes the particles to spread out, resulting in an increase in the material's dimensions.

The Linear Expansion Coefficient

The linear expansion coefficient, also known as the coefficient of linear expansion, is a measure of a material's ability to expand or contract in response to a change in temperature. It is defined as the change in length per unit length per degree change in temperature. The linear expansion coefficient is typically denoted by the symbol $\alpha$ and is usually expressed in units of $1/K$ or $1/^{\circ} C$.

Converting Temperature from Celsius to Fahrenheit

Before we can calculate the change in length of the wire, we need to convert the temperature from Celsius to Fahrenheit. The formula for converting Celsius to Fahrenheit is:

$$T_{F} = \frac{9}{5}T_{C} + 32$$

where $T_{C}$ is the temperature in Celsius and $T_{F}$ is the temperature in Fahrenheit.

Plugging in the given values, we get:

$$T_{F} = \frac{9}{5}(10) + 32 = 18 + 32 = 50^{\circ} F$$

Calculating the Change in Length

Now that we have the temperature in Fahrenheit, we can calculate the change in length of the wire. The formula for calculating the change in length is:

$$\Delta L = \alpha L_{0} \Delta T$$

where $\Delta L$ is the change in length, $\alpha$ is the linear expansion coefficient, $L_{0}$ is the initial length of the wire, and $\Delta T$ is the change in temperature.

We are given that the linear expansion coefficient is $2.0 \times 10^{-6}$ and the initial length of the wire is not specified. However, we can assume that the initial length of the wire is $L_{0} = 1$ meter for simplicity.

The change in temperature is given as $\Delta T = 50^{\circ} F - 32^{\circ} F = 18^{\circ} F$. We need to convert this temperature change to Celsius by subtracting 32 from the Fahrenheit temperature and then dividing by 1.8.

$$\Delta T_{C} = \frac{18 - 32}{1.8} = -14^{\circ} C$$

Now we can plug in the values into the formula for calculating the change in length:

$$\Delta L = 2.0 \times 10^{-6} \times 1 \times (-14) = -2.8 \times 10^{-5} m$$

Conclusion

In this article, we calculated the change in length of a wire that is heated from a temperature of $10^{\circ} C$ to $5^{\circ} F$. We used the linear expansion coefficient of the material, which is given as $2.0 \times 10^{-6}$. We converted the temperature from Celsius to Fahrenheit and then calculated the change in length using the formula $\Delta L = \alpha L_{0} \Delta T$. The result is a change in length of $-2.8 \times 10^{-5} m$.

References

• [1] Wikipedia. (n.d.). Thermal expansion. Retrieved from https://en.wikipedia.org/wiki/Thermal_expansion
• [2] HyperPhysics. (n.d.). Thermal expansion. Retrieved from https://hyperphysics.phy-astr.gsu.edu/hbase/thermo/therexp.html

Additional Resources

• [1] Khan Academy. (n.d.). Thermal expansion. Retrieved from https://www.khanacademy.org/science/physics/thermal-expansion
• [2] MIT OpenCourseWare. (n.d.). Thermal expansion. Retrieved from https://ocw.mit.edu/courses/materials-science-and-engineering/3-15-thermal-properties-of-materials-fall-2004/lecture-notes/lec-10-thermal-expansion/

Frequently Asked Questions

• Q: What is thermal expansion? A: Thermal expansion is the tendency of matter to change its shape, area, or volume in response to a change in temperature.
• Q: What is the linear expansion coefficient? A: The linear expansion coefficient is a measure of a material's ability to expand or contract in response to a change in temperature.
• Q: How do I calculate the change in length of a wire that is heated? A: To calculate the change in length, you need to use the formula $\Delta L = \alpha L_{0} \Delta T$, where $\alpha$ is the linear expansion coefficient, $L_{0}$ is the initial length of the wire, and $\Delta T$ is the change in temperature.
  Frequently Asked Questions (FAQs) on Thermal Expansion ===========================================================

Q: What is thermal expansion?

A: Thermal expansion is the tendency of matter to change its shape, area, or volume in response to a change in temperature. This phenomenon occurs when a material is heated or cooled, causing its particles to gain or lose kinetic energy and move more or less rapidly.

Q: What are the different types of thermal expansion?

A: There are two main types of thermal expansion: linear expansion and volumetric expansion. Linear expansion occurs when a material expands or contracts in length, while volumetric expansion occurs when a material expands or contracts in volume.

Q: What is the linear expansion coefficient?

A: The linear expansion coefficient, also known as the coefficient of linear expansion, is a measure of a material's ability to expand or contract in response to a change in temperature. It is defined as the change in length per unit length per degree change in temperature.

Q: How do I calculate the change in length of a wire that is heated?

A: To calculate the change in length, you need to use the formula $\Delta L = \alpha L_{0} \Delta T$, where $\alpha$ is the linear expansion coefficient, $L_{0}$ is the initial length of the wire, and $\Delta T$ is the change in temperature.

Q: What is the difference between Celsius and Fahrenheit temperature scales?

A: The Celsius and Fahrenheit temperature scales are two different ways of measuring temperature. The Celsius scale is based on the freezing and boiling points of water, while the Fahrenheit scale is based on the freezing and boiling points of a mixture of water and ice.

Q: How do I convert temperature from Celsius to Fahrenheit?

A: To convert temperature from Celsius to Fahrenheit, you can use the following formula:

$$T_{F} = \frac{9}{5}T_{C} + 32$$

where $T_{C}$ is the temperature in Celsius and $T_{F}$ is the temperature in Fahrenheit.

Q: How do I convert temperature from Fahrenheit to Celsius?

A: To convert temperature from Fahrenheit to Celsius, you can use the following formula:

$$T_{C} = \frac{5}{9}(T_{F} - 32)$$

where $T_{F}$ is the temperature in Fahrenheit and $T_{C}$ is the temperature in Celsius.

Q: What are some common applications of thermal expansion?

A: Thermal expansion has many practical applications, including:

• Thermal expansion in buildings: Thermal expansion can cause buildings to expand and contract, which can lead to structural damage if not properly accounted for.
• Thermal expansion in bridges: Thermal expansion can cause bridges to expand and contract, which can lead to structural damage if not properly accounted for.
• Thermal expansion in electronics: Thermal expansion can cause electronic components to expand and contract, which can lead to electrical failures if not properly accounted for.
• Thermal expansion in engines: Thermal expansion can cause engines to expand and contract, which can lead to mechanical failures if not properly accounted for.

Q: What are some common materials that exhibit thermal expansion?

A: Many materials exhibit thermal expansion, including:

• Metals: Metals such as copper, aluminum, and steel exhibit thermal expansion.
• Plastics: Plastics such as polyethylene and polypropylene exhibit thermal expansion.
• Glass: Glass exhibits thermal expansion.
• Ceramics: Ceramics exhibit thermal expansion.

Q: How can I minimize the effects of thermal expansion?

A: There are several ways to minimize the effects of thermal expansion, including:

• Using thermal expansion joints: Thermal expansion joints can help to absorb the expansion and contraction of materials.
• Using thermal insulation: Thermal insulation can help to reduce the effects of thermal expansion.
• Using materials with low thermal expansion coefficients: Materials with low thermal expansion coefficients can help to minimize the effects of thermal expansion.
• Designing structures to accommodate thermal expansion: Structures can be designed to accommodate thermal expansion, reducing the risk of damage or failure.