Problem 9 - College Physics

Question

\(\cdot\) A metal rod is 40.125 \(\mathrm{cm}\) long at \(20.0^{\circ} \mathrm{C}\) and 40.148 \(\mathrm{cm}\) long at \(45.0^{\circ} \mathrm{C} .\) Calculate the average coefficient of linear expansion of the rod's material for this temperature range.

Step-by-Step Solution

Answer

The coefficient of linear expansion is approximately \(2.294 \times 10^{-5} \, \mathrm{per} \, ^{\circ} \mathrm{C}\).

Step 1: Identify Given Values

First, write down the known information:- Initial length of the rod, \(L_0 = 40.125\, \mathrm{cm}\)- Final length of the rod, \(L = 40.148\, \mathrm{cm}\)- Initial temperature, \(T_0 = 20.0^{\circ} \mathrm{C}\)- Final temperature, \(T = 45.0^{\circ} \mathrm{C}\)

Step 2: Understand the Formula

The formula for linear expansion is:\[\Delta L = \alpha \times L_0 \times \Delta T\]where \(\Delta L\) is the change in length, \(\alpha\) is the coefficient of linear expansion, \(L_0\) is the initial length, and \(\Delta T\) is the change in temperature. We need to solve for \(\alpha\).

Step 3: Calculate Change in Length

Calculate the change in length, \(\Delta L\):\[ \Delta L = L - L_0 = 40.148\, \mathrm{cm} - 40.125\, \mathrm{cm} = 0.023\, \mathrm{cm}\]

Step 4: Calculate Change in Temperature

Find the change in temperature, \(\Delta T\):\[ \Delta T = T - T_0 = 45.0^{\circ} \mathrm{C} - 20.0^{\circ} \mathrm{C} = 25.0^{\circ} \mathrm{C}\]

Step 5: Solve for Coefficient of Linear Expansion

Rearrange the formula to solve for \(\alpha\):\[\alpha = \frac{\Delta L}{L_0 \times \Delta T}\]Substitute in the known values:\[\alpha = \frac{0.023\, \mathrm{cm}}{40.125\, \mathrm{cm} \times 25.0^{\circ} \mathrm{C}} = \frac{0.023}{1003.125}\]Calculate the value:\[\alpha \approx 2.294 \times 10^{-5} \, \mathrm{per} \, ^{\circ} \mathrm{C}\]

Step 6: Conclusion

The average coefficient of linear expansion of the rod's material for this temperature range is approximately \(2.294 \times 10^{-5} \, \mathrm{per} \, ^{\circ} \mathrm{C}\).

Key Concepts

Linear Expansion

Linear expansion refers to the increase in length of a material as it becomes warmer. In simpler terms, when materials, such as metal, heat up, they expand slightly and become longer. This is a common occurrence in everyday life and is especially significant in engineering and construction.
The change in length, denoted as \( \Delta L \), is calculated using the formula \( \Delta L = \alpha \times L_0 \times \Delta T \). Here, \( L_0 \) is the initial length of the material, \( \Delta T \) is the change in temperature, and \( \alpha \) is the coefficient of linear expansion. \( \alpha \) is unique for each material and indicates how much a material will expand per degree of temperature change.

Understanding linear expansion is crucial because it helps us predict how much a material will expand when subjected to heat. Engineers use this concept to design structures that can accommodate expansion and contraction without damage.

Thermal Expansion

Thermal expansion is a broad concept that includes not only linear expansion but also volumetric and area expansion. It is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature.

Expanding on the idea of linear expansion, thermal expansion is critical in different realms:

• Volumetric Expansion: This occurs when the entire volume of a solid, liquid, or gas expands with temperature increase. It affects how substances like liquids in a closed container behave when heated.
• Area Expansion: Similar to linear expansion but applied to two-dimensional areas. It's less common in practical applications than linear or volumetric expansion but still helpful in some engineering calculations.

When engineers and scientists talk about thermal expansion, they often consider these factors holistically to ensure safety and functionality in various applications, from building bridges to developing everyday appliances.

Temperature Change

Temperature change is the difference between the initial and final temperatures in any expansion process. This change is a vital part of the linear expansion and thermal expansion calculation as it drives the extent to which materials expand or contract.
In practical scenarios, knowing the temperature change helps in:

• Predicting Material Behavior: Helps in determining how a certain temperature increase or decrease will affect materials used in construction or manufacturing.
• Designing Structures: Assists engineers in designing buildings, railways, and roads that withstand temperature fluctuations without significant damage.

In the context of the exercise given, the temperature change \( \Delta T = 45.0^{\circ} \mathrm{C} - 20.0^{\circ} \mathrm{C} = 25.0^{\circ} \mathrm{C} \) is crucial for calculating how much the metal rod will expand when the temperature rises from 20°C to 45°C.