Problem 9 - College Physics

Question

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 \( 2.29 \times 10^{-5} \mathrm{~^{\circ}C^{-1}} \).

Step 1: Understand the Concept

The coefficient of linear expansion, denoted by \( \alpha \), is a measure of how much a material expands per degree change in temperature. The formula to calculate \( \alpha \) is \[ \alpha = \frac{\Delta L}{L_0 \Delta T} \] where \( \Delta L \) is the change in length, \( L_0 \) is the original length, and \( \Delta T \) is the temperature change.

Step 2: Identify Known Values

Given in the problem: \( L_0 = 40.125 \) cm at \( 20.0^{\circ} \mathrm{C} \), \( L = 40.148 \) cm at \( 45.0^{\circ} \mathrm{C} \). So, \( \Delta L = 40.148 - 40.125 \), and \( \Delta T = 45.0 - 20.0 \).

Step 3: Calculate Change in Length, \( \Delta L \)

Calculate \( \Delta L \), which is the difference in length: \( \Delta L = 40.148 \mathrm{~cm} - 40.125 \mathrm{~cm} = 0.023 \mathrm{~cm} \).

Step 4: Calculate Change in Temperature, \( \Delta T \)

Calculate \( \Delta T \) as the difference in temperature: \( \Delta T = 45.0^{\circ} \mathrm{C} - 20.0^{\circ} \mathrm{C} = 25.0^{\circ} \mathrm{C} \).

Step 5: Substitute Values into the Formula

Substitute the known values into the coefficient of linear expansion formula: \[ \alpha = \frac{\Delta L}{L_0 \Delta T} = \frac{0.023 \mathrm{~cm}}{40.125 \mathrm{~cm} \times 25.0^{\circ} \mathrm{C}} \]

Step 6: Perform the Calculation

Calculate \( \alpha \): \[ \alpha = \frac{0.023}{40.125 \times 25} \approx 2.29 \times 10^{-5} \mathrm{~^{\circ}C^{-1}} \]

Step 7: Interpret the Result

The average coefficient of linear expansion for the rod's material over the given temperature range is \( 2.29 \times 10^{-5} \mathrm{~^{\circ}C^{-1}} \).

Key Concepts

Thermal Expansion

Thermal expansion refers to the change in size of a material when its temperature changes. When heated, most materials expand, and when cooled, they contract. This phenomenon is due to the increase in the vibration of atoms as the temperature rises, which causes them to take up more space. The concept is crucial in understanding how materials will behave under different temperature conditions.

The amount a material expands is usually characterized by the coefficient of linear expansion, denoted by \( \alpha \). This coefficient tells you how much the length of a material will change for each degree change in temperature. It's typically measured in units of \( \mathrm{^{\circ} C^{-1}} \).

Understanding thermal expansion is important for practical applications, as it can affect the integrity of structural components and everyday items such as pipes and bridges. For instance, gaps between sections of metal rails on train tracks can be explained by thermal expansion theory, as it allows the rails to expand in the heat without bending.

Physics Problem Solving

Physics problem-solving involves applying established principles and formulas to find a solution to a given problem. It's a vital skill that requires understanding the underlying physics concepts and being able to manipulate mathematical equations to reach a logical conclusion.

When tackling a problem related to thermal expansion, like calculating the coefficient of linear expansion, here are some steps to consider:

• Identify the known variables such as initial length, change in length, and temperature change.
• Understand and write down the formula that applies to the problem at hand.
• Substitute the known values into the formula correctly to ensure precision.
• Perform calculations carefully, checking each step to avoid errors.
  
• Interpret the result meaningfully to align with the context of the problem.

Solving physics problems can be challenging, but with practice and a systematic approach, it becomes more manageable and rewarding.

Materials Science

Materials science is the study of the properties and applications of materials of all types. By understanding materials science, one gains insight into why certain materials behave the way they do, especially under varying conditions like temperature changes.

Key factors considered in materials science, particularly with regard to thermal properties, include:

• The structure of the material at atomic or molecular scales, which influences how it reacts to temperature changes.
• The type of bonding between atoms, be it metallic, covalent, or ionic, as this affects expansion characteristics.
• The material's thermal expansion properties, which help in selecting the appropriate material for specific applications.

By studying these elements, engineers and scientists can predict how different materials will behave in diverse environments, ensuring that the best materials are chosen for safety, efficiency, and longevity in various applications.