Problem 21 - University Physics with Modern 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 for this temperature range.

Step-by-Step Solution

Answer

The average coefficient of linear expansion is approximately 0.000022857 °C⁻¹.

Step 1: Understanding the problem

We need to find the average coefficient of linear expansion for the metal rod given two measurements of its length at different temperatures.

Step 2: Identify the given data

The rod is 40.125 cm long at 20.0°C and 40.148 cm long at 45.0°C. The temperature change ( ΔT ) is 45.0°C - 20.0°C = 25.0°C, and the change in length ( ΔL ) is 40.148 cm - 40.125 cm = 0.023 cm.

Step 3: Understanding the formula

The formula for linear expansion is given by ΔL = αL₀ΔT, where α is the average coefficient of linear expansion, L₀ is the initial length, ΔL is the change in length, and ΔT is the temperature change.

Step 4: Rearrange the formula

To find the coefficient of linear expansion, α, rearrange the formula to get α = ΔL / (L₀ΔT).

Step 5: Plug in the values

Substitute the given values into the equation: α = 0.023 cm / (40.125 cm × 25.0°C).

Step 6: Calculate the coefficient

Perform the calculation: α = 0.023 / (40.125 × 25.0) = 0.000022857.

Key Concepts

Understanding Linear Expansion

Linear expansion refers to the increase in length of an object as the temperature increases. This change occurs because molecules within the material move more when heated, causing them to take up more space.
Generally, this can be observed in solid materials like metals, which expand on heating and contract when cooled back to their original temperature.
The formula that governs linear expansion is:
\[ ΔL = αL_0ΔT \]
Where:

• \( ΔL \) is the change in length,
• \( α \) is the coefficient of linear expansion,
• \( L_0 \) is the original length,
• \( ΔT \) is the change in temperature.

The coefficient of linear expansion \( α \) is unique to each material and provides an indication of how much the material will expand with a unit change in temperature.
Materials with a larger \( α \) will expand more than those with a smaller \( α \) for the same temperature change.

Principles of Thermal Expansion

Thermal expansion is a broader concept that covers the change in size, area, volume, or shape of a material due to temperature changes. This phenomenon is important in various engineering applications, as it helps in designing structures that can accommodate temperature-induced changes.
It is important to consider that different materials will react differently when exposed to changes in temperature.

• Solids, like metals and ceramics, primarily exhibit linear expansion.
• Liquids generally expand in volume when heated.
• Gases, following the ideal gas law, expand significantly more than solids and liquids.

Understanding thermal expansion is crucial for ensuring safety and integrity in engineering projects. For instance, bridges and railroads employ expansion joints to absorb the expansion and contraction due to weather changes, preventing potential deformations or structural failures.

Role of Thermal Physics

Thermal physics deals with the study of heat and temperature and their relation to energy and work. It encompasses understanding how materials expand when heated. Thermal physics explores the principles behind thermal expansion and provides the theoretical basis for the coefficient of linear expansion.
The behavior of particles within a material under the influence of temperature is one of the key areas of focus.

• Temperature is a measure of the average kinetic energy of the particles in a material, leading us to understand how temperature changes influence particle motion.
• Increased kinetic energy results in particles moving further apart, which manifests as expansion in solids.

By understanding these principles, engineers and scientists can predict how different materials will behave under varying thermal conditions, allowing them to design better and more reliable products.

Impact of Temperature Change

Temperature change is a critical factor when considering the expansion or contraction of materials. As the temperature rises, the kinetic energy of particles increases, leading to greater inter-particle distances. This increase in distance among particles results in the expansion of materials, primarily noted in the dimensions (length, area, or volume) depending on the state of the matter.

• A positive temperature change usually indicates expansion.
• A negative temperature change or cooling is associated with contraction, where particles lose kinetic energy and draw closer.

In our exercise, the rod expands because of a temperature increase from \(20.0^{\circ} \mathrm{C}\) to \(45.0^{\circ} \mathrm{C}\), demonstrating how even a relatively small temperature change can lead to noticeable alterations in material length.
Calculations related to temperature change and its impact on materials involve determining the coefficient of linear expansion to predict changes across different temperature ranges accurately.