Problem 20 - University Physics with Modern Physics

Question

(a) If an area measured on the surface of a solid body is \(A_{0}\) at some initial temperature and then changes by \(\Delta A\) when the temperature changes by \(\Delta T,\) show that $$\Delta A=(2 \alpha) A_{0} \Delta T$$ where \(\alpha\) is the coefficient of linear expansion. (b) A circular sheet of aluminum is 55.0 \(\mathrm{cm}\) in diameter at \(15.0^{\circ} \mathrm{C}\) . By how much does the area of one side of the sheet change when the temperature increases to \(27.5^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

The area increases by approximately 14.3 cm² when the temperature rises from 15.0°C to 27.5°C.

Step 1: Understand the Concept

The problem involves thermal expansion, which refers to the increase in size of a material as its temperature rises. The coefficient of linear expansion, \( \alpha \), is used to quantify the change in length of a material as its temperature changes. The problem requires us to find the change in area due to temperature change on a solid body.

Step 2: Derive the Formula for Area Expansion

The linear expansion formula is given by \( \Delta L = \alpha L_{0} \Delta T \), where \( L_{0} \) is the initial length. For area expansion, the change \( \Delta A \) can be derived similarly, considering the body’s dimensions expand linearly in two directions (length and width). Hence, \( \Delta A = (2 \alpha) A_{0} \Delta T \).

Step 3: Setting Up For Part (b)

Given: Diameter of aluminum sheet = 55.0 cm at 15.0°C. We need to calculate the change in area when temperature increases to 27.5°C. First, calculate the initial area, \( A_{0} \).

Step 4: Calculate Initial Area

The initial area of the circular sheet is \( A_{0} = \pi \left(\frac{d}{2}\right)^{2} \), where \( d = 55.0 \) cm. Calculate \( A_{0} \) using this formula.

Step 5: Determine Temperature Change

Calculate the change in temperature: \( \Delta T = 27.5°C - 15.0°C = 12.5°C \).

Step 6: Apply Area Expansion Formula

Use the derived formula \( \Delta A = (2 \alpha) A_{0} \Delta T \). For aluminum, \( \alpha = 24 \times 10^{-6} /°C \). Substitute \( A_{0} \), \( \alpha \), and \( \Delta T \) to find \( \Delta A \).

Step 7: Final Calculation

Perform the calculation for \( \Delta A \) using all the substituted values from the previous steps, ensuring you keep track of units and significant figures.

Key Concepts

Coefficient of Linear Expansion

The coefficient of linear expansion, denoted as \( \alpha \), is a fundamental concept in thermal dynamics. It quantifies how much a material's length changes per degree change in temperature. Specifically, \( \alpha \) is defined as the fractional change in length per unit change in temperature. This property is crucial for understanding how materials expand or contract when heated or cooled.

When a material undergoes a change in temperature, its new length can be evaluated using the formula:

• \( \Delta L = \alpha L_0 \Delta T \)

where \( \Delta L \) is the change in length, \( L_0 \) is the original length, and \( \Delta T \) is the temperature change. This formula applies to linear dimensions, such as the length of a rod or the height of a building.

It is important to recognize that different materials have different coefficients of linear expansion. For example, aluminum expands more per degree than steel, so knowing the material-specific \( \alpha \) is essential for precise calculations in engineering and construction.

Area Expansion

Area expansion refers to the change in area of a solid body when exposed to a temperature change. When we consider thermal expansion in two dimensions, we are interested in how both the length and width of a material increase due to heating.

The formula to calculate the change in area, \( \Delta A \), when the temperature changes is:

• \( \Delta A = (2 \alpha) A_{0} \Delta T \)

where \( A_{0} \) is the initial area, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature. Notice that the factor \(2\) is introduced because we consider expansion in both dimensions (length and width).

This phenomenon is crucial for surfaces like metal sheets or glass panes. In the exercise, for instance, the circular aluminum sheet experiences a temperature increase, resulting in area expansion. Using the correct \( \alpha \) value for aluminum ensures an accurate calculation of the increase in area.

Temperature Change

Temperature change, noted as \( \Delta T \), plays a pivotal role in calculating thermal expansion. It is simply the difference between the final temperature and the initial temperature. In mathematical terms, it's given by:

• \( \Delta T = T_{final} - T_{initial} \)

where \( T_{final} \) is the temperature after heating, and \( T_{initial} \) is the starting temperature. Instructions on using this change are straightforward; multiplying \( \Delta T \) directly affects how much a material expands.

In our exercise, the circular aluminum sheet is initially at \( 15.0^{\circ} \mathrm{C} \) and is heated to \( 27.5^{\circ} \mathrm{C} \). Therefore, the temperature change is calculated as \( \Delta T = 27.5^{\circ} \mathrm{C} - 15.0^{\circ} \mathrm{C} = 12.5^{\circ} \mathrm{C} \). This \( \Delta T \) is then used in the area expansion formula to determine how much the area increases due to the temperature rise.