Problem 19 - University Physics with Modern Physics

Question

A machinist bores a hole of diameter 1.35 cm in a steel plate that is at 25.0\(^\circ\)C. What is the cross-sectional area of the hole (a) at 25.0\(^\circ\)C and (b) when the temperature of the plate is increased to 175\(^\circ\)C? Assume that the coefficient of linear expansion remains constant over this temperature range.

Step-by-Step Solution

Answer

(a) 1.4305 cm² at 25°C; (b) 1.4353 cm² at 175°C.

Step 1: Calculate Initial Cross-Sectional Area

At the initial temperature of 25°C, the diameter of the hole is 1.35 cm. The cross-sectional area \( A \) of a circular hole is given by the formula \( A = \pi r^2 \), where \( r \) is the radius. First, we need to calculate the radius: \( r = \frac{1.35}{2} = 0.675 \) cm. Hence, the initial area is \( A = \pi (0.675)^2 = 1.4305 \) cmeat{2}.

Step 2: Calculate Diameter Expansion

To find the diameter at 175°C, we apply the linear expansion formula for length: \( \Delta L = \alpha L_0 \Delta T \), where \( \alpha \) is the coefficient of linear expansion, \( L_0 \) is the initial length (diameter here), and \( \Delta T \) is the change in temperature. Assuming a typical value for steel, \( \alpha \approx 11 \times 10^{-6} \text{°C}^{-1} \). We have \( L_0 = 1.35 \text{ cm} \) and \( \Delta T = 175 - 25 = 150 \text{°C} \). Thus, \( \Delta L = 11 \times 10^{-6} \times 1.35 \times 150 \approx 0.0022 \text{ cm} \). The new diameter is \( 1.3522 \text{ cm} \).

Step 3: Calculate New Cross-Sectional Area

With the new diameter of 1.3522 cm at 175°C, find the new radius: \( r = \frac{1.3522}{2} = 0.6761 \text{ cm} \). Now, calculate the new area: \( A = \pi (0.6761)^2 \approx 1.4353 \text{ cm}^2 \).

Step 4: Summarize Results

The cross-sectional area at 25°C is approximately 1.4305 cm². At 175°C, after thermal expansion, the cross-sectional area is approximately 1.4353 cm².

Key Concepts

Linear Expansion

Linear expansion is a phenomenon that occurs when materials expand in size due to an increase in temperature. As a material heats up, its molecules move more vigorously, causing an increase in the distance between them. As a result, the material expands along its dimensions. Linear expansion specifically refers to the change in length or diameter of a material, which can be calculated using a straightforward formula:

\[ \Delta L = \alpha L_0 \Delta T \]

where:

• \( \Delta L \) is the change in length or diameter.
• \( \alpha \) is the coefficient of linear expansion, which varies for different materials.
• \( L_0 \) is the original length or diameter.
• \( \Delta T \) is the change in temperature.

This relationship helps us understand how the diameter of a hole in a steel plate can change with temperature. By applying the formula, we can determine that increasing the temperature of the plate from 25°C to 175°C results in a slight increase in diameter from 1.35 cm to 1.3522 cm. These small changes can have significant implications in engineering and construction, where precise dimensions are critical.

Cross-Sectional Area

The cross-sectional area refers to the surface area of a two-dimensional slice perpendicular to the length of an object. In the case of a circular hole, we calculate this area using the formula for the area of a circle. Since the cross-section of the hole is circular, the area is given by:

\[ A = \pi r^2 \]

Here, \( r \) is the radius of the hole, which is half of the diameter. Calculating the cross-sectional area involves determining the radius from the given diameter, then substituting this value into the formula. At 25°C, for a hole with a diameter of 1.35 cm, the radius is 0.675 cm, resulting in a cross-sectional area of approximately 1.4305 cm².

When the temperature increases and the diameter changes due to thermal expansion, we repeat the calculation with the new radius. This demonstrates how sensitive the cross-sectional area is to changes in dimensions, reflecting the effects of linear expansion on physical properties.

Coefficient of Linear Expansion

The coefficient of linear expansion, \( \alpha \), is a crucial factor in understanding how materials respond to temperature changes. It quantifies the extent to which a material expands per degree Celsius of temperature change. Each material has its own specific coefficient, derived from its molecular structure and bonding properties.

In our exercise, we used a typical value for steel, which is \( 11 \times 10^{-6} \text{ °C}^{-1} \). This means that for every 1°C increase in temperature, steel will expand by 11 millionths of its original length. Knowing this coefficient allows engineers and designers to predict changes in material dimensions under varying thermal conditions.

The coefficient is essential for calculating changes in length, diameter, and subsequently, the cross-sectional area. Without understanding \( \alpha \), it would be challenging to ensure structural integrity and functionality of components that experience temperature fluctuations, particularly in environments that experience extreme temperature variations.