Problem 83 - University Physics with Modern Physics

Question

A metal rod that is 30.0 \(\mathrm{cm}\) long expands by 0.0650 \(\mathrm{cm}\) when its temperature is raised from \(0.0^{\circ} \mathrm{C}\) to \(100.0^{\circ} \mathrm{C} .\) A rod of a different metal and of the same length expands by 0.0350 \(\mathrm{cm}\) for the same rise in temperature. A third rod, also 30.0 \(\mathrm{cm}\) long, is made up of pieces of each of the above metals placed end to end and expands 0.0580 \(\mathrm{cm}\) between \(0.0^{\circ} \mathrm{C}\) and \(100.0^{\circ} \mathrm{C} .\) Find the length of each portion of the composite rod.

Step-by-Step Solution

Answer

The composite rod consists of 26.0 cm of the first metal and 4.0 cm of the second metal.

Step 1: Understanding Thermal Expansion

Thermal expansion of a material can be described by the formula \( \Delta L = L \cdot \alpha \cdot \Delta T \), where \( \Delta L \) is the change in length, \( L \) is the initial length, \( \alpha \) is the coefficient of linear expansion, and \( \Delta T \) is the change in temperature.

Step 2: Calculate Coefficients of Expansion

For the first metal rod: \( 0.0650 = 30.0 \cdot \alpha_1 \cdot 100 \). Solving for \( \alpha_1 \), we get \( \alpha_1 = \frac{0.0650}{30.0 \cdot 100} \). For the second metal rod: \( 0.0350 = 30.0 \cdot \alpha_2 \cdot 100 \), solving for \( \alpha_2 \), we get \( \alpha_2 = \frac{0.0350}{30.0 \cdot 100} \).

Step 3: Calculate Expansion of Composite Rod

For the composite rod made from lengths \( x \) of the first metal and \( y \) of the second metal such that \( x + y = 30.0 \), the total expansion \( 0.0580 = x \cdot \alpha_1 \cdot 100 + y \cdot \alpha_2 \cdot 100 \).

Step 4: Solve the System of Equations

From \( x + y = 30.0 \) and substituting \( y = 30 - x \) into the expansion equation gives \( 0.0580 = x \cdot \alpha_1 \cdot 100 + (30 - x) \cdot \alpha_2 \cdot 100 \). Substitute the values of \( \alpha_1 \) and \( \alpha_2 \) from Step 2 and solve for \( x \).

Step 5: Calculate Lengths of Each Portion

Substituting back, we find \( x = 26.0 \text{ cm} \) and \( y = 4.0 \text{ cm} \). These represent the length of the first and second metal, respectively, in the composite rod.

Key Concepts

Linear Expansion Coefficient

The linear expansion coefficient, denoted as \( \alpha \), is a fundamental property that quantifies how much a material expands per degree of temperature change. To calculate \( \alpha \) for any material, we use the formula: \[ \Delta L = L \cdot \alpha \cdot \Delta T \] where:

• \( \Delta L \) is the change in length of the material.
• \( L \) is the original length of the material before temperature change.
• \( \Delta T \) is the change in temperature.

For instance, if we have a metal rod with an initial length of 30.0 cm that expands by 0.0650 cm when heated from 0°C to 100°C, the linear expansion coefficient can be calculated as: \[ \alpha = \frac{\Delta L}{L \cdot \Delta T} = \frac{0.0650}{30.0 \cdot 100} \]. This coefficient is specific to each material, effectively serving as a fingerprint for how a material responds to heat.

Composite Materials

Composite materials integrate multiple distinct materials to leverage the benefits of each. In the case of our rods, we have two different metals combined end to end, creating a composite rod. A unique challenge with composites is predicting how they behave under temperature changes due to the different linear expansion coefficients. For instance, if we have a composite rod made by joining two rods of different metals, it will expand by a combination of each metal's expansion:

• The formula used is: \[ 0.0580 = x \cdot \alpha_1 \cdot 100 + y \cdot \alpha_2 \cdot 100 \] where \( x \) and \( y \) are lengths of each metal in the composite.
• This accounts for the length contributions from each metal to the total expansion based on their specific \( \alpha \) values.

This composite approach helps in designing materials that exploit individual strengths while minimizing weaknesses, a common practice in engineering.

Temperature Change Effects

Temperature changes impact materials by causing them to expand or contract. When a metal is heated, its particles gain energy and move more, causing an increase in length. This property, while useful, requires careful consideration when designing materials needing precision, such as machinery components or engines. In the exercise, the metal rods expand differently when subject to the same temperature rise, illustrating the variance caused by their unique \( \alpha \) values.

• For the metal rod with a higher \( \alpha \), a greater expansion is observed, while the rod with a lower \( \alpha \) expands less.
• Temperature-induced expansion is crucial in scenarios like bridge construction where expansion joints are required to accommodate changes.

Understanding these effects helps in planning artifacts and structures to ensure they can withstand temperature fluctuations without structural failure.