Understanding Temperature Expansion: A Practical Example with Brass Ring

In this article, we will explore the practical applications of the linear expansivity concept using a brass ring. We will solve a real-world problem involving the expansion of a brass ring due to an increase in temperature, and in the process, we will understand the underlying principles and their significance in the field of materials science.

Problem Statement

The diameter of a brass ring is 50.0 cm at a temperature of 30 degrees Celsius. To achieve a diameter of 50.29 cm, we need to find the required temperature. The linear expansivity of brass is given as 0.000019 K-1.

Understanding Linear Expansivity

Linear expansivity, denoted as αL, is a material's property that indicates how much its linear dimensions change with a small increase in temperature. It is typically measured in units such as cm/K or m/K, representing the change in length per degree Kelvin. In this example, we will use cm/K.

The formula for linear expansivity is given by the following equation:

ΔL / L0 αL ΔT

Here, ΔL is the change in length, L0 is the original length, αL is the linear expansivity, and ΔT is the change in temperature.

Solving the Problem

We need to solve for the required temperature to achieve the desired diameter increase. Let's start by rewriting the problem statement using the given values:

50.0 cm at 30°C becomes 50.29 cm

Change in diameter, ΔD 50.29 cm - 50.0 cm 0.29 cm

Linear expansivity, αL 0.000019 K-1

Initial temperature, T0 30°C

We need to find the final temperature, T, such that the change in diameter is 0.29 cm. The equation can be rearranged to solve for the change in temperature, ΔT:

ΔT ΔD / (L0 × αL)

Substituting the given values:

ΔT 0.29 cm / (50.0 cm × 0.000019 K-1)

Calculate ΔT:

ΔT 0.29 / (50.0 × 0.000019) 305.6 K

To find the final temperature, T, we add this change in temperature to the initial temperature:

T 30 305.6 335.6°C

Thus, the brass ring must be heated to approximately 335.6°C to achieve the desired diameter of 50.29 cm.

Key Concepts

Linear Expansivity (αL): A material's property indicating how much its linear dimensions change with a small increase in temperature. It is typically measured in units such as cm/K or m/K. Temperature Scale Conversion: It is important to understand the difference between Celsius and Kelvin. Both scales have the same increment in heat content per degree. Celsius is set to zero at the freezing point of water, while Kelvin is set to zero at the absolute zero point, the point at which all molecular motion ceases. Thermal Expansion: The increase in the size of a material due to an increase in temperature. This concept is crucial in materials science and engineering for designing and manufacturing processes.

Conclusion

Understanding the linear expansivity of materials is essential in various applications, including the design of precise engineering components, temperature sensing devices, and even everyday objects. By solving practical problems such as the one discussed in this article, we can deepen our understanding of the underlying principles and their real-world implications.

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Note: When writing expressions using SI units, always include a space between the coefficient and the unit symbol (e.g., 50.29 cm), as this is a standard convention in scientific notation.