Thermal Expansion of Iron: Calculating Rod Length at 50°C

The question of determining the length of an iron rod at 50°C, given its length at 0°C and 100°C, is a common problem in both textbooks and practical scenarios involving materials science and thermodynamics. Let's explore this question in depth and understand why the problem is often approached differently from the initial assumption.

The Given Data and Initial Assumptions

The iron rod is 50 cm long at 0°C and 50.06 cm long at 100°C. The question asks for its length at 50°C. The initial argument that the difference in length (0.06 cm) cannot be directly attributed to the temperature change due to insufficient precision is a valid point.

Significant Figures and Precision

When dealing with measurements and calculations, significant figures play a crucial role in determining the accuracy of the results. In this case, if the measuring device is accurate to the nearest 0.01 cm, it would be more appropriate to consider the measurements with the same precision. The difference of 0.06 cm can be attributed to the temperature change if we assume linearity, but this assumption requires justification and context.

Linear Expansion and Thermal Coefficient

Iron, like most metals, exhibits a more or less linear thermal expansion behavior over small temperature ranges. However, without the exact coefficients, we can make an educated guess based on the provided data.

Linear Interpolation Method

To find the length of the iron rod at 50°C, we can use linear interpolation. The linear expansion formula for a material can be expressed as:

L L0 (1 αΔT)

Where:

L is the length at the new temperature, L0 is the original length at the reference temperature (0°C), α is the coefficient of thermal expansion, ΔT is the change in temperature.

Given that the rod is 50 cm at 0°C and 50.06 cm at 100°C, we can estimate the coefficient of expansion:

α (50.06 cm - 50 cm) / (100°C - 0°C) 0.0006 cm/°C

Using linear interpolation, we can estimate the length at 50°C:

L50°C 50 cm (0.0006 cm/°C) * (50°C - 0°C) 50 cm 0.03 cm 50.03 cm

Explanation and Discussion

The linear expansion coefficient of 0.0006 cm/°C implies that for every degree Celsius increase, the rod expands by 0.0006 cm. Therefore, over a 50°C increase, the rod would expand by 0.03 cm, leading to an overall length of 50.03 cm at 50°C.

Assumptions and Precision

It's important to note that this approximation works well for small temperature changes and for materials like iron that have relatively linear thermal expansion behavior. However, in more precise engineering and materials science applications, the actual coefficient of expansion might vary slightly, and more accurate methods would be used.

Conclusion

In summary, given the linear nature of iron's thermal expansion and the provided data, the length of the iron rod at 50°C would be approximately 50.03 cm. This result is derived using a simplified linear interpolation method based on the available temperature data.

This article aims to provide a clear understanding of how to approach such problems, emphasizing the importance of context, precision, and the linear approximation in thermal expansion calculations.

References

Linear Expansion of Materials Linear Expansion Formula