To tackle the problem of finding the ratio of the lengths of an iron rod and an aluminum rod such that the difference in their lengths remains constant regardless of temperature changes, we need to delve into the concept of linear expansion. The coefficients of linear expansion for iron and aluminum are given as 12 x 10^-6 /°C and 23 x 10^-6 /°C, respectively. Let's break this down step by step.
Understanding Linear Expansion
Linear expansion refers to the way materials change in length when subjected to temperature variations. The change in length (ΔL) of a rod can be expressed using the formula:
ΔL = L₀ * α * ΔT
- L₀ = original length of the rod
- α = coefficient of linear expansion
- ΔT = change in temperature
For our two rods, we can denote the original lengths as L₁ for iron and L₂ for aluminum. The changes in length due to temperature changes can be expressed as:
- ΔL₁ = L₁ * (12 x 10^-6) * ΔT
- ΔL₂ = L₂ * (23 x 10^-6) * ΔT
Setting Up the Equation
According to the problem, we want the difference in lengths of the two rods to remain constant as the temperature changes. This means:
L₁ + ΔL₁ - (L₂ + ΔL₂) = constant
Substituting the expressions for ΔL₁ and ΔL₂, we have:
L₁ + L₁ * (12 x 10^-6) * ΔT - (L₂ + L₂ * (23 x 10^-6) * ΔT) = constant
Rearranging this gives us:
(L₁ - L₂) + (L₁ * 12 x 10^-6 - L₂ * 23 x 10^-6) * ΔT = constant
Analyzing the Temperature Independence
For the difference in lengths to be independent of temperature, the coefficient of ΔT must equal zero. This leads us to the equation:
L₁ * 12 x 10^-6 = L₂ * 23 x 10^-6
Finding the Ratio
From the equation above, we can derive the ratio of the lengths:
L₁ / L₂ = (23 x 10^-6) / (12 x 10^-6)
By simplifying this, we find:
L₁ / L₂ = 23 / 12
Conclusion
Thus, the ratio of the lengths of the iron rod to the aluminum rod, ensuring that the difference in lengths remains independent of temperature changes, is 23:12. This means that for every 23 units of length of the iron rod, there should be 12 units of length of the aluminum rod. Understanding the role of temperature independence is crucial, as it allows us to set up the equation that leads to this ratio, ensuring that the expansion characteristics of both materials balance each other out perfectly.
