To tackle this problem, we need to delve into the concepts of linear expansion and how it relates to the lengths of two rods made from different materials. The key idea here is that if the lengths of the rods are inversely proportional to their coefficients of linear expansion, we can derive a relationship that holds true at all temperatures. 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 changes. The formula for linear expansion is given by:
ΔL = L₀ * α * ΔT
Where:
- ΔL = change in length
- L₀ = original length
- α = coefficient of linear expansion
- ΔT = change in temperature
Setting Up the Problem
Let’s denote the lengths of the steel and brass rods at 0ºC as L₁ and L₂, respectively. Their coefficients of linear expansion will be α₁ for steel and α₂ for brass. According to the problem, we have:
L₁ ∝ 1/α₁ and L₂ ∝ 1/α₂
This means we can express the lengths as:
L₁ = k/α₁ and L₂ = m/α₂
Where k and m are constants of proportionality. The difference in length between the two rods at any temperature T can be expressed as:
ΔL = L₁ - L₂
Calculating the Difference in Length
Now, substituting our expressions for L₁ and L₂ into the difference:
ΔL = (k/α₁) - (m/α₂)
As temperature changes, both rods will expand according to their respective coefficients. The change in length for each rod can be expressed as:
ΔL₁ = L₁ * α₁ * ΔT and ΔL₂ = L₂ * α₂ * ΔT
Thus, the new lengths at temperature T will be:
L₁(T) = L₁ + ΔL₁ = L₁ + L₁ * α₁ * ΔT
L₂(T) = L₂ + ΔL₂ = L₂ + L₂ * α₂ * ΔT
Substituting these into the difference gives:
ΔL(T) = (L₁ + L₁ * α₁ * ΔT) - (L₂ + L₂ * α₂ * ΔT)
Rearranging this, we can see that the terms involving ΔT will cancel out if we maintain the relationship between k and m such that:
k/α₁ - m/α₂ = constant
This shows that the difference in length remains constant at all temperatures, provided the initial lengths are set correctly based on their coefficients of linear expansion.
Finding Specific Lengths for Steel and Brass
Now, let’s determine the lengths of the steel and brass rods at 0ºC so that their difference in length is 0.30 m. We know:
ΔL = L₁ - L₂ = 0.30 m
Assuming the coefficients of linear expansion are approximately:
- α₁ (steel) ≈ 11 × 10⁻⁶ /ºC
- α₂ (brass) ≈ 19 × 10⁻⁶ /ºC
Using the relationship we established earlier, we can set:
L₁ = k/α₁ and L₂ = m/α₂
To satisfy the condition that L₁ - L₂ = 0.30 m, we can express this as:
k/α₁ - m/α₂ = 0.30
To find specific values for L₁ and L₂, we can choose a value for k and m that satisfies this equation. For instance, if we let:
k = 0.30 * α₁ * α₂
m = 0.30 * α₂
Substituting these values into the equation will yield the lengths of the rods. After some calculations, you can find that:
L₁ = 0.30 * (19 × 10⁻⁶) / (11 × 10⁻⁶) + 0.30
L₂ = 0.30
Thus, you can derive the specific lengths for both rods that will maintain a constant difference of 0.30 m across all temperatures. This approach illustrates the beauty of physics in understanding material properties and their interactions with temperature changes.
