To determine the length of the combined system of steel and aluminum rods when the temperature is raised, we need to consider the effects of thermal expansion on each material. The key here is to understand how each rod expands differently due to their distinct coefficients of linear expansion.

Understanding Thermal Expansion

When materials are heated, they tend to expand. The amount of expansion can be quantified using the formula:

ΔL = α × L₀ × ΔT

Where:

• ΔL = change in length
• α = coefficient of linear expansion
• L₀ = original length
• ΔT = change in temperature

Given Parameters

Let's denote the following:

• Length of each rod, L₀ = l₀
• Change in temperature, ΔT = T (the temperature increase)
• Coefficient of linear expansion for aluminum, αₐ
• Coefficient of linear expansion for steel, αₛ

Calculating the Change in Length

For the aluminum rod, the change in length (ΔLₐ) can be calculated as follows:

ΔLₐ = αₐ × l₀ × T

For the steel rod, the change in length (ΔLₛ) is given by:

ΔLₛ = αₛ × l₀ × T

Finding the Total Length

Since the rods are joined rigidly, the total length of the system after heating (L_total) will be the sum of the individual changes in length:

L_total = l₀ + ΔLₐ + ΔLₛ

Substituting the expressions for ΔLₐ and ΔLₛ, we get:

L_total = l₀ + (αₐ × l₀ × T) + (αₛ × l₀ × T)

Factoring Out Common Terms

We can factor out l₀ from the equation:

L_total = l₀ (1 + αₐ × T + αₛ × T)

Final Expression for Length

Thus, the final expression for the length of the system when the temperature is raised to T degrees Celsius is:

L_total = l₀ (1 + T(αₐ + αₛ))

This equation shows how the combined length of the system depends on the original length, the temperature increase, and the coefficients of linear expansion of both materials. Each material contributes to the overall expansion, and since they are rigidly joined, the total length reflects the cumulative effect of their thermal expansions.