To determine the change in length of the system when the temperature is raised, we need to consider the properties of each material involved: the two steel rods and the aluminum rod. Since they are joined rigidly and initially at zero tension at 0 °C, we can analyze how each material expands with temperature and how that affects the overall length of the system.

Understanding Linear Expansion

Linear expansion refers to the increase in length of a material when its temperature rises. 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 of the material
• ΔT = change in temperature

Coefficients of Linear Expansion

For our scenario, let’s denote:

• α₁ = a for aluminum
• α₂ = s for steel

Calculating Changes in Length

Assuming the original length of each rod is L, we can calculate the change in length for each rod when the temperature increases by ΔT.

Change in Length of Each Rod

The change in length for the aluminum rod will be:

ΔL₁ = L * a * ΔT

For the first steel rod, the change in length will be:

ΔL₂ = L * s * ΔT

For the second steel rod, the change in length will also be:

ΔL₃ = L * s * ΔT

Overall Change in Length of the System

Since the rods are arranged in a vertical stack (aluminum in the middle), the total change in length of the system can be expressed as:

ΔL_total = ΔL₁ + ΔL₂ + ΔL₃

Substituting the individual changes in length, we have:

ΔL_total = (L * a * ΔT) + (L * s * ΔT) + (L * s * ΔT)

This simplifies to:

ΔL_total = L * (a + 2s) * ΔT

Final Length of the System

The final length of the system after the temperature increase can be calculated by adding the original length to the total change in length:

L_final = L + ΔL_total

Substituting for ΔL_total gives:

L_final = L + L * (a + 2s) * ΔT

Factoring out L, we find:

L_final = L * (1 + (a + 2s) * ΔT)

Conclusion

Thus, the final length of the system when the temperature is raised is given by the formula:

L_final = L * (1 + (a + 2s) * ΔT)

This formula allows you to calculate the new length based on the original length, the coefficients of linear expansion for aluminum and steel, and the change in temperature. Make sure to substitute the appropriate values for L, a, s, and ΔT to find the specific final length for your scenario.