To solve this problem, we are addressing the expansion or contraction of materials (aluminum and steel) based on the given temperatures, using their thermal expansion coefficients.

Given Information:

1. Initial temperature: $21^\circ C$
2. Final temperatures:
   • $80^\circ C$ (Expansion)
   • $-30^\circ C$ (Contraction)
3. Material properties:
   • Aluminum (Al) and steel (Fe) have different coefficients of linear thermal expansion ($\alpha$).
   • $\alpha_{\text{Al}} = 23 \times 10^{-6} \, \text{°C}^{-1}$
   • $\alpha_{\text{Fe}} = 12 \times 10^{-6} \, \text{°C}^{-1}$
4. Length of material: $L = 0.65 \, \text{m}$

Formula:

The change in length ($\Delta L$) is given by: $\Delta L = \alpha \cdot L \cdot \Delta T$ Where:

• $\alpha$: Coefficient of thermal expansion
• $L$: Initial length
• $\Delta T$: Temperature change

Step-by-Step Solution:

(a) Temperature = $80^\circ C$:

$\Delta T = 80 - 21 = 59 \, \text{°C}$

Expansion of Aluminum: $\Delta L_{\text{Al}} = \alpha_{\text{Al}} \cdot L \cdot \Delta T$ $\Delta L_{\text{Al}} = (23 \times 10^{-6}) \cdot 0.65 \cdot 59 = 0.00088135 \, \text{m} = 0.881 \, \text{mm}$

Expansion of Steel: $\Delta L_{\text{Fe}} = \alpha_{\text{Fe}} \cdot L \cdot \Delta T$ $\Delta L_{\text{Fe}} = (12 \times 10^{-6}) \cdot 0.65 \cdot 59 = 0.0004602 \, \text{m} = 0.460 \, \text{mm}$

Difference: $\Delta L_{\text{diff}} = \Delta L_{\text{Al}} - \Delta L_{\text{Fe}}$ $\Delta L_{\text{diff}} = 0.881 - 0.460 = 0.421 \, \text{mm}$

(b) Temperature = $-30^\circ C$:

$\Delta T = -30 - 21 = -51 \, \text{°C}$

Contraction of Aluminum: $\Delta L_{\text{Al}} = \alpha_{\text{Al}} \cdot L \cdot \Delta T$ $\Delta L_{\text{Al}} = (23 \times 10^{-6}) \cdot 0.65 \cdot (-51) = -0.00076185 \, \text{m} = -0.762 \, \text{mm}$

Contraction of Steel: $\Delta L_{\text{Fe}} = \alpha_{\text{Fe}} \cdot L \cdot \Delta T$ $\Delta L_{\text{Fe}} = (12 \times 10^{-6}) \cdot 0.65 \cdot (-51) = -0.0003978 \, \text{m} = -0.398 \, \text{mm}$

Difference: $\Delta L_{\text{diff}} = \Delta L_{\text{Fe}} - \Delta L_{\text{Al}}$ $\Delta L_{\text{diff}} = -0.762 - (-0.398) = -0.364 \, \text{mm}$

Final Results:

• (a) At $80^\circ C$: Aluminum is 0.421 mm longer than steel.
• (b) At $-30^\circ C$: Aluminum is 0.364 mm shorter than steel.

Would you like additional clarification or details about this? Here are five questions to consider:

1. How are the thermal expansion coefficients derived for different materials?
2. Why does aluminum expand more than steel?
3. How can we account for cumulative temperature cycling effects?
4. What are practical implications of this length difference in construction?
5. How does the thickness of the material affect the expansion?

Tip: Always verify the coefficients of expansion for precise calculations as they might slightly vary depending on the alloy or material purity.