Let's solve the problem step by step: ### **Given Data:** - Length of rod, \( L = 2 \) m \( = 2000 \) mm - Gap, \( \delta = 0.25 \) mm - Temperature rise, \( \Delta T = 100^\circ C \) - Coefficient of thermal expansion, \( \alpha = 18 \times 10^{-6} /^\circ C \) - Young's modulus, \( E = 1 \times 10^5 \) N/mm² --- ### **Step 1: Free Expansion Due to Temperature Rise** The free expansion if there was no restraint: \[ \Delta L_{\text{free}} = L \cdot \alpha \cdot \Delta T = 2000 \times 18 \times 10^{-6} \times 100 = 2000 \times 0.0018 = 3.6 \text{ mm} \] --- ### **Step 2: Actual Expansion** Since there is a gap, the rod will first expand by 0.25 mm to close the gap, and then compressive stress will develop for the remaining expansion. Expansion causing stress: \[ \Delta L_{\text{stress}} = \Delta L_{\text{free}} - \delta = 3.6 - 0.25 = 3.35 \text{ mm} \] --- ### **Step 3: Compressive Stress Developed** The compressive strain, \( \epsilon \), is: \[ \epsilon = \frac{\Delta L_{\text{stress}}}{L} = \frac{3.35}{2000} = 0.001675 \] The stress, \( \sigma \), is: \[ \sigma = E \cdot \epsilon = 1 \times 10^5 \times 0.001675 = 167.5 \text{ N/mm}^2 \] --- ### **Step 4: Correct Answer** The correct answer is: \[ \boxed{167.50 \text{ N/mm}^2} \] So, select **167.50 N/mm²**.