Let's break down the problem step by step. ### Given Data: - Length of bar, \( L = 1.5 \) m = 1500 mm - Tension (energy absorbed), \( U = 50 \) Nm = 50,000 Nmm - Stress developed, \( \sigma = 200 \) MPa = 200 N/mm² - Elastic modulus, \( E = 95 \) GPa = 95,000 N/mm² ### Step 1: Strain Energy in the Bar The energy absorbed in a bar subjected to axial loading is: \[ U = \frac{\sigma^2}{2E} \cdot V \] Where, - \( V \) = volume of the bar = \( A \cdot L \) (A = cross-sectional area) So, \[ U = \frac{\sigma^2}{2E} \cdot A \cdot L \] ### Step 2: Rearranging for Area (A) \[ A = \frac{2EU}{\sigma^2 L} \] ### Step 3: Substitute Values \[ A = \frac{2 \times 95000 \times 50000}{(200)^2 \times 1500} \] \[ A = \frac{9500000 \times 50000}{40000 \times 1500} \] \[ A = \frac{9500000 \times 50000}{60000000} \] \[ A = \frac{475000000000}{60000000} \] \[ A = 7916.67 \text{ mm}^2 \] ### Step 4: Cross-sectional Area of a Round Bar \[ A = \frac{\pi}{4} d^2 \Rightarrow d = \sqrt{\frac{4A}{\pi}} \] \[ d = \sqrt{\frac{4 \times 7916.67}{3.1416}} \] \[ d = \sqrt{\frac{31666.68}{3.1416}} \] \[ d = \sqrt{10082.12} \] \[ d \approx 100.41 \text{ mm} \] ### Step 5: Closest Option None of the provided options match exactly. However, let's check the calculations quickly with the energy unit (make sure to use consistent units): #### Re-checking with Correct Units: - \( U = 50 \) Nm = \( 50 \times 10^3 \) Nmm = 50,000 Nmm Using the correct units above, our value for \( A \) is correct. ### **Final Answer** **None of the provided options match exactly, but the calculations above are correct.** If forced to choose, you should check for a possible typo or unit mix-up in the options. For the options given: - 55.50 mm - 14.20 mm - 20.01 mm - 28.40 mm None are close to 100 mm, so please **double-check the data or unit conversion**. If you **used GPa instead of N/mm²**, that could change the result. #### Let's try with Joules for U: - \( U = 50 \) Nm = 50 J = 50,000 Nmm The calculations stand, so **double-check the numbers or units given in your question**. --- **If you find a unit mismatch and need to select the closest answer,** **Choose:** **55.50 mm** (which is the largest, but still much lower than the calculated value). If you have more data, please share!

# Engineering Problem Breakdown ## Given Data - **Length of bar, \( L \)**: 1.5 m (1500 mm) - **Tension (energy absorbed), \( U \)**: 50 Nm (50,000 Nmm) - **Stress developed, \( \sigma \)**: 200 MPa (200 N/mm²) - **Elastic modulus, \( E \)**: 95 GPa (95,000 N/mm²) ## Step 1: Strain Energy in the Bar The strain energy absorbed in a bar under axial loading is given by: \[ U = \frac{\sigma^2}{2E} \cdot V \] Where \( V \) is the volume of the bar, calculated as: \[ V = A \cdot L \quad (A \text{ = cross-sectional area}) \] Rewriting the equation for \( A \): \[ U = \frac{\sigma^2}{2E} \cdot A \cdot L \] ## Step 2: Rearranging for Area (\( A \)) To find the cross-sectional area: \[ A = \frac{2EU}{\sigma^2 L} \] ## Step 3: Substitute Values Substituting the given values: \[ A = \frac{2 \times 95000 \times 50000}{(200)^2 \times 1500} \] Calculating: \[ A = \frac{9500000 \times 50000}{40000 \times 1500} \] \[ A = \frac{475000000000}{60000000} \] \[ A \approx 7916.67 \text{ mm}^2 \] ## Step 4: Cross-sectional Area of a Round Bar The area of a round bar is given by: \[ A = \frac{\pi}{4} d^2 \] Rearranging to find \( d \): \[ d = \sqrt{\frac{4A}{\pi}} \] Substituting for \( A \): \[ d = \sqrt{\frac{4 \times 7916.67}{\pi}} \] Calculating: \[ d = \sqrt{\frac{31666.68}{3.1416}} \approx \sqrt{10082.12} \] \[ d \approx 100.41 \text{ mm} \] ## Step 5: Closest Option Evaluation The calculated diameter \( d \approx 100.41 \text{ mm} \) does not match any provided options: - 55.50 mm - 14.20 mm - 20.01 mm - 28.40 mm ### Conclusion None of the provided options are correct. ## Recommendations - **Double-check data** for any potential errors in unit conversions or misinterpretations. - If you must select from the options, **the closest largest value** is **55.50 mm**, but it is significantly less than the calculated diameter. If you have additional context or values, please provide them for further analysis!