To determine the Young's modulus of the wire, we need to use the formula for Young's modulus (E), which is defined as the ratio of stress (σ) to strain (ε): \[ E = \frac{\sigma}{\epsilon} \] ### Step 1: Identify the Given Values - **Stress (σ)**: Given as \(10^7 \, \text{N/m}^2\) - **Original Length (L₀)**: \(10.00 \, \text{cm} = 0.10 \, \text{m}\) - **Stretched Length (L)**: \(10.05 \, \text{cm} = 0.1005 \, \text{m}\) ### Step 2: Calculate the Change in Length (ΔL) The change in length (ΔL) can be calculated as: \[ \Delta L = L - L_0 = 0.1005 \, \text{m} - 0.10 \, \text{m} = 0.0005 \, \text{m} \] ### Step 3: Calculate the Strain (ε) Strain (ε) is defined as the change in length divided by the original length: \[ \epsilon = \frac{\Delta L}{L_0} = \frac{0.0005 \, \text{m}}{0.10 \, \text{m}} = 0.005 \] ### Step 4: Calculate Young's Modulus (E) Now we can substitute the values of stress and strain into the Young's modulus formula: \[ E = \frac{\sigma}{\epsilon} = \frac{10^7 \, \text{N/m}^2}{0.005} \] Calculating this gives: \[ E = \frac{10^7}{0.005} = 2 \times 10^9 \, \text{N/m}^2 \] ### Conclusion The Young's modulus of the wire is \(2 \times 10^9 \, \text{N/m}^2\). ### Correct Option Thus, the correct answer is **D. \(2.0 \times 10^9 \, \text{N/m}^2\)**. ### Explanation of Other Options - **A. \(5.0 \times 10^4 \, \text{N/m}^2\)**: This value is too low and does not reflect the calculated Young's modulus. It likely results from a misunderstanding of the stress or strain values. - **B. \(5.0 \times 10^5 \, \text{N/m}^2\)**: This is also too low. It may arise from incorrect calculations or misinterpretation of the stress or strain. - **C. \(2.0 \times 10^8 \, \text{N/m}^2\)**: This value is closer but still significantly lower than the calculated Young's modulus. It suggests a miscalculation in the strain or stress. ### Common Pitfalls - **Unit Conversion**: Always ensure that units are consistent (e.g., converting cm to m). - **Understanding Stress and Strain**: Stress is force per unit area, while strain is a dimensionless ratio of change in length to original length. - **Calculation Errors**: Double-check arithmetic operations, especially when dividing or multiplying by small numbers. ### Revision Summary - Young's modulus (E) is calculated using the formula \(E = \frac{\sigma}{\epsilon}\). - Stress (σ) is given, and strain (ε) is calculated from the change in length. - Ensure all units are consistent, particularly when converting lengths. - The correct answer for the Young's modulus in this case is \(2.0 \times 10^9 \, \text{N/m}^2\).