SEP Use Mathematics \( A \) wire with a length of 3.0 m and a cross- sectional area of \( 0.0020 \mathrm{~m}^{2} \) is stretched 0.10 cm by a \( 98-\mathrm{kg} \) hanging weight. Determine Young's modulus for the wire.

To find Young's modulus (E) for the wire, we can use the formula: \[ E = \frac{\text{stress}}{\text{strain}} \] First, calculate the stress (force per unit area) applied to the wire. The force (F) is equal to the weight of the hanging object: \[ F = mg = 98 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 962.58 \, \text{N} \] Now, stress is defined as force divided by the cross-sectional area (A): \[ \text{Stress} = \frac{F}{A} = \frac{962.58 \, \text{N}}{0.0020 \, \text{m}^2} = 481290 \, \text{N/m}^2 \] Next, we need to calculate strain, which is the change in length (ΔL) divided by the original length (L): \[ \Delta L = 0.10 \, \text{cm} = 0.0010 \, \text{m} \] \[ \text{Strain} = \frac{\Delta L}{L} = \frac{0.0010 \, \text{m}}{3.0 \, \text{m}} \approx 0.000333 \] Now we can plug the stress and strain back into the formula for Young's modulus: \[ E = \frac{481290 \, \text{N/m}^2}{0.000333} \approx 1.44 \times 10^6 \, \text{N/m}^2 \] So, Young's modulus for the wire is approximately \( 1.44 \times 10^6 \, \text{Pa} \).