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 determine **Young's modulus (Y)** for the wire, we'll use the fundamental relationship: \[ Y = \frac{\text{Stress}}{\text{Strain}} \] Where: - **Stress** is the force applied per unit area. - **Strain** is the relative change in length. ### Given Data: - **Length of the wire, \( L \)**: 3.0 meters - **Cross-sectional area, \( A \)**: \( 0.0020 \, \text{m}^2 \) - **Change in length, \( \Delta L \)**: 0.10 cm = 0.0010 meters - **Mass hanging, \( m \)**: 98 kg - **Acceleration due to gravity, \( g \)**: \( 9.8 \, \text{m/s}^2 \) ### Step 1: Calculate the Force (F) The force exerted by the hanging mass is given by: \[ F = m \times g = 98 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 960.4 \, \text{N} \] ### Step 2: Calculate Stress \[ \text{Stress} = \frac{F}{A} = \frac{960.4 \, \text{N}}{0.0020 \, \text{m}^2} = 480200 \, \text{N/m}^2 \, \text{or} \, 4.802 \times 10^5 \, \text{Pa} \] ### Step 3: Calculate Strain \[ \text{Strain} = \frac{\Delta L}{L} = \frac{0.0010 \, \text{m}}{3.0 \, \text{m}} = 3.333 \times 10^{-4} \] ### Step 4: Calculate Young's Modulus (Y) \[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{4.802 \times 10^5 \, \text{Pa}}{3.333 \times 10^{-4}} \approx 1.44 \times 10^9 \, \text{Pa} \] ### Final Answer: Young’s modulus for the wire is approximately 1.44 × 10⁹ Pascals.