Question

A wire of length \(5.00 \mathrm{m}\) with a cross-sectional area of $0.100 \mathrm{cm}^{2}\( stretches by \)6.50 \mathrm{mm}\( when a load of \)1.00 \mathrm{kN}$ is hung from it. What is the Young's modulus for this wire?

Short answer

Question: Calculate the Young's modulus for a material given the following information: the length of the wire is 4 m, its cross-sectional area is 2 cm², an elongation of 3 mm occurred under a load of 3 kN. Solution: Step 1: Calculate Stress Force (F) = 3 kN * 1000 = 3000 N Area (A) = 2 cm² * (0.01 m/cm)^2 = 0.0002 m² Stress (σ) = F / A = 3000 N / 0.0002 m² = 15,000,000 N/m² or 15,000,000 Pa Step 2: Calculate Strain Elongation (ΔL) = 3 mm * 0.001 m/mm = 0.003 m Original length (L) = 4 m Strain (ε) = ΔL / L = 0.003 m / 4 m = 0.00075 Step 3: Calculate Young's Modulus Young's modulus (Y) = σ / ε = 15,000,000 Pa / 0.00075 = 20,000,000,000 Pa or 20 GPa The Young's modulus for the material is 20 GPa.

Step-by-step solution

Calculate Stress

Stress is given as the force applied to a material divided by its cross-sectional area. In this case, the force applied is given in kilonewtons (kN) and the cross-sectional area is given in square centimeters (cm²). Convert these values into the standard SI units: newtons (N) and square meters (m²). The stress (σ) can be calculated using the formula: $$ \sigma = \frac{F}{A} $$ where F is the applied force and A is the cross-sectional area.

Calculate Strain

Strain (ε) is defined as the change in length (ΔL) divided by the original length (L). In this case, the elongation is given in millimeters (mm), so convert this value into meters (m). The original length is already given in meters. The strain can be calculated using the formula: $$ ε = \frac{\Delta L}{L} $$

Calculate Young's Modulus

With the stress and strain values from steps 1 and 2, the Young's modulus (Y) can be calculated with the formula: $$ Y = \frac{\sigma}{ε} $$ Plug in the values for stress and strain, and solve for Young's modulus. Ensure that the final Young's modulus is in SI units (Pa, or N/m²).