A copper wire 2 m long and 0.5m in diameter supports a mass of 10kg It is stretched by 2.38 mm . Calculate the Young's modulus of the wire.

Answer

To calculate the Young's modulus of the copper wire, we will follow these steps:

### Step 1: Identify the given values
- Length of the wire (L) = 2 m
- Diameter of the wire (d) = 0.5 mm = 0.5 × 10^-3 m
- Radius of the wire (r) = d/2 = 0.25 mm = 0.25 × 10^-3 m
- Mass supported (m) = 10 kg
- Extension (ΔL) = 2.38 mm = 2.38 × 10^-3 m
- Acceleration due to gravity (g) = 9.8 m/s²

### Step 2: Calculate the force (F) acting on the wire
The force exerted by the mass is given by:
\[ F = m \cdot g \]
\[ F = 10 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 98 \, \text{N} \]

### Step 3: Calculate the cross-sectional area (A) of the wire
The area of a circle is given by:
\[ A = \pi r^2 \]
\[ A = \pi (0.25 \times 10^{-3} \, \text{m})^2 \]
\[ A = \pi (0.0625 \times 10^{-6} \, \text{m}^2) \]
\[ A \approx 1.9635 \times 10^{-7} \, \text{m}^2 \]

### Step 4: Calculate the stress (σ)
Stress is defined as the force per unit area:
\[ \sigma = \frac{F}{A} \]
\[ \sigma = \frac{98 \, \text{N}}{1.9635 \times 10^{-7} \, \text{m}^2} \]
\[ \sigma \approx 4.99 \times 10^{8} \, \text{N/m}^2 \]

### Step 5: Calculate the strain (ε)
Strain is defined as the change in length divided by the original length:
\[ \epsilon = \frac{\Delta L}{L} \]
\[ \epsilon = \frac{2.38 \times 10^{-3} \, \text{m}}{2 \, \text{m}} \]
\[ \epsilon = 1.19 \times 10^{-3} \]

### Step 6: Calculate Young's modulus (Y)
Young's modulus is defined as the ratio of stress to strain:
\[ Y = \frac{\sigma}{\epsilon} \]
\[ Y = \frac{4.99 \times 10^{8} \, \text{N/m}^2}{1.19 \times 10^{-3}} \]
\[ Y \approx 4.19 \times 10^{11} \, \text{N/m}^2 \]

### Final Answer
The Young's modulus of the copper wire is approximately \( 4.19 \times 10^{11} \, \text{N/m}^2 \).

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