A wire of length 15m and diameter 4 mm elongates 0.3mm when stretched by weigth of 2.20kg. Calulate the Young's modulus of the material of the wire.

Explanation

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

### Step 1: Gather the Given Data
- Length of the wire (L) = 15 m
- Diameter of the wire (D) = 4 mm = 4 × 10⁻³ m
- Elongation (ΔL) = 0.3 mm = 0.3 × 10⁻³ m
- Mass (m) = 2.2 kg

### Step 2: Calculate the Radius of the Wire
The radius (r) is half of the diameter.
\[ r = \frac{D}{2} = \frac{4 \times 10^{-3}}{2} = 2 \times 10^{-3} \text{ m} \]

### Step 3: Calculate the Force (F) Applied to the Wire
The force is calculated using the weight of the mass:
\[ F = m \cdot g = 2.2 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 21.56 \, \text{N} \]

### Step 4: Calculate the Cross-sectional Area (A) of the Wire
The area (A) can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 = \pi (2 \times 10^{-3})^2 = \pi (4 \times 10^{-6}) \approx 12.5664 \times 10^{-6} \, \text{m}^2 \]

### Step 5: Calculate the Strain (ε)
Strain is defined as the change in length divided by the original length:
\[ \text{Strain} (\epsilon) = \frac{\Delta L}{L} = \frac{0.3 \times 10^{-3}}{15} = 2 \times 10^{-5} \]

### Step 6: Calculate the Young's Modulus (Y)
Young's modulus is defined as the ratio of stress to strain:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{\frac{F}{A}}{\epsilon} \]
Substituting the values:
\[ Y = \frac{\frac{21.56}{12.5664 \times 10^{-6}}}{2 \times 10^{-5}} \]

Calculating the stress:
\[ \text{Stress} = \frac{21.56}{12.5664 \times 10^{-6}} \approx 1.717 \times 10^6 \, \text{N/m}^2 \]
Now substituting back to find Young's modulus:
\[ Y = \frac{1.717 \times 10^6}{2 \times 10^{-5}} \approx 8.585 \times 10^{10} \, \text{N/m}^2 \]

### Final Answer
The Young's modulus of the material of the wire is approximately:
\[ Y \approx 8.585 \times 10^{10} \, \text{N/m}^2 \]