A load of 2 kg produces an extension of 1 mm in a wire 3 m in length and 1 mm in diameter.l Calculate Young's modulus of elesticity of wire.

Answer

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

### Step 1: Identify the given values
- Load (Force) \( F = 2 \, \text{kg} \)
- Extension (Elongation) \( \Delta L = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Original length of the wire \( L = 3 \, \text{m} \)
- Diameter of the wire \( d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)

### Step 2: Convert the load from kg to Newtons
To convert the load into Newtons, we use the relation:
\[ F = m \cdot g \]
where \( g \approx 9.81 \, \text{m/s}^2 \). Thus,
\[ F = 2 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 19.62 \, \text{N} \]

### Step 3: Calculate the cross-sectional area of the wire
The area \( A \) of the wire can be calculated using the formula for the area of a circle:
\[ A = \pi \left( \frac{d}{2} \right)^2 \]
Substituting the diameter:
\[ A = \pi \left( \frac{1 \times 10^{-3}}{2} \right)^2 = \pi \left( 0.5 \times 10^{-3} \right)^2 \]
\[ A = \pi \times (0.5^2) \times (10^{-6}) = \pi \times 0.25 \times 10^{-6} \]
\[ A \approx 0.7854 \times 10^{-6} \, \text{m}^2 \]

### Step 4: Calculate the strain
Strain \( \epsilon \) is defined as the ratio of extension to the original length:
\[ \epsilon = \frac{\Delta L}{L} = \frac{1 \times 10^{-3}}{3} = \frac{1}{3000} \]

### Step 5: Calculate the stress
Stress \( \sigma \) is defined as the force per unit area:
\[ \sigma = \frac{F}{A} = \frac{19.62}{0.7854 \times 10^{-6}} \]
Calculating this gives:
\[ \sigma \approx 2.5 \times 10^{7} \, \text{N/m}^2 \]

### Step 6: Calculate Young's modulus
Young's modulus \( Y \) is defined as the ratio of stress to strain:
\[ Y = \frac{\sigma}{\epsilon} = \frac{2.5 \times 10^{7}}{\frac{1}{3000}} \]
Calculating this gives:
\[ Y = 2.5 \times 10^{7} \times 3000 = 7.5 \times 10^{10} \, \text{N/m}^2 \]

### Final Answer
Thus, the Young's modulus of elasticity of the wire is approximately:
\[ Y \approx 7.5 \times 10^{10} \, \text{N/m}^2 \]

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