Find the work done in stretching a wire Im long and area of cross section 0.03 cm under a load of 100 N if Young.s modulus of the material of the wire is `2 xx 10^(11)N//m^(2)`

Answer

To find the work done in stretching a wire, we can use the formula for work done in terms of force and change in length. The steps to solve the problem are as follows:

### Step 1: Understand the given values
- Length of the wire (L) = 1 m
- Cross-sectional area (A) = 0.03 cm² = 0.03 × 10⁻⁴ m² (convert cm² to m²)
- Load (Force, F) = 100 N
- Young's modulus (Y) = 2 × 10¹¹ N/m²

### Step 2: Calculate the change in length (ΔL)
Using the formula for Young's modulus:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]

Rearranging the formula gives:
\[ \Delta L = \frac{F \cdot L}{A \cdot Y} \]

Substituting the values:
\[ \Delta L = \frac{100 \, \text{N} \cdot 1 \, \text{m}}{0.03 \times 10^{-4} \, \text{m}^2 \cdot 2 \times 10^{11} \, \text{N/m}^2} \]

### Step 3: Calculate ΔL
Calculating the denominator:
\[ 0.03 \times 10^{-4} \cdot 2 \times 10^{11} = 0.03 \cdot 2 \cdot 10^{7} = 0.06 \times 10^{7} = 6 \times 10^{5} \]

Now substituting back:
\[ \Delta L = \frac{100}{6 \times 10^{5}} \]

Calculating:
\[ \Delta L = \frac{100}{600000} = \frac{1}{6000} \approx 1.67 \times 10^{-4} \, \text{m} \]

### Step 4: Calculate the work done (W)
The work done in stretching the wire is given by:
\[ W = \frac{1}{2} \cdot F \cdot \Delta L \]

Substituting the values:
\[ W = \frac{1}{2} \cdot 100 \cdot 1.67 \times 10^{-4} \]

Calculating:
\[ W = 50 \cdot 1.67 \times 10^{-4} = 8.35 \times 10^{-3} \, \text{J} \]

### Final Answer
The work done in stretching the wire is approximately:
\[ W \approx 8.35 \times 10^{-3} \, \text{J} \]

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