Find the percentage increase in length of a wire subjected to a stress of 1 gm. Wt `// m m^(2)` . Young's modulus of the material of the wire = 100 Gpa.

Explanation

To find the percentage increase in length of a wire subjected to a stress of 1 gram weight per mm², given that the Young's modulus of the material of the wire is 100 GPa, we can follow these steps:

### Step 1: Convert the stress from gram weight per mm² to Newton per meter²
1. **Given Stress (σ)**: 1 gm/mm²
2. **Convert grams to kilograms**:
\[ 1 \text{ gm} = 0.001 \text{ kg} \]
3. **Convert mm² to m²**:
\[ 1 \text{ mm}^2 = 1 \times 10^{-6} \text{ m}^2 \]
4. **Calculate the stress in Newtons**:
\[ \text{Weight} = \text{mass} \times g = 0.001 \text{ kg} \times 9.81 \text{ m/s}^2 = 0.00981 \text{ N} \]
5. **Stress in N/m²**:
\[ \sigma = \frac{0.00981 \text{ N}}{1 \times 10^{-6} \text{ m}^2} = 9.81 \times 10^3 \text{ N/m}^2 \]

### Step 2: Use Young's modulus to find strain
1. **Young's Modulus (Y)**: 100 GPa = \(100 \times 10^9 \text{ N/m}^2\)
2. **Using the formula**:
\[ Y = \frac{\sigma}{\epsilon} \Rightarrow \epsilon = \frac{\sigma}{Y} \]
3. **Substituting the values**:
\[ \epsilon = \frac{9.81 \times 10^3 \text{ N/m}^2}{100 \times 10^9 \text{ N/m}^2} = \frac{9.81}{100 \times 10^6} \]

### Step 3: Calculate the percentage increase in length
1. **Percentage increase in length**:
\[ \frac{\Delta L}{L} \times 100 = \epsilon \times 100 \]
2. **Substituting the strain**:
\[ \frac{\Delta L}{L} \times 100 = \left(\frac{9.81}{100 \times 10^6}\right) \times 100 \]
3. **Simplifying**:
\[ = \frac{9.81}{10^6} \text{ percent} \]
4. **Final result**:
\[ = 9.81 \times 10^{-6} \text{ percent} \]

### Final Answer:
The percentage increase in length of the wire is \(9.81 \times 10^{-6}\) percent.

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