A uniform wire of length 10m and diameter 0.8 mm stretched 5 mm under a certain force . If the change in diameter is `10 xx 10^(-8) m`, calculate the Poissson's ratio.

Explanation

To calculate Poisson's ratio for the given wire, we will follow these steps:

### Step 1: Identify the given values
- Length of the wire (L) = 10 m
- Diameter of the wire (D) = 0.8 mm = 0.8 × 10^(-3) m = 8 × 10^(-4) m
- Change in length (ΔL) = 5 mm = 5 × 10^(-3) m
- Change in diameter (ΔD) = 10 × 10^(-8) m = 10^(-7) m

### Step 2: Calculate the longitudinal strain
Longitudinal strain is defined as the change in length divided by the original length:
\[ \text{Longitudinal Strain} = \frac{\Delta L}{L} = \frac{5 \times 10^{-3}}{10} = 0.0005 \]

### Step 3: Calculate the lateral strain
Lateral strain is defined as the change in diameter divided by the original diameter:
\[ \text{Lateral Strain} = \frac{\Delta D}{D} = \frac{10 \times 10^{-8}}{8 \times 10^{-4}} = \frac{10^{-7}}{8 \times 10^{-4}} = \frac{10^{-7}}{8 \times 10^{-4}} = 1.25 \times 10^{-4} \]

### Step 4: Calculate Poisson's ratio (σ)
Poisson's ratio is defined as the ratio of lateral strain to longitudinal strain:
\[ \sigma = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = \frac{1.25 \times 10^{-4}}{0.0005} \]
Calculating this gives:
\[ \sigma = \frac{1.25 \times 10^{-4}}{5 \times 10^{-4}} = 0.25 \]

### Final Answer:
The Poisson's ratio for the given wire is **0.25**.

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