The Youngg's modulus of a material is `6 xx 10^(10) Nm^(-2)` and its bulk modulus is `2.8 xx 10^(10) Nm^(-2)` . What is its Poisson's ratio ?

Explanation

To find the Poisson's ratio (σ) given the Young's modulus (E) and the bulk modulus (K), we can use the relationship between these quantities. The formula relating Young's modulus, bulk modulus, and Poisson's ratio is:

\[ E = 3K(1 - 2\sigma) \]

### Step-by-Step Solution:

1. **Write down the given values:**
- Young's modulus (E) = \( 6 \times 10^{10} \, \text{N/m}^2 \)
- Bulk modulus (K) = \( 2.8 \times 10^{10} \, \text{N/m}^2 \)

2. **Substitute the values into the formula:**
\[ 6 \times 10^{10} = 3 \times (2.8 \times 10^{10}) \times (1 - 2\sigma) \]

3. **Calculate \( 3K \):**
\[ 3K = 3 \times (2.8 \times 10^{10}) = 8.4 \times 10^{10} \]

4. **Rearrange the equation:**
\[ 6 \times 10^{10} = 8.4 \times 10^{10} \times (1 - 2\sigma) \]

5. **Divide both sides by \( 8.4 \times 10^{10} \):**
\[ \frac{6 \times 10^{10}}{8.4 \times 10^{10}} = 1 - 2\sigma \]
\[ \frac{6}{8.4} = 1 - 2\sigma \]

6. **Calculate \( \frac{6}{8.4} \):**
\[ \frac{6}{8.4} = 0.7142857143 \approx 0.71 \]

7. **Set the equation:**
\[ 0.71 = 1 - 2\sigma \]

8. **Rearranging gives:**
\[ 2\sigma = 1 - 0.71 \]
\[ 2\sigma = 0.29 \]

9. **Solve for σ:**
\[ \sigma = \frac{0.29}{2} = 0.145 \]

### Final Answer:
The Poisson's ratio (σ) is approximately \( 0.145 \).