The Young's modulus of a material is `10^(11)N//m^(2)` and its Poisson's ratio is 0.2 . The modulus of rigidity of the material is

Options

A. `0.42xx10^(11)N//m^(2)`
B. `0.56xx10^(11)N//m^(2)`
C. `0.2xx10^(11)N//m^(2)`
D. `5.6xx10^(11)N//m^(2)`

Explanation

To find the modulus of rigidity (η) of a material given its Young's modulus (Y) and Poisson's ratio (σ), we can use the relationship between these quantities:

### Step-by-Step Solution:

1. **Identify the given values:**
- Young's modulus (Y) = \(10^{11} \, \text{N/m}^2\)
- Poisson's ratio (σ) = 0.2

2. **Use the formula relating Young's modulus, modulus of rigidity, and Poisson's ratio:**
\[ Y = 2η(1 + σ) \]
Here, Y is the Young's modulus, η is the modulus of rigidity, and σ is the Poisson's ratio.

3. **Rearrange the formula to solve for η:**
\[ η = \frac{Y}{2(1 + σ)} \]

4. **Substitute the known values into the equation:**
\[ η = \frac{10^{11}}{2(1 + 0.2)} \]

5. **Calculate the denominator:**
\[ 1 + 0.2 = 1.2 \]
Therefore,
\[ 2(1 + 0.2) = 2 \times 1.2 = 2.4 \]

6. **Now substitute this back into the equation for η:**
\[ η = \frac{10^{11}}{2.4} \]

7. **Perform the division:**
\[ η = \frac{10^{11}}{2.4} \approx 4.1667 \times 10^{10} \, \text{N/m}^2 \]

8. **Final answer:**
\[ η \approx 4.17 \times 10^{10} \, \text{N/m}^2 \]