If the young's modulus of the material is 6 times it modulus of rigidity, then its volume elasticity will be

Options

A. `(2Y)/(3)`
B. `oo`
C. `(Y)/(9)`
D. `(-Y)/(9)`

Explanation

To solve the problem, we need to find the volume elasticity (bulk modulus) of a material given that its Young's modulus (Y) is 6 times its modulus of rigidity (G).

### Step-by-Step Solution:

1. **Identify the relationship between Young's modulus (Y), modulus of rigidity (G), and Poisson's ratio (σ)**:
The relationship is given by the formula:
\[ Y = 2G(1 + \sigma) \]

2. **Substitute the given relationship into the formula**:
Since we know that \( Y = 6G \), we can substitute this into the equation:
\[ 6G = 2G(1 + \sigma) \]

3. **Simplify the equation**:
Divide both sides by \( G \) (assuming \( G \neq 0 \)):
\[ 6 = 2(1 + \sigma) \]

4. **Solve for Poisson's ratio (σ)**:
Rearranging the equation gives:
\[ 6 = 2 + 2\sigma \]
\[ 4 = 2\sigma \]
\[ \sigma = 2 \]

5. **Use the value of σ to find the bulk modulus (B)**:
The bulk modulus (B) can be expressed in terms of Young's modulus (Y) and Poisson's ratio (σ) using the formula:
\[ B = \frac{Y}{3(1 - 2\sigma)} \]

6. **Substitute the value of σ into the bulk modulus formula**:
Substituting \( \sigma = 2 \):
\[ B = \frac{Y}{3(1 - 2 \cdot 2)} = \frac{Y}{3(1 - 4)} = \frac{Y}{3(-3)} = -\frac{Y}{9} \]

7. **Substitute Y = 6G into the bulk modulus formula**:
Since \( Y = 6G \):
\[ B = -\frac{6G}{9} = -\frac{2G}{3} \]

### Final Answer:
The volume elasticity (bulk modulus) of the material is:
\[ B = -\frac{2G}{3} \]