If Y, K and `eta` are the values of Young's modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

Options

A. `Y = (9K eta)/(3K - eta) N//m^2`
B. `eta = (3YK)/(9K + Y) N//m^2`
C. `Y = (9K eta)/(2 eta + 3K) N//m^2`
D. `K = (Yn)/(9 eta - 3Y) N//m^2`

Answer

To solve the problem, we need to establish the relationships between Young's modulus (Y), bulk modulus (K), and modulus of rigidity (η) for a given material.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- Young's Modulus (Y) measures the tensile stiffness of a solid material.
- Bulk Modulus (K) measures a substance's response to uniform pressure.
- Modulus of Rigidity (η) measures a material's response to shear stress.

2. **Establishing the Relationships**:
- The relationship between Young's modulus (Y) and bulk modulus (K) can be expressed as:
\[ Y = 3K(1 - 2\sigma) \]
where σ is the Poisson's ratio.
- The relationship between Young's modulus (Y) and modulus of rigidity (η) is given by:
\[ Y = 2η(1 + \sigma) \]

3. **Finding the Value of σ**:
- From the first equation, rearranging gives:
\[ \sigma = \frac{1 - \frac{Y}{3K}}{2} \]
- Substitute this expression for σ into the second equation.

4. **Substituting σ into the Second Equation**:
- Substitute σ into the equation:
\[ Y = 2η\left(1 + \frac{1 - \frac{Y}{3K}}{2}\right) \]
- Simplifying this gives:
\[ Y = 2η\left(\frac{2 + 1 - \frac{Y}{3K}}{2}\right) \]
- This simplifies further to:
\[ Y = η(3 + \frac{Y}{3K}) \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ Y - \frac{Yη}{3K} = 3η \]
- Factoring out Y:
\[ Y\left(1 - \frac{η}{3K}\right) = 3η \]
- Finally, solving for K gives:
\[ K = \frac{9η}{3Y + η} \]

6. **Conclusion**:
- The derived relationship between Y, K, and η is:
\[ K = \frac{9η}{3Y + η} \]
- Therefore, the correct relation among the parameters is established.