Relation between `Y,etaandK is `

Options

A. `(Y)/(3)=(3)/(K)+(1)/(eta)`
B. `(9)/(Y)=(eta)/(3)+(1)/(K)`
C. `(3)/(Y)=(1)/(eta)+(1)/(3K)`
D. `(Y)/(3)=(3)/(eta)+(1)/(K)`

Answer

To find the relation between Young's modulus (Y), modulus of rigidity (η), and bulk modulus (K), we can use the known formulas for these moduli in terms of Poisson's ratio (μ).

### Step-by-Step Solution:

1. **Identify the Definitions**:
- Young's Modulus (Y): It measures the ability of a material to withstand changes in length when under lengthwise tension or compression.
- Modulus of Rigidity (η): It measures the material's response to shear stress (the ratio of shear stress to shear strain).
- Bulk Modulus (K): It measures a substance's response to uniform pressure applied in all directions.

2. **Use the Relationship Formulas**:
- The relationship between Young's modulus (Y), modulus of rigidity (η), and Poisson's ratio (μ) is given by:
\[ Y = 2η(1 + μ) \]
- The relationship between Young's modulus (Y), bulk modulus (K), and Poisson's ratio (μ) is given by:
\[ Y = 3K(1 - 2μ) \]

3. **Set Up the Equations**:
- From the first equation, we can express μ in terms of Y and η:
\[ μ = \frac{Y}{2η} - 1 \]
- From the second equation, we can express μ in terms of Y and K:
\[ μ = \frac{3K - Y}{2Y} \]

4. **Equate the Two Expressions for μ**:
- Set the two expressions for μ equal to each other:
\[ \frac{Y}{2η} - 1 = \frac{3K - Y}{2Y} \]

5. **Cross Multiply and Simplify**:
- Cross-multiplying gives:
\[ 2Y\left(\frac{Y}{2η} - 1\right) = 3K - Y \]
- This simplifies to:
\[ \frac{Y^2}{η} - 2Y = 3K - Y \]
- Rearranging gives:
\[ \frac{Y^2}{η} + Y - 3K = 2Y \]

6. **Final Relation**:
- Rearranging the equation leads to:
\[ \frac{Y^2}{η} = 3K - Y \]
- This can be further manipulated to find a relation between Y, η, and K.

7. **Conclusion**:
- The final relation can be expressed as:
\[ \frac{1}{Y} = \frac{1}{3K} + \frac{1}{η} \]