The relationship between Young's modulus Y, Bulk modulus K and modulus of rigidity `eta` is

Options

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

Answer

To find the relationship between Young's modulus (Y), Bulk modulus (K), and Modulus of rigidity (η), we can follow these steps:

### Step 1: Write down the definitions of the moduli
- Young's modulus (Y) relates stress to strain in a material.
- Bulk modulus (K) measures a substance's response to uniform pressure.
- Modulus of rigidity (η) describes a material's response to shear stress.

### Step 2: Write the equations for Young's modulus and modulus of rigidity
1. **Young's modulus (Y)**:
\[ Y = 2η(1 + σ) \]
where σ is the applied stress.

2. **Bulk modulus (K)**:
\[ Y = 3K(1 - 2σ) \]

### Step 3: Rearrange the equation for modulus of rigidity to express σ
From the equation for Young's modulus:
\[ σ = \frac{Y}{2η} - 1 \]

### Step 4: Substitute σ into the Bulk modulus equation
Substituting the expression for σ into the Bulk modulus equation:
\[ Y = 3K\left(1 - 2\left(\frac{Y}{2η} - 1\right)\right) \]

### Step 5: Simplify the equation
1. Distribute the terms:
\[ Y = 3K\left(1 - \left(\frac{Y}{η} - 2\right)\right) \]
\[ Y = 3K\left(3 - \frac{Y}{η}\right) \]

2. Expand and rearrange:
\[ Y = 9K - \frac{3KY}{η} \]

### Step 6: Collect like terms
Bringing all terms involving Y to one side:
\[ Y + \frac{3KY}{η} = 9K \]

### Step 7: Factor out Y
\[ Y\left(1 + \frac{3K}{η}\right) = 9K \]

### Step 8: Solve for Y
\[ Y = \frac{9K}{1 + \frac{3K}{η}} \]

### Final Relationship
The relationship between Young's modulus (Y), Bulk modulus (K), and Modulus of rigidity (η) can be expressed as:
\[ Y = \frac{9Kη}{η + 3K} \]