The relation between Y. `eta` and B is

Options

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

Explanation

To find the relation between Young's modulus (Y), modulus of rigidity (η), and bulk modulus (B), we can follow these steps:

### Step 1: Understand the Definitions
- **Young's Modulus (Y)**: It measures the tensile or compressive stiffness of a solid material.
- **Modulus of Rigidity (η)**: It measures the material's response to shear stress.
- **Bulk Modulus (B)**: It measures a material's response to uniform pressure applied in all directions.

### Step 2: Write the Relationships
From the video transcript, we have two important relationships:
1. \( Y = 2η(1 - μ) \)
2. \( B = \frac{Y}{3(1 - 2μ)} \)

Where \( μ \) is the Poisson's ratio.

### Step 3: Rearranging the Equations
From the first equation, we can express \( η \):
\[ η = \frac{Y}{2(1 - μ)} \]

From the second equation, we can express \( Y \) in terms of \( B \):
\[ Y = 3B(1 - 2μ) \]

### Step 4: Substitute and Rearrange
Now, substituting the expression for \( Y \) from the second equation into the first equation:
\[ η = \frac{3B(1 - 2μ)}{2(1 - μ)} \]

### Step 5: Finding the Relation
To find a relation between \( Y \), \( η \), and \( B \), we can manipulate these equations further.

Using the relationships derived:
1. \( Y = 2η(1 - μ) \)
2. \( Y = 3B(1 - 2μ) \)

We can equate the two expressions for \( Y \):
\[ 2η(1 - μ) = 3B(1 - 2μ) \]

### Step 6: Final Relation
This can be rearranged to find a relation involving \( Y \), \( η \), and \( B \):
\[ \frac{9}{Y} = \frac{3}{η} + \frac{1}{B} \]

Thus, the final relation between Young's modulus (Y), modulus of rigidity (η), and bulk modulus (B) is:
\[ \frac{9}{Y} = \frac{3}{η} + \frac{1}{B} \]

### Conclusion
The correct answer is option 4, which states the derived relation.

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