For most of the material , Young's modulus (Y) and rigidity modulus (G) are related as

Options

A. G=3Y
B. `G=(Y)/(3)`
C. `G=(3)/(2)Y`
D. `G=(Y)/(8)`

Explanation

To solve the question regarding the relationship between Young's modulus (Y) and rigidity modulus (G) for most materials, we can follow these steps:

### Step 1: Understand the Definitions
- **Young's Modulus (Y)**: It measures the ability of a material to withstand changes in length when under lengthwise tension or compression.
- **Rigidity Modulus (G)**: It measures the material's response to shear stress, indicating how much it deforms under shear strain.

### Step 2: Recall the Relationship Between Moduli
The relationship between Young's modulus (Y), rigidity modulus (G), and Poisson's ratio (σ) is given by the formula:
\[ G = \frac{Y}{2(1 + \sigma)} \]

### Step 3: Assume a Value for Poisson's Ratio
For most materials, the Poisson's ratio (σ) is approximately equal to \( \frac{1}{2} \).

### Step 4: Substitute the Value of σ into the Formula
Substituting \( \sigma = \frac{1}{2} \) into the equation:
\[ G = \frac{Y}{2(1 + \frac{1}{2})} \]
\[ G = \frac{Y}{2 \times \frac{3}{2}} \]
\[ G = \frac{Y}{3} \]

### Step 5: Conclusion
Thus, we find that:
\[ G \approx \frac{Y}{3} \]
This means that for most materials, the rigidity modulus (G) is approximately one-third of Young's modulus (Y).

### Final Answer
The relationship is:
\[ G \approx \frac{Y}{3} \]

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