The metal cube of side 10 cm is subjected to a shearing stress of `10^(4) N m^(-2)`. The modulus of rigidity if the top of the cube is displaced by 0.05 cm with respect to its bottom is

Options

A. `2xx10^(6) N m^(-2)`
B. `10x^(5) Nm^(-2)`
C. `1xx10^(7) N m^(-2)`
D. `4xx 10^(5)N m^(-2)`

Explanation

To find the modulus of rigidity (also known as shear modulus) of the metal cube, we can follow these steps:

### Step 1: Identify the given values
- Side of the cube (L) = 10 cm = 10 × 10^(-2) m = 0.1 m
- Shearing stress (σ) = 10^4 N/m²
- Displacement (ΔL) = 0.05 cm = 5 × 10^(-2) cm = 5 × 10^(-4) m

### Step 2: Calculate the strain
Strain (γ) is defined as the ratio of the change in length (displacement) to the original length. Therefore, we can calculate strain using the formula:
\[ \text{Strain} (\gamma) = \frac{\Delta L}{L} \]
Substituting the values:
\[ \gamma = \frac{5 \times 10^{-4} \text{ m}}{0.1 \text{ m}} = \frac{5 \times 10^{-4}}{10^{-1}} = 5 \times 10^{-3} \]

### Step 3: Calculate the modulus of rigidity
The modulus of rigidity (G) is defined as the ratio of shear stress (σ) to shear strain (γ). Thus, we can use the formula:
\[ G = \frac{\sigma}{\gamma} \]
Substituting the values:
\[ G = \frac{10^4 \text{ N/m}^2}{5 \times 10^{-3}} = \frac{10^4}{5 \times 10^{-3}} = 2 \times 10^6 \text{ N/m}^2 \]

### Step 4: Conclusion
The modulus of rigidity of the metal cube is:
\[ G = 2 \times 10^6 \text{ N/m}^2 \]