A metal cube of side `10 cm` is subjected to a shearing stress of `10^(6)N//m^(2)`. Calculate the modulus of rigidity if the of the cube is displaced by `0.05 cm` with respect to its bottom.

Options

A. `1 xx 10^8`
B. `2 xx 10^8`
C. `3 xx 10^8`
D. `4 xx 10^8`

Explanation

To solve the problem, we need to calculate the modulus of rigidity (also known as shear modulus) using the given values. Here’s the step-by-step solution:

### Step 1: Identify the given values
- Side of the cube (L) = 10 cm = 0.1 m (conversion from cm to m)
- Shearing stress (τ) = \(10^6 \, \text{N/m}^2\)
- Displacement (Δx) = 0.05 cm = 0.0005 m (conversion from cm to m)

### Step 2: Calculate the shear strain (γ)
Shear strain (γ) is defined as the ratio of the displacement (Δx) to the original length (L).

\[ \gamma = \frac{\Delta x}{L} \]

Substituting the values:

\[ \gamma = \frac{0.0005 \, \text{m}}{0.1 \, \text{m}} = 0.005 \]

### Step 3: Use the formula for modulus of rigidity (G)
The modulus of rigidity (G) is defined as the ratio of shear stress (τ) to shear strain (γ):

\[ G = \frac{\tau}{\gamma} \]

### Step 4: Substitute the values into the formula
Now we substitute the values of shear stress and shear strain into the formula:

\[ G = \frac{10^6 \, \text{N/m}^2}{0.005} \]

### Step 5: Perform the calculation
Calculating the above expression:

\[ G = \frac{10^6}{0.005} = 2 \times 10^8 \, \text{N/m}^2 \]

### Conclusion
The modulus of rigidity of the metal cube is \(2 \times 10^8 \, \text{N/m}^2\).

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