A metalic cube whose side is 10cm is subjected to a shearing force of 100 N. The top face is displaced through 0.25cm with respect to the bottom . Calculate the shearing stress, strain and shear modulus.

Explanation

To solve the problem step by step, we will calculate the shearing stress, strain, and shear modulus for the metallic cube subjected to a shearing force.

### Step 1: Calculate the Shearing Stress

**Formula:**
\[ \text{Shearing Stress} (\tau) = \frac{\text{Force} (F)}{\text{Area} (A)} \]

**Given:**
- Side of the cube, \( s = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Shearing force, \( F = 100 \, \text{N} \)

**Area of the face of the cube:**
\[ A = s^2 = (0.1 \, \text{m})^2 = 0.01 \, \text{m}^2 \]

**Calculate Shearing Stress:**
\[ \tau = \frac{F}{A} = \frac{100 \, \text{N}}{0.01 \, \text{m}^2} = 10,000 \, \text{N/m}^2 = 10^4 \, \text{N/m}^2 \]

### Step 2: Calculate the Shearing Strain

**Formula:**
\[ \text{Shearing Strain} (\gamma) = \frac{\text{Displacement} (x)}{\text{Original Length} (L)} \]

**Given:**
- Displacement, \( x = 0.25 \, \text{cm} = 0.0025 \, \text{m} \)
- Original length (side of the cube), \( L = 10 \, \text{cm} = 0.1 \, \text{m} \)

**Calculate Shearing Strain:**
\[ \gamma = \frac{x}{L} = \frac{0.0025 \, \text{m}}{0.1 \, \text{m}} = 0.025 \]

### Step 3: Calculate the Shear Modulus

**Formula:**
\[ \text{Shear Modulus} (G) = \frac{\text{Shearing Stress} (\tau)}{\text{Shearing Strain} (\gamma)} \]

**Using previously calculated values:**
- Shearing Stress, \( \tau = 10^4 \, \text{N/m}^2 \)
- Shearing Strain, \( \gamma = 0.025 \)

**Calculate Shear Modulus:**
\[ G = \frac{\tau}{\gamma} = \frac{10^4 \, \text{N/m}^2}{0.025} = 4 \times 10^5 \, \text{N/m}^2 \]

### Final Results:
- Shearing Stress: \( 10^4 \, \text{N/m}^2 \)
- Shearing Strain: \( 0.025 \)
- Shear Modulus: \( 4 \times 10^5 \, \text{N/m}^2 \)

---