A metallic cube whose each side is 10 cm is subjected to a shearing force of 100 kgf. The top face is displaced through 0.25 cm 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, shearing strain, and shear modulus for the metallic cube.

### Step 1: Calculate the Area of the Top Face
The area \( A \) of the top face of the cube can be calculated using the formula for the area of a square:
\[ A = \text{side}^2 \]
Given that each side of the cube is 10 cm, we convert this to meters:
\[ \text{side} = 10 \, \text{cm} = 0.1 \, \text{m} \]
Now, calculate the area:
\[ A = (0.1 \, \text{m})^2 = 0.01 \, \text{m}^2 \]

### Step 2: Convert the Shearing Force to Newtons
The shearing force \( F \) is given as 100 kgf. We need to convert this to Newtons using the conversion factor \( 1 \, \text{kgf} = 9.8 \, \text{N} \):
\[ F = 100 \, \text{kgf} \times 9.8 \, \text{N/kgf} = 980 \, \text{N} \]

### Step 3: Calculate Shearing Stress
Shearing stress \( \tau \) is defined as the force per unit area:
\[ \tau = \frac{F}{A} \]
Substituting the values we have:
\[ \tau = \frac{980 \, \text{N}}{0.01 \, \text{m}^2} = 98000 \, \text{N/m}^2 = 9.8 \times 10^4 \, \text{N/m}^2 \]

### Step 4: Calculate Shearing Strain
Shearing strain \( \gamma \) is defined as the ratio of the displacement \( \Delta L \) to the original length \( L \):
\[ \gamma = \frac{\Delta L}{L} \]
Given that the displacement \( \Delta L \) is 0.25 cm, we convert this to meters:
\[ \Delta L = 0.25 \, \text{cm} = 0.0025 \, \text{m} \]
The original length \( L \) is 10 cm, which is:
\[ L = 0.1 \, \text{m} \]
Now, substituting the values:
\[ \gamma = \frac{0.0025 \, \text{m}}{0.1 \, \text{m}} = 0.025 \]

### Step 5: Calculate Shear Modulus
The shear modulus \( G \) is defined as the ratio of shearing stress to shearing strain:
\[ G = \frac{\tau}{\gamma} \]
Substituting the values we calculated:
\[ G = \frac{9.8 \times 10^4 \, \text{N/m}^2}{0.025} = 3.92 \times 10^6 \, \text{N/m}^2 \]

### Final Results
- Shearing Stress \( \tau = 9.8 \times 10^4 \, \text{N/m}^2 \)
- Shearing Strain \( \gamma = 0.025 \)
- Shear Modulus \( G = 3.92 \times 10^6 \, \text{N/m}^2 \)