A 8cm cube has its upper face displaced by 0.5 mm by a tangential force of 10KN. What is the shear modulus of the cube.

Options

A. `1 xx 10^(10)N//m^(2)`
B. `10^(10)N//m^(2)`
C. `0.25 xx 10^(9)N//m^(2)`
D. `1N//m^(2)`

Explanation

To find the shear modulus of the cube, we can follow these steps:

### Step 1: Convert all measurements to SI units
- The side length of the cube is given as 8 cm. Convert this to meters:
\[ \text{Side length} = 8 \, \text{cm} = 8 \times 10^{-2} \, \text{m} \]
- The displacement of the upper face is given as 0.5 mm. Convert this to meters:
\[ \text{Displacement} = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m} \]
- The tangential force is given as 10 kN. Convert this to Newtons:
\[ \text{Force} = 10 \, \text{kN} = 10 \times 10^{3} \, \text{N} \]

### Step 2: Calculate the area of the upper face of the cube
- The area \( A \) of the upper face of the cube can be calculated as:
\[ A = \text{side length}^2 = (8 \times 10^{-2})^2 = 64 \times 10^{-4} \, \text{m}^2 \]

### Step 3: Calculate the shear stress
- Shear stress \( \tau \) is defined as the force divided by the area:
\[ \tau = \frac{\text{Force}}{A} = \frac{10 \times 10^{3}}{64 \times 10^{-4}} = \frac{10^{4}}{64 \times 10^{-4}} = \frac{10^{4}}{6.4 \times 10^{-3}} = 1562.5 \, \text{N/m}^2 \]

### Step 4: Calculate the shear strain
- Shear strain \( \gamma \) is defined as the displacement divided by the original length (the side length of the cube):
\[ \gamma = \frac{\text{Displacement}}{\text{Original length}} = \frac{0.5 \times 10^{-3}}{8 \times 10^{-2}} = \frac{0.5 \times 10^{-3}}{0.08} = 0.00625 \]

### Step 5: Calculate the shear modulus
- The shear modulus \( G \) is defined as the ratio of shear stress to shear strain:
\[ G = \frac{\tau}{\gamma} = \frac{1562.5}{0.00625} = 250000 \, \text{N/m}^2 = 2.5 \times 10^{5} \, \text{N/m}^2 \]

### Final Result
The shear modulus of the cube is:
\[ G = 2.5 \times 10^{5} \, \text{N/m}^2 \]