The edges of an aluminum cube are `10 cm` long. One face of the cube is firmly fixed to a vertical wall. A mass of `100 kg` is then attached to the opposite face of the cube. Shear modulus of aluminum is `25 xx 10^(9)` Pa, the vertical deflection in the face to which mass is attached is

Options

A. `4xx10^(-4)m`
B. `4xx10^(-7)m`
C. `25xx10^(-6)m`
D. `6x10^(-7)m`

Explanation

To find the vertical deflection \( x \) in the face of the aluminum cube when a mass is attached, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Edge length of the cube, \( L = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Mass attached, \( m = 100 \, \text{kg} \)
- Shear modulus of aluminum, \( \eta = 25 \times 10^9 \, \text{Pa} \)

2. **Calculate the Force (Weight) Acting on the Cube:**
\[ F = m \cdot g \]
where \( g \) (acceleration due to gravity) is approximately \( 9.81 \, \text{m/s}^2 \).
\[ F = 100 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 981 \, \text{N} \]

3. **Calculate the Area of Cross Section of the Cube:**
Since the cube has a square face, the area \( A \) is given by:
\[ A = L^2 = (0.1 \, \text{m})^2 = 0.01 \, \text{m}^2 \]

4. **Use the Shear Modulus Formula:**
The shear modulus \( \eta \) is defined as:
\[ \eta = \frac{\text{Shearing Stress}}{\text{Shearing Strain}} = \frac{F/A}{\tan(\theta)} \]
For small angles, \( \tan(\theta) \approx \frac{x}{L} \), where \( x \) is the vertical deflection. Therefore:
\[ \eta = \frac{F}{A} \cdot \frac{L}{x} \]

5. **Rearranging for Vertical Deflection \( x \):**
Rearranging the equation gives:
\[ x = \frac{F \cdot L}{A \cdot \eta} \]

6. **Substituting the Values:**
\[ x = \frac{981 \, \text{N} \cdot 0.1 \, \text{m}}{0.01 \, \text{m}^2 \cdot 25 \times 10^9 \, \text{Pa}} \]
\[ x = \frac{98.1 \, \text{N m}}{0.01 \, \text{m}^2 \cdot 25 \times 10^9 \, \text{Pa}} = \frac{98.1}{250000000} = 3.924 \times 10^{-7} \, \text{m} \]

7. **Final Answer:**
The vertical deflection \( x \) is approximately:
\[ x \approx 3.924 \times 10^{-7} \, \text{m} \approx 4 \times 10^{-7} \, \text{m} \]