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. `4 xx 10^(-5) m`
B. `4 xx 10^(-6) m`
C. `4 xx 10^(-7) m`
D. `4 xx 10^(-8) m`

Explanation

To find the vertical deflection in the face of the aluminum cube to which the mass is attached, we can follow these steps:

### Step 1: Identify Given Values
- Length of the cube edge (L) = 10 cm = 0.1 m
- Mass (m) = 100 kg
- Shear modulus (G) = 25 x 10^9 Pa
- Gravitational acceleration (g) = 9.8 m/s²

### Step 2: Calculate the Force (F) Applied
The force applied due to the mass can be calculated using:
\[ F = m \cdot g \]
Substituting the values:
\[ F = 100 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 980 \, \text{N} \]

### Step 3: Calculate the Area (A) of the Face
The area of one face of the cube can be calculated as:
\[ A = L^2 \]
Substituting the value of L:
\[ A = (0.1 \, \text{m})^2 = 0.01 \, \text{m}^2 \]

### Step 4: Calculate Shear Stress (τ)
Shear stress (τ) can be calculated using:
\[ \tau = \frac{F}{A} \]
Substituting the values:
\[ \tau = \frac{980 \, \text{N}}{0.01 \, \text{m}^2} = 98000 \, \text{Pa} \]

### Step 5: Relate Shear Stress to Shear Strain
Using the relationship between shear modulus (G), shear stress (τ), and shear strain (γ):
\[ G = \frac{\tau}{\gamma} \]
Rearranging gives:
\[ \gamma = \frac{\tau}{G} \]
Substituting the values:
\[ \gamma = \frac{98000 \, \text{Pa}}{25 \times 10^9 \, \text{Pa}} = 3.92 \times 10^{-6} \]

### Step 6: Calculate the Vertical Deflection (Δl)
The vertical deflection (Δl) can be calculated using the formula:
\[ \Delta l = \gamma \cdot L \]
Substituting the values:
\[ \Delta l = 3.92 \times 10^{-6} \cdot 0.1 \, \text{m} = 3.92 \times 10^{-7} \, \text{m} \]
This can be approximated to:
\[ \Delta l \approx 4 \times 10^{-7} \, \text{m} \]

### Final Answer
The vertical deflection in the face to which the mass is attached is approximately:
\[ \Delta l \approx 4 \times 10^{-7} \, \text{m} \]

---