The terminal speed attained by an aluminium sphere of radius 1 mm falling through water at `20^@C` will be close to

Options

A. `9.2 ms^(-1)`
B. `6.9ms^(-1)`
C. `4.6ms^(-1)`
D. `2.3ms^(-1)`

Explanation

To find the terminal speed attained by an aluminium sphere of radius 1 mm falling through water at 20°C, we can use the following formula for terminal velocity (VT):

\[ V_T = \frac{2}{9} \cdot \frac{g \cdot r^2 \cdot (\sigma - \rho)}{\eta} \]

Where:
- \( g \) = acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \))
- \( r \) = radius of the sphere (in meters)
- \( \sigma \) = density of the solid (aluminium)
- \( \rho \) = density of the fluid (water)
- \( \eta \) = dynamic viscosity of the fluid (water at 20°C)

### Step-by-step Solution:

1. **Identify the values**:
- Radius of the sphere \( r = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m} \)
- Density of aluminium \( \sigma = 2700 \, \text{kg/m}^3 \)
- Density of water \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)
- Dynamic viscosity of water at 20°C \( \eta = 9.1 \times 10^{-4} \, \text{Pa.s} \)

2. **Substitute the values into the formula**:
\[ V_T = \frac{2}{9} \cdot \frac{10 \cdot (1 \times 10^{-3})^2 \cdot (2700 - 1000)}{9.1 \times 10^{-4}} \]

3. **Calculate \( r^2 \)**:
\[ r^2 = (1 \times 10^{-3})^2 = 1 \times 10^{-6} \, \text{m}^2 \]

4. **Calculate the difference in densities**:
\[ \sigma - \rho = 2700 - 1000 = 1700 \, \text{kg/m}^3 \]

5. **Substitute \( r^2 \) and the density difference back into the formula**:
\[ V_T = \frac{2}{9} \cdot \frac{10 \cdot 1 \times 10^{-6} \cdot 1700}{9.1 \times 10^{-4}} \]

6. **Calculate the numerator**:
\[ 10 \cdot 1 \times 10^{-6} \cdot 1700 = 17 \times 10^{-3} = 0.017 \, \text{m}^3/\text{s}^2 \]

7. **Calculate the entire fraction**:
\[ V_T = \frac{2}{9} \cdot \frac{0.017}{9.1 \times 10^{-4}} \approx \frac{2}{9} \cdot 18.68 \approx 4.15 \, \text{m/s} \]

8. **Final result**:
The terminal speed \( V_T \) is approximately \( 4.15 \, \text{m/s} \). The closest value is \( 4.6 \, \text{m/s} \).

### Conclusion:
The terminal speed attained by the aluminium sphere of radius 1 mm falling through water at 20°C will be close to \( 4.6 \, \text{m/s} \).