The terminal velocity of a rain drop is `30cm//s`. If the viscosity of air is `1.8xx10^(-5)Nsm^(-2)`.The radius of rain drop is

Options

A. `1mum`
B. `0.5mm`
C. `0.05mm`
D. `1mm`

Explanation

To find the radius of a raindrop given its terminal velocity and the viscosity of air, we can use the formula for terminal velocity in a viscous fluid. Here’s a step-by-step solution:

### Step 1: Understand the formula for terminal velocity
The terminal velocity \( V_t \) of a sphere falling through a viscous medium is given by the formula:
\[ V_t = \frac{2}{9} \cdot \frac{r^2 (\rho - \sigma) g}{\eta} \]
Where:
- \( r \) = radius of the raindrop
- \( \rho \) = density of the liquid (water)
- \( \sigma \) = density of the gas (air)
- \( g \) = acceleration due to gravity
- \( \eta \) = viscosity of the fluid (air)

### Step 2: Simplify the formula
Since the density of air (\( \sigma \)) is much smaller than the density of water (\( \rho \)), we can neglect \( \sigma \). Thus, the formula simplifies to:
\[ V_t = \frac{2}{9} \cdot \frac{r^2 \rho g}{\eta} \]

### Step 3: Rearrange the formula to solve for \( r \)
Rearranging the formula to solve for the radius \( r \):
\[ r^2 = \frac{9 \eta V_t}{2 \rho g} \]
Taking the square root of both sides gives:
\[ r = \sqrt{\frac{9 \eta V_t}{2 \rho g}} \]

### Step 4: Substitute the known values
Given:
- \( V_t = 30 \, \text{cm/s} = 0.3 \, \text{m/s} \)
- \( \eta = 1.8 \times 10^{-5} \, \text{N s/m}^2 \)
- \( \rho = 1000 \, \text{kg/m}^3 \) (density of water)
- \( g = 9.8 \, \text{m/s}^2 \)

Substituting these values into the equation:
\[ r = \sqrt{\frac{9 \cdot (1.8 \times 10^{-5}) \cdot (0.3)}{2 \cdot 1000 \cdot 9.8}} \]

### Step 5: Calculate the value
Calculating the numerator:
\[ 9 \cdot (1.8 \times 10^{-5}) \cdot (0.3) = 5.4 \times 10^{-5} \]
Calculating the denominator:
\[ 2 \cdot 1000 \cdot 9.8 = 19600 \]
Now substituting these into the equation for \( r \):
\[ r = \sqrt{\frac{5.4 \times 10^{-5}}{19600}} = \sqrt{2.7551 \times 10^{-9}} \approx 5.25 \times 10^{-5} \, \text{m} \]

### Step 6: Convert to millimeters
To convert meters to millimeters:
\[ r \approx 5.25 \times 10^{-5} \, \text{m} = 0.0525 \, \text{mm} \approx 0.05 \, \text{mm} \]

### Final Answer
The radius of the raindrop is approximately \( 0.05 \, \text{mm} \).

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