The speed of a ball of radius 2 cm in a viscous liquid is 20 cm/s. Then the speed of ball of radius I cm in the same liquid is

Options

A. `5 cm s^(-1)`
B. `10 cm s^(-1)`
C. `40 cm s^(-1)`
D. `80 cm s^(-1)`

Answer

To solve the problem, we will use the relationship between the speed of a sphere falling through a viscous liquid and its radius. According to Stokes' law, the speed of a sphere in a viscous medium is proportional to the square of its radius.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Speed of the first ball (V1) = 20 cm/s
- Radius of the first ball (R1) = 2 cm
- Radius of the second ball (R2) = 1 cm

2. **Use the Proportionality Relation:**
According to Stokes' law, the speed (V) of a sphere in a viscous liquid is proportional to the square of its radius (R):
\[ V \propto R^2 \]
This means we can write the ratio of the speeds of the two balls as:
\[ \frac{V_2}{V_1} = \frac{R_2^2}{R_1^2} \]

3. **Substitute the Known Values:**
Substitute the known values into the equation:
\[ \frac{V_2}{20} = \frac{1^2}{2^2} \]
This simplifies to:
\[ \frac{V_2}{20} = \frac{1}{4} \]

4. **Solve for V2:**
To find \( V_2 \), multiply both sides by 20:
\[ V_2 = 20 \times \frac{1}{4} = 5 \text{ cm/s} \]

5. **Final Answer:**
The speed of the ball of radius 1 cm in the same liquid is:
\[ V_2 = 5 \text{ cm/s} \]