The terminal velocity of a sphere of radius R, falling in a viscous fluid, is proportional to

Options

A. R
B. `(1)/(R)`
C. `R^(2)`
D. `(1)/(R^(2))`

Answer

To determine how the terminal velocity \( V \) of a sphere of radius \( R \) falling in a viscous fluid is proportional to the radius, we can use the formula for terminal velocity.

### Step-by-step Solution:

1. **Understand the Formula for Terminal Velocity**:
The terminal velocity \( V \) of a sphere falling through a viscous fluid is given by the equation:
\[ V = \frac{2}{9} \frac{(R^2)(\rho - \sigma)g}{\eta} \]
where:
- \( R \) is the radius of the sphere,
- \( \rho \) is the density of the sphere,
- \( \sigma \) is the density of the fluid,
- \( g \) is the acceleration due to gravity,
- \( \eta \) is the viscosity of the fluid.

2. **Identify the Proportionality**:
From the formula, we can see that the terminal velocity \( V \) is directly proportional to \( R^2 \) (the square of the radius). This means that as the radius increases, the terminal velocity increases with the square of that increase.

3. **Conclusion**:
Therefore, we can conclude that the terminal velocity \( V \) is proportional to \( R^2 \).

### Final Answer:
The terminal velocity of a sphere of radius \( R \) falling in a viscous fluid is proportional to \( R^2 \).