A small metal sphere of radius a is falling with a velocity `upsilon` through a vertical column of a viscous liquid. If the coefficient of viscosity of the liquid is `eta`, then the sphere encounters an opposing force of

Options

A. `6 pi eta a^(2)upsilon`
B. `(6eta upsilon)/(pia)`
C. `6pi eta a upsilon`
D. `(pi eta upsilon)/(6a^(3))`

Explanation

To find the opposing force encountered by a small metal sphere of radius \( a \) falling with a velocity \( v \) through a viscous liquid with a coefficient of viscosity \( \eta \), we can use Stokes' law.

### Step-by-Step Solution:

1. **Understand Stokes' Law**: Stokes' law states that the viscous force \( F \) acting on a sphere moving through a viscous fluid is given by the formula:
\[ F = 6 \pi \eta r v \]
where \( r \) is the radius of the sphere, \( \eta \) is the coefficient of viscosity of the fluid, and \( v \) is the velocity of the sphere.

2. **Identify the Parameters**: In this case, we have:
- Radius of the sphere \( r = a \)
- Coefficient of viscosity \( \eta \)
- Velocity of the sphere \( v = \upsilon \)

3. **Substitute the Parameters into Stokes' Law**: Plugging in the values into the formula, we get:
\[ F = 6 \pi \eta a \upsilon \]

4. **Conclusion**: Therefore, the opposing force encountered by the sphere as it falls through the viscous liquid is:
\[ F = 6 \pi \eta a \upsilon \]

### Final Answer:
The sphere encounters an opposing force of \( F = 6 \pi \eta a \upsilon \).