Viscous force on a small sphere of radius r moving in a fluid varies directly with

Options

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

Answer

To solve the question regarding the relationship of viscous force on a small sphere moving in a fluid, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Viscous Force**:
The viscous force is the resistance experienced by an object moving through a fluid. It arises due to the internal friction in the fluid.

2. **Stokes' Law**:
According to Stokes' Law, the viscous force \( F \) acting on a small sphere moving through a viscous fluid is given by the formula:
\[ F = 6 \pi \eta r v \]
where:
- \( F \) is the viscous force,
- \( \eta \) is the viscosity of the fluid,
- \( r \) is the radius of the sphere,
- \( v \) is the velocity of the sphere.

3. **Identifying Direct Proportionality**:
In the formula \( F = 6 \pi \eta r v \), we can see that the viscous force \( F \) is directly proportional to the radius \( r \) of the sphere when the viscosity \( \eta \) and velocity \( v \) are held constant.

4. **Conclusion**:
Therefore, we conclude that the viscous force on a small sphere moving in a fluid varies directly with the radius of the sphere.

### Final Answer:
The viscous force on a small sphere of radius \( r \) moving in a fluid varies directly with the radius \( r \).

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