Spherical balls of radius R are falling in a viscous fluid of velocity v .The retarding viscous force acting on the spherical ball is

Options

A. directly proportional to R but inversely proportional v.
B. directly proportional to both radius R and velocity v.
C. inversely proportional to both radius R and velocity v.
D. inversely proportional to R but directly proportional to velocity v.

Explanation

To solve the problem of finding the retarding viscous force acting on a spherical ball falling in a viscous fluid, we can follow these steps:

### Step 1: Understand the Concept of Viscous Force
The retarding force acting on an object moving through a viscous fluid is known as the viscous drag force. For a spherical object, this force can be derived from Stokes' law.

### Step 2: Apply Stokes' Law
According to Stokes' law, the viscous force \( F \) acting on a spherical object moving through a viscous fluid is given by the formula:
\[ F = 6 \pi \eta R v \]
where:
- \( F \) is the retarding viscous force,
- \( \eta \) is the coefficient of viscosity of the fluid,
- \( R \) is the radius of the spherical ball,
- \( v \) is the velocity of the ball relative to the fluid.

### Step 3: Identify the Variables
From the formula, we can identify the variables:
- \( R \) (radius of the ball) is a factor that affects the force.
- \( v \) (velocity of the ball) is also a factor that affects the force.
- \( \eta \) (coefficient of viscosity) is a property of the fluid that also affects the force.

### Step 4: Analyze the Proportional Relationships
From the equation \( F = 6 \pi \eta R v \), we can see that:
- The force \( F \) is directly proportional to the radius \( R \) of the ball.
- The force \( F \) is directly proportional to the velocity \( v \) of the ball.
- The force \( F \) is also directly proportional to the coefficient of viscosity \( \eta \).

### Step 5: Conclusion
Thus, we conclude that the retarding viscous force acting on the spherical ball is directly proportional to both the radius of the ball and its velocity in the fluid.

### Final Answer
The retarding viscous force acting on the spherical ball is given by:
\[ F = 6 \pi \eta R v \]

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