A small ball of mass M and density `rho` is dropped in a viscous liquid of density `rho_0` . After some time, the ball falls with a constant velocity. What is the viscous force on the ball?

Options

A. `F =Mg(1+ (rho_0)/rho)`
B. `F =Mg(1+ (rho)/rho_0)`
C. `F =Mg(1-(rho_0)/rho)`
D. `F =Mg(1pmrhorho_0)`

Answer

To solve the problem, we need to analyze the forces acting on the ball when it reaches a constant velocity (terminal velocity) while falling through the viscous liquid.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Ball:**
When the ball is dropped into the viscous liquid, there are two main forces acting on it:
- The gravitational force (weight) acting downwards: \( F_g = mg \)
- The viscous force acting upwards: \( F_v \)

2. **Understand Terminal Velocity:**
When the ball falls with a constant velocity, it means that the net force acting on the ball is zero. This condition is known as terminal velocity. Therefore, we can write:
\[ F_g - F_v = 0 \]
This implies:
\[ F_g = F_v \]

3. **Express the Gravitational Force:**
The gravitational force acting on the ball can be expressed as:
\[ F_g = mg \]
where \( m \) is the mass of the ball and \( g \) is the acceleration due to gravity.

4. **Calculate the Mass of the Ball:**
The mass \( m \) of the ball can be expressed in terms of its density \( \rho \) and its volume \( V \):
\[ m = \rho V \]

5. **Calculate the Volume of the Ball:**
The volume \( V \) of the ball can be expressed as:
\[ V = \frac{m}{\rho} \]

6. **Express the Viscous Force:**
The viscous force \( F_v \) can be expressed in terms of the density of the liquid \( \rho_0 \) and the volume of the ball:
\[ F_v = \rho_0 V g \]
Substituting \( V \) from the previous step, we get:
\[ F_v = \rho_0 \left(\frac{m}{\rho}\right) g \]

7. **Set the Forces Equal:**
Since \( F_g = F_v \), we can set the two expressions equal to each other:
\[ mg = \rho_0 \left(\frac{m}{\rho}\right) g \]

8. **Simplify the Equation:**
Dividing both sides by \( g \) (assuming \( g \neq 0 \)):
\[ m = \frac{\rho_0 m}{\rho} \]
Rearranging gives:
\[ 1 = \frac{\rho_0}{\rho} \]

9. **Calculate the Viscous Force:**
The viscous force can now be expressed as:
\[ F_v = mg - F_v \]
Rearranging gives:
\[ F_v = mg - \left(1 - \frac{\rho_0}{\rho}\right) mg \]
Thus, the viscous force on the ball is:
\[ F_v = mg \left(1 - \frac{\rho_0}{\rho}\right) \]

### Final Answer:
The viscous force on the ball is:
\[ F_v = mg \left(1 - \frac{\rho_0}{\rho}\right) \]