The velocity of small ball of mass `M` and density `d_(1)` when dropped in a container filled with glycerin becomes constant after sometime. If the density glycerin of is `d_(2)`, the viscous force acting on ball is

Options

A. `(Md_(1)g)/(d_(2))`
B. `Mg(1-(d_(2))/(d_(1)))`
C. `(M(d_(1)+d_(2)))/g`
D. `Md_(1)d_(2)`

Explanation

To find the viscous force acting on a small ball of mass \( M \) and density \( d_1 \) when it is dropped in glycerin (with density \( d_2 \)), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces Acting on the Ball**:
- When the ball is dropped into the glycerin, three forces act on it:
- The weight of the ball (\( W \)) acting downwards, given by \( W = M \cdot g \), where \( g \) is the acceleration due to gravity.
- The buoyant force (\( F_b \)) acting upwards, which is given by Archimedes' principle.
- The viscous force (\( F_v \)) acting upwards, opposing the motion of the ball.

2. **Setting Up the Equation**:
- Since the ball reaches a constant velocity, it means that the net force acting on it is zero. Therefore, we can write the equation:
\[ W = F_b + F_v \]
- This implies:
\[ M \cdot g = F_b + F_v \]

3. **Calculating the Buoyant Force**:
- The buoyant force can be calculated using the formula:
\[ F_b = V \cdot d_2 \cdot g \]
- Where \( V \) is the volume of the ball. The volume of the ball can be expressed in terms of its mass and density:
\[ V = \frac{M}{d_1} \]
- Substituting this into the buoyant force equation gives:
\[ F_b = \left(\frac{M}{d_1}\right) \cdot d_2 \cdot g \]

4. **Substituting Back into the Equation**:
- Now, substituting \( F_b \) back into the equation \( M \cdot g = F_b + F_v \):
\[ M \cdot g = \left(\frac{M}{d_1} \cdot d_2 \cdot g\right) + F_v \]

5. **Solving for the Viscous Force**:
- Rearranging the equation to solve for \( F_v \):
\[ F_v = M \cdot g - \left(\frac{M}{d_1} \cdot d_2 \cdot g\right) \]
- Factoring out \( M \cdot g \):
\[ F_v = M \cdot g \left(1 - \frac{d_2}{d_1}\right) \]

### Final Result:
The viscous force acting on the ball is:
\[ F_v = M \cdot g \left(1 - \frac{d_2}{d_1}\right) \]