A sphere is dropped gently into a medium of infinite extent. As the sphere falls, the force acting downwards on it.

Options

A. remain constant throughout
B. increases for sometimes and then becomes constant
C. decreases for sometimes and then becomes zero
D. increases for sometime and then decreases.

Explanation

To solve the problem of the forces acting on a sphere that is dropped gently into a medium of infinite extent, we can break down the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Forces Acting on the Sphere**:
When the sphere is dropped into the medium, two main forces act on it: the gravitational force (weight) acting downwards and the buoyant force acting upwards.

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

3. **Buoyant Force**:
The buoyant force (\( F_b \)) acting on the sphere can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the sphere. It can be expressed as:
\[ F_b = \rho V g \]
where \( \rho \) is the density of the medium, \( V \) is the volume of the sphere, and \( g \) is the acceleration due to gravity.

4. **Net Force Acting on the Sphere**:
As the sphere falls, the net force (\( F_{net} \)) acting on it can be expressed as:
\[ F_{net} = F_g - F_b \]
Initially, when the sphere is dropped, the buoyant force is less than the gravitational force, resulting in a downward acceleration.

5. **Equilibrium Condition**:
As the sphere continues to fall, it displaces more fluid, which increases the buoyant force. Eventually, the buoyant force will increase to equal the gravitational force. At this point, the sphere reaches a state of equilibrium, and the net force becomes zero:
\[ F_g = F_b \]
This means that the sphere will stop accelerating and will fall at a constant velocity, known as terminal velocity.

6. **Conclusion**:
Thus, the force acting downwards on the sphere decreases as it falls due to the increasing buoyant force until it reaches equilibrium.

### Final Answer:
The force acting downwards on the sphere decreases until it reaches equilibrium, at which point the net force becomes zero.