A horizontal platform is rotating with uniform angular velcity around the vertical axis passing through its centre. At some instant of time a viscous fluid of mass `m` is dropped at the centre and is allowed to spread out and finally fall. The angular velocity during this period :

Options

A. decreases continuously
B. decreases initially and increases again
C. remains unaltered
D. increases continuously

Explanation

To solve the problem, we need to analyze the situation using the principles of angular momentum and the effects of adding mass to a rotating system.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a horizontal platform rotating with a uniform angular velocity (ω) around a vertical axis.
- At a certain moment, a viscous fluid of mass 'm' is dropped at the center of the platform.

2. **Moment of Inertia**:
- The moment of inertia (I) of the system will change when the fluid is added. Initially, the moment of inertia is I₀ (the moment of inertia of the platform alone).
- When the fluid is added and spreads out, the total moment of inertia increases because the fluid has mass and it contributes to the overall distribution of mass in the system.

3. **Conservation of Angular Momentum**:
- According to the conservation of angular momentum, the initial angular momentum (L₀) of the system must equal the final angular momentum (L) after the fluid is added.
- Mathematically, this can be expressed as:
\[ I₀ \cdot ω = I \cdot ω' \]
where ω' is the new angular velocity after the fluid is added.

4. **Effect of Increasing Moment of Inertia**:
- Since the moment of inertia (I) increases due to the addition of the fluid, and angular momentum is conserved, we can see that:
\[ I₀ \cdot ω = I \cdot ω' \]
implies that if I increases, ω' must decrease to keep the equation balanced.

5. **Final Behavior of Angular Velocity**:
- Initially, as the fluid spreads out, the moment of inertia increases, causing the angular velocity to decrease.
- After the fluid has spread out and starts to fall, the moment of inertia may decrease (as the mass is no longer contributing to the rotation), which could lead to an increase in angular velocity.
- Therefore, we can conclude that the angular velocity decreases initially and then may increase again after the fluid has settled.

### Conclusion:
The correct answer is that the angular velocity decreases initially and then increases again.