A small tiny lead shot is gently dropped on the surface of a viscous liquid

Options

A. the lead shot will fall will an acceleration equal to g at that place
B. the velocity of the lead shot will decrease with time
C. the velocity of the lead shot will increase continuously
D. the velocity of the lead shot will reach a steady value after sometime

Answer

To solve the problem of a small tiny lead shot being dropped into a viscous liquid, we need to analyze the forces acting on it and understand the motion of the lead shot over time. Here’s a step-by-step breakdown:

### Step 1: Identify the Forces Acting on the Lead Shot
When the lead shot is dropped into the viscous liquid, two primary forces act on it:
- **Weight (W)**: This is the force due to gravity acting downwards, given by \( W = mg \), where \( m \) is the mass of the lead shot and \( g \) is the acceleration due to gravity.
- **Drag Force (F_d)**: This is the resistive force exerted by the viscous liquid acting upwards, which is proportional to the velocity of the lead shot. It can be expressed as \( F_d = 6 \pi \eta r v \), where \( \eta \) is the viscosity of the liquid, \( r \) is the radius of the lead shot, and \( v \) is the velocity.

### Step 2: Analyze the Motion of the Lead Shot
Initially, when the lead shot is dropped, its velocity \( v \) is zero. As it begins to fall, the drag force \( F_d \) starts at zero and increases gradually as the velocity increases.

### Step 3: Determine Terminal Velocity
The lead shot will continue to accelerate until the drag force equals the weight of the lead shot. At this point, the net force acting on the lead shot becomes zero, and it reaches a constant velocity known as the **terminal velocity**. This can be expressed mathematically as:
\[ mg = F_d \]
\[ mg = 6 \pi \eta r v_t \]
where \( v_t \) is the terminal velocity.

### Step 4: Analyze the Options
Now, let’s evaluate the provided options based on our analysis:

1. **The lead shot will fall with an acceleration equal to zero.**
- This is incorrect at the moment of dropping since the lead shot initially accelerates due to gravity until it reaches terminal velocity.

2. **The velocity of the shot will decrease with time.**
- This is incorrect. The velocity will initially increase until it reaches terminal velocity.

3. **The velocity of the lead shot will increase continuously.**
- This is incorrect because the velocity will not increase indefinitely; it will reach a steady state (terminal velocity).

4. **The velocity of the lead shot will reach a steady state after some time.**
- This is correct. Eventually, the lead shot will reach terminal velocity where the forces balance out.

### Conclusion
The correct option is that **the velocity of the lead shot will reach a steady state after some time**.