Sand particles drop vertically at the rate of `2kgs^-1` on a conveyor belt moving horizontally with a velocity of `0.2ms^-1`.
The extra force required to keep the belt moving is

Options

A. 0.08 W
B. 0.04 W
C. 4 W
D. 1 W

Answer

To solve the problem, we need to determine the extra force required to keep the conveyor belt moving when sand particles drop onto it. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Rate of sand falling (mass flow rate), \( \frac{dm}{dt} = 2 \, \text{kg/s} \)
- Velocity of the conveyor belt, \( v = 0.2 \, \text{m/s} \)

### Step 2: Understand the concept of momentum
When sand particles fall onto the conveyor belt, they have a vertical momentum. As they land on the moving conveyor belt, they need to be accelerated to the speed of the belt (horizontal velocity). This change in momentum requires an additional force.

### Step 3: Calculate the extra force required
The extra force required to keep the belt moving can be calculated using the formula for force due to the change in momentum:
\[ F = \frac{d}{dt}(m \cdot v) \]
Since the velocity \( v \) is constant, we can rewrite the force as:
\[ F = v \cdot \frac{dm}{dt} \]
Substituting the known values:
\[ F = 0.2 \, \text{m/s} \cdot 2 \, \text{kg/s} \]
\[ F = 0.4 \, \text{N} \]

### Step 4: Conclusion
The extra force required to keep the conveyor belt moving is \( 0.4 \, \text{N} \).