Sand is being dropped on a conveyor belt at the rate of `Mkg//s` . The force necessary to kept the belt moving with a constant with a constant velocity of `vm//s` will be.

Options

A. `(Mv)/(2)` newton
B. zero
C. Mv newton
D. 2 Mv newton

Explanation

To solve the problem, we need to determine the force required to keep a conveyor belt moving at a constant velocity when sand is being dropped onto it at a certain rate.

### Step-by-Step Solution:

1. **Understand the problem**: We have a conveyor belt moving at a constant velocity \( v \) m/s, and sand is being added to it at a rate of \( M \) kg/s. We need to find the force required to maintain this constant velocity.

2. **Identify the forces involved**: The force required to keep the conveyor belt moving at a constant velocity can be derived from Newton's second law of motion, which states that force is equal to mass times acceleration (\( F = ma \)). However, in this case, since the velocity is constant, the acceleration is zero.

3. **Use the concept of momentum**: The force can also be expressed in terms of the rate of change of momentum. The momentum \( p \) of an object is given by \( p = mv \). If mass \( m \) is changing with time, we can express the force as:
\[ F = \frac{dp}{dt} = \frac{d(mv)}{dt} \]

4. **Apply the product rule**: Using the product rule of differentiation, we can expand this as:
\[ F = v \frac{dm}{dt} + m \frac{dv}{dt} \]
Since the velocity \( v \) is constant, \( \frac{dv}{dt} = 0 \). Therefore, the equation simplifies to:
\[ F = v \frac{dm}{dt} \]

5. **Substitute the known values**: We know that \( \frac{dm}{dt} = M \) kg/s (the rate at which sand is being added). Substituting this into the equation gives:
\[ F = v \cdot M \]

6. **Final expression for force**: Thus, the force necessary to keep the belt moving at a constant velocity is:
\[ F = Mv \text{ Newtons} \]

### Conclusion:
The force necessary to keep the conveyor belt moving at a constant velocity of \( v \) m/s while sand is being dropped at a rate of \( M \) kg/s is \( F = Mv \) Newtons.