Question

A conveyor belt is moving with constant velocity \( V \). Sand is being dropped on the belt at the rate of \( M \, \text{kg/s} \). The force necessary to keep the belt moving with a constant velocity \( V \, \text{m/s} \) will be

• A. \( \frac{M}{V} \, \text{N} \)
• B. \( 2MV \, \text{N} \)
• C. Zero \( \text{N} \)
• D. \( MV \, \text{N} \)

Hint:

The force required to keep a conveyor belt moving at constant velocity is related to the rate at which mass is added to the system and the velocity of the conveyor belt.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Force is defined as the rate of change of momentum. If mass is changing while velocity is constant, a force must be applied.
Step 2: Key Formula or Approach:
Newton's Second Law: \( F = \frac{d(mv)}{dt} = v \frac{dm}{dt} + m \frac{dv}{dt} \).
Step 3: Detailed Explanation:
Given:
Velocity \( v = V \) (constant), so \( \frac{dv}{dt} = 0 \).
Rate of change of mass \( \frac{dm}{dt} = M \text{ kg/s} \).
Substitute into the force formula:
\[ F = V \cdot M + m \cdot 0 \]
\[ F = MV \text{ Newtons} \]
Step 4: Final Answer:
The necessary force is \( MV \) Newtons.