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 problem.
In this problem, sand is being dropped onto a conveyor belt that is moving with a constant velocity \( V \). The rate at which sand is being added to the belt is \( M \, \text{kg/s} \). The question asks us to determine the force required to keep the belt moving at constant velocity.

Step 2: The momentum principle.
The key concept in this problem is the change in momentum of the sand. As the sand is dropped onto the belt, it is accelerated from zero velocity to the velocity of the belt, which is \( V \). The rate at which momentum is added to the system (the sand being added to the belt) is the force required to maintain the belt’s motion. The momentum imparted to the sand per second is given by: \[ F = \Delta p = \frac{d}{dt} (m \cdot v). \]

Step 3: Calculating the force required.
Since the sand is being added at a rate of \( M \, \text{kg/s} \), the force required to accelerate the sand to the velocity of the belt is: \[ F = M \cdot V. \]

Step 4: Final Answer.
Thus, the force necessary to keep the belt moving at constant velocity \( V \) is: \[ \boxed{MV \, \text{N}}. \]