Sand is being dropped on a conveyor belt at the rate of $M ext{ kg/s}$. The force necessary to keep the belt

Question

Sand is being dropped on a conveyor belt at the rate of $M 	ext{ kg/s}$. The force necessary to keep the belt moving with a constant velocity of $v 	ext{ m/s}$ will be:

Accepted Answer

1. **Concept:** According to Newton's Second Law of Motion, the net force acting on a body is equal to the rate of change of its momentum ($F = \frac{dp}{dt}$) [NCERT Class 11, Physics Part I, Sec 5.5].
2. **Variable Mass System:** Here, the mass is changing with time, but the velocity is constant. The momentum $p = mv$.
   $$ F = \frac{d(mv)}{dt} = m\frac{dv}{dt} + v\frac{dm}{dt} $$
3. **Application:**
   - Since the velocity $v$ is constant, acceleration $\frac{dv}{dt} = 0$.
   - The rate of change of mass is given as $\frac{dm}{dt} = M 	ext{ kg/s}$.
   - Therefore, the term $m\frac{dv}{dt}$ vanishes, leaving:
     $$ F = v \left( \frac{dm}{dt} ight) = v \cdot M = Mv 	ext{ N} $$
4. **Conclusion:** The force required to maintain constant velocity is $Mv$ newton.

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