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Step-by-Step Solution
Step 1: Identify the physical situation
Sand is continuously dropped onto a conveyor belt at a constant rate of mass flow (M kg/s). The belt moves with a constant velocity $v$. We need to find the force required to maintain this constant velocity as new sand is added.
Step 2: Recall the momentum principle
Force can be expressed as the rate of change of momentum:
$ F = \frac{d}{dt}(mv) $
In this problem, the mass $m$ increases with time (because sand is being added), while the velocity $v$ remains constant.
Step 3: Set up the rate of change of momentum
Let $m(t)$ be the mass of sand that has landed on the belt up to time $t$. Since the belt travels at a constant velocity $v$, its momentum at any instant is $p(t) = m(t)\,v$.
The force is then:
$ F = \frac{d}{dt}\bigl(m(t)\,v\bigr) = v \frac{dm(t)}{dt} + m(t)\,\frac{dv}{dt}. $
Step 4: Simplify using constant velocity
Because the conveyor belt's velocity $v$ is constant, $ \frac{dv}{dt} = 0 $. Hence, the force simplifies to:
$ F = v \frac{dm(t)}{dt}. $
Step 5: Substitute the mass flow rate
We are told sand is dropped at a constant rate of $M$ kg/s, meaning:
$ \frac{dm(t)}{dt} = M\ \text{(kg/s)}. $
Thus,
$ F = v \times M = M\,v. $
Step 6: Final answer
Therefore, the force required to keep the conveyor belt moving at the constant speed $v$ is:
$ F = M\,v $ newton.
