Question

As shown in the figure, sand is falling from a stationary hopper onto a freight car which is moving with a uniform velocity $v_0=10\text{m/s}.$ The sand falls at the rate of $\mu =5\text{kg/s}.$ How much force is needed to keep the freight car moving at the speed $v_0$?

• A. $50\text{N}$
• B. $0\text{N}$
• C. $25\text{N}$
• D. $20\text{N}$

Answer 1

The correct option is A $50\text{N}$

As we know, thrust force acting on the car in the horizontal direction is given by
$F_{th}=\mu v_r=\left(v_r\frac{dm}{dt}\right)$
Here, relative velocity of sand loading $v_r=-v_0$
i.e $F_{th}=-\mu v_0=-5\times 10=-50\text{N}$
(in opposite direction of velocity of car)

and thrust force acting in vertical direction
$F_{tv}=\mu v$ (Balanced by Normal force)
where $v$ is the velocity of sand falling into the car.

So we need, $F=50\text{N}$ force in the direction of the car to move it with same speed).