Question

Sand falls vertically at the rate of $2kg/s$ on a conveyor belt moving horizontally with a velocity of $0.2m/s$ . The extra force required to keep the belt moving is:-
(A) $0.4N$
(B) $0.08N$
(C) $0.04N$
(D) $0.2N$

Answer 1

The formula of the force in terms of mass and velocity has to be known. By considering the 2nd law of motion of Newton the relation between the force and momentum can be found. Since the momentum is the product of the mass and the velocity of an object, the relation between the force, mass and velocity can be understood. Note that, the movement of the sand here is along the vertical and also is uniform.

Complete step-by-step solution:
According to the Question, let us consider
$F =$ the external force applied on the body
According to Newton’s second law of motion,
$F = \dfrac{{dP}}{{dt}}$
$P =$ the momentum of the sand.
$P = M \times v$
$M =$ the mass of the body
$\therefore F = \dfrac{{d(mv)}}{{dt}}$
$\Rightarrow F = v\dfrac{{dm}}{{dt}} + m\dfrac{{dv}}{{dt}}$
Since, the sand falls with a uniform velocity, the term $\dfrac{{dv}}{{dt}} = 0$
$\therefore F = v\dfrac{{dm}}{{dt}}$
Given that, $\dfrac{{dm}}{{dt}} = 2kg/s$ and, Velocity $v = 0.2{\text{ }}m/s$

$\Rightarrow F = 0.2 \times 2 = 0.4N$
So, the extra force required to keep the belt moving is $0.4{\text{ }}N$

Note: According to Newton’s second law of motion, the rate of change of linear momentum is directly proportional to the external force applied to the body. Further, the momentum of the body happens to be in the direction where the force is exerted.
For an object already in motion, the direction of the applied force matters to determine its state.
If the external force is applied in the direction in which the object is moving, the acceleration of the body increases and vice versa.
Force is numerically equal to the product of the mass and the acceleration of the body.
Force and acceleration are vector quantities.
Multiple forces can act on the body at the same time.