Question

A cart of mass $M$ has a block of mass $m$ in contact with it as shown in the figure. The coefficient of friction between the block and the cart is $\mu$ . What should be the minimum acceleration of the cart so that the block of mass $m$ does not fall?

Answer 1

In this question, let us assume the cart of mass $M$ moves with an acceleration $a{\text{ m/}}{{\text{s}}^2}$. Thus, a pseudo force $ma$ acts on the small block of mass $m$ and it must be equal to the normal force between the blocks for horizontal equilibrium. The force of friction $\mu N$ will act in the upward direction and must be equal to $mg$ so that the block does not fall. From this condition, we can find out the minimum acceleration required.

Formula used:
$F = ma$
Where $F$ is the total force, $m$ is the mass of the system and $a$ is the net acceleration of the system.
$f = \mu N$
Where, $f$ is the limiting frictional force, $\mu$ is the coefficient of friction and $N$ is the normal force acting.

Complete step by step answer:
Let us assume the cart of mass $M$ is moving with an acceleration of $a{\text{ m/}}{{\text{s}}^2}$. It is in contact with a block of mass $m$ such that the coefficient of friction between the cart and block is $\mu$. A normal force $N$ acts on the block in the horizontal direction.As the cart is accelerating a pseudo force $F = ma$ acts in the opposite direction of acceleration
For horizontal equilibrium,
$N = F = ma$

Frictional force $f = \mu N$ acts in the upward direction. For block to be in equilibrium frictional force must balance its weight,
$f \geqslant mg$
$\Rightarrow \mu N \geqslant mg$
Substituting $N = F = ma$,
$\Rightarrow \mu ma \geqslant mg$
$\therefore a \geqslant \dfrac{g}{\mu }$

Thus, minimum acceleration so that the block does not fall must be $\dfrac{g}{\mu }{\text{ m/}}{{\text{s}}^2}$.

Note: Pseudo force is an imaginary force like frictional force and comes in effect when the frame of reference has started acceleration compared to a non-accelerating frame. It always acts in the opposite direction of the motion. The work done by a pseudo force is zero as it acts to appear on the body.