Two blocks $A$ and $B$ of masses $6\, kg$ and $3\, kg$ are placed on a smooth horizontal surface as shown in the figure. If the coefficient of friction between $A$ and $B$ is $0.4$,find the maximum horizontal force $F$ that can be applied to block $A$ such that they move together without separation. (Take $g = 10\, m/s^2$)

$A$ $40 \,kg$ slab rests on a frictionless floor. $A$ $10 \,kg$ block rests on top of the slab. The coefficient of friction between the block and the slab is $0.40$. The $10 \,kg$ block is acted upon by a horizontal force of $100 \,N$. If $g=10 \,m/s^2$,the resulting acceleration of the slab will be ............. $m/s^2$.

$A$ block of mass $M$ is placed on a horizontal surface and is tied with an inextensible string to a block of mass $m$,as shown in the figure. $A$ block of mass $m_0$ is also placed on $M$. The horizontal surface is frictionless. Find the minimum value of the coefficient of friction $\mu$ between the blocks $M$ and $m_0$ such that all three blocks move together.

The maximum value of $F$ which can be applied on the system shown in the figure so that both blocks move together with the same acceleration is ..........

Consider a system of two masses and a pulley shown in the figure. The coefficient of friction between the two blocks and also between the bottom block and the table is $\mu = 0.1$. Find the force $F$ that must be applied to the $0.8 \text{ kg}$ block such that it attains an acceleration of $5 \text{ m/s}^2$. (Assume acceleration due to gravity, $g = 10 \text{ m/s}^2$.) (in $\text{ N}$)

Determine the time in which the smaller block reaches the other end of the bigger block as shown in the figure. $(g = 10\ m s^{-2})$.