$A$ pen of mass $m$ is lying on a piece of paper of mass $M$ placed on a rough table. If the coefficients of friction between the pen and paper and the paper and the table are $\mu_1$ and $\mu_2$,respectively,then the minimum horizontal force with which the paper has to be pulled for the pen to start slipping is given by:

Consider the system shown below. $A$ horizontal force $F$ is applied to a block $X$ of mass $8 \,kg$,such that the block $Y$ of mass $2 \,kg$ adjacent to it does not slip downwards under gravity. There is no friction between the horizontal plane and the base of the block $X$. The coefficient of friction between the surfaces of blocks $X$ and $Y$ is $0.5$. The minimum value of $F$ is ............ $N$ (take acceleration due to gravity to be $10 \,ms^{-2}$).

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 $g = 10\, m/s^2$)

Two blocks $(A)$ of $2\,kg$ and $(B)$ of $5\,kg$ rest one over the other on a smooth horizontal plane. The coefficient of static and dynamic friction between $(A)$ and $(B)$ is the same and equal to $0.60$. Find the maximum horizontal force that can be applied to $(B)$ such that both $(A)$ and $(B)$ do not have any relative motion. $(g = 10\,m/s^2)$

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 $\text{ N}$)

$A$ 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 $M$. If friction exists between the block $M$ and the block $m_0$ and there is no friction between the block $M$ and the horizontal surface,then the minimum value of $\mu$ for which the block $m_0$ remains stationary with respect to $M$ is: