Three blocks $A$,$B$,and $C$ of equal mass $m$ are placed on a smooth surface as shown. The coefficient of friction between any two blocks $A$,$B$,and $C$ is $\mu$. The maximum value of mass $M$ of block $D$ such that blocks $A$,$B$,and $C$ move without slipping over each other is:

$A$ block of mass $m_{2}$ is placed on a horizontal table and another block of mass $m_{1}$ is placed on top of it. An increasing horizontal force $F=\alpha t$ is exerted on the upper block,but the lower block never moves as a result. If the coefficient of friction between the blocks is $\mu_{1}$ and that between the lower block and the table is $\mu_{2}$,then what is the maximum possible value of $\mu_{1} / \mu_{2}$?

The rear side of a truck is open and a box of $40 \;kg$ mass is placed $5 \;m$ away from the open end as shown in Figure. The coefficient of friction between the box and the surface below it is $0.15$. On a straight road,the truck starts from rest and accelerates with $2 \;m s^{-2}$. At what distance from the starting point does the box fall off the truck? (Ignore the size of the box).

Two blocks $A$ and $B$ of masses $6 \, kg$ and $3 \, kg$ rest 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. (in $, N$)

Two blocks ($m = 0.5\, kg$ and $M = 4.5\, kg$) are arranged on a horizontal frictionless table as shown in the figure. The coefficient of static friction between the two blocks is $\frac{3}{7}$. Find the maximum horizontal force $F$ that can be applied on the larger block so that the blocks move together. (Round off to the nearest integer) [Take $g = 9.8\, m/s^2$]

Two blocks of masses $M$ and $m$ are placed on one another on a smooth horizontal surface as shown in the figure. The force $F$ is acting on the mass $M$ horizontally during time interval $t$. Assuming no relative sliding between the blocks,the work done by friction on the blocks is .......... .