A solid cube of mass $5 \mathrm{kg}$ is placed on a rough horizontal surface in $xy-$plane as shown. The friction coefficient between the surface and the cube is $0.4$. An external force $\overrightarrow{F}=6\hat{\mathrm{i}}+8\hat{\mathrm{j}}+20\hat{\mathrm{k}} \mathrm{N}$ is applied on the cube (use $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

If the coefficient of friction between wedge $A$ and block $B$ is $\mathrm{\mu }$, the maximum horizontal acceleration of the wedge $A$ for which the block $B$ will remain at rest with respect to the wedge is

A block of mass$M=5 \mathrm{kg}$ is resting on a rough horizontal surface for which the coefficient of friction is $0.2$. When the force $F=40 \mathrm{N}$ is applied, the acceleration of the block will be

A force of $750 \mathrm{N}$ is applied to a block of mass $102 \mathrm{kg}$ to prevent it from sliding on a plane with an inclination angle $30°$ with the horizontal. If the coefficients of static friction and kinetic friction between the block and the plane are $0.4$ and $0.3$, respectively, then the frictional force acting on the block is

Two blocks of masses $m_1$ and $m_2$ connected by a string are placed gently over a fixed inclined plane, such that the tension in the connecting string is initially zero. The coefficient of the friction between $m_1$ and inclined plane is ${\mu }_1$, between $m_2$ and the inclined plane is ${\mu }_2$. The tension in the string shall remain zero if