A block of mass $M$ rests on a rough horizontal surface. The coefficient of friction between the block and the surface is $\mu$. A force $F=Mg$, acting at an angle $\mathrm{\theta }$ with the vertical side of the block, pulls it. In which of the following cases, the block can be pulled along the surface?

Blocks $A$ and $B$ are arranged as shown in the figure. The pulley is frictionless. The mass of $A$ is $10 \mathrm{kg}$. The coefficient of friction between the block $A$ and the horizontal surface is $0.20$. The minimum mass of $B$, to start the motion, will be

The minimum value of $F$, such that the block remain at rest, is

The minimum value of $F$, such that, the block is at rest, is $(\tan \text{θ}>\mu )$

A block of mass $m$ is in contact with the cart $C$, as shown in the figure. The coefficient of static friction between the block and the cart is $\mu$. The acceleration $\left(a\right)$ of the cart that will prevent the block from falling satisfies

Consider the situation shown in the figure. The wall is smooth but the surfaces of $A$ and $B$, in contact, are rough. The friction on $B$ to $A$, in equilibrium, 

A block of mass $m$, lying on a rough horizontal plane, is acted upon by a horizontal force $P$ and another force $Q$, inclined at an angle $\mathrm{\theta }$ to the vertical. The block will remain in equilibrium, if the coefficient of friction between the block and the surface is