A man of mass $75 \mathrm{kg}$ is pushing a heavy box on a flat floor. The coefficient of kinetic and static friction between the floor and the box is $0.20,$ and the coefficient of static friction between the man's shoes and the floor is $0.80.$ If the man pushes horizontally (see figure), what is the maximum mass (in $\times \mathrm{kg}$) of the box he can move? 

In the following figure shows an accelerating conveyor belt inclined at an angle $37°$ above horizontal. The coefficient of friction between the belt and block is $1.$ Find the least time (in sec) in which block can reach the top, starting from rest at the bottom.

In the figure shown, find the minimum force $F$(in $\times {10}^2 \mathrm{N}$) to be applied perpendicular to the incline so that the block does not slide.

A block of $7 \mathrm{kg}$ is placed on a rough horizontal surface and is pulled through a variable force $F$ (in $\mathrm{N}$)$=5t,$ where $t$ is time in second, at an angle of $37°$ with the horizontal as shown in figure. The coefficient of static friction of the block with the surface is one. If the force starts acting at $t=0 \mathrm{s},$ if the time (in $\sec$) at which the block starts to slide is found to be $10n.$ Find.

A block of mass $m_1=4 \mathrm{kg}$ is placed on a wedge of an angle $\mathrm{\theta }=45°$, as shown. The block is moving over the inclined surface of the wedge of mass $m_2=5 \mathrm{kg}$. The friction coefficient between the block and the wedge is ${\mu }_1=\frac{1}{2}$, whereas it is ${\mu }_2$ between the wedge and the horizontal surface. If the minimum value of ${\mu }_2$, so that the wedge remains stationary on the surface is found to be $\frac{1}{n}$. Find the value of $n$. Take, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$.

Two wedges, each of mass $m$, are placed next to each other on a flat horizontal floor. A cube of mass $M$ is balanced on the wedges as shown in the figure. Assume no friction between the cube and the wedges, but a coefficient of static friction $\mu =\frac{1}{3}$ between the wedges and the floor. What is the largest ratio of $\frac{M}{m}$, so that the system can be balanced without the motion of the wedges?

The three flat blocks in the figure are positioned on the $37°$ incline and a force parallel to the inclined plane is applied to the middle block. The upper block is prevented from moving by a wire which attaches it to the fixed support. The masses of three blocks in $\mathrm{kg}$ and coefficient of static friction for each of the three pairs of contact surfaces are shown in the figure. If the maximum value which force $P$ may have before slipping takes place anywhere is found to be $4x \mathrm{N}$. Find the value of $x$.

A block of mass $m=4 \mathrm{kg}$ is placed over a rough inclined plane as shown in the figure. The coefficient of friction between the block and the plane is $\mu =0.5.$ A force $F=10 \mathrm{N}$ is applied on the block at an angle of $30°.$ Find the contact force (in $N$) between the block and the plane.

A side view of a simplified form of the vertical latch $B$ of mass $m=0.6 \mathrm{kg}$ is as shown in the figure. The lower member $A$ can be pushed forward in its horizontal channel. The sides of the channels are smooth, but at the interfaces of $A$ and $B,$ which are at $45°$ with the horizontal, there exists a static coefficient of friction $\mu =0.4.$ What is the minimum force $F$ (in $N$) that must be applied horizontally to $A$ to start the motion of the latch $B.$