A uniform hemisphere with radius $R=10 \mathrm{m}$ is placed on a horizontal smooth surface. Its centre of mass $C$ is below the centre $O.$ An object $\text{'}A\text{'}$ is placed on the flat surface of the hemisphere at point $O.$ The mass of the object is $1/8^{\mathrm{th}}$ of mass of the hemisphere. The coefficient of friction between the object and the surface of the hemisphere is $\mu =0.2.$ Now the object $\text{'}A\text{'}$ is displaced slowly along the flat surface of the hemisphere towards its outer rim. Find the maximum distance (in $m$) of the object from the centre $O$ without slipping on the hemispherc.

hFigure show a uniform block of mass $m=\sqrt{\frac{3}{13}} \mathrm{kg}$ having square base and equilateral triangular cross section placed at the edge of the floor of a truck which is under constant acceleration $a=\frac{g\sqrt{3}}{4} \mathrm{m}/{\mathrm{s}}^2$ to the right. A horizontal force on the apex of the triangle $F$ is just enough to just tilt the block. Find the force (in newton) on the block at point $A$ where the block is in contact with the step edge when $F$ is applied. (Take $g=10 \mathrm{m}/{\mathrm{s}}^2$)

A semicircular disc of radius $R$ and mass $M$ is pulled by a horizontal force $F$ so that it moves with uniform velocity. The coefficient of friction between disc and ground is $\mu =\frac{4}{3\pi }.$ Find the value of $\theta$ in degree.

In the arrangement shown in figure the cylinder of mass $M=9 \mathrm{kg}$ is at rest on an incline. The plane is inclined at angle $37°$ with horizontal. The string between the cylinder and the pulley $\left(P\right)$ is horizontal. Find the minimum the mass of the block (in $\mathrm{kg}$) to keep the system in equilibrium.

In the instant shown in the figure, the board is moving up vertically with constant velocity $\left(v\right).$ The drum winds up at a constant rate $\omega =15 \mathrm{rad}/\mathrm{s}.$ If the radius of the drum is $R=10 \mathrm{cm}$ and the board always remains horizontal. Find its velocity of board at this moment (in $m/s$).

A weightless rod of length $l$ with a small load of mass $m$ at the end is hinged at point $A$ as shown and occupies a strictly vertical position, touching a block of mass $M.$ A light jerk sets the system in motion. For what mass ratio $M/m$ will the rod form an angle $\theta =\pi /6$ with the horizontal at the moment of the separation from the block?