A disc is in pure rolling motion with a velocity $v$ on a rough horizontal surface. The resultant velocity of a point $P$ at an angle $\theta$ with the horizontal would be

A square plate of side $a=20 \mathrm{cm}$ and mass $M=2 \mathrm{kg}$ is lying on the horizontal plane. What will be the moment of inertia of the plate about an axis in the plane of the plate and at an angle of $\theta =18.6°$ from one of its sides?

Two masses $\left(M\right)$ are moving perpendicular to the rod of mass $M$ with velocities $v\text{ and} 2v$ as shown in the figure. The length of the rod is $L$. Find the ratio of the final velocity of the rod and its angular velocity if the masses stop at their place after collision.