The velocity of the centre of mass of a solid sphere of radius $R$ rotating with angular velocity $\omega$ about an axis passing through its centre of mass is

In the figure given below, a bullet of mass $m$ moving with speed $u$ strikes a rod $AB$ of mass $M$ and length $L$ at the end $B$ and gets embedded into it. If initially the rod is vertical and hinged at end $A$, then angular speed of the system just after collision will be

In the figure given below, a child of mass $4 \mathrm{kg}$ is standing on a wooden plank of mass $40 \mathrm{kg}$ which is kept at rest on a horizontal smooth ground. If child walks on the plank with a velocity of $10 \mathrm{m} {\mathrm{s}}^{-1}$ with respect to plank, then the speed of the plank w.r.t. ground will be

Three rings each of mass $m$ and radius $r$ are so placed that they touch each other as shown in the figure. The moment of inertia of the system about the axis $O{O}^{\text{'}}$ is

In the figure shown below, a meter stick of mass $M$ is supported in a horizontal position by two light strings attached at points $A$ and $B$ of the rod. The initial angular acceleration of the stick if one of the strings is cut, is [Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$]