Four rods of equal mass $m$ and length $L$ forms a square $ABCD$ as shown in figure. The moment of inertia of $ABCD$ about the axis $O{O}^{\text{'}}$ passing, through the point $A$ lying in the plane of square is

 

From a uniform sphere of mass $M$ and radius $R$ a cavity of diameter $R$ is created as shown. Find the ratio of moment of inertia of the sphere left about $A{A}^{\text{'}}$ and $B{B}^{\text{'}}$.

A force $\overrightarrow{F}=\left(3\hat{\mathrm{i}}+4\hat{\mathrm{j}}\right)\mathrm{N}$ is acting on a point mass $m=\left(\frac{1}{2}\right)\mathrm{kg}$ at a point $A\left(2\mathrm{m},2\mathrm{m}\right)$. Find the angular acceleration of the line $OA$ at this instant.

A ring of mass $m$ and radius$R$ is acted upon by a force $F$ as shown in the figure. There is sufficient friction between the ring and the ground. The force of friction necessary for pure rolling is

A solid cylinder is rolling without slipping with velocity of its centre of mass $v$ and angular velocity about its centre of mass $\omega$ on a horizontal frictionless surface as shown in figure. If it collides with a frictionless vertical wall $X$, then after collision its velocity and angular velocity respectively become

The minimum horizontal force applied to the centre of wheel of radius $R$ and weight $w$ to pull it over a step of height $R/2$

A rod of mass $m$ and length $l$ hinged at its centre is placed on a horizontal surface. A bullet of mass $m$ moving with velocity $v$ strikes the end $A$ of the rod and gets embedded in it. The angular velocity with which the system rotates after the bullet strikes the rod