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

A uniform rod of length $6\mathrm{a}$ and mass $8\mathrm{m}$ lies on a smooth horizontal table. Two particles of masses $m$ and $2\mathrm{m}$ moving in the same horizontal plane with speed $2\mathrm{v}$ and $v$ respectively, strike the rod and stick to the rod after collision as shown in the figure. The velocity of centre of mass and angular velocity about centre of mass just after collision are respectively.