A thin hollow sphere of mass $m$ is completely filled with an ideal liquid of mass $m$. when the sphere rolls with a velocity $v$. Kinetic energy of the system is equal to

At shown instant a thin uniform rod $\mathrm{AB}$ of length $L=1 \mathrm{m}$ and mass $m=1 \mathrm{kg}$ is coincident with $y-$axis such that centre of rod is at origin. The velocity of end $A$ and centre $O$ of rod at shown instant are ${\overrightarrow{v}}_{\mathrm{A}}=-2\hat{\mathrm{i}} \mathrm{m} {\mathrm{s}}^{-1}$ and ${\overrightarrow{v}}_0=10\hat{\mathrm{i}} \mathrm{m} {\mathrm{s}}^{-1}$ respectively. Then the kinetic energy of rod at the shown instant is:

The moment of inertia of an elliptical disc of uniform mass distribution of mass $m$, major axis $r$ and minor axis $d$, about its axis is

A uniform disc of radius $R$ lies in the $x-y$ plane, with its centre at the origin. Its moment of inertia about $z$-axis is equal to its moment of inertia about the line$y=x+c.$ The value of $c$will be

A square plate of edge $\frac{a}{2}$ is cut out from a uniform square plate of edge $a$, as shown in the figure. The mass of the remaining portion is $M.$The moment of inertia of the shaded portion about an axis passing through $O$ (centre of the square of side $a$) and perpendicular to plane of the plate is

Moment of inertia of a uniform disc about the axis $\mathrm{O} \mathrm{O}\text{'}$ is:

Seven identical circular planar disks, each of mass $M$ and radius $R$ are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point $\mathrm{P}$ is :