$ABC$ is a plane lamina of the shape of an equilateral triangle. $D, E$ are mid-points of $AB, AC$ and $G$ is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through $G$ and perpendicular to the plane $ABC$ is $I_0$. If part $ADE$ is removed, the moment of inertia of the remaining part about the same axis is $\frac{N{I}_0}{16}$ where $N$ is an integer. Value of $N$ is:

 

For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through $\mathrm{O}$ (the centre of mass) and $\mathrm{O}$' (corner point) is:

A thin rod of mass $0.9 \mathrm{kg}$ and length $1 \mathrm{m}$ is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of mass $0.1 \mathrm{kg}$ moving in a straight line with velocity $80 \mathrm{m} {\mathrm{s}}^{-1}$ hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in $\mathrm{rad} {\mathrm{s}}^{-1}$) of the rod immediately after the collision will be …………

Shown in the figure is a hollow ice-cream cone (it is open at top). If its mass is $\mathrm{M}$, radius of its top is $\mathrm{R}$ and height, $\mathrm{H}$, then its moment of inertia about its axis is

Four point masses, each of mass $\mathrm{m}$, are fixed at the corners of a square of side I. The square is rotating with angular frequency $\omega ,$ about an axis passing through one of the corners of the square and parallel to the diagonal, as shown in the figure. The angular momentum of the square about the axis is