Four holes of radius $\mathrm{R}$ are cut from a thin square plate of side $4\mathrm{R}$ and mass $\mathrm{M}$. The moment of inertia of the remaining portion about $\mathrm{z}$-axis 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 :

In figure $\left(A\right)$, half of the metre scale is made of wood while the other half of steel. The wooden part is pivoted at $O$. A force $F$ is applied at the end of steel part. In figure $\left(B\right)$, the steel part is pivoted at $O\text{'}$ and the same force is applied at the wooden end, then

A ring of mass $m$ and radius $R$ is rolling down a rough inclined plane of angle $\mathrm{\theta }$ with horizontal. Plot the angular momentum of the ring about the point of contact of the ring and the plane as a function of time.

A uniform disc of mass $M$ and radius $R$ is released from rest in the shown position. $PQ$ is a string, $OP$ is a horizontal line, $O$ is the centre of the disc and distance $OP$ is $\frac{R}{2}.$ Then tension in the string just after the disc is released will be: