$A$ quarter part is cut from a uniform circular disc of radius $R$. The mass of this cut part is $M$. It rotates about an axis passing through the center of the original disc and perpendicular to the plane of the disc. What will be its moment of inertia about the axis of rotation?

$A$ non-uniform rod $OM$ (of length $l$) is kept along the $x$-axis and is rotating about an axis $AB$,which is perpendicular to the rod as shown in the figure. The rod has a linear mass density that varies with the distance $r$ from the left end $O$ of the rod according to $\lambda = \lambda_0 \left( \frac{r^3}{l^3} \right)$,where $\lambda_0$ is a constant. If the axis $AB$ is at a distance $x$ from the end $O$,what is the value of $x$ such that the moment of inertia of the rod about the axis $AB$ $(I_{AB})$ is minimum?

What is the moment of inertia of a solid sphere of radius $R$ and density $\rho$ about its diameter?

Match List-$I$ with List-$II$:

List-$I$	List-$II$
$(A)$ Moment of inertia of solid sphere of radius $R$ about any tangent	$(I)$ $\frac{5}{3} MR^2$
$(B)$ Moment of inertia of hollow sphere of radius $R$ about any tangent	$(II)$ $\frac{7}{5} MR^2$
$(C)$ Moment of inertia of circular ring of radius $R$ about its diameter	$(III)$ $\frac{1}{4} MR^2$
$(D)$ Moment of inertia of circular disc of radius $R$ about any diameter	$(IV)$ $\frac{1}{2} MR^2$

Choose the correct answer from the options given below:

$A$ wheel of mass $10 \ kg$ has a moment of inertia of $160 \ kg \ m^2$ about its axis. Its radius of gyration is ...... $m$.

Two circular loops $P$ and $Q$ are made from a uniform wire. The radii of $P$ and $Q$ are $R_1$ and $R_2$ respectively. The moments of inertia about their own axis are $I_{P}$ and $I_{Q}$ respectively. If $\frac{I_{P}}{I_{Q}}=\frac{1}{8}$,then $\frac{R_2}{R_1}$ is