Two rods each of mass $m$ and length $\ell$ are joined at their centers to form a cross. What is the moment of inertia of the cross about an axis passing through the common center and perpendicular to the plane formed by them?

The moment of inertia of a sphere of mass $M$ and radius $R$ about an axis passing through its centre is $\frac{2}{5} M R^2$. The radius of gyration of the sphere about a parallel axis to the above and tangent to the sphere is

$A$ 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?

$A$ flywheel is constructed such that its entire mass is concentrated at the rim because:

The moment of inertia of a thin circular plate of mass $1\,kg$ and diameter $0.2\,m$ about one of its diameters is:

$A$ thin uniform rectangular plate of mass $2 \, kg$ is placed in the $xy$-plane as shown in the figure. The moment of inertia about the $x$-axis is $I_{x}=0.2 \, kg \cdot m^{2}$ and the moment of inertia about the $y$-axis is $I_{y}=0.3 \, kg \cdot m^{2}$. The radius of gyration of the plate about the axis passing through $O$ and perpendicular to the plane of the plate is (in $ \, cm$)