Consider two uniform discs of the same thickness and different radii $R_{1} = R$ and $R_{2} = \alpha R$ made of the same material. If the ratio of their moments of inertia $I_{1}$ and $I_{2}$,respectively,about their axes is $I_{1} : I_{2} = 1 : 16$,then the value of $\alpha$ is:

The ratio of the radius of gyration of a hollow sphere to that of a solid cylinder of equal mass,for the moment of inertia about their diameter axis $AB$ as shown in the figure,is $\sqrt{\frac{8}{x}}$. The value of $x$ 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$)

Four identical point masses (each of mass $m$) are placed on the circumference of a disc. The disc has mass $M$ and radius $R$. What will be the moment of inertia about an axis passing through the center $O$ and perpendicular to the plane of the disc?

If $I_1, I_2$ and $I_3$ are moments of inertia of a solid sphere,a hollow sphere,and a ring of the same mass $M$ and radius $R$ about their respective geometrical axes,which of the following statements holds true?

If a spherical planet of mass $M$ and radius $R$ suddenly shrinks to half its size,and its mass reduces to half,what is the new moment of inertia of the planet about its diameter?