Three thin uniform rods each of mass $M$ and length $L$ are placed along the three axes of a Cartesian coordinate system with one end of all the rods at the origin. The moment of inertia of the system of the rods about the $z$-axis is:

Consider a sphere of mass $M$ and radius $R$ centered at the origin. The density of the material of the sphere is $\rho = A r^\alpha$,where $r$ is the radial distance,and $\alpha$ and $A$ are constants. If the moment of inertia of the sphere about the axis passing through the centre is $\frac{6}{7} M R^2$,then the value of $\alpha$ is

$A$ thin wire of length $L$ and uniform linear mass density $\lambda$ is bent into a circular ring. The moment of inertia of the ring about a tangential axis in its plane is:

The radius of gyration of a circular disc of radius $R$ and mass $m$ rotating about its diameter as an axis is:

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

Write the practical uses of moment of inertia.