Calculate the moment of inertia of the system of particles shown in the figure about the axis of rotation $XX'$.

$A$ slender rod of mass $M$ and length $L$ is hinged at one end to swing freely in a vertical plane. Its density is non-uniform and varies linearly from the hinged end to the free end,doubling its value. The moment of inertia of the rod about the rotation axis passing through the hinge point is:

Two loops $P$ and $Q$ of radii $R_1$ and $R_2$ are made from uniform metal wire of the same material. If $I_P$ and $I_Q$ are the moments of inertia of loops $P$ and $Q$ respectively,then the ratio $R_1 / R_2$ is (Given $I_P / I_Q = 27$) (in $:1$)

$A$ solid sphere and a thin-walled hollow sphere have the same mass and the same material. Which of them has a greater moment of inertia about its diameter? ($I_{h} =$ moment of inertia of a hollow sphere about an axis coinciding with its diameter,$I_{s} =$ moment of inertia of a solid sphere about an axis coinciding with its diameter)

$A$ long slender rod is welded to a thin circular disc of diameter $0.5 \ m$ at a point on its circumference. The rod is in the same plane as that of the disc and forms a tangent to the disc. The radius of gyration of the disc about the rod (in $m$) is

What is the moment of inertia of a thin circular disc about its diameter? ($M$ is the mass and $R$ is the radius of the disc.)