$A$ thin rod has a length of $1 \ m$ and a mass of $0.6 \ kg$. The moment of inertia of the rod about an axis perpendicular to its length and passing through a point $20 \ cm$ from one end is ................... $kg \cdot m^2$ (the width of the rod is negligible).

$A$ square plate of mass $M$ and edge $L$ is shown in the figure. The moment of inertia of the plate about the axis in the plane of the plate passing through one of its vertices making an angle $15^{\circ}$ with the horizontal is:

$A$ system of five solid spheres, each of mass $m$ and radius $r$, is rotating about an axis $AA'$. The moment of inertia of the system about the axis of rotation $AA'$ is: (in $mr^2$)

$A$ thin square plate $ABCD$ has a uniform thickness. What is the moment of inertia about an axis passing through its center and perpendicular to its plane?

The moment of inertia of a rod of mass $40 \text{ kg}$ and length $3 \text{ m}$ about an axis $AB$ passing through its end and perpendicular to its length is equal to the moment of inertia of a solid sphere of mass $10 \text{ kg}$ and radius $R$ about an axis parallel to the $AB$ axis, with a separation of $3 \text{ m}$ between the axes, as shown in the figure. The value of $R$ is given as $\sqrt{\frac{\alpha}{2}}$. The value of $\alpha$ is . . . . . . .

From a disc of radius $R$,a concentric circular portion of radius $r$ is cut out so as to leave an annular disc of mass $M$. The moment of inertia of this annular disc about the axis perpendicular to its plane and passing through its centre of gravity is