$M$ and $R$ are the mass and radius of a disc. $A$ small disc of radius $R/3$ is removed from the bigger disc as shown in the figure. The moment of inertia of the remaining part of the bigger disc about an axis $\text{AB}$ passing through the centre $O$ and perpendicular to the plane of the disc is $\frac{4}{x} MR^2$. The value of $x$ is . . . . . . .

The moment of inertia of a uniform square plate about an axis perpendicular to its plane and passing through the centre is $\frac{Ma^2}{6}$,where $M$ is the mass and $a$ is the side of the square plate. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

Two hollow spheres each of mass $M$ and radius $\frac{R}{2}$ are connected with a massless rod of length $2R$ as shown in the figure. What will be the moment of inertia of the system about an axis $yy'$ passing through the center of one of the spheres and perpendicular to the rod?

With reference to the figure of a cube of edge $a$ and mass $m$, state which of the following options are correct ($O$ is the centre of the cube):
$(a)$ The moment of inertia of the cube about the $z$-axis is $I_z = I_x + I_y$
$(b)$ The moment of inertia of the cube about the $A$-axis is $I_A = I_z + \frac{ma^2}{2}$
$(c)$ The moment of inertia of the cube about the $B$-axis is $I_B = I_z + \frac{ma^2}{2}$
$(d)$ $I_x = I_z$

If the moment of inertia of a uniform solid cylinder about the axis of the cylinder is $\frac{1}{n}$ times its moment of inertia about an axis passing through its midpoint and perpendicular to its length, then the ratio of the length and radius of the cylinder is

The $X$ and $Z$ axes are mutually perpendicular in the plane of a disc,and the $Y$ axis is perpendicular to the plane of the disc. If the moments of inertia of the object about the $X$ and $Y$ axes are $30 \ kg \ m^2$ and $40 \ kg \ m^2$ respectively,then the moment of inertia about the $Z$ axis will be ....... $kg \ m^2$.