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

Three spheres,each of mass $m$ and radius $r$,are placed as shown in the figure. Consider an axis $YY^1$,which is tangent to two spheres (labeled $2$ and $3$) and passes through the diameter of the third sphere (labeled $1$). The moment of inertia of the system consisting of these three spheres about the $YY^1$ axis is:

$ABC$ is a plane lamina in the shape of an equilateral triangle. $D$ and $E$ are the midpoints of $AB$ and $AC$,respectively,and $G$ is the centroid of the lamina. The moment of inertia of the lamina about an axis passing through $G$ and perpendicular to the plane $ABC$ is $I_{0}$. If the part $ADE$ is removed,the moment of inertia of the remaining part about the same axis is $\frac{NI_{0}}{16}$,where $N$ is an integer. The value of $N$ 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 ....... $kg \ m^2$.

Two spheres each of mass $M$ and radius $R$ are connected with a massless rod of length $4R$. The moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular to the rod will be

The moment of inertia of a ring of mass $m = 3 \ gm$ and radius $r = 1 \ cm$ about an axis passing through its edge and parallel to its natural axis is: