The moment of inertia of a thin square plate $PQRS$ of uniform thickness,about an axis passing through the centre '$O$' and perpendicular to the plane of the plate is (where $I_1, I_2, I_3, I_4$ are respectively the moments of inertia about axes $1, 2, 3, 4$ which are in the plane of the plate as shown in the figure):

The radius of gyration $K$ of a hollow sphere of mass $M$ and radius $R$ about an axis $XY$ is equal to $R$. The distance of that axis from the center of the sphere is $h$. The value of $h$ is

The figure shows an isosceles triangular plate of mass $M$ and base $L$. The angle at the apex is $90^o$. The apex lies at the origin and the base is parallel to the $X$-axis. The moment of inertia of the plate about its base parallel to the $x$-axis is

Let $I$ be the moment of inertia of a uniform square plate about an axis $AB$ that passes through its centre and is parallel to two of its sides. $CD$ is a line in the plane of the plate that passes through the centre of the plate and makes an angle $\theta$ with $AB$. The moment of inertia of the plate about the axis $CD$ is then equal to:

Each spherical shell has a radius $R$ and mass $M$. They are connected by a light massless rod. Calculate the moment of inertia about the axis $xx'$.

$(a)$ Find the moment of inertia of a sphere about a tangent to the sphere,given the moment of inertia of the sphere about any of its diameters to be $2 M R^{2} / 5,$ where $M$ is the mass of the sphere and $R$ is the radius of the sphere.
$(b)$ Given the moment of inertia of a disc of mass $M$ and radius $R$ about any of its diameters to be $M R^{2} / 4,$ find its moment of inertia about an axis normal to the disc and passing through a point on its edge.