$A$ solid sphere of radius $r = 5 \text{ cm}$ is rotating about an axis at a distance $d = 10 \text{ cm}$ from its center, as shown in the figure. If the radius of gyration about that axis is $\sqrt{x} \text{ cm}$, then the value of $x$ is:

If $I_1$ is the moment of inertia of a thin rod of mass $M$ and length $l$ about an axis perpendicular to its length and passing through its centre of mass,and $I_2$ is the moment of inertia of the ring formed by bending the rod about an axis passing through its centre and perpendicular to its plane,then:

The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through its center is $k$. Given that the length of the rod is $10 \sqrt{3} \ m$,find the value of $k$ in meters.

We have a rectangular plate of uniform thickness. $E, F, G, H$ are the midpoints of $AB, BC, CD,$ and $AD$ respectively. About which axis will the moment of inertia be minimum?

Two rings of the same radius and mass are placed such that their centers are at the same point and their planes are mutually perpendicular. The moment of inertia of this system about an axis passing through the center and perpendicular to the plane of one of the rings is (mass of ring = $m$,radius = $r$):

$A$ circular disc is to be made by using iron and aluminium,so that it acquires maximum moment of inertia about its geometrical axis. This is possible with: