$A$ thin wire of length $l$ and mass $m$ is bent in the form of a semicircle. The moment of inertia about an axis perpendicular to its plane and passing through the end of the wire is .............

The figure shows the variation of the moment of inertia of a uniform rod about an axis passing through its centre and inclined at an angle $\theta$ to the length of the rod. The moment of inertia of the rod about an axis passing through one of its ends and making an angle $\theta = \frac{\pi}{3}$ with the rod will be ........ $kg-m^2$.

Two iron solid discs of negligible thickness have radii $R_1$ and $R_2$ and moments of inertia $I_1$ and $I_2$,respectively. For $R_2 = 2 R_1$,the ratio of $I_1$ and $I_2$ would be $1 / x$,where $x = $ . . . . . . .

If $I_1, I_2$ and $I_3$ are moments of inertia of a solid sphere,a hollow sphere,and a ring of the same mass $M$ and radius $R$ about their respective geometrical axes,which of the following statements holds true?

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 = $r$):

$A$ rigid body can be hinged about any point on the $x$-axis. When it is hinged such that the hinge is at $x$,the moment of inertia is given by $I = 2x^2 - 12x + 27$. The $x$-coordinate of the centre of mass is: