$A$ uniform solid cylinder with radius $R$ and length $L$ has a moment of inertia $I_1$ about its central axis. $A$ concentric solid cylinder of radius $R' = \frac{R}{2}$ and length $L' = \frac{L}{2}$ is carved out of the original cylinder. If $I_2$ is the moment of inertia of the carved-out portion of the cylinder about the same axis,then $\frac{I_1}{I_2} = ..........$

The moment of inertia of a solid disc rotating about its diameter is $2.5$ times higher than the moment of inertia of a ring rotating in a similar way. The moment of inertia of a solid sphere which has the same radius as the disc and is rotating in a similar way,is $n$ times higher than the moment of inertia of the given ring. Here,$n=$ . . . . . . . Consider all the bodies have equal masses.

Two rings of radius $R$ and $nR$ made of the same material have a ratio of moment of inertia about an axis passing through their centers and perpendicular to their planes as $1:8$. The value of $n$ is (mass per unit length $= \lambda$).

$A$ thin wire of length $\ell$ and mass $m$ is bent in the form of a semicircle as shown. Its moment of inertia about an axis joining its free ends will be ...........

$A$ solid sphere of radius $10 \ cm$ is rotating about an axis which is at a distance $15 \ cm$ from its centre. The radius of gyration about this axis is $\sqrt{n} \ cm$. The value of $n$ is

The moment of inertia of a uniform thin rod of length $L$ and mass $M$ about an axis passing through a point at a distance of $\frac{L}{3}$ from one of its ends and perpendicular to the rod is