$A$ solid sphere $A$ is rotating about an axis $PQ$. If the radius of the sphere is $5 \, cm$ and the distance of its center from the axis $PQ$ is $10 \, cm$,then its radius of gyration about $PQ$ will be $\sqrt{x} \, cm$. The value of $x$ is $................$.

Two discs $A$ and $B$ of same material and thickness have radii $R$ and $3R$ respectively. Their moments of inertia about their axis will be in the ratio

$A$ thin metal rod of mass $M$ and length $L$ is cut into four equal parts by cutting it perpendicular to its length. If the moment of inertia of the rod about an axis passing through its centre and perpendicular to its length is $I$,then the moment of inertia of each part about an axis passing through its own centre and perpendicular to its length is:

The mass per unit area of a circular disc of radius $a$ depends on the distance $r$ from its centre as $\sigma(r) = A + Br$. The moment of inertia of the disc about the axis perpendicular to the plane and passing through its centre is

From a uniform wire,two circular loops are made: $(i)\ P$ of radius $r$ and $(ii)\ Q$ of radius $nr$. If the moment of inertia of $Q$ about an axis passing through its center and perpendicular to its plane is $8$ times that of $P$ about a similar axis,the value of $n$ is (diameter of the wire is very much smaller than $r$ or $nr$).

An iron circular disc $X$ has radius $R$ and thickness $t$. Another iron circular disc $Y$ has radius $4R$ and thickness $t/4$. The relationship between the moments of inertia $I_X$ and $I_Y$ of these two discs is .......