If the length of a thin uniform rod is $L$ and the radius of gyration of the rod about an axis perpendicular to its length and passing through one end is $K$, then $K: L=$

The linear mass density of a thin rod $AB$ of length $L$ varies from $A$ to $B$ as $\lambda(x) = \lambda_{0}(1 + \frac{x}{L})$,where $x$ is the distance from $A$. If $M$ is the mass of the rod,then its moment of inertia about an axis passing through $A$ and perpendicular to the rod is $......ML^{2}$.

Two circular loops $P$ and $Q$ of radii $r$ and $nr$ are made respectively from a uniform wire. The moment of inertia of loop $Q$ about its axis is four times that of loop $P$ about its axis. The value of $n$ is

If the radius of a solid sphere is $35\,cm,$ calculate the radius of gyration when the axis is along a tangent.

$A$ disc $X$ of radius $R$ and thickness $t$ is made of an iron plate. Another disc $Y$ of radius $4R$ and thickness $t/4$ is also made of the same iron plate. What is the relationship between their moments of inertia $I_x$ and $I_y$?

Two rods of equal length $(l)$ and equal mass $M$ are kept along the $x$ and $y$ axes respectively,such that their center of mass lies at the origin. The moment of inertia about the line $y = x$ is ...........