The moment of inertia of a uniform thin rod of mass $m$ and length $\ell$ about a perpendicular axis passing through one end is $I_{1}$. The same rod is bent into a ring and its moment of inertia about a diameter is $I_{2}$. If $\frac{I_{1}}{I_{2}} = \frac{x \pi^{2}}{3}$,then the value of $x$ will be ...............

The figure shows a uniform rod of length $L$ and mass $M$ lying along the $x$-axis with one end at the origin $O$. The locus of all points $(x, y)$ in the $xy$-plane,about which the moment of inertia of the rod is the same as that about $O$,is:

$A$ wheel of mass $10 \ kg$ has a moment of inertia of $160 \ kg \ m^2$ about its axis. Its radius of gyration is ...... $m$.

Let $M$ and $L$ be the mass and length of a thin uniform rod,respectively. In the $1^{\text{st}}$ case,the axis of rotation passes through the center and is perpendicular to its length. In the $2^{\text{nd}}$ case,the axis of rotation passes through one end and is perpendicular to its length. The ratio of the radius of gyration in the first case to the second case 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 $nr$).

If a wheel rotating with a constant angular speed experiences a break during its motion,what happens to its radius of gyration?