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 .......

The moment of inertia of a solid sphere about its diameter is $I$. It is then recast into $27$ small spheres of the same diameter. The moment of inertia of each small sphere about its diameter is:

Two identical thin rods of mass $M \ kg$ and length $L \ m$ are connected as shown in the figure. The moment of inertia of the combined rod system about an axis passing through point $P$ and perpendicular to the plane of the rods is $\frac{x}{12} M L^2 \ kg \ m^2$. The value of $x$ is . . . . . . .

Two masses $m_1$ and $m_2$ are placed at a distance $r$ from each other. Find the moment of inertia of the system about an axis passing through the center of mass and perpendicular to the line joining the masses.

$A$ circular disc $X$ of radius $R$ is made from an iron plate of thickness $t$,and another disc $Y$ of radius $4R$ is made from an iron plate of thickness $\frac{t}{4}$. Then the relation between the moment of inertia $I_x$ and $I_y$ is

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 $4$ times that of loop $P$ about its axis. The value of $n$ is: