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 $= \lambda$).

Three point masses $m_1, m_2, m_3$ are placed at the vertices of an equilateral triangle of side $a$. The moment of inertia of the system about an axis passing through the vertex $m_1$ and perpendicular to the plane of the triangle is not what is asked; rather,the axis passes through $m_1$ and bisects the opposite side $m_2m_3$. What is the moment of inertia of the system about this axis?

$A$ 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 $\frac{t}{4}$. The ratio of the moment of inertia $\frac{I_Y}{I_X}$ is:

Moment of inertia of a solid sphere about its diameter is $I$. It is then casted into $27$ small spheres of same diameter. The moment of inertia of each new sphere is

Let $M$ and $L$ be the mass and length of a thin uniform rod, respectively. In the first case, the axis of rotation passes through the centre and is perpendicular to the length of the rod. In the second case, the axis of rotation passes through one end and is perpendicular to the length of the rod. The ratio of the radius of gyration in the first case to the second case is:

Two rings have their moments of inertia in the ratio $4 : 1$ and their diameters are in the ratio $4 : 1$. The ratio of their masses is