The moment of inertia of a thin semicircular disc (mass $= M$ and radius $= R$) about an axis passing through point $O$ and perpendicular to the plane of the disc is given by:

The moment of inertia of a meter scale of mass $0.6 \ kg$ about an axis perpendicular to the scale and located at the $20 \ cm$ position on the scale in $kg-m^2$ is: (Breadth of the scale is negligible)

$A$ uniform solid cylinder with radius $R$ and length $L$ has a moment of inertia $I_1$ about its central axis. $A$ concentric solid cylinder of radius $R' = \frac{R}{2}$ and length $L' = \frac{L}{2}$ is carved out of the original cylinder. If $I_2$ is the moment of inertia of the carved-out portion of the cylinder about the same axis,then $\frac{I_1}{I_2} = ..........$

Consider a sphere of mass $M$ and radius $R$ centered at the origin. The density of the material of the sphere is $\rho = A r^\alpha$,where $r$ is the radial distance,and $\alpha$ and $A$ are constants. If the moment of inertia of the sphere about the axis passing through the centre is $\frac{6}{7} M R^2$,then the value of $\alpha$ is

Four masses are fixed on a massless rod as shown in the figure. The moment of inertia about the axis $P$ is about ....... $kg \cdot m^2$.

The ratio of the moments of inertia of two uniform rings of radii $R$ and $nR$ about an axis passing through their centers and perpendicular to their planes is $1 : 8$. What is the value of $n$?