$A$ one-quarter sector is cut from a uniform circular disc of radius $R$. This sector has mass $M$. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

Four point masses (each of mass $m$) are arranged in the $X-Y$ plane at coordinates $(a, 0)$,$(0, a)$,$(-a, 0)$,and $(0, 2a)$. What is the moment of inertia of this system about the $Y$-axis?

$A$ cylinder of mass $500 \ g$ and radius $10 \ cm$ has a moment of inertia about an axis passing through its centre and parallel to its length equal to:

$A$ disc of radius $R$ and thickness $\frac{R}{6}$ has moment of inertia $I$ about an axis passing through its centre and perpendicular to its plane. The disc is melted and recast into a solid sphere. The moment of inertia of the sphere about its diameter is

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

The ratio of the radii of two solid spheres of same mass is $2:3$. The ratio of the moments of inertia of the spheres about their diameters is