Three point masses $m_1, m_2, m_3$ are located at the vertices of an equilateral triangle of side length $a$. The moment of inertia of the system about an axis along the altitude of the triangle passing through $m_1$ is:

Three particles,each of mass $m \; g$,are situated at the vertices of an equilateral triangle $ABC$ of side $l \; cm$ (as shown in the figure). The moment of inertia of the system about a line $AX$ perpendicular to $AB$ and in the plane of $ABC$ in $g \cdot cm^2$ units will be:

$A$ thin rod of length $L$ and mass $M$ is bent at its midpoint $O$ at an angle of $90^{\circ}$. The moment of inertia of the bent rod about an axis passing through the point $O$ and perpendicular to the plane of the bent rod is:

The moment of inertia of a uniform semicircular disc of mass $M$ and radius $r$ about an axis perpendicular to the plane of the disc passing through its center is:

Five particles of mass $2 \ kg$ each are attached to the rim of a circular disc of radius $0.1 \ m$ and negligible mass. The moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane is ........ $kg \ m^2$.

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