Moment of inertia of a circular wire of mass $M$ and radius $R$ about its diameter is

The moment of inertia of a cylinder of mass $M$,length $L$,and radius $R$ about an axis passing through its centre and perpendicular to the axis of the cylinder is $I = M \left(\frac{R^2}{4} + \frac{L^2}{12}\right)$. If such a cylinder is to be made for a given mass of material,the ratio $L/R$ for it to have the minimum possible $I$ is

The ratio of masses and radii of two circular rings are $1 : 2$ and $2 : 1$ respectively. What is the ratio of their moments of inertia about their central axes?

The figure shows an isosceles triangular plate of mass $M$ and base $L$. The angle at the apex is $90^o$. The apex lies at the origin and the base is parallel to the $X$-axis. The moment of inertia of the plate about the $z$-axis is

$A$ thin uniform wire of mass $m$ and linear mass density $\rho$ is bent in the form of a circular loop. The moment of inertia of the loop about its diameter is

Four rods are arranged in the form of a square. Calculate the moment of inertia about an axis passing through the center and perpendicular to the plane. (Mass $M$ and length $L$ for each rod)