$A$ thin wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular loop with centre at $O$ as shown. The moment of inertia of the loop about the axis $XX'$ is

According to the parallel axis theorem,$I = I_C + Mx^2$. Which of the following graphs of $I$ versus $x$ is correct?

The moment of inertia of a solid cylinder about its own axis is the same as its moment of inertia about an axis passing through its centre of gravity and perpendicular to its length. The relation between its length $L$ and radius $R$ is

Identical rings are arranged in a hexagonal plane pattern such that each ring touches its neighbouring rings. Each ring has mass $M$ and radius $R$. The moment of inertia of the system of seven rings about an axis passing through the centre of the central ring and normal to the plane of all rings is: (in $MR^2$)

The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through which of the following points?

Suppose there is a uniform circular disc of mass $M$ and radius $r$ shown in the figure. Two shaded circular regions, each of radius $r/4$, are cut out from the disc. The centers of these cut-out discs are at a distance of $3r/4$ from the center of the original disc. The moment of inertia of the remaining part about the axis $A$ (passing through the center of the disc and perpendicular to its plane) is given by $\frac{x}{256} Mr^2$. The value of $x$ is . . . . . . .