Moment of inertia of a disc of mass $M$ and radius $R$ about any of its diameter is $MR^2/4$. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be $(x/2)MR^2$. The value of $x$ is

The string of a simple pendulum is replaced by a uniform rod of length $L$ and mass $M$. If the mass of the bob of the pendulum is $m$,then for small oscillations its time period would be (assume radius of bob $r << L$):

The moment of inertia of a uniform thin rod of length $L$ and mass $M$ about an axis passing through a point at a distance of $\frac{L}{4}$ from one of its ends and perpendicular to the rod is:

$A$ 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

Let $I_1, I_2$ and $I_3$ be the moment of inertia of a uniform square plate of mass $M$ and side length $L$ about axes $AOC, xDx'$ and $yBy'$ respectively as shown in the figure. The ratio of the moments of inertia $I_1 : I_2 : I_3$ is:

$A$ solid cylinder of length $L = 80 \, \text{cm}$ and mass $M$ has a radius $r = 20 \, \text{cm}$. Calculate the density of the material used if the moment of inertia of the cylinder about an axis $CD$ parallel to the central axis $AB$ (as shown in the figure) is $2.7 \, \text{kg m}^2$.