Moment of Inertia

From a triangular lamina of side a and mass $m$ , a small triangular Lamina made by joining mid-points of sides of triangular lamina is cut. If Moment of Inertia of complete Lamina about COM is $I_o$ and Moment of Inertia of remaining Lamina about centroid is $\frac{x{I}_0}{16}$, then find the value of $x$.

Four point masses $($each of mass $m$$)$ are arranged in the $X-Y$ plane. The moment of inertia of this array of masses about $Y$-axis is 

The moment of inertia of hollow cylinder of mass $M$ and inner radius $R_1$ and outer radius $R_2$ about its central axis,

M.O.I is the ratio of 

A solid sphere, a thin ring and a cylinder of same mass and radius, held at rest are released simultaneously to roll down on an inclined plane. Which one will reach the bottom first?

Point masses $1, 2, 3$ and $4 \mathrm{kg}$ are lying at the points $(0, 0, 0), (2, 0, 0), (0, 3, 0)$ and $(-2, -2, 0)$, respectively. The moment of inertia of this system about the $x$-axis will be,

Four similar point masses (each of mass $m$) are placed on the circumference of a disk of mass $M$ and radius $R$. The moment of inertia of the system about the normal axis through the center $O$ will be,

On the marks of $20 \mathrm{cm}$ and $70 \mathrm{cm}$ of a light meter scale, two weights, $1 \mathrm{kg}$ and $4 \mathrm{kg}$, respectively, are placed. The moment of inertia (in $\mathrm{kg} {\mathrm{m}}^2$) about the vertical axis through the $100 \mathrm{cm}$ mark will be,

M.O.I is the ratio of 

M.O.I. of a disc is minimum about an axis 

A circular disc is to be made by using iron and aluminium, so that it acquires maximum moment of inertia about its geometrical axis. It is possible with

The Moment of inertia of a uniform angular with mass $m$, thickness $t$ and inner and outer radius as $R_1$ and $R_2$ respectively can be expressed as

The moment of inertia of a solid cylinder of diameter $D$ and length $L$ about an axis perpendicular to its length and passing through the centre of gravity will be,

A solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown below.

The cube is free to rotate about the fixed axis $AB$. A bullet of mass $m\left(<<M\right)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height $h$ above the surface to impart the cube an angular speed ${\omega }_c$, so that the cube just topples over. Then, ${\omega }_c$ is (Note the moment of inertia of the cube about an axis perpendicular to the face and passing through the centre of mass is $\frac{2M{a}^3}{3}$)

A flywheel is used in an automobile engine due to its:

The moment of inertia of a uniform solid disc of mass $M$ and radius $R$ about an axis normal to the disk and passing through its center is $\frac{M{R}^2}{2}.$ What is the moment of inertia of the same disc about an axis lying in its plane and tangent to it (as shown in the figure) ?

The moment of inertia of a thin uniform rod of length $L$ and mass $M$ about an axis passing through a point at a distance of $\frac{L}{3}$ from one of its ends and perpendicular to the rod is

The moment of inertia of a uniform cylinder of length $l$ and radius $R$ about its perpendicular bisector is $I.$ What is the ratio $\frac{l}{R}$ such that the moment of inertia is minimum?

A hoop of mass M and radius R is hung from a support fixed in a wall. Its moment of inertia about the support is

The ratio of the dimensions of Planck’s constant of that of moment of inertia is the dimension of