Moment of Inertia

A circular disc is to be made using iron and aluminium. To keep its moment of inertia maximum about a geometrical axis, it should be so prepared that

A thin rod of length $L$ and mass $M$ is bent at the middle point $O$ as shown in the figure. Consider an axis passing through two middle the point $O$ and perpendicular to the plane of the bent rod. Then the moment of inertia about this axis is

Two rings have their moments of inertia in the ratio $2:1$ and their diameters are in the ratio $2:1$. The ratio of their masses will be 

The mass of a solid circular disc is $500 \mathrm{g}$, the circumference is $3.14 \mathrm{m}$ and thickness is $5 \mathrm{mm}$. The moment of Inertia of the disc about an axis tangential to the rim and perpendicular to the plane of rotation is

The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is,

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

A thin uniform rectangular plate of mass $2 \mathrm{kg}$ is placed in $X-Y$ plane as shown in the figure. The moment of inertia about $x$-axis is $I_x=0.2 \mathrm{kg} {\mathrm{m}}^2$ and the moment of inertia about $y$ - axis is $I_y=0.3 \mathrm{kg} {\mathrm{m}}^2$. The radius of gyration of the plate about the axis passing through $O$ and perpendicular to the plane of the plate is

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?

A square plate of side $a=20 \mathrm{cm}$ and mass $M=2 \mathrm{kg}$ is lying on the horizontal plane. What will be the moment of inertia of the plate about an axis in the plane of the plate and at an angle of $\theta =18.6°$ from one of its sides?

A disc of mass $M$ and radius $R$ is lying on the $x\text{-}y$ plane. The locus of all points on the $x\text{-}y$ plane about which the moment of inertia of the rod is same as that about $0$ will be,

The moment of Inertia of a semicircular ring about the centre is $M{R}^2$. Find the $MI$ about a radially perpendicular point $\left(P\right)$ and the axis perpendicular to the plane of the ring as shown in the diagram.

A rigid body can be hinged about any point on the axis. When it is hinged in a such a way that the hinged position is at $x$, the moment of inertia is given by, $I=3x^2-24x+17$. Then, the $x$-coordinate of centre of mass is,

A flywheel of mass $10 \mathrm{kg}$ and radius $10 \mathrm{cm}$ is revolving at a speed of $240 \mathrm{rpm}$. Its kinetic energy is

The $\mathrm{M}.\mathrm{I}.$ of a uniform disc about the diameter is $I$. Its $\mathrm{M}.\mathrm{I}.$ about an axis perpendicular to its plane and passing through a point on its $\mathrm{rim}$ is

The moment of inertia of uniform circular disc about an axis passing through its centre is $6 \mathrm{kg} {\mathrm{m}}^2$. Its $\mathrm{M}.\mathrm{I}.$ about an axis perpendicular to its plane and just touching the $\mathrm{rim}$ will be

There are four-point masses $m$ each on the corners of a square of side length$l$ about one of its diagonals, the moment of inertia of the system is

A solid sphere of radius $R$ has a moment of inertia $I$ about its geometrical axis. If it is melted into a disc of radius $r$ and thickness $t$. If it's moment of inertia about the tangential axis (which is perpendicular to the plane of the disc), is also equal to $I$, then the value of $r$ is equal to

Moment of inertia of disc about the tangent parallel to plane is $I$. The moment of inertia of disc about tangent and perpendicular to its plane is

M.I. of a thin uniform rod about the axis passing through its centre and ${\perp }^{\mathrm{er}}$ to its length is $\frac{M{L}^2}{12}$. The rod is cut transversely into two halves, which are then riveted end to end. M.I. of the composite rod about the axis passing through its centre and ${\perp }^{\mathrm{er}}$ to its length will be?
 

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 $\frac{L}{3}$ from one of its ends and perpendicular to the rod is