Moment of Inertia

Three co-planar uniform identical discs, each of mass$M$and radius $R$, are joined as shown in figure. The moment of inertia about indicated axis is

The ratio of radius of gyration of a solid sphere of mass $M$ and radius $R$ about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is :

A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively $\left(k_{sph}:k_{cyl}\right)$ is $2:\sqrt{x}$. The value of $x$ is _______.

The moment of inertia of a semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is$\frac{1}{\mathrm{x}} {\mathrm{MR}}^2$, where $\mathrm{R}$ is the radius and $\mathrm{M}$ is the mass of the semicircular ring. The value of $\mathrm{x}$ will be $\_\_\_\_\_\_\_$.

A ring and a solid sphere rotating about an axis passing through their centres have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is $\sqrt{\frac{2}{x}}$. The value of $x$ is _____.

A pulley of radius $1.5 \mathrm{m}$ is rotated about its axis by a force $F=12t-3t^2 \mathrm{N}$ applied tangentially (while $t$ is measured in seconds). If moment of inertia of the pulley about its axis of rotation is $4.5 \mathrm{kg} {\mathrm{m}}^2$, the number of rotations made by the pulley before its direction of motion is reversed, will be$\frac{K}{\pi }$. The value of $K$ is

A small hole is made in a disc of mass$M$ and radius $R$ at a distance$\frac{R}{4}$ from centre. The disc is supported on a horizontal peg through this hole. The moment of inertia of the disc about horizontal peg is

The moment of inertia of semi-circular ring of mass $M$ and radius $R$ about an axis passing through centre and perpendicular to the plane of ring is

A solid sphere and a ring have equal masses and equal radius of gyration. If sphere is rotating about its diameter and ring about an axis passing through the centre and perpendicular to its plane, then the ratio of radius is $\sqrt{\frac{x}{2}}$, then find the value of $x$.

Two circular loops $A\&B$ of radii $R$ and $2R$  respectively are made of the similar wire. Their moments of inertia about the axis passing through the centre of perpendicular to their plane

A thin uniform rod is rotating about an axis passing through its centre and perpendicular to its length, moment of inertia is $I_0$. Moment of inertia of rod about an axis through one end and perpendicular to the rod will be

A plate is oscillating with time period $T$. Suddenly, another plate is put on the first plate, then time period

The M.I. of a disc about its diameter is $2$ units. Its M.I. about axis through a point on its rim and in the plane of the disc is

A Ring of mass $6 \mathrm{kg}$ and radius $40 \mathrm{cm}$ is revolving at the rate of $300$ revolutions per minute. Its moment of inertia will be

The moment of inertia of a square lamina about an axis passing through its centre of mass perpendicular to plane is $30 \mathrm{kg} {\mathrm{m}}^2$. Then, its moment of inertia about an axis touching its side and in the plane of the lamina will be :-

Four solid rigid balls each of mass $m$ and radius $r$ are fixed on a rigid-ring of radius $2r$and mass $2m$. The system is whirled about $O$ as shown. The radius of gyration of the system is :

The moment of inertia of a uniform circular disc about its diameter is $200 \mathrm{gm} {\mathrm{cm}}^2$. Then its moment of inertia about an axis passing through its center and perpendicular to its circular face is
 

In the figure shown below, a meter stick of mass$M$ is supported in a horizontal position by two light strings attached at points$A$and $B$ of the rod. The initial angular acceleration of the stick if one of the strings is cut, is [Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$] 

If the moment of inertia of a disc about an axis which is along its diameter is$I$ then the moment of inertia about the axis passing through its centre and perpendicular to plane is

Find the MOI of a system of two particles having the same mass m separated by a distance $d$ about the Centre$O$.