Rolling Motion

A uniform disc of mass $0.5 \mathrm{kg}$ and radius $r$ is projected with velocity $18 \mathrm{m} {\mathrm{s}}^{-1}$ at $t=0 \mathrm{s}$ on a rough horizontal surface. It starts off with a purely sliding motion at$t=0 \mathrm{s}$. After $2 \mathrm{s}$ it acquires a purely rolling motion (see figure). The total kinetic energy of the disc after $2 \mathrm{s}$ will be-------- J (given, coefficient of friction is$0.3$ and $g=10 \mathrm{m} {\mathrm{s}}^{-2}$).

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

A hollow spherical ball of uniform density rolls up a curved surface with an initial velocity $3 \mathrm{m} {\mathrm{s}}^{-1}$ (as shown in figure). Maximum height with respect to the initial position covered by it will be _____ $\mathrm{cm}$
(take, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A solid sphere of mass $500 \mathrm{g}$ radius $5 \mathrm{cm}$ is rotated about one of its diameter with angular speed of $10 \mathrm{rad} {\mathrm{s}}^{-1}$. If the moment of inertia of the sphere about its tangent is $x\times {10}^{-2}$ times its angular momentum about the diameter. Then the value of $x$ will be

A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is $\pi :22$ then, the value of its angular speed will be$\mathrm{\_\_\_\_\_\_\_\_\_rad} {\mathrm{s}}^{-1}.$

For rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is $\frac{x}{5}$. The value of $x$ is _____.

Two identical solid spheres each of mass $2 \mathrm{kg}$ and radii $10 \mathrm{cm}$are fixed at the ends of a light rod. The separation between the centres of the spheres is $40 \mathrm{cm}.$ The moment of inertia of the system about an axis perpendicular to the rod passing through its middle point is$\mathrm{\_\_\_\_\_\_}\times {10}^{–3} \mathrm{kg} {\mathrm{m}}^2.$

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 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

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

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

 A uniform rod$(M,l)$ hinged at its one end is released from horizontal position in vertical plane. Acceleration of centre of mass of rod just after releasing the rod will be $\frac{3g}{n}$. Value of $n$ is