Rigid Body Rotation

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 $g=10 \mathrm{m} {\mathrm{s}}^{-2}$).

A circular plate is rotating in horizontal plane, about an axis passing through its centre and perpendicular to the plate, with an angular velocity $\omega$. A person sits at the centre having two dumbbells in his hands. When he stretched out his hands, the moment of inertia of the system becomes triple. If $E$ be the initial Kinetic energy of the system, then final Kinetic energy will be $\frac{E}{x}$. The value of $x$ 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 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 _____.

A body is rotating with kinetic energy $E$. If angular velocity of body is increased to three times of initial angular velocity, the kinetic energy becomes $nE$. Find $n$.

Assertion $\left(A\right):$ Fan spins even after switch is off

Reason $\left(R\right):$ Fan in rotation has rotational inertia

The velocity of the centre of mass of a solid sphere of radius $R$rotating with angular velocity about an axis passing through its centre of mass is

A disk of mass $2 \mathrm{kg}$ and diameter $20 \mathrm{cm}$ is rotating about an axis passing through its center perpendicular to its plans with an angular speed that varies with time as $2t^2+3t+2$, find the torque experienced by the disk at the instant $t=2 \mathrm{s}$.

 A solid sphere having mass$m$ and radius$r$rolls down an inclined plane. Then its kinetic energy is 

A solid sphere is rolling down an inclined plane without slipping. If the inclined plane has inclination $\theta$ with the horizontal, then the coefficient of friction between the sphere and the inclined plane should be

A hollow sphere is rolling (without slipping) on horizontal surface with velocity $v$. It then moves up a curved track. If sliding does not occur, then the maximum height moved-up by sphere is

A light thread is wound on a disk of mass $m$ and other end of the thread is connected to a block of mass $m$, which is placed on a rough ground as shown in the diagram. Find the minimum value of the coefficient of friction for which the block remains at rest:

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Two cylinders $P$ and $Q$ are released from the top of an inclined plane such that $P$ rolls down without slipping and $Q$ slips down without rolling which cylinder reaches the base first?

A uniform rod $AB$ of length $2l$ and mass $2m$ has a particle of $m$ attached at point $B$. The rod is free to rotate in a vertical plane about a horizontal axis passing through $A$. When the rod is hanging at rest with $B$ below $A$ it is given angular velocity ${\omega }_0=\frac{7}{2}\sqrt{\frac{g}{l}}$. Find the reaction at the axis, when the rod becomes horizontal first. (Given, $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

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An $L=70 \mathrm{m}$ long thin tape wound on a spool of radius $r_0=10\mathrm{mm}$ makes a tape roll of outer radius $R=25\mathrm{mm}$. A motor used to wind the tape rotates the spool at a constant angular velocity and takes $T=165 \mathrm{s}$ to complete the winding. Calculate length of the tape, which has been wound in $t=110 \mathrm{s}$ from the beginning of the winding.

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A solid cylinder of mass $m$ and radius $R$ rolls down an inclined plane of height $30 \mathrm{m}$ without slipping. The speed of its centre of mass when the cylinder reaches the bottom is [use $g=10 \mathrm{m} {\mathrm{s}}^{-2}$]

A uniform thin circular ring of mass $m\left(m=0.4 \mathrm{kg}\right)$ and radius $R$ has a small particle of the same mass $m$ fixed on it as shown in the figure. The line joining the particle to centre is initially horizontal. The ground is frictionless. Find the contact force (magnitude) exerted by the ground on the ring, when the system is released from rest.

A particle is rotating m circle with speed $10\pi \mathrm{rad} {\mathrm{s}}^{-1}$ with acc $-2\pi \mathrm{rad} {\mathrm{s}}^{-2}$, then calculate:

(i) Time after which it comes to rest.

(ii) After how many rotations will it come to rest.

A disc of radius $R$ is rolling on a rough horizontal surface. Velocity of centre of mass is $v$. The magnitude of velocity of the point $P$ (as shown in the figure) on the periphery of disc is