Disc A has moment of inertia $I$ and angular velocity $\omega$. Disc B has moment of inertia $2I$ and is initially at rest. When disc B is brought in contact with disc A, they acquire a common angular velocity in time $t$. The average frictional torque on one disc by the other during this period is
(A) $\frac{2I\omega}{3t}$
(B) $\frac{9I\omega}{2t}$
(C) $\frac{9I\omega}{4t}$
(D) $\frac{3I\omega}{2t}$

The moment of inertia of a uniform semicircular disc of mass $M$ and radius $r$ about a line perpendicular to the plane of the disc through the centre is
(1) $\frac{1}{4}\mathrm{Mr}^2$
(2) $\frac{2}{5}\mathrm{Mr}^2$
(3) $\mathrm{Mr}^2$
(4) $\frac{1}{2}\mathrm{Mr}^2$

A 'T' shaped object with dimensions shown in the figure, is lying on a smooth floor. A force $F$ is applied at the point $P$ parallel to $AB$, such that the object has only the translational motion without rotation. Find the location of $P$ with respect to $C$
(1) $\frac{2}{3}\ell$
(2) $\frac{3}{2}\ell$
(3) $\frac{4}{3}\ell$
(4) $\ell$

A force of $-F\hat{k}$ acts on $O$, the origin of the coordinate system. The torque about the point $(1, -1)$ is
(1) $-F(\hat{i} - \hat{j})$
(2) $F(\hat{i} - \hat{j})$
(3) $-F(\hat{i} + \hat{j})$
(4) $F(\hat{i} + \hat{j})$

Consider a uniform square plate of side ' $a$ ' and mass ' $m$ '. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is
(1) $\frac { 5 } { 6 } m a ^ { 2 }$
(2) $\frac { 1 } { 12 } m a ^ { 2 }$
(3) $\frac { 7 } { 12 } m a ^ { 2 }$
(4) $\frac { 2 } { 3 } m a ^ { 2 }$

A thin uniform rod of length $\ell$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$. Its centre of mass rises to a maximum height of
(1) $\frac { 1 } { 3 } \frac { \ell ^ { 2 } \omega ^ { 2 } } { g }$
(2) $\frac { 1 } { 6 } \frac { \ell \omega } { g }$
(3) $\frac { 1 } { 2 } \frac { \ell ^ { 2 } \omega ^ { 2 } } { g }$
(4) $\frac { 1 } { 6 } \frac { \ell ^ { 2 } \omega ^ { 2 } } { g }$

A pulley of radius 2 m is rotated about its axis by a force $\mathrm{F} = \left(20\mathrm{t} - 5\mathrm{t}^{2}\right)$ Newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $10\,\mathrm{kg\,m}^2$, the number of rotations made by the pulley before its direction of motion is reversed, is:
(1) more than 3 but less than 6
(2) more than 6 but less than 9
(3) more than 9
(4) less than 3

A circular hole of radius $\frac { R } { 4 }$ is made in a thin uniform disc having mass $M$ and radius $R$, as shown in figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point $O$ and perpendicular to the plane of the disc is-
(1) $\frac { 219 M R ^ { 2 } } { 256 }$
(2) $\frac { 237 M R ^ { 2 } } { 512 }$
(3) $\frac { 197 M R ^ { 2 } } { 256 }$
(4) $\frac { 19 M R ^ { 2 } } { 512 }$

A thin uniform bar of length L and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m moving in the same horizontal plane from opposite sides of the bar with speeds 2 v and $v$ respectively. The masses stick to the bar after collision at a distance $\frac { \mathrm { L } } { 3 }$ and $\frac { \mathrm { L } } { 6 }$ respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be: [Figure]
(1) $\frac { \mathrm { v } } { 6 \mathrm {~L} }$
(2) $\frac { 6 \mathrm { v } } { 5 \mathrm {~L} }$
(3) $\frac { 3 \mathrm { v } } { 5 \mathrm {~L} }$
(4) $\frac { \mathrm { v } } { 5 \mathrm {~L} }$

A slab is subjected to two forces $\overrightarrow { \mathrm { F } } _ { 1 }$ and $\overrightarrow { \mathrm { F } } _ { 2 }$ of same magnitude $F$ as shown in the figure. Force $\overrightarrow { \mathrm { F } _ { 2 } }$ is in XY plane while force $\mathrm { F } _ { 1 }$ acts along $z$-axis at the point $( 2 \vec { i } + 3 \vec { j } )$. The moment of these forces about point O will be:
(1) $( 3 \hat { i } - 2 \hat { j } + 3 \hat { k } ) \mathrm { F }$
(2) $( 3 \hat { i } - 2 \hat { j } - 3 \hat { k } ) \mathrm { F }$
(3) $( 3 \hat { i } + 2 \hat { j } - 3 \hat { k } ) \mathrm { F }$
(4) $( 3 \hat { i } + 2 \hat { j } + 3 \hat { k } ) \mathrm { F }$

The radius of gyration of a uniform rod of length $l$, about an axis passing through a point $\frac { l } { 4 }$ away from the centre of the rod, and perpendicular to it, is:
(1) $\frac { 1 } { 4 } l$
(2) $\frac { 1 } { 8 } l$
(3) $\sqrt { \frac { 7 } { 48 } } l$
(4) $\sqrt { \frac { 3 } { 8 } } l$

Mass per unit area of a circular disc of radius a depends on the distance $r$ from its centre as $\sigma ( r ) = A + B r$. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is:
(1) $2 \pi a ^ { 4 } \left( \frac { A } { 4 } + \frac { a B } { 5 } \right)$
(2) $2 \pi a ^ { 4 } \left( \frac { a A } { 4 } + \frac { B } { 5 } \right)$
(3) $\pi a ^ { 4 } \left( \frac { A } { 4 } + \frac { a B } { 5 } \right)$
(4) $2 \pi \mathrm { a } ^ { 4 } \left( \frac { \mathrm {~A} } { 4 } + \frac { \mathrm { B } } { 5 } \right)$

Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side of length $d$. The ratio $\frac { I _ { 0 } } { I _ { A } }$ of moment of inertia $I _ { 0 }$ of the system about an axis passing the centroid and about center of any of the spheres $I _ { A }$ and perpendicular to the plane of the triangle is:
(1) $\frac { 13 } { 23 }$
(2) $\frac { 15 } { 13 }$
(3) $\frac { 23 } { 13 }$
(4) $\frac { 13 } { 15 }$

One end of a straight uniform $1 m$ long bar is pivoted on horizontal table. It is released from rest when it makes an angle $30 ^ { \circ }$ from the horizontal (see figure). Its angular speed when it hits the table is given as $\sqrt { n } \mathrm { rad } s ^ { - 1 }$, where $n$ is an integer. The value of $n$ is $\_\_\_\_$

A force $\vec { F } = ( \hat { i } + 2 \hat { j } + 3 \hat { k } ) \mathrm { N }$ acts at a point $( 4 \hat { i } + 3 \hat { j } - \widehat { k } ) \mathrm { m }$. Then the magnitude of torque about the point $( \hat { i } + 2 \hat { j } + \widehat { k } ) \mathrm { m }$ will be $\sqrt { x } \mathrm {~N} - \mathrm { m }$. The value of $x$ is.

A thin uniform rod of length 2 m, cross sectional area $A$ and density $d$ is rotated about an axis passing through the centre and perpendicular to its length with angular velocity $\omega$. If value of $\omega$ in terms of its rotational kinetic energy $E$ is $\sqrt{\frac{\alpha E}{Ad}}$, then the value of $\alpha$ is $\_\_\_\_$.

A light rope is wound around a hollow cylinder of mass 5 kg and radius 70 cm. The rope is pulled with a force of 52.5 N. The angular acceleration of the cylinder will be $\_\_\_\_$ rad $s ^ { - 2 }$.

A string is wrapped around the rim of a wheel of moment of inertia $0.40 \mathrm { kgm } ^ { 2 }$ and radius 10 cm. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of 40 N. The angular velocity of the wheel after 10 s is $x \mathrm { rad } / \mathrm { s }$, where $x$ is $\_\_\_\_$

The coordinates of a particle with respect to origin in a given reference frame is $(1,1,1)$ meters. If a force of $\overrightarrow{\mathrm{F}} = \hat{i} - \hat{j} + \hat{k}$ acts on the particle, then the magnitude of torque (with respect to origin) in the $z$-direction is \_\_\_\_ .

Two thin circular rings are lying in the same plane and are touching each other at a single point. The first ring has mass 5 kg and radius 10 cm , while the second ring has mass 10 kg and radius 20 cm . Find the moment of inertia of the combined system about a straight line passing through the point of contact and lying in the plane of the rings.
(A) $\frac { 27 } { 50 } \mathrm {~kg} \mathrm {~m} ^ { 2 }$
(B) $\frac { 24 } { 40 } \mathrm {~kg} \mathrm {~m} ^ { 2 }$
(C) $\frac { 27 } { 40 } \mathrm {~kg} \mathrm {~m} ^ { 2 }$
(D) $\frac { 17 } { 12 } \mathrm {~kg} \mathrm {~m} ^ { 2 }$

On releasing the system 400 g mass fall down by 81 cm in 9 s , then determine the determine the moment of inertia of pulley