Rotational Work and Energy

Q. A ring, a solid sphere and a thin disc of different masses rotate with the same kinetic energy. Equal torques are applied to stop them. Which will make the least number of rotations before coming to rest

1. Disc
2. Ring
3. Solid sphere
4. All will make same number of rotations

Q. A body of moment of inertia of $3kg-m^2$ is rotating with an angular velocity of 2 rad/sec and has the same kinetic energy as a mass of 12 kg moving with a velocity of

1. 8 m/s
2. 0.5 m/s
3. 2 m/s
4. 1 m/s

Q. A small solid sphere of radius $r$ rolls down an incline without slipping which ends into a vertical loop of radius $R$. Find the height above the base so that it just loops the loop.

1. $\frac{5}{2}R$
2. $\frac{5}{2}(R-r)$
3. $\frac{25}{10}(R-r)$
4. $\frac{27}{10}R-\frac{17r}{10}$

Q. A solid cylinder is rolling without sliding. What fraction of its total kinetic energy is associated to rotational motion ?

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{1}{4}$
4. $\frac{1}{5}$

Q. A solid sphere of mass M and radius R is rolling on a horizontal surface without sliding with a velocity v. The ratio of its rotational and linear kinetic energies is

1. 2 : 5
2. 5 : 2
3. 7 : 10
4. 2 : 7

Q. A constant force of $10\text{N}$ applied on the flywheel of radius $20\text{cm}$ as shown in figure. If the flywheel covers an angular displacement of ${60}^{\circ }$. Calculate the work done on the flywheel.
(Assume $\pi =3$)

1. $0.67\text{J}$
2. $1.4\text{J}$
3. $2\text{J}$
4. $4\text{J}$

Q. A thin hollow sphere of mass $m$ is completely filled with non-viscous liquid of mass $m$. When the sphere rolls on horizontal ground such that centre moves with velocity $v$, kinetic energy of the system is equal to :

1. $m{v}^2$
2. $\frac{4}{3}m{v}^2$
3. $\frac{4}{5}m{v}^2$
4. None of these

Q. A rigid body is made up of three identical thin rods $A,B$ and $C$, each of length $L$ fastened together in the form of letter $H$. The body is free to rotate about a horizontal axis that runs along the length of one of the legs ($A$) of the body $H$. The body is allowed to fall from rest from a position in which the plane of $H$ is horizontal. What is the angular speed of the body when the plane of $H$ becomes vertical?

1. $\sqrt{\frac{g}{L}}$
2. $\frac{1}{2}\sqrt{\frac{g}{L}}$
3. $\frac{3}{2}\sqrt{\frac{g}{L}}$
4. $2\sqrt{\frac{g}{L}}$

Q. A flywheel of moment of inertia $0.32kg-m^2$ is rotated steadily at 120 rad/sec by an electric motor. The kinetic energy of the flywheel is

1. 4608 J
2. 1152 J
3. 2304 J
4. 6912 J

Q. A ring of radius 0.5 m and mass 10 kg is rotating about its diameter with an angular velocity of 20 rad/s. Its kinetic energy is

1. 10 J
2. 100 J
3. 500 J
4. 250 J

Q. If the angular momentum of a rotating body is increased by $200\%,$ then its kinetic energy of rotation will be increased by

1. $400\%$
2. $800\%$
3. $200\%$
4. $100\%$

Q. A uniform rod of length $l$ & mass $M$ is free to rotate about frictionless pin through one end. The rod is released from rest in a horizontal position. Find momentum of system (rod) when rod is at its lowest position.

1. $M\sqrt{5gl}$
2. $M\sqrt{3gl}$
3. $M\sqrt{2gl}$
4. $\frac{M}{2}\sqrt{3gl}$

Q. The ratio of kinetic energy of two bodies is $2:1$ and their angular momentum are in the ratio of $1:2$. Then the ratio of their moment of inertia will be (assume pure rotational motion)

1. $1:4$
2. $4:1$
3. $8:1$
4. $1:8$

Q. The torque of the force $\overrightarrow{F}$ = (2$\hat{i}$ - 3$\hat{j}$ + 4$\hat{k}$) N acting at the point $\overrightarrow{r}$ = (3$\hat{i}$ + 2$\hat{j}$ + 3$\hat{k}$) m about the origin be

1. 6$\hat{i}$ - 6$\hat{j}$ + 12$\hat{k}$
2. 17$\hat{i}$ - 6$\hat{j}$ - 13$\hat{k}$
3. -6$\hat{i}$ + 6$\hat{j}$ - 12$\hat{k}$
4. -17$\hat{i}$ + 6$\hat{j}$ + 13$\hat{k}$

Q. Two identical hollow spheres having mass $m=3\text{kg}$ and radius $r=1\text{m}$ are performing combined translational and rotational motion as shown in the figure. If $\omega =2\text{rad/s}$, $v_{com}=3\text{m/s}$ and ${\omega }^{\prime }=3\text{rad/s},v_{com}^{{}^{\prime }}=2\text{m/s}$, find the ratio of their kinetic energies.

1. $\frac{3}{7}$
2. $\frac{7}{6}$
3. $\frac{8}{7}$
4. $\frac{3}{5}$

Q. A disc of mass $m$ and radius $R$ is free to rotate in a horizontal plane about a vertical smooth fixed axis passing through its centre. There is a smooth groove along the diameter of the disc and two small balls of mass $m/2$ each, are placed in it on either side of the centre of the disc as shown in the figure. The disc is given an initial angular velocity ${\omega }_0$ and released.

The speed of each ball relative to the ground just after they leave the disc is

1. $\frac{R{\omega }_0}{\sqrt{3}}$
2. $\frac{R{\omega }_0}{\sqrt{2}}$
3. $\frac{2R{\omega }_0}{3}$
4. None of these

Q. A disc of moment of inertia $\frac{9.8}{{\pi }^2}{\text{kg m}}^2$ is rotating at $600\text{rpm}$. If the frequency of rotation changes from $600\text{rpm}$ to $300\text{rpm}$, then what is the work done approximately?

1. $1467\text{J}$
2. $1452\text{J}$
3. $1567\text{J}$
4. $1632\text{J}$

Q. A metal stick of mass $1$ kg and length 1 metre is held vertically with one end on the floor and is then allowed to fall. All values are in SI units. Take approximations.
$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(a) Initial mechanical energy}& \text{(p)}5.425\\ {\text{(b)(Angular velocity)}}^2\text{just before hitting the ground}& \text{(q) 29.429}\\ \text{(c) Linear velocity of the free end just before hitting the ground}& \text{(r) 4.9}\\ \text{(d) Moment of inertia}& \text{(s) 0.333}\end{array}$

1. $a-p;b-q;c-r;d-p$
2. $a-r;b-q;c-p;d-s$
3. $a-q;b-r;c-p;d-s$
4. $a-r;b-q;c-s;d-p$

Q. A solid cylinder of mass ($M=1\text{kg}$) and radius $R=0.5\text{m}$ is pivoted at its centre The axis of rotation of the cylinder is horizontal. Three small particles of mass ($m=0.1\text{kg}$) are mounted along its surface as shown in figure. The system is initially at rest.

The angular speed of cylinder, when it has rotated through ${90}^{\circ }$ in anticlockwise direction.
(Take $g=10{\text{m/s}}^2$)

1. $\sqrt{5}\text{rad/s}$
2. $2\sqrt{3}\text{rad/s}$
3. $\sqrt{10}\text{rad/s}$
4. $4\sqrt{2}\text{rad/s}$

Q. A tangential force of 10 N is applied on the periphery of a ring due to which it starts rotating about an axis through its centre and perpendicular to its plane. What is the work done by this force in rotating the ring by ${45}^o$? {R = 2m}.

1. $4\pi J$
2. $5\pi J$
3. $10\pi J$
4. $15\pi J$

Q. A small object of uniform density rolls up a curved surface with an initial velocity $V$. It reaches up to a maximum height of $\frac{3V^2}{4g}$ w.r.t. the initial position [neglect initial potential energy]. The object is a

1. ring
2. solid sphere
3. hollow sphere
4. disc

Q. If each point mass of a rigid ring covers a distance of $\frac{\pi }{2}m,$ by how much angle (in radians) the ring has been rotated? (R = 2m)

1. $\frac{\pi }{2}radians$
2. $\frac{\pi }{4}radians$
3. $\pi$ radians
4. $\frac{3\pi }{4}radians$

Q. A solid cylinder of mass ($M=1\text{kg}$) and radius $R=0.5\text{m}$ is pivoted at its centre The axis of rotation of the cylinder is horizontal. Three small particles of mass ($m=0.1\text{kg}$) are mounted along its surface as shown in figure. The system is initially at rest.

The angular speed of cylinder, when it has rotated through ${90}^{\circ }$ in anticlockwise direction.
(Take $g=10{\text{m/s}}^2$)

1. $\sqrt{5}\text{rad/s}$
2. $2\sqrt{3}\text{rad/s}$
3. $\sqrt{10}\text{rad/s}$
4. $4\sqrt{2}\text{rad/s}$

Q. Two bodies rotating with the same angular momentum have M.I. $I_{1}and{I}_2$ respectively such as $I_1>I_2$ respectively, then

1. $E_2>E_1$
2. $E_1>E_2$
3. $E_2=E_1$
4. $E_1=2E_2$

Q. In the figure shown, a ring $A$ is initially rolling without sliding with a velocity $v$ on a horizontal surface of the body $B$ (of same mass as $A$). All surfaces are smooth. $B$ has no initial velocity. What will be the maximum height reached by $A$ on $B$?

1. $\frac{3v^2}{4g}$
2. $\frac{v^2}{4g}$
3. $\frac{v^2}{2g}$
4. $\frac{v^2}{3g}$

Q. Two plates 1 and 2 move with velocities $v$ and $2v$ as shown. If the sphere does not slide relative to the plates and assuming the mass of each body as $m$, find the kinetic energy of the system (plates + sphere).

1. $\frac{123}{40}m{v}^2$
2. $\frac{230}{40}m{v}^2$
3. $\frac{123}{40}m{v}^3$
4. $\frac{120}{50}m{v}^2$

Q. Two bodies rotating with the same angular momentum have M.I. $I_{1}and{I}_2$ respectively such as $I_1>I_2$ respectively, then

1. $E_2>E_1$
2. $E_1>E_2$
3. $E_2=E_1$
4. $E_1=2E_2$