Kinetic Energy of a Rigid Body

Q.

If momentum (P), area (A), and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is .

Q. A body rolls down an inclined plane. If its kinetic energy of rotation is $40\%$ of it's kinetic energy of translation motion, then the body is

1. Hollow cylinder
2. Ring
3. Disc
4. Solid sphere

Q. A hollow sphere of mass $M$ and radius $R$ is rolling without slipping on a rough horizontal surface. Then the percentage of it's total $KE$ which is translational, will be

1. $28\%$
2. $60\%$
3. $40\%$
4. $72\%$

Q. A ring and a disc of same mass and radius roll without slipping along a horizontal surface with the same velocity. If the $K.E.$ of the ring is $8\text{J}$, then $K.E.$ of the disc is

1. $2\text{J}$
2. $16\text{J}$
3. $4\text{J}$
4. $6\text{J}$

Q.

To maintain a rotor at a uniform angular speed of $200rad{s}^{-1}$, an engine needs to transmit a torque of 180 Nm. What is the power required by the engine?
(Note: uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque). Assume that the engine is 100 % efficient.

Q.

A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of 40 rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 2/5 times the initial value? Assume that the turntable rotates without friction.
Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?

Q. A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is

1. $\frac{2}{5}$
2. $\frac{2}{7}$
3. $\frac{1}{5}$
4. $\frac{7}{10}$

Q. The speed of a homogeneous solid sphere after rolling down an inclined plane of vertical height $h$ from rest without sliding is

1. $\sqrt{\frac{10}{7}gh}$
2. $\sqrt{gh}$
3. $\sqrt{\frac{6}{5}gh}$
4. $\sqrt{\frac{4}{3}gh}$

Q.

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds ${\omega }_1$ and ${\omega }_2$ are brought into contact face to face with their axes of rotation coincident.

(a) What is the angular speed of the two-disc system?

Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take ${\omega }_1\ne {\omega }_2$

Q.

A hoop of radius 2 m weighs 100 kg. It rolls along a horizontal floor so that its centre of mass has a speed of 20 cm/s. How much work has to be done to stop it?

Q.

The oxygen molecule has a mass of $5.30\times {10}^{-26}kg$ and a moment of inertia of $1.94\times {10}^{-46}kg{m}^2$ about an axis through its centre perpendicular to the lines joining the two atoms.
Suppose the mean speed of such a molecule in a gas is 500 m/s and that its kinetic energy of rotation is two thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

Q. A quarter disc of radius $R$ and mass $m$ is rotating about an axis $O{O}^{\prime }$ which is perpendicular to the plane of the disc as shown in figure. Rotational kinetic energy of the disc is

1. $\frac{1}{4}m{R}^2{\omega }^2$
2. $\frac{1}{8}m{R}^2{\omega }^2$
3. $\frac{1}{2}m{R}^2{\omega }^2$
4. $\frac{1}{16}m{R}^2{\omega }^2$

Q. The moment of inertia of a body about a given axis is $1.2\text{kg}\times {\text{m}}^2$. Initially, the body is at rest. In order to produce a rotational KE of 1500 joule, an angular acceleration of $25\frac{\text{rad}}{{\text{s}}^2}$ must be applied about that axis for a duration of

1. 4 seconds
2. 2 seconds
3. 8 seconds
4. 10 seconds

Q. A solid cylinder of mass $20\text{kg}$ rotates about its axis with an angular speed of $100\text{rad/s}$. The radius of the cylinder is $0.25\text{m}$. The kinetic energy associated with the rotation of the cylinder is

1. $3125\text{J}$
2. $725\text{J}$
3. $31.25\text{J}$
4. $7.25\text{J}$

Q. A solid sphere of mass $2\text{kg}$ and diameter $10\text{cm}$ is rolling without slipping at a speed of $5{\text{ms}}^{-1}.$ The rotational kinetic energy of the sphere will be

1. $10\text{J}$
2. $30\text{J}$
3. $50\text{J}$
4. $70\text{J}$

Q.

A hollow spherical ball rolls on a table without slipping. Ratio of its rotational kinetic energy to its total kinetic energy is

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

Q. To maintain a rotor, at a uniform angular speed of $100{\text{rad s}}^{-1}$, an engine needs to transmit a uniform torque of $100\text{N-m}$. The average power of the engine is -

1. $10\text{kW}$
2. $12\text{kW}$
3. $10\text{GW}$
4. $10\text{MW}$

Q.

A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute.
The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to $7.6kg{m}^2$.

What is his new angular speed? (Neglect friction.)
Is kinetic energy conserved in the process? If not, from where does the change come about?

Q. If a solid sphere of mass $1\text{kg}$ and radius $0.1\text{m}$ rolls without slipping at a uniform velocity of $1\text{m/s}$ along a straight line on a horizontal floor, then its kinetic energy is

1. $\frac{7}{5}\text{J}$
2. $\frac{2}{5}\text{J}$
3. $\frac{7}{10}\text{J}$
4. $1\text{J}$

Q. Moment of inertia of a body about a given axis is $1.5{\text{kg-m}}^2$. Initially the body is at rest. In order to produce a rotational kinetic energy of $1200\text{J}$, the angular acceleration of $20{\text{rad/s}}^2$ must be applied about the axis for a duration of

1. $2\text{s}$
2. $5\text{s}$
3. $2.5\text{s}$
4. $3\text{s}$

Q. A flywheel is rotating about a fixed axis passing through its COM. It has a kinetic energy of $420\text{Joules}$ when its angular speed is $20\text{rad/s}$. The moment of inertia of the wheel about the fixed axis is

1. $2.1{\text{kg m}}^2$
2. $1.44{\text{kg m}}^2$
3. $1.9{\text{kg m}}^2$
4. $0.48{\text{kg m}}^2$

Q.

Which of the following compounds has pseudo inert gas configuration?

1. ${\mathrm{K}}^{+}$

2. ${\mathrm{S}}^{2-}$

3. ${\mathrm{Cu}}^{+}$

4. ${\mathrm{Na}}^{+}$

Q. A horizontal $90\text{kg}$ merry-go-round is a solid disc of radius $1.50\text{m}$ is started from rest by a constant horizontal force of $50\text{N}$ applied tangentially to the edge of the disc. The kinetic energy of the disc after $3\text{s}$ is

1. $125\text{J}$
2. $250\text{J}$
3. $500\text{J}$
4. $150\text{J}$

Q. A small solid sphere of radius $r$ is released coaxially from point $A$ inside the fixed cylindrical bowl of radius $R$ as shown in figure. If the friction between the small sphere and the larger cylinder is sufficient enough to prevent any slipping. Find the ratio of total kinetic energy and rotational kinetic energy, when solid sphere reaches the bottom of the larger cylinder.

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

Q. A disk of mass $m$ and radius $r$ rotates about an axis passing through its center and perpendicular to its plane with angular velocity $\omega$. Find the percentage change in the kinetic energy when an identical disc is placed over the first disc and both the discs rotate about an axis passing through their center and perpendicular to the plane with same angular velocity $\omega$. Assume there is no friction between the surfaces.

1. $25\%$
2. $50\%$
3. $75\%$
4. $100\%$

Q. A circular disc of radius ${}^{\prime }{r}^{\prime }$ and mass ${}^{\prime }{m}^{\prime }$ is rotating about an axis passing through its COM perpendicular to its plane with an angular speed $\omega$. If the radius of the disc is halved and the angular speed is doubled, which of the following options is correct?

1. $K.E_1=\frac{K.E_2}{2}$
2. $K.E_1=\frac{K.E_2}{4}$
3. $K.E_1=2K.E_2$
4. $K.E_1=K.E_2$

Q.

A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad $s^{-1}$. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder? What is the magnitude of angular momentum of the cylinder about its axis?

Q. A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $\left(E_{\text{sphere}}/E_{\text{cylinder}}\right)$ will be

1. 3 : 1
2. 2 : 3
3. 1 : 5
4. 1 : 4

Q. If $x$ is the ratio of rotational kinetic energy and translational kinetic energy of a rolling body and considering friction to be sufficient enough to prevent any slipping, which of the following statement is true

1. $x\ge 1$
2. $x=\frac{1}{2}$
3. $x\le 1$
4. $x=1$

Q.

A solid cylinder of mass $50$ $kg$ and radius $0.5$ $m$ is free to rotate about the axis passing through its centre. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of $2rad{s}^{-2}$ is

1. $78.5$ $N$
2. $157$ $N$
3. $25$ $N$
4. $50$ $N$