Kinetic Energy of a Rigid Body

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

Q. A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. (a) Will it reach the bottom with the same speed in each case? (b) Will it take longer to roll down one plane than the other? (c) If so, which one and why?

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

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. Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by υ² = [ 2gh / (1 + k ² / R ²)] using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane

Q. The oxygen molecule has a mass of 5.30 × 10¯²⁶ kg and a moment of inertia of 1.94×10-¯⁴⁶ kg m2 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 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 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 $\left(E_{sphere}/E_{cylinder}\right)$ will be

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

Q. A solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown in the figure. The cube is free to rotate about the fixed axis $AB$. A bullet of mass $m(<<M)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height $h$ above the surface to impart the cube an angular speed ${\omega }_c$ so that the cube just topples over. Then ${\omega }_c$ is (note: the moment of inertia of the cube about an axis perpendicular to the face and passing through the centre of mass is $2M{a}^3/3)$.

1. $\sqrt{3gM/2ma}$
2. $\sqrt{3g/4h}$
3. $\sqrt{3g(\sqrt{2}-1)/2a}$
4. $\sqrt{3g(\sqrt{2}-1)/4a}$

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

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. The moment of inertia of a flywheel having kinetic energy 1800J and angular speed of 30 radians/ sec is:

1. $8kg{m}^2$
2. $4kg{m}^2$
3. $20kg{m}^2$
4. $6kg{m}^2$

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

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

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

Q. A uniform sphere of mass $m$ and radius $r$ rolls without sliding over a horizontal plane, rotating about a horizontal axle OA (figure). In the process, the centre of the sphere moves with velocity $v$ along a circle of radius $R$. Find the kinetic energy of the sphere.

Q. A circular disc rolls down an inclined plane without slipping. What fraction of its total energy is translational energy?

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

Q. Two discs of moments of inertia I₁ and I₂ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds ω ₁ and ω₂ are brought into contact face to face with their axes of rotation coincident. (a) What is the angular speed of the two-disc system? (b) 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 ω₁ ≠ ω₂.

Q. A body is rolling without slipping on a horizontal floor. The translation kinetic energy is $\frac{2}{3}rd$ of the total kinetic energy. Identify the body

1. Hollow Cylinder
2. Solid Sphere
3. Hollow Sphere
4. Solid Cylinder

Q. A hollow spherical ball of radius $2\text{m}$ and mass $3\text{kg}$ is rotating about a fixed axis passing through its COM. If it has kinetic energy of $576\text{J}$, the angular speed of the hollow spherical ball about the fixed axis is

1. $12\text{rad/s}$
2. $9.8\text{rad/s}$
3. $6\text{rad/s}$
4. $15.5\text{rad/s}$

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

Q. Two uniform spheres of mass M have radii R and 2R. Each sphere is rotating about a fixed axis through a diameter. The rotational kinetic energies of the spheres are identical. What is the ratio of the magnitude of the angular momenta of these spheres? That is, $\frac{L_{2R}}{L_R}=$

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

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

Q. A hollow spherical ball of radius $2\text{m}$ and mass $3\text{kg}$ is rotating about a fixed axis passing through its COM. If it has kinetic energy of $576\text{J}$, the angular speed of the hollow spherical ball about the fixed axis is

1. $12\text{rad/s}$
2. $9.8\text{rad/s}$
3. $6\text{rad/s}$
4. $15.5\text{rad/s}$

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