Kinetic Energy of a Rigid Body

Q. A solid cylinder of mass $3\text{kg}$ is rolling without slipping on a smooth horizontal surface with velocity $4\text{m/s}$. It collides with a horizontal spring of spring constant $200{\text{Nm}}^{-1}$. The maximum compression produced in the spring will be

1. $0.7\text{m}$
2. $0.2\text{m}$
3. $0.5\text{m}$
4. $0.6\text{m}$
   

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

Q. The rotational kinetic energy of a rigid body is proportional to

1. Angular speed squared
2. Angular speed
3. Angular speed cubed
4. Independent of Angular speed

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. 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 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. Point masses $m_1$ and $m_2$ are placed at the opposite ends of a rigid rod of length $L$, and negligible mass. The rod is to be set rotating about an axis
The Position of point $P$ on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity $\omega$ is minimum, is given by

1. $x=\frac{m_2}{m_1}L$
2. $x=\frac{m_{2}L}{m_1+m_2}$
3. $x=\frac{m_1}{m_2}L$
4. $x=\frac{m_{1}L}{m_1+m_2}$

Q. A solid sphere of mass $M$ and radius $R$ is rotating about its diameter. A hollow sphere of same mass and same external radius is also rotating with an angular speed twice that of solid sphere. The ratio of their kinetic energies of rotation $(E_S:E_H)$ will be

1. $3:20$
2. $1:10$
3. $20:3$
4. $10:1$

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. 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. 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. $3\text{s}$