Moment of Inertia of a Ring

Q. Two light identical springs of spring constant $k$ are attached horizontally at the two ends of a uniform horizontal rod $\text{AB}$ of length $l$ and mass $m.$ The rod is pivoted at its centre $\text{O}$ and can rotate freely in horizontal plane. The other ends of two springs are fixed to rigid supports as shown in the figure. The rod is gently pushed through a small angle and released, the linear frequency of the resulting oscillation is :

1. $\frac{1}{2\pi }\sqrt{\frac{3k}{m}}$
2. $\frac{1}{2\pi }\sqrt{\frac{2k}{m}}$
3. $\frac{1}{2\pi }\sqrt{\frac{6k}{m}}$
4. $\frac{1}{2\pi }\sqrt{\frac{k}{m}}$

Q. A uniform disc of mass 2 kg and radius 1 m is mounted on an axle supported on fixed frictionless bearings. A light chord is wrapped around the rim of the disc and mass of 1 kg is tied to the free end. If it is released from rest-

1. the tension in the chord is 5 N
2. in first four seconds the angular displacement of the disc is 40 rad
3. the work done by the torque on the disc in first four seconds is 200 J
4. the increase in the kinetic energy of the disc in the first four seconds is 200 J

Q. Which of the following has the highest moment of inertia when each of them has the same mass and the same outer radius.

1. A ring about its axis, perpendicular to the plane of the ring.
2. A disc about its axis, perpendicular to the plane of the ring.
3. A solid sphere about one of its diameters
4. A spherical shell about one of its diameters.

Q. Two rings of same radius and mass are placed such that their centres are at a common point and their planes are perpendicular to each other. The moment of inertia of the system about an axis passing through the centre and perpendicular to the plane of one of the rings is
(Mass of each ring $=m$, radius $=r$)

1. $m{r}^2$
2. $\frac{3}{2}m{r}^2$
3. $2m{r}^2$
4. $\frac{1}{2}m{r}^2$

Q. A string is wound around a hollow cylinder of mass $5\text{kg}$ and radius $0.5\text{m}$. If the string is now pulled with a horizontal force of $40\text{N}$ and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string)

1. $10{\text{rad/s}}^2$
2. $16{\text{rad/s}}^2$
3. $20{\text{rad/s}}^2$
4. $12{\text{rad/s}}^2$

Q. Two identical spherical balls of mass $M$ and radius $R$ each are stuck on two ends of a rod of length $2R$ and mass $M$ (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is

1. $\frac{137}{15}M{R}^2$
2. $\frac{17}{15}M{R}^2$
3. $\frac{209}{15}M{R}^2$
4. $\frac{152}{15}M{R}^2$

Q. The moment of inertia of a uniform cylinder of length $l$ and radius $R$ about its perpendicular bisector is $I$. What is the ratio $l/R$ such that the moment of inertia is minimum ?

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

Q. A thin uniform rod of mass $M$ and length $L$ has its moment of inertia $I_{rod}$ about its perpendicular bisector. This rod is bend in the form of a circular shape of radius $r$. Now, circle has its moment of inertia $I_{\text{circle}}$ about its centre and perpendicular to its plane. Then, the ratio of $I_{rod}:I_{\text{circle}}$ is $\frac{{\pi }^2}{n}$ where $n$ is

Q.

The radius of gyration of a uniform disc about a line perpendicular to the disc equals its radius. Find the distance of the line from the centre.

Q.

The moment of inertia of a circular ring about an axis passing through its center and normal to its plane is $200gm-c{m}^2$. Then its moment of inertia about a diameter is

1. 40 gm cm²

2. 60gm cm²

3. 80gm cm²

4. 100gm cm²

Q. A small ball of mass $m$ is released from rest from top of smooth track which has circular part at other end as shown in figure. The normal force exerted by track on the ball when it reaches to point $A,$ is

1. $2mg$
2. $\frac{mg}{2}$
3. zero
4. $mg$

Q. The uniform rod of mass $20\text{kg}$ and length $1.6\text{m}$ is pivoted at one of its ends and is held in horizontal position, as shown in the diagram. What will be its initial angular acceleration, if it is just released from this position?

1. $\frac{15g}{16}$
2. $\frac{17g}{16}$
3. $\frac{19g}{16}$
4. $\frac{21g}{16}$

Q. The figure shows two solid discs with radius $R$ and $r$ respectively. If mass per unit area is same for both, what is the ratio of $\text{MI}$ of bigger disc around axis $\text{AB}$ $($ which is $\perp$ to the plane of the disc and passing through its centre $)$ of $\text{MI}$ of smaller disc around one of its diameters lying on its plane ? Given ${}^{\prime }{M}^{\prime }$ is the mass of the larger disc. ($\text{MI}$ stands for moment of inertia)

1. $R^2:r^2$
2. $2R^4:r^4$
3. $2r^4:R^4$
4. $2R^2:r^2$

Q.

The moment of inertia of a cylinder of mass$M$, length$L$, and radius$R$ about an axis passing through its center and perpendicular to the axis of the cylinder is$I=M\left(\frac{R^2}{4}+\frac{L^2}{12}\right)$. If such a cylinder is to be made for a given mass of a material, the ratio $\frac{L}{R}$ for it to have the minimum possible I is:

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

2. $\frac{3}{2}$

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

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

Q. A ring of mass $M$ and radius $R$ lies in the $x-y$ plane with its centre at the origin as shown. The mass distribution of the ring is non-uniform such that at any point $\text{P}$ on the ring, the mass per unit length is given by $\lambda ={\lambda }_0{\cos }^2\theta$ (where ${\lambda }_0$ is a positive constant). Then the moment of inertia of the ring about $z-$ axis is:

1. $\frac{1}{2}M{R}^2$
2. $M{R}^2$
3. $\frac{1}{\pi }\frac{M}{{\lambda }_0}R$
4. $\frac{1}{2}\frac{M}{{\lambda }_0}R$

Q. A particle of mass $m$ is suspended at the lower end of a thin rod of negligible mass. The upper end of the rod is free to rotate in the plane of the page about a horizontal axis passing through the point $\text{O}$. The time period of a system when it is slightly displaced from its mean position is $\pi \sqrt{\frac{L}{n}}$ then $n$ is _____.


Take $k=\frac{9mgL}{l^2}$ and $g=10{\text{m/s}}^2$

Q. A rod of length $l$ and mass $m$ is bent in shape of a ring. The moment of inertia of the ring about any of its diameter is:

1. $\frac{m{l}^2}{{\pi }^2}$
2. $\frac{m{l}^2}{4{\pi }^2}$
3. $\frac{m{l}^2}{2{\pi }^2}$
4. $\frac{m{l}^2}{8{\pi }^2}$

Q. $I_A,I_B$ and $I_C$ are the moments of inertia of a ring, semicircular ring and an arc of the same mass about the axes as shown below. Which of the following options is correct?

1. $I_A>I_B>I_C$
2. $I_A<I_B<I_C$
3. $I_A=I_B=I_C$
4. $I_A>I_B=I_C$

Q. A rod of length $l$ and mass $m$ is bent in shape of a ring. The moment of inertia of the ring about any of its diameter is:

1. $\frac{m{l}^2}{4{\pi }^2}$
2. $\frac{m{l}^2}{2{\pi }^2}$
3. $\frac{m{l}^2}{{\pi }^2}$
4. $\frac{m{l}^2}{8{\pi }^2}$

Q. A circular ring of radius $5\text{m}$ has moment of inertia of $100{\text{kg-m}}^2$ about its natural axis. This ring is converted into a circular disc having same moment of inertia about the disc’s natural axis. What is the radius of the circular disc?

1. $10\text{m}$
2. $\sqrt{2}\text{m}$
3. $5\sqrt{2}\text{m}$
4. $5\text{m}$

Q. An equilateral triangle $ABC$ is cut from a thin solid sheet of wood. (See figure) $D,E\text{and}F$ are the mid-points of its sides as shown and $G$ is the centre of the triangle. The moment of inertia of the triangle about an axis passing through $G$ and perpendicular to the plane of the triangle is $I_0$. If the smaller triangle $DEF$ is removed from $ABC$, the moment of inertia of the remaining figure about the same axis is $I$. Then:

1. $I=\frac{15}{16}{I}_0$
2. $I=\frac{3}{4}{I}_0$
3. $I=\frac{9}{16}{I}_0$
4. $I=\frac{I_0}{4}$

Q. A uniform rod $AB$ is hinged at end $A$ in horizontal position as shown in figure. The other end is connected to a block through a massless pulley- string arrangement as shown in figure. The acceleration of block just after release from the shown position is

1. $\frac{6g}{13}$
2. $\frac{3g}{8}$
3. $\frac{g}{4}$
4. $\frac{7g}{13}$

Q.

Calculate the moment of inertia of a thin ring of mass $\mathrm{m}$ and radius $\mathrm{r}$.

Q. Two concentric rings, one with radius of $0.5\text{m}$ and mass $1\text{kg}$ and other having a radius of $1\text{m}$ and mass $2\text{kg}$ are placed together on a horizontal table. Find out net moment of inertia of the system, about an axis perpendicular to their plane and passing through their common centre.

1. $1.75{\text{kg-m}}^2$
2. $1.2{\text{kg-m}}^2$
3. $2.25{\text{kg-m}}^2$
4. $1.5{\text{kg-m}}^2$

Q. A circular ring of radius $5\text{m}$ has moment of inertia of $100{\text{kg-m}}^2$ about its natural axis. This ring is converted into a circular disc having same moment of inertia about the disc’s natural axis. What is the radius of the circular disc?

1. $10\text{m}$
2. $\sqrt{2}\text{m}$
3. $5\sqrt{2}\text{m}$
4. $5\text{m}$

Q. A ring of radius $5\text{m}$ and linear mass density $\lambda =\frac{0.1}{\pi }\text{kg/m}$ is spinning about an axis passing through its COM perpendicular to its plane. Find the moment of inertia of the ring about this axis.

1. $25{\text{kg-m}}^2$
2. $42{\text{kg-m}}^2$
3. $13{\text{kg-m}}^2$
4. $31{\text{kg-m}}^2$

Q. A spool is pulled at an angle $\theta$ with the horizontal on a rough horizontal surface as shown in the figure. If the spool remains at rest, the angle $\theta$ is

1. ${60}^{\circ }$
2. ${45}^{\circ }$
3. ${20}^{\circ }$
4. ${30}^{\circ }$

Q. A thin uniform rod of mass $M$ and length $L$ has its moment of inertia $I_{rod}$ about its perpendicular bisector. This rod is bend in the form of a circular shape of radius $r$. Now, circle has its moment of inertia $I_{\text{circle}}$ about its centre and perpendicular to its plane. Find the ratio of $I_{rod}:I_{\text{circle}}$.

1. $\frac{{\pi }^2}{3}$
2. $\frac{{\pi }^2}{12}$
3. $1$
4. $\frac{{\pi }^2}{4}$

Q.
A uniform rod of length $l$ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed $\omega$ the rod makes an angle $\theta$ with it (see figure). To find $\theta$, equate the rate of change of angular momentum (direction going into the paper) $\frac{m{l}^2}{12}{\omega }^2\sin \theta \cos \theta$ about the centre of mass $\left(CM\right)$ to the torque provided by the horizontal and vertical forces $F_H\text{and}{F}_V$ about the $CM$. The value of $\theta$ is then such that:

1. $\cos \theta =\frac{2g}{3l{\omega }^2}$
2. $\cos \theta =\frac{g}{2l{\omega }^2}$
3. $\cos \theta =\frac{g}{l{\omega }^2}$
4. $\cos \theta =\frac{3g}{2l{\omega }^2}$

Q. A system of a circular disc of mass $900\text{g}$ and radius $4\text{m}$ and a circular ring of mass $2\text{kg}$ and radius $5\text{m}$ are stacked about their common centre as shown in figure. What is the moment of inertia of the system about an axis passing through their common centre perpendicular to the plane of the disc and the ring?

1. $65.4{\text{kgm}}^2$
2. $54.2{\text{kgm}}^2$
3. $50{\text{kgm}}^2$
4. $57.2{\text{kgm}}^2$