Moment of Inertia of a Ring

Q.

The surface mass density $\left(\frac{mass}{area}\right)$ of a circular disc of radius a depends on the distance from the centre as $\rho \left(r\right)$ = $A+Br$. Find its moment of inertia about the line perpendicular to the plane of the disc and passing through its centre

1. $2\pi a^4[\left(\frac{B}{4}\right)+\left(\frac{Aa}{5}\right)]$

2. $2\pi a^4[\left(\frac{A}{4}\right)+\left(\frac{Ba}{5}\right)]$

3. $2\pi a^4[\left(\frac{A}{5}\right)+\left(\frac{aB}{4}\right)]$

4. $2\pi a^4[\left(\frac{A}{4}\right)+\left(\frac{aB}{4}\right)]$

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

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. From a circular disc of radius $R$ and mass $9M$, a small disc of mass $M$ and radius $R/3$ is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is

1. $\frac{4}{9}M{R}^2$
2. $4M{R}^2$
   
3. $M{R}^2$
4. $\frac{40}{9}M{R}^2$

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

Q.

The surface mass density $\left(\frac{mass}{area}\right)$ of a circular disc of radius a depends on the distance from the centre as $\rho \left(r\right)$ = $A+Br$. Find its moment of inertia about the line perpendicular to the plane of the disc and passing through its centre

1. $2\pi a^4[\left(\frac{B}{4}\right)+\left(\frac{Aa}{5}\right)]$

2. $2\pi a^4[\left(\frac{A}{4}\right)+\left(\frac{Ba}{5}\right)]$

3. $2\pi a^4[\left(\frac{A}{5}\right)+\left(\frac{aB}{4}\right)]$

4. $2\pi a^4[\left(\frac{A}{4}\right)+\left(\frac{aB}{4}\right)]$

Q. A disc has a mass of 10 kg distribute uniformly and has a radius of 2m. What will be its moment of inertia about an axis passing through its centre and perpendicular to its plane?

1. 20 $kg-m^2$
2. 10 $kg-m^2$
3. 40 $kg-m^2$
4. None of these

Q. Four identical rods of mass $M=6\text{kg}$ each are welded at their ends to form a square and then welded to a massive ring having mass $M=4\text{kg}$ and radius $R=1\text{m}$. If the system is allowed to roll down the incline of inclination $\theta ={30}^{\circ }$.


The acceleration of the system will be

1. $\frac{g}{2}$
2. $\frac{g}{4}$
3. $\frac{7g}{24}$
4. $\frac{g}{8}$

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

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. $1.5{\text{kg-m}}^2$