Moment of Inertia of Hollow Sphere

Q. Three identical spherical shells, each of mass $m$ and radius $r$ are placed as shown in figure. Consider an axis $X{X}^{\prime }$, which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about $X{X}^{\prime }$ axis is

1. $\frac{11}{5}m{r}^2$
2. $4m{r}^2$
3. $\frac{16}{5}m{r}^2$
4. $3m{r}^2$

Q.

(a) Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be $2M{R}^2/5$, where M is the mass of the sphere and R is the radius of the sphere.
(b) Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be $M{R}^2/4$, find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

Q.

Calculate the moment of Inertia of uniform solid sphere of mass $M$ and Radius $R$, about its diameter

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

2. $\frac{2}{5}M{R}^2$

3. $\frac{5}{3}M{R}^2$

4. $\frac{5}{2}M{R}^2$

Q.

Two spheres each of mass $M$ and radius $\frac{R}{2}$ are connected with a massless rod of length $R$ as shown in the figure. The moment of inertia of the system about an axis $\left(Y{Y}^{\prime }\right)$ passing through the centre of one of the spheres and perpendicular to the rod is

1. $\frac{21}{5}M{R}^2$
2. $\frac{2}{5}M{R}^2$
3. $\frac{5}{2}M{R}^2$
4. $\frac{5}{21}M{R}^2$

Q. From a solid sphere of mass M and radius R, a cube of maximum possible volume is cut. The moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is

1. $\frac{M{R}^2}{16\sqrt{2}\pi }$
2. $\frac{4M{R}^2}{9\sqrt{3}\pi }$
3. $\frac{4M{R}^2}{3\sqrt{3}\pi }$
4. $\frac{M{R}^2}{32\sqrt{2}\pi }$

Q. A solid sphere of radius R has moment of inertia I about its geometrical axis. If it is melted into a disc of radius r and thickness t. If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc), is also equal to I , then the value of r is equal to a) (2/√15)R. b)(2/√5)R c) (3√15)R d)(√3/√15)R

Q.

What is the moment of inertia in a thin hollow sphere touching the edge (tangent)???

Q. The moment of inertia of a solid sphere of density $\rho$ and radius R about its diameter is

1. 
2. 
3. 
4. 

Q.

If the radius of a solid sphere is 35 cm. The radius of gyration when the axis along a tangent is

1. 

2. 

3. 

4. 

Q.

State four uses to which a simple machine is put?

Q. The moment of inertia of a solid sphere about an axis passing through its centre is $0.4{\text{kg-m}}^2$. If mass of the sphere is kept same, but its density is increased eight times, find the new moment of inertia of the sphere about the same axis.

1. ${\text{0.4 kg-m}}^2$
2. ${\text{0.1 kg-m}}^2$
3. ${\text{0.3 kg-m}}^2$
4. ${\text{0.2 kg-m}}^2$

Q.

A thin rod of mass M and length L is rotating about an axis perpendicular to length of rod and passing through its centre. If half of the total rod that lies at one side of axis is cut and removed then the moment of inertia of remaining rod is

1. 

2. 

3. 

4. 

Q. The moment of inertia of a solid sphere about an axis passing through its centre is $0.4{\text{kg-m}}^2$. If mass of the sphere is kept same, but its density is increased eight times, find the new moment of inertia of the sphere about the same axis.

1. ${\text{0.3 kg-m}}^2$
2. ${\text{0.2 kg-m}}^2$
3. ${\text{0.4 kg-m}}^2$
4. ${\text{0.1 kg-m}}^2$

Q.

A cube of side $4cm$ is painted on all its sides. If it is sliced in $1cubiccm$ cubes, then number of such cubes that will have exactly two of their faces painted is __________.

Q.

A machine can help us in doing ……..work.

Q.

A mosquito is sitting on an L.P. record disc rotating on a turn table at $33\frac{1}{3}$ revolutions per minute. The distance of the mosquito from the centre of the turn table is 10 cm. Show that the friction coefficient between the record and the mosquito is greater than $\pi \frac{2}{81.}$ Take $g=10m/s^2.$

Q. The densities of two solid spheres A and B of the same radii $R$ vary with radial distance r as ${\rho }_A\left(r\right)=k\left(\frac{r}{R}\right)$ and ${\rho }_B\left(r\right)=k{\left(\frac{r}{R}\right)}^5$, respectively, where $k$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are $I_A$ and $I_B$, respectively. If $\frac{I_B}{I_A}=\frac{n}{10}$, the value of $n$ is

Q. A solid sphere of radius R has moment of inertia =I about its diameter. What will be moment of inertia of Shell of same mass and same radius about its diameter??

Q. A cylinder of mass M has length L that is $\sqrt{3}$ times its radius. What is the ratio of its moment of ineria about its own axis and that about an axis passing through its centre and perpendicular to its axis ?

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

Q. What is the moment of inertia of a hollow sphere of radius $3\text{m}$ and mass $2\text{kg}$ about an axis passing through the geometric centre of the sphere?

1. ${\text{20 kg-m}}^2$
2. ${\text{40 kg-m}}^2$
3. ${\text{12 kg-m}}^2$
4. ${\text{24 kg-m}}^2$

Q. Three identical spherical shells, each of mass $m$ and radius $r$ are placed as shown in figure. Consider an axis $X{X}^{\prime }$, which is touching two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about $X{X}^{\prime }$ axis is :

1. $\frac{11}{5}m{r}^2$
2. $4m{r}^2$
3. $\frac{16}{5}m{r}^2$
4. $3m{r}^2$

Q.

A composite disc is to be made using equal masses of aluminum and iron so that it has as high a moment of inertia as possible. This is possible when

1. The surfaces of the discs are made of iron with aluminum inside

2. The whole of aluminum is kept in the core and the iron at the outer rim of the disc

3. The whole of the iron is kept in the core and the aluminum at the outer rim of the disc

4. The whole disc is made with thin alternate sheets of iron and aluminum

Q. Two lenses of same mass and radius are given one is convex and the other is concave .Which one will have greater momentum of inertia while rotating about an Axis perpendicular to the plane and passing through the optical Centre? Why it is so?

Q. Find the radius of gyration and the moment of inertia of a rod of mass 200 g and length 200cm about an axis through it's centre and perpendicular to it's length.

Q.

The radius of gyration of a uniform rod of length $l$ about an axis passing through a point $\frac{l}{4}$ away from the center of the rod, and perpendicular to it, is

1. $\sqrt{\frac{7}{48}}l$

2. $\sqrt{\frac{3}{8}}l$

3. $\frac{1}{4}l$

4. $\frac{1}{8}l$

Q. In the figure shown below, a solid sphere of radius $5\text{m}$ is arranged concentrically with a hollow sphere of radius $9\text{m}$. Find the moment of inertia of the system about an axis passing through their centre. (Take $m_s=$ mass of solid sphere $=5\text{kg},m_h=$ mass of hollow sphere $=3\text{kg}$).

1. $162{\text{kg-m}}^2$
2. $202{\text{kg-m}}^2$
3. $212{\text{kg-m}}^2$
4. $81{\text{kg-m}}^2$

Q.

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

1. $10M{R}^2$

2. $\frac{37}{9}M{R}^2$

3. $4M{R}^2$

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

Q.

A balloon is pumped at the rate of $a$ cm³/minute. The rate of increase of its surface area when the radius is $b$ cm is

1. $\frac{2a^2}{b^4}$ ${\mathrm{cm}}^2/\min$

2. $\frac{a^2}{2b^2}$${\mathrm{cm}}^2/\min$

3. $\frac{2a}{b}$ ${\mathrm{cm}}^2/\min$

4. None of these

Q. ntFrom a disc of radius R and mass M , a circular hole of diameter R , whose rim passes through the centre is cut . What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre ?n

Q.

A bent wire AB carrying a current I is placed in a region of uniform magnetic field $\overrightarrow{B}$. The force on the wire AB is

1. BIL

2. zero

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

4. $\frac{BIL}{2}$