Moment of Inertia of a Disc

Q.

Example of inertia of direction.

Q.

Why spokes are needed for bicycles?

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

1. $4M{R}^2$
2. $\frac{40}{4}M{R}^2$
3. $10M{R}^2$
4. $\frac{37}{9}M{R}^2$

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 axis passing through their centres are $I_A$ and $I_B$ respectively. If $\frac{I_B}{I_A}=\frac{n}{10}$, then value of $n$ is-

Q.

The moment of inertia of a uniform semicircular disc of mass $M$ and radius $r$ about a line perpendicular to the plane of the disc through the centre is

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

Q. A thin disc of mass M and radius R has mass per unit area $\sigma \left(r\right)=k{r}^2$, where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is

1. $\frac{M{R}^2}{2}$
2. $\frac{M{R}^2}{6}$
3. $\frac{M{R}^2}{3}$
4. $\frac{2M{R}^2}{3}$

Q.

Moment of inertia of a thin rod of mass M and length L about an axis passing through centre is ML^{​​​​​2}/12. Its moment of inertia about a parallel axis at a distance of L/4 from the axis is given by?

A. ML2/48

B. ML3/48

C. ML2/12

D. 7ML2/48

Q. Three objects $A$ (solid sphere), $B$ (thin circular disk) and $C$ (circular ring) each have the same mass $M$ and radius $R$. They all spin with the same angular speed $\omega$ about their own symmetry axes. The amounts of work ($W$) required to bring them to rest, would satisfy the relation

1. $W_B>W_A>W_C$
2. $W_C>W_B>W_A$
3. $W_A>W_C>W_B$
4. $W_A>W_B>W_C$

Q.

Mass per unit area of a circular disc of radius$a$ depends on the distance$r$ from its center as$\sigma \left(r\right)=A+Br$. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its center is

1. $2{\mathrm{\pi a}}^4\left(\frac{\mathrm{A}}{4}+\frac{\mathrm{aB}}{5}\right)$

2. $2{\mathrm{\pi a}}^4\left(\frac{\mathrm{aA}}{4}+\frac{B}{5}\right)$

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

4. $2{\mathrm{\pi a}}^4\left(\frac{\mathrm{A}}{4}+\frac{B}{5}\right)$

Q. A solid sphere of mass M and radius R having a moment of inertia I about its diameter is recast into a solid disc fo radius r and thickness t. The moment of inertia of the disc about an axis passing the edge and perpendicular to the plane remains I. Then R and r are related as

1. $r=\sqrt{\frac{2}{15}}R$
2. $r=\sqrt{\frac{2}{15}}R$
3. $r=\frac{2}{15}R$
4. $r=\frac{2}{\sqrt{15}}R$

Q. Two circular discs $A$ and $B$ are of equal masses and thickness but made of metals with densities $d_A$ and $d_B(d_A>d_B)$ . If their moments of inertia about an axis passing through their centres and perpendicular to the circular faces are $I_A$ and $I_B$ then

1. $I_A=I_B$
2. $I_A>I_B$
3. $I_A<I_B$
4. $I_A\ge I_B$

Q.

Why are shockers used in scooters and cars? Explain

Q. One end of a horizontal thick copper wire of length 2L and radius 2R is welded to an end of another horizontal thin copper wire of length L and radius R. When the arrangement is stretched by applying forces at two ends, the ration of the elongation in the thin wire to that in the thick wire is

1. 4.00
2. 0.50
3. 0.25
4. 2.00

Q. The moment of inertia of a uniform cylinder of length $l$ and radius $R$ about it's perpendicular bisector is $I$. What is the ratio $\frac{l}{R}$ 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. moment of inertia of a uniform disc of mass M and radius R about an axis passing through its edge and perpendicular to its plane is I. Its moment of inertia about its diameter will be

Q. A solid sphere of mass M and radius R is divided into two unequal parts. The first part has a mass of $\frac{7M}{8}$ and is converted into a uniform disc of radius $2\text{R}$. The second part is converted into a uniform solid sphere. Let $I_1$ be the moment of inertia of the disc about its axis and $I_2$ be the moment of inertia of the new sphere about its axis. The ratio $\frac{I_1}{I_2}$ is given by

Q. In the figure shown, the plank is being pulled to the right with a constant speed $V$. If the cylinder does not slip then,

$\left(A\right)$ the speed of the centre of mass of the cylinder is $2V$
$\left(B\right)$ the speed of the centre of mass of the cyllinder is zero
$\left(C\right)$ the angular velocity of the cylinder is $V/R$
$\left(D\right)$ the angular velocity of the cylinder is zero

1. $A$ and $B$ are correct
2. $B$ and $C$ are correct
3. $C$ and $D$ are correct
4. All are wrong

Q.

A circular disc is to be made by using iron and aluminum so that it acquired maximum moment of inertia about geometrical axis. It is possible if

1. aluminum at the interior and iron surround to it

2. iron at interior and aluminum surround to it

3. using iron and aluminum layers in alternate order

4. sheet of iron is used at both external surface and aluminum sheet as internal layers

Q. Two discs have same mass and thickness. Their materials are of densities $d_1$ and $d_2$. The ratio of their moments of inertia about an axis passing through the centre and perpnedicular to the plane is

1. $d_2:d_1$
2. $d_1:d_2$
3. $\left(\frac{d_1}{d_2}\right)$
4. ${\left(\frac{d_2}{d_1}\right)}^2$

Q. The moment of inertia of a uniform circular disc of radius ${}^{\prime }{R}^{\prime }$ and mass ${}^{\prime }{M}^{\prime }$ about an axis passing from the edge of the disc and normal to the disc is

1. $M{R}^2$
2. $\frac{1}{2}M{R}^2$
3. $\frac{3}{2}M{R}^2$
4. $\frac{7}{2}M{R}^2$

Q. Find the moment of inertia of an equilateral triangular lamina of mass $1\text{kg}$ and having sides $3\text{m}$ about the axis passing through its centre of mass and perpendicular to the plane of triangular lamina.

1. $0.75{\text{kg-m}}^2$
2. $1.5{\text{kg-m}}^2$
3. $1{\text{kg-m}}^2$
4. $3{\text{kg-m}}^2$

Q. Five particles of mass 2 kg are attached to the rim of a circular disc of radius 0.1 m and negligible mass. Moment of inertia of the system about the axis passing through the center of the disc and perpendicular to its plane is.

1. $1kg{m}^2$
2. $0.1kg{m}^2$
3. $2kg{m}^2$
4. $0.2kg{m}^2$

Q. The uniform disc shown in the figure has a moment of inertia of $0.6{\text{kg-m}}^2$ around the axis that passes through $O$ and is perpendicular to the plane. If a segment is cut out from the disc as shown, what is the moment of inertia of the remaining disc in ${\text{kg-m}}^2$ ?

Q. For a system of particles, if mass shifts towards the axis of rotation, then moment of inertia of system will

1. decrease
2. depends upon mass of particle
3. increase
4. remain same

Q. A thin uniform disc of mass $M$ and radius $R$ has concentric hole of radius $r$. Find the moment of inertia of the disc about an axis passing through its centre and perpendicular to its plane.

1. $\frac{1}{2}M(R^2-r^2)$
2. $\frac{1}{3}M(R^2+r^2)$
3. $\frac{1}{2}M(R^2+r^2)$
4. $\frac{1}{2R}M(R^3+r^3)$

Q.

A football and a stone have the same mass

1. both have the same inertia

2. both have the same momentum

3. both have different inertia

4. both have different momentum

Q. A circular disc of radius $1\text{m}$ is rotating about an axis passing through its $COM$ and perpendicular to its plane. The moment of inertia about this axis is given by $I_{com}={\text{2 kg-m}}^2$. If the disc is rotated about an axis along its diameter, then the moment of inertia along that axis is $I_{dia}$. The value of $(I_{com}-I_{dia})$ is equal to

1. ${\text{2 kg-m}}^2$
2. ${\text{1 kg-m}}^2$
3. ${\text{3 kg-m}}^2$
4. ${\text{0.5 kg-m}}^2$

Q. Two disc of the same material and thickness have radii 0.2m and 0.6m. Their moments of inertia about their axes will be in the ratio

1. 1 : 81
2. 1 : 3
3. 1 : 27
4. 1 : 9

Q. The surface mass density of a disc of radius $a$ varies with radial distance $r$ as $\sigma =A+Br$ where A & B are positive constants. Then, moment of inertia of the disc about an axis passing through its centre and perpendicular to the plane (in ${\text{kg m}}^2$) is:

1. $2\pi a^4(\frac{A}{4}+\frac{Ba}{5})$
2. $2\pi a^4(\frac{Aa}{4}+\frac{B}{5})$
3. $\pi a^4(\frac{A}{4}+\frac{Ba}{5})$
4. $2\pi a^4(\frac{A}{5}+\frac{Ba}{4})$

Q. Two discs of mass $2\text{kg}$ and $3\text{kg}$ of radius $1\text{m}$ and $2\text{m}$ respectively placed together are rotated about an axis through their common centre as shown in figure. Find the net moment of inertia of the system about given axis.

1. $6{\text{kg-m}}^2$
2. $5{\text{kg-m}}^2$
3. $1{\text{kg-m}}^2$
4. $7{\text{kg-m}}^2$