Rotational Inertia

Q.

The moment of inertia of a body is independent of

1. Mass of the body

2. Position of axis

3. Nature of distribution of mass about axis of rotation

4. Angular velocity

Q.

What happens when moment of inertia increases?

Q. A uniform thin rod of length $L$ and mass $M$ is bent at the middle point $O$ as shown in figure. Consider an axis passing through its middle point $O$ and perpendicular to the plane of the bent rod. Then moment of inertia about this axis is :

1. $\frac{2}{3}m{L}^2$
2. $\frac{1}{3}m{L}^2$
3. $\frac{1}{12}m{L}^2$
4. dependent on $\theta$

Q. Find the MOI of the uniform rod about the given axis

1. $\frac{m{L}^2}{12}$
2. $\frac{m{L}^2}{12}{\cos }^2\theta$
3. $\frac{m{L}^2}{12}{\sin }^2\theta$
4. $\frac{m{L}^2}{12}\sin \theta$

Q. Three particles each of mass $2\text{kg}$ are placed at the corners of an equilateral triangle of side $6\text{m}$. What is the moment of inertia of the system about an axis passing through the height of the triangle on the same plane as the particles?

1. $12{\text{kgm}}^2$
2. $36{\text{kgm}}^2$
3. $48{\text{kgm}}^2$
4. $40{\text{kgm}}^2$

Q. Moment of inertia of a continuous body depends on

1. Axis of rotation
2. Density of material of the body
3. Shape and size of the body
4. All of these.

Q. Column I of the following table gives a list of bodies and column II gives their moments of inertia about the axis of symmetry passing through their centre of mass. Assume mass of each of the bodies is $M$. Then identify the correct matching.

Column-I	Column-II
i. Thin uniform hollow cylinder of radius $R$	p. $\frac{2}{5}M{R}^2$
ii. Uniform solid sphere of radius $R$	q. $\frac{2}{3}M{R}^2$
iii. Uniform rectangular lamina of length $l$ and breadth $b$	r. $M{R}^2$
iv. Thin uniform hollow sphere of radius $R$	s. $M\left(\frac{l^2+b^2}{12}\right)$

1. $i-r,ii-q,iii-s,iv-p$
2. $i-r,ii-p,iii-s,iv-q$
3. $i-r,ii-p,iii-s,iv-p$
4. $i-r,ii-q,iii-s,iv-q$

Q. One quarter sector is cut from a uniform circular disc of radius $R$. This sector has mass $M$. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

1. $\frac{M{R}^2}{8}$
2. $\frac{M{R}^2}{2}$
3. $\frac{M{R}^2}{4}$
4. $M{R}^2$

Q. Four particles of masses $1\text{g},2\text{g},3\text{g}$ and $4\text{g}$ are kept at points $(0,0),(0,3\text{cm}),(3\text{cm},3\text{cm}),(3\text{cm},0\text{cm})$ respectively. Find the moment of inertia of the system of particles (in ${\text{g-cm}}^2$) about $y$ axis.

1. $63{\text{g-cm}}^2$
2. $21{\text{g-cm}}^2$
3. $10.5{\text{g-cm}}^2$
4. $16.67{\text{g-cm}}^2$

Q. A system of particles is shown in figure. Point mass particles of same mass $m$ are present at each corner of the hexagon whose each side is equal to $a$. Find the $MOI$ about an axis passing through its $COM$ and perpendicular to its plane.

1. $6m{a}^2$
2. $3m{a}^2$
3. $4m{a}^2$
4. $2m{a}^2$

Q.

What is the definition of moment of inertia?

Q. A rectangular lamina of mass $6\text{kg}$ is placed in such a way that the centre of rectangular lamina coincides with the origin($\text{O}$) as shown in figure. If $I_x$ and $I_y$ are the moment of inertia of rectangular lamina about x - axis and y - axis respectively, then find the ratio $\frac{I_x}{I_y}$.

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

Q. A thin square plate of side $4\text{cm}$ and mass $m$ has a moment of inertia $I_{\text{Sq}}$ about the line $AB$ as shown in figure. It is then reformed into a rectangle of length $8\text{cm}$ and breadth $2\text{cm}$ with a moment of inertia $I_{\text{Rect}}$ about the line $CD$ as shown in the figure. What is the value of the ratio $\frac{I_{\text{Sq}}}{I_{\text{Rect}}}?$

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

Q.

The moment of Inertia of a solid cylinder of mass '$M$' and radius $R$ about its Axis(as shown) is.

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

Q. A rectangular lamina of mass $6\text{kg}$, breadth $4\text{m}$ and length $6\text{m}$ is taken as shown in figure. If $I_x$ and $I_y$ are the moments of inertia of the rectangular lamina about $x-$ axis and $y-$ axis respectively, then the value $I_y+I_x$ is

1. $26{\text{kg-m}}^2$
2. $10{\text{kg-m}}^2$
3. $8{\text{kg-m}}^2$
4. $18{\text{kg-m}}^2$

Q. A particle of mass $m$ is located at point O. Find $I_A,I_B,I_{AB}$ and $I_{A{A}^{\prime }}$ respectively.

1. $m{d}^2,m{d}^2{\tan }^2\theta ,m{d}^2,m{d}^2{\sin }^2\theta$
2. $m{d}^2,m{d}^2{\tan }^2\theta ,m{d}^2{\sin }^2\theta ,m{d}^2$
3. $0,m{d}^2{\tan }^2\theta ,m{d}^2,0$
4. $m{d}^2\sin \theta ,0,m{d}^2,0$

Q. Three particles of masses $2\text{kg},3\text{kg}$ and $5\text{kg}$ are situated at the vertices of a right angled triangle $\Delta ABC$ as shown in figure. Find the moment of inertia of the system about line $PQ$ perpendicular to the $BC$ and in the plane of $\Delta ABC$.

1. $40{\text{kg-m}}^2$
2. $80{\text{kg-m}}^2$
3. $32{\text{kg-m}}^2$
4. $48{\text{kg-m}}^2$

Q. A square lamina of mass density $\left(\sigma \right)=10{\text{kg/m}}^2$ and side length $2\text{m}$ is rotated about an axis passing through one of it’s side as shown in figure. Find the moment of inertia of the lamina about this axis.

1. ${\text{26.66 kg-m}}^2$
2. ${\text{16.66 kg-m}}^2$
3. ${\text{26.33 kg-m}}^2$
4. ${\text{53.33 kg-m}}^2$

Q. A system consists of particles of mass $4\text{kg}$ placed at $(4,0),6\text{kg}$ placed at $(0,-4)$ and $3\text{kg}$ placed at $(-3,4)$. How far from the origin must a particle of mass $10\text{kg}$ be placed so that the moment of inertia of the system about the origin becomes $300{\text{kgm}}^2$?

1. $25\text{m}$
2. $2.5\text{m}$
3. $10\text{m}$
4. $1\text{m}$

Q.

The moment of Inertia of a solid cylinder of mass '$M$' and radius $R$ about its Axis(as shown) is.

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

Q. Let $I_A$ and $I_B$ be the moments of inertia of two solid cylinders $A\&B$ of identical dimensions and size, about their axis of symmetry. Solid cylinders $A\&B$ are made of aluminium and iron respectively, then the relation between $I_A$ and $I_B$ is

1. $I_A<I_B$
2. $I_A=I_B$
3. $I_A>I_B$
4. Relation between $I_A$ and $I_B$ cannot be predicted as it depends on the frame of reference.

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 centre 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. A system is shown in the figure. Find the moment of inertia about an axis passing through its $COM$ and perpendicular to the plane. (Assume masses are connected with light inextensible strings)

1. $\frac{M{L}^2}{18}$
2. $\frac{17M{L}^2}{6}$
3. $\frac{25M{L}^2}{18}$
4. $\frac{7M{L}^2}{6}$

Q. A rectangular lamina of mass $2\text{kg}$ is having side $a=6\text{m}$ and $b=3\text{m}$. What is the moment of inertia of the rectangular lamina about an axis passing through the side $3\text{m}$ as shown in figure below?

1. $6{\text{kg-m}}^2$
2. $15{\text{kg-m}}^2$
3. $24{\text{kg-m}}^2$
4. $14.8{\text{kg-m}}^2$

Q. Three particles each of mass $1\text{kg}$ are placed at the vertices of an equilateral triangle $ABC$ of side $2\text{m}$. Find the moment of inertia of system about a line perpendicular to the plane of system and passing through middle point ($D$) of side $BC$ of the triangle.

1. $5{\text{kg-m}}^2$
2. $3{\text{kg-m}}^2$
3. $2.5{\text{kg-m}}^2$
4. $1.1{\text{kg-m}}^2$

Q. There is a flat uniform triangular plate ABC such that $AB=4\text{cm},BC=3\text{cm}$ and $\angle ABC={90}^{\circ }$ as shown in the figure. The moment of inertia of the plate about AB, BC and CA is $I_1,I_2$ and $I_3$ respectively. The incorrect statement is

1. $I_3<I_2$
2. $I_2>I_1$
3. $I_3<I_1$
4. $I_3>I_2$

Q. Two rods of equal length $\left(L\right)$ and equal mass $\left(M\right)$ are kept along $x$ and $y$ axis respectively such that their centres of mass lie at the origin. The moment of inertia about the line $y=x$, is

1. $\frac{M{L}^2}{3}$
2. $\frac{M{L}^2}{12}$
3. $\frac{M{L}^2}{4}$
4. $\frac{M{L}^2}{6}$

Q. A system is shown in the figure. Find the moment of inertia about an axis passing through its $COM$ and perpendicular to the plane. (Assume masses are connected with light inextensible strings)

1. $\frac{M{L}^2}{18}$
2. $\frac{17M{L}^2}{6}$
3. $\frac{25M{L}^2}{18}$
4. $\frac{7M{L}^2}{6}$

Q. Point masses of $1,2,3$ and $4\text{kg}$ are lying at the points $(0,0,0),(2,0,0),(0,3,0)\text{and}(-2,-2,0)$ respectively. What will be the moment of inertia of the system about $x$ - axis?

1. $51{\text{kg m}}^2$
2. $43{\text{kg m}}^2$
3. $67{\text{kg m}}^2$
4. $40{\text{kg m}}^2$

Q. One quarter sector is cut from a uniform circular disc of radius $R$. This sector has mass $M$. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

1. $\frac{M{R}^2}{8}$
2. $\frac{M{R}^2}{2}$
3. $\frac{M{R}^2}{4}$
4. $M{R}^2$