Rotational Inertia

Q.

What is the dimensional formula for the torque?

Q. The moment of inertia of a body does not depend on

1. Mass of the body
2. The distribution of the mass in the body
3. Position of axis of rotation of the body.
4. None of the above

Q.

Consider a uniform square plate of side a and mass m. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is:

Q.

What is moment of inertia in simple terms?

Q.

The ABC is a triangular plate of uniform thickness. The sides are in the ratio shown in the figure. I_AB, I_BC and I_CA are the moments of inertia of the plate about AB, BC and CA respectively. Which one of the following relations is correct ?

1. $I_{AB}>I_{BC}$

2. $I_{BC}>I_{AB}$

3. $I_{AB}+I_{BC}=I_{CA}$

4. $I_{CA}ismaximum$

Q.

Particles of masses 1 g, 2 g, 3 g, ...., 100 g are kept at the marks 1 cm, 2 cm, 3 cm, ...., 100 cm respectively on a metre scale. Find the moment of inertia of the system of particles about a perpendicular bisector of the metre scale.

Q.

Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?

Q.

Derive an expression for torque in terms of the moment of inertia.

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

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

Q.

What are the different types of moments of inertia?

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

Q. The moment of inertia of HCl molecule about an axis passing through its centre of mass and perpendicular to the line joining the $H^{+}$ and $C{l}^{-}$ ions will be, if the interatomic distance is 1 $\stackrel{o}{A}$

1. 
2. 
3. 
4. 0

Q. Density of a solid sphere is given by the expression $\rho ={\rho }_0(1+\frac{x}{R})$ where $x$ is the distance from the centre of the sphere and $0\le x\le R$, $R$ is the radius of the solid sphere. Moment of inertia of the solid sphere about an axis passing through its centre will be: (Given ${\rho }_0$ is a constant)

1. $\frac{44}{45}\pi {\rho }_0{R}^5$
2. $\frac{44}{45}\pi {\rho }_0{R}^2$
3. $\frac{44}{35}\pi {\rho }_0{R}^2$
4. $\frac{48}{38}\pi {\rho }_0{R}^5$

Q. Three point masses $m_1,m_2$ and $m_3$ are located at the vertices of an equilateral triangle of side ${}^{\prime }{a}^{\prime }$. What is the moment of inertia of the system about an axis along the altitude of the triangle passing through $m_1$?

1. $(m_1+m_3)\frac{a^2}{4}$
2. $(m_1+m_2+m_3)\frac{a^2}{4}$
3. $(m_1+m_2)\frac{a^2}{4}$
   
4. $(m_2+m_3)\frac{a^2}{4}$

Q. Four spheres, each of mass 'M' and radius 'r' are situated at the four corners of square of side 'R'. The moment of inertia of the system about an axis perpendicular to the plane of square and passing through its centre will be

1. 
2. 
3. 
4. 

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.

Solve the previous problem if the pulley has a moment of inertial I about its axis and the string does not slip over it.

Q.

A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to $10\pi rad{s}^{-1}$. Which of the two will start to roll earlier? The co-efficient of kinetic friction is ${\mu }_k=0.2$.

Q. A child stands on the edge of a merry-go-round as shown below. The rotational inertia of the merry-go-round about its rotation axis is $150{\text{kg-m}}^2$. The child catches a ball of mass $1.0\text{kg}$ thrown by a friend. What is the angular speed of the merry-go-round just after the ball is caught?

1. $0.05\text{rad/s}$
2. $0.07\text{rad/s}$
3. $0.03\text{rad/s}$
4. $0.09\text{rad/s}$

Q.

Why is it easier to revolve a stone using a shorter string than a longer string?

Q. A massless equilateral triangle $EFG$ of side $a$ (As shown in figure) has three particles of mass m situated at its vertices. The moment of inertia of the system about the line $EX$ perpendicular to $EG$ in the plane of $EFG$ is $\frac{N}{20}m{a}^2$ where $N$ is an integer. The value of $N$ is .

Q. A uniform rectangular lamina is shown in figure. Find the moment of inertia $I_{\text{com}}$ about $z$ axis which passes through its centre and perpendicular to its plane. Here $b=10\text{m}$, $l=20\text{m}\&M=12\text{kg}$.

1. $100{\text{kg-m}}^2$
2. $1200{\text{kg-m}}^2$
3. $400{\text{kg-m}}^2$
4. $500{\text{kg-m}}^2$

Q.

What Is Meant By Twisting Couple?

Q. A soild cylinder and a hollow cylinder is having same mass $M$. Hollow cylinder is having the outer radius twice to that of solid cylinder. If $I_s$ and $I_h$ are the moment of inertia about the axis, along the height of the solid cylinder and hollow cylinder respectively passing through their centre, find the ratio of $I_h:I_s$.

1. $1:1$
2. $8:1$
3. $4:1$
4. $2:1$

Q. Two uniform circular thin rods of mass $m$ and length $l$ are connected with each other at their mid points and perpendicular to each other. A ring is also connected to both of the rods as shown in figure. Masses of the ring and both rods are the same. Find the moment of inertia of the system about the axis passing through the mid point of the rods and perpendicular to the plane of ring.

1. $\frac{9}{12}m{l}^2$
2. $\frac{3}{12}m{l}^2$
3. $\frac{5}{12}m{l}^2$
4. $\frac{7}{12}m{l}^2$

Q. Find the moment of inertia of a non-uniform rod having mass per unit length $\lambda ={\lambda }_0(1+\frac{x}{L})$ about an axis $\perp$ to the length of the rod and passing through an end point $L$. Length of the rod is $L$ and $m$ is mass of the rod.

1. $\frac{m{L}^2}{3}$
2. $\frac{m{L}^2}{18}$
3. $\frac{7m{L}^2}{12}$
4. $\frac{7m{L}^2}{18}$

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. The distance of a system of particles from its axis of rotation is increased. How will this affect the moment of inertia of the system?

1. The moment of inertia of the system increases
2. The moment of inertia of the system remains same
3. None of these
4. The moment of inertia of the system decreases

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