Additional Questions - Chapter 5 - Motion of System of Particles and Rigid Bodies - 11th Science Guide Samacheer Kalvi Solutions

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Additional Questions Solved
MultipleChoice Questions
Question 1.
The changes produced by the deforming forces in a rigid body are -
(a) very large
(b) infinity
(c) negligibly small
(d) small
Answer:
(c) negligibly small
Question 2.
When a rigid body moves all particles that constitute the body follows-
(a) same path
(b) different paths
(c) either same or different path
(d) circular path
Answer:
(b) different path
Question 3.
For bodies of regular shape and uniform mass distribution, the center of mass is at (a) the comers
(b) inside the objects
(c) the point where the diagonals meet
(d) the geometric center
Answer:
(d) the geometric center

Question 4.
For square and rectangular objects center of mass lies at -
(a) the point where the diagonals meet
(b) at the comers
(c) on the center surface
(d) any point
Answer:
(a) the point where the diagonals meet
Question 5.
Center of mass may lie -
(a) within the body
(b) outside the body
(c) both (a) and (b)
(d) only at the center
Answer:
(c) both (a) and (b)
Question 6.
The dimension of point mass is -
(a) positive
(b) negative
(c) zero
(d) infinity
Answer:
(c) zero
Question 7.
The motion of center of mass of a system of two particles is unaffected by their internal forces -
(a) irrespective of the actual directions of the internal forces
(b) only if they are along the line joining the particles
(c) only if acts perpendicular to each other
(d) only if acting opposite
Answer:
(a) irrespective of the actual directions of the internal forces
Question 8.
A circular plate of diameter $10 \mathrm{~cm}$ is kept in contact with a square plate of side $10 \mathrm{~cm}$. The density of the material and the thickness are same everywhere. The center of mass of the system will be
(a) inside the circular plate
(b) inside the square plate
(c) At the point of contact
(d) outside the system

Answer:
(6) inside the square plate
Question 9.
The center of mass of a system of particles does not depend on
(a) masses of particles
(b) position of the particles
(c) distribution of masses
(d) forces acting on the particles
Answer:
(d) forces acting on the particles
Question 10.
The center of mass of a solid cone along the line from the center of the base to the vertex is at -
(a) $\frac{1}{2}$ th of its height
(b) $\frac{1}{3}$ of its height
(c) $\frac{1}{4}$ th of its height
(d) $\frac{1}{5}$ th of its height
Answer:
(d) $\frac{1}{5}$ th of its height
Question 11.
All the particles of a body are situated at a distance of $\mathrm{X}$ from origin. The distance of the center of mass from the origin is -
(a) $\geq r$
(b) $\leq r$
(c) $=r$
(d) $>r$

Answer:

(b) $\leq r$
Question 12
A free falling body breaks into three parts of unequal masses. The center of mass of the three parts taken together shifts horizontally towards -
(a) heavier piece
(b) lighter piece
(c) does not shift horizontally
(d) depends on vertical velocity
Answer:
(c) does not shift horizontally
Question 13.
The distance between the center of carbon and oxygen atoms in the gas molecule is $1.13 \mathrm{~A}$. The center of mass of the molecule relative to oxygen atom is -
(a) $0.602 Å$
(b) $0.527 Å$

(c) $1.13 Å$
(d) $0.565 Å$
Answer:

(b) $0.527 Å$
Given,
Inter atomic distance $=1.13 \AA$
Mass of carbon atom $=14$
Mass of oxygen atom $=16$
Let C.M. of molecule lies at a distance of $\mathrm{X}$ from oxygen atom-
$
\begin{aligned}
& \text { i.e. } \mathrm{m}_1 \mathrm{r}_1=\mathrm{m}_2 \mathrm{r}_2 \\
& 16 \mathrm{X}=14(1.13-\mathrm{X}) \\
& 30 \mathrm{X}=15.82 \\
& \mathrm{X}=0.527 \AA
\end{aligned}
$
Question 14 .
The unit of position vector of center of mass is-
(a) $\mathrm{kg}$
(b) $\mathrm{kg} \mathrm{m}^2$
(c) $\mathrm{m}$
(d) $\mathrm{m}^2$
Answer:
(c) $\mathrm{m}$
Question 15.
The sum of moments of masses of all the particles in a system about the center of mass is-
(a) minimum
(b) maximum
(c) zero
(d) infinity
Answer:
(c) zero

Question 16 .
The motion of center of mass depends on-
(a) external forces acting on it
(b) internal forces acting within it
(c) both (a) and (b)
(d) neither (a) nor (b)
Answer:
(a) external forces acting on it
Question 17.
Two particles $\mathrm{P}$ and $\mathrm{Q}$ move towards with each other from rest with the velocities of $10 \mathrm{~ms}^{-1}$ and $20 \mathrm{~ms}^{-1}$ under the mutual force of attraction. The velocity of center of mass is-
(a) $15 \mathrm{~ms}^{-1}$
(b) $20 \mathrm{~ms}^{-1}$
(c) $30 \mathrm{~ms}^{-1}$
(d) zero
Answer:
(d) zero
Question 18 .
The reduced mass of the system of two particles of masses $2 \mathrm{~m}$ and $4 \mathrm{~m}$ will be -
(a) $2 \mathrm{~m}$
(b) $\frac{2}{3} \mathrm{y} \mathrm{m}$
(c) $\frac{3}{2} \mathrm{y} \mathrm{m}$
(d) $\frac{4}{3} \mathrm{~m}$
Answer:
(d) $\frac{4}{3} \mathrm{~m}$

Question 19.
The motion of the center of mass of a system consists of many particles describes its -
(a) rotational motion
(b) vibratory motion
(c) oscillatory motion
(d) translator y motion
Answer:
(c) oscillatory motion
Question 20.
The position of center of mass can be written in the vector form as -
(a) $\sum m_i \overrightarrow{r_i}$
(b) $\sum m_i \vec{r}_i^2$
(c) $\frac{\Sigma m_i \vec{r}_i}{\mathrm{M}}$
(d) $\frac{\Sigma m_i r_i^2}{\mathrm{M}}$
Answer:
(c) $\frac{\Sigma m_i \vec{r}_i}{\mathrm{M}}$
Question 21.
The positions of two masses $\mathrm{m}_1$ and $\mathrm{m}_2$ are $\mathrm{x}_1$ and $\mathrm{x}_2$. The position of center of mass is -

(a) $\frac{m_1 m_2}{m_1 x_1+m_2 x_2}$
(b) $m_1 x_1+m_2 x_2$
(c) $\frac{m_1 x_1+m_2 x_2}{m_1+m_2}$
(d) $\frac{m\left(x_1+x_2\right)}{m_1+m_2}$
Answer:
(c) $\frac{m_1 x_1+m_2 x_2}{m_1+m_2}$
Question 22.
In a two particle system, one particle lies at origin another one lies at a distance of $\mathrm{X}$. Then the position of center of mass of these particles of equal mass is -
(a) $\frac{m_2 \mathrm{X}_2}{m_1+m_2}$
(b) $\frac{\mathrm{X}}{2}$
(c) $\frac{m \mathrm{X}}{m_1+m_2}$
(d) $\frac{m_1+m_2}{m \mathrm{X}}$
Answer:
(a) $\frac{X}{2}$
Question 23.
Principle of moments is -
(a) $m_1 \mathrm{X}_2=m_2 \mathrm{X}_1$
(b) $\frac{m_1}{m_2}=\frac{\boldsymbol{\Lambda}_2}{\mathrm{X}_1}$
(c) $\frac{m_1}{m_2}=\frac{\mathrm{X}_1}{\mathrm{X}_2}$
(d) $\frac{m_1 \mathrm{X}_1}{m_2 \mathrm{X}_2}=0$
Answer:
(b) $\frac{m_1}{m_2}=\frac{\mathrm{X}_2}{\mathrm{X}_1}$

Question 24.
Infinitesimal quantity means -
(a) collective particles
(b) extremely small
(c) nothing
(d) extremely larger
Answer:
(b) extremely small
Question 25.
In the absence of external forces the center of mass will be in a state of -
(a) rest
(b) uniform motion
(c) may be at rest or in uniform motion
(d) vibration
Answer:
(c) may be at rest or in uniform motion
Question 26.
The activity of the force to produce rotational motion in a body is called as -
(a) angular momentum
(b) torque
(c) spinning
(d) drive force
Answer:
(b) torque
Question 27.
The moment of the external applied force about a point or axis of rotation is known as -
(a) angular momentum
(b) torque
(c) spinning
(d) drive force
Answer:

(b) torque
Question 28.
Torque is given as -
(a) $\vec{r} \cdot \vec{F}$
(b) $\vec{r} \times \vec{F}$
(c) $\vec{F} \times \vec{r}$
(d) $r F \cos \theta$
Answer:
(b) $\vec{r} \times \vec{F}$
Question 29.
The magnitude of torque is -
(a) $\mathrm{rF} \sin \theta$
(b) $\mathrm{rF} \cos \theta$
(c) $\mathrm{rF} \tan \theta$
(d) $\mathrm{rF}$
Answer:
(a) $\mathrm{rF} \sin \theta$
Question 30.
The direction of torque ácts -
(a) along $\vec{F}$
(b) along $\vec{r} \& \vec{F}$
(c) Perpendicular to $\vec{r}$
(d) Perpendicular to both $\vec{r} \& \vec{F}$
Answer:
(d) Perpendicular to both $\vec{r} \& \vec{F}$
Question 31.
The unit of torque is -
(a) is
(b) $\mathrm{Nm}^{-2}$

(c) $\mathrm{Nm}$
(d) $\mathrm{Js}^{-1}$
Answer:
(c) $\mathrm{Nm}$
Question 32.
The direction of torque is found using -
(a) left hand rule
(b) right hand rule
(c) palm rule
(d) screw rule
Answer:
(b) right hand rule
Question 33.
if the direction of torque is out of the paper then the rotation produced by the torque is -
(a) clockwise
(b) anticlockwise
(c) straight line
(d) random direction
Answer:
(a) clockwise
Question 34.
If the direction of the torque is inward the paper then the rotation is -
(a) clockwise
(b) anticlockwise
(c) straight line
(d) random direction
Answer:
(a) clockwise

Question 35.
if $\vec{r}$ and $\vec{F}$ are parallel or anti parallel, then the torque is -
(a) zero
(b) minimum
(c) maximum
(d) infinity
Answer:
(a) zero
Question 36.
The maximum possible value of torque is -
(a) zero
(b) infinity
(c) $\vec{r}+\vec{F}$
(d) $\mathrm{rF}$
Answer:
(d) $\mathrm{rF}$
Question 37.
The relation between torque and angular acceleration is -
(a) $\vec{\tau}=\frac{1}{\alpha}$
(b) $\vec{\alpha}=\frac{\vec{\tau}}{\mathrm{I}}$
(c) $\vec{\alpha}=\mathrm{I} \vec{\tau}$
(d) $\vec{\tau}=\frac{\vec{\alpha}}{\mathrm{I}}$
Answer:
(b) $\vec{\alpha}=\frac{\vec{\tau}}{I}$
Question 38 .
Angular momentum is -
(a) $\vec{P} \times \vec{r}$
(b) $\vec{r} \times \vec{P}$
(c) $\frac{\vec{r}}{\vec{p}}$
(d) $\vec{r} \cdot \vec{P}$
Answer:
(b) $\vec{r} \times \vec{P}$
Question 39.
The magnitude of angular momentum is given by -
(a) $\mathrm{rp}$
(b) $\mathrm{rp} \sin \theta$
(c) $r \mathrm{p} \cos \theta$

(d) $\operatorname{rp} \tan \theta$
Answer:
(b) $1 \mathrm{p} \sin \theta$
Question 40.
Angular momentum is associated with -
(a) rotational motion
(b) linear motion
(c) both (a) and (b)
(d) circular motion only
Answer:
(c) both (a) and (b)
Question 41.
Angular momentum acts perpendicular to -
(a) $\vec{r}$
(b) $\vec{P}$
(c) both $\vec{r}$ and $\vec{P}$
(d) plane of the paper
Answer:
(c) both $\vec{r}$ and $\vec{P}$
Question 42 .
Angular momentum is given by -
(a) $\frac{I}{\omega}$
(b) $\tau \omega$
(c) $\mathrm{I} \omega$
(d) $\frac{\omega I}{2}$
Answer:
(c) $\mathrm{I} \omega$
Question 43.
The rate of change of angular momentum is -
(a) Torque
(b) angular velocity
(c) centripetal force
(d) centrifugal force
Answer:
(a) Torque
Question 44.
The forces acting on a body when it is at rest -
(a) is gravitational force
(b) Normal force

(c) both gravitational as well as normal force
(d) No force is acting
Answer:
(c) both gravitational as well as normal force
Question 45.
The net force acting on a body when it is at rest is -
(a) gravitational force
(b) Normal force
(c) Sum of gravitational and normal force
(d) zero
Answer:
(d) zero
Question 46.
If net force acting on a body is zero, then the body is in -
(a) transnational equilibrium
(b) rotational equilibrium
(c) both (a) and (b)
(d) none
Answer:
(a) transnational equilibrium
Question 47.
If the net torque acting on the body is zero, then the body is in -
(a) transnational equilibrium
(b) rotational equilibrium
(c) mechanical equilibrium
(d) none
Answer:
(b) rotational equilibrium

Question 48.
when the net force and net torque acts on the body is zero then the body is in -
(a) transnational equilibrium
(b) rotational equilibrium
(c) mechanical equilibrium
(d) none
Answer:
(d) none
Question 49.
When the net force and net torque acts on the body is zero then the body is in -
(a) static equilibrium
(b) Dynamic equilibrium
(c) both (a) and (b)
(d) transnational equilibrium
Answer:
(c) both (a) and (b)
Question 50.
When two equal and opposite forces acting on the body at two different points, it may give -
(a) net force
(b) torque
(c) stable equilibrium
(d) none
Answer:
(b) torque
Question 51.
The torque in rotational motion is analogous to in transnational motion -
(a) linear momentum
(b) mass
(c) couple
(d) force
Answer:

(d) force
Question 52.
Which of the following example does not constitute a couple?
(a) steering a car
(b) turning a pen cap
(c) ball rolls on the floor
(d) closing the door
Answer:
(c) ball rolls on the floor
Questioner 53.
If the linear momentum and angular momentum are zero, then the object is said to be in -
(a) stable equilibrium
(b) unstable equilibrium
(c) neutral equilibrium
(d) all the above
Answer:
(d) all the above
Question 54.
When the body is disturbed, the potential energy remains same, then the body is in -
(a) stable equilibrium
(b) unstable equilibrium
(c) neutral equilibrium
(d) all the above
Answer:
(c) neutral equilibrium
Question 55
The point where the entire weight of the body acts is called as -
(a) center of mass
(b) center of gravity
(c) both (a) and (b)
(d) pivot

Answer:
(b) center of gravity
Question 56.
The forces acting on a cyclist negotiating a circular Level road is /are -
(a) gravitational force
(b) centrifugal force
(c) frictional force
(d) all the above
Answer:
(d) all the above
Question 57.
While negotiating a circular level road a cyclist has to bend by an angle $\theta$ from vertical to stay in an equilibrium is-
(a) $\tan \theta=\frac{r g}{r^2}$
(b) $\theta=\tan ^{-1}\left(\frac{v^2}{r g}\right)$
(c) $\theta=\sin ^{-1}\left(\frac{r g}{r^2}\right)$
(d) zero
Answer:
(b) $\theta=\tan ^{-1}\left(\frac{v^2}{r g}\right)$
Question 58 .
Moment of inertia for point masses -
(a) $\mathrm{m}^2 \mathrm{r}$
(b) $1 w^2$
(c) $\mathrm{mr}^2$
(d) zero
Answer:
(c) $\mathrm{mr}^2$
Question 59.
Moment of inertia for bulk object -
(a) $\mathrm{rm}^2$
(b) $1 \mathrm{w}^2$
(c) $m_i r_i^2$
(d) $\Sigma m_i r_i^2$
Answer:
(d) $\Sigma m_i r_i^2$

Question 60 .
For rotational motion, moment of inertia is a measure of -
(a) transnational inertia
(b) mass
(c) rotational inertia
(d) invariable quantity
Answer:
(c) rotational inertia
Question 61.
Unit of moment of inertia -
(a) $\mathrm{kgm}$
(b) $\mathrm{mkg}^{-2}$
(c) $\mathrm{kgm}^2$
(d) $\mathrm{kgm}^{-1}$
Answer:
(c) $\mathrm{kgm}^2$
Question 62.
Dimensional formula for moment of inertia is -
(a) $\left[\mathrm{ML}^{-2}\right]$
(b) $\left[\mathrm{M}^2 \mathrm{~L}^{-1}\right]$
(c) $\left[\mathrm{M}^{-2}\right]$
(d) $\left[\mathrm{ML}^2\right]$
Answer:
(d) $\left[\mathrm{ML}^2\right]$
Question 63.
Moment of inertia of a body is a -
(a) variable quantity
(b) invariable quantity
(c) constant quantity

(d) measure of torque
Answer:
(a) variable quantity
Question 64.
Moment of inertia of a thin uniform rod about an axis passing through the center of mass and perpendicular to the length is -
(a) $\frac{1}{3} \mathrm{Ml}^2$
(b) $\frac{1}{12} \mathrm{Ml}^2$
(c) $\frac{1}{2} \mathrm{M}\left(1^2+b^2\right)$
(d) $\mathrm{Ml}^2$
Answer:
(b) $\frac{1}{12} \mathrm{Ml}^2$
Question 65 .
Moment of inertia ofa thin uniform rod about an axis passing through one end and perpendicular to the length is-
(a) $\frac{1}{3} \mathrm{Ml}^2$
(b) $\frac{1}{12} \mathrm{Ml}^2$
(c) $\frac{1}{2} \mathrm{M}\left(\mathrm{l}^2+\mathrm{b}^2\right)$
(d) $\mathrm{Ml}^2$
Answer:
(a) $\frac{1}{3} \mathrm{Ml}^2$
Question 66 .
Moment of inertia of a thin uniform rectangular sheet about an axis passing through the center of mass and perpendicular to the plane of the sheet is-
(a) $\frac{1}{3} \mathrm{Ml}^2$
(b) $\frac{1}{12} \mathrm{Ml}^2$

(c) $\frac{1}{2} \mathrm{M}\left(\mathrm{l}^2+\mathrm{b}^2\right)$
(d) $\mathrm{Ml}^2$
Answer:
(c) $\frac{1}{2} \mathrm{M}\left(1^2+b^2\right)$
Question 67.
Moment of inertia of a thin uniform ring about an axis passing through the center of gravity and perpendicular to the plane is -
(a) $\mathrm{MR}^2$
(b) $2 \mathrm{MR}^2$
(c) $\frac{1}{2} \mathrm{MR}^2$
(d) $\frac{3}{2} \mathrm{MR}^2$
Answer:
(a) $\mathrm{MR}^2$
Question 68.
Moment of inertia of a thin uniform ring about an axis passing through the center and lying on the plane (along diameter) is -
(a) $\mathrm{MR}^2$
(b) $2 \mathrm{MR}^2$
(c) $\frac{1}{2} \mathrm{MR}^2$
(d) $\frac{2}{3} \mathrm{MR}^2$
Answer:
(c) $\frac{1}{2} \mathrm{MR}^2$

Question 69.
Moment of inertia of a thin uniform disc about an axis passing through the center and perpendicular to the plane is -
(a) $\mathrm{MR}^2$
(b) $2 \mathrm{MR}^2$
(c) $\frac{1}{2} \mathrm{MR}^2$
(d) $\frac{2}{3} \mathrm{MR}^2$
Answer:
(c) $\frac{1}{2} \mathrm{MR}^2$
Question 70.
Moment of inertia of a thin uniform disc about an axis passing through the center lying on the plane (along diameter is)
(a) $\mathrm{MR}^2$
(b) $\frac{1}{2} \mathrm{MR}^2$
(c) $\frac{3}{2} \mathrm{MR}^2$
(d) $\frac{1}{4} \mathrm{MR}^2$
Answer:
(d) $\frac{1}{4} \mathrm{MR}^2$
Question 71.
Moment of inertia of a thin uniform hollow cylinder about an axis of the cylinder is -
(a) $\mathrm{MR}^2$
(b) $\frac{1}{2} \mathrm{MR}^2$
(c) $\frac{3}{2} \mathrm{MR}^2$
(d) $\frac{1}{4} \mathrm{MR}^2$
Answer:
(a) $\mathrm{MR}^2$
Question 72.
Moment of inertia of a thin uniform hollow cylinder about an axis of the cylinder is -
(a) $\mathrm{MR}^2$
(b) $\mathrm{M}\left(\frac{\mathrm{R}^2}{2}+\frac{l^2}{12}\right)$
(c) $\frac{1}{2} \mathrm{MR}^2$
(d) $\mathrm{M}\left(\frac{\mathrm{R}^2}{4}+\frac{l^2}{12}\right)$
Answer:
(b) $\mathrm{M}\left(\frac{\mathrm{R}^2}{2}+\frac{l^2}{12}\right)$

Question 73.
Moment of inertia of a uniform solid cylinder about an axis passing through the center and along
the axis of the cylinder is -
(a) $\mathrm{MR}^2$
(b) $\mathrm{M}\left(\frac{\mathrm{R}^2}{2}+\frac{l^2}{12}\right)$
(c) $\frac{1}{2} \mathrm{MR}^2$
(d) $\mathrm{M}\left(\frac{\mathrm{R}^2}{4}+\frac{l^2}{12}\right)$
Answer:
(c) $\frac{1}{2} \mathrm{MR}^2$
Question 74.
Moment of inertia of a uniform solid cylinder about as axis passing perpendicular to the length and passing through the center is -
(a) $\mathrm{MR}^2$
(b) $\mathrm{M}\left(\frac{\mathrm{R}^2}{2}+\frac{l^2}{12}\right)$
(c) $\frac{1}{2} \mathrm{MR}^2$
(d) $\mathrm{M}\left(\frac{\mathrm{R}^2}{4}+\frac{l^2}{12}\right)$
Answer:
(d) $\mathrm{M}\left(\frac{\mathrm{R}^2}{4}+\frac{l^2}{12}\right)$
Question 75.
Moment of inertia of a thin hollow sphere about an axis passing through the center along its diameter is
(a) $\frac{2}{3} \mathrm{MR}^2$
(b) $\frac{5}{3} \mathrm{MR}^2$
(c) $\frac{7}{5} \mathrm{MR}^2$
(d) $\frac{2}{5} \mathrm{MR}^2$

Answer:
(a) $\frac{2}{3} \mathrm{MR}^2$
Question 76.
Moment of inertia of a thin hollow sphere about an axis passing through the edge along its tangent is -
(a) $\frac{2}{3} \mathrm{MR}^2$
(b) $\frac{5}{3} \mathrm{MR}^2$
(c) $\frac{7}{5} \mathrm{MR}^2$
(d) $\frac{2}{5} \mathrm{MR}^2$
Answer:
(b) $\frac{5}{3} \mathrm{MR}^2$
Question 77.
torment of inertia of a uniform solid sphere about an axis passing through the center along its diameter is -
(a) $\frac{2}{3} \mathrm{MR}^2$
(b) $\frac{5}{3} \mathrm{MR}^2$
(c) $\frac{7}{5} \mathrm{MR}^2$
(d) $\frac{2}{5} \mathrm{MR}^2$
Answer:
(d) $\frac{2}{5} \mathrm{MR}^2$
Question 78.
Moment of inertia of a uniform solid sphere about an axis passing through the edge along its tangent is -
(a) $\frac{2}{3} \mathrm{MR}^2$
(b) $\frac{5}{3} \mathrm{MR}^2$

(c) $\frac{7}{5} \mathrm{MR}^2$
(d) $\frac{2}{5} \mathrm{MR}^2$
Answer:
(c) $\frac{7}{5} \mathrm{MR}^2$
Question 79.
The ratio of $\mathrm{K}^2 / \mathrm{R}^2$ of a thin uniform ring about an axis passing through the center and perpendicular to the plane is-
(a) 1
(b) 2
(c) $\frac{7}{5}$
(d) $\frac{3}{2}$
Answer:
(a) 1
Question 80.
The ratio of $\mathrm{K}^2 / \mathrm{R}^2$ of a thin uniform disc about an axis passing through the center and perpendicular to the plane is -
(a) 1
(b) 2
(c) $\frac{1}{2}$
(d) $\frac{3}{2}$
Answer:
(c) $\frac{1}{2}$
Question 81 .
When no external torque acts on the body, the net angular momentum of a rotating body.
(a) increases
(b) decreases
(c) increases or decreases
(d) remains constant

Answer:
(d) remains constant
Question 82.
Moment of inertia of a body is proportional to -
(a) $\omega$
(b) $\frac{1}{\omega}$
(c) $\omega^2$
(d) $\frac{1}{\omega^2}$
Answer:
(b) $\frac{1}{\omega}$
Question 83.
When the hands are brought closer to the body, the angular velocity of the ice dancer -
(a) decreases
(b) increases
(c) constant
(d) may decrease or increase
Answer:
(b) increases
Question 84.
When the hands are stretched out from the body, the moment of inertia of the ice dancer -
(a) decreases
(b) increases
(c) constant
(d) may decrease or increase
Answer:
(b) increases
Question 85.
The work done by the torque is -
(a) F. ds
(b) F. $d \theta$
(c) $\tau \mathrm{d} \theta$
(d) r.d $\theta$
Answer:
(c) $\tau \mathrm{d} \theta$
Question 86.
Rotational Kinetic energy of a body is -
(a) $\frac{1}{2} \mathrm{mr}$
(b) $\frac{1}{2} \mathrm{I} \omega^2$

(c) $\frac{1}{2} \mathrm{Iv}^2$
(d) $\frac{1}{2} \mathrm{~m} \omega^2$
Answer:
(b) $\frac{1}{2} \mathrm{I} \omega^2$
Question 87.
Rotational kinetic energy is given by -
(a) $\frac{1}{2} \mathrm{mr}$
(b) $\frac{1}{2} \mathrm{Iv}^2$
(c) $\frac{\mathrm{L}^2}{2 \mathrm{I}}$
(d) $\frac{2 \mathrm{I}}{\mathrm{L}^2}$
Answer:
(c) $\frac{\mathrm{L}^2}{2 \mathrm{I}}$
Question 88.
If $\mathrm{E}$ is a rotational kinetic energy then angular momentum is-
(a) $\sqrt{2 \mathrm{IE}}$
(b) $\frac{\mathrm{E}^2}{2 \mathrm{I}}$
(c) $\frac{2 \mathrm{I}}{\mathrm{E}^2}$
(d) $\frac{E}{I^2 \omega^2}$
Answer:
(a) $\sqrt{2 \mathrm{IE}}$
Question 89.
The product of torque acting on a body and angular velocity is -
(a) Energy
(b) power
(c) work done
(d) kinetic energy
Answer:
(b) power
Question 90.
The work done per unit time in rotational motion is given by -
(a) $\vec{F} \cdot \mathrm{v}$
(b) $\frac{d \theta}{d t}$
(c) $\tau \omega$
(d) I $\omega$
Answer:
(c) $\tau \omega$

Question 91.
While rolling, the path of center of mass of an object is -
(a) straight line
(b) parabola
(c) hyperbola
(d) circle
Answer:
(a) straight line
Question 92.
In pure rolling, the velocity of the point of the rolling object which comes in contact with the surface is -
(a) maximum
(b) minimum
(c) zero
(d) $2 \mathrm{~V}_{\mathrm{CM}}$
Answer:
(c) zero
Question 93.
In pure rolling velocity of center of mass is equal to -
(a) zero
(b) $\mathrm{R} \omega$
(c) $\frac{\omega}{R}$
(d) $\frac{R}{\omega}$
Answer:
(b) $\mathrm{R} \omega$
Question 94.
In pure rolling, rotational velocity of points at its edges is equal to-
(a) $\mathrm{R} \omega$
(b) velocity of center of mass
(c) transnational velocity
(d) all the above

Answer:
(a) $\mathrm{R} \omega$
Question 95.
Sliding of the object occurs when -
(a) $V_{\text {trans }} (b) $\mathrm{V}_{\text {trans }}=\mathrm{V}_{\text {rot }}$
(c) $\mathrm{V}_{\text {trans }}>\mathrm{V}_{\text {rot }}$
(d) $\mathrm{V}_{\text {trans }}=0$
Answer:
(c) $\mathrm{V}_{\text {trans }}>\mathrm{V}_{\text {rot }}$
Question 96.
Sliding of the object occurs while -
(a) $V_{\text {trans }}=\mathrm{V}_{\text {rot }}$
(b) $\mathrm{V}_{\mathrm{CM}}=\mathrm{R} \omega$
(c) $\mathrm{V}_{\mathrm{CM}}<\mathrm{R} \omega$
(d) $\mathrm{V}_{\mathrm{CM}}>\mathrm{R} \omega$
Answer:
(d) $\mathrm{V}_{\mathrm{CM}}>\mathrm{R} \omega$
Question 97.
Slipping of the object occurs when -
(a) $V_{\text {trans }} (b) $\mathrm{V}_{\text {trans }}=V_{\text {rot }}$
(c) $V_{\text {trans }}>V_{\text {rot }}$
(d) $\mathrm{V}_{\text {trans }}=0$
Answer:
(a) $\mathrm{V}_{\text {trans }}<\mathrm{V}_{\text {rot }}$

Question 98.
Slipping of the object occurs when -
(a) $\mathrm{V}_{\text {trans }}=\mathrm{V}_{\text {rot }}$
(b) $\mathrm{V}_{\mathrm{CM}}=\mathrm{R} \omega$
(c) $\mathrm{V}_{\mathrm{CM}}<\mathrm{R} \omega$
(d) $\mathrm{V}_{\mathrm{CM}}>\mathrm{R} \omega$
Answer:
(c) $\mathrm{V}_{\mathrm{CM}}<\mathrm{R} \omega$
Question 99.
In sliding, the resultant velocity of a point of contact acts along -
(a) forward direction
(b) backward direction
(c) either (a) or (b)
(d) tangential direction
Answer:
(a) forward direction
Question 100.
In slipping, the resultant velocity of a point of contact acts along -
(a) forward direction
(b) backward direction
(c) either (a) or (b)
(d) tangential direction
Answer:
(b) backward direction
Question 101.
When a solid sphere is undergoing pure rolling, the ratio of transnational kinetic energy to rotational kinetic - energy is -
(a) $2: 5$
(b) $5: 2$
(c) $1: 5$
(d) $5: 1$
Answer:
(b) $5: 2$
Question 102 .
Time taken by the rolling object in inclined plane to reach its bottom is -
(a) $\sqrt{\frac{1+\frac{k^2}{\mathrm{R}^2}}{g \sin ^2 \theta}}$
(b) $\sqrt{\frac{2 g h}{1+\frac{k^2}{\mathrm{R}^2}}}$
(c) $\sqrt{\frac{2 h\left(1+\frac{k^2}{\mathrm{R}^2}\right)}{g \sin ^2 \theta}}$
(d) $\sqrt{\frac{2 h\left(1+\frac{\mathrm{R}^2}{k^2}\right)}{g \sin ^2 \theta}}$
Answer:
(c) $\sqrt{\frac{2 h\left(1+\frac{k^2}{\mathrm{R}^2}\right)}{g \sin ^2 \theta}}$
Question 103
The velocity of the rolling object on inclined plane at the bottom of inclined plane is -

(a) $\sqrt{\frac{1+\frac{k^2}{\mathrm{R}^2}}{g \sin ^2 \theta}}$
(b) $\sqrt{\frac{2 g h}{1+\frac{k^2}{\mathrm{R}^2}}}$
(c) $\sqrt{\frac{2 h\left(1+\frac{k^2}{\mathrm{R}^2}\right)}{g \sin ^2 \theta}}$
(d) $\sqrt{\frac{2 h\left(1+\frac{\mathrm{R}^2}{k^2}\right)}{g \sin ^2 \theta}}$
Answer:
(b) $\sqrt{\frac{2 g h}{1+\frac{k^2}{\mathrm{R}^2}}}$
Question 104.
Moment of inertia of an annular disc about an axis passing through the centre and perpendicular to the plane of disc is -
(a) $\frac{\mathrm{M}}{2}\left(\mathrm{R}_1^2+\mathrm{R}_2^2\right)$
(b) $\frac{\mathrm{M}}{2}\left(\mathrm{R}_1^2-\mathrm{R}_2^2\right)$
(c) $\frac{2}{\mathrm{M}}\left(\mathrm{R}_1^2+\mathrm{R}_2^2\right)$
(d) $\frac{2}{\mathrm{M}}\left(\mathrm{R}_1^2-\mathrm{R}_2^2\right)$

Answer:
(a) $\frac{\mathrm{M}}{2}\left(\mathrm{R}_1^2+\mathrm{R}_2^2\right)$
Question 105 .
Moment of inertia of a cube about an axis passing through the center of mass and perpendicular to face is -
(a) $\frac{\mathrm{Ma}^2}{6}$
(b) $\frac{1}{3} \mathrm{Ma}^2$
(c) $\frac{M a}{6}$
(d) $\frac{\mathrm{Ma}^2}{12}$
Answer:
(a) $\frac{\mathrm{Ma}^2}{6}$
Question 106.
Moment of inertia of a rectangular plane sheet about an axis passing through center of mass and perpendicular to side $b$ in its plane is -
(a) $\frac{\mathrm{MI}}{12}$
(b) $\frac{\mathrm{Ma}^2}{12}$
(c) $\frac{\mathrm{Mb}^2}{12}$
(d) $\frac{\mathrm{M} \mathrm{I}^2}{6}$
Answer:
(c) $\frac{\mathrm{Mb}^2}{12}$

Question 107.
Rotational kinetic energy can be calculated by using -
(a) $\frac{1}{2} \mathrm{I} \omega^2$
(b) $\frac{\mathrm{L}^2}{2 I}$
(c) $\frac{1}{2} \mathrm{~L} \omega$
(d) all the above
Answer:
(b) $\frac{\mathrm{L}^2}{2 I}$ )
Question 108.
The radius of gyration of a solid sphere of radius $r$ about a certain axis is $r$. The distance of that axis from the center of the sphere is -
(a) $\frac{2}{5} \mathrm{r}$
(b) $\sqrt{\frac{2}{5}} \mathrm{r}$
(c) $\sqrt{0.6 r}$
(d) $\sqrt{\frac{5}{3}}$
Answer:
(c) $\sqrt{0.6 r}$
From parallel axis theorem
$
\mathrm{I}=\mathrm{I}_{\mathrm{G}}+\mathrm{Md}^2
$
$
\begin{aligned}
& \mathrm{mr}^2=\frac{2}{5} \mathrm{mr}^2+\mathrm{md}^2 \\
& \mathrm{~d}=\sqrt{\frac{3}{5}} \mathrm{r}=\sqrt{0.6 r}
\end{aligned}
$

Question 109.
A wheel is rotating with angular velocity $2 \mathrm{rad} / \mathrm{s}$. It is subjected to a uniform angular acceleration 2 $\mathrm{rad} / \mathrm{s}^2$ then the angular velocity after $10 \mathrm{~s}$ is
(a) $12 \mathrm{rad} / \mathrm{s}$
(b) $20 \mathrm{rad} / \mathrm{s}$
(c) $22 \mathrm{rad} / \mathrm{s}$
(d) $120 \mathrm{rad} / \mathrm{s}$
Answer:
(c) $22 \mathrm{rad} / \mathrm{s}$
$\omega=\omega_0+\alpha t$
Here $\omega_0=2 \mathrm{rad} / \mathrm{s}$
$\alpha=2 \mathrm{rad} / \mathrm{s} 2$
$\omega=10 \mathrm{~s}$
$\omega=2+2 \times 10=22 \mathrm{rad} / \mathrm{s}$
Question 110.
Two rotating bodies $A$ and $B$ of masses $m$ and $2 m$ with moments of inertia $I_A$ and $I_B\left(I_b>I_A\right)$ have equal kinetic energy of rotation. If $\mathrm{L}_{\mathrm{A}}$ and $\mathrm{L}_{\mathrm{B}}$ be their angular momenta respectively,
then,
(a) $\mathrm{L}_{\mathrm{B}}>\mathrm{L}_{\mathrm{A}}$
(b) $\mathrm{L}_{\mathrm{A}}>\mathrm{L}_{\mathrm{B}}$
(c) $\mathrm{L}_{\mathrm{A}}=\frac{L_B}{2}$
(d) $\mathrm{L}_{\mathrm{A}}=2 \mathrm{~L}_{\mathrm{B}}$
Answer:
(a) $\mathrm{L}_{\mathrm{B}}>\mathrm{L}_{\mathrm{A}}$
Question 111.
Three identical particles lie in $\mathrm{x}, \mathrm{y}$ plane. The $(\mathrm{x}, \mathrm{y})$ coordinates of their positions are $(3,2),(1,1)$, $(5,3)$ respectively. The $(x, y)$ coordinates of the center of mass are -
(a) $(\mathrm{a}, \mathrm{b})$
(b) $(1,2)$
(c) $(3,2)$
(d) $(2,1)$
Answer

(c) The $\mathrm{X}$ and $\mathrm{Y}$ coordinates of the center of mass are
$
\begin{aligned}
& \mathrm{X}=\frac{m_1 \mathrm{X}_1+m_2 \mathrm{X}_2+m_3 \mathrm{X}_3}{m_1+m_2+m_3} \\
& =\frac{1}{3}\left(\mathrm{X}_1+\mathrm{X}_2+\mathrm{X}_3\right) \\
& =\frac{1}{3}(3+1+5)=3 \\
& \mathrm{Y}=\frac{1}{3}(2+1+3)=2
\end{aligned}
$
Question 112.
A solid cylinder of mass $3 \mathrm{~kg}$ and radius $10 \mathrm{~cm}$ is rotating about its axis with a frequency of $20 / \pi$. The rotational kinetic energy of the cylinder
(a) $10 \pi \mathrm{J}$
(b) $12 \mathrm{~J}$
(c) $\frac{6 \times 10^2}{\pi} \mathrm{J}$
(d) $3 \mathrm{~J}$
Answer:
(b) $12 \mathrm{~J}$
Given,
$
\begin{aligned}
& \mathrm{M}=3 \mathrm{~kg} \\
& \mathrm{R}=0.1 \mathrm{~m} \\
& \mathrm{v}=20 / \pi
\end{aligned}
$
Angular frequency $\omega=2 \pi \mathrm{V}=\frac{2 \pi x 20}{\pi}=40 \mathrm{rad} / \mathrm{s}^{-1}$
Moment of inertia of the cylinder about its axis $=\mathrm{I}=\frac{1}{2} \mathrm{mR}^2=\frac{1}{2} \times 3 \times(0.1)^2=0.015 \mathrm{~kg} \mathrm{~m}^2$
$
\mathrm{K} . \mathrm{E} .=\frac{1}{2} \mathrm{I} \omega^2=\frac{1}{2} \times 0.015 \times(40)^2=12 \mathrm{~J}
$
Question 113 .
A circular disc is rolling down in an inclined plane without slipping. The percentage of rotational energy in its total energy is
(a) $66.61 \%$
(b) $33.33 \%$
(c) $22.22 \%$
(d) $50 \%$
Answer:
(b) $33.33 \%$
Rotational K.E. $=\frac{1}{2} \mathrm{I}^2 \frac{1}{2}\left(\frac{1}{2} \mathrm{MR}^2\right) \omega^2=\frac{1}{4} \mathrm{MR}^2 \omega^2$
Transnational $\mathrm{K} \cdot \mathrm{E} .=\frac{1}{2} \mathrm{MV}^2=\frac{1}{2} \mathrm{M}(\mathrm{R} \omega)^2=\frac{1}{2} \mathrm{MR}^2 \omega^2$

Total kinetic energy $=\mathrm{E}_{\text {rot }}+\mathrm{E}_{\text {trans }}=\frac{1}{4} \mathrm{MR}^2 \omega^2 \frac{1}{2} \mathrm{M}(\mathrm{R} \omega)^2=\frac{3}{4} \mathrm{MR}^2 \omega^2$ $\%$ of $\mathrm{E}_{\mathrm{rot}}=\frac{E_{\text {rot }}}{E_{\text {Tot }}} \times 100 \%=33.33 \%$
Question 114 .
A sphere rolls down in an inclined plane without slipping. The percentage of transnational energy in its total energy is
(a) $29.6 \%$
(b) $33.4 \%$
(c) $71.4 \%$
(d) $50 \%$
Answer:
(c) $71.4 \%$
Rotational $\mathrm{K} . \mathrm{e} . \mathrm{E}_{\text {rot }}=$ $=\frac{1}{2} I \omega^2=\frac{1}{2}\left(\frac{2}{5} \mathrm{MR}^2\right) \omega^2=\frac{1}{5} \mathrm{MR}^2 \omega^2$
Translational K.E. $\left(\mathrm{E}_{\text {trans }}\right)=\frac{1}{2} \mathrm{M} v^2=\frac{1}{2} \mathrm{MR}^2 \omega^2$
$
\begin{aligned}
& \mathrm{E}_{\mathrm{Tot}}=\mathrm{E}_{\text {rot }}+\mathrm{E}_{\mathrm{trans}}=\frac{1}{5} \mathrm{MR}^2 \omega^2+\frac{1}{2} \mathrm{MR}^2 \omega^2=\frac{7}{10} \mathrm{MR}^2 \omega^2 \\
& \% \text { of } \mathrm{E}_{\mathrm{Trans}}=\frac{\mathrm{E}_{\mathrm{Trans}}}{\mathrm{E}_{\mathrm{Tot}}} \times 100 \%=\frac{5}{7} \times 100 \%=71.4 \%
\end{aligned}
$
Question 115.
Two blocks of masses $10 \mathrm{~kg}$ and $4 \mathrm{~kg}$ are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives a velocity of $14 \mathrm{~m} / \mathrm{s}$ to the heavier block in the direction of the lighter block. The velocity of the center of mass is -
(a) $30 \mathrm{~m} / \mathrm{s}$
(b) $20 \mathrm{~m} / \mathrm{s}$

(c) $10 \mathrm{~m} / \mathrm{s}$
(d) $5 \mathrm{~m} / \mathrm{s}$
Answer:
(c) According to law of conservation of linear momentum
$
\begin{aligned}
& \mathrm{MV}=(\mathrm{M}+\mathrm{M}) \mathrm{V}_{\mathrm{CM}} \\
& \mathrm{V}_{\mathrm{CM}}=\frac{M V}{M+M}=\frac{10 \times 10}{10+4}=10 \mathrm{~ms}^{-1}
\end{aligned}
$
Question 116.
A mass is whirled in a circular path with constant angular velocity and its angular momentum is $\mathrm{L}$. If the string is now halved keeping the angular velocity the same, the angular momentum is -
(a) $\frac{L}{4}$
(b) $\frac{L}{2}$
(c) $\mathrm{L}$
(d) $2 \mathrm{~L}$
Answer:
(a) $\frac{L}{4}$
We know that
angular momentum $\mathrm{L}=\mathrm{Mr}^2$
Here, $m$ and co are constants $L \alpha r^2$
If $r$ becomes $\frac{r}{2}$ angular momentum becomes $\frac{1}{4}$ th of its initial value.
Question 117.
The moment of inertia of a thin uniform ring of mass $1 \mathrm{~kg}$ and radius $20 \mathrm{~cm}$ rotating about the axis passing through the center and perpendicular to the plane of the ring is -
(a) $4 \times 10^{-2} \mathrm{~kg} \mathrm{~m}^2$
(b) $1 \times 10^{-2} \mathrm{~kg} \mathrm{~m}^2$
(c) $20 \times 10^{-2} \mathrm{~kg} \mathrm{~m}^2$
(d) $10 \times 10^{-2} \mathrm{~kg} \mathrm{~m}^2$
Answer:
(b) Moment of inertia $\mathrm{I}=\mathrm{MR}^2=1 \times\left(10 \times 10^{-2}\right)^2=1 \times 10^{-2} \mathrm{~kg} \mathrm{~m}^2$.

Question 118.
A solid sphere is rolling down in the inclined plane, from rest without slipping. The angle of inclination with horizontal is $30^{\circ}$. The linear acceleration of the sphere is -
(a) $28 \mathrm{~ms}^{-2}$
(b) $3.9 \mathrm{~ms}^{-2}$
(c) $\frac{25}{7} \mathrm{~ms}^{-2}$
(d) $\frac{1}{20} \mathrm{~ms}^{-2}$
Answer:
(c) $\frac{25}{7} \mathrm{~ms}^{-2}$
We know that, $\mathrm{a}=$
$
\begin{aligned}
&= \frac{g \sin \theta}{1+\frac{k^2}{\mathrm{R}^2}}=\frac{g \sin 30^{\circ}}{1+\frac{2}{5}} \\
& \because \frac{k^2}{\mathrm{R}^2} \text { for solid sphere }=\frac{5}{5} \\
& a= \frac{10 \times \frac{1}{2}}{\frac{7}{5}}=\frac{25}{7}=3.6 \mathrm{~ms}^{-2}
\end{aligned}
$
Question 119 .
An electron is revolving in an orbit of radius $2 \mathrm{~A}$ with a speed of $4 \times 10^5 \mathrm{~m} / \mathrm{s}$. The angular momentum of the electron is $\left[\mathrm{Me}=9 \times 10^{-31} \mathrm{~kg}\right]$
(a) $2 \times 10^{-35} \mathrm{~kg} \mathrm{~m}^2 \mathrm{~s}^{-1}$
(b) $72 \times 10^{-36} \mathrm{~kg} \mathrm{~m}^2 \mathrm{~s}^{-1}$
(c) $7.2 \times 10^{-34} \mathrm{~kg} \mathrm{~m}^2 \mathrm{~s}^{-1}$
(d) $0.72 \times 10^{-37} \mathrm{~kg} \mathrm{~m}^2 \mathrm{~s}^{-1}$

Answer:
(b) Angular momentum $\mathrm{L}=\mathrm{mV} \times \mathrm{r}=9 \times 10^{-31} \times 4 \times 10^5 \times 2 \times 10^{-10}=72 \times 10^{-36} \mathrm{~kg} \mathrm{~m}^2 \mathrm{~s}^{-1}$
Question 120 .
A raw egg and hard boiled egg are made to spin on a table with the same angular speed about the same axis. The ratio of the time taken by the eggs to stop is -
(a) $=1$
(b) $<1$
(c) $>1$
(d) none of these
Answer:
(d) When a raw egg spins, the fluid inside comes towards its side.
$\therefore$ " 1 " will increase in - turn it decreases $\omega$. Therefore it takes lesser time than boiled egg.
$
\therefore \frac{\text { timefirrawegg }}{\text { timeforboiledegg }}<1
$
Short Answer Questions (1 Mark)
Question 1.
What is a rigid body?
Answer:
A rigid body is the one which maintains its definite and fixed shape even when an external force acts on it.
Question 2.
When an object will have procession? Give one example.
Answer:
bout the axis will rotate the object about it and the torque perpendicular to the axis will turn the axis of rotation when both exist simultaneously on a rigid body the body will have a procession.
Example:
The spinning top when it is about to come to rest.
Question 3.
Define angular momentum. Give an expression for it.
Answer:
The angular momentum of a point mass is defined as the moment of its linear momentum. $\vec{L}=\vec{r} \times \vec{p}$ or $\mathrm{L}=\mathrm{rp} \sin \theta$
Question 4.
When an angular momentum of the object will be zero?
Answer:
If the straight path of the particle passes through the origin, then the angular momentum is zero, which is also a constant.

Question 5.
When an object be in-mechanical equilibrium?
Answer:
A rigid body is said to be in mechanical equilibrium when both its linear momentum and angular momentum remain constant.
Question 6.
Derive an expression for the power delivered by torque.
Answer:
Power delivered is the work done per unit time. IF we differentiate the expression for work done with respect to time, we get the instantaneous
power (P).
$
\begin{aligned}
& \mathrm{p}=\frac{d w}{d t}=\tau \frac{d \theta}{d t} \\
& \mathrm{p}=\tau \mathrm{d} \omega
\end{aligned}
$
Question 7.
A boy sits near the edge of revolving circular disc
1. What will be the change in the motion of a disc?
2. If the boy starts moving from edge to the center of the disc, what will happen?
Answer:
1. As we know $\mathrm{L}=\mathrm{I} \omega=$ constant if the boy sits on the edge of revolving disc, its $\mathrm{I}$ will be increased in turn it reduces angular velocity.
2. If the boy starts moving towards the center of the disc, its I will decrease in turn that increases its angular velocity.
Question 8 .
Are moment of inertia and radius of gyration of a body constant quantities?
Answer:
No, moment of inertia and radius of gyration depends on axis of rotation and also on the distribution of mass of the body about its axis..
Question 9.
A cat is able to land on its feet after a fall. Which principle of physics is being used? Explain.
Answer:
A cat is able to land on its feet after a fall. This is based on law of conservation of angular momentum. When the cat is about to fall, it curls its body to decrease the moment of inertia and increase its angular velocity. When it lands it stretches out its limbs. By which it increases its moment of inertia and inturn it decreases its angular velocity. Hence, the cat lands safety.
Question 10.
About which axis a uniform cube will have minimum moment of inertia?
Answer:
It will be about an axis passing through the center of the cube and connecting the opposite comers.

Question 11 .
State the principle of moments of rotational equilibrium.
Answer:
$
\sum=\bar{\tau}=0
$
Question 12.
Write down the moment of inertia of a disc of radius $\mathrm{R}$ and mass $\mathrm{m}$ about an axis in its plane at a distance $\mathrm{R} / 2$ from its center.
Answer:
$
\frac{1}{2} \mathrm{MR}^2
$
Question 13.
Can the couple acting on a rigid body produce translator motion?
Answer:
No. It can produce only rotatory motion.
Question 14 .
Which component of linear momentum does not contribute to angular momentum?
Answer:
Radial Component.
Question 15 .
A system is in stable equilibrium. What can we say about its potential energy?
Answer:
PE. is minimum.
Question 16.
Is radius of gyration a constant quantity?
Answer:
No, it changes with the position of axis of rotation.
Question 17.
Two solid spheres of the same mass are made of metals of different densities. Which of them has a large moment of inertia about the diameter?
Answer:

$
\mathrm{K}=\frac{\mathrm{L}^2}{2 \mathrm{I}} \Rightarrow \mathrm{K}_{\mathrm{A}}>\mathrm{K}_{\mathrm{A}}
$
Question 19.
A particle moves on a circular path with decreasing speed. What happens to its angular momentum?
Answer:
As $\vec{L}=\vec{r} \times \mathrm{m} \vec{v}$ i.e., $\vec{L}$ magnitude decreases but direction remains constant.
Question 20 .
What is the value of instantaneous speed of the point of contact during pure rolling ?
Answer:
Zero.
Question 21.
Which physical quantity is conserved when a planet revolves around the sun?
Answer:
Angular momentum of planet.
Question 22.
What is the value of torque on the planet due to the gravitational force of sun?
Answer:
Zero.
Question 23.
If no external torque acts on a body, will its angular velocity be constant?
Answer:
No.
Question 24.
Why there are two propellers in a helicopter ?
Answer:
Due to conservation of angular momentum.

Question 25.
A child sits stationary at one end of a long trolley moving uniformly with speed $\mathrm{V}$ on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, then what is the effect of the speed of the centre of mass of the (trolley + child) system?
Answer:
No change in speed of system as no external force is working.
Short Answer Questions (2 Marks)
Question 26.
State the factors on which the moment of inertia of a body depends.
Answer:
- Mass of body
- Size and shape of body
- Mass distribution w.r.t. axis of rotation
- Position and orientation of rotational axis
Question 27.
On what factors does radius of gyration of body depend?
Answer:
Mass distribution.
Question 28.
Why the speed of whirl wind in a Tornado is alarmingly high?
Answer:
In this, air from nearly regions get concentrated in a small space, so I decreases considerably. Since $\mathrm{I} \omega=$ constant so $\omega$ increases so high.
Question 29.
Can a body be in equilibrium while in motion? If yes, give an example.
Answer:
Yes, if body has no linear and angular acceleration then a body in uniform straight line of motion will be in equilibrium.

Question 30 .
There is a stick half of which is wooden and half is of steel, (i) it is pivoted at the wooden end and a force is applied at the steel end at right angle to its length (ii) it is pivoted at the steel end and the same force is applied at the wooden end. In which case is the angular acceleration more and why?
Answer:
I (first case) $>1$ (Second case)
$\therefore \tau \mathrm{r}=1 \alpha$
$\Rightarrow \alpha$ (first case) $<\alpha$ (second case)
Question 31.
If earth contracts to half of its present radius what would be the length of the day at equator? Answer:
$
\begin{aligned}
\mathrm{I}_1 & =\frac{2}{5} \mathrm{MR}^2 \Rightarrow \mathrm{I}_2=\frac{2}{5} \mathrm{M}\left(\frac{\mathrm{R}}{2}\right)^2 \Rightarrow \mathrm{I}_2=\frac{\mathrm{I}_1}{4} \\
\mathrm{~L} & =\mathrm{I}_1 \omega_1=\mathrm{I}_2 \omega_2
\end{aligned}
$
or
$I\left(\frac{2 \pi}{T_1}\right)=\frac{I}{4}\left(\frac{2 \pi}{T_2}\right) \quad$ or $\quad T_2=\frac{T_1}{4}=\frac{24}{4}=6$ hours
Question 32.
An internal force cannot change the state of motion of center of mass of a body. Flow does the internal force of the brakes bring a vehicle to rest?
Answer:
In this case the force which bring the vehicle to rest is friction, and it is an external force.
Question 33 .
When does a rigid body said to be in equilibrium? State the necessary condition for a body to be in equilibrium.
Answer:
For translation equilibrium

$
\sum F_{\mathrm{ext}}=0
$
For rotational equilibrium
$
\sum \bar{\tau}_{\mathrm{ext}}=0
$
Question 34.
How will you distinguish between a hard boiled egg and a raw egg by spinning it on a table top' Answer:
For same external torque, angular acceleration of raw egg will be small than that of Hard boiled egg.
Question 35.
Equal torques are applied on a cylinder and a sphere. Both have same mass and radius. Cylinder rotates about its axis and sphere rotates about one of its diameter. Which will acquire greater speed and why?
Answer:
$
\tau=\mathrm{I} \alpha \alpha=\frac{\tau}{I}
$
$\alpha$ in cylinder, $\alpha_C=\frac{\tau}{I_C}$
$\alpha$ in sphere, $\alpha_S=\frac{\tau}{I_S}$
$
\frac{\alpha_{\mathrm{C}}}{\alpha_{\mathrm{S}}}=\frac{\mathrm{I}_{\mathrm{S}}}{\mathrm{I}_{\mathrm{C}}}=\frac{\frac{2}{5} \mathrm{MR}^2}{\mathrm{MR}^2}=\frac{2}{5}
$
Question 36.
In which condition a body lying in gravitational field is in stable equilibrium?
Answer:
When vertical line through center of gravity passes through the base of the body.
Question 37.
Give the physical significance of moment of inertia. Explain the need of fly wheel in Engine.
Answer:
It plays the same role in rotatory motion as the mass does in translator y motion.
Short Answer Questions (3 Marks)
Question 38.
Three mass point $m_1, m_2, m_3$ are located at the vertices of equilateral $\mathrm{A}$ of side ' $a$ '. What is the moment of inertia of system about an axis along the altitude of A passing through mi?

Answer:
$
\begin{aligned}
\mathbf{I} & =\sum_{i=1}^n m_i r_i^2 \\
& =m_1 \times 0+m_2 \times(\mathrm{BD})^2+m_3 \times(\mathrm{DC})^2 \\
& =0+m_2\left(\frac{a}{2}\right)^2+m_3\left(\frac{a}{2}\right)^2 \\
\mathrm{I} & =\frac{1}{4}\left(m_2+m_3\right) a^2
\end{aligned}
$
Question 39.
A disc rotating about its axis with angular speed $\omega_0$ is placed lightly (without any linear push) on a perfectly friction less table. The radius of the disc is $R$. What are the linear velocities of the points $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ on the disc shown in figure. Will the disc roll?
Answer:

For $A V_A=\mathrm{R} \omega_0$ in forward direction
For $B=V_B=R \omega_0$ in backward direction $R$
For $\mathrm{C}, \mathrm{V}_{\mathrm{C}}=\frac{R}{2} \omega_0$ in forward direction disc will not roll.
Question 40 .
Find the torque of a force $7 \hat{i}-3 \hat{j}-5 \hat{k}$ about the origin which acts on a particle whose position vector is $\hat{j}+\hat{j}-\hat{j}$
Answer:
$
\vec{\tau}=\vec{r} \times \overrightarrow{\mathrm{F}}=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 1 & -1 \\
7 & -3 & -5
\end{array}\right|=-8 \hat{i}-2 \hat{j}-10 \hat{k}
$