Objective Physics : NEET 2020 Examination textbook Gurukul Oswal

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Chapter 5

SYSTEM OF PARTICLES AND ROTATIONAL MOTION - all with Video Answers

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Section 1

System of Particles and Rotational Motion

Problem 1

The centre of mass of a body:
(a) lies always outside the body
(b) may lie within, outside or on the surface of the body
(c) lies always inside the body
(d) lies always on the surface of the body

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Narayan Hari

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Problem 2

Three identical spheres, each of mass $1 \mathrm{~kg}$ are kept as shown in figure, touching each other, with their centres on a straight line. If their centres are marked $P$, $Q$, and $R$ respectively, the distance of centre of mass of the system from $P$ is:
(a) $\frac{P Q+P R+Q R}{3}$
(b) $\frac{P Q+P R}{3}$
(c) $\frac{P Q+Q R}{3}$
(d) $\frac{P R+Q R}{3}$

Narayan Hari

Numerade Educator

Problem 3

Two identical particles are located at $\vec{x}$ and $\vec{y}$ with reference to the origin of three-dimensional coordinate system. The position vector of centre of mass of
the system is given by:
$\begin{array}{ll}\text { (a) } \vec{x}-\vec{y} & \text { (b) } \frac{\vec{x}+\vec{y}}{2}\end{array}$
(c) $(\vec{x}-\vec{y})$
(d) $\frac{\vec{x}-\vec{y}}{2}$

Narayan Hari

Numerade Educator

Problem 4

The centre of mass of three bodies each of mass $1 \mathrm{~kg}$ located at the points $(0,0),(3,0)$, and $(0,4)$ in the $X Y$ plane is:
(a) $\left(\frac{4}{3}, 1\right)$
(b) $\left(\frac{1}{3}, \frac{2}{3}\right)$
(c) $\left(\frac{1}{2}, \frac{1}{2}\right)$
(d) $\left(1, \frac{4}{3}\right)$

Narayan Hari

Numerade Educator

Problem 5

Two spheres $A$ and $B$ of masses $m$ and $2 m$ and radii $2 R$ and $R$ respectively are placed in contact as shown. The CM of the system lies:
(a) inside $A$
(b) inside $B$
(c) at the point of contact
(d) None of these

Narayan Hari

Numerade Educator

Problem 6

The motion of a rigid body which is not pivoted or fixed in some way is either a pure ___A___ or a combination of translation and rotation. The motion of a rigid body which is pivoted or fixed in some way is ___B___ Here, A and B refer to:
(a) rotation and translation
(b) translation and rotation
(c) translation and the combination of rotation and translation
(d) None of the above

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Problem 7

In rotation of a rigid body about a fixed axis, every ___A___ of the body moves in a ___B___, which lies in a plane ___C___ to the axis and has its centre on
the axis. Here, A, B, and C refer to:
(a) particle, perpendicular, and circle
(b) circle, particle, and perpendicular
(c) particle, circle, and perpendicular
(d) particle, perpendicular, and perpendicular

Narayan Hari

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Problem 8

Consider the following statements and choose for correct option.
I. Position vector of centre of mass of two particles of equal mass is equal to the position vector of either particle.
II. Centre of mass is always at the mid-point of the line joining two particles.
III. Centre of mass of a body can lie where there is no mass.
(a) I and II
(b) II only
(c) III only
(d) I, II, and III

Dheeraj Sharma

Numerade Educator

Problem 9

The motion of binary stars, $S_{1}$, and $S_{2}$ is the combination of $X_{ }$ and $Y_{-}$ Here, $X$ and $Y$ refer to:
(a) motion of the CM and motion about the $\mathrm{CM}$
(b) motion about the CM and motion of one star
(c) position of the CM and motion of the $\mathrm{CM}$
(d) motion about CM and position of one star

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 10

Three masses are placed on the $x$ -axis: $300 \mathrm{~g}$ at origin, $500 \mathrm{~g}$ at $x=40 \mathrm{~cm}$ and $400 \mathrm{~g}$ at $x=70 \mathrm{~cm}$. The distance of the centre of mass from the origin is:
(a) $40 \mathrm{~cm}$
(b) $45 \mathrm{~cm}$
(c) $50 \mathrm{~cm}$
(d) $30 \mathrm{~cm}$

Narayan Hari

Numerade Educator

Problem 11

A body $A$ of mass $M$ while falling vertically downwards under gravity breaks into two parts; a body $B$ of mass $M / 3$ and a body $C$ of mass $2 / 3 M$. The centre of mass of bodies $B$ and $C$ taken together shifts compared to that of body $A:$
(a) does not shift
(b) depends on height of breaking
(c) towards body $B$
(d) towards body $C$

Narayan Hari

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Problem 12

A system consists of three particles, each of mass $m$ and located at $(1,1),(2,2)$, and $(3,3) .$ The coordinates of the centre of mass are:
(a) $(1,1)$
(b) $(2,2)$
(c) $(3,3)$
(d) $(6,6)$

Narayan Hari

Numerade Educator

Problem 13

Position vector of centre of mass of two particles system is given by:
(a) $\vec{R}=\frac{m_{1} \vec{r}_{1}-m_{2} \vec{r}_{2}}{m_{1}+m_{2}}$
(b) $\vec{R}=\frac{m_{1} \vec{r}_{1} \cdot m_{2} \vec{r}_{2}}{\vec{r}_{1}+\vec{r}_{2}}$
(c) $\vec{R}=\frac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{\vec{r}_{1}+\vec{r}_{2}}$
(d) $\vec{R}=\frac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}}{m_{1}+m_{2}}$

Dheeraj Sharma

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Problem 14

The position of centre of mass of a system of particles does not depend upon the:
(a) mass of particles
(b) symmetry of the body
(c) position of the particles
(d) relative distance between the particles

Narayan Hari

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Problem 15

The mass per unit length of a non-uniform rod of length $L$ varies $m=\lambda x$ where $\lambda$ is constant. The centre of mass of the rod will be at:
(a) $\frac{2}{3} L$
(b) $\frac{3}{2} L$
(c) $\frac{1}{2} L$
(d) $\frac{4}{3} L$

Narayan Hari

Numerade Educator

Problem 16

The motion of the centre of mass depends on:
(a) total external forces (b) total internal forces
(c) Sum of (a) and (b)
(d) None of these

Narayan Hari

Numerade Educator

Problem 17

A pulley fixed to the ceiling carries a string with blocks of mass $m$ and $3 m$ attached to its ends. The masses of string and pulley are negligible. When the system is released, its centre of mass moves with what acceleration?
(a) 0
(b) $-\frac{g}{4}$
(c) $\frac{\mathrm{g}}{2}$
(d) $-\frac{\mathrm{g}}{2}$

Narayan Hari

Numerade Educator

Problem 18

In rotatory motion, linear velocities of all the particles of the body are:
(a) same
(b) different
(c) zero
(d) Cannot say

Narayan Hari

Numerade Educator

Problem 19

A wheel of moment of inertia $2.5 \mathrm{~kg} \mathrm{~m}^{2}$ has an initial angular velocity of $40 \mathrm{rad} / \mathrm{s}$. A constant torque of $10 \mathrm{~N} \mathrm{~m}$ acts on the wheel. The time during which the wheel is accelerated to $60 \mathrm{rad} / \mathrm{s}^{2}$ is:
(a) $4.4 \mathrm{~s}$
(b) $6 \mathrm{~s}$
(c) $5 \mathrm{~s}$
(d) $2.5 \mathrm{~s}$

Narayan Hari

Numerade Educator

Problem 20

A rod PQ of mass $M$ and length $L$ is hinged at end
$P$. The rod kept horizontal by a massless string tied to point $Q$ as shown in the figure. When string is cut, the initial angular acceleration of the $\operatorname{rod}$ is:
(a) $\frac{3 g}{2 L}$
(b) $\frac{\mathrm{g}}{L}$
(c) $\frac{2 g}{L}$
(d) $\frac{2 g}{3 L}$

Narayan Hari

Numerade Educator

Problem 21

A solid cylinder of mass $50 \mathrm{~kg}$ and radius $0.5 \mathrm{~m}$ is, free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 revolution $\mathrm{s}^{-2}$ is:
(a) $25 \mathrm{~N}$
(b) $50 \mathrm{~N}$
(c) $78.5 \mathrm{~N}$
(d) $157 \mathrm{~N}$

Narayan Hari

Numerade Educator

Problem 22

A wheel rotates with a constant acceleration of $2.0 \mathrm{rad} / \mathrm{s}^{2}$. If the wheel starts from rest, the number of revolutions it makes in first 10 seconds is:
(a) 8
(b) 16
(c) 24
(d) 32

Narayan Hari

Numerade Educator

Problem 23

A rigid body rotates about a fixed axis with variable angular velocity equal to $(a-b t)$ at time $t$ where $a$ and $b$ are constants. The angle through which it rotates before it comes to rest is:
(a) $\frac{a^{2}}{b}$
(b) $\frac{a^{2}}{2 b}$
(c) $\frac{a^{2}}{4 b}$
(d) $\frac{a^{2}}{2 b^{2}}$

Narayan Hari

Numerade Educator

Problem 24

A round disc of moment of inertia $I_{2}$ about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia $I_{1}$ rotating with an angular velocity $\omega$ about the same axis. The final angular velocity of the combination of disc is:
(a) $\frac{I_{2} \omega}{I_{1}+I_{2}}$
(b) $\omega$
(c) $\frac{I_{1} \omega}{I_{1}+I_{2}}$
(d) $\frac{\left(I_{1}+I_{2}\right) \omega}{I_{1}}$

Dheeraj Sharma

Numerade Educator

Problem 25

An athlete throws a discus from rest to a final angular velocity of $15 \mathrm{rad} / \mathrm{s}$ in $0.270 \mathrm{~s}$ before releasing it. During acceleration, discus moves a circular arc of radius $0.810 \mathrm{~m}$. Acceleration of discus before it is released is:
(a) $45 \mathrm{~m} / \mathrm{s}^{2}$
(b) $182 \mathrm{~m} / \mathrm{s}^{2}$
(c) $187 \mathrm{~m} / \mathrm{s}^{2}$
(d) $192 \mathrm{~m} / \mathrm{s}^{2}$

Narayan Hari

Numerade Educator

Problem 26

Two bodies of masses $2 \mathrm{~kg}$ and $4 \mathrm{~kg}$ are moving with velocities $2 \mathrm{~m} / \mathrm{s}$ and $10 \mathrm{~m} / \mathrm{s}$ respectively along same direction. Then the velocity of their centre of mass will be:
(a) $8.1 \mathrm{~m} / \mathrm{s}$
(b) $7.3 \mathrm{~m} / \mathrm{s}$
(c) $6.4 \mathrm{~m} / \mathrm{s}$
(d) $5.3 \mathrm{~m} / \mathrm{s}$

Narayan Hari

Numerade Educator

Problem 27

In the figure shown, $\mathrm{ABC}$ is a uniform wire. If centre of mass of wire lies vertically below point $\mathrm{A}$, then $\frac{\mathrm{BC}}{\mathrm{AB}}$ is close to:
(a) $1.85$
(b) $1.5$
(c) $1.37$
(d) 3

Narayan Hari

Numerade Educator

Problem 28

A circular disc of radius $R$ is removed from a bigger circular disc of radius $2 R$ such that the circumferences of the discs coincide. The centre of mass of the new disc is $\alpha / R$ from the centre of the bigger disc. The value of $\alpha$ is:
(a) $1 / 4$
(b) $1 / 3$
(c) $1 / 2$
(d) $1 / 6$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

Which of the following is incorrect?
(a) $\vec{v}=\vec{\omega} \times \vec{r}$
(b) $\vec{a}=\vec{v} \times \vec{r}$
(c) $a=\frac{d \omega}{d t}$
(d) None of these

Narayan Hari

Numerade Educator

Problem 30

Concrete mixture is made by mixing cement, stone and sand in a rotating cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute (rpm) to ensure proper mixing is close to (take the radius of the drum to be $1.25 \mathrm{~m}$ and its axle to be horizontal):
(a) 27
(b) $0.4$
(c) $1.3$
(d) 8

Narayan Hari

Numerade Educator

Problem 31

Angular velocity of each particle of a rotating rigid body about axis of rotation is:
(a) same
(b) different
(c) depends on relative position
(d) None of these

Narayan Hari

Numerade Educator

Problem 32

A wheel rotates with a constant acceleration of $4.0 \mathrm{rad} / \mathrm{s}^{2}$. If the wheel starts from rest, the number of revolutions it makes in the first 10 seconds will be approximately:
(a) 8
(b) 31
(c) 24
(d) 32

Narayan Hari

Numerade Educator

Problem 33

A cord is wound over the rim of a flywheel of mass $20 \mathrm{~kg}$ and radius $25 \mathrm{~cm}$. A mass $2.5 \mathrm{~kg}$ attached to the cord can fall under gravity. The angular acceleration of the flywheel is:
(a) $25 \mathrm{rad} / \mathrm{s}^{2}$
(b) $20 \mathrm{rad} / \mathrm{s}^{2}$
(c) $10 \mathrm{rad} / \mathrm{s}^{2}$
(d) $5 \mathrm{rad} / \mathrm{s}^{2}$

Dheeraj Sharma

Numerade Educator

Problem 34

Initial angular velocity of a circular disc of mass $M$ is $\omega_{1} .$ Then, two small spheres of $m$ are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc?
(a) $\left(\frac{M+m}{M}\right) \omega_{1}$
(b) $\left(\frac{M+m}{m}\right) \omega_{1}$
(c) $\left(\frac{M}{M+4 m}\right) \omega_{1}$
(d) $\left(\frac{M}{M+2 m}\right) \omega_{1}$

Narayan Hari

Numerade Educator

Problem 35

A rope of negligible mass is wound around a hollow cylinder of mass $3 \mathrm{~kg}$ and radius $40 \mathrm{~cm} .$ What is the angular acceleration of the cylinder, if the rope is pulled with a force of $30 \mathrm{~N}$ ? Assume that there is no slipping:
(a) $10 \mathrm{rad} / \mathrm{s}^{2}$
(b) $15 \mathrm{rad} / \mathrm{s}^{2}$
(c) $20 \mathrm{rad} / \mathrm{s}^{2}$
(d) $25 \mathrm{rad} / \mathrm{s}^{2}$

Narayan Hari

Numerade Educator

Problem 36

When the disc rotates with uniform angular velocity. Which of the following is not true?
(a) The sense of rotation remains same.
(b) The orientation of the axis of rotation remains same.
(c) The speed of rotation is non-zero and remains same.
(d) The angular acceleration is non-zero and remains same.

Narayan Hari

Numerade Educator

Problem 37

A flywheel of moment of inertia $3 \times 10^{2} \mathrm{kgm}^{2}$ is rotating with uniform angular speed of $4.6 \mathrm{rad} / \mathrm{s}$. If a torque of $6.9 \times 10^{2} \mathrm{~N} \mathrm{~m}$ retards the wheel, then the time in which the wheel comes to rest is:
(a) $1.5 \mathrm{~s}$
(b) $2 \mathrm{~s}$
(c) $0.5 \mathrm{~s}$
(d) $1 \mathrm{~s}$

Narayan Hari

Numerade Educator

Problem 38

A thin circular ring of mass $m$ and radius $r$ is rotating about its axis with a constant angular momentum, $\omega$. Four objects each of mass, $m$ are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be:
(a) $\left(\frac{M \omega}{M+4 m}\right)$
(b) $\frac{(M+4 m) \omega}{M}$
(c) $\frac{(M-4 m) \omega}{M+4 m}$
(d) $\left(\frac{M \omega}{4 m}\right)$

Narayan Hari

Numerade Educator

Problem 39

If the frequency of the rotating platform is $v$ and the distance of a boy from the centre is $r$, what is the area swept out per second by the line connecting the boy to the centre?
(a) $\pi r v$
(b) $2 \pi r v$
(c) $\pi r^{2} v$
(d) $2 \pi r^{2} v$

Narayan Hari

Numerade Educator

Problem 40

A thin and circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity $\omega$. If another disc of same dimensions but of mass $M / 4$ is placed gently on the first disc coaxially, then the new angular velocity of the system is:
(a) $5 \omega / 4$
(b) $2 \omega / 3$
(c) $4 \omega / 5$
(d) $3 \omega / 2$

Narayan Hari

Numerade Educator

Problem 41

A uniform rod of length $l$ and mass $\mathrm{m}$ is free to rotate in a vertical plane about $A$. the rod initially in horizontal position is released. The initial angular acceleration of rod is:
(a) $\frac{3 \mathrm{~g}}{2 l}$
(b) $\frac{2 l}{3 g}$
(c) $\frac{3 g}{2 l^{2}}$
(d) $m g \frac{l}{2}$

Narayan Hari

Numerade Educator

Problem 42

When a ceiling fan is switched on, it makes 10 revolutions in the first $3 \mathrm{~s}$. Assuming a uniform angular acceleration, how many rotations it will make in the next $3 \mathrm{~s} ?$
(a) 10
(b) 20
(c) 30
(d) 40

Dheeraj Sharma

Numerade Educator

Problem 43

A uniform rod of mass $M$ and length $L$ is free to rotate in $X Y$ -plane, i.e., about $y$ -axis. If a force of $F=(3 i+2 j+6 k) \mathrm{N}$ is acting on $(L / 2,0,0)$ in the situation as shown in figure. The angular acceleration of rod is: (Take, $M=6 \mathrm{~kg}, L=4 \mathrm{~m}$ )
(a) $-\frac{3}{2} \hat{i}+\frac{1}{2} \hat{k}$
(b) $-\frac{3}{2} \hat{j}$
(c) $\frac{1}{2} \hat{k}$
(d) $4 \hat{j}$

Narayan Hari

Numerade Educator

Problem 44

A tube of length $L$ is filled completely with an incompressible liquid of mass $m$ and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity, $\omega$. The force exerted by the liquid at the other end is:
(a) $M \omega^{2} L / 2$
(b) $M \omega^{2} L$
(c) $M \omega^{2} L / 4$
(d) $M \omega^{2} L^{2} / 2$

Narayan Hari

Numerade Educator

Problem 45

If $2 \mathrm{~kg}$ mass is rotating on a circular path of radius $0.8 \mathrm{~m}$ with angular velocity of $44 \mathrm{rad} / \mathrm{sec}$. If radius of the path becomes $1 \mathrm{~m}$, then what will be the value of angular velocity?
(a) $28.16 \mathrm{rad} / \mathrm{sec}$
(b) $19.28 \mathrm{rad} / \mathrm{sec}$
(c) $8.12 \mathrm{rad} / \mathrm{sec}$
(d) $35.26 \mathrm{rad} / \mathrm{sec}$

Dheeraj Sharma

Numerade Educator

Problem 46

Angular momentum of a body is defined as the product of:
(a) mass and angular velocity
(b) centripetal force and radius
(c) linear velocity and angular velocity
(d) moment of inertia and angular velocity

Dheeraj Sharma

Numerade Educator

Problem 47

The drive shaft of an automobile rotates at 3600 rpm and transmits $80 \mathrm{HP}$ up from the engine to the rear wheels. The torque developed by the engine is:
(a) $16.58 \mathrm{Nm}$
(b) $0.022 \mathrm{Nm}$
(c) $158.31 \mathrm{Nm}$
(d) $141.6 \mathrm{Nm}$

Narayan Hari

Numerade Educator

Problem 48

A particle of mass $1 \mathrm{~kg}$ is moving along the line $y=x+2$ (here $x$ and $y$ are in metres) with speed 2 $\mathrm{m} / \mathrm{s}$. The magnitude of angular momentum of particle about origin is:
(a) $4 \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-1}$
(b) $2 \sqrt{2} \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-1}$
(c) $4 \sqrt{2} \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-1}$
(d) $2 \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-1}$

Dheeraj Sharma

Numerade Educator

Problem 49

In an orbital motion, the angular momentum vector is:
(a) along the radius vector
(b) parallel to the linear momentum
(c) in the orbital plane
(d) perpendicular to the orbital plane

Dheeraj Sharma

Numerade Educator

Problem 50

A rigid horizontal smooth rod $A B$ of mass $0.75 \mathrm{~kg}$ and length $40 \mathrm{~cm}$ can rotate freely about a fixed vertical axis through its mid-point $\mathrm{O}$. Two rings each of mass $1 \mathrm{~kg}$ initially at rest are placed at a distance of $10 \mathrm{~cm}$ from $\mathrm{O}$ on either side of the rod. The rod is set in rotation with an angular velocity of 30 radian per sec and when the rings reach the ends of the rod, the angular velocity in $\mathrm{rad} / \mathrm{sec}$ is:
(a) 5
(b) 10
(c) 15
(d) 20

Narayan Hari

Numerade Educator

Problem 51

A couple consisting of two forces $F_{1}$ and $F_{2}$ each equal to $5 \mathrm{~N}$ is acting at the rim of a disc of mass 2 $\mathrm{kg}$ and radius $1 / 2 \mathrm{~m}$ for $5 \mathrm{sec}$. Initially, the disc is at rest, the final angular momentum of the disc is:
(a) 15
(b) 20
(c) 25
(d) 30

Narayan Hari

Numerade Educator

Problem 52

A particle is moving along a straight line parallel to $x$ -axis with constant velocity. Its angular momentum about the origin:
(a) decreases with time
(b) increases with time
(c) remains constant
(d) is zero

Dheeraj Sharma

Numerade Educator

Problem 53

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?
(a) Moment of inertia
(b) Angular momentum
(c) Angular velocity
(d) Rotational kinetic energy

Vysakh M

Numerade Educator

Problem 54

The radius vector and linear momentum are respectively given by vectors $2 \hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}-3 \hat{j}+\hat{k}$. Then the angular momentum is:
(a) $2 i-4 \hat{k}$
(b) $4 \hat{i}-8 \hat{k}$
(c) $2 \hat{i}-4 \hat{j}+2 k$
(d) $4 i-8 j$

Narayan Hari

Numerade Educator

Problem 55

A mass is whirled in a circular path with constant angular velocity and its angular momentum is $L$. If the string is now halved keeping the angular velocity the same, the angular momentum is:
(a) $L / 4$
(b) $L / 2$
(c) $\bar{L}$
(d) $2 L$

Narayan Hari

Numerade Educator

Problem 56

The position of a particle is given $\hat{j}$ by $\vec{r}=\hat{i}+2 \hat{j}-\hat{k}$ and its linear momentum is given by $\vec{p}=3 \hat{i}+4 \hat{j}-2 \hat{k}$. Then its angular momentum, about the origin is perpendicular to:
(a) $y$ z-plane
(b) $z$ -axis
(c) $y$ -axis
(d) $x$ -axis

Dheeraj Sharma

Numerade Educator

Problem 57

If $r$ denotes the distance between the Sun and the Earth, then the angular momentum of the Earth around the Sun is proportional to:
(a) $r^{3} / r$
(b) $r$
(c) $\sqrt{r}$
(d) $r^{2}$

Narayan Hari

Numerade Educator

Problem 58

If $I$ is the moment of inertia and $E$ is the kinetic energy of rotation of a body, then its angular momentum will be:
(a) $\sqrt{(E I)}$
(b) $2 E I$
(c) $E / I$
(d) $\sqrt{(2 E I)}$

Narayan Hari

Numerade Educator

Problem 59

A particle of mass $\mathrm{m}$ in the XY-plane with a velocity $v$ along the straight line AB. If the angular momentum of the particle with respect to origin $\mathrm{O}$ is $L_{\mathrm{A}}$ when it is at B, then:
(a) $L_{A}>L_{B}$
(b) $L_{\Lambda}=L_{B}$
(c) The relationship between $L_{A}$ and $L_{B}$ depends upon the slope of the line $\mathrm{AB}$
(d) $L_{\Lambda}<L_{B}$

Narayan Hari

Numerade Educator

Problem 60

Total angular momentum of a rotating body remains constant, if the net torque acting on the body is:
(a) zero
(b) maximum
(c) minimum
(d) unity

Dheeraj Sharma

Numerade Educator

Problem 61

Four equal and parallel forces are acting on a rod of length $100 \mathrm{~cm}$, as shown in figure, at distances of $20 \mathrm{~cm}, 40 \mathrm{~cm}, 60 \mathrm{~cm}$, and $80 \mathrm{~cm}$ respectively from one end of the rod. Under the influence of these forces, the rod: (neglecting its weight)
(a) experiences no torque
(b) experiences torque
(c) experiences a linear motion
(d) experiences torque and also a linear motion

Narayan Hari

Numerade Educator

Problem 62

A uniform horizontal metre scale of mass $m$ is suspended by two vertical strings attached to its two ends. A body of mass $2 m$ is placed on the $75 \mathrm{~cm}$ mark. The tensions in the two strings are in the ratio is:
(a) $1: 2$
(b) $1: 3$
(c) $2: 3$
(d) $3: 4$

Narayan Hari

Numerade Educator

Problem 63

A metre stick is balanced on a knife edge at its centre. When two coins, each of mass $5 \mathrm{~g}$ are put one on top of the other at the $12.0 \mathrm{~cm}$ mark, the stick is found to be balanced at $45.0 \mathrm{~cm}$. What is the mass of the metre stick?
(a) $56 \mathrm{~g}$
(b) $66 \mathrm{~g}$
(c) $76 \mathrm{~g}$
(d) $86 \mathrm{~g}$

Dheeraj Sharma

Numerade Educator

Problem 64

A rigid rod of length $2 L$ is acted upon by some forces. All forces labelled $F$ have the same magnitude. Which cases have a non-zero net torque acting on the rod about its centre?
(a) I and II only
(b) II and III only
(c) I and III only
(d) The net torque is zero in all cases

Dheeraj Sharma

Numerade Educator

Problem 65

The moment of inertia of a ___A___ body about an axis ___B___ to its plane is equal to the sum of its moments of inertia about two ___C___ axes concurrent with perpendicular axis and lying in the plane of the body. Here, A, B, and C refer to:
(a) three dimensional, perpendicular, and perpendicular
(b) planar, perpendicular, and parallel
(c) planar, perpendicular, and perpendicular
(d) three dimensional, parallel, and perpendicular

Dheeraj Sharma

Numerade Educator

Problem 66

A thin rod of length $l$ and mass $m$ is bent at midpoint $O$ at angle of $60^{\circ}$. The moment of inertia of the rod about an axis passing through $O$ and perpendicular to the plane of the rod will be:
(a) $\frac{m l^{2}}{3}$
(b) $\frac{m l^{2}}{6}$
(c) $\frac{m l^{2}}{8}$
(d) $\frac{m l^{2}}{12}$

Dheeraj Sharma

Numerade Educator

Problem 67

Three-point masses, each of mass, $m$ are placed at the corner of an equilateral triangle of side, $l$. Then the moment of inertia of this system about an axis along one side of the triangle is:
(a) $3 m l^{2}$
(b) $m l^{2}$
(c) $\frac{3}{4} m l^{2}$
(d) $\frac{3}{2} m l^{2}$

Narayan Hari

Numerade Educator

Problem 68

The moment of inertia of a uniform circular disc of radius, $R$ and mass, $M$ about an axis passing from the edge of the disc and normal to the disc is:
(a) $M R^{2}$
(b) $\frac{1}{2} M R^{2}$
(c) $\frac{3}{2} M R^{2}$
(d) $\frac{7}{2} M R^{2}$

Dheeraj Sharma

Numerade Educator

Problem 69

Moment of inertia of a hollow cylinder of mass $M$ and radius $r$ about its own axis is:
(a) $\frac{2}{3} M r^{2}$
(b) $\frac{2}{5} M r^{2}$
(c) $\frac{1}{3} M r^{2}$
(d) $\mathrm{Mr}^{2}$

Narayan Hari

Numerade Educator

Problem 70

A constant torque of $3.14 \mathrm{~N} \mathrm{~m}$ is exerted on a pivoted wheel. If the angular acceleration of the wheel is $4 \pi \mathrm{rad} / \mathrm{s}^{2}$, then the moment of inertia of the wheel is:
(a) $0.25 \mathrm{~kg} \mathrm{~m}^{2}$
(b) $2.5 \mathrm{~kg} \mathrm{~m}^{2}$
(c) $4.5 \mathrm{~kg} \mathrm{~m}^{2}$
(d) $25 \mathrm{~kg} \mathrm{~m}^{2}$

Narayan Hari

Numerade Educator

Problem 71

Which of the following has the highest moment of inertia when each of them has the same mass and the same outer radius:
(a) a ring about its axis, perpendicular to the plane of the ring.
(b) a disc about its axis, perpendicular to the plane of the disc.
(c) a solid sphere about one of its diameters.
(d) a spherical shell about one of its diameters.

Narayan Hari

Numerade Educator

Problem 72

The moment of inertia of a thin uniform rod of mass $M$ and length $l$ about an axis perpendicular to the rod through its centre is $I$. The moment of inertia of the rod thyough its end point is:
(a) $\frac{1}{4}$
(b) $\frac{I}{2}$
(c) $2 I$
(d) $4 I$

Dheeraj Sharma

Numerade Educator

Problem 73

The correct relation between moment of inertia $I$, radius of gyration $K$ and mass $M$ of the body is:
(a) $K=I^{2} M$
(b) $K=I M^{2}$
(c) $K=\sqrt{\frac{M}{I}}$
(d) $K=\sqrt{\frac{I}{M}}$

Dheeraj Sharma

Numerade Educator

Problem 74

The radius of gyration of a uniform rod of length $L$ about an axis passing through its centre of mass is:
(a) $\frac{L}{\sqrt{12}}$
(b) $\frac{L^{2}}{\sqrt{12}}$
(c) $\frac{L}{\sqrt{3}}$
(d) $\frac{L}{\sqrt{2}}$

Narayan Hari

Numerade Educator

Problem 75

A thin wire of length $l$ and mass $m$ is bent in the form of a semicircle as shown in the figure. Its moment of inertia about an axis joining its free ends will be:
(a) $\mathrm{ml}^{2}$
(b) 0
(c) $m l^{2} / \pi^{2}$
(d) $m l^{2} / 2 \pi^{2}$

Narayan Hari

Numerade Educator

Problem 76

If two circular discs $A$ and $B$ are of same mass but of radii $r$ and $2 r$ respectively, then the moment of inertia of $A$ is:
(a) the same as that of $B$.
(b) twice that of $B$.
(c) four times that of $B$.
(d) $1 / 4$ that of $B$.

Narayan Hari

Numerade Educator

Problem 77

For a given mass and size, moment of inertia of a solid disc is:
(a) more than that of a ring
(b) less than that of a ring
(c) equal to that of a ring
(d) depend on the material of ring and disc

Dheeraj Sharma

Numerade Educator

Problem 78

A solid cylinder of mass $20 \mathrm{~kg}$ rotates about its axis with angular speed $100 \mathrm{rad} / \mathrm{s}$. The radius of the cylinder is $0.25 \mathrm{~m}$. The kinetic energy associated with the rotation of the cylinder is:
(a) $3025 \mathrm{~J}$
(b) $3225 \mathrm{~J}$
(c) $3250 \mathrm{~J}$
(d) $3125 \mathrm{~J}$

Dheeraj Sharma

Numerade Educator

Problem 79

A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is $K$. The child now stretches his arms so that the moment of inertia of the system becomes doubled. The kinetic energy of the system now is:
(a) $2 K$
(b) $\mathrm{K} / 2$
(c) $K / 4$
(d) $4 K$

Dheeraj Sharma

Numerade Educator

Problem 80

A circular disc rolls down an inclined plane. The ratio of rotational kinetic energy to total kinetic energy is:
(a) $1 / 2$
(b) $1 / 3$
(c) $2 / 3$
(d) $3 / 4$

Dheeraj Sharma

Numerade Educator

Problem 81

A mass $m$ moves in a circle on a smooth horizontal plane with velocity $v_{0}$ at a radius $R_{0}$. The mass is attached to a string which passes through a smooth hold in the plane as shown in figure:
The tension in the string is increased gradually and finally $m$ moves in a circle of radius $\frac{R_{0}}{2}$. The final
value of the kinetic energy is:
(a) $\frac{1}{4} m v_{0}^{2}$
(b) $2 m v_{0}^{2}$
(c) $\frac{1}{2} m v_{0}^{2}$
(d) $m v_{0}^{2}$

Dheeraj Sharma

Numerade Educator

Problem 82

A tangential force $F$ acts at the top of a thin spherical shell of mass $m$ and radius $R .$ The acceleration of the shell, if it rolls without slipping is:
(a) $\frac{5 F}{6 m}$
(b) $\frac{6 F}{5 m}$
(c) $\frac{7}{2} \frac{m}{F}$
(d) $\frac{2 m}{7} \frac{m}{F}$

Dheeraj Sharma

Numerade Educator

Problem 83

A solid sphere, disc and solid cylinder all of the same mass and made up of same material are allowed to roll down (from rest) on inclined plane, then:
(a) solid sphere reaches the bottom first
(b) solid sphere reaches the bottom late
(c) disc will reach the bottom first
(d) all of them reach the bottom at the same time

Dheeraj Sharma

Numerade Educator

Problem 84

An inclined plane makes an angle $30^{\circ}$ with the horizontal. A solid sphere rolling down this inclined plane from rest without slipping has a linear acceleration equal to:
(a) $\mathrm{g} / 3$
(b) $2 \mathrm{~g} / 3$
(c) $5 \mathrm{~g} / 7$
(d) $5 \mathrm{~g} / 14$

Dheeraj Sharma

Numerade Educator

Problem 85

A uniform solid cylindrical roller of mass $m$ is being pulled on a horizontal surface with force $F$ parallel to the surface and applied at its centre. If the acceleration of the cylinder is $a$ and it is rolling without slipping, then the value of $F$ is:
(a) $m a$
(b) $5 / 3 \mathrm{ma}$
(c) $3 / 2 \mathrm{ma}$
(d) $2 \mathrm{ma}$

Dheeraj Sharma

Numerade Educator

Problem 86

A body rolls down an inclined plane. If its kinetic energy of rotation is $40 \%$ of its kinetic energy of translation motion, then the body is:
(a) hollow cylinder
(b) ring
(c) solid disc
(d) solid sphere

Dheeraj Sharma

Numerade Educator

Problem 87

A solid cylinder of mass $2 \mathrm{~kg}$ and radius $0.1 \mathrm{~m}$ rolls down an inclined plane of height $3 \mathrm{~m}$ without slipping. Its rotational kinetic energy when it reaches the foot of the plane would be:
(a) $22.7 \mathrm{~J}$
(b) $19.6 \mathrm{~J}$
(c) $10.2 \mathrm{~J}$
(d) $9.8 \mathrm{~J}$

Dheeraj Sharma

Numerade Educator

Problem 88

The acceleration of a disc (mass $m$ and radius $R$ ) rolling down an incline of angle $\theta$ without slipping is:
(a) $2 / 3 \mathrm{~g} \sin \theta$
(b) $5 / 7 \mathrm{~g} \sin \theta$
(c) $1 / 2 \mathrm{~g} \sin \theta$
(d) $7 / 5 \mathrm{~g} \sin \theta$

Dheeraj Sharma

Numerade Educator

Problem 89

A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its centre of mass is $K$. If radius of the ball be $R$, then the fraction of total energy associated with its rotational energy will be:
(a) $\frac{K^{2}}{R^{2}}$
(b) $\frac{K^{2}}{K^{2}+R^{2}}$
(c) $\frac{R^{2}}{K^{2}+R^{2}}$
(d) $\frac{K^{2}+R^{2}}{R^{2}}$

Dheeraj Sharma

Numerade Educator

Problem 90

A sphere rolls down on an inclined plane of inclination $6 .$ What is the acceleration as the sphere reaches the bottom?
(a) $\frac{5}{7} g \sin \theta$
(b) $\frac{3}{5} g \sin \theta$
(c) $\frac{2}{7} g \sin \theta$
(d) $\frac{2}{5} g \sin \theta$

Dheeraj Sharma

Numerade Educator

Problem 91

A hollow smooth uniform sphere $A$ of mass $m$ rolls without sliding on a smooth horizontal surface. It collides head on elastically with another stationary smooth solid sphere $B$ of the same mass $m$ and same radius. The ratio of kinetic energy of $B$ to that of $A$ just after the collision is:
(a) $1: 1$
(b) $2: 3$
(c) $3: 2$
(d) None of these

Saman Zulfiqar

Numerade Educator

Problem 92

A solid sphere, disc, and solid cylinder all of the same mass and made of the same material are allowed to roll down (from rest) on an inclined plane, then
(a) solid sphere reaches the bottom first.
(b) solid sphere reaches the bottom last.
(c) disc will reach the bottom first.
(d) all reach the bottom at the same time.

Dheeraj Sharma

Numerade Educator

Problem 93

A solid cylinder of mass $m$ and radius $R$ rolls down inclined plane without slipping. The speed of its CM when it reaches the bottom is:
(a) $\sqrt{2 g h}$
(b) $\sqrt{4 \mathrm{gh} / 3}$
(c) $\sqrt{3 / 4 g h}$
(d) $\sqrt{4 g h}$

Dheeraj Sharma

Numerade Educator

Problem 94

Match the following:
A (i) Rotational work done by the friction is negative till pure rolling begins
B (ii) Translational work done is positive till pure rolling starts
C (iii) When pure rolling begins velocity of centre of mass is minimum
D (iv) Takes maximum time for
pure rolling to begin

Surendra Kumar

Numerade Educator

Problem 95

The ratio of the acceleration for a solid sphere (mass $m$ and radius $R$ ) rolling down an incline of angle $\theta$ without slipping and slipping down the incline without rolling is:
(a) $5: 7$
(b) $2: 3$
(c) $2: 5$
(d) $7: 5$

Dheeraj Sharma

Numerade Educator

Problem 96

A small object of uniform density rolls up a curved surface with an initial velocity $v .$ It reaches up to a maximum height of $3 v^{2} / 4 \mathrm{~g}$ with respect to the initial position. The object is:
(a) ring
(b) solid sphere
(c) hollow sphere
(d) disc

Vysakh M

Numerade Educator

Problem 97

Two identical uniform solid spherical ball $A$ and $B$ of mass $m$ each are placed on the fixed wedge as shown in figure. Ball $B$ is kept at rest and it is released just before two ball collides. Ball $A$ rolls down without slipping on inclined plane and collide elastically with ball $B$. The kinetic energy of ball $A$ just after the collision with $B$ is:
(a) $\frac{m g h}{7}$
(b) $\frac{m g h}{2}$
(c) $\frac{2 m g h}{5}$
(d) $\frac{7 m g h}{5}$

Dheeraj Sharma

Numerade Educator

Problem 98

A cylinder of mass $M_{c}$ and sphere of mass $M_{s}$ are placed at points $A$ and $B$ of two inclines, respectively (see figure). If they roll on the incline without sipping such that their accelerations are the same, then the ratio
$$
\frac{\sin \theta_{c}}{\sin \theta_{s}} \text { is: }
$$
(a) $\sqrt{\frac{8}{7}}$
(b) $\sqrt{\frac{15}{14}}$
(c) $\frac{8}{7}$
(d) $\frac{15}{14}$

Saman Zulfiqar

Numerade Educator