Objective NCERT at your FINGERTIPS: Physics

NCERT

Chapter 7

System of Particles and Rotational Motion - all with Video Answers

Educators

Chapter Questions

Problem 1

In a spinning top, axis moves around the vertical through its point of contact with the ground sweeping out a cone. This movement of the axis of around the vertical is known as the top around translation (b) translation
(a) rotation
(d) rolling
(c) precession

Dheeraj Sharma

Dheeraj Sharma

Numerade Educator

Problem 2

The centre of mass of a body
(a) lies always at the geometrical centre
(b) lies always inside the body
(c) lies always outside the body
(d) may lie within or outside the body

Dheeraj Sharma

Numerade Educator

Problem 3

The position of the centre of mass of a cube of uniform mass density will be at
(a) the centre of one face
(b) the centre of the interaction of diagonals of one face.
(c) the geometric centre of the cube
(d) the edge of a cube

Dheeraj Sharma

Numerade Educator

Problem 4

The reduce mass of two particles having masses $m$ and
$2 m$ is
(a) $2 \mathrm{~m}$
(b) $3 m$
(c) $\frac{2 m}{3}$
(d) $\frac{m}{2}$

Dheeraj Sharma

Numerade Educator

Problem 5

Three particles of masses $1 \mathrm{~kg}, \frac{3}{2} \mathrm{~kg}$, and $2 \mathrm{~kg}$ are located at the vertices of an equilateral triangle of side $a$. The $x_{1} y$ coordinates of the centre of mass
are
(a) $\left(\frac{5 a}{9}, \frac{2 a}{3 \sqrt{3}}\right)$
(b) $\left(\frac{2 a}{3 \sqrt{3}}, \frac{5 a}{9}\right)$
(c) $\left(\frac{5 a}{9}, \frac{2 a}{\sqrt{3}}\right)$
(d) $\left(\frac{2 a}{\sqrt{3}}, \frac{5 a}{9}\right)$

Dheeraj Sharma

Numerade Educator

Problem 6

The $x, y$ coordinates of the centre of mass of a uniform $L$-shaped lamina of mass $3 \mathrm{~kg}$ is
(a) $\left(\frac{5}{6} \mathrm{~m}_{1} \frac{5}{6} \mathrm{~m}\right)$
(b) $(1 \mathrm{~m}, 1 \mathrm{~m})$
(c) $\left(\frac{6}{5} \mathrm{~m}, \frac{6}{5} \mathrm{~m}\right)$
(d) $(2 \mathrm{~m}, 2 \mathrm{~m})$

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 7

The centre of mass of a system of two particles of masses $m_{1}$ and $m_{2}$ is at a distance $d_{1}$ from $m_{1}$ and at a distance $d_{2}$ from mass $m_{2}$ such that
(a) $\frac{d_{1}}{d_{2}}=\frac{m_{2}}{m_{1}}$
(b) $\frac{d_{1}}{d_{2}}=\frac{m_{1}}{m_{2}}$
(c) $\frac{d_{1}}{d_{2}}=\frac{m_{1}}{m_{1}+m_{2}}$
(d) $\frac{d_{1}}{d_{2}}=\frac{m_{2}}{m_{1}+m_{2}}$

Dheeraj Sharma

Numerade Educator

Problem 8

Centre of mass of three particles of masses $1 \mathrm{~kg}$, $2 \mathrm{~kg}$ and $3 \mathrm{~kg}$ lies at the point $(1,2,3)$ and centre of mass of another system of particles $3 \mathrm{~kg}$ and $2 \mathrm{~kg}$ lies at the point $(-1,3,-2)$. Where should we put a particle of mass $5 \mathrm{~kg}$ so that the centre of mass of entire system lies at the centre of mass of first system?
(a) $(0,0,0)$
(b) $(1,3,2)$
(c) $(-1,2,3)$
(d) $(3,1,8)$

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 9

Two particles of mass $1 \mathrm{~kg}$ and $3 \mathrm{~kg}$ have position vectors $2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $-2 \hat{i}+3 \hat{j}-4 \hat{k}$ respectively.
The centre of mass has a position vector
(a) $\hat{i}+3 \hat{j}-2 \hat{k}$
(b) $-\hat{i}-3 \hat{j}-2 \hat{k}$
(c) $-\hat{i}+3 \hat{j}+2 \hat{k}$
(d) $-\hat{i}+3 \hat{j}-2 \hat{k}$

Dheeraj Sharma

Numerade Educator

Problem 10

When an explosive shell travelling in a parabolic path under the effect of gravity explodes in the mid air, the centre of mass of the fragments will move
(a) vertically downwards
(b) along the original parabolic path
(c) vertically upwards and then vertically downwards
(d) horizontally followed by parabolic path

Dheeraj Sharma

Numerade Educator

Problem 11

The velocity of centre of mass of the system remains constant, if the total external force acting on the
system is (b) maximum
(a) minimum
(c) unity
(d) $\mathrm{zero}$

Dheeraj Sharma

Numerade Educator

Problem 12

Two particles of equal mass have velocities $\vec{v}_{1}=2 \hat{i} \mathrm{~m} \mathrm{~s}^{-1}$ and $\bar{v}_{2}=2 \hat{j} \mathrm{~m} \mathrm{~s}^{-1}$, First particle
has an acceleration $\vec{a}_{1}=(3 \hat{i}+3 \hat{j}) \mathrm{m} \mathrm{s}^{-2}$ while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of
(a) straight line
(b) parabola
(c) circle
(d) ellipse

Dheeraj Sharma

Numerade Educator

Problem 13

A child is standing at one end of a long trolley moving with a speed $v$ on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u$, the centre of mass of the system (trolley + child) will move with a speed
(a) zero
(b) $(v+u)$
(c) $(v-u)$
(d) $v$

Dheeraj Sharma

Numerade Educator

Problem 14

Two masses $m_{1}=1 \mathrm{~kg}$ and $m_{2}=2 \mathrm{~kg}$ are connected by
a light inextensible string and suspended by means of a weightless pulley as shown in the figure. Assuming that both the $1 \mathrm{k}$
masses start from rest. the distance travelled by the centre of mass in two seconds is $\left(\right.$ Take $\left.g=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$
(a) $\frac{20}{9} \mathrm{~m}$
(b) $\frac{40}{9} \mathrm{~m}$
(c) $\frac{2}{3} \mathrm{~m}$
(d) $\frac{1}{3} \mathrm{~m}$

Dheeraj Sharma

Numerade Educator

Problem 15

Which of the following is the correct relation between linear relocity $\bar{v}$ and angular velocity $\vec{\omega}$ of a particle?
(a) $\overline{\boldsymbol{v}}=\overline{\bar{F}} \times \bar{\omega}$
(b) $\vec{v}=\overrightarrow{(\dot{a}} \times \overline{\boldsymbol{r}}$
(c) $\overrightarrow{\mathbf{u}}=\bar{r} \times \vec{v}$
(d) $\overrightarrow{\mathbf{g}}=\overline{\bar{v}} \times \vec{r}$

Dheeraj Sharma

Numerade Educator

Problem 16

The direction of the angular velocity vector is al $_{\mathrm{O} \mathrm{h} \text { og }}$
(a) the tangent to the circular path
(b) the inward radius
(c) the outward radius
(d) the axis of rotation

Dheeraj Sharma

Numerade Educator

Problem 17

Which of the following statements is correct?
(a) For a general translation motion, momentu_\eta] $\hat{p}$ and velocity $\vec{v}$ need not parallel.
(b) For a general rotational motion, angul_st momentum $\vec{L}$ and angular velocity $\left.(\overrightarrow{1}) \mathrm{a} / \mathrm{w}_{\mathrm{d}}\right)_{\text {s }}$ be parallel.
(c) For a general translation motion, acceleration $\vec{a}$ and velocity $\vec{v}$ are always parallel.
(d) For a general rotational motion, angular momentum $\vec{L}$ and angular velocity $\vec{\omega}$ need not be parallel.

Dheeraj Sharma

Numerade Educator

Problem 18

A body is rotating with angular velocity $\bar{\omega}=(3 \hat{i}-4 \hat{j}+\hat{k})$. The linear velocity of a point having position vector $\vec{r}=(5 \hat{i}-6 \hat{j}+6 \hat{k})$ is
(a) $6 \hat{i}+2 \hat{j}-3 \hat{k}$
(b) $18 \hat{i}+3 \hat{j}-2 \hat{k}$
(c) $-18 \hat{i}-13 \hat{j}+2 \hat{k}$
(d) $6 \hat{i}-2 \hat{j}+8 \hat{k}$

Dheeraj Sharma

Numerade Educator

Problem 19

A disc rotating about its axis with angular speed $\omega_{0}$ is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is $R$. Let $\mathrm{k}, v_{8}$ and $\mathrm{\psi}$ be the magnitudes of linear velocities of the points $A, B$ and $C$ on the disc as shown. Then
(a) $v_{A}>v_{B}>v_{C}$
(b) $v_{A}<v_{B}<v_{C}$
(c) $v_{A}=v_{B}<v_{C}$
(d) $v_{A}=v_{B}>v_{C}$

Dheeraj Sharma

Numerade Educator

Problem 20

Figure shows a lamina in $x-y$ plane. Two axes $z$ and $z^{\prime}$ pass perpendicular to its plane.
A force $F$ acts in the plane of lamina at point $P$ as shown. Which of the following statements is incorrect? (The point $P$ is closer to $z^{\prime}$-axis than the $z$-axis).
(a) Torque t caused by $F$ about $z$ axis is along $\hat{k}$.
(b) Torque $\tau^{\prime}$ caused by $F$ about $z^{\prime}$ axis is along $-\hat{k}$.
'c) Torque caused by $F$ about $z$ axis is greater in magnitude than that about $z^{\prime}$ axis.
d) Total torque is given by $\tau=\tau+\tau^{\prime}$.

Akshaya Rs

Numerade Educator

Problem 21

When a torque acting upon a system is zero. Which of the following will be constant?
(a) Force
(b) Linear impulse
(c) Linear momentum
(d) Angular momentum

Dheeraj Sharma

Numerade Educator

Problem 22

eefi of $f^{\infty}$ acting on a particle having position force acting orgue of this force about the If $\vec{F}$ is the force the torque vector then $\vec{F} \cdot \bar{\tau}<0$
origin, then 0 and $\vec{r} \cdot \vec{\tau}>0$
$\vec{F}, \tau=0$
$\vec{r} \cdot \vec{\tau}=0$
(c) $\vec{r}^{\prime} \vec{\tau} \neq 0$ and $\vec{F} \cdot \vec{\tau}=0$

Dheeraj Sharma

Numerade Educator

Problem 23

Angular mon constant due to central constant torque
(a) corstant force constant force
(b) constant
(c) corbrorque (d) zero toryut $7 \hat{j}+3 \hat{j}-5 \hat{k}$ acts on a par

Dheeraj Sharma

Numerade Educator

Problem 24

force $7 i+3 j$ is $\hat{i}-\hat{j}+\hat{k}$. What is the torque of a The foro vector is $t^{2}$ the origin? position vecteforce about $\hat{k}$ (b) $2 \hat{i}+10 \hat{j}+12 \hat{k}$
given
(a) $2 \hat{i}+12 \hat{j}+10 \hat{k}$.
(d) $10 \hat{i}+2 \hat{j}+\hat{k}$
$2 \hat{i}+10 \hat{j}+10 \hat{k}$

Dheeraj Sharma

Numerade Educator

Problem 25

A disc is rotating with angant whose nosition vector 31 axis. A force $\vec{F}$ acts at a point whose position vector with respect to the axis of rotation is $\vec{r}$. The power associated with the torque due to the force is given by
(a) $(\vec{r} \times \vec{F}) \cdot \vec{\omega}$
(b) $(\vec{r} \times \vec{F}) \times \vec{\omega}$
(c) $\bar{r} \cdot(\vec{F} \times \vec{\omega})$
(d) $\bar{r} \times(\vec{F} \cdot \vec{\omega})$

Dheeraj Sharma

Numerade Educator

Problem 26

A mass $M$ is moving with a constal to $x$-axis. Its angular momentum w.r.t. origin
(a) is zero
(b) remains constant
(c) goes on increasing
(d) goes on decreasing
$=$

Dheeraj Sharma

Numerade Educator

Problem 27

A small mass $m$ is attached to a massless string whose other end is fixed at $P$ as shown in figure. The mass is undergoing circular motion in $x-y$ plane with centre $O$ ? and constant angular speed $\omega$. If the angular momentum of the calculated about $O$ and Pand denoted by $\vec{L}_{O}$ and $\vec{L}_{P}$ respectively, then
(a) $\vec{L}_{Q}$ and $\vec{L}_{p}$ do not vary with time.
(b) $\vec{L}_{O}$ varies with time while $\vec{L}_{P}$ remains
constant.
(c) $\vec{L}_{O}$ remains constant while $\vec{L}_{P}$ varies with
time.
(d) $\vec{L}_{O}$ and $\vec{L}_{p}$ both vary with time.

Dheeraj Sharma

Numerade Educator

Problem 28

The position of a particle is given by $\vec{r}=\hat{i}+2 \hat{j}-k$ 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) $x$-axis
(b) $y$-axis
(c) $z$-axis
(d) $y z$-plane

Dheeraj Sharma

Numerade Educator

Problem 29

The $z$ component of the angular momentum of a particle whose position vector is $\vec{r}$ with components $x_{1} y$ and $z$ and linear momentum is $\vec{p}$ with components $p_{x} p_{y}$ and $p_{z}$ is
(a) $x p_{y}-y p_{x}$
(b) $y p_{z}-z p_{y}$
(c) $z p_{x}-x p_{z}$
(d) $x p_{y}+y p_{x}$

Dheeraj Sharma

Numerade Educator

Problem 30

Consider a particle of mass $m$ having linear momentum $\vec{p}$ at position $\bar{r}$ relative to the origin $O$. Let $\vec{L}$ be the angular momentum of the particle with respect to the origin. Which of the following equations correctly relate(s) $\vec{r}, \vec{p}$ and $\vec{L} ?$
(a) $\frac{d \vec{L}}{d t}+\vec{r} \times \frac{d \vec{p}}{d t}=0$
(b) $\frac{d \vec{L}}{d t}+\frac{d \bar{r}}{d t} \times \vec{p}=0$
(c) $\frac{d \vec{L}}{d t}-\frac{d \vec{r}}{d t} \times \vec{p}=0$
(d) $\frac{d \vec{L}}{d t}-\vec{r} \times \frac{d p}{d t}=0$

Dheeraj Sharma

Numerade Educator

Problem 31

A rigid body is said to be in partial equilibrium, when it is in
(a) translational equilibrium only
(b) rotational equilibrium only
(c) either (a) or (b)
(d) neither (a) nor (b)

Dheeraj Sharma

Numerade Educator

Problem 32

Moment of couple is called
(a) angular momentum (b) force
(c) torque
(d) impulse

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 33

A couple produces
(a) purely translational motion
(b) purely rotational motion
(c) both translational and rotational motion
(d) no motion

Dheeraj Sharma

Numerade Educator

Problem 34

Which of the following statements is incorrect?
(a) A pair of equal and opposite forces with different lines of action is known as couple.
(b) A couple produces rotation without translation.
(c) When we open the lid of a bottle by turning it, our fingers apply a couple to the lid.
(d) Moment of a couple depends on the point about which we take the moment.

Akshaya Rs

Numerade Educator

Problem 35

Which of the following relations is correct?
(a) Mechanical advantage $=\frac{\text { Effort }}{\text { Load }}$
(b) Load arm $\times$ Effort $=$ Effort arm $\times$ Load
(c) Load arm $\times$ Load $=$ Effort arm $\times$ Effort
(d) None of these

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 36

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) 1 and II only
(b) II and III only
(c) I and III only
(d) The net torque is zero in all cases.
4

Dheeraj Sharma

Numerade Educator

Problem 37

(1) Centre of gravity of a body is the point at which the weight of the body acts.
(2) Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius.
(3) To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its centre of gravity.
(4) The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the centre of gravity of the body to the axis. Which one of the following pairs of statements is correct?
(a) $(1)$ and $(4)$
(b) $(1)$ and $(2)$
(c) $(2)$ and $(3)$
(d) (3) and (4)

Akshaya Rs

Numerade Educator

Problem 38

A non-uniform bar of weight $W$ and length $L$ is $\lambda$ suspended by two strings of negligible weight as shown in figure. The angles $\hat{x} \rightarrow d^{n}$ made by the strings with the vertical are $\theta_{1}$ and $\theta_{2}$ respectively. The distance $d$ of the centre of gravity of the bar from its left end is
(a) $L\left(\frac{\tan \theta_{1}+\tan \theta_{2}}{\tan \theta_{1}}\right)$
(b) $L\left(\frac{\tan \theta_{1}}{\tan \theta_{1}+\tan \theta_{2}}\right)$
(c) $L\left(\frac{\tan \theta_{2}}{\tan \theta_{1}+\tan \theta_{2}}\right)$
(d) $L\left(\frac{\tan \theta_{1}+\tan \theta_{2}}{\tan \theta_{2}}\right)$

Dheeraj Sharma

Numerade Educator

Problem 39

A uniform rod of length $1 \mathrm{~m}$ and $\mathrm{mas}$ $m_{s}$
supported on two knife-edges placed $10 \mathrm{~cm}_{\mathrm{g}}$ cach end. A $60 \mathrm{~N}$ weight is suspended at $30 \mathrm{~cm} \mathrm{c}_{\mathrm{t}} \mathrm{c}$ one end. The reactions at the knife ed ges is
(a) $60 \mathrm{~N}, 40 \mathrm{~N}$
(b) $75 \mathrm{~N}, 25 \mathrm{~N}$
(c) $65 \mathrm{~N}, 35 \mathrm{~N}$
(d) $55 \mathrm{~N}, 45 \mathrm{~N}$

Dheeraj Sharma

Numerade Educator

Problem 40

A car weighs $1800 \mathrm{~kg}$. The distance between its $\mathrm{ft}_{\text {aly }}$ and back axles is $1.8 \mathrm{~m}$. Its centre of gravity is behind the front axle. The force exerted by the $\mathrm{l}_{\mathrm{m}}$ ground on each front wheel and each back wheel ground on (Take $g=10 \mathrm{~ms}^{-2}$ )
(a) $4000 \mathrm{~N}$ on each front wheel, $5000 \mathrm{~N}$ on $\mathrm{e}_{2 \mathrm{Ch}}$
back wheel
(b) $5000 \mathrm{~N}$ on each front wheel, $4000 \mathrm{~N}$ an each back wheel
(c) $4500 \mathrm{~N}$ on each front wheel, $4500 \mathrm{~N}$ on each back wheel
(d) $3000 \mathrm{~N}$ on each front wheel, $6000 \mathrm{~N}$ on each back wheel

Dheeraj Sharma

Numerade Educator

Problem 41

A uniform ladder $3 \mathrm{~m}$ long weighing $20 \mathrm{~kg}$ leans against a frictionless wall. Its foot rest on a rough floor $1 \mathrm{~m}$ from the wall. The reaction forces of the wall and floor are
(a) $25 \sqrt{2} \mathrm{~N}, 203 \mathrm{~N}$
(b) $50 \sqrt{2} \mathrm{~N}, 230 \mathrm{~N}$
(c) $203 \mathrm{~N}, 25 \sqrt{2} \mathrm{~N}$
(d) $230 \mathrm{~N}, 50 \sqrt{2} \mathrm{~N}$

Dheeraj Sharma

Numerade Educator

Problem 42

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 $12 \mathrm{~cm} \mathrm{mark}$, the stick is found to be balanced at $45 \mathrm{~cm}$. The mass of the metre stick is
(a) $56 \mathrm{~g}$
(b) $66 \mathrm{~g}$
(c) $76 \mathrm{~g}$
(d) $86 \mathrm{~g}$

Dheeraj Sharma

Numerade Educator

Problem 43

A uniform cube of mass $M$ and side $a$ is placed on a mh frictionless horizontal surface. A vertical force $F$ is applied to the edge as shown in figure. Match the Column I with Column II. \begin{tabular}{|l|l|l|l|}
\hline \multicolumn{2}{|c|} { Column I } & \multicolumn{1}{c|} { Column II } \\
\hline (A) & $\frac{M g}{4}<F<\frac{M g}{2}$ & (p) & Cube will move up. \\
\hline (B) & $F>\frac{M g}{2}$ & (q) & Cube will not exhibit motion. \\
\hline (C) & $F>M g$ & (r) & Cube will begin to \\
(D) & $F=\frac{M g}{4}$ & & rotate and slip at $A .$ \\
\hline
\end{tabular}
(a) $A=p, b=c, c-s$
(b) $A-r, B-s, C-q, D-p$
(c) $A-q, B-r, C-p, D-s$
of $\mathrm{A}-5, \mathbb{B}-\mathrm{p}, \mathrm{C}-\mathrm{r}, \mathrm{D}-\mathrm{q}$

Akshaya Rs

Numerade Educator

Problem 44

of mass in rotational motion is
(a) moment of inertia radius of gyration
(d) angular momentum
(c)

Dheeraj Sharma

Numerade Educator

Problem 45

The morm of the body
(a) mass
(b) axis of rotation of the body
(c) shape and size of the body
(d) all of these

Dheeraj Sharma

Numerade Educator

Problem 46

which of the following has the highest moment of inertia when each of them has the same mass and radius? the same
(a) A ring about any of its diameter.
(b) A disc about any of its diameter.
(c) A hollow sphere about any of its diameter.
(d) A solid sphere about any of its diameter.

Dheeraj Sharma

Numerade Educator

Problem 47

A person is standing on a rotating table with metal spheres in his hands. If he withdraws his hands to his chest, then the effect on his angular velocity will be
(a) increase
(b) decrease
(c) remain same
(d) can't say

Dheeraj Sharma

Numerade Educator

Problem 48

A solid cylinder of mass $M$ and radius $R$ rotates about its axis with angular speed $\omega$. Its rotational kinetic energy is
(a) $\frac{1}{2} M R^{2} \omega^{2}$
(b) $M R^{2} \omega^{2}$
(c) $\frac{1}{4} M R^{2} \omega^{2}$
(d) $\frac{1}{8} M R^{2} \omega^{2}$

Dheeraj Sharma

Numerade Educator

Problem 49

Match the Column I with Column II. \begin{tabular}{|l|l|l|l|}
\hline \multicolumn{1}{|c|} { Column I } & \multicolumn{2}{c|} { Column II } \\
\hline (A) & For translational equilibrium & (p) & $M k^{2}$ \\
\hline (B) & For rotational equilibrium & (q) & Angular acceleration \\
\hline (C) & Moment of inertia of a body & (r) & $\sum \vec{F}=0$ \\
\hline (D) & Torque is required to produce & (s) & $\sum \vec{\tau}=0$ \\
\hline
\end{tabular}
(a) $\mathrm{A}-\mathrm{p}, \mathrm{B}-\mathrm{q}, \mathrm{C}-\mathrm{r}, \mathrm{D}-\mathrm{s}$
(b) $\mathrm{A}-\mathrm{q}, \mathrm{B}-\mathrm{r}, \mathrm{C}-\mathrm{s}, \mathrm{D}-\mathrm{p}$
(c) $\mathrm{A}-\mathrm{r}, \mathrm{B}-\mathrm{q}, \mathrm{C}-\mathrm{p}, \mathrm{D}-\mathrm{s}$
(d) $\mathrm{A}-\mathrm{r}, \mathrm{B}-\mathrm{s}, \mathrm{C}-\mathrm{p}, \mathrm{D}-\mathrm{q}$

Dheeraj Sharma

Numerade Educator

Problem 50

The radius of gyration of an uniform rod of length l about an axis passing through one of its ends and perpendicular to its length is
(a) $\frac{1}{\sqrt{2}}$
(b) $\frac{l}{3}$
(c) $\frac{1}{\sqrt{3}}$
(d) $\frac{1}{2}$

Dheeraj Sharma

Numerade Educator

Problem 51

Two masses each of mass $M$ are attached to the end of a rigid massless rod of length $L$. The moment of inertia of the system about an axis passing centre of mass and perpendicular to its length is
(a) $\frac{M L^{2}}{4}$
(b) $\frac{M L^{2}}{2}$
(c) $M L^{2}$
(d) $2 M L^{2}$

Dheeraj Sharma

Numerade Educator

Problem 52

From a circular disc of radius $R$ and mass $9 M$, a small disc of radius $\frac{R}{3}$ is removed as shown in figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through $O$ is
(a) $4 M R^{2}$
(b) $\frac{40}{9} \mathrm{MR}^{2}$
(c) $40 \mathrm{MR}^{2}$
(d) $\frac{37}{9} \mathrm{MR}^{2}$

Dheeraj Sharma

Numerade Educator

Problem 53

A uniform rectangular plate $R$ of sides $a$ and $b$ and a uniform square plate $S$ of side c have same masses and areas as shown in the figure. Then,
(i) $\frac{I_{x R}}{I_{x S}}<1$
(ii) $\frac{I_{g R}}{I_{y S}}>1$
Which of the above relations is correct?
(a) (i) only
(b) (ii) only
(c) Both (i) and (ii)
(d) Neither (i) nor (ii)

Dheeraj Sharma

Numerade Educator

Problem 54

The moment of inertia of a solid sphere of mass $\mathrm{M}$ and radius $R$ about a tangent to the sphere is
(a) $\frac{2}{5} \mathrm{MR}^{2}$
(b) $\frac{6}{5} \mathrm{MR}^{2}$
(c) $\frac{4}{5} \mathrm{MR}^{2}$
(d) $\frac{7}{5} M R^{2}$

Dheeraj Sharma

Numerade Educator

Problem 55

With reference to figure of a cube of edge $a$ and mass $m_{1}$ which of the following is the incorrect statement?
(O is the centre of the cube)
(a) The moment of inertia of cube about $z^{\prime}$ is
$$
I_{z}^{\prime}=I_{z}+\frac{m a^{2}}{2}
$$
(b) The moment of inertia of cube about $z^{\prime \prime}$ is
$I_{z}^{\prime \prime}=I_{z}+\frac{m a^{2}}{2}$
(c) $I_{x}=I_{y}$
(d) None of these

Nikhil Choudhary

Nikhil Choudhary

Numerade Educator

Problem 56

An athlete throws a discus from rest to a final angular velocity of $15 \mathrm{rad} \mathrm{s}^{-1}$ 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
$\begin{array}{ll}\text { released is } \\ -2 & \text { (b) } 182 \mathrm{~ms}^{-2}\end{array}$
(a) $45 \mathrm{~m} \mathrm{~s}^{-2}$
(d) $192 \mathrm{~m} \mathrm{~s}^{-2}$
(c) $187 \mathrm{~m} \mathrm{~s}^{-2}$

Dheeraj Sharma

Numerade Educator

Problem 57

A flywheel rotating at 420 rpm slows down as constant rate of $2 \mathrm{rad} \mathrm{s}^{-2}$. The time required to stop
the flywheel is
$\begin{array}{llll}\text { the } & \text { (c) } 44 s & \text { (d) } 125\end{array}$
(a) $22 \mathrm{~s}$
(b) $11 \mathrm{~s}$

Dheeraj Sharma

Numerade Educator

Problem 58

The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds. The angular acceleration of the motor wheel is
(a) $2 \pi \mathrm{rad} \mathrm{s}^{-2}$
(b) $4 \pi \mathrm{rad} 5^{-2}$
(c) $6 \pi \mathrm{rad} \mathrm{s}^{-2}$
(d) $8 \pi \operatorname{rad} s^{-2}$

Dheeraj Sharma

Numerade Educator

Problem 59

An automobile engine develops $100 \mathrm{~kW}$ when rotating at a speed of $1800 \mathrm{rpm}$. The torque delivered by the engine is
(a) $\frac{10^{2}}{6 \pi} \mathrm{N} \mathrm{m}$
(b) $\frac{10^{4}}{6 \pi} \mathrm{Nm}$
(c) $\frac{10^{6}}{6 \pi} \mathrm{N} \mathrm{m}$
(d) $\frac{10^{8}}{6 \pi} \mathrm{N} \mathrm{m}$

Dheeraj Sharma

Numerade Educator

Problem 60

A grindstone has a moment of inertia of $6 \mathrm{~kg} \mathrm{~m}^{2}$. A constant torque is applied and the grindstone is found to have a speed of $150 \mathrm{rpm}, 10$ seconds after starting from rest. The torque is
(a) $3 \pi \mathrm{Nm}$
(b) $3 \mathrm{~N} \mathrm{~m}$
(c) $\frac{\pi}{3} \mathrm{~N} \mathrm{~m}$
(d) $4 \pi \mathrm{N} \mathrm{m}$

Dheeraj Sharma

Numerade Educator

Problem 61

The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta(t)=2 t^{3}-6 t^{2}$. The torque on the wheel becomes zero at
(a) $t=15$
(b) $t=0.5 \mathrm{~s}$
(c) $t=0.25 \mathrm{~s}$
(d) $t=2 s$

Dheeraj Sharma

Numerade Educator

Problem 62

A rope of negligible mass is wound round a hollow cylinder of mass $3 \mathrm{~kg}$ and radius $40 \mathrm{~cm}$. If the rope with a force of $30 \mathrm{~N}$, then the angulat is pulled acceleration produced in the cylinder is
(b) $20 \mathrm{rad} \mathrm{s}^{-2}$
(a) $15 \mathrm{rad} s$
(d) $30 \mathrm{rad} \mathrm{s}^{-2}$
(c) 25 rad $s^{-2}$, her 62 , the linear acceleration

Dheeraj Sharma

Numerade Educator

Problem 63

In the question number the rope is
(b) $10 \mathrm{~ms}^{-2}$
(a) $5 \mathrm{~m} 5^{-2}$
(d) $20 \mathrm{~ms}^{-2}$
(c) $15 \mathrm{~m} \mathrm{~s}^{-2}$. $\mathrm{H}$ and radius $R$ is rotating

Dheeraj Sharma

Numerade Educator

Problem 64

A hollow cylinder about its axis of symmetry and a solid sphere of same and radius is rotating about an axis passing mass and radostre. If torques of equal magnitude through its centre. If toryen the ratio of angular are applied to them, accelerations produced is
(c) $\frac{5}{4}$
(d) $\frac{4}{5}$
(a) $\frac{2}{5}$ (b) $\frac{5}{2}$ (c) 4 andorm angular speed

Dheeraj Sharma

Numerade Educator

Problem 65

To maintain a rotor at a uni 00
needs to transmit torque of 100 rad $s^{-1}$, an engine needs $100 \mathrm{~N} \mathrm{~m}$. The power of the cro
(b) $100 \mathrm{~kW}$
(a) $10 \mathrm{~kW}$
(d) $100 \mathrm{MW}$
(c) $10 \mathrm{MW}$

Dheeraj Sharma

Numerade Educator

Problem 66

A cord of negligible mass is w L01
a flywheel of mass $20 \mathrm{~kg}$ and radius $20 \mathrm{~cm}$. A steady pull of $25 \mathrm{~N}$ is applied on the cord. The work done by the pull when $2 \mathrm{~m}$ of the cord is unwound is
(a) $20 \mathrm{~J}$
(b) $25 \mathrm{~J}$
(c) $45 \mathrm{~J}$
(d) $50 \mathrm{~J}$

Dheeraj Sharma

Numerade Educator

Problem 67

number 66 , if wheel starts rrom rest, In the question what is the kinetic energy of the wheel when $2 \mathrm{~m}$ of the cord is unwound?
(a) $20 \mathrm{~J}$
(b) $25 \mathrm{~J}$
(c) $45 \mathrm{~J}$
(d) $50 \mathrm{~J}$

Dheeraj Sharma

Numerade Educator

Problem 68

A uniform disc of mass $M$ and
its rim. The coefficient of friction radius $R$, is resting on a table on
between disc and table is $\mu$. Now the disc is pulled with a force $F$ as shown in the figure. What is the maximum value of $F$ for which the disc rolls without
slipping?
(a) $\mu \mathrm{Mg}$
(b) $2 \mu M g$
c) $3 \mu \mathrm{Mg}$
(d) $4 \mu \mathrm{Mg}$

Dheeraj Sharma

Numerade Educator

Problem 69

Which of the following principles a circus acrobat employs in his performance?
(a) Conservation of energy
(b) Conservation of linear momentum
(c) Conservation of mass
(d) Conservation of angular momentum

Dheeraj Sharma

Numerade Educator

Problem 70

Total ang if the net torque conserve.
(b) maximum
(d) unity
(a) zero (c) minimum mass is rotating in a plane about a fixed . directed along

Dheeraj Sharma

Numerade Educator

Problem 71

When a mass is momentum is directed along point its angular moment (a) the radius (b) the tangent the orbit the line at angle of $45^{\circ}$ to the plane of rotation the line af rotation
(d)

Dheeraj Sharma

Numerade Educator

Problem 72

Figure shows two articles 1 and 2 , each of moving in opposite nass $m$, movith same speed directions witlitsallel lines.
Which of the following is position vectors of the parallel lines. plane of the patement? the correct statement Angular momentum $\vec{L}_{1}$ of particle 1 about $A$ is (a) Ang $\vec{L}_{1}=m v \vec{r}_{1} \odot$
(b) Angular momentum $\vec{L}_{2}$ of particle 2 about $A$ is $\vec{L}_{2}=m v \vec{r}_{2}$
Total angular momentum of the system about $A$ (c) Tot $\vec{L}=m v\left(\vec{r}_{1}+\vec{r}_{2}\right)$
Total angular momentum of the system about $A$ (d) Total ang $\overrightarrow{\text { is }} \vec{L}=m v\left(d_{2}-d_{1}\right) \otimes$
$\otimes$ represents a unit vector going into the page. Q represents a unit vector coming out of the
page?

Nikhil Choudhary

Nikhil Choudhary

Numerade Educator

Problem 73

A solid cylinder of mass $20 \mathrm{~kg}$ rotates about its axis with a angular speed $100 \mathrm{rad} \mathrm{s}^{-1}$. of the cylinder about its The angular momentum
$\begin{array}{ll}\text { axis is } & \text { (b) } 400 \mathrm{~J} \mathrm{~s}\end{array}$
(a) $40 \mathrm{Js}$
(d) $200 \mathrm{Js}$
$(c) 20] s$

Dheeraj Sharma

Numerade Educator

Problem 74

Two bodies have their moments of inertia la $2 I$ respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the ratio
(a) $1: 2$
(b) $\sqrt{2}: 1$
(c) $1: \sqrt{2}$
(d) $2: 1$

Dheeraj Sharma

Numerade Educator

Problem 75

A child is standing with his two arms outstretched at the centre of a turntable that is rotating about its central axis with an angular speed $\omega_{0}$. Now, the child folds his hands back so that moment of inertia becomes 3 times the initial value. The new angular speed is
(a) $3 \omega_{0}$
(b) $\frac{\omega_{0}}{3}$
(c) $6(1)_{0}$
(d) $\frac{\omega_{0}}{6}$

Dheeraj Sharma

Numerade Educator

Problem 76

A circular platform is mounted on a vertical frictionless axle. Its radius is $r=2 \mathrm{~m}$ and its moment of inertia $I=200 \mathrm{~kg} \mathrm{~m}^{2}$. It is initially at rest. A $70 \mathrm{~kg}$ man stands on the edge of the platform and begins to walk along the edge at speed $v_{0}=1 \mathrm{~m} \mathrm{~s}^{-1}$ relative to the ground. The angular velocity of the platform is
(a) $1.2 \mathrm{rad} \mathrm{s}^{-1}$
(b) $0.4 \mathrm{rad} \mathrm{s}^{-1}$
(c) $0.7 \mathrm{rad} \mathrm{s}^{-1}$
(d) $2 \mathrm{rad} \mathrm{s}^{-1}$

Dheeraj Sharma

Numerade Educator

Problem 77

A man stands on a rotating platform with his arms stretched holding a $5 \mathrm{~kg}$ weight in each hand. The angular speed of the platform is $1.2$ rev $\mathrm{s}^{-1}$. The moment of inertia of the man together with the platform may be taken to be constant and equal to $6 \mathrm{~kg} \mathrm{~m}^{2}$. If the man brings his arms close to his chest with the distance of each weight from the axis changing from $100 \mathrm{~cm}$ to $20 \mathrm{~cm}$. The new angular speed of the platform is
(a) 2 rev $s^{-1}$
(b) 3 rev $s^{-1}$
(c) $5 \mathrm{rev} \mathrm{s}^{-1}$
(d) 6 rev $\mathrm{s}^{-1}$

Dheeraj Sharma

Numerade Educator

Problem 78

Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes, rotating with angular frequencies $\omega_{1}$ and $\omega_{2}$ respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be
(a) $\frac{I_{1} \omega_{1}+I_{2} \omega_{2}}{I_{1}+I_{2}}$
(b) $\frac{I_{2} \omega_{1}+I_{1} \omega_{2}}{I_{1}+I_{2}}$
(c) $\frac{I_{1} \omega_{1}-I_{2} \omega_{2}}{I_{1}-I_{2}}$
(d) $\frac{I_{2} \omega_{1}-I_{1} \omega_{2}}{I_{1}-I_{2}}$

Dheeraj Sharma

Numerade Educator

Problem 79

A ballet dancer, dancing on a smooth floor is spinning about a vertical axis with her arms folded with an angular velocity of $20 \mathrm{rad} / \mathrm{s}$. When she stretches her arms fully, the spinning speed decrease in $10 \mathrm{rad} / \mathrm{s}$. If $I$ is the initial moment of inertia of the dancer, the new moment of inertia is
(a) $2 I$
(b) $3 I$
(c) $I / 2$
(d) $1 / 3$

Dheeraj Sharma

Numerade Educator

Problem 80

Angular momentum $L$ and rotational kinetic energy $K_{R}$ of a rigid body are related to each other by the relation. (I = moment of inertia)
(a) $K_{R}=2 I L$
(b) $K_{R}=\frac{L^{2}}{2 I}$
(b) $K_{R}=\frac{2 I}{L}$
(d) $K_{R}=\frac{L^{2}}{I}$

Dheeraj Sharma

Numerade Educator

Problem 81

A person, with outstretched arms, is spinning on a rotating stool. He suddenly brings his arms down to his sides. Which of the following is true about his kinetic energy $K$ and angular momentum $L ?$
(a) Both $\mathrm{K}$ and $L$ increase
(b) Both $\mathrm{K}$ and $L$ remain unchanged
(c) $\mathrm{N}$ remains constant, $L$ increases
(d) $\lambda$ increases but $L$ remains constant

Dheeraj Sharma

Numerade Educator

Problem 82

A child is standing with folded hands at the centre of a plattorm rotating about its central axis, The kinetic energy of the system is $K$. Now, the child stretches his arms so that moment of inertia of the system doubled. Now, the kinetic energy of the system is
(a) $\frac{K^{*}}{4}$
(b) $\frac{K}{2}$
(c) $2 \mathrm{~K}$
(d) $4 K^{*}$

Dheeraj Sharma

Numerade Educator

Problem 83

A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its. geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $\left(E_{\text {setere }} / E_{\text {colinder }}\right)$ will be
(a) $2: 3$
(b) $1: \overline{5}$
(c) $1: 4$
(d) $3: 1$

Dheeraj Sharma

Numerade Educator

Problem 84

Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes (normal to the disc and passing through the centre), and rotating with angular speed $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident. What is the loss in kinetic energy of the system in the process?
(a) $\frac{I_{1} I_{2}\left(\omega_{1}-\omega_{2}\right)^{2}}{2\left(I_{1}+I_{2}\right)}$
(b) $\frac{I_{1} I_{2}\left(\omega_{1}-\omega_{2}\right)^{2}}{\left(I_{1}+I_{2}\right)}$
(c) $\frac{I_{1} I_{2}\left(\omega_{1}+\omega_{2}\right)^{2}}{\left(I_{1}-I_{2}\right)}$
(d) $\frac{I_{1} I_{2}\left(\omega_{1}+\omega_{2}\right)^{2}}{2\left(I_{1}-I_{2}\right)}$

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 85

A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. In both cases
(a) the speed and time of descend will be same.
(b) the speed will be same but time of descend will be different.
(c) the speed will be different but time of descend will be same.
(d) speed and time of descend both are different.

Dheeraj Sharma

Numerade Educator

Problem 86

Which of the following statements is incorrect?
(a) Torque is the rotational analogue of force.
(b) Rolling motion of a cylinder down an inclined plane is combination of translation and rotational motion.
(c) If the effort arm is larger than the load arm, the mechanical advantage is lesser than one.
(d) For the extended body, the centre of mass and centre of gravity do not coincide.

Dheeraj Sharma

Numerade Educator

Problem 87

Which of the following statements is not correcy?
(a) During rolling, the instantaneous speed of of contact is zero.
(b) During rolling, the instantaneous acceleratis of the point of contact is zero.
friction is zero. (d) $\mathrm{A}$ wheel moving down a perfectly frictiogl $_{\mathrm{C}}$ w inclined plane will slip but not roll on the plane.

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 88

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 fractio of total energy associated with its rotation will be
(a) $\frac{k^{2}+R^{2}}{R^{2}}$
(b) $\frac{k^{2}}{R^{2}}$
(c) $\frac{k^{2}}{1 \cdot 2+R^{2}}$
(d) $\frac{R^{2}}{k^{2}+R^{2}}$

Dheeraj Sharma

Numerade Educator

Problem 89

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down an inclined plane making an angle $\theta$ with the horizontal. Then its acceleration is
(a) $\frac{1}{3} g \sin \theta$
(b) $\frac{2}{3} g \sin \theta$
(c) $\frac{2}{5} g \sin \theta$
(d) $\frac{2}{7} g \sin \theta$

Dheeraj Sharma

Numerade Educator

Problem 90

In the question number 89, the force of friction acting on the cylinder is
(a) $\frac{2}{3} M g \sin \theta$
(b) $\frac{1}{3} M g \sin \theta$
(c) $\frac{2}{5} M g \sin \theta$
(d) $\frac{2}{7} M g \sin \theta$

Dheeraj Sharma

Numerade Educator

Problem 91

A solid cylinder rolls up an inclined plane of inclination $\theta$ with an initial velocity $v$. How far does the cylinder go up the plane?
(a) $\frac{3 v^{2}}{2 g \sin \theta}$
(b) $\frac{v^{2}}{4 g \sin \theta}$
(c) $\frac{3 v^{2}}{g \sin \theta}$
(d) $\frac{3 v^{2}}{4 g \sin \theta}$

Dheeraj Sharma

Numerade Educator

Problem 92

The solid cylinder is rolling without slipping on a plane having inclination $\theta$ and the coefficient of static friction $\mu_{s}$. The relation between $\theta$ and $\mu_{s}$ is
(a) $\tan \theta>3 \mu_{s}$
(b) $\tan \theta \leq 3 \mu_{j}$
(c) $\tan \theta<3 \mu_{s}^{2}$
(d) none of these

Dheeraj Sharma

Numerade Educator

Problem 93

A ring of radius $R$ is rotating with an angular speed $\omega_{0}$ about a horizontal axis. It is placed on a rough horizontal table. The coefficient of kinetic friction is $\mu_{k}$. The time after which it starts rolling is
(a) $\frac{\omega_{0}+_{k} R}{2 \xi}$
(b) $\frac{\omega_{0} \mathrm{~S}}{2 \mu_{k} R}$
(c) $\frac{2 \mathrm{co}_{\mathrm{g}} \mathrm{R}}{\|_{1} \mathrm{~g}}$
(d) $\frac{\left(a_{0} R\right.}{2 \mu_{k} g}$

Dheeraj Sharma

Numerade Educator

Problem 94

solid sphere rolis re dewn an when a solane making an angle 0 with the horizontal, inclined plane accelcration of its centre of mass is $a$. If the same without friction, its acceleration $a$ ' will the acte slides sphere bc - (b) $\frac{5}{7} a$
(a) $\frac{7}{2} a$
(c) $\frac{7}{5} a$
(d) $\frac{5}{2} a$

Dheeraj Sharma

Numerade Educator

Problem 95

A uniform sphere of mass M and radius $R$ is placed on
$a$
rough horizontal surface (Figure). The sphere is struck horizontally at a height $h$ the floor. Match the Column I with Column Il. front \begin{tabular}{|l|l|l|l|}
\hline Column I & \multicolumn{3}{|c|} { Column II } \\
\hline (A) & $h=\frac{R}{2}$ & (p) & Sphere rolls without slipping with a constant velocity and no loss of energy \\
\hline (B) & $h=R$ & (q) & Sphere spins clockwise, loses energy by friction. \\
\hline (C) & $h=\frac{3}{2} R$ & (r) & Sphere spins anti-clockwise, loses energy by friction. \\
\hline (D) & $h=\frac{7}{5} R$ & (s) & Sphere has only a translational motion, loses energy by friction. \\
\hline
\end{tabular}
(a) $A-r_{v} B-s, C-q, D-p$
(b) $A-5, B-p, C-r, D-q$
(c) $A-q, B-r, C-p, D-s$
(d) $A-p, B-q, C-s, D-r$

Nikhil Choudhary

Nikhil Choudhary

Numerade Educator

Problem 96

equal, then
(a) Kinetic energy of $A=$ Kinetic energy of $B$
(b) Kinetic energy of $A>$ Kinetic energy of $B$
(c) Kinctic energy of $A<$ Kinetic energy of $B$
(d) Kinetic energy of the two bodies cannot be compared with the given data

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 97

A body is rolling down an inclined plane. If kinetic energy of rotation is $40 \%$ of kinetic energy in translatory state, then the body is a
(a) ring
(b) cylinder
(c) hollow ball
(d) solid ball

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 98

A wheel of mass $5 \mathrm{~kg}$ and radius $0.40 \mathrm{~m}$ is rolling on a road without sliding with angular velocity $10 \mathrm{rad} \mathrm{s}^{-1}$. The moment of inertia of the whed about the axis of rotation is $0.65 \mathrm{~kg} \mathrm{~m}^{2}$. The percentage of kinetic energy of rotation in the total kinetic energy of the wheel is
(a) $22.4 \%$
(b) $11.2 \%$
(c) $88.8 \%$
(d) $44.8 \%$

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 99

Three bodies, a ring, a solid cylinder and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of the bodies are identical. Which of the bodies reaches the ground with maximum velocity?
(a) Ring
(b) Solid cylinder
(c) Solid sphere
(d) All reach the ground with same velocity

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator

Problem 100

A hoop of radius $2 \mathrm{~m}$ weighs $100 \mathrm{~kg}$. It rolls along a horizontal floor so that its centre of mass has a speed of $20 \mathrm{~cm} \mathrm{~s}^{-1}$. How much work has to be done to stop it?
(a) 2]
(b) $4 \mathrm{~T}$
(c) $6 \mathrm{~J}$
(d) $8 \mathrm{~J}$

Mohammed Nadhir

Mohammed Nadhir

Numerade Educator