Objective Physics for NEET

Abhay Kumar

Chapter 6

Centre of Mass and Collision - all with Video Answers

Educators

Chapter Questions

Problem 1

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

Dheeraj Sharma

Numerade Educator

Problem 2

The centre of mass of a body
(a) depends on the choice of coordinate system
(b) is independent of the choice of coordinate system
(c) may or may not depend on the choice of coordinate system
(d) None of these

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Numerade Educator

Problem 3

If the origin of co-ordinate system lies at the centre of mass, the sum of the moments of the masses of the system about the centre of mass
(a) may be greater than zero
(b) may be less than zero
(c) may be equal to zero
(d) is always zero

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Numerade Educator

Problem 4

The centre of mass of a body is defined as the point at which the whole of its mass is supposed to be concentrated, while centre of gravity of a body is defined as the point at which whole of its weight is supposed to be concentrated then,
(a) the centre of gravity always coincides with the centre of mass
(b) the centre of gravity may lie slightly below the centre of mass
(c) the centre of gravity may lie slightly above the centre of mass
(d) None of these

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Numerade Educator

Problem 5

The centre of mass of a system of particles does not depend on
(a) masses of the particles
(b) internal forces acting on the particles
(c) position of the particles
(d) relative distances between the particles

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Numerade Educator

Problem 6

The motion of the centre of mass of system of two particles is not affected by the internal forces
(a) irrespective of their directions
(b) only when they act along the line joining the particles
(c) only when the forces are perpendicular to each other
(d) when the angle between the lines of action of the forces lies between $0^{\circ}$ and $90^{\circ}$

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Numerade Educator

Problem 7

Two particles of masses $m_{1}$ and $m_{2}$ separated by a distance $d$ are at rest initially. If they move towards each other under mutual interaction (say electric, gravitational or elastic), where will they meet?
(a) At the centre of line joining the two particles
(b) Anywhere in between two masses
(c) At the centre of mass of the system of two particles
(d) None of these

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Numerade Educator

Problem 8

Two bodies $A$ and $B$ are attracted towards each other due to gravitation. Given that $A$ is much heavier than $B$, which of the following correctly describes the relative motion of the centre of mass of the bodies?
(a) It moves towards $A$
(b) It remains at rest with respect to $A$ as well as $B$
(c) It moves towards $B$
(d) It moves perpendicular to the line joining the particles

Saman Zulfiqar

Numerade Educator

Problem 9

The position of centre of mass of a system consisting of two particles of masses $m_{1}$ and $m_{2}$ separated by a distance $L$ apart, from $m_{1}$ will be
(a) $\frac{m_{1} L}{m_{1}+m_{2}}$
(b) $\frac{m_{2} L}{m_{1}+m_{2}}$
(c) $\frac{m_{2}}{m_{1}} L$
(d) $\frac{L}{2}$

Dheeraj Sharma

Numerade Educator

Problem 10

A system consists of mass $M$ and $m(\ll \angle M)$. The centre of mass of the system is
(a) at the middle
(b) nearer to $M$
(c) nearer to $\bar{m}$
(d) at the position of larger mass

Dheeraj Sharma

Numerade Educator

Problem 11

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

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Numerade Educator

Problem 12

Three particles of masses $1 \mathrm{~kg}, 2 \mathrm{~kg}$ and $3 \mathrm{~kg}$ are situated at the corners of an equilateral triangle of side $b$. The coordinates of the centre of mass are
(a) $\left[0, \frac{7 b}{12}, \frac{3 \sqrt{3} b}{12}\right]$
(b) $\left[\frac{3 \sqrt{3} b}{12}, \frac{7 b}{12}, 0\right]$
(c) $\left[\frac{7 b}{12}, \frac{3 \sqrt{3} b}{12}, 0\right]$
(d) $\left[\frac{7 b}{12}, 0, \frac{3 \sqrt{3} b}{12}\right]$

Dheeraj Sharma

Numerade Educator

Problem 13

3 particles of masses $2 \mathrm{~kg}$ each are placed such that 1 st one lies on $-$ ve $x$ -axis, 2 nd one lies on -ve $y$ -axis and the third one lies on $+$ ve $z$ -axis at distances of $2 \mathrm{~m}, 3 \mathrm{~m}$ and $1 \mathrm{~m}$ respectively from the origin. Then the square of the distance of centre of mass of the system from the origin is
(a) $1.55 \mathrm{~m}^{2}$
(b) $\sqrt{1.55} \mathrm{~m}^{2}$
(c) $1.55 \mathrm{~m}$
(d) $1.25 \mathrm{~m}^{2}$

Dheeraj Sharma

Numerade Educator

Problem 14

Two particles of masses $4 \mathrm{~kg}$ and $8 \mathrm{~kg}$ are separated by a distance of $12 \mathrm{~m}$. If they are moving towards each other under the influence of a mutual force of attraction, then the two particles will meet each other at a distance of
(a) $6 \mathrm{~m}$ from $8 \mathrm{~kg}$ mass
(b) $2 \mathrm{~m}$ from $8 \mathrm{~kg}$ mass
(c) $4 \mathrm{~m}$ from $8 \mathrm{~kg}$ mass
(d) $8 \mathrm{~m}$ from $8 \mathrm{~kg}$ mass

Dheeraj Sharma

Numerade Educator

Problem 15

A man of mass $m$ climbs a rope of length $L$ suspended below a balloon of mass $M$. The balloon is stationary with respect to ground. If the man begins to climb up the rope at a speed $v_{\text {rel }}$ (relative to rope) in what direction and with what speed (relative to ground) will the balloon move?
(a) $\vec{V}=\frac{m}{M} \vec{v}_{\mathrm{rel}}$
(b) $\vec{V}=-\frac{m}{M} \vec{v}_{\text {rel }}$
(c) $\vec{V}=-\frac{m}{m+M} \vec{v}_{\mathrm{rel}}$
(d) $\vec{V}=+\frac{m}{m+M} \vec{v}_{\text {rel }}$

Dheeraj Sharma

Numerade Educator

Problem 16

Two particles of masses $m_{1}$ and $m_{2}\left(m_{1}>m_{2}\right)$ attract each other with a force inversely proportional to the square of the distance between them. The particles are initially held at rest and then released. Which one is correct?
(a) The CM moves towards $m_{1}$
(b) The CM moves towards $m_{2}$
(c) The CM remains at rest
(d) The CM moves at right angles to the line joining $m_{1}$ and $m_{2}$

Saman Zulfiqar

Numerade Educator

Problem 17

Two bodies of mass $10 \mathrm{~kg}$ and $2 \mathrm{~kg}$ are moving with velocities $2 \hat{i}-7 \hat{j}+3 \hat{k}$ and $-10 \hat{i}+35 \hat{j}-3 \hat{k} \mathrm{~m} / \mathrm{s}$ respec-
tively. The velocity of their centre of mass is
(a) $2 \hat{i} \mathrm{~m} / \mathrm{s}$
(b) $2 \hat{k} \mathrm{~m} / \mathrm{s}$
(c) $(2 \hat{j}+2 \hat{k}) \mathrm{m} / \mathrm{s}$
(d) $(2 \hat{i}+2 \hat{j}+2 \hat{k}) \mathrm{m} / \mathrm{s}$

Dheeraj Sharma

Numerade Educator

Problem 18

The particles attract each other and are permitted to move towards each other along the line joining their centres of mass. At a particular moment of time their speeds are $v$ and $2 v$. What is the speed, if their common centre of mass at this instant?
(a) zero
(b) $v$
(c) $1.5 v$
(d) $3 v$

Saman Zulfiqar

Numerade Educator

Problem 19

Two spheres of masses $2 M$ and $M$ are initially at rest at a distance $R$ apart. Due to mutual force of attraction they approach each other. When they are at separation $R / 2$, the acceleration of the centre of mass of sphere would be
(a) zero
(b) $g \mathrm{~m} / \mathrm{s}^{2}$
(c) $3 g \mathrm{~m} / \mathrm{s}^{2}$
(d) $12 g \mathrm{~m} / \mathrm{s}^{2}$

Saman Zulfiqar

Numerade Educator

Problem 20

A circular ring of mass $6 \mathrm{~kg}$ and radius $a$ is placed such that its centre lies at the origin. Two particles of masses $2 \mathrm{~kg}$ each are placed at the intersecting points of the circle with $+$ ve $x$ -axis and +ve $y$ -axis. Then the angle made by the position vector of centre of mass of entire system with $x$ -axis is
(a) $45^{\circ}$
(b) $60^{\circ}$
(c $\tan ^{-1}(4 / 5)$
(d) $30^{\circ}$

Dheeraj Sharma

Numerade Educator

Problem 21

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 1 st system?
(a) $(0,0,0)$
(b) $(1,3,2)$
(c) $(-1,2,3)$
(d) $(3,1,8)$

Dheeraj Sharma

Numerade Educator

Problem 22

Mass is distributed uniformly over a thin square plate. If two end points of a diagonal are $(-2,0)$ and $(2,2)$, what are the co-ordinates of the centre of mass of plate?
(a) $(2,1)$
(b) $(2,2)$
(c) $(1,0)$
(d) $(0,1)$

Dheeraj Sharma

Numerade Educator

Problem 23

Mass is distributed uniformly over a thin triangular plate and positions of two vertices are given by $(1,3)$ and $(2,-4) .$ What is the position of 3 rd vertex if centre of mass of the plate lies at the origin?
(a) $(1,-2)$
(b) $(-2,4)$
(c) $(-3,1)$
(d) $(1,2)$

Dheeraj Sharma

Numerade Educator

Problem 24

If the net external force acting on the system of particles is zero, then which of the following may vary?
(a) Momentum of the system
(b) Kinetic energy of the system
(c) Velocity of centre of mass
(d) Position of centre of mass

Saman Zulfiqar

Numerade Educator

Problem 25

The two bodies of masses $m_{1}$ and $m_{2}\left(m_{1}>m_{2}\right)$ respectively are tied to the ends of a string which passes over a light frictionless pulley. The masses are initially at rest and released. The acceleration of the centre of mass is
(a) $\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2} g$
(b) $\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right) g$
(c) $g$
(d) zero

Saman Zulfiqar

Numerade Educator

Problem 26

Two particles of equal masses have velocities $\vec{v}_{1}=2 \hat{i} \mathrm{~m} / \mathrm{s}$ and $\vec{v}_{2}=2 \hat{j} \mathrm{~m} / \mathrm{s}$. The 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
(a) circle
(b) parabola
(c) straight line
(d) ellipse

Dheeraj Sharma

Numerade Educator

Problem 27

A man of mass $80 \mathrm{~kg}$ is riding on a small cart of mass $40 \mathrm{~kg}$ which is rolling along a level floor at a speed of 2 $\mathrm{m} / \mathrm{s}$. He is running on the cart so that his velocity relative to the cart is $3 \mathrm{~m} / \mathrm{s}$ in the direction opposite to the motion of the cart. What is the speed of the centre of mass of the system?
(a) $1.5 \mathrm{~m} / \mathrm{s}$
(b) $1 \mathrm{~m} / \mathrm{s}$
(c) $3 \mathrm{~m} / \mathrm{s}$
(d) Zero

Dheeraj Sharma

Numerade Educator

Problem 28

Mass is non-uniformly distributed on the circumference of a ring of radius $a$ and centre at origin. Let $b$ be the distance of the centre of mass of the ring from origin. Then
(a) $b=a$
(b) $0 \leq \underline{b} \leq a$
(c) $b<a$
(d) $b>a$

Dheeraj Sharma

Numerade Educator

Problem 29

The distance of the centre of mass of the T-shaped plate from $O$ is
(a) $7 \mathrm{~m}$
(b) $2.7 \mathrm{~m}$
(c) $4 \mathrm{~m}$
(d) $1 \mathrm{~m}$

Saman Zulfiqar

Numerade Educator

Problem 30

A circular plate of uniform thickness has a diameter of $56 \mathrm{~cm}$. A circular portion of diameter $42 \mathrm{~cm}$ is removed from one edge as shown in the figure. The centre of mass of the remaining portion from the centre of plate will be
(a) $5 \mathrm{~cm}$
(b) $7 \mathrm{~cm}$
(c) $9 \mathrm{~cm}$
(d) $11 \mathrm{~cm}$

Saman Zulfiqar

Numerade Educator

Problem 31

A block $Q$ of mass $M$ is placed on a horizontal frictionless surface $A B$ and a body $P$ of mass $m$ is released on its frictionless slope. As $P$ slides by a length $L$ on this slope of inclination $\theta$, the block $Q$ would slide by a distance
(a) $\frac{m}{M} L \cos \theta$
(b) $\frac{m}{M+m} L$
(c) $\frac{M+m}{m L \cos \theta}$
(d) $\frac{m L \cos \theta}{m+M}$

Saman Zulfiqar

Numerade Educator

Problem 32

If linear density of a rod of length $3 \mathrm{~m}$ varies as $\lambda=2+$ $x$, then the position of the centre of mass of the rod is
(a) $\frac{7}{3} \mathrm{~m}$
(b) $\frac{12}{7} \mathrm{~m}$
(c) $\frac{10}{7} \mathrm{~m}$
(d) $\frac{9}{7} \mathrm{~m}$

Saman Zulfiqar

Numerade Educator

Problem 33

Four point masses $P, Q, R$ and $S$ with respective masses $1 \mathrm{~kg}, 1 \mathrm{~kg}, 2 \mathrm{~kg}$ and $2 \mathrm{~kg}$ from the corners of a square of side $a$. The centre of mass of the system will be farthest from
(a) $P$ only
(b) $R$ ans $S$
(c) $R$ only
(d) $P$ and $Q$

Saman Zulfiqar

Numerade Educator

Problem 34

Particles of masses $m, 2 m, 3 m, \ldots, n m$ grams are placed on the same line at distances $l, 2 l, 3 l, \ldots, n l \mathrm{~cm}$ from a fixed point. The distance of centre of mass of the particles from the fixed point in centimetres is
(a) $\frac{(2 n+1) l}{3}$
(b) $\frac{l}{n+1}$
(c) $\frac{n\left(n^{2}+1\right) l}{3}$
(d) $\frac{2 l}{n\left(n^{2}+1\right)}$

Saman Zulfiqar

Numerade Educator

Problem 35

A sphere $P$ of mass $\mathrm{m}$ and velocity $\vec{v}$ undergoes an oblique and perfectly elastic collision with an identical sphere $Q$ initially at rest. The angle $\theta$ between the velocities of the spheres after the collision shall be
(a) 0
(b) $45^{\circ}$
(c) $90^{\circ}$
(d) $180^{\circ}$

Saman Zulfiqar

Numerade Educator

Problem 36

A billiards player hits a stationary ball by an identical ball to pocket the target ball in a corner pocket that is at an angle of $35^{\circ}$ with respect to the direction of motion of the first ball. Assuming the collision as elastic and that friction and rotational motion are not important, the angle made by the target ball with respect to the incoming ball is
(a) $35^{\circ}$
(b) $50^{\circ}$
(c) $55^{\circ}$
(d) $60^{\circ}$

Saman Zulfiqar

Numerade Educator

Problem 37

A particle falls from a height $h$ upon a fixed horizontal plane and rebounds. If $e$ is the coefficient of restitution, the total distance travelled before rebounding has stopped is
(a) $h\left(\frac{1+e^{2}}{1-e^{2}}\right)$
(b) $h\left(\frac{1-e^{2}}{1+e^{2}}\right)$
(c) $\frac{h}{2}\left(\frac{1-e^{2}}{1+e^{2}}\right)$
(d) $\frac{h}{2}\left(\frac{1+e^{2}}{1-e^{2}}\right)$

Saman Zulfiqar

Numerade Educator

Problem 38

A body of mass $M_{1}$ collides elastically with another mass $M_{2}$ at rest. There is maximum transfer of energy when
(a) $M_{1}>M_{2}$
(b) $M_{1}<M_{2}$
(c) $M_{1}=M_{2}$
(d) Same for all values of $M_{1}$ and $M_{2}$

Saman Zulfiqar

Numerade Educator

Problem 39

A lorry and a car moving with the same $\mathrm{KE}$ are brought to rest by applying the same retarding force, then
(a) Lorry will come to rest in a shorter distance
(b) Car will come to rest in a shorter distance
(c) Both come to rest in a same distance
(d) None of these

Saman Zulfiqar

Numerade Educator

Problem 40

An open knife edge of mass $m$ is dropped from a height $h$ on a wooden floor. If the blade penetrates upto the depth $d$ into the wood, the average resistance offered by the wood to the knife edge is
(a) $m g$
(b) $m g\left(1-\frac{h}{d}\right)$
(c) $m g\left(1+\frac{h}{d}\right)$
(d) $m g\left(1+\frac{h}{d}\right)^{2}$

Saman Zulfiqar

Numerade Educator