Centre of mass questions practice NEET Notes | EduRev

NEET : Centre of mass questions practice NEET Notes | EduRev

 Page 1


Topic Page No.
Theory 01 - 02
Exercise - 1 04 - 13
Exercise - 2 13 - 20
Exercise - 3 21 - 25
Exercise - 4   25
Answer Key 26 - 27
Contents
Syllabus
CENTRE OF MASS
Name : ____________________________ Contact No. __________________
Systems of particles ; Centre of mass and its motion ; Impulse ;
Elastic and inelastic collisions.
ETOOSINDIA.COM
India's No.1 Online Coaching for JEE Main & Advanced
3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
Page 2


Topic Page No.
Theory 01 - 02
Exercise - 1 04 - 13
Exercise - 2 13 - 20
Exercise - 3 21 - 25
Exercise - 4   25
Answer Key 26 - 27
Contents
Syllabus
CENTRE OF MASS
Name : ____________________________ Contact No. __________________
Systems of particles ; Centre of mass and its motion ; Impulse ;
Elastic and inelastic collisions.
ETOOSINDIA.COM
India's No.1 Online Coaching for JEE Main & Advanced
3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
CENTRE OF MASS (Advanced) # 1
CENTRE OF MASS
CENTRE OF MASS MOMENTUM & COLLISION
The action of force with respect to time is defined in terms of Impulse, that is,
I = 
?
Fdt = mv
f
 ? mv
i
 =?p
In the absence of a net external force, the momentum of a system is conserved.
i.e.
dt
dP
= F
ext
 = 0
p = p
1
 + p
2
 + ............+ p
N
 = constant
1. Collision is a kind of interaction between two or more bodies which come in contact with each other for a
very short time interval.
2. Types of collision: Elastic and Inelastic
Collisions may be either elastic or inelastic. Linear momentum is conserved in both cases.
(i) A perfectly elastic collision is defined as one in which the total kinetic energy of the system is conserved.
(ii) In an inelastic collision, the total kinetic energy of the system changes.
(iii) In a completely inelastic collision, the two bodies couple or stick togehter.
3. Coefficient of Restitution : It is defined as the ratio of the velocity of separation to the velocity of approach
of the two colliding bodies.
e = 
approach of velocity . rel
separation of velocity . rel
For a perfectly elastic collision, e = 1
For an inelastic collision, 0 < e < 1
For completely inelastic collision, e = 0
Note that the velocity of approach and the velocity of separation are always taken along the normal to the
striking surface.
CENTRE OF MASS
1. Discrete System : The position vector of the centre of mass is
r
c 
= 
n 2 1
n n 2 2 1 1
m ......... m m
r m ......... r m r m
? ?
? ? ?
where 
n 2 1
r ,..., r , r
? ? ?
are the position vectors of masses m
1
, m
2
, ...., m
n
 respectively. .
The components of the position vector of centre of mass are defined as
x
c
 = 
M
x m
i i ?
; y
c 
= 
M
y m
i i ?
; z
c 
= 
M
z m
i i ?
2. Continuous system : The centre of mass of a continuous body is defined as
?
? dm r
M
1
r
c
?
In the component form
x
c
 = 
?
dm x
M
1
; y
c
 = 
?
dm y
M
1
; z
c
 = 
?
dm z
M
1
ETOOSINDIA.COM
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3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
Page 3


Topic Page No.
Theory 01 - 02
Exercise - 1 04 - 13
Exercise - 2 13 - 20
Exercise - 3 21 - 25
Exercise - 4   25
Answer Key 26 - 27
Contents
Syllabus
CENTRE OF MASS
Name : ____________________________ Contact No. __________________
Systems of particles ; Centre of mass and its motion ; Impulse ;
Elastic and inelastic collisions.
ETOOSINDIA.COM
India's No.1 Online Coaching for JEE Main & Advanced
3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
CENTRE OF MASS (Advanced) # 1
CENTRE OF MASS
CENTRE OF MASS MOMENTUM & COLLISION
The action of force with respect to time is defined in terms of Impulse, that is,
I = 
?
Fdt = mv
f
 ? mv
i
 =?p
In the absence of a net external force, the momentum of a system is conserved.
i.e.
dt
dP
= F
ext
 = 0
p = p
1
 + p
2
 + ............+ p
N
 = constant
1. Collision is a kind of interaction between two or more bodies which come in contact with each other for a
very short time interval.
2. Types of collision: Elastic and Inelastic
Collisions may be either elastic or inelastic. Linear momentum is conserved in both cases.
(i) A perfectly elastic collision is defined as one in which the total kinetic energy of the system is conserved.
(ii) In an inelastic collision, the total kinetic energy of the system changes.
(iii) In a completely inelastic collision, the two bodies couple or stick togehter.
3. Coefficient of Restitution : It is defined as the ratio of the velocity of separation to the velocity of approach
of the two colliding bodies.
e = 
approach of velocity . rel
separation of velocity . rel
For a perfectly elastic collision, e = 1
For an inelastic collision, 0 < e < 1
For completely inelastic collision, e = 0
Note that the velocity of approach and the velocity of separation are always taken along the normal to the
striking surface.
CENTRE OF MASS
1. Discrete System : The position vector of the centre of mass is
r
c 
= 
n 2 1
n n 2 2 1 1
m ......... m m
r m ......... r m r m
? ?
? ? ?
where 
n 2 1
r ,..., r , r
? ? ?
are the position vectors of masses m
1
, m
2
, ...., m
n
 respectively. .
The components of the position vector of centre of mass are defined as
x
c
 = 
M
x m
i i ?
; y
c 
= 
M
y m
i i ?
; z
c 
= 
M
z m
i i ?
2. Continuous system : The centre of mass of a continuous body is defined as
?
? dm r
M
1
r
c
?
In the component form
x
c
 = 
?
dm x
M
1
; y
c
 = 
?
dm y
M
1
; z
c
 = 
?
dm z
M
1
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3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
CENTRE OF MASS (Advanced) # 2
3. Centre of Mass of Some Common Systems :
(i) A system of two point masses.
The centre of mass lie closer to the heavier mass.
(ii) A circular cone
y
c 
= 
4
h
(iii) A semi-circular ring
y
c
 = 
?
R 2
 ;  x
c 
= 0
(iv) A semi-circular disc
y
c
 = 
? 3
R 4
 ; x
c
 = 0
(v) A hemispherical shell
y
c
 = 
2
R
 ; x
c
 = 0
(vi) A solid hemisphere
y
c 
= 
8
R 3
 ; x
c
 = 0
4. Motion of the centre of mass :
(i) Velocity : The instantaneous velocity of the centre of mass is defined as
v
c
 = 
M
v m
i i ?
(ii) Acceleration : The acceleration of the centre of mass is defined as
a
c
 = 
M
a m
i i ?
(iii) Momentum : The total momentum of a system of particles is
p = Mv
c
(iv) Kinetic Energy : The kinetic energy of a system of particles consisits of two parts.
K = K
c
 + K?
where K
c
 = 
2
c
Mv
2
1
, kinetic energy due to motion of c.m. relative to the fixed origin O,
and K? = 
?
2
i i
v m
2
1
, kinetic energy of the particles relative to the c.m.
Note that the term K? may involve translational, rotational or vibrational energies relative to the centre of
mass.
5. Newon?s Laws of a system of particles : The first and second laws of motion for a system of particles are
modified as :
First law : The centre of mass of an isolated system is at rest or moves with constant velocity.
Second law : The net  external force acting on a system of total of mass M is related to the acceleration of
centre of mass of the system.
cm ext
a M F
?
?
?
?
ETOOSINDIA.COM
India's No.1 Online Coaching for JEE Main & Advanced
3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
Page 4


Topic Page No.
Theory 01 - 02
Exercise - 1 04 - 13
Exercise - 2 13 - 20
Exercise - 3 21 - 25
Exercise - 4   25
Answer Key 26 - 27
Contents
Syllabus
CENTRE OF MASS
Name : ____________________________ Contact No. __________________
Systems of particles ; Centre of mass and its motion ; Impulse ;
Elastic and inelastic collisions.
ETOOSINDIA.COM
India's No.1 Online Coaching for JEE Main & Advanced
3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
CENTRE OF MASS (Advanced) # 1
CENTRE OF MASS
CENTRE OF MASS MOMENTUM & COLLISION
The action of force with respect to time is defined in terms of Impulse, that is,
I = 
?
Fdt = mv
f
 ? mv
i
 =?p
In the absence of a net external force, the momentum of a system is conserved.
i.e.
dt
dP
= F
ext
 = 0
p = p
1
 + p
2
 + ............+ p
N
 = constant
1. Collision is a kind of interaction between two or more bodies which come in contact with each other for a
very short time interval.
2. Types of collision: Elastic and Inelastic
Collisions may be either elastic or inelastic. Linear momentum is conserved in both cases.
(i) A perfectly elastic collision is defined as one in which the total kinetic energy of the system is conserved.
(ii) In an inelastic collision, the total kinetic energy of the system changes.
(iii) In a completely inelastic collision, the two bodies couple or stick togehter.
3. Coefficient of Restitution : It is defined as the ratio of the velocity of separation to the velocity of approach
of the two colliding bodies.
e = 
approach of velocity . rel
separation of velocity . rel
For a perfectly elastic collision, e = 1
For an inelastic collision, 0 < e < 1
For completely inelastic collision, e = 0
Note that the velocity of approach and the velocity of separation are always taken along the normal to the
striking surface.
CENTRE OF MASS
1. Discrete System : The position vector of the centre of mass is
r
c 
= 
n 2 1
n n 2 2 1 1
m ......... m m
r m ......... r m r m
? ?
? ? ?
where 
n 2 1
r ,..., r , r
? ? ?
are the position vectors of masses m
1
, m
2
, ...., m
n
 respectively. .
The components of the position vector of centre of mass are defined as
x
c
 = 
M
x m
i i ?
; y
c 
= 
M
y m
i i ?
; z
c 
= 
M
z m
i i ?
2. Continuous system : The centre of mass of a continuous body is defined as
?
? dm r
M
1
r
c
?
In the component form
x
c
 = 
?
dm x
M
1
; y
c
 = 
?
dm y
M
1
; z
c
 = 
?
dm z
M
1
ETOOSINDIA.COM
India's No.1 Online Coaching for JEE Main & Advanced
3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
CENTRE OF MASS (Advanced) # 2
3. Centre of Mass of Some Common Systems :
(i) A system of two point masses.
The centre of mass lie closer to the heavier mass.
(ii) A circular cone
y
c 
= 
4
h
(iii) A semi-circular ring
y
c
 = 
?
R 2
 ;  x
c 
= 0
(iv) A semi-circular disc
y
c
 = 
? 3
R 4
 ; x
c
 = 0
(v) A hemispherical shell
y
c
 = 
2
R
 ; x
c
 = 0
(vi) A solid hemisphere
y
c 
= 
8
R 3
 ; x
c
 = 0
4. Motion of the centre of mass :
(i) Velocity : The instantaneous velocity of the centre of mass is defined as
v
c
 = 
M
v m
i i ?
(ii) Acceleration : The acceleration of the centre of mass is defined as
a
c
 = 
M
a m
i i ?
(iii) Momentum : The total momentum of a system of particles is
p = Mv
c
(iv) Kinetic Energy : The kinetic energy of a system of particles consisits of two parts.
K = K
c
 + K?
where K
c
 = 
2
c
Mv
2
1
, kinetic energy due to motion of c.m. relative to the fixed origin O,
and K? = 
?
2
i i
v m
2
1
, kinetic energy of the particles relative to the c.m.
Note that the term K? may involve translational, rotational or vibrational energies relative to the centre of
mass.
5. Newon?s Laws of a system of particles : The first and second laws of motion for a system of particles are
modified as :
First law : The centre of mass of an isolated system is at rest or moves with constant velocity.
Second law : The net  external force acting on a system of total of mass M is related to the acceleration of
centre of mass of the system.
cm ext
a M F
?
?
?
?
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3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
CENTRE OF MASS (Advanced) # 3
PART - I : OBJECTIVE QUESTIONS
* Marked Questions are having more than one correct option.
SECTION (A) : CALCULATION OF CENTRE OF MASS
A-1. A thin uniform wire is bent to form the two equal sides AB and AC of triangle ABC, where
AB = AC = 5 cm. The third side BC, of length 6cm, is made from uniform wire of twice the density of the
first. The distance of centre of mass from A is :
(A) 
11
34
cm (B) 
34
11
 cm (C) 
9
34
cm (D) 
45
11
 cm
A-2. All the particles of a system are situated at a distance r from the origin. The distance of the centre of
mass of the system from the origin is
(A) = r (B) ? r (C) > r (D) ? r
A-3. A hemisphere and a solid cone have a common base. The centre of mass of the common structure
coincides with the centre of the common base. If R is the radius of hemisphere and h is height of the
cone, then
(A) 
3
h
R
?
(B) 
1
3
h
R
?
(C) 
3
h
R
?
(D) 
1
3
h
R
?
A-4. Five homogeneous bricks, each of length L, are arranged as shown in figure. Each brick is displaced
with respect to the one in contact by L/5. Find the x-coordinate of the centre of mass relative to the
origin O shown.
(A) 
33 L
25
(B)
11 L
25
(C)
22 L
25
(D) 
50
L 33
A-5. ABC is a part of ring having radius R
2
 and ADC is a part of disc having inner radius R
1
 and outer R
2
. Part
ABC and ADC have same mass. Then center of mass will be located, from the centre O.
(A) 
2 1 1 2
1 2
(R ? R )(2R R )
3 (R R )
?
? ?
 (above) (B) 
2 1 1 2
1 2
(R ? R )(2R R )
3 (R R )
?
? ?
(below)
(C) 
1 2
2R R
3
?
?
(above) (D) 
1 2
2R R
3
?
?
(below)
ETOOSINDIA.COM
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Page 5


Topic Page No.
Theory 01 - 02
Exercise - 1 04 - 13
Exercise - 2 13 - 20
Exercise - 3 21 - 25
Exercise - 4   25
Answer Key 26 - 27
Contents
Syllabus
CENTRE OF MASS
Name : ____________________________ Contact No. __________________
Systems of particles ; Centre of mass and its motion ; Impulse ;
Elastic and inelastic collisions.
ETOOSINDIA.COM
India's No.1 Online Coaching for JEE Main & Advanced
3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
CENTRE OF MASS (Advanced) # 1
CENTRE OF MASS
CENTRE OF MASS MOMENTUM & COLLISION
The action of force with respect to time is defined in terms of Impulse, that is,
I = 
?
Fdt = mv
f
 ? mv
i
 =?p
In the absence of a net external force, the momentum of a system is conserved.
i.e.
dt
dP
= F
ext
 = 0
p = p
1
 + p
2
 + ............+ p
N
 = constant
1. Collision is a kind of interaction between two or more bodies which come in contact with each other for a
very short time interval.
2. Types of collision: Elastic and Inelastic
Collisions may be either elastic or inelastic. Linear momentum is conserved in both cases.
(i) A perfectly elastic collision is defined as one in which the total kinetic energy of the system is conserved.
(ii) In an inelastic collision, the total kinetic energy of the system changes.
(iii) In a completely inelastic collision, the two bodies couple or stick togehter.
3. Coefficient of Restitution : It is defined as the ratio of the velocity of separation to the velocity of approach
of the two colliding bodies.
e = 
approach of velocity . rel
separation of velocity . rel
For a perfectly elastic collision, e = 1
For an inelastic collision, 0 < e < 1
For completely inelastic collision, e = 0
Note that the velocity of approach and the velocity of separation are always taken along the normal to the
striking surface.
CENTRE OF MASS
1. Discrete System : The position vector of the centre of mass is
r
c 
= 
n 2 1
n n 2 2 1 1
m ......... m m
r m ......... r m r m
? ?
? ? ?
where 
n 2 1
r ,..., r , r
? ? ?
are the position vectors of masses m
1
, m
2
, ...., m
n
 respectively. .
The components of the position vector of centre of mass are defined as
x
c
 = 
M
x m
i i ?
; y
c 
= 
M
y m
i i ?
; z
c 
= 
M
z m
i i ?
2. Continuous system : The centre of mass of a continuous body is defined as
?
? dm r
M
1
r
c
?
In the component form
x
c
 = 
?
dm x
M
1
; y
c
 = 
?
dm y
M
1
; z
c
 = 
?
dm z
M
1
ETOOSINDIA.COM
India's No.1 Online Coaching for JEE Main & Advanced
3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
CENTRE OF MASS (Advanced) # 2
3. Centre of Mass of Some Common Systems :
(i) A system of two point masses.
The centre of mass lie closer to the heavier mass.
(ii) A circular cone
y
c 
= 
4
h
(iii) A semi-circular ring
y
c
 = 
?
R 2
 ;  x
c 
= 0
(iv) A semi-circular disc
y
c
 = 
? 3
R 4
 ; x
c
 = 0
(v) A hemispherical shell
y
c
 = 
2
R
 ; x
c
 = 0
(vi) A solid hemisphere
y
c 
= 
8
R 3
 ; x
c
 = 0
4. Motion of the centre of mass :
(i) Velocity : The instantaneous velocity of the centre of mass is defined as
v
c
 = 
M
v m
i i ?
(ii) Acceleration : The acceleration of the centre of mass is defined as
a
c
 = 
M
a m
i i ?
(iii) Momentum : The total momentum of a system of particles is
p = Mv
c
(iv) Kinetic Energy : The kinetic energy of a system of particles consisits of two parts.
K = K
c
 + K?
where K
c
 = 
2
c
Mv
2
1
, kinetic energy due to motion of c.m. relative to the fixed origin O,
and K? = 
?
2
i i
v m
2
1
, kinetic energy of the particles relative to the c.m.
Note that the term K? may involve translational, rotational or vibrational energies relative to the centre of
mass.
5. Newon?s Laws of a system of particles : The first and second laws of motion for a system of particles are
modified as :
First law : The centre of mass of an isolated system is at rest or moves with constant velocity.
Second law : The net  external force acting on a system of total of mass M is related to the acceleration of
centre of mass of the system.
cm ext
a M F
?
?
?
?
ETOOSINDIA.COM
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3rd Floor, H.No.50 Rajeev Gandhi Nagar, Kota, Rajasthan 324005 
HelpDesk : Tel. 092142 33303
CENTRE OF MASS (Advanced) # 3
PART - I : OBJECTIVE QUESTIONS
* Marked Questions are having more than one correct option.
SECTION (A) : CALCULATION OF CENTRE OF MASS
A-1. A thin uniform wire is bent to form the two equal sides AB and AC of triangle ABC, where
AB = AC = 5 cm. The third side BC, of length 6cm, is made from uniform wire of twice the density of the
first. The distance of centre of mass from A is :
(A) 
11
34
cm (B) 
34
11
 cm (C) 
9
34
cm (D) 
45
11
 cm
A-2. All the particles of a system are situated at a distance r from the origin. The distance of the centre of
mass of the system from the origin is
(A) = r (B) ? r (C) > r (D) ? r
A-3. A hemisphere and a solid cone have a common base. The centre of mass of the common structure
coincides with the centre of the common base. If R is the radius of hemisphere and h is height of the
cone, then
(A) 
3
h
R
?
(B) 
1
3
h
R
?
(C) 
3
h
R
?
(D) 
1
3
h
R
?
A-4. Five homogeneous bricks, each of length L, are arranged as shown in figure. Each brick is displaced
with respect to the one in contact by L/5. Find the x-coordinate of the centre of mass relative to the
origin O shown.
(A) 
33 L
25
(B)
11 L
25
(C)
22 L
25
(D) 
50
L 33
A-5. ABC is a part of ring having radius R
2
 and ADC is a part of disc having inner radius R
1
 and outer R
2
. Part
ABC and ADC have same mass. Then center of mass will be located, from the centre O.
(A) 
2 1 1 2
1 2
(R ? R )(2R R )
3 (R R )
?
? ?
 (above) (B) 
2 1 1 2
1 2
(R ? R )(2R R )
3 (R R )
?
? ?
(below)
(C) 
1 2
2R R
3
?
?
(above) (D) 
1 2
2R R
3
?
?
(below)
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CENTRE OF MASS (Advanced) # 4
A-6. From the uniform disc of radius R; an equilateral triangle of side R 3 is cut as shown in the figure. The
new position of centre of mass is -
(A) (0,0) (B) (0, R) (C) 
3 R
0,
2
? ?
? ?
? ?
? ?
(D) None of these
SECTION (B) : MOTION OF CENTRE OF MASS
B-1. An object A is dropped from rest from the top of a 30 m high building and at the same moment another
object B is projected vertically upwards with an initial speed of 15 m/s from the base of the building.
Mass of the object A is 2 kg while mass of the object B is 4 kg. The maximum height above the ground
level attained by the centre of mass of the A and B system is (take g = 10 m/s
2
) :
(A) 15 m (B) 25 m (C) 30 m (D) 35 m
B-2. Two particles having mass ratio n : 1 are interconnected by a light inextensible string that passes over a
smooth pulley. If the system is released, then the acceleration of the centre of mass of the system is :
(A) (n ? 1)
2
 g (B) 
g
1 n
1 n
2
?
?
?
?
?
?
?
?
(C) 
g
1 n
1 n
2
?
?
?
?
?
?
?
?
(D) 
g
1 n
1 n
?
?
?
?
?
?
?
?
B-3. Inside a smooth spherical shell of radius R a ball of the same mass is released from the shown position
(fig.) Find the distance travelled by the shell on the horizontal floor when the ball comes to the just
opposite position of itself with respect to its initial position in the shell.
(A) 
3R
5
(B) 
R
4
(C) 
4
R 3
(D) 
5R
4
B-4. A block of mass M is tied to one end of a massless rope. The other end of the rope is in the hands of
a man of mass 2M as shown in the figure. the block and the man are resting on a rough wedge of mass
M as shown in the figure. The whole system is resting on a smooth horizontal surface. The man pulls
the rope. Pulley is massless and frictionless. What is the displacement of the wedge when the block
meets the pulley. (Man does not leave his position during the pull)
(A) 0.5 m (B) 1 m (C) Zero (D) 2 3 m
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