Circular Motion (NCERT) - Class 11 PDF Download

 Page 1


Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
When a particle moves in a plane such that its distance from a fixed (or moving) point remains constant
then its motion is called as the circular motion with respect to that fixed (or moving) point. That fixed
point is called centre and the distance between fixed point and particle is called radius.
v v v
?
1
?
2
B
A
The car is moving in a straight line with respect to the man A. But the man B continuously rotate his face
to see the car. So with respect to man A 
d
dt
?
? 0
But with respect to man B 
d
dt
?
? 0
Therefore we conclude that with respect to A the motion of car is straight line but for man B it has some
angular velocity
2. KINEMATICS OF CIRCULAR MOTION :
2.1 Variables of Motion :
(a) Angular Position :
The angle made by the position vector with given line (reference line) is called
angular position Circular motion is a two dimensional motion or motion in a plane.
Suppose a particle P is moving in a circle of radius r and centre O. The position of   
P'
P
r
x
O
?
? ?
Y
the particle P at a given instant may be described by the angle ? between OP and OX.
This angle ? is called the angular position of the particle. As the particle moves on the
circle its angular position ? change. Suppose the point rotates an angle ? ? in time ?t.
(b) Angular Displacement :
Definition :
Angle rotated by a position vector of the moving particle in a given time interval with some
 reference line is called its angular displacement.
______________________________________________________________________________________________
Important point :
• It is dimensionless and has proper unit SI unit radian while other units are degree or revolution 2 ? rad
= 360° = 1 rev
• Infinitely small angular displacement is a vector quantity but finite angular displacement is not because
the addition of the small angular displacement is cummutative while for large is not.
d
?
?
1
 + d
?
?
2
 = d
?
?
2
 + d
?
?
1
  but  ?
1
 + ?
2
 ?  ?
2
 + ?
1
• Direction of small angular displacement is decided by right hand thumb rule. When the fingers are
directed along the motion of the point then thumb will represents the direction of angular displacement.
• Angular displacement can be different for different observers
______________________________________________________________________________________________
CIRCULAR MOTION CIRCULAR MOTION
Page 2


Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
When a particle moves in a plane such that its distance from a fixed (or moving) point remains constant
then its motion is called as the circular motion with respect to that fixed (or moving) point. That fixed
point is called centre and the distance between fixed point and particle is called radius.
v v v
?
1
?
2
B
A
The car is moving in a straight line with respect to the man A. But the man B continuously rotate his face
to see the car. So with respect to man A 
d
dt
?
? 0
But with respect to man B 
d
dt
?
? 0
Therefore we conclude that with respect to A the motion of car is straight line but for man B it has some
angular velocity
2. KINEMATICS OF CIRCULAR MOTION :
2.1 Variables of Motion :
(a) Angular Position :
The angle made by the position vector with given line (reference line) is called
angular position Circular motion is a two dimensional motion or motion in a plane.
Suppose a particle P is moving in a circle of radius r and centre O. The position of   
P'
P
r
x
O
?
? ?
Y
the particle P at a given instant may be described by the angle ? between OP and OX.
This angle ? is called the angular position of the particle. As the particle moves on the
circle its angular position ? change. Suppose the point rotates an angle ? ? in time ?t.
(b) Angular Displacement :
Definition :
Angle rotated by a position vector of the moving particle in a given time interval with some
 reference line is called its angular displacement.
______________________________________________________________________________________________
Important point :
• It is dimensionless and has proper unit SI unit radian while other units are degree or revolution 2 ? rad
= 360° = 1 rev
• Infinitely small angular displacement is a vector quantity but finite angular displacement is not because
the addition of the small angular displacement is cummutative while for large is not.
d
?
?
1
 + d
?
?
2
 = d
?
?
2
 + d
?
?
1
  but  ?
1
 + ?
2
 ?  ?
2
 + ?
1
• Direction of small angular displacement is decided by right hand thumb rule. When the fingers are
directed along the motion of the point then thumb will represents the direction of angular displacement.
• Angular displacement can be different for different observers
______________________________________________________________________________________________
CIRCULAR MOTION CIRCULAR MOTION
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
(c) Angular Velocity ?
(i) Average Angular Velocity
?
av
  = 
Total Angle of Rotation
Total time taken
 ;
?
av
 = 
? ?
2 1
2 1
–
– t t
 = 
? ?
?t
where ?
1
 and ?
2
 are angular position of the particle at time t
1
 and t
2
 respectively.
(ii)  Instantaneous Angular Velocity
The rate at which the position vector of a particle with respect to the centre rotates, is called as
instantaneous angular velocity with respect to the centre.
? = 
lim
?t ?0
 
? ?
?t
 = 
d
dt
?
_______________________________________________________________________________________________
Important points :
• It is an axial vector with dimensions [T
–1
] and SI unit rad/s.
• For a rigid body as all points will rotate through same angle in same time, angular velocity is a
characteristic of the body as a whole, e.g., angular velocity of all points of earth about its own axis
is (2 ?/24) rad/hr.
• If a body makes ‘n’ rotations in ‘t’ seconds then angular velocity in radian per second will be
?
av
 = 
2 ?n
t
If T is the period and ‘f’ the frequency of uniform circular motion
?
av
 = 
2 1 ? ?
T
 = 2 ?f
• If ? ? ? = a – bt + ct
2
 then ? = 
d
dt
?
 = – b + 2ct
Relation between speed and angular velocity :
? = 
lim
?
? ?
? t t ? ?
 = 
d
dt
?
The rate of change of angular velocity is called the angular acceleration ( ?). Thus,
? = 
d
dt
?
 = 
2
2
t d
d ?
The linear distance PP’ travelled by the particle in time ?t is
Y
X
O
P
P'
? ?
r
?s = r ? ?   or  
lim
?t ?0
 
?
?
S
t
= 
r
t
lim
? ?0
? ?
? t
or  
?
?
s
t
 = r
d
dt
?
or   v = r ?
Here, v is the linear speed of the particle
It is only valid for circular motion
v = r ? is a scalar quantity (
?
?
? ?
v
r
)
Page 3


Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
When a particle moves in a plane such that its distance from a fixed (or moving) point remains constant
then its motion is called as the circular motion with respect to that fixed (or moving) point. That fixed
point is called centre and the distance between fixed point and particle is called radius.
v v v
?
1
?
2
B
A
The car is moving in a straight line with respect to the man A. But the man B continuously rotate his face
to see the car. So with respect to man A 
d
dt
?
? 0
But with respect to man B 
d
dt
?
? 0
Therefore we conclude that with respect to A the motion of car is straight line but for man B it has some
angular velocity
2. KINEMATICS OF CIRCULAR MOTION :
2.1 Variables of Motion :
(a) Angular Position :
The angle made by the position vector with given line (reference line) is called
angular position Circular motion is a two dimensional motion or motion in a plane.
Suppose a particle P is moving in a circle of radius r and centre O. The position of   
P'
P
r
x
O
?
? ?
Y
the particle P at a given instant may be described by the angle ? between OP and OX.
This angle ? is called the angular position of the particle. As the particle moves on the
circle its angular position ? change. Suppose the point rotates an angle ? ? in time ?t.
(b) Angular Displacement :
Definition :
Angle rotated by a position vector of the moving particle in a given time interval with some
 reference line is called its angular displacement.
______________________________________________________________________________________________
Important point :
• It is dimensionless and has proper unit SI unit radian while other units are degree or revolution 2 ? rad
= 360° = 1 rev
• Infinitely small angular displacement is a vector quantity but finite angular displacement is not because
the addition of the small angular displacement is cummutative while for large is not.
d
?
?
1
 + d
?
?
2
 = d
?
?
2
 + d
?
?
1
  but  ?
1
 + ?
2
 ?  ?
2
 + ?
1
• Direction of small angular displacement is decided by right hand thumb rule. When the fingers are
directed along the motion of the point then thumb will represents the direction of angular displacement.
• Angular displacement can be different for different observers
______________________________________________________________________________________________
CIRCULAR MOTION CIRCULAR MOTION
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
(c) Angular Velocity ?
(i) Average Angular Velocity
?
av
  = 
Total Angle of Rotation
Total time taken
 ;
?
av
 = 
? ?
2 1
2 1
–
– t t
 = 
? ?
?t
where ?
1
 and ?
2
 are angular position of the particle at time t
1
 and t
2
 respectively.
(ii)  Instantaneous Angular Velocity
The rate at which the position vector of a particle with respect to the centre rotates, is called as
instantaneous angular velocity with respect to the centre.
? = 
lim
?t ?0
 
? ?
?t
 = 
d
dt
?
_______________________________________________________________________________________________
Important points :
• It is an axial vector with dimensions [T
–1
] and SI unit rad/s.
• For a rigid body as all points will rotate through same angle in same time, angular velocity is a
characteristic of the body as a whole, e.g., angular velocity of all points of earth about its own axis
is (2 ?/24) rad/hr.
• If a body makes ‘n’ rotations in ‘t’ seconds then angular velocity in radian per second will be
?
av
 = 
2 ?n
t
If T is the period and ‘f’ the frequency of uniform circular motion
?
av
 = 
2 1 ? ?
T
 = 2 ?f
• If ? ? ? = a – bt + ct
2
 then ? = 
d
dt
?
 = – b + 2ct
Relation between speed and angular velocity :
? = 
lim
?
? ?
? t t ? ?
 = 
d
dt
?
The rate of change of angular velocity is called the angular acceleration ( ?). Thus,
? = 
d
dt
?
 = 
2
2
t d
d ?
The linear distance PP’ travelled by the particle in time ?t is
Y
X
O
P
P'
? ?
r
?s = r ? ?   or  
lim
?t ?0
 
?
?
S
t
= 
r
t
lim
? ?0
? ?
? t
or  
?
?
s
t
 = r
d
dt
?
or   v = r ?
Here, v is the linear speed of the particle
It is only valid for circular motion
v = r ? is a scalar quantity (
?
?
? ?
v
r
)
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
Ex.1 If ? depends on time t in following way
? = 2t
2
 + 3 then
(a) Find out ? average upto 3 sec. (b) ? at 3 sec
Sol. ?
avg
 = 
1 2
i f
t – t
–
time total
nt displaceme angular Total ? ?
?
?
f
 = 2 (3)
2
 + 3 = 21 rad
?
i 
= 2 (0) + 3 = 3 rad.
So, ?
avg
 = 
21 3
3
–
 = 6 rad/sec
?
instantaneous
 = 
d
dt
?
 = 4t
?
at t = 3 sec 
 = 4 × 3 = 12 rad/sec
(d) Relative Angular Velocity
Angular velocity is defined with respect to the point from which the position vector of the moving particle
is drawn Here angular velocity of the particle w.r.t. ‘O’ and ‘A’ will be different
O
P'
P
lin Ref
A
?
?
PO
d
dt
? ;  ?
?
PA
d
dt
?
Definition :
Relative angular velocity of a particle ‘A’ with respect to the other moving particle ‘B’ is the angular
velocity of the position vector of ‘A’ with respect to ‘B’. That means it is the rate at which position vector
of ‘A’ with respect to ‘B’ rotates at that instant
A
r
B
V
B
V
A
?
AB
AB
AB
V
r
?
?
( )
here 
V
AB ?
?
 Relative velocity 
?
 to position vector AB
     
B and A between Seperation
AB line to lar perpendicu B . t . r . w A of velocity lative Re
?
( ) sin sin V V V
AB A B ?
? ? ? ?
1 2
r r
AB
?
?
? ?
AB
A B
V V
r
?
? sin sin
1 2
_______________________________________________________________________________________
Page 4


Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
When a particle moves in a plane such that its distance from a fixed (or moving) point remains constant
then its motion is called as the circular motion with respect to that fixed (or moving) point. That fixed
point is called centre and the distance between fixed point and particle is called radius.
v v v
?
1
?
2
B
A
The car is moving in a straight line with respect to the man A. But the man B continuously rotate his face
to see the car. So with respect to man A 
d
dt
?
? 0
But with respect to man B 
d
dt
?
? 0
Therefore we conclude that with respect to A the motion of car is straight line but for man B it has some
angular velocity
2. KINEMATICS OF CIRCULAR MOTION :
2.1 Variables of Motion :
(a) Angular Position :
The angle made by the position vector with given line (reference line) is called
angular position Circular motion is a two dimensional motion or motion in a plane.
Suppose a particle P is moving in a circle of radius r and centre O. The position of   
P'
P
r
x
O
?
? ?
Y
the particle P at a given instant may be described by the angle ? between OP and OX.
This angle ? is called the angular position of the particle. As the particle moves on the
circle its angular position ? change. Suppose the point rotates an angle ? ? in time ?t.
(b) Angular Displacement :
Definition :
Angle rotated by a position vector of the moving particle in a given time interval with some
 reference line is called its angular displacement.
______________________________________________________________________________________________
Important point :
• It is dimensionless and has proper unit SI unit radian while other units are degree or revolution 2 ? rad
= 360° = 1 rev
• Infinitely small angular displacement is a vector quantity but finite angular displacement is not because
the addition of the small angular displacement is cummutative while for large is not.
d
?
?
1
 + d
?
?
2
 = d
?
?
2
 + d
?
?
1
  but  ?
1
 + ?
2
 ?  ?
2
 + ?
1
• Direction of small angular displacement is decided by right hand thumb rule. When the fingers are
directed along the motion of the point then thumb will represents the direction of angular displacement.
• Angular displacement can be different for different observers
______________________________________________________________________________________________
CIRCULAR MOTION CIRCULAR MOTION
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
(c) Angular Velocity ?
(i) Average Angular Velocity
?
av
  = 
Total Angle of Rotation
Total time taken
 ;
?
av
 = 
? ?
2 1
2 1
–
– t t
 = 
? ?
?t
where ?
1
 and ?
2
 are angular position of the particle at time t
1
 and t
2
 respectively.
(ii)  Instantaneous Angular Velocity
The rate at which the position vector of a particle with respect to the centre rotates, is called as
instantaneous angular velocity with respect to the centre.
? = 
lim
?t ?0
 
? ?
?t
 = 
d
dt
?
_______________________________________________________________________________________________
Important points :
• It is an axial vector with dimensions [T
–1
] and SI unit rad/s.
• For a rigid body as all points will rotate through same angle in same time, angular velocity is a
characteristic of the body as a whole, e.g., angular velocity of all points of earth about its own axis
is (2 ?/24) rad/hr.
• If a body makes ‘n’ rotations in ‘t’ seconds then angular velocity in radian per second will be
?
av
 = 
2 ?n
t
If T is the period and ‘f’ the frequency of uniform circular motion
?
av
 = 
2 1 ? ?
T
 = 2 ?f
• If ? ? ? = a – bt + ct
2
 then ? = 
d
dt
?
 = – b + 2ct
Relation between speed and angular velocity :
? = 
lim
?
? ?
? t t ? ?
 = 
d
dt
?
The rate of change of angular velocity is called the angular acceleration ( ?). Thus,
? = 
d
dt
?
 = 
2
2
t d
d ?
The linear distance PP’ travelled by the particle in time ?t is
Y
X
O
P
P'
? ?
r
?s = r ? ?   or  
lim
?t ?0
 
?
?
S
t
= 
r
t
lim
? ?0
? ?
? t
or  
?
?
s
t
 = r
d
dt
?
or   v = r ?
Here, v is the linear speed of the particle
It is only valid for circular motion
v = r ? is a scalar quantity (
?
?
? ?
v
r
)
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
Ex.1 If ? depends on time t in following way
? = 2t
2
 + 3 then
(a) Find out ? average upto 3 sec. (b) ? at 3 sec
Sol. ?
avg
 = 
1 2
i f
t – t
–
time total
nt displaceme angular Total ? ?
?
?
f
 = 2 (3)
2
 + 3 = 21 rad
?
i 
= 2 (0) + 3 = 3 rad.
So, ?
avg
 = 
21 3
3
–
 = 6 rad/sec
?
instantaneous
 = 
d
dt
?
 = 4t
?
at t = 3 sec 
 = 4 × 3 = 12 rad/sec
(d) Relative Angular Velocity
Angular velocity is defined with respect to the point from which the position vector of the moving particle
is drawn Here angular velocity of the particle w.r.t. ‘O’ and ‘A’ will be different
O
P'
P
lin Ref
A
?
?
PO
d
dt
? ;  ?
?
PA
d
dt
?
Definition :
Relative angular velocity of a particle ‘A’ with respect to the other moving particle ‘B’ is the angular
velocity of the position vector of ‘A’ with respect to ‘B’. That means it is the rate at which position vector
of ‘A’ with respect to ‘B’ rotates at that instant
A
r
B
V
B
V
A
?
AB
AB
AB
V
r
?
?
( )
here 
V
AB ?
?
 Relative velocity 
?
 to position vector AB
     
B and A between Seperation
AB line to lar perpendicu B . t . r . w A of velocity lative Re
?
( ) sin sin V V V
AB A B ?
? ? ? ?
1 2
r r
AB
?
?
? ?
AB
A B
V V
r
?
? sin sin
1 2
_______________________________________________________________________________________
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 4
Important points :
• If two particles are moving on the same circle or different coplanar concentric circles in same direction
with different uniform angular speed ?
A
 and ?
B
 respectively, the rate of change of angle between 
OA
?
and 
OB
?
 is
d
dt
B A
?
? ? ? ?
O
A
B
Initial line
O
A
B
Initial line
So the time taken by one to complete one revolution around O w.r.t. the other
T
T T
T T
rel
? ?
?
?
?
2 2
2 1
1 2
1 2
?
?
?
? ?
• If two particles are moving on two different concentric circles with different velocities then angular velocity
of B relative to A as observed by A will depend on their positions and velocities. consider the case when
A and B are closest to each other moving in same direction as shown in figure. In this situation
v v v v v
rel B A B A
? ? ? ? | |
? ?
r r r r r
rel B A B A
? ? ? ? | |
? ?
v
A
v
B
A
B
r
O
r
A
r
B
so,
?
BA
rel
rel
B A
B A
v
r
v v
r r
? ?
?
?
?
( )
( ) v
rel ?
= Relative velocity perpendicular to position vector
_______________________________________________________________________________________
Ex.2 Two particles move on a circular path (one just inside and the other just outside) with angular
velocities ? and 5 ? starting from the same point. Then, which is incorrect.
(a) they cross each other at regular intervals of time 
? ?
? ?
 when their angular velocities are
oppositely directed
(b) they cross each other at points on the path subtending an angle of 60° at the centre if their
angular velocities are oppositely directed
(c) they cross at intervals of time 
? ?
?
 if their angular velocities are oppositely directed
(d) they cross each other at points on the path subtending 90° at the centre if their angular
velocities are in the same sense
Sol. If the angular velocities are oppositely directed, they meet at intervals of
time t = 
2 ?
?
rel
 = 
2
6
?
?
 = 
?
? 3
Angle subtended at the centre by the crossing points
? = ?t = 
?
3
 = 60°
When their angular velocities are in the same direction,
t’ = 
2 ?
?
rel
 = 
2
4
?
?
 = 
?
? 2
 and ?’ = 
?
?
?
2
? = 
?
2
Ans. (a)
Page 5


Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 0
When a particle moves in a plane such that its distance from a fixed (or moving) point remains constant
then its motion is called as the circular motion with respect to that fixed (or moving) point. That fixed
point is called centre and the distance between fixed point and particle is called radius.
v v v
?
1
?
2
B
A
The car is moving in a straight line with respect to the man A. But the man B continuously rotate his face
to see the car. So with respect to man A 
d
dt
?
? 0
But with respect to man B 
d
dt
?
? 0
Therefore we conclude that with respect to A the motion of car is straight line but for man B it has some
angular velocity
2. KINEMATICS OF CIRCULAR MOTION :
2.1 Variables of Motion :
(a) Angular Position :
The angle made by the position vector with given line (reference line) is called
angular position Circular motion is a two dimensional motion or motion in a plane.
Suppose a particle P is moving in a circle of radius r and centre O. The position of   
P'
P
r
x
O
?
? ?
Y
the particle P at a given instant may be described by the angle ? between OP and OX.
This angle ? is called the angular position of the particle. As the particle moves on the
circle its angular position ? change. Suppose the point rotates an angle ? ? in time ?t.
(b) Angular Displacement :
Definition :
Angle rotated by a position vector of the moving particle in a given time interval with some
 reference line is called its angular displacement.
______________________________________________________________________________________________
Important point :
• It is dimensionless and has proper unit SI unit radian while other units are degree or revolution 2 ? rad
= 360° = 1 rev
• Infinitely small angular displacement is a vector quantity but finite angular displacement is not because
the addition of the small angular displacement is cummutative while for large is not.
d
?
?
1
 + d
?
?
2
 = d
?
?
2
 + d
?
?
1
  but  ?
1
 + ?
2
 ?  ?
2
 + ?
1
• Direction of small angular displacement is decided by right hand thumb rule. When the fingers are
directed along the motion of the point then thumb will represents the direction of angular displacement.
• Angular displacement can be different for different observers
______________________________________________________________________________________________
CIRCULAR MOTION CIRCULAR MOTION
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
(c) Angular Velocity ?
(i) Average Angular Velocity
?
av
  = 
Total Angle of Rotation
Total time taken
 ;
?
av
 = 
? ?
2 1
2 1
–
– t t
 = 
? ?
?t
where ?
1
 and ?
2
 are angular position of the particle at time t
1
 and t
2
 respectively.
(ii)  Instantaneous Angular Velocity
The rate at which the position vector of a particle with respect to the centre rotates, is called as
instantaneous angular velocity with respect to the centre.
? = 
lim
?t ?0
 
? ?
?t
 = 
d
dt
?
_______________________________________________________________________________________________
Important points :
• It is an axial vector with dimensions [T
–1
] and SI unit rad/s.
• For a rigid body as all points will rotate through same angle in same time, angular velocity is a
characteristic of the body as a whole, e.g., angular velocity of all points of earth about its own axis
is (2 ?/24) rad/hr.
• If a body makes ‘n’ rotations in ‘t’ seconds then angular velocity in radian per second will be
?
av
 = 
2 ?n
t
If T is the period and ‘f’ the frequency of uniform circular motion
?
av
 = 
2 1 ? ?
T
 = 2 ?f
• If ? ? ? = a – bt + ct
2
 then ? = 
d
dt
?
 = – b + 2ct
Relation between speed and angular velocity :
? = 
lim
?
? ?
? t t ? ?
 = 
d
dt
?
The rate of change of angular velocity is called the angular acceleration ( ?). Thus,
? = 
d
dt
?
 = 
2
2
t d
d ?
The linear distance PP’ travelled by the particle in time ?t is
Y
X
O
P
P'
? ?
r
?s = r ? ?   or  
lim
?t ?0
 
?
?
S
t
= 
r
t
lim
? ?0
? ?
? t
or  
?
?
s
t
 = r
d
dt
?
or   v = r ?
Here, v is the linear speed of the particle
It is only valid for circular motion
v = r ? is a scalar quantity (
?
?
? ?
v
r
)
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 2
Ex.1 If ? depends on time t in following way
? = 2t
2
 + 3 then
(a) Find out ? average upto 3 sec. (b) ? at 3 sec
Sol. ?
avg
 = 
1 2
i f
t – t
–
time total
nt displaceme angular Total ? ?
?
?
f
 = 2 (3)
2
 + 3 = 21 rad
?
i 
= 2 (0) + 3 = 3 rad.
So, ?
avg
 = 
21 3
3
–
 = 6 rad/sec
?
instantaneous
 = 
d
dt
?
 = 4t
?
at t = 3 sec 
 = 4 × 3 = 12 rad/sec
(d) Relative Angular Velocity
Angular velocity is defined with respect to the point from which the position vector of the moving particle
is drawn Here angular velocity of the particle w.r.t. ‘O’ and ‘A’ will be different
O
P'
P
lin Ref
A
?
?
PO
d
dt
? ;  ?
?
PA
d
dt
?
Definition :
Relative angular velocity of a particle ‘A’ with respect to the other moving particle ‘B’ is the angular
velocity of the position vector of ‘A’ with respect to ‘B’. That means it is the rate at which position vector
of ‘A’ with respect to ‘B’ rotates at that instant
A
r
B
V
B
V
A
?
AB
AB
AB
V
r
?
?
( )
here 
V
AB ?
?
 Relative velocity 
?
 to position vector AB
     
B and A between Seperation
AB line to lar perpendicu B . t . r . w A of velocity lative Re
?
( ) sin sin V V V
AB A B ?
? ? ? ?
1 2
r r
AB
?
?
? ?
AB
A B
V V
r
?
? sin sin
1 2
_______________________________________________________________________________________
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 4
Important points :
• If two particles are moving on the same circle or different coplanar concentric circles in same direction
with different uniform angular speed ?
A
 and ?
B
 respectively, the rate of change of angle between 
OA
?
and 
OB
?
 is
d
dt
B A
?
? ? ? ?
O
A
B
Initial line
O
A
B
Initial line
So the time taken by one to complete one revolution around O w.r.t. the other
T
T T
T T
rel
? ?
?
?
?
2 2
2 1
1 2
1 2
?
?
?
? ?
• If two particles are moving on two different concentric circles with different velocities then angular velocity
of B relative to A as observed by A will depend on their positions and velocities. consider the case when
A and B are closest to each other moving in same direction as shown in figure. In this situation
v v v v v
rel B A B A
? ? ? ? | |
? ?
r r r r r
rel B A B A
? ? ? ? | |
? ?
v
A
v
B
A
B
r
O
r
A
r
B
so,
?
BA
rel
rel
B A
B A
v
r
v v
r r
? ?
?
?
?
( )
( ) v
rel ?
= Relative velocity perpendicular to position vector
_______________________________________________________________________________________
Ex.2 Two particles move on a circular path (one just inside and the other just outside) with angular
velocities ? and 5 ? starting from the same point. Then, which is incorrect.
(a) they cross each other at regular intervals of time 
? ?
? ?
 when their angular velocities are
oppositely directed
(b) they cross each other at points on the path subtending an angle of 60° at the centre if their
angular velocities are oppositely directed
(c) they cross at intervals of time 
? ?
?
 if their angular velocities are oppositely directed
(d) they cross each other at points on the path subtending 90° at the centre if their angular
velocities are in the same sense
Sol. If the angular velocities are oppositely directed, they meet at intervals of
time t = 
2 ?
?
rel
 = 
2
6
?
?
 = 
?
? 3
Angle subtended at the centre by the crossing points
? = ?t = 
?
3
 = 60°
When their angular velocities are in the same direction,
t’ = 
2 ?
?
rel
 = 
2
4
?
?
 = 
?
? 2
 and ?’ = 
?
?
?
2
? = 
?
2
Ans. (a)
Circular Motion – Nirmaan TYCRP
 97/1, 3F, Adhchini, Sri Aurobindo Marg, Near NCERT, New Delhi |  011-32044009 4
Ex.3 Two moving particles P and Q are 10 m apart at a certain instant. The velocity of P is 8 m/s making
30° with the line joining P and Q and that of Q is 6 m/s making 30° with PQ in the figure. Then
the angular velocity of Q with respect to P in rad/s at that instant is
30°
6 m/s
10 m
Q
30°
8 m/s
P
(A) 0 (B) 0.1 (C) 0.4 (D) 0.7
Sol.
30°
6 m/s
10 m
Q
30°
8 m/s
P
Angular velocity of Q relative to P = 
Projection of V perpendicular to the line PQ
Separation between P and Q
QP
V
Q P
V
PQ
sin – sin ? ?
2 1
 = 
6 30 30
10
sin –(–8 sin ) ? ?
 = 0.7 rad/s
?  (D)
(e) Angular Acceleration ? :
(i)   Average Angular Acceleration :
Let ?
1
 and ?
2
 be the instantaneous angular speeds at times t
1
 and t
2
 respectively, then the average
angular acceleration ?
av 
is defined as
?
av
 = 
? ?
2 1
2 1
–
– t t
 = 
?
?
?
t
(ii)   Instantaneous Angular Acceleration :
It is the limit of average angular acceleration as ?t approaches zero, i.e.,
? = 
lim
?t ?0
? ?
?t
 = 
d
dt
?
 = ? 
d
d
?
?
_______________________________________________________________________________________________
Important points :
• It is also an axial vector with dimension [T
–2
] and unit rad/s
2
• If ? = 0, circular motion is said to be uniform.
• As ? = 
d
dt
?
,  ? = 
d
dt
?
 = 
d
dt
2
2
?
,
i.e., second derivative of angular displacement w.r.t time gives angular acceleration.
• ? is a axial vector and direction of ? is along ?
if ? increases and opposite to ? if ? decreases
_______________________________________________________________________________________