Circular Motion | Physics Optional Notes for UPSC PDF Download

 Page 1


 
Circular Motion 
LECTURE NOTES 
 
What is Circular Motion?  
Circular motion refers to the movement of an object along the circumference of a circle 
or a circular path. It's a type of rotational motion with several key characteristics, 
including constant distance from a fixed point (the center of the circle) and, in uniform 
circular motion, constant speed. However, even in a uniform circular motion, the 
direction of the velocity vector changes continuously, which means there is a constant 
change in velocity, indicating the presence of acceleration. 
This acceleration, called centripetal acceleration, is always directed toward the center of 
the circle and is necessary to keep the object moving in a circular path. The force that 
provides this centripetal acceleration is called the centripetal force. Examples of 
centripetal force include gravitational force (for planets orbiting a star), tension (in the 
case of a string attached to a rotating object), or friction (between the tires of a car and 
the road as the car turns). 
 
 
Page 2


 
Circular Motion 
LECTURE NOTES 
 
What is Circular Motion?  
Circular motion refers to the movement of an object along the circumference of a circle 
or a circular path. It's a type of rotational motion with several key characteristics, 
including constant distance from a fixed point (the center of the circle) and, in uniform 
circular motion, constant speed. However, even in a uniform circular motion, the 
direction of the velocity vector changes continuously, which means there is a constant 
change in velocity, indicating the presence of acceleration. 
This acceleration, called centripetal acceleration, is always directed toward the center of 
the circle and is necessary to keep the object moving in a circular path. The force that 
provides this centripetal acceleration is called the centripetal force. Examples of 
centripetal force include gravitational force (for planets orbiting a star), tension (in the 
case of a string attached to a rotating object), or friction (between the tires of a car and 
the road as the car turns). 
 
 
1. KINEMATICS OF CIRCULAR MOTION 
Angular Displacement : 
Angular displacement is the angle between the initial and final positions of an object, 
with respect to a fixed point.   
??? = angular displacement ?  angle =
arc
 radius 
 ? ??? =
arc ????
?? 
Angular displacement is defined as ??? =
 arc-length 
 radius 
 
 
nit is radian 
? 360
°
= 2?? radian 
(e.g) 
Page 3


 
Circular Motion 
LECTURE NOTES 
 
What is Circular Motion?  
Circular motion refers to the movement of an object along the circumference of a circle 
or a circular path. It's a type of rotational motion with several key characteristics, 
including constant distance from a fixed point (the center of the circle) and, in uniform 
circular motion, constant speed. However, even in a uniform circular motion, the 
direction of the velocity vector changes continuously, which means there is a constant 
change in velocity, indicating the presence of acceleration. 
This acceleration, called centripetal acceleration, is always directed toward the center of 
the circle and is necessary to keep the object moving in a circular path. The force that 
provides this centripetal acceleration is called the centripetal force. Examples of 
centripetal force include gravitational force (for planets orbiting a star), tension (in the 
case of a string attached to a rotating object), or friction (between the tires of a car and 
the road as the car turns). 
 
 
1. KINEMATICS OF CIRCULAR MOTION 
Angular Displacement : 
Angular displacement is the angle between the initial and final positions of an object, 
with respect to a fixed point.   
??? = angular displacement ?  angle =
arc
 radius 
 ? ??? =
arc ????
?? 
Angular displacement is defined as ??? =
 arc-length 
 radius 
 
 
nit is radian 
? 360
°
= 2?? radian 
(e.g) 
 
It starts from A and complete one circular path and come to rest at B. 
angle ? 2?? + ?? (Anti clock wise) 
Frequency (n) : Number of revolutions described by particle per second is its frequency. 
Its unit is revolutions per second (rps) or revolutions per minute (rpm). 
Note : 1rps = 60rpm 
Time Period (T) : It is the time taken by particle to complete one revolution. i.e. ?? =
1
?? 
Angular Velocity ( ?? ) : It is defined as the rate of change of angular displacement of a 
moving particle, w.r.t. to time. 
?? =
 angle traced 
 time taken 
= Lim
??? ?0
 
??? ?t
=
d ?? dt
 
? Its unit is rad /s and dimensions is [T
-1
] 
Relation between linear and Angular velocity : 
Angle ( ??? ) =
 arc 
 radius 
=
??? ?? ? ??? = ?? ??? 
Page 4


 
Circular Motion 
LECTURE NOTES 
 
What is Circular Motion?  
Circular motion refers to the movement of an object along the circumference of a circle 
or a circular path. It's a type of rotational motion with several key characteristics, 
including constant distance from a fixed point (the center of the circle) and, in uniform 
circular motion, constant speed. However, even in a uniform circular motion, the 
direction of the velocity vector changes continuously, which means there is a constant 
change in velocity, indicating the presence of acceleration. 
This acceleration, called centripetal acceleration, is always directed toward the center of 
the circle and is necessary to keep the object moving in a circular path. The force that 
provides this centripetal acceleration is called the centripetal force. Examples of 
centripetal force include gravitational force (for planets orbiting a star), tension (in the 
case of a string attached to a rotating object), or friction (between the tires of a car and 
the road as the car turns). 
 
 
1. KINEMATICS OF CIRCULAR MOTION 
Angular Displacement : 
Angular displacement is the angle between the initial and final positions of an object, 
with respect to a fixed point.   
??? = angular displacement ?  angle =
arc
 radius 
 ? ??? =
arc ????
?? 
Angular displacement is defined as ??? =
 arc-length 
 radius 
 
 
nit is radian 
? 360
°
= 2?? radian 
(e.g) 
 
It starts from A and complete one circular path and come to rest at B. 
angle ? 2?? + ?? (Anti clock wise) 
Frequency (n) : Number of revolutions described by particle per second is its frequency. 
Its unit is revolutions per second (rps) or revolutions per minute (rpm). 
Note : 1rps = 60rpm 
Time Period (T) : It is the time taken by particle to complete one revolution. i.e. ?? =
1
?? 
Angular Velocity ( ?? ) : It is defined as the rate of change of angular displacement of a 
moving particle, w.r.t. to time. 
?? =
 angle traced 
 time taken 
= Lim
??? ?0
 
??? ?t
=
d ?? dt
 
? Its unit is rad /s and dimensions is [T
-1
] 
Relation between linear and Angular velocity : 
Angle ( ??? ) =
 arc 
 radius 
=
??? ?? ? ??? = ?? ??? 
 
?
??? ??? =
?? ??? ??? if ??? ? 0 then 
????
????
= ?? ????
????
? ?? = ???? 
In vector form ?? ? = ?? ? ? ? × ?? ? (direction of ?? ? is according to right hand thumb rule) 
Here,  ?? ? = Linear velocity  [ Tangential vector] 
?? ? ? ? = Angular velocity [Axial vector] 
r ? = Radius vector or position vector 
Average Angular Velocity ( ?? av
) : 
?? av
=
 total angle of rotation 
 total time taken 
=
?? 2
- ?? 1
t
2
- t
1
=
??? ?t
 
where ?? 1
 and ?? 2
 are the angular positions of the particle at instants ?? 1
 and ?? 2
. 
 
 
Page 5


 
Circular Motion 
LECTURE NOTES 
 
What is Circular Motion?  
Circular motion refers to the movement of an object along the circumference of a circle 
or a circular path. It's a type of rotational motion with several key characteristics, 
including constant distance from a fixed point (the center of the circle) and, in uniform 
circular motion, constant speed. However, even in a uniform circular motion, the 
direction of the velocity vector changes continuously, which means there is a constant 
change in velocity, indicating the presence of acceleration. 
This acceleration, called centripetal acceleration, is always directed toward the center of 
the circle and is necessary to keep the object moving in a circular path. The force that 
provides this centripetal acceleration is called the centripetal force. Examples of 
centripetal force include gravitational force (for planets orbiting a star), tension (in the 
case of a string attached to a rotating object), or friction (between the tires of a car and 
the road as the car turns). 
 
 
1. KINEMATICS OF CIRCULAR MOTION 
Angular Displacement : 
Angular displacement is the angle between the initial and final positions of an object, 
with respect to a fixed point.   
??? = angular displacement ?  angle =
arc
 radius 
 ? ??? =
arc ????
?? 
Angular displacement is defined as ??? =
 arc-length 
 radius 
 
 
nit is radian 
? 360
°
= 2?? radian 
(e.g) 
 
It starts from A and complete one circular path and come to rest at B. 
angle ? 2?? + ?? (Anti clock wise) 
Frequency (n) : Number of revolutions described by particle per second is its frequency. 
Its unit is revolutions per second (rps) or revolutions per minute (rpm). 
Note : 1rps = 60rpm 
Time Period (T) : It is the time taken by particle to complete one revolution. i.e. ?? =
1
?? 
Angular Velocity ( ?? ) : It is defined as the rate of change of angular displacement of a 
moving particle, w.r.t. to time. 
?? =
 angle traced 
 time taken 
= Lim
??? ?0
 
??? ?t
=
d ?? dt
 
? Its unit is rad /s and dimensions is [T
-1
] 
Relation between linear and Angular velocity : 
Angle ( ??? ) =
 arc 
 radius 
=
??? ?? ? ??? = ?? ??? 
 
?
??? ??? =
?? ??? ??? if ??? ? 0 then 
????
????
= ?? ????
????
? ?? = ???? 
In vector form ?? ? = ?? ? ? ? × ?? ? (direction of ?? ? is according to right hand thumb rule) 
Here,  ?? ? = Linear velocity  [ Tangential vector] 
?? ? ? ? = Angular velocity [Axial vector] 
r ? = Radius vector or position vector 
Average Angular Velocity ( ?? av
) : 
?? av
=
 total angle of rotation 
 total time taken 
=
?? 2
- ?? 1
t
2
- t
1
=
??? ?t
 
where ?? 1
 and ?? 2
 are the angular positions of the particle at instants ?? 1
 and ?? 2
. 
 
 
Example. A particle completes 1 circle in 30sec with constant  ? find  ? 
Sol. In 30sec? 2?? rad . 
1sec?
2?? 30
rad /sec
?? 15
rad /sec .
 
? RPM: Revolution per minute. 
1RPM? 2?? in 1 min . 
1RPM=
?? 30
rad /sec . 
Angular Acceleration ( ?? ) : 
Rate of change of angular velocity is called angular acceleration. 
                                    i.e.    ?? ? = Lim
??? ?0
 
???? ? ??
??? =
?? ??? ? ??
????
 
Average Angular Acceleration : 
?? ?
avg 
=
 change in angular velocity 
 time taken 
=
??? ? ? ?
??? 
2.3 Equations of circular motion 
Translatory / Linear Motion Rotational Motion 
- Initial velocity ( u) Initial angular velocity ( ?? 0
) 
- Final velocity ( v) Final angular velocity ( ?? ) 
- Displacement ( s) Angular displacement ( ?? ) 
- Acceleration ( a) Angular Acceleration ( ?? ) 
If  a = constant, then If ?? = constant, then 
v = u + at ?? = ?? 0
+ ?? t 
s = ut +
1
2
?? t
2
 ?? = ?? 0
t +
1
2
?? t
2
 
v
2
= u
2
+ 2as ?? 2
= ?? 0
 
2
+ 2????