Simple Harmonic Motion | Physics Optional Notes for UPSC PDF Download

 Page 1


 
SIMPLE HARMONIC MOTION 
PERIODIC MOTION 
When an object repeats the same motion over and over at regular intervals, we call it 
periodic motion. The time it takes for one complete repetition of this motion is called the 
time period of the motion. 
The number of periods comprised in one second is called the frequency, has, if T is the 
period, and n is the frequency of a periodic motion, we have n = 1/T. 
The motion of the hands of a clock is periodic, the period of the motion of the minute 
hand being one hour, or 3600 seconds. The bob of a pendulum moves periodically, the 
period being equal to the time of one complete (to and fro) oscillation. Periodic motion 
can be along any path. 
OSCILLATORY MOTION 
When an object moves back and forth along the same path repeatedly, we call it oscillatory 
motion. This type of motion happens around a stable position called the equilibrium 
position. When the object is pushed away from this position and then released, it 
experiences a force or torque that pulls it back toward equilibrium. As it moves back, the 
restoring force or torque does positive work on it, giving it kinetic energy. This causes the 
object to overshoot the equilibrium position, come to a stop, and then move back in the 
opposite direction. An example is a ball rolling back and forth in a round bowl or a 
pendulum swinging past its lowest point and then back again. 
If the particle moves to & fro on the same path then the motion is called oscillatory 
motion. 
 
Periodic but not oscillatory 
 
Periodic & oscillatory 
 
Page 2


 
SIMPLE HARMONIC MOTION 
PERIODIC MOTION 
When an object repeats the same motion over and over at regular intervals, we call it 
periodic motion. The time it takes for one complete repetition of this motion is called the 
time period of the motion. 
The number of periods comprised in one second is called the frequency, has, if T is the 
period, and n is the frequency of a periodic motion, we have n = 1/T. 
The motion of the hands of a clock is periodic, the period of the motion of the minute 
hand being one hour, or 3600 seconds. The bob of a pendulum moves periodically, the 
period being equal to the time of one complete (to and fro) oscillation. Periodic motion 
can be along any path. 
OSCILLATORY MOTION 
When an object moves back and forth along the same path repeatedly, we call it oscillatory 
motion. This type of motion happens around a stable position called the equilibrium 
position. When the object is pushed away from this position and then released, it 
experiences a force or torque that pulls it back toward equilibrium. As it moves back, the 
restoring force or torque does positive work on it, giving it kinetic energy. This causes the 
object to overshoot the equilibrium position, come to a stop, and then move back in the 
opposite direction. An example is a ball rolling back and forth in a round bowl or a 
pendulum swinging past its lowest point and then back again. 
If the particle moves to & fro on the same path then the motion is called oscillatory 
motion. 
 
Periodic but not oscillatory 
 
Periodic & oscillatory 
 
Periodic oscillatory 
Imp. Characteristics of oscillatory motion: 
1. When a particle in stable equilibrium is disturbed, then it has tendency to return 
to the position of equilibrium and this tendency is exhibited as oscillatory 
motion. 
2. The force on the body acts towards the mean position i.e. Force is always opposite 
to the displacement vector of the particle w.r.t. mean position. (This force is 
known as restoring force) 
?? ˆ
= -??ˆ,??ˆ = -?? ˆ
 
Where ??ˆ and ?? ˆ
 are displacements and angular displacement from the mean position. 
3. Energy is also conserved. If energy is not conserved then the particle will not be 
able to repeat the parameters of the motion. 
Optional Explanation of why we study SHM : 
Sinusoidal Vibrations: 
Our focus will primarily be on sinusoidal vibrations, which arise when the net force or 
torque experienced by an oscillating object is proportional to its distance or angular 
displacement from the mean position. This relationship is fundamental to our study for 
two reasons: one physical and one mathematical, both crucial to the subject. 
The physical reason is that purely sinusoidal vibrations are commonly observed in 
various mechanical systems. This type of motion is almost always achievable when the 
displacements are small enough. For instance, if we have a body attached to a spring, the 
force exerted on it at a displacement x from equilibrium is directly proportional to x. 
?? ( ?? )= -( ?? 1
?? + ?? 2
?? 2
+ ?? 3
?? 3
+ ?……0 
Where?? 1
,?? 2
,?? 3
, etc., are a set of constants, and we can always find a range of values of ?? 
within which the sum of the terms in ?? 2
,?? 3
, etc., is negligible, compared to the term 
-?? 1
?? . If the body is of mass ?? and the mass of the spring is negligible, the equation of 
motion of the body then becomes 
m
?? 2
?? ?? ?? 2
= -k
1
x 
It is easy to verify, that the above equation is satisfied by an equation of the form 
?? = ?? sin ? ( ???? + ?? 0
) 
Page 3


 
SIMPLE HARMONIC MOTION 
PERIODIC MOTION 
When an object repeats the same motion over and over at regular intervals, we call it 
periodic motion. The time it takes for one complete repetition of this motion is called the 
time period of the motion. 
The number of periods comprised in one second is called the frequency, has, if T is the 
period, and n is the frequency of a periodic motion, we have n = 1/T. 
The motion of the hands of a clock is periodic, the period of the motion of the minute 
hand being one hour, or 3600 seconds. The bob of a pendulum moves periodically, the 
period being equal to the time of one complete (to and fro) oscillation. Periodic motion 
can be along any path. 
OSCILLATORY MOTION 
When an object moves back and forth along the same path repeatedly, we call it oscillatory 
motion. This type of motion happens around a stable position called the equilibrium 
position. When the object is pushed away from this position and then released, it 
experiences a force or torque that pulls it back toward equilibrium. As it moves back, the 
restoring force or torque does positive work on it, giving it kinetic energy. This causes the 
object to overshoot the equilibrium position, come to a stop, and then move back in the 
opposite direction. An example is a ball rolling back and forth in a round bowl or a 
pendulum swinging past its lowest point and then back again. 
If the particle moves to & fro on the same path then the motion is called oscillatory 
motion. 
 
Periodic but not oscillatory 
 
Periodic & oscillatory 
 
Periodic oscillatory 
Imp. Characteristics of oscillatory motion: 
1. When a particle in stable equilibrium is disturbed, then it has tendency to return 
to the position of equilibrium and this tendency is exhibited as oscillatory 
motion. 
2. The force on the body acts towards the mean position i.e. Force is always opposite 
to the displacement vector of the particle w.r.t. mean position. (This force is 
known as restoring force) 
?? ˆ
= -??ˆ,??ˆ = -?? ˆ
 
Where ??ˆ and ?? ˆ
 are displacements and angular displacement from the mean position. 
3. Energy is also conserved. If energy is not conserved then the particle will not be 
able to repeat the parameters of the motion. 
Optional Explanation of why we study SHM : 
Sinusoidal Vibrations: 
Our focus will primarily be on sinusoidal vibrations, which arise when the net force or 
torque experienced by an oscillating object is proportional to its distance or angular 
displacement from the mean position. This relationship is fundamental to our study for 
two reasons: one physical and one mathematical, both crucial to the subject. 
The physical reason is that purely sinusoidal vibrations are commonly observed in 
various mechanical systems. This type of motion is almost always achievable when the 
displacements are small enough. For instance, if we have a body attached to a spring, the 
force exerted on it at a displacement x from equilibrium is directly proportional to x. 
?? ( ?? )= -( ?? 1
?? + ?? 2
?? 2
+ ?? 3
?? 3
+ ?……0 
Where?? 1
,?? 2
,?? 3
, etc., are a set of constants, and we can always find a range of values of ?? 
within which the sum of the terms in ?? 2
,?? 3
, etc., is negligible, compared to the term 
-?? 1
?? . If the body is of mass ?? and the mass of the spring is negligible, the equation of 
motion of the body then becomes 
m
?? 2
?? ?? ?? 2
= -k
1
x 
It is easy to verify, that the above equation is satisfied by an equation of the form 
?? = ?? sin ? ( ???? + ?? 0
) 
where?? = ( k/m)
1/2
. Thus sinusoidal vibration or simple harmonic motion is likely a 
possibility in small vibrations. But we should remember that in general, it is only an 
approximation (although perhaps a very close one) to the true motion. 
The second reason is the mathematical one. The actual importance of purely sinusoidal 
vibrations is proved by a famous theorem given by the French mathematician J.B. 
Fourier in 1807. According to Fourier's theorem, any periodic function with a period ?? 
can be considered as the sum of pure sinusoidal vibrations of periods T,T/2, T/3, etc., 
with appropriately chosen amplitudes. A thorough familiarity with sinusoidal vibrations 
will be a stepping stone for our understanding of every conceivable problem involving 
periodic phenomena. 
TYPES OF SHM 
(a) Linear SHM: 
When a particle moves to and fro about an equilibrium point, along a straight line. A and 
B are extreme positions. M is the mean position AM= MB= Amplitude. 
 
(b) Angular SHM: 
When the body/particle is free to rotate about a given axis executing angular oscillations. 
SMM 
It is a special kind of periodic oscillation in which a force (restoring) acts on the body 
toward an equilibrium position. The magnitude of this force is directly proportional to 
the distance of the body from the equilibrium position. 
Condition for SHM:  
(1) Periodic 
(2) Oscillatory 
(3) |?? | ? |?? | 
 
 
F = -kx;k is called force constant. 
 
Page 4


 
SIMPLE HARMONIC MOTION 
PERIODIC MOTION 
When an object repeats the same motion over and over at regular intervals, we call it 
periodic motion. The time it takes for one complete repetition of this motion is called the 
time period of the motion. 
The number of periods comprised in one second is called the frequency, has, if T is the 
period, and n is the frequency of a periodic motion, we have n = 1/T. 
The motion of the hands of a clock is periodic, the period of the motion of the minute 
hand being one hour, or 3600 seconds. The bob of a pendulum moves periodically, the 
period being equal to the time of one complete (to and fro) oscillation. Periodic motion 
can be along any path. 
OSCILLATORY MOTION 
When an object moves back and forth along the same path repeatedly, we call it oscillatory 
motion. This type of motion happens around a stable position called the equilibrium 
position. When the object is pushed away from this position and then released, it 
experiences a force or torque that pulls it back toward equilibrium. As it moves back, the 
restoring force or torque does positive work on it, giving it kinetic energy. This causes the 
object to overshoot the equilibrium position, come to a stop, and then move back in the 
opposite direction. An example is a ball rolling back and forth in a round bowl or a 
pendulum swinging past its lowest point and then back again. 
If the particle moves to & fro on the same path then the motion is called oscillatory 
motion. 
 
Periodic but not oscillatory 
 
Periodic & oscillatory 
 
Periodic oscillatory 
Imp. Characteristics of oscillatory motion: 
1. When a particle in stable equilibrium is disturbed, then it has tendency to return 
to the position of equilibrium and this tendency is exhibited as oscillatory 
motion. 
2. The force on the body acts towards the mean position i.e. Force is always opposite 
to the displacement vector of the particle w.r.t. mean position. (This force is 
known as restoring force) 
?? ˆ
= -??ˆ,??ˆ = -?? ˆ
 
Where ??ˆ and ?? ˆ
 are displacements and angular displacement from the mean position. 
3. Energy is also conserved. If energy is not conserved then the particle will not be 
able to repeat the parameters of the motion. 
Optional Explanation of why we study SHM : 
Sinusoidal Vibrations: 
Our focus will primarily be on sinusoidal vibrations, which arise when the net force or 
torque experienced by an oscillating object is proportional to its distance or angular 
displacement from the mean position. This relationship is fundamental to our study for 
two reasons: one physical and one mathematical, both crucial to the subject. 
The physical reason is that purely sinusoidal vibrations are commonly observed in 
various mechanical systems. This type of motion is almost always achievable when the 
displacements are small enough. For instance, if we have a body attached to a spring, the 
force exerted on it at a displacement x from equilibrium is directly proportional to x. 
?? ( ?? )= -( ?? 1
?? + ?? 2
?? 2
+ ?? 3
?? 3
+ ?……0 
Where?? 1
,?? 2
,?? 3
, etc., are a set of constants, and we can always find a range of values of ?? 
within which the sum of the terms in ?? 2
,?? 3
, etc., is negligible, compared to the term 
-?? 1
?? . If the body is of mass ?? and the mass of the spring is negligible, the equation of 
motion of the body then becomes 
m
?? 2
?? ?? ?? 2
= -k
1
x 
It is easy to verify, that the above equation is satisfied by an equation of the form 
?? = ?? sin ? ( ???? + ?? 0
) 
where?? = ( k/m)
1/2
. Thus sinusoidal vibration or simple harmonic motion is likely a 
possibility in small vibrations. But we should remember that in general, it is only an 
approximation (although perhaps a very close one) to the true motion. 
The second reason is the mathematical one. The actual importance of purely sinusoidal 
vibrations is proved by a famous theorem given by the French mathematician J.B. 
Fourier in 1807. According to Fourier's theorem, any periodic function with a period ?? 
can be considered as the sum of pure sinusoidal vibrations of periods T,T/2, T/3, etc., 
with appropriately chosen amplitudes. A thorough familiarity with sinusoidal vibrations 
will be a stepping stone for our understanding of every conceivable problem involving 
periodic phenomena. 
TYPES OF SHM 
(a) Linear SHM: 
When a particle moves to and fro about an equilibrium point, along a straight line. A and 
B are extreme positions. M is the mean position AM= MB= Amplitude. 
 
(b) Angular SHM: 
When the body/particle is free to rotate about a given axis executing angular oscillations. 
SMM 
It is a special kind of periodic oscillation in which a force (restoring) acts on the body 
toward an equilibrium position. The magnitude of this force is directly proportional to 
the distance of the body from the equilibrium position. 
Condition for SHM:  
(1) Periodic 
(2) Oscillatory 
(3) |?? | ? |?? | 
 
 
F = -kx;k is called force constant. 
 
  
 x F Acceleration Velocity 
1 + - - + 
2 + - - - 
3 - + + - 
4 - + + + 
 
EQUATION OF SHM 
The necessary and sufficient condition for SHM is 
?? = -???? 
Where ?k = positive constant for a SHM = Force constant x= displacement from mean 
position 
or ?? ?? 2
?? ?? ?? 2
= -???? ?
?? 2
?? ?? ?? 2
+
?? ?? ?? = 0? [differential equation of SHM] 
? ?
d
2
x
dt
2
+ ?? 2
x= 0? Where ?? = v
k
m
 
Its solution is x= Asin ? ( ???? + ?? ) 
This means if the displacement of the particle is a sinusoidal function, it will perform 
SHM. 
DERIVATION 
F = -kx 
a = -
k
m
x? put 
k
m
= ?? 2
 
?? = -?? 2
?? 
?
0
?? ??????? = -?
?? ?? ??? 2
?????? ?
?? 2
2
= -
?? 2
2
[?? 2
- ?? 2
] 
?? = ±?? v
?? 2
- ?? 2
?
????
????
= ±?? v
?? 2
- ?? 2
 
Page 5


 
SIMPLE HARMONIC MOTION 
PERIODIC MOTION 
When an object repeats the same motion over and over at regular intervals, we call it 
periodic motion. The time it takes for one complete repetition of this motion is called the 
time period of the motion. 
The number of periods comprised in one second is called the frequency, has, if T is the 
period, and n is the frequency of a periodic motion, we have n = 1/T. 
The motion of the hands of a clock is periodic, the period of the motion of the minute 
hand being one hour, or 3600 seconds. The bob of a pendulum moves periodically, the 
period being equal to the time of one complete (to and fro) oscillation. Periodic motion 
can be along any path. 
OSCILLATORY MOTION 
When an object moves back and forth along the same path repeatedly, we call it oscillatory 
motion. This type of motion happens around a stable position called the equilibrium 
position. When the object is pushed away from this position and then released, it 
experiences a force or torque that pulls it back toward equilibrium. As it moves back, the 
restoring force or torque does positive work on it, giving it kinetic energy. This causes the 
object to overshoot the equilibrium position, come to a stop, and then move back in the 
opposite direction. An example is a ball rolling back and forth in a round bowl or a 
pendulum swinging past its lowest point and then back again. 
If the particle moves to & fro on the same path then the motion is called oscillatory 
motion. 
 
Periodic but not oscillatory 
 
Periodic & oscillatory 
 
Periodic oscillatory 
Imp. Characteristics of oscillatory motion: 
1. When a particle in stable equilibrium is disturbed, then it has tendency to return 
to the position of equilibrium and this tendency is exhibited as oscillatory 
motion. 
2. The force on the body acts towards the mean position i.e. Force is always opposite 
to the displacement vector of the particle w.r.t. mean position. (This force is 
known as restoring force) 
?? ˆ
= -??ˆ,??ˆ = -?? ˆ
 
Where ??ˆ and ?? ˆ
 are displacements and angular displacement from the mean position. 
3. Energy is also conserved. If energy is not conserved then the particle will not be 
able to repeat the parameters of the motion. 
Optional Explanation of why we study SHM : 
Sinusoidal Vibrations: 
Our focus will primarily be on sinusoidal vibrations, which arise when the net force or 
torque experienced by an oscillating object is proportional to its distance or angular 
displacement from the mean position. This relationship is fundamental to our study for 
two reasons: one physical and one mathematical, both crucial to the subject. 
The physical reason is that purely sinusoidal vibrations are commonly observed in 
various mechanical systems. This type of motion is almost always achievable when the 
displacements are small enough. For instance, if we have a body attached to a spring, the 
force exerted on it at a displacement x from equilibrium is directly proportional to x. 
?? ( ?? )= -( ?? 1
?? + ?? 2
?? 2
+ ?? 3
?? 3
+ ?……0 
Where?? 1
,?? 2
,?? 3
, etc., are a set of constants, and we can always find a range of values of ?? 
within which the sum of the terms in ?? 2
,?? 3
, etc., is negligible, compared to the term 
-?? 1
?? . If the body is of mass ?? and the mass of the spring is negligible, the equation of 
motion of the body then becomes 
m
?? 2
?? ?? ?? 2
= -k
1
x 
It is easy to verify, that the above equation is satisfied by an equation of the form 
?? = ?? sin ? ( ???? + ?? 0
) 
where?? = ( k/m)
1/2
. Thus sinusoidal vibration or simple harmonic motion is likely a 
possibility in small vibrations. But we should remember that in general, it is only an 
approximation (although perhaps a very close one) to the true motion. 
The second reason is the mathematical one. The actual importance of purely sinusoidal 
vibrations is proved by a famous theorem given by the French mathematician J.B. 
Fourier in 1807. According to Fourier's theorem, any periodic function with a period ?? 
can be considered as the sum of pure sinusoidal vibrations of periods T,T/2, T/3, etc., 
with appropriately chosen amplitudes. A thorough familiarity with sinusoidal vibrations 
will be a stepping stone for our understanding of every conceivable problem involving 
periodic phenomena. 
TYPES OF SHM 
(a) Linear SHM: 
When a particle moves to and fro about an equilibrium point, along a straight line. A and 
B are extreme positions. M is the mean position AM= MB= Amplitude. 
 
(b) Angular SHM: 
When the body/particle is free to rotate about a given axis executing angular oscillations. 
SMM 
It is a special kind of periodic oscillation in which a force (restoring) acts on the body 
toward an equilibrium position. The magnitude of this force is directly proportional to 
the distance of the body from the equilibrium position. 
Condition for SHM:  
(1) Periodic 
(2) Oscillatory 
(3) |?? | ? |?? | 
 
 
F = -kx;k is called force constant. 
 
  
 x F Acceleration Velocity 
1 + - - + 
2 + - - - 
3 - + + - 
4 - + + + 
 
EQUATION OF SHM 
The necessary and sufficient condition for SHM is 
?? = -???? 
Where ?k = positive constant for a SHM = Force constant x= displacement from mean 
position 
or ?? ?? 2
?? ?? ?? 2
= -???? ?
?? 2
?? ?? ?? 2
+
?? ?? ?? = 0? [differential equation of SHM] 
? ?
d
2
x
dt
2
+ ?? 2
x= 0? Where ?? = v
k
m
 
Its solution is x= Asin ? ( ???? + ?? ) 
This means if the displacement of the particle is a sinusoidal function, it will perform 
SHM. 
DERIVATION 
F = -kx 
a = -
k
m
x? put 
k
m
= ?? 2
 
?? = -?? 2
?? 
?
0
?? ??????? = -?
?? ?? ??? 2
?????? ?
?? 2
2
= -
?? 2
2
[?? 2
- ?? 2
] 
?? = ±?? v
?? 2
- ?? 2
?
????
????
= ±?? v
?? 2
- ?? 2
 
 
?
????
v?? 2
- ?? 2
= ±?? ????? 
sin
-1
??? /?? = ±???? + ?? ? at ??? = 0??? = ?? 0
 
x= Asin ? {?? t+ ?? } 
Amplitude: It is the maximum distance of particle from its mean position. In other 
words displacement between mean & extreme position is amplitude. In equation??? =
?? sin ? ( ???? + ?? ) , amplitude is ?? . 
Time period: It is the minimum time after which particle repeats its motion. 
t = 0 x
1
= Asin ? ( ?? ( 0)+ ?? )
t = T x
2
= Asin ? ( ?? T+ ?? )
t = T ( ?? T+ ?? )- ( ?? ( 0)+ ?? )= 2?? ?? T = 2?? T =
2?? ?? 
CHARACTERISTICS OF SHM 
Note: In the figure shows, path of the particle is on a straight line. 
(a) Displacement: 
It is defined as the distance of the particle from the mean position at that instant. 
Displacement in SHM at time is given by x= Asin ? ( ?? t+ ?? ) 
(b) Amplitude: 
It is the maximum value of displacement of the particle from its equilibrium position 
 
 
Amplitude = 1/2 (distance between extreme points/position) 
It depends on energy of the system.