II Semester B.Sc. Physics : Unit – 1 Oscillations (SHM) and Elasticity
Syllabus : OSCILLATIONS : SHM ; Differential equation of SHM and its solutions, Kinetic
and Potential energy, Simple and compound pendulum; oscillations of two masses connected
by a spring; damped oscillations – over damped, under damped and un-damped oscillations;
forced oscillations - concept of resonance; Coupled Oscillators - in phase and out of phase
oscillations- energy transfer
Periodic Motion : A motion that repeats itself in equal intervals of time is called periodic
motion. It is also called harmonic motion.
Examples. 1. Revolution of earth round the sun or about its own axis, 2. Revolution of moon and
other artificial satellites around the earth, 3. Rotation of the electrons round the nucleus of an
atom, 4. The motion of pendulum of a clock, 5. Motion of prongs of a tuning fork and 6. Oscillations
of a loaded spring.
The two common periodic motions are the simple harmonic motion and uniform circular
motion.
Oscillatory Motion : The motion of a particle is oscillatory if it moves back and forth
or to and fro about the same path at equal intervals of time.
Examples. 1. Motion of a pendulum, 2. Motion of mass attached to a suspended spring, 3. Motion
of atoms in a molecule or a lattice, 4. Motion of particles of the medium through which sound
travels, 5. Up and down motion of a floating object on water when waves propagate through it, 6.
Motion of prongs of a tuning fork.
All oscillatory motions are periodic in nature but all periodic motions are not
oscillatory. For example, revolution of earth round the sun is periodic but not oscillatory
and Rotation of the electrons round the nucleus of an atom is periodic but not oscillatory.
Definitions:
1. The smallest time interval in which the motion repeats is called the period (T). It is the time
taken for one oscillation.
2. The number of repetitions of motion that occur per second is called the frequency (f) of the
1
periodic motion. Frequency is equal to the reciprocal of period. 𝑓 = 𝑇
.
3. The maximum displacement of a particle from its mean position during an oscillation is called
amplitude (A) of oscillation.
4. Phase : It represents the state of motion of a particle in periodic motion. It is expressed in
terms of fraction of period T or the fraction of angle 2 measured from the instant when the
body has crossed the mean position in the positive direction.
Dr. K S Suresh, Associate professor, Vijaya College Page 1
, II Semester B.Sc. Physics : Unit – 1 Oscillations (SHM) and Elasticity
Simple Harmonic Motion (SHM) x
Consider a particle executing oscillatory motion about O P
the mean position (E – M – E) with E as the extreme
E M E
position and M is the mean position) with amplitude A.
Let 𝑥 be the displacement of the particle at an instant of A A
time t. A restoring force F acts on the particle to bring it
back to its mean position. This force is directly
proportional to the displacement.
Mathematically 𝑭 ∝ 𝒙 or 𝑭 = −𝒌𝒙 ….(1) where k is a constant called the force constant.
The negative sign indicates that F acts opposite to the direction of motion of the particle.
𝑘
As 𝐹 = 𝑚𝑎 …..(2) From (1) and (2) 𝑚𝑎 = −𝑘𝑥 𝑜𝑟 𝑎 = −( ) 𝑥 𝑜𝑟 𝒂 ∝ 𝒙 .
𝑚
A particle is said to execute simple harmonic motion if the acceleration of the
particle is directly proportional to the displacement of the particle from the mean
position and it is directed towards the mean or equilibrium position.
Examples of SHM : Motion of a pendulum, motion of mass attached to a suspended
spring, Vibrations of a guitar string, etc…
Differential form of SHM : Consider a particle executing SHM. If 𝑥 is the displacement
of the particle from the mean position at an instant of time t,
Then, the restoring force is 𝐹 ∝ 𝑥 or 𝐹 = −𝑘𝑥 ….(1)
𝑘
As 𝐹 = 𝑚𝑎, ….(2) Comparing (1) and (2) m𝑎 = −𝑘𝑥 𝑜𝑟 𝑎 = − (𝑚 ) 𝑥 ….(3)
𝑑2 𝑥 𝑑2 𝑥 𝑘 𝑑2 𝑥 𝑘
As 𝑎 = , equation (3) is, = − (𝑚 ) 𝑥 ……(4) or + (𝑚 ) 𝑥 = 0 …..(5)
𝑑𝑡 2 𝑑𝑡 2 𝑑𝑡 2
𝑘
let 𝑚 = 𝜔2 where 𝜔 is called the angular velocity or angular frequency of oscillating particle.
𝒅𝟐 𝒙
Now, equation (5) is + 𝝎𝟐 𝒙 = 𝟎 ………(6) Equation (6) is called differential form of
𝒅𝒕𝟐
SHM.
Solution of the differential equation of SHM
1. Expression for displacement of the particle
𝒅𝟐 𝒙
The differential form of SHM is + 𝝎𝟐 𝒙 = 𝟎 …….(1)
𝒅𝒕𝟐
Dr. K S Suresh, Associate professor, Vijaya College Page 2
, II Semester B.Sc. Physics : Unit – 1 Oscillations (SHM) and Elasticity
𝒅𝟐 𝒙 𝑑𝑥
or = − 𝝎𝟐 𝒙 Multiplying both the sides of this equation by 2 𝑑𝑡 , we get
𝒅𝒕𝟐
𝑑𝑥 𝒅𝟐 𝒙 𝑑𝑥 𝑑 𝑑𝑥 2 𝑑
2 𝑑𝑡 = − 2 𝑑𝑡 𝝎𝟐 𝒙 . This equation can be expressed as ( 𝑑𝑡 ) = − 𝜔2 (𝑥)2 …(2)
𝒅𝒕𝟐 𝑑𝑡 𝑑𝑡
𝑑𝑥 2
Integrating equation (2) we get ( 𝑑𝑡 ) = − 𝜔2 𝑥 2 + 𝐶 …..(3) where C is the constant of
integration.
𝑑𝑥
When the displacement is maximum, i.e. 𝑥 = 𝐴, the velocity of the particle =𝜐=0
𝑑𝑡
Putting this condition in (3), 0 = − 𝜔2 𝐴2 + 𝐶 or 𝐶 = 𝜔2 𝐴2
𝑑𝑥 2
Now the equation (3) becomes ( 𝑑𝑡 ) = − 𝜔2 𝑥 2 + 𝜔2 𝐴2
𝑑𝑥 2 𝒅𝒙
or ( 𝑑𝑡 ) = 𝜔2 (𝐴2 − 𝑥 2 ) or = 𝝎 √(𝑨𝟐 − 𝒙𝟐 ) ….(4) (This is also the expression for the
𝒅𝒕
velocity of the particle executing SHM)
𝑑𝑥
Rewriting equation (4) as, = 𝜔 𝑑𝑡 .
√(𝐴2 − 𝑥 2 )
𝑑𝑥 𝑥
Integrating this equation, ∫ √(𝐴2 = 𝜔 ∫ 𝑑𝑡 gives 𝑠𝑖𝑛−1 (𝐴) = 𝜔𝑡 + 𝜑
− 𝑥2)
where 𝜑 is the constant called the initial phase of the particle executing SHM. It is also
called epoch.
𝑥
or (𝐴) = 𝑠𝑖𝑛 (𝜔𝑡 + 𝜑) or 𝒙 = 𝑨 𝒔𝒊𝒏 (𝝎𝒕 + 𝝋) ….(5)
This is the solution of differential equation of SHM (eqn. (1)). Equation (5) is the expression
for the displacement of the particle at time t.
2. Expression for Velocity of the particle executing SHM
The equation of SHM is given by 𝑥 = 𝐴 𝑠𝑖𝑛 (𝜔𝑡 + 𝜑)
If the initial position from where time is measured is the mean position, then 𝜑 = 0
Thus 𝑥 = 𝐴 𝑠𝑖𝑛𝜔𝑡 . The velocity of the particle at a given instant of time t is given by
𝑑𝑥 𝑑𝑥
differentiating the above equation, i.e. = 𝐴𝜔 𝑐𝑜𝑠𝜔𝑡 or = 𝐴𝜔 √1 − 𝑠𝑖𝑛2 𝜔𝑡
𝑑𝑡 𝑑𝑡
𝑥 𝑑𝑥 𝑥2 𝑑𝑥 𝐴2 −𝑥 2
As 𝑠𝑖𝑛𝜔𝑡 = , we have = 𝐴𝜔 √1 − 𝐴2 or = 𝐴𝜔 √
𝐴 𝑑𝑡 𝑑𝑡 𝐴2
𝒅𝒙
Thus the velocity of the particle is 𝒗= = 𝝎 √𝑨𝟐 − 𝒙𝟐
𝒅𝒕
Dr. K S Suresh, Associate professor, Vijaya College Page 3
Syllabus : OSCILLATIONS : SHM ; Differential equation of SHM and its solutions, Kinetic
and Potential energy, Simple and compound pendulum; oscillations of two masses connected
by a spring; damped oscillations – over damped, under damped and un-damped oscillations;
forced oscillations - concept of resonance; Coupled Oscillators - in phase and out of phase
oscillations- energy transfer
Periodic Motion : A motion that repeats itself in equal intervals of time is called periodic
motion. It is also called harmonic motion.
Examples. 1. Revolution of earth round the sun or about its own axis, 2. Revolution of moon and
other artificial satellites around the earth, 3. Rotation of the electrons round the nucleus of an
atom, 4. The motion of pendulum of a clock, 5. Motion of prongs of a tuning fork and 6. Oscillations
of a loaded spring.
The two common periodic motions are the simple harmonic motion and uniform circular
motion.
Oscillatory Motion : The motion of a particle is oscillatory if it moves back and forth
or to and fro about the same path at equal intervals of time.
Examples. 1. Motion of a pendulum, 2. Motion of mass attached to a suspended spring, 3. Motion
of atoms in a molecule or a lattice, 4. Motion of particles of the medium through which sound
travels, 5. Up and down motion of a floating object on water when waves propagate through it, 6.
Motion of prongs of a tuning fork.
All oscillatory motions are periodic in nature but all periodic motions are not
oscillatory. For example, revolution of earth round the sun is periodic but not oscillatory
and Rotation of the electrons round the nucleus of an atom is periodic but not oscillatory.
Definitions:
1. The smallest time interval in which the motion repeats is called the period (T). It is the time
taken for one oscillation.
2. The number of repetitions of motion that occur per second is called the frequency (f) of the
1
periodic motion. Frequency is equal to the reciprocal of period. 𝑓 = 𝑇
.
3. The maximum displacement of a particle from its mean position during an oscillation is called
amplitude (A) of oscillation.
4. Phase : It represents the state of motion of a particle in periodic motion. It is expressed in
terms of fraction of period T or the fraction of angle 2 measured from the instant when the
body has crossed the mean position in the positive direction.
Dr. K S Suresh, Associate professor, Vijaya College Page 1
, II Semester B.Sc. Physics : Unit – 1 Oscillations (SHM) and Elasticity
Simple Harmonic Motion (SHM) x
Consider a particle executing oscillatory motion about O P
the mean position (E – M – E) with E as the extreme
E M E
position and M is the mean position) with amplitude A.
Let 𝑥 be the displacement of the particle at an instant of A A
time t. A restoring force F acts on the particle to bring it
back to its mean position. This force is directly
proportional to the displacement.
Mathematically 𝑭 ∝ 𝒙 or 𝑭 = −𝒌𝒙 ….(1) where k is a constant called the force constant.
The negative sign indicates that F acts opposite to the direction of motion of the particle.
𝑘
As 𝐹 = 𝑚𝑎 …..(2) From (1) and (2) 𝑚𝑎 = −𝑘𝑥 𝑜𝑟 𝑎 = −( ) 𝑥 𝑜𝑟 𝒂 ∝ 𝒙 .
𝑚
A particle is said to execute simple harmonic motion if the acceleration of the
particle is directly proportional to the displacement of the particle from the mean
position and it is directed towards the mean or equilibrium position.
Examples of SHM : Motion of a pendulum, motion of mass attached to a suspended
spring, Vibrations of a guitar string, etc…
Differential form of SHM : Consider a particle executing SHM. If 𝑥 is the displacement
of the particle from the mean position at an instant of time t,
Then, the restoring force is 𝐹 ∝ 𝑥 or 𝐹 = −𝑘𝑥 ….(1)
𝑘
As 𝐹 = 𝑚𝑎, ….(2) Comparing (1) and (2) m𝑎 = −𝑘𝑥 𝑜𝑟 𝑎 = − (𝑚 ) 𝑥 ….(3)
𝑑2 𝑥 𝑑2 𝑥 𝑘 𝑑2 𝑥 𝑘
As 𝑎 = , equation (3) is, = − (𝑚 ) 𝑥 ……(4) or + (𝑚 ) 𝑥 = 0 …..(5)
𝑑𝑡 2 𝑑𝑡 2 𝑑𝑡 2
𝑘
let 𝑚 = 𝜔2 where 𝜔 is called the angular velocity or angular frequency of oscillating particle.
𝒅𝟐 𝒙
Now, equation (5) is + 𝝎𝟐 𝒙 = 𝟎 ………(6) Equation (6) is called differential form of
𝒅𝒕𝟐
SHM.
Solution of the differential equation of SHM
1. Expression for displacement of the particle
𝒅𝟐 𝒙
The differential form of SHM is + 𝝎𝟐 𝒙 = 𝟎 …….(1)
𝒅𝒕𝟐
Dr. K S Suresh, Associate professor, Vijaya College Page 2
, II Semester B.Sc. Physics : Unit – 1 Oscillations (SHM) and Elasticity
𝒅𝟐 𝒙 𝑑𝑥
or = − 𝝎𝟐 𝒙 Multiplying both the sides of this equation by 2 𝑑𝑡 , we get
𝒅𝒕𝟐
𝑑𝑥 𝒅𝟐 𝒙 𝑑𝑥 𝑑 𝑑𝑥 2 𝑑
2 𝑑𝑡 = − 2 𝑑𝑡 𝝎𝟐 𝒙 . This equation can be expressed as ( 𝑑𝑡 ) = − 𝜔2 (𝑥)2 …(2)
𝒅𝒕𝟐 𝑑𝑡 𝑑𝑡
𝑑𝑥 2
Integrating equation (2) we get ( 𝑑𝑡 ) = − 𝜔2 𝑥 2 + 𝐶 …..(3) where C is the constant of
integration.
𝑑𝑥
When the displacement is maximum, i.e. 𝑥 = 𝐴, the velocity of the particle =𝜐=0
𝑑𝑡
Putting this condition in (3), 0 = − 𝜔2 𝐴2 + 𝐶 or 𝐶 = 𝜔2 𝐴2
𝑑𝑥 2
Now the equation (3) becomes ( 𝑑𝑡 ) = − 𝜔2 𝑥 2 + 𝜔2 𝐴2
𝑑𝑥 2 𝒅𝒙
or ( 𝑑𝑡 ) = 𝜔2 (𝐴2 − 𝑥 2 ) or = 𝝎 √(𝑨𝟐 − 𝒙𝟐 ) ….(4) (This is also the expression for the
𝒅𝒕
velocity of the particle executing SHM)
𝑑𝑥
Rewriting equation (4) as, = 𝜔 𝑑𝑡 .
√(𝐴2 − 𝑥 2 )
𝑑𝑥 𝑥
Integrating this equation, ∫ √(𝐴2 = 𝜔 ∫ 𝑑𝑡 gives 𝑠𝑖𝑛−1 (𝐴) = 𝜔𝑡 + 𝜑
− 𝑥2)
where 𝜑 is the constant called the initial phase of the particle executing SHM. It is also
called epoch.
𝑥
or (𝐴) = 𝑠𝑖𝑛 (𝜔𝑡 + 𝜑) or 𝒙 = 𝑨 𝒔𝒊𝒏 (𝝎𝒕 + 𝝋) ….(5)
This is the solution of differential equation of SHM (eqn. (1)). Equation (5) is the expression
for the displacement of the particle at time t.
2. Expression for Velocity of the particle executing SHM
The equation of SHM is given by 𝑥 = 𝐴 𝑠𝑖𝑛 (𝜔𝑡 + 𝜑)
If the initial position from where time is measured is the mean position, then 𝜑 = 0
Thus 𝑥 = 𝐴 𝑠𝑖𝑛𝜔𝑡 . The velocity of the particle at a given instant of time t is given by
𝑑𝑥 𝑑𝑥
differentiating the above equation, i.e. = 𝐴𝜔 𝑐𝑜𝑠𝜔𝑡 or = 𝐴𝜔 √1 − 𝑠𝑖𝑛2 𝜔𝑡
𝑑𝑡 𝑑𝑡
𝑥 𝑑𝑥 𝑥2 𝑑𝑥 𝐴2 −𝑥 2
As 𝑠𝑖𝑛𝜔𝑡 = , we have = 𝐴𝜔 √1 − 𝐴2 or = 𝐴𝜔 √
𝐴 𝑑𝑡 𝑑𝑡 𝐴2
𝒅𝒙
Thus the velocity of the particle is 𝒗= = 𝝎 √𝑨𝟐 − 𝒙𝟐
𝒅𝒕
Dr. K S Suresh, Associate professor, Vijaya College Page 3
