OSCILLATIONS

Chapter-14 OSCILLATIONS

Periodic motion: A motion that repeats itself at regular intervals of time is called periodic
motion.
Ex: Motion of planets in solar system, uniform circular motion.

Oscillatory motion: A motion in which a body moves to and fro between two extreme
positions about an equilibrium position.
Ex: boat tossing up and down, piston of a steam engine, motion of simple pendulum.

Equilibrium (Mean) position: It is the position of a body during oscillatory motion at

which the net external force acting on the body is zero.
It is the position, at which if it is at rest, it remains at rest forever.

Oscillations or vibrations: The motion of a body between two extreme positions forms
oscillations or vibrations.

Note: (i) There is no significant difference between oscillations and vibrations. When the
frequency is small we call it oscillation, while the frequency is high we call it vibrations.
(ii) Every oscillatory motion is periodic; but every periodic motion need not be oscillatory.

Importance of oscillatory motion: This motion is basic to physics. In musical

instruments we come across vibrating strings, membranes in drums and diaphragms in telephone
and speaker system vibrate, vibrations of air molecule, vibrations of atoms in solid include
oscillatory motion. The concepts of oscillatory motion are required to understand many physical
phenomena listed above.

Description of oscillatory motion: The description of oscillatory motion requires some

fundamental concepts like period, frequency, displacement, amplitude and phase.

Period or Time period (T): The smallest interval of time after which a periodic motion repeats
is called period.
In case of oscillation, the time taken by the body to complete one oscillation is called period. SI
unit of period is 𝑠𝑒𝑐𝑜𝑛𝑑.

Frequency (𝝂): Number of times a periodic motion repeats per unit time is called frequency.
In case of oscillations, number of oscillations per unit time is called frequency. SI unit of
frequency is ℎ𝑒𝑟𝑡𝑧 (𝐻𝑧). 1𝐻𝑧 = 1 oscillation per second.

Note: Relation between period and frequency is given by, 𝑇 = or 𝜈 = 1⁄

1⁄ 𝑇
𝜈

Displacement (x or y): The term displacement refers to change of physical quantity with
time. In periodic motion displacement may be linear as well as angular.

Linear displacement: The straight line distance travelled by a

particle from its equilibrium position.

Angular displacement: It is the angle through which position vector of the body rotates in a
given time.
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Amplitude (A): The maximum displacement of the particle from its equilibrium
position is called amplitude.

Periodic function: Any function which repeats itself after a regular interval of
time is called periodic function.
In periodic motion displacement is periodic function and it can be represented by a mathematical
function of time. The simplest of these functions is given by, 𝑓(𝑡) = 𝐴 cos 𝜔𝑡.
If 𝜔𝑡 is increased by an integral multiple of 2𝜋 radian, the value of the function remains same and
𝑓(𝑡) is periodic.

If 𝑇 2𝜋
= 𝜔
𝑓(𝑡 + 𝑇) = 𝐴 cos 𝜔(𝑡 + 𝑇)
2𝜋
𝑓(𝑡 + 𝑇) = 𝐴 cos 𝜔 (𝑡 + )
𝜔
𝑓(𝑡 + 𝑇) = 𝐴 cos(𝜔𝑡 + 2𝜋)
𝑓(𝑡 + 𝑡) = 𝐴 cos 𝜔𝑡 = 𝑓(𝑡)

Note: (i) In cos 𝜔𝑡, the term 𝜔 is called angular frequency.

(ii) The function 𝑓(𝑡) = 𝐴 sin 𝜔𝑡 is also periodic.
(iii) The linear combination of both sine and cosine function is also periodic and it is represented
by 𝑓(𝑡) = 𝐴 sin 𝜔𝑡 + 𝐵 cos 𝜔𝑡 and it is called Fourier series.
By putting, 𝐴 = 𝐷 cos 𝜙 and 𝐵 = 𝐷 sin 𝜙
𝑓(𝑡) = 𝐷 sin 𝜔𝑡 cos 𝜙 + 𝐷 cos 𝜔𝑡 sin 𝜙
𝑓(𝑡) = 𝐷 sin(𝜔𝑡 + 𝜙)
𝐵
where 𝐷 = √𝐴2 + 𝐵2 and 𝜙 = 𝑡𝑎𝑛−1 ( )
𝐴

Simple harmonic motion (SHM): The oscillatory motion is said to be simple harmonic, if
the displacement of the particle from the origin varies with time as;
𝑥(𝑡) = 𝐴 cos(𝜔𝑡 + 𝜙) or 𝑦(𝑡) = 𝐴 sin(𝜔𝑡 + 𝜙).
Simple harmonic motion is a periodic motion in which displacement is a sinusoidal function of
time.

Note: The simplest kind of periodic motion is simple harmonic motion.

Consider a particle oscillating back and forth about the origin along 𝑥 − 𝑎𝑥𝑖𝑠 between the limits
Analysis of simple harmonic motion:

+𝐴
and −𝐴 as shown.

Figure shows graph of 𝑥 versus 𝑡 which gives the values of displacements as function of time.

Phase: During the periodic motion, the position and velocity of the particle at any time 𝑡
is determined by the term (𝜔𝑡 + 𝜙) in cosine function. This quantity is called phase of the motion.

Phase constant (Phase angle):

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The value of phase at 𝑡 = 0 is 𝜙 and it is called the phase constant or phase angle.
OSCILLATIONS

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 OSCILLATIONS

Consider a particle moving with a uniform sped along the circumference of circle of radius 𝐴.
Simple harmonic motion and uniform circular motion:

Let the particle start from the point 𝑋 with a constant speed 𝜔.
After some time it reaches to 𝑃.
Draw 𝑃𝑀 perpendicular to 𝑦 −
𝑎𝑥𝑖𝑠.
𝑂𝑀 represents the projection of position vector of the particle on 𝑦 − 𝑎𝑥𝑖𝑠.

When the particle moves from 𝑋 to 𝑌 its projection of the position

vector moves from 𝑂 to 𝑌. As the particle moves from 𝑌 to 𝑋′, its
projection moves from 𝑌 to 𝑂. Similarly the particle moves from 𝑋′
to
𝑋 via 𝑌′, its projection moves from 𝑂 to 𝑌′ and 𝑌′ to 𝑂. This
shows that if the particle moves uniformly on a circle, its projection
on the diameter (𝑦 − 𝑎𝑥𝑖𝑠) of the circle executes SHM.

The position of the particle on the circle is given by, 𝑥(𝑡) = 𝐴 cos(𝜔𝑡 + 𝜙)
The displacement of the projection on 𝑦 − 𝑎𝑥𝑖𝑠 is given by, 𝑦(𝑡) = 𝐴 sin(𝜔𝑡 + 𝜙) which is
also SHM with same amplitude but different in phase by 𝜋⁄2.

Consider a particle moving on a circle of radius 𝐴 with uniform velocity 𝜔.

Equation of SHM:

Let the particle start from 𝑋 and subtend an angle 𝜃 in time 𝑡 and reaches
𝑃.
Angular velocity, 𝜔 𝜃
=
𝑡
𝜃 = 𝜔𝑡
The projection of the particle on 𝑥 − 𝑎𝑥𝑖𝑠 is, 𝑂𝑁 = 𝑥

In Δ𝑂𝑃𝑀, cos 𝑂𝑁
𝜃= 𝑥
=
𝑂𝑃
𝐴
𝑥 = 𝐴 cos 𝜃
𝒙 = 𝑨 𝐜𝐨𝐬 𝝎𝒕
If the particle starts from 𝑄, 𝒙 = 𝑨 𝐜𝐨𝐬 (𝝎𝒕 + 𝝓)

We have, 𝑥 = 𝐴 cos(𝜔𝑡 + 𝜙)
Velocity of the particle:

𝑑𝑥
𝑣 = = −𝐴𝜔 sin(𝜔𝑡 + 𝜙)
𝑑𝑡
Further, 𝑣 = −𝐴𝜔√1 − cos2 𝜔𝑡

𝑣 = √1 −𝑥(
2
−𝐴𝜔 )
𝐴

𝐴 2 − 𝑥2
𝑣 = −𝐴𝜔√
𝐴2

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 𝒗 = −𝝎√𝑨𝟐 − 𝒙𝟐
OSCILLATIONS

Negative sign shows that 𝑣 has a direction opposite to the positive direction of 𝑥 − 𝑎𝑥𝑖𝑠.

The above equation tells that,

(i) When 𝑥 = 0, 𝑣 = 𝜔𝐴 - velocity is maximum, velocity is maximum at equilibrium
(mean) position
(ii) When 𝑥 = 𝐴, 𝑣 = 0 - velocity is minimum, velocity is minimum at extreme position.

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 OSCILLATIONS

We have 𝑥 = 𝐴 cos(𝜔𝑡 + 𝜙)
Acceleration:

𝑑𝑥
𝑣 = = −𝐴𝜔 sin(𝜔𝑡 + 𝜙)
𝑑
𝑡
𝑎 = 𝑑2
𝑥 = −𝐴𝜔 cos (𝜔𝑡 + 𝜙)
2

𝑑𝑡
2

𝒂 = −𝝎𝟐𝒙
Negative sign indicates that the direction of displacement and acceleration are opposite to each
other.
(i) When 𝑥 = 0, 𝑎 = 0, acceleration is minimum at mean position.
(ii) when 𝑥 = 𝐴, |𝑎| = 𝜔2𝐴, acceleration is maximum at extreme position.

𝑑2
𝐍𝐨𝐭 𝑥 + 𝜔2𝑥 = 0, is called differntial equation of SHM.
𝐞:
𝑑𝑡
2

Acceleration of a particle executing SHM is given by, 𝑎 = −𝜔2𝑥

Force law for SHM:

From Newton’s second law, 𝐹 = 𝑚𝑎

𝐹 = 𝑚(−𝜔2𝑥)
𝐹 = −𝑚𝜔2𝑥
𝑭= 𝑤ℎ𝑒𝑟𝑒 𝑘 = 𝑚𝜔2
−𝒌𝒙
Negative sign indicates that force and displacement are oppositely directed.

Note: (i) A particle oscillating under a force given by 𝐹 = −𝑘𝑥 is called linear harmonic
oscillator

𝑘 (ii)We have 𝑘 =2𝑚𝜔 , 𝜔 =

√
𝑚

Energy in SHM:
A particle executing SHM possess,
(i) Kinetic energy - because it is moving.
(ii) Potential energy - because it is subjected to conservative force 𝐹 = −𝑘𝑥

Kinetic energy,
𝐾= 𝑚𝑣2
1 2 22
( )
𝐾 = 𝑚𝐴 si 𝜔𝑡 + 𝜙
𝜔2 n
1
𝐾 = 𝑘 𝐴2 sin2(𝜔𝑡 +
𝜙) 2

Potential energy, 1
𝑈= 𝑘𝑥2
2
1
𝑈 = 𝑘 𝐴2 cos2(𝜔𝑡 +
𝜙) 2
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 1 1
2 2( ) 2 2
( )
OSCILLATIONS

Total energy, 𝐸 = 𝐾 𝑘 si 𝜔𝑡 + + 𝑘 co 𝜔𝑡 + 𝜙
+𝑈= 𝐴2 n 𝜙 𝐴2 s
1 2[ ) 2
( )]
2(
𝐸 = 𝑘 si 𝜔𝑡 + + 𝜔𝑡 + 𝜙
𝐴2 n 𝜙 cos
𝟏
𝑬= 𝒌
𝑨𝟐 𝟐

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 OSCILLATIONS

1 1
Variation of Kinetic energy and potential energy of
𝑚𝜔 = 𝑘
𝐴 𝐴2 2
oscillator:
2
(i)
When the particle is at mean position, 𝑥 = 0, 𝑈= 2 2
0 and 𝐾 =
At mean position kinetic energy is maximum and potential energy is zero.

When the particle is at extreme position, 𝑥 1

𝑘 𝐴2 and 𝐾 = 0
(ii)

= 𝐴, 𝑈
2
=
At the extreme positions kinetic energy is zero and potential energy is maximum.

Some systems executing SHM:

There are no practical examples for absolutely pure simple harmonic motion. But under certain
conditions, some systems can be considered as approximately simple harmonic.

(i) Oscillations due to spring (Expression for Time period of oscillating

Consider a block of mass 𝑚 attached to a spring. The other end of the spring is rigidly fixed.
string):

If the block is pulled and released, it executes to and fro motion.

Let 𝑥 = 0 be the mean position of the block.
The restoring force of the block is given by, 𝐹 = −𝑘𝑥
𝑚𝑎 = −𝑘𝑥
𝑘
𝑎=−()𝑥 − − − (1)
�

The standard equation for SHM is, 𝑎 = −𝜔2𝑥 − − − (2)

�

𝑘
On comparing, 𝜔2 =
𝑚
𝜔 𝑘
=√
𝑚
𝟐𝝅 �
Time period of the block is, 𝑻 =√ =
𝟐𝝅
𝝎 𝒌
�

Consider a simple pendulum of mass 𝑚 tied to a string of length 𝐿.

(ii) Simple pendulum (Expression for time period of Simple pendulum):

Let the bob is set into oscillations.

Let 𝑃 be the position of the bob at time 𝑡.
Let 𝜃 be the angle made by the string with the vertical.
Force acting on the bob are,
(i) weight of the bob 𝑚𝑔, vertically downwards, which can be
resolved into two components; 𝑚𝑔 cos 𝜃 along the string and
𝑚𝑔 sin 𝜃 perpendicular to the string.
(ii) Tension 𝑇 in the string towards point of suspension.
The bob has two accelerations (i) radial acceleration
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 OSCILLATIONS
(ii) tangential acceleration
Radial acceleration provided by 𝑇 − 𝑚𝑔 cos 𝜃
Tangential acceleration provided by 𝑚𝑔 sin 𝜃

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 OSCILLATIONS
Radial force gives zero torque.
Therefore, Torque on the bob |𝜏⃗| = |𝑟⃗ × 𝐹⃗|
𝜏 = −𝐿𝑚𝑔 sin 𝜃
Negative sign indicates that the restoring torque tends to reduce angular displacement.
By Newton’s second law, 𝜏 = 𝐼𝛼
𝐼𝛼 = −𝐿𝑚𝑔 sin 𝜃
𝑚𝑔𝐿
𝛼 =− sin
� 𝜃
𝑚𝑔𝐿
If 𝜃 is small, sin 𝜃 ≈ 𝜃 , 𝛼 = − 𝜃
�
𝐼
Comparing with, 𝛼 = −𝜔2𝜃
𝑚𝑔𝐿
𝜔2 = 𝐼
𝑚𝑔𝐿
𝜔=√ 𝐼

2𝜋 𝑚𝑔𝐿
=√
𝑇 𝐼

𝐼
𝑇 = 2𝜋√
𝑚𝑔
𝐿

𝑚𝐿 (𝐼 = 𝑚𝐿2)
2

𝑇 = 2𝜋√
𝑚𝑔
𝐿

�
𝑻=
𝟐𝝅√ �
�

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