Page 1
CHAPTER THIRTEEN
OSCILLATIONS
13.1 INTRODUCTION
In our daily life we come across various kinds of motions.
You have already learnt about some of them, e.g., rectilinear
motion and motion of a projectile. Both these motions are
non-repetitive. We have also learnt about uniform circular
motion and orbital motion of planets in the solar system. In
these cases, the motion is repeated after a certain interval of
time, that is, it is periodic. In your childhood, you must have
enjoyed rocking in a cradle or swinging on a swing. Both
these motions are repetitive in nature but different from the
periodic motion of a planet. Here, the object moves to and fro
about a mean position. The pendulum of a wall clock executes
a similar motion. Examples of such periodic to and fro
motion abound: a boat tossing up and down in a river, the
piston in a steam engine going back and forth, etc. Such a
motion is termed as oscillatory motion. In this chapter we
study this motion.
The study of oscillatory motion is basic to physics; its
concepts are required for the understanding of many physical
phenomena. In musical instruments, like the sitar, the guitar
or the violin, we come across vibrating strings that produce
pleasing sounds. The membranes in drums and diaphragms
in telephone and speaker systems vibrate to and fro about
their mean positions. The vibrations of air molecules make
the propagation of sound possible. In a solid, the atoms vibrate
about their equilibrium positions, the average energy of
vibrations being proportional to temperature. AC power
supply give voltage that oscillates alternately going positive
and negative about the mean value (zero).
The description of a periodic motion, in general, and
oscillatory motion, in particular, requires some fundamental
concepts, like period, frequency, displacement, amplitude
and phase. These concepts are developed in the next section.
13.1 Introduction
13.2 Periodic and oscillatory
motions
13.3 Simple harmonic motion
13.4 Simple harmonic motion
and uniform circular
motion
13.5 Velocity and acceleration
in simple harmonic motion
13.6 Force law for simple
harmonic motion
13.7 Energy in simple harmonic
motion
13.8 The simple pendulum
Summary
Points to ponder
Exercises
2024-25
Page 2
CHAPTER THIRTEEN
OSCILLATIONS
13.1 INTRODUCTION
In our daily life we come across various kinds of motions.
You have already learnt about some of them, e.g., rectilinear
motion and motion of a projectile. Both these motions are
non-repetitive. We have also learnt about uniform circular
motion and orbital motion of planets in the solar system. In
these cases, the motion is repeated after a certain interval of
time, that is, it is periodic. In your childhood, you must have
enjoyed rocking in a cradle or swinging on a swing. Both
these motions are repetitive in nature but different from the
periodic motion of a planet. Here, the object moves to and fro
about a mean position. The pendulum of a wall clock executes
a similar motion. Examples of such periodic to and fro
motion abound: a boat tossing up and down in a river, the
piston in a steam engine going back and forth, etc. Such a
motion is termed as oscillatory motion. In this chapter we
study this motion.
The study of oscillatory motion is basic to physics; its
concepts are required for the understanding of many physical
phenomena. In musical instruments, like the sitar, the guitar
or the violin, we come across vibrating strings that produce
pleasing sounds. The membranes in drums and diaphragms
in telephone and speaker systems vibrate to and fro about
their mean positions. The vibrations of air molecules make
the propagation of sound possible. In a solid, the atoms vibrate
about their equilibrium positions, the average energy of
vibrations being proportional to temperature. AC power
supply give voltage that oscillates alternately going positive
and negative about the mean value (zero).
The description of a periodic motion, in general, and
oscillatory motion, in particular, requires some fundamental
concepts, like period, frequency, displacement, amplitude
and phase. These concepts are developed in the next section.
13.1 Introduction
13.2 Periodic and oscillatory
motions
13.3 Simple harmonic motion
13.4 Simple harmonic motion
and uniform circular
motion
13.5 Velocity and acceleration
in simple harmonic motion
13.6 Force law for simple
harmonic motion
13.7 Energy in simple harmonic
motion
13.8 The simple pendulum
Summary
Points to ponder
Exercises
2024-25
PHYSICS 260
13.2 PERIODIC AND OSCILLATORY MOTIONS
Fig. 13.1 shows some periodic motions. Suppose
an insect climbs up a ramp and falls down, it
comes back to the initial point and repeats the
process identically. If you draw a graph of its
height above the ground versus time, it would
look something like Fig. 13.1 (a). If a child climbs
up a step, comes down, and repeats the process
identically, its height above the ground would
look like that in Fig. 13.1 (b). When you play the
game of bouncing a ball off the ground, between
your palm and the ground, its height versus time
graph would look like the one in Fig. 13.1 (c).
Note that both the curved parts in Fig. 13.1 (c)
are sections of a parabola given by the Newton’s
equation of motion (see section 2.6),
2
1
2
+ gt h = ut
for downward motion, and
2
1
2
– gt h = ut
for upward motion,
with different values of u in each case. These
are examples of periodic motion. Thus, a motion
that repeats itself at regular intervals of time is
called periodic motion.
Fig. 13.1 Examples of periodic motion. The period T
is shown in each case.
Very often, the body undergoing periodic
motion has an equilibrium position somewhere
inside its path. When the body is at this position
no net external force acts on it. Therefore, if it is
left there at rest, it remains there forever. If the
body is given a small displacement from the
position, a force comes into play which tries to
bring the body back to the equilibrium point,
giving rise to oscillations or vibrations. For
example, a ball placed in a bowl will be in
equilibrium at the bottom. If displaced a little
from the point, it will perform oscillations in the
bowl. Every oscillatory motion is periodic, but
every periodic motion need not be oscillatory.
Circular motion is a periodic motion, but it is
not oscillatory.
There is no significant difference between
oscillations and vibrations. It seems that when
the frequency is small, we call it oscillation (like,
the oscillation of a branch of a tree), while when
the frequency is high, we call it vibration (like,
the vibration of a string of a musical instrument).
Simple harmonic motion is the simplest form
of oscillatory motion. This motion arises when
the force on the oscillating body is directly
proportional to its displacement from the mean
position, which is also the equilibrium position.
Further, at any point in its oscillation, this force
is directed towards the mean position.
In practice, oscillating bodies eventually
come to rest at their equilibrium positions
because of the damping due to friction and other
dissipative causes. However, they can be forced
to remain oscillating by means of some external
periodic agency. We discuss the phenomena of
damped and forced oscillations later in the
chapter.
Any material medium can be pictured as a
collection of a large number of coupled
oscillators. The collective oscillations of the
constituents of a medium manifest themselves
as waves. Examples of waves include water
waves, seismic waves, electromagnetic waves.
We shall study the wave phenomenon in the next
chapter.
13.2.1 Period and frequency
We have seen that any motion that repeats itself
at regular intervals of time is called periodic
motion. The smallest interval of time after
which the motion is repeated is called its
period. Let us denote the period by the symbol
T. Its SI unit is second. For periodic motions,
(a)
(b)
(c)
2024-25
Page 3
CHAPTER THIRTEEN
OSCILLATIONS
13.1 INTRODUCTION
In our daily life we come across various kinds of motions.
You have already learnt about some of them, e.g., rectilinear
motion and motion of a projectile. Both these motions are
non-repetitive. We have also learnt about uniform circular
motion and orbital motion of planets in the solar system. In
these cases, the motion is repeated after a certain interval of
time, that is, it is periodic. In your childhood, you must have
enjoyed rocking in a cradle or swinging on a swing. Both
these motions are repetitive in nature but different from the
periodic motion of a planet. Here, the object moves to and fro
about a mean position. The pendulum of a wall clock executes
a similar motion. Examples of such periodic to and fro
motion abound: a boat tossing up and down in a river, the
piston in a steam engine going back and forth, etc. Such a
motion is termed as oscillatory motion. In this chapter we
study this motion.
The study of oscillatory motion is basic to physics; its
concepts are required for the understanding of many physical
phenomena. In musical instruments, like the sitar, the guitar
or the violin, we come across vibrating strings that produce
pleasing sounds. The membranes in drums and diaphragms
in telephone and speaker systems vibrate to and fro about
their mean positions. The vibrations of air molecules make
the propagation of sound possible. In a solid, the atoms vibrate
about their equilibrium positions, the average energy of
vibrations being proportional to temperature. AC power
supply give voltage that oscillates alternately going positive
and negative about the mean value (zero).
The description of a periodic motion, in general, and
oscillatory motion, in particular, requires some fundamental
concepts, like period, frequency, displacement, amplitude
and phase. These concepts are developed in the next section.
13.1 Introduction
13.2 Periodic and oscillatory
motions
13.3 Simple harmonic motion
13.4 Simple harmonic motion
and uniform circular
motion
13.5 Velocity and acceleration
in simple harmonic motion
13.6 Force law for simple
harmonic motion
13.7 Energy in simple harmonic
motion
13.8 The simple pendulum
Summary
Points to ponder
Exercises
2024-25
PHYSICS 260
13.2 PERIODIC AND OSCILLATORY MOTIONS
Fig. 13.1 shows some periodic motions. Suppose
an insect climbs up a ramp and falls down, it
comes back to the initial point and repeats the
process identically. If you draw a graph of its
height above the ground versus time, it would
look something like Fig. 13.1 (a). If a child climbs
up a step, comes down, and repeats the process
identically, its height above the ground would
look like that in Fig. 13.1 (b). When you play the
game of bouncing a ball off the ground, between
your palm and the ground, its height versus time
graph would look like the one in Fig. 13.1 (c).
Note that both the curved parts in Fig. 13.1 (c)
are sections of a parabola given by the Newton’s
equation of motion (see section 2.6),
2
1
2
+ gt h = ut
for downward motion, and
2
1
2
– gt h = ut
for upward motion,
with different values of u in each case. These
are examples of periodic motion. Thus, a motion
that repeats itself at regular intervals of time is
called periodic motion.
Fig. 13.1 Examples of periodic motion. The period T
is shown in each case.
Very often, the body undergoing periodic
motion has an equilibrium position somewhere
inside its path. When the body is at this position
no net external force acts on it. Therefore, if it is
left there at rest, it remains there forever. If the
body is given a small displacement from the
position, a force comes into play which tries to
bring the body back to the equilibrium point,
giving rise to oscillations or vibrations. For
example, a ball placed in a bowl will be in
equilibrium at the bottom. If displaced a little
from the point, it will perform oscillations in the
bowl. Every oscillatory motion is periodic, but
every periodic motion need not be oscillatory.
Circular motion is a periodic motion, but it is
not oscillatory.
There is no significant difference between
oscillations and vibrations. It seems that when
the frequency is small, we call it oscillation (like,
the oscillation of a branch of a tree), while when
the frequency is high, we call it vibration (like,
the vibration of a string of a musical instrument).
Simple harmonic motion is the simplest form
of oscillatory motion. This motion arises when
the force on the oscillating body is directly
proportional to its displacement from the mean
position, which is also the equilibrium position.
Further, at any point in its oscillation, this force
is directed towards the mean position.
In practice, oscillating bodies eventually
come to rest at their equilibrium positions
because of the damping due to friction and other
dissipative causes. However, they can be forced
to remain oscillating by means of some external
periodic agency. We discuss the phenomena of
damped and forced oscillations later in the
chapter.
Any material medium can be pictured as a
collection of a large number of coupled
oscillators. The collective oscillations of the
constituents of a medium manifest themselves
as waves. Examples of waves include water
waves, seismic waves, electromagnetic waves.
We shall study the wave phenomenon in the next
chapter.
13.2.1 Period and frequency
We have seen that any motion that repeats itself
at regular intervals of time is called periodic
motion. The smallest interval of time after
which the motion is repeated is called its
period. Let us denote the period by the symbol
T. Its SI unit is second. For periodic motions,
(a)
(b)
(c)
2024-25
OSCILLATIONS 261
which are either too fast or too slow on the scale
of seconds, other convenient units of time are
used. The period of vibrations of a quartz crystal
is expressed in units of microseconds (10
–6
s)
abbreviated as µs. On the other hand, the orbital
period of the planet Mercury is 88 earth days.
The Halley’s comet appears after every 76 years.
The reciprocal of T gives the number of
repetitions that occur per unit time. This
quantity is called the frequency of the periodic
motion. It is represented by the symbol ?. The
relation between ? and T is
? = 1/T (13.1)
The unit of ? is thus s
–1
. After the discoverer of
radio waves, Heinrich Rudolph Hertz (1857–1894),
a special name has been given to the unit of
frequency. It is called hertz (abbreviated as Hz).
Thus,
1 hertz = 1 Hz =1 oscillation per second =1 s
–1
(13.2)
Note, that the frequency, ?, is not necessarily
an integer.
u Example 13.1 On an average, a human
heart is found to beat 75 times in a minute.
Calculate its frequency and period.
Answer The beat frequency of heart = 75/(1 min)
= 75/(60 s)
= 1.25 s
–1
= 1.25 Hz
The time period T = 1/(1.25 s
–1
)
= 0.8 s ?
13.2.2 Displacement
In section 3.2, we defined displacement of a
particle as the change in its position vector. In
this chapter, we use the term displacement
in a more general sense. It refers to change
with time of any physical property under
consideration. For example, in case of rectilinear
motion of a steel ball on a surface, the distance
from the starting point as a function of time is
its position displacement. The choice of origin
is a matter of convenience. Consider a block
attached to a spring, the other end of the spring
is fixed to a rigid wall [see Fig.13.2(a)]. Generally,
it is convenient to measure displacement of the
body from its equilibrium position. For an
oscillating simple pendulum, the angle from the
vertical as a function of time may be regarded
as a displacement variable [see Fig.13.2(b)]. The
term displacement is not always to be referred
Fig. 13.2(a) A block attached to a spring, the other
end of which is fixed to a rigid wall. The
block moves on a frictionless surface. The
motion of the block can be described in
terms of its distance or displacement x
from the equilibrium position.
Fig.13.2(b) An oscillating simple pendulum; its
motion can be described in terms of
angular displacement ? from the vertical.
in the context of position only. There can be
many other kinds of displacement variables. The
voltage across a capacitor, changing with time
in an AC circuit, is also a displacement variable.
In the same way, pressure variations in time in
the propagation of sound wave, the changing
electric and magnetic fields in a light wave are
examples of displacement in different contexts.
The displacement variable may take both
positive and negative values. In experiments on
oscillations, the displacement is measured for
different times.
The displacement can be represented by a
mathematical function of time. In case of periodic
motion, this function is periodic in time. One of
the simplest periodic functions is given by
f (t) = A cos ?t (13.3a)
If the argument of this function, ?t, is
increased by an integral multiple of 2p radians,
the value of the function remains the same. The
2024-25
Page 4
CHAPTER THIRTEEN
OSCILLATIONS
13.1 INTRODUCTION
In our daily life we come across various kinds of motions.
You have already learnt about some of them, e.g., rectilinear
motion and motion of a projectile. Both these motions are
non-repetitive. We have also learnt about uniform circular
motion and orbital motion of planets in the solar system. In
these cases, the motion is repeated after a certain interval of
time, that is, it is periodic. In your childhood, you must have
enjoyed rocking in a cradle or swinging on a swing. Both
these motions are repetitive in nature but different from the
periodic motion of a planet. Here, the object moves to and fro
about a mean position. The pendulum of a wall clock executes
a similar motion. Examples of such periodic to and fro
motion abound: a boat tossing up and down in a river, the
piston in a steam engine going back and forth, etc. Such a
motion is termed as oscillatory motion. In this chapter we
study this motion.
The study of oscillatory motion is basic to physics; its
concepts are required for the understanding of many physical
phenomena. In musical instruments, like the sitar, the guitar
or the violin, we come across vibrating strings that produce
pleasing sounds. The membranes in drums and diaphragms
in telephone and speaker systems vibrate to and fro about
their mean positions. The vibrations of air molecules make
the propagation of sound possible. In a solid, the atoms vibrate
about their equilibrium positions, the average energy of
vibrations being proportional to temperature. AC power
supply give voltage that oscillates alternately going positive
and negative about the mean value (zero).
The description of a periodic motion, in general, and
oscillatory motion, in particular, requires some fundamental
concepts, like period, frequency, displacement, amplitude
and phase. These concepts are developed in the next section.
13.1 Introduction
13.2 Periodic and oscillatory
motions
13.3 Simple harmonic motion
13.4 Simple harmonic motion
and uniform circular
motion
13.5 Velocity and acceleration
in simple harmonic motion
13.6 Force law for simple
harmonic motion
13.7 Energy in simple harmonic
motion
13.8 The simple pendulum
Summary
Points to ponder
Exercises
2024-25
PHYSICS 260
13.2 PERIODIC AND OSCILLATORY MOTIONS
Fig. 13.1 shows some periodic motions. Suppose
an insect climbs up a ramp and falls down, it
comes back to the initial point and repeats the
process identically. If you draw a graph of its
height above the ground versus time, it would
look something like Fig. 13.1 (a). If a child climbs
up a step, comes down, and repeats the process
identically, its height above the ground would
look like that in Fig. 13.1 (b). When you play the
game of bouncing a ball off the ground, between
your palm and the ground, its height versus time
graph would look like the one in Fig. 13.1 (c).
Note that both the curved parts in Fig. 13.1 (c)
are sections of a parabola given by the Newton’s
equation of motion (see section 2.6),
2
1
2
+ gt h = ut
for downward motion, and
2
1
2
– gt h = ut
for upward motion,
with different values of u in each case. These
are examples of periodic motion. Thus, a motion
that repeats itself at regular intervals of time is
called periodic motion.
Fig. 13.1 Examples of periodic motion. The period T
is shown in each case.
Very often, the body undergoing periodic
motion has an equilibrium position somewhere
inside its path. When the body is at this position
no net external force acts on it. Therefore, if it is
left there at rest, it remains there forever. If the
body is given a small displacement from the
position, a force comes into play which tries to
bring the body back to the equilibrium point,
giving rise to oscillations or vibrations. For
example, a ball placed in a bowl will be in
equilibrium at the bottom. If displaced a little
from the point, it will perform oscillations in the
bowl. Every oscillatory motion is periodic, but
every periodic motion need not be oscillatory.
Circular motion is a periodic motion, but it is
not oscillatory.
There is no significant difference between
oscillations and vibrations. It seems that when
the frequency is small, we call it oscillation (like,
the oscillation of a branch of a tree), while when
the frequency is high, we call it vibration (like,
the vibration of a string of a musical instrument).
Simple harmonic motion is the simplest form
of oscillatory motion. This motion arises when
the force on the oscillating body is directly
proportional to its displacement from the mean
position, which is also the equilibrium position.
Further, at any point in its oscillation, this force
is directed towards the mean position.
In practice, oscillating bodies eventually
come to rest at their equilibrium positions
because of the damping due to friction and other
dissipative causes. However, they can be forced
to remain oscillating by means of some external
periodic agency. We discuss the phenomena of
damped and forced oscillations later in the
chapter.
Any material medium can be pictured as a
collection of a large number of coupled
oscillators. The collective oscillations of the
constituents of a medium manifest themselves
as waves. Examples of waves include water
waves, seismic waves, electromagnetic waves.
We shall study the wave phenomenon in the next
chapter.
13.2.1 Period and frequency
We have seen that any motion that repeats itself
at regular intervals of time is called periodic
motion. The smallest interval of time after
which the motion is repeated is called its
period. Let us denote the period by the symbol
T. Its SI unit is second. For periodic motions,
(a)
(b)
(c)
2024-25
OSCILLATIONS 261
which are either too fast or too slow on the scale
of seconds, other convenient units of time are
used. The period of vibrations of a quartz crystal
is expressed in units of microseconds (10
–6
s)
abbreviated as µs. On the other hand, the orbital
period of the planet Mercury is 88 earth days.
The Halley’s comet appears after every 76 years.
The reciprocal of T gives the number of
repetitions that occur per unit time. This
quantity is called the frequency of the periodic
motion. It is represented by the symbol ?. The
relation between ? and T is
? = 1/T (13.1)
The unit of ? is thus s
–1
. After the discoverer of
radio waves, Heinrich Rudolph Hertz (1857–1894),
a special name has been given to the unit of
frequency. It is called hertz (abbreviated as Hz).
Thus,
1 hertz = 1 Hz =1 oscillation per second =1 s
–1
(13.2)
Note, that the frequency, ?, is not necessarily
an integer.
u Example 13.1 On an average, a human
heart is found to beat 75 times in a minute.
Calculate its frequency and period.
Answer The beat frequency of heart = 75/(1 min)
= 75/(60 s)
= 1.25 s
–1
= 1.25 Hz
The time period T = 1/(1.25 s
–1
)
= 0.8 s ?
13.2.2 Displacement
In section 3.2, we defined displacement of a
particle as the change in its position vector. In
this chapter, we use the term displacement
in a more general sense. It refers to change
with time of any physical property under
consideration. For example, in case of rectilinear
motion of a steel ball on a surface, the distance
from the starting point as a function of time is
its position displacement. The choice of origin
is a matter of convenience. Consider a block
attached to a spring, the other end of the spring
is fixed to a rigid wall [see Fig.13.2(a)]. Generally,
it is convenient to measure displacement of the
body from its equilibrium position. For an
oscillating simple pendulum, the angle from the
vertical as a function of time may be regarded
as a displacement variable [see Fig.13.2(b)]. The
term displacement is not always to be referred
Fig. 13.2(a) A block attached to a spring, the other
end of which is fixed to a rigid wall. The
block moves on a frictionless surface. The
motion of the block can be described in
terms of its distance or displacement x
from the equilibrium position.
Fig.13.2(b) An oscillating simple pendulum; its
motion can be described in terms of
angular displacement ? from the vertical.
in the context of position only. There can be
many other kinds of displacement variables. The
voltage across a capacitor, changing with time
in an AC circuit, is also a displacement variable.
In the same way, pressure variations in time in
the propagation of sound wave, the changing
electric and magnetic fields in a light wave are
examples of displacement in different contexts.
The displacement variable may take both
positive and negative values. In experiments on
oscillations, the displacement is measured for
different times.
The displacement can be represented by a
mathematical function of time. In case of periodic
motion, this function is periodic in time. One of
the simplest periodic functions is given by
f (t) = A cos ?t (13.3a)
If the argument of this function, ?t, is
increased by an integral multiple of 2p radians,
the value of the function remains the same. The
2024-25
PHYSICS 262
function f (t) is then periodic and its period, T,
is given by
?
p 2
= T
(13.3b)
Thus, the function f (t) is periodic with period T,
f (t) = f (t+T )
The same result is obviously correct if we
consider a sine function, f (t ) = A sin ?t. Further,
a linear combination of sine and cosine functions
like,
f (t) = A sin ?t + B cos ?t (13.3c)
is also a periodic function with the same period
T. Taking,
A = D cos f and B = D sin f
Eq. (13.3c) can be written as,
f (t) = D sin (?t + f ) , (13.3d)
Here D and f are constant given by
2 2 1
and tan ? =
–
D = A + B
B
A
?
?
?
?
?
?
The great importance of periodic sine and
cosine functions is due to a remarkable result
proved by the French mathematician, Jean
Baptiste Joseph Fourier (1768–1830): Any
periodic function can be expressed as a
superposition of sine and cosine functions
of different time periods with suitable
coefficients.
u Example 13.2 Which of the following
functions of time represent (a) periodic and
(b) non-periodic motion? Give the period for
each case of periodic motion [? is any
positive constant].
(i) sin ?t + cos ?t
(ii) sin ?t + cos 2 ?t + sin 4 ?t
(iii) e
–?t
(iv) log (?t)
Answer
(i) sin ?t + cos ?t is a periodic function, it can
also be written as 2 sin (?t + p/4).
Now 2 sin (?t + p/4)=
2
sin (?t + p/4+2p)
=
2
sin [? (t + 2p/?) + p/4]
The periodic time of the function is 2p/?.
(ii) This is an example of a periodic motion. It
can be noted that each term represents a
periodic function with a different angular
frequency. Since period is the least interval
of time after which a function repeats its
value, sin ?t has a period T
0
= 2p/? ; cos 2 ?t
has a period p/? =T
0
/2; and sin 4 ?t has a
period 2p/4? = T
0
/4. The period of the first
term is a multiple of the periods of the last
two terms. Therefore, the smallest interval
of time after which the sum of the three
terms repeats is T
0
, and thus, the sum is a
periodic function with a period 2p/?.
(iii) The function e
–?t
is not periodic, it
decreases monotonically with increasing
time and tends to zero as t ? 8 and thus,
never repeats its value.
(iv) The function log(?t) increases
monotonically with time t. It, therefore,
never repeats its value and is a non-
periodic function. It may be noted that as
t ? 8, log(?t) diverges to 8. It, therefore,
cannot represent any kind of physical
displacement. ?
13.3 SIMPLE HARMONIC MOTION
Consider a particle oscillating back and forth
about the origin of an x-axis between the limits
+A and –A as shown in Fig. 13.3. This oscillatory
motion is said to be simple harmonic if the
displacement x of the particle from the origin
varies with time as :
x (t) = A cos (? t + f ) (13.4)
Fig. 13.3 A particle vibrating back and forth about
the origin of x-axis, between the limits +A
and –A.
where A, ? and f are constants.
Thus, simple harmonic motion (SHM) is not
any periodic motion but one in which
displacement is a sinusoidal function of time.
Fig. 13.4 shows the positions of a particle
executing SHM at discrete value of time, each
interval of time being T/4, where T is the period
of motion. Fig. 13.5 plots the graph of x versus t,
which gives the values of displacement as a
continuous function of time. The quantities A,
2024-25
Page 5
CHAPTER THIRTEEN
OSCILLATIONS
13.1 INTRODUCTION
In our daily life we come across various kinds of motions.
You have already learnt about some of them, e.g., rectilinear
motion and motion of a projectile. Both these motions are
non-repetitive. We have also learnt about uniform circular
motion and orbital motion of planets in the solar system. In
these cases, the motion is repeated after a certain interval of
time, that is, it is periodic. In your childhood, you must have
enjoyed rocking in a cradle or swinging on a swing. Both
these motions are repetitive in nature but different from the
periodic motion of a planet. Here, the object moves to and fro
about a mean position. The pendulum of a wall clock executes
a similar motion. Examples of such periodic to and fro
motion abound: a boat tossing up and down in a river, the
piston in a steam engine going back and forth, etc. Such a
motion is termed as oscillatory motion. In this chapter we
study this motion.
The study of oscillatory motion is basic to physics; its
concepts are required for the understanding of many physical
phenomena. In musical instruments, like the sitar, the guitar
or the violin, we come across vibrating strings that produce
pleasing sounds. The membranes in drums and diaphragms
in telephone and speaker systems vibrate to and fro about
their mean positions. The vibrations of air molecules make
the propagation of sound possible. In a solid, the atoms vibrate
about their equilibrium positions, the average energy of
vibrations being proportional to temperature. AC power
supply give voltage that oscillates alternately going positive
and negative about the mean value (zero).
The description of a periodic motion, in general, and
oscillatory motion, in particular, requires some fundamental
concepts, like period, frequency, displacement, amplitude
and phase. These concepts are developed in the next section.
13.1 Introduction
13.2 Periodic and oscillatory
motions
13.3 Simple harmonic motion
13.4 Simple harmonic motion
and uniform circular
motion
13.5 Velocity and acceleration
in simple harmonic motion
13.6 Force law for simple
harmonic motion
13.7 Energy in simple harmonic
motion
13.8 The simple pendulum
Summary
Points to ponder
Exercises
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PHYSICS 260
13.2 PERIODIC AND OSCILLATORY MOTIONS
Fig. 13.1 shows some periodic motions. Suppose
an insect climbs up a ramp and falls down, it
comes back to the initial point and repeats the
process identically. If you draw a graph of its
height above the ground versus time, it would
look something like Fig. 13.1 (a). If a child climbs
up a step, comes down, and repeats the process
identically, its height above the ground would
look like that in Fig. 13.1 (b). When you play the
game of bouncing a ball off the ground, between
your palm and the ground, its height versus time
graph would look like the one in Fig. 13.1 (c).
Note that both the curved parts in Fig. 13.1 (c)
are sections of a parabola given by the Newton’s
equation of motion (see section 2.6),
2
1
2
+ gt h = ut
for downward motion, and
2
1
2
– gt h = ut
for upward motion,
with different values of u in each case. These
are examples of periodic motion. Thus, a motion
that repeats itself at regular intervals of time is
called periodic motion.
Fig. 13.1 Examples of periodic motion. The period T
is shown in each case.
Very often, the body undergoing periodic
motion has an equilibrium position somewhere
inside its path. When the body is at this position
no net external force acts on it. Therefore, if it is
left there at rest, it remains there forever. If the
body is given a small displacement from the
position, a force comes into play which tries to
bring the body back to the equilibrium point,
giving rise to oscillations or vibrations. For
example, a ball placed in a bowl will be in
equilibrium at the bottom. If displaced a little
from the point, it will perform oscillations in the
bowl. Every oscillatory motion is periodic, but
every periodic motion need not be oscillatory.
Circular motion is a periodic motion, but it is
not oscillatory.
There is no significant difference between
oscillations and vibrations. It seems that when
the frequency is small, we call it oscillation (like,
the oscillation of a branch of a tree), while when
the frequency is high, we call it vibration (like,
the vibration of a string of a musical instrument).
Simple harmonic motion is the simplest form
of oscillatory motion. This motion arises when
the force on the oscillating body is directly
proportional to its displacement from the mean
position, which is also the equilibrium position.
Further, at any point in its oscillation, this force
is directed towards the mean position.
In practice, oscillating bodies eventually
come to rest at their equilibrium positions
because of the damping due to friction and other
dissipative causes. However, they can be forced
to remain oscillating by means of some external
periodic agency. We discuss the phenomena of
damped and forced oscillations later in the
chapter.
Any material medium can be pictured as a
collection of a large number of coupled
oscillators. The collective oscillations of the
constituents of a medium manifest themselves
as waves. Examples of waves include water
waves, seismic waves, electromagnetic waves.
We shall study the wave phenomenon in the next
chapter.
13.2.1 Period and frequency
We have seen that any motion that repeats itself
at regular intervals of time is called periodic
motion. The smallest interval of time after
which the motion is repeated is called its
period. Let us denote the period by the symbol
T. Its SI unit is second. For periodic motions,
(a)
(b)
(c)
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OSCILLATIONS 261
which are either too fast or too slow on the scale
of seconds, other convenient units of time are
used. The period of vibrations of a quartz crystal
is expressed in units of microseconds (10
–6
s)
abbreviated as µs. On the other hand, the orbital
period of the planet Mercury is 88 earth days.
The Halley’s comet appears after every 76 years.
The reciprocal of T gives the number of
repetitions that occur per unit time. This
quantity is called the frequency of the periodic
motion. It is represented by the symbol ?. The
relation between ? and T is
? = 1/T (13.1)
The unit of ? is thus s
–1
. After the discoverer of
radio waves, Heinrich Rudolph Hertz (1857–1894),
a special name has been given to the unit of
frequency. It is called hertz (abbreviated as Hz).
Thus,
1 hertz = 1 Hz =1 oscillation per second =1 s
–1
(13.2)
Note, that the frequency, ?, is not necessarily
an integer.
u Example 13.1 On an average, a human
heart is found to beat 75 times in a minute.
Calculate its frequency and period.
Answer The beat frequency of heart = 75/(1 min)
= 75/(60 s)
= 1.25 s
–1
= 1.25 Hz
The time period T = 1/(1.25 s
–1
)
= 0.8 s ?
13.2.2 Displacement
In section 3.2, we defined displacement of a
particle as the change in its position vector. In
this chapter, we use the term displacement
in a more general sense. It refers to change
with time of any physical property under
consideration. For example, in case of rectilinear
motion of a steel ball on a surface, the distance
from the starting point as a function of time is
its position displacement. The choice of origin
is a matter of convenience. Consider a block
attached to a spring, the other end of the spring
is fixed to a rigid wall [see Fig.13.2(a)]. Generally,
it is convenient to measure displacement of the
body from its equilibrium position. For an
oscillating simple pendulum, the angle from the
vertical as a function of time may be regarded
as a displacement variable [see Fig.13.2(b)]. The
term displacement is not always to be referred
Fig. 13.2(a) A block attached to a spring, the other
end of which is fixed to a rigid wall. The
block moves on a frictionless surface. The
motion of the block can be described in
terms of its distance or displacement x
from the equilibrium position.
Fig.13.2(b) An oscillating simple pendulum; its
motion can be described in terms of
angular displacement ? from the vertical.
in the context of position only. There can be
many other kinds of displacement variables. The
voltage across a capacitor, changing with time
in an AC circuit, is also a displacement variable.
In the same way, pressure variations in time in
the propagation of sound wave, the changing
electric and magnetic fields in a light wave are
examples of displacement in different contexts.
The displacement variable may take both
positive and negative values. In experiments on
oscillations, the displacement is measured for
different times.
The displacement can be represented by a
mathematical function of time. In case of periodic
motion, this function is periodic in time. One of
the simplest periodic functions is given by
f (t) = A cos ?t (13.3a)
If the argument of this function, ?t, is
increased by an integral multiple of 2p radians,
the value of the function remains the same. The
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PHYSICS 262
function f (t) is then periodic and its period, T,
is given by
?
p 2
= T
(13.3b)
Thus, the function f (t) is periodic with period T,
f (t) = f (t+T )
The same result is obviously correct if we
consider a sine function, f (t ) = A sin ?t. Further,
a linear combination of sine and cosine functions
like,
f (t) = A sin ?t + B cos ?t (13.3c)
is also a periodic function with the same period
T. Taking,
A = D cos f and B = D sin f
Eq. (13.3c) can be written as,
f (t) = D sin (?t + f ) , (13.3d)
Here D and f are constant given by
2 2 1
and tan ? =
–
D = A + B
B
A
?
?
?
?
?
?
The great importance of periodic sine and
cosine functions is due to a remarkable result
proved by the French mathematician, Jean
Baptiste Joseph Fourier (1768–1830): Any
periodic function can be expressed as a
superposition of sine and cosine functions
of different time periods with suitable
coefficients.
u Example 13.2 Which of the following
functions of time represent (a) periodic and
(b) non-periodic motion? Give the period for
each case of periodic motion [? is any
positive constant].
(i) sin ?t + cos ?t
(ii) sin ?t + cos 2 ?t + sin 4 ?t
(iii) e
–?t
(iv) log (?t)
Answer
(i) sin ?t + cos ?t is a periodic function, it can
also be written as 2 sin (?t + p/4).
Now 2 sin (?t + p/4)=
2
sin (?t + p/4+2p)
=
2
sin [? (t + 2p/?) + p/4]
The periodic time of the function is 2p/?.
(ii) This is an example of a periodic motion. It
can be noted that each term represents a
periodic function with a different angular
frequency. Since period is the least interval
of time after which a function repeats its
value, sin ?t has a period T
0
= 2p/? ; cos 2 ?t
has a period p/? =T
0
/2; and sin 4 ?t has a
period 2p/4? = T
0
/4. The period of the first
term is a multiple of the periods of the last
two terms. Therefore, the smallest interval
of time after which the sum of the three
terms repeats is T
0
, and thus, the sum is a
periodic function with a period 2p/?.
(iii) The function e
–?t
is not periodic, it
decreases monotonically with increasing
time and tends to zero as t ? 8 and thus,
never repeats its value.
(iv) The function log(?t) increases
monotonically with time t. It, therefore,
never repeats its value and is a non-
periodic function. It may be noted that as
t ? 8, log(?t) diverges to 8. It, therefore,
cannot represent any kind of physical
displacement. ?
13.3 SIMPLE HARMONIC MOTION
Consider a particle oscillating back and forth
about the origin of an x-axis between the limits
+A and –A as shown in Fig. 13.3. This oscillatory
motion is said to be simple harmonic if the
displacement x of the particle from the origin
varies with time as :
x (t) = A cos (? t + f ) (13.4)
Fig. 13.3 A particle vibrating back and forth about
the origin of x-axis, between the limits +A
and –A.
where A, ? and f are constants.
Thus, simple harmonic motion (SHM) is not
any periodic motion but one in which
displacement is a sinusoidal function of time.
Fig. 13.4 shows the positions of a particle
executing SHM at discrete value of time, each
interval of time being T/4, where T is the period
of motion. Fig. 13.5 plots the graph of x versus t,
which gives the values of displacement as a
continuous function of time. The quantities A,
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OSCILLATIONS 263
? and f which characterize a given SHM have
standard names, as summarised in Fig. 13.6.
Let us understand these quantities.
The amplitutde A of SHM is the magnitude
of maximum displacement of the particle.
[Note, A can be taken to be positive without
any loss of generality]. As the cosine function
of time varies from +1 to –1, the displacement
varies between the extremes A and – A. Two
simple harmonic motions may have same ?
and f but different amplitudes A and B, as
shown in Fig. 13.7 (a).
While the amplitude A is fixed for a given
SHM, the state of motion (position and velocity)
of the particle at any time t is determined by the
Fig. 13.4 The location of the particle in SHM at the
discrete values t = 0, T/4, T/2, 3T/4, T,
5T/4. The time after which motion repeats
itself is T. T will remain fixed, no matter
what location you choose as the initial (t =
0) location. The speed is maximum for zero
displacement (at x = 0) and zero at the
extremes of motion.
Fig. 13.5 Displacement as a continuous function of
time for simple harmonic motion.
Fig. 13.7 (b) A plot obtained from Eq. (13.4). The
curves 3 and 4 are for f = 0 and -p/4
respectively. The amplitude A is same for
both the plots.
Fig. 13.7 (a) A plot of displacement as a function of
time as obtained from Eq. (14.4) with
f = 0. The curves 1 and 2 are for two
different amplitudes A and B.
x (t) : displacement x as a function of time t
A : amplitude
? : angular frequency
?t + f : phase (time-dependent)
f : phase constant
Fig. 13.6 The meaning of standard symbols
in Eq. (13.4)
argument (?t + f) in the cosine function. This
time-dependent quantity, (?t + f) is called the
phase of the motion. The value of plase at t = 0
is f and is called the phase constant (or phase
angle). If the amplitude is known, f can be
determined from the displacement at t = 0. Two
simple harmonic motions may have the same A
and ? but different phase angle f, as shown in
Fig. 13.7 (b).
Finally, the quantity ? can be seen to be
related to the period of motion T. Taking, for
simplicity, f = 0 in Eq. (13.4), we have
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