CHAPTER FOURTEEN

OSCILLATIONS

14.1 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.

14.1 Introduction

14.2 Periodic and oscillatory

motions

14.3 Simple harmonic motion

14.4 Simple harmonic motion

and uniform circular

motion

14.5 Velocity and acceleration

in simple harmonic motion

14.6 Force law for simple

harmonic motion

14.7 Energy in simple harmonic

motion

14.8 Some systems executing

simple harmonic motion

14.9 Damped simple harmonic

motion

14.10 Forced oscillations and

resonance

Summary

Points to ponder

Exercises

Additional Exercises

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14.2 PERIODIC AND OSCILLATORY MOTIONS

Fig. 14.1 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. 14.1 (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. 14.1 (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. 14.1 (c).

Note that both the curved parts in Fig. 14.1 (c)

are sections of a parabola given by the Newton’s

equation of motion (see section 3.6),

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. 14.1 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.

14.2.1 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 343

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

µ

s. 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

ν

. The

relation between

ν

and T is

ν

= 1/T (14.1)

The unit of ν 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 =1s

–1

(14.2)

Note, that the frequency,

ν

, is not necessarily

an integer.

u Example 14.1 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 t

14.2.2 Displacement

In section 4.2, 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.14.2(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.14.2(b)]. The

term displacement is not always to be referred

Fig. 14.2(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.14.2(b) An oscillating simple pendulum; its

motion can be described in terms of

angular displacement

θ

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

ω

t (14.3a)

If the argument of this function,

ω

t, is

increased by an integral multiple of 2

π

radians,

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