14.

Oscillations
Periodic Motion
A motion that repeats itself at regular intervals of time is called periodic motion. Eg.
bouncing ball (without loss of energy) represented in x-t graph.

Periodic motion is classified into oscillatory motion and non-oscillatory.

1) Oscillatory/oscillation: Motion which has an equilibrium/mean position in its path
about which it undergoes to and fro motion. These rae also known as vibrations
(oscillations have small frequencies whereas vibrations have high frequencies). Simple
harmonic motion is a type of oscillatory motion.
2) Non-oscillatory: Motion that does not have a mean position and hence does not have a
back and forth motion but are periodic. Circular motion is of this type.
Oscillatory Motion – Characteristics
Since oscillation is a repeated motion, it repeats after a certain time interval. Hence, two
characteristics arise out of this repeated motion.
 Time period (T) - interval of time after which the motion repeats (S.I unit – s)
 Frequency (𝝂) - no. of repetitions/oscillations per unit time (S.I unit – Hz)
 Relation between 𝝂 and T: 𝝂 = 1/T

Simple Harmonic Motion

Motion is said to be simple harmonic if its displacement (from the mean position) can be
represented sinusoidally i.e. displacement is a function of time. It is of the form
x(t) = A cos (ωt + )
Interpretation –

 Displacement [x(t)] – position of the object with respect to mean position which varies
with time.
 Amplitude (A) – maximum displacement of the object from the mean position
 Cosine function – signifies the motion of the object between a maximum and minimum
value
 Angular frequency (ω) – signifies the oscillating nature of the cosine function (between
0 to 2π ). It is related to time period (T) as
ω = 2π/T. S.I unit is rad/s
 Initial Phase ( ) – indicates the initiation of the oscillation i.e. that the stage of
oscillation when t= 0

Simple Harmonic Motion from Circular Motion

Starting from the horizontal axis, as the OP rotates with
some angular frequency ω, the projection of P on x-axis
(Q) traces a path along the x-axis. This gives the
displacement x(t). Maximum displacement (amplitude) is
when OP of radius equal to amplitude (A) lies along the
horizontal axis - +A on the positive x-axis and –A on the
negative axis. If t = 0 at an angle , then is the initial
phase
x(t) = A cos (ωt + )
Note –
 Anti-clockwise is taken as the standard rotation which
gives a positive . Clockwise rotation gives a negative .
 If the projection is taken on y-axis, then equation takes the
form y(t) = A sin (ωt + )

Velocity & Acceleration of SHM

Velocity: If the position at some time t is given as x(t) = A cos (ωt + ), the velocity v(t)

v(t) =
 = [A cos (ωt + )]

=A (cos (ωt + )) (∵ A is a constant)

= A [- sin (ωt + ) . ω] (∵ = - sin t)(differential of cosine term x differential of

argument – because both contain ‘t’)

v(t) = - ωA sin (ωt + )

This gives the expression for instantaneous velocity at some time t
Acceleration: Instantaneous acceleration at some time t is the time variation of
instantaneous velocity & given as: a(t) =

= (v(t))

= (- ωA sin (ωt + ) )

= - ωA (sin (ωt + ) ) (∵ ωA is a constant)

= - ωA [cos (ωt + ) . ω] (∵ = cos t)

(differential of cosine term x differential of argument – because both contain ‘t’)

a(t) = -ω2A cos (ωt + )
or a(t) = - ω2 x(t)
Note –

 Note that instantaneous velocity v(t) is a function of sine instead of cosine. This
implies that its initiation is from zero instead of maximum displacement (amplitude)
 Instantaneous acceleration a(t) is same as x(t) except that it has an associated negative
sign.
 The coefficient of the sine/cosine function is taken as the amplitude in a simple
harmonic equation. Thus, the amplitude of v(t) is ωA and amplitude of a(t) is ω 2A.
 v(t) has phase difference of π/2 with x(t) whereas a(t) has a phase difference of π with
x(t) i.e. a(t) is said to be out of phase with x(t)
 Graphs of instantaneous velocity and acceleration vs. time (in comparison to x vs. t
graph) are given below:

Force Law for Simple Harmonic Motion

Using Newton’s second law of motion, the force acting on a particle of mass m in SHM
is F = ma. However, for a particle executing simple harmonic motion, acceleration is
given as
 a (t) = -ω2 x(t)
⇒ F = - mω2 x(t)
or F = - k x
where k is a force constant (= mω2) and indicates the stiffness of an oscillation
It follows from here that angular frequency, ω, can be expressed as

Note –

 Expression of force for SHM has a negative sign. This implies that this force is always
acting opposite to the displacement, effectively trying to bring the particle to the mean
position. Such a type of force which is directed towards the mean position is also
known as restoring force.
 Force is linearly proportional to x(t). A particle oscillating under such a force is called
a linear oscillator.
 In the real world, the force may contain small additional terms proportional to x 2 , x3 ,
etc. These then are called non-linear oscillators.

Energy in Simple Harmonic Motion

A particle executing simple harmonic motion has a kinetic energy (K) associated with its
velocity. Its expression is given by
K = ½ mv2
∵ velocity in SHM is v = - ωA sin (ωt + )

⇒ K = ½ m [- ωA sin (ωt + )]2

= ½ m [ω2A2 sin2 (ωt + )]

 = ½ (mω2) A2 sin2 (ωt + )
K = ½ kA2 sin2 (ωt + )
This expression shows that kinetic energy varies sinusoidally as a function of square of
sine.
The concept of potential energy is possible only for conservative forces. Since force of
SHM is a restoring force, it is a conservative force and has an associated potential energy
given as: U = ½ kx2
∵ displacement in SHM is x = cos (ωt + )

⇒ U = ½ k [A cos (ωt + )]2

= ½ k [A2 cos2 (ωt + )]

U = ½ k A2 cos2 (ωt + )
This expression shows that potential energy varies sinusoidally as a function of square of
cosine.
∴ Total energy in SHM, E = K +U
E = ½ kA2 sin2 (ωt + ) + ½ kA2 cos2 (ωt + )
E = ½ kA2 [sin2 (ωt + ) + cos2 (ωt + ) ]
E = ½ kA2 (1) , since sin2 + cos2 = 1
E = ½ kA2
i.e. total energy is a constant throughout simple harmonic motion which is expected as
per law of energy conservation
Graph representation:

 Energy vs. time

 Energy vs. position

Note - Both kinetic energy and potential energy peak twice during each period of SHM.
For x = 0, the energy is kinetic; at the extremes x = ±A, potential energy dominates. In
the course of motion between these limits, kinetic energy increases at the expense of
potential energy or vice-versa.
Total energy E remains constant i.e. no dissipation of energy in the form of heat, air
resistance, friction etc.
Systems executing Simple Harmonic Motion

1) The Simple Pendulum

Consider simple pendulum — a small bob of

mass m tied to an inextensible massless string of
length L. The other end of the string is fixed to a
rigid support. The bob oscillates in a plane about
the vertical line through the support. Let θ be the
angle made by the string with the vertical, as
given in the figure. When the bob is at the mean position,
θ = 0. For rotational motion about the rigid support, maximum torque is observed when
distance to the centre is perpendicular to the force causing the torque (τ = r x F = rF for
angle between r and F as 90o). Since the radial force (mg cos θ) gives zero torque, torque
τ about the support is entirely provided by the tangental component of force (mg sin θ).
As r = L, torque is given as
= - L (mg sin θ) -------- 1
This is the restoring torque that tends to reduce angular displacement and directed
towards the mean position, hence the negative sign. By Newton’s law of rotational
motion, torque is directly proportional to the angular acceleration,
=I ----------- 2
where I is the moment of inertia of the system about the support.

∴ from equations 1 and 2,

- L (mg sin θ) = I

For small θ, sin θ θ

⇒
However, this equation for angular acceleration ( ) is mathematically analogous to the
expression for acceleration in SHM – a = -ω2 x

⇒ angular frequency, ω2 =

or

This shows that for small displacement θ, a simple pendulum executes simple harmonic
motion with angular frequency ω and a time period T given by

T = 2π
ω

Now since the string of the simple pendulum is massless, the moment of inertia I = mL2
and thus, time period of simple pendulum is dependent on the length of the string as

2) Mass on a spring
A block of mass m is placed on a frictionless horizontal surface. If the block is pulled on
one side and is released, it then executes a to and fro motion about the mean position. Let
x = 0, indicate the position of the centre of the block
when the spring is in equilibrium. The positions
marked as –A and +A indicate the maximum
displacements to the left and the right of the mean
position. Springs have special properties; when
deformed, are subject to a restoring force, the
magnitude of which is proportional to the deformation or the displacement and acts in
opposite direction. This is known as Hooke’s law. It holds good for displacements small
in comparison to the length of the spring. At any time t, if the displacement of the block
from its mean position is x, the restoring force F acting on the block is,

F (x) = –k x

The value of constant of proportionality, k, called the spring constant, is governed by the
elastic properties of the spring. This force is of the same form as the force law for SHM
and therefore it can be concluded that the system of mass on a spring executes a simple
harmonic motion.
Thus, it follows from above conclusion that the mass undergoes SHM with a frequency

and the oscillator has a time period of T given by the equation.

Note:- A stiff spring has large k and a soft spring has small k