Oscillations

13.1 Introduction

In our daily lives, we encounter various types of motion. This chapter focuses on oscillatory
motion, where an object moves to and fro about a mean position. This motion is
fundamental to understanding many physical phenomena.

Examples of oscillatory motion include:

 A boat tossing on a river

 A piston in a steam engine

 The pendulum of a wall clock

13.2 Periodic and Oscillatory Motions 🔄

Periodic Motion: A motion that repeats itself at regular intervals of time.

Oscillatory Motion: A periodic motion where a body moves to and fro about a mean
position.

 Every oscillatory motion is periodic, but not every periodic motion is oscillatory.

 Example: Circular motion is periodic but not oscillatory.

There is no significant difference between oscillations and vibrations, except that

oscillations refer to lower frequencies (e.g., a tree branch), while vibrations refer to higher
frequencies (e.g., a musical instrument string).

Simple Harmonic Motion (SHM) is the simplest form of oscillatory motion, where the force
on the oscillating body is directly proportional to its displacement from the mean position.

In practice, oscillating bodies come to rest due to damping from friction and other
dissipative causes, unless forced to remain oscillating by an external periodic agency.

13.2.1 Period and Frequency

Period (T): The smallest interval of time after which the motion is repeated. Its SI unit is the
second (s).

Frequency (ν): The number of repetitions that occur per unit time. It is the reciprocal of the
period.

ν=1Tν=T1

The unit of frequency is hertz (Hz), where 1 Hz = 1 oscillation per second = 1 s⁻¹.

Example 13.1: A human heart beats 75 times in a minute. Calculate its frequency and
period.
  Frequency = 75 beats / 60 s = 1.25 Hz

 Period = 1 / 1.25 Hz = 0.8 s

13.2.2 Displacement 📏

Displacement: Refers to the change with time of any physical property under consideration.

The displacement is measured from an object's equilibrium position. The displacement

variable may take both positive and negative values.

The displacement can be represented by a mathematical function of time. For periodic

motion, this function is periodic in time.

One of the simplest periodic functions is:

f(t)=Acos⁡ωtf(t)=Acosωt

The period T is given by:

ωT=2πωT=2π

Thus, the function f(t) is periodic with period T, f(t)=f(t+T)f(t)=f(t+T).

A linear combination of sine and cosine functions like,

f(t)=Asin⁡ωt+Bcos⁡ωtf(t)=Asinωt+Bcosωt

is also a periodic function with the same period T. This can be written as:

f(t)=Dsin⁡(ωt+ϕ)f(t)=Dsin(ωt+ϕ),

where D=A2+B2D=A2+B2 and ϕ=tan⁡−1(BA)ϕ=tan−1(AB).

Example 13.2: Determine whether the following functions of time represent periodic or
non-periodic motion, and find the period for each periodic motion:

(i) sin⁡ωt+cos⁡ωtsinωt+cosωt (ii) sin⁡ωt+cos⁡2ωt+sin⁡4ωtsinωt+cos2ωt+sin4ωt (iii) e−ωte−ωt (iv)

log⁡(ωt)log(ωt)

 (i) Periodic function; can be written as 2sin⁡(ωt+π4)2sin(ωt+4π). Periodic

time T=2πωT=ω2π

 (ii) Periodic function. Period is T0=2πωT0=ω2π.

 (iii) Non-periodic; it decreases monotonically with increasing time and never repeats
its value.

 (iv) Non-periodic; it increases monotonically with time and never repeats its value. It
also diverges to −∞−∞ as t→0t→0.

13.3 Simple Harmonic Motion ⚖️

Simple Harmonic Motion (SHM): Oscillatory motion in which the displacement x of the
particle from the origin varies with time as:

x(t)=Acos⁡(ωt+ϕ)x(t)=Acos(ωt+ϕ)

where A, ω, and φ are constants. SHM is a sinusoidal function of time.

Key parameters of SHM:

Parameter Description

Amplitude (A) Magnitude of maximum displacement of the particle

Angular Frequency (ω) Determines the period of the motion

Phase (ωt + φ) Time-dependent quantity that determines the state of motion (position a

Phase Constant (φ) Value of the phase at t = 0

Since the motion has a period T, x(t) is equal to x(t + T):

ω=2πTω=T2π

13.4 Simple Harmonic Motion and Uniform Circular Motion ⭕

The projection of uniform circular motion on a diameter of the circle follows simple
harmonic motion.

If a particle P is moving uniformly on a circle of radius A with angular speed ω, the position
of P on the x-axis is given by:

x(t)=Acos⁡(ωt+ϕ)x(t)=Acos(ωt+ϕ)

This shows that if P moves uniformly on a circle, its projection P' on a diameter of the circle
executes SHM.

 Reference Particle: Particle P moving on the circle.

 Reference Circle: The circle on which the particle moves.

Example 13.4: Given two circular motions, determine the simple harmonic motions of the x-
projection of the radius vector of the rotating particle P in each case.

(a) x(t)=Acos⁡(2πTt+π4)x(t)=Acos(T2πt+4π) for T=4sT=4s (b) x(t)=Bsin⁡(2πTt)x(t)=Bsin(T2πt) for

T=30sT=30s

13.5 Velocity and Acceleration in Simple Harmonic Motion 🏃

The speed of a particle v in uniform circular motion is its angular speed times the radius of
the circle A:

v=Aωv=Aω

The velocity of the projection particle P' at time t is:

v(t)=−Aωsin⁡(ωt+ϕ)v(t)=−Aωsin(ωt+ϕ)

The instantaneous acceleration of a particle undergoing SHM is:

a(t)=−ω2Acos⁡(ωt+ϕ)=−ω2x(t)a(t)=−ω2Acos(ωt+ϕ)=−ω2x(t)

Example 13.5: A body oscillates with SHM according to the

equation x=5cos⁡[2πt+π4]x=5cos[2πt+4π]. At t = 1.5 s, calculate:

(a) Displacement: x=−3.535mx=−3.535m (b) Speed: v=22m/sv=22m/s (c)

Acceleration: a=140m/s2a=140m/s2

13.6 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(t)=ma=−mω2x(t)F(t)=ma=−mω2x(t)

i.e.,

F(t)=−kx(t)F(t)=−kx(t)

where

k=mω2k=mω2

or

ω=kmω=mk

Force is always directed towards the mean position. Simple harmonic motion can be defined
by the displacement equation or by its force law.

Example 13.6: Two identical springs of spring constant k are attached to a block of mass m.
Show that when the mass is displaced, it executes SHM and find the period of oscillations.

The net force acting on the mass is F=−2kxF=−2kx, and the time period of oscillations is:

T=2πm2kT=2π2km

13.7 Energy in Simple Harmonic Motion ⚡

Both kinetic and potential energies of a particle in SHM vary between zero and their
maximum values.

Kinetic energy (K) of a particle executing SHM:

K=12mv2=12mω2A2sin⁡2(ωt+ϕ)=12kA2sin⁡2(ωt+ϕ)K=21mv2=21mω2A2sin2(ωt+ϕ)=21
kA2sin2(ωt+ϕ)

Potential energy (U) of a particle executing simple harmonic motion:

U(x)=12kx2=12kA2cos⁡2(ωt+ϕ)U(x)=21kx2=21kA2cos2(ωt+ϕ)

The total energy (E) of the system is:

E=U+K=12kA2E=U+K=21kA2

Energy of Simple Harmonic Oscillators 🧮

The total mechanical energy (E) of a harmonic oscillator remains constant over time under
a conservative force. It's the sum of kinetic energy (K) and potential energy (U).

E=K+UE=K+U

E=12kA2E=21kA2

Where:

 kk is the spring constant

 AA is the amplitude

Example: Block and Spring System 🧱

Consider a 1kg block attached to a spring with a spring constant of 50 N/m. The block is
pulled 10 cm from equilibrium.

1. Angular frequency ($ω$):

ω=km=50 N/m1 kg=7.07 rad/sω=mk=1kg50N/m=7.07rad/s

2. Displacement ($x(t)$):

x(t)=0.1cos⁡(7.07t)x(t)=0.1cos(7.07t)

3. At x = 5 cm:

0.05=0.1cos⁡(7.07t)0.05=0.1cos(7.07t)

cos⁡(7.07t)=0.5cos(7.07t)=0.5

sin⁡(7.07t)=0.866sin(7.07t)=0.866

4. Velocity ($v$):

v=0.1⋅7.07⋅0.866=0.61 m/sv=0.1⋅7.07⋅0.866=0.61m/s

5. Kinetic Energy (K.E.):

K.E.=12mv2=12⋅1 kg⋅(0.61 m/s)2=0.19 JK.E.=21mv2=21⋅1kg⋅(0.61m/s)2=0.19J

 6. Potential Energy (P.E.):

P.E.=12kx2=12⋅50 N/m⋅(0.05 m)2=0.0625 JP.E.=21kx2=21⋅50N/m⋅(0.05m)2=0.0625J

7. Total Energy:

E=K.E.+P.E.=0.19 J+0.0625 J=0.25 JE=K.E.+P.E.=0.19J+0.0625J=0.25J At maximum

displacement, K.E. is zero, and total energy equals P.E.:

E=12⋅50 N/m⋅(0.1 m)2=0.25 JE=21⋅50N/m⋅(0.1m)2=0.25J

The total energy remains consistent, illustrating the conservation of energy.

Simple Pendulum

A simple pendulum consists of a mass (bob) suspended from a fixed point by a string or rod
of negligible weight.

Restoring Torque

The restoring torque ($τ$) is given by:

τ=−L(mgsin⁡θ)τ=−L(mgsinθ)

Where:

 LL is the length of the pendulum

 mm is the mass of the bob

 gg is the acceleration due to gravity

 θθ is the angular displacement

Equation of Motion

Using Newton's law for rotational motion ($τ = Iα$), where I is the moment of inertia and α
is the angular acceleration:

Iα=−mgLsin⁡θIα=−mgLsinθ

For a simple pendulum, $I = mL^2$, thus:

mL2α=−mgLsin⁡θmL2α=−mgLsinθ

α=−gLsin⁡θα=−Lgsinθ

Small Angle Approximation

For small angles, $\sin \theta \approx \theta$, so:

α=−gLθα=−Lgθ

This equation is mathematically identical to that of simple harmonic motion.

Time Period

The time period ($T$) of a simple pendulum is:

T=2πLgT=2πgL

Example: Calculating Pendulum Length 📏

For a pendulum that "ticks seconds" (T = 2 s):

L=gT24π2L=4π2gT2

L=9.8 m/s2⋅(2 s)24π2=1 mL=4π29.8m/s2⋅(2s)2=1m

Table of Sine Values at small angles

Angle (degrees) Angle (radians)

0 0

5 0.087

10 0.174

15 0.259

20 0.342