GROUP OF STARS

Group members-
1. Nikita Pandey
2. Arya Tripathi
3. Shivangi Modanwal
4. Jyoti Tiwari
5. Mahima Sharma
 OSCILLATION
Our hearts beat, Our lungs oscillate, we
shiver when we are cold, we sometimes
snore, we can hear and speak because our
eardrums and larynges vibrate.We can not
even say “Vibration” properly without the tip
of tongue oscillating.
During an Earthquake ,buildings may be set oscillating so strongly
That they are shaken apart.When an airplane is in flight, wings may
oscillate due to turbulence of the air ,resulting in metal fatigue and
even failure.when a tall building sways slowly, we may not even notice
it.But it becomes annoying if it repeats more than 10 times/second.
Some mechanisms are used to decrease sway of tall buildings.
e.g., a large ball(5.4 x 10^5 kg) hangs on the 92^nd floor of one of the
world’s tallest buildings. How the sway of building is countered?
Aircrafts designers ensures that none of the natural angular frequencies
at which a wing can oscillate matches the angular frequency of the
engine in flight .
The study and control of oscillations are two of the main goals of both Physics and
engineering.
DIFFERENT TYPES OF MOTION

Motion of Bus in a straight

path(rectilinear motion) and a body
released from the window of moving
bus (projectile motion) such motions
do not repeat with time and are called
non-repetitive motion.
If an object repeats its motion after a
definite interval of time and are called
repetitive motion or periodic motion.
PERIODIC MOTION
 Periodic Motion of a body is that motion which is
repeated identically after a fixed interval of time.
 The fixed interval of time after which the motion is
repeated is called period of motion.
 OR
 Any motion in which all the parameters of motion
 are repeated after a definite time interval is called
periodic motion.
 For instance, The motion of Earth around the Sun is
periodic.
 The needles of clock also performs
 periodic motion.
• OSCILLATORY OR VIBRATORY MOTION IS THAT MOTION IN WHICH THE
BODY MOVES TO AND FRO OR BACK AND FORTH REPEATEDLY ABOUT A
FIXED POINT (CALLED MEAN POSITION OR EQUILLIBRIUM POSITION OR
NATURAL POSITION OR STABLE POSITION) IN A DEFINITE INTERVAL OF TIME
• IN SUCH A MOTION , THE BODY IS CONFINED WITHIN WELL DEFINED LIMITS
(CALLED EXTREME POSITION) ON EITHER SIDE OF THE MEAN POSITION.
• A PERIODIC AND BOUNDED MOTION OF A BODY ABOUT A FIXED POINT IS
CALLED AN OSCILLATORY MOTION.
• FOR EXAMPLE, A SIMPLE PENDULUM PERFORMING TO AND FRO MOTION IS
OSCILLATORY.
A body that undergoes oscillatory motion always does
about a stable equillibrium position. When it is moved
away from this position and released ,it experiences a
net force or torque to pull it back toward equillibrium
position But by the time it gets there ,the restoring
force/torque would have done some positive work on
it.Thus,it must gained some kinetic energy, so it
overshoots the equillibrium position.Now it stops
somewhere on the other side, and is again pulled back
towards equillibrium position.
 EQUATION OF OSCILLATION
 F=-kx^n
 Where n is any odd number that is 1,3,5,7,9_ _ _
 If n =1 that is F=-kx
 This is the simplest form of oscillation.
 And this type of oscillation is called the simple harmonic
motion.
 If n is even then the motion can not be oscillatory.
 Difference between periodic
motion and oscillatory motion:
PERIODIC MOTION CAN BE ALONG ANY PATH
BUT IN OSCILLATORY MOTION , THE PATH
SHOULD BE SAME.
IN 99% CASE, EVERY OSCILLATION IS
PERIODIC. BUT, EVERY PERIODIC
MOTION IS NOT OSCILLATORY.

WHEN THE TO AND FRO MOTION OF A BODY ABOUT A FIXED

POSITION HAS SMALL FREQUENCY , WE WILL CALL IT
OSCILLATION. WHEN THE TO AND FRO MOTION OF A BODY HAS
HIGH FREQUENCY , WE CALL IT VIBRATORY MOTION.
 HARMONIC OSCILLATION

Harmonic oscillation is that oscillation which can be expressed in terms of single

harmonic function that is sine function or cosine function.

A harmonic oscillation of constant amplitude and of single frequency is called

simple harmonic oscillation.
 NON-HARMONIC OSCILLATION

Non-Harmonic oscillation is that oscillation which can

not be expressed in terms of single harmonic function.
A Non-Harmonic oscillation is a combination of two or
more than two harmonic oscillations.
y=asinωt+bsin2ωt
IMPORTANT CHARACTERISTICS OF OSCILLATORY MOTION

When a particle in stable equilibrium is disturbed ,it has tendency

to return to equilibrium position and this tendency is exhibited as
oscillatory motion.
The force on the body acts towards the mean position ,that is force
is always opposite to the displacement vector of the particle with
respect to mean position.(this force is known as restoring force.)
 F=-r and Γ=-Ɵ
Energy is also conserved in the motion .If energy is not conserved
then the particle will not be able to repeat the parameters of the
motion.
SIMPLE HARMONIC MOTION (SHM)
• SINUSOIDAL VIBRATIONS:
• Sinusoidal Vibrations arise when net force experienced by an oscillating body
has magnitude proportional to the distance from the mean position or torque is
directly proportional to the angular displacements from mean position .
• There are two reasons for this one physical, and one mathematical, and both
basic to the whole subject .The physical reason is that purely sinusoidal
vibrations are common in many types of mechanical systems. Such motion is
almost always possible if the displacements are small enough.
• If ,for example ,we have a body attached to a spring, the force exerted on it at a
displacement x from equillibrium is actually
• F(x)=-k1x+k2x2+k3x3+--------
• Where k1,k2,k3,etc. are a set of constants, and we can always find a range of
values of x within which the sum of the terms in X2 ,X3,--- is negligible
,compared to the term –k1x.
 SHM
• Simple harmonic motion is a special type of
periodic motion, in which a particle moves to and
fro or back and forth repeatedly about mean(i.e.,
stable equilibrium or natural) position under a
restoring force, which always directed towards
mean position and whose magnitude at any
instant is directly proportional to the
displacement of the particle from the mean
position at that instant.
• Consider a particle oscillating back and forth about
the origin of x-axis between the limits +A and –
A.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)=Acos(ωt+ɸ)
• Where A, ω and ɸ are constants.
• Thus, simple harmonic motion (SHM) is not any
periodic motion but one in which displacement is a
sinusoidal function of time.
The restoring force acting on the particle at that
instant is:
F=-kx
Thus, a system will be performing SHM if its
motion obey the above relation.
A system performing oscillations, in which the force
F is proportional to displacement x, rather than to
some other power of x, is called
a linear harmonic oscillator. The system in which
the restoring force is non-linear function of x, is
termed as non-linear harmonic oscillator or
anharmonic oscillator.
 CHARACTERISTICS OF SHM
• DISPLACEMENT- In general,the name displacement
given to a change in physical quantity under consideration with time in a
periodic motion.
• Thus,displacement represents change in physical quantity with time
such as position,
angle, pressure, electric and magnetic fields, etc.
Examples: (i)In a loaded spring, when a body is oscillating under the
action of a spring displacement variable is its deviation from the mean
position of the oscillation, with time.
(ii) During the propagation of sound wave in air, the displacement
variables is the local change in pressure with time.
(iii)During the propagation of electromagnetic waves ,the displacement
variables are electric and magnetic fields which vary periodically, with
time.
The displacement is always away from the mean position.
 AMPLITUDE

• The maximum displacement

on either side of mean
position is called amplitude of
SHM.
• In SHM, the maximum value
of sinɵ or cosɵ=1
• Therefore, from equation
given earlier, the maximum
value of (y or x) will be A.
• If S is the span of SHM, then
• Amplitude A =S/2. Here , a is the amplitude.
 VELOCITY

• The velocity in SHM at an • The velocity in SHM is not

instant is the projection of uniform throughout the
velocity vector of particle of motion.
reference on y-axis or x-axis • The maximum value of
at that instant. velocity is called velocity
• y= Aωcos(ωt+ɸ) amplitude in SHM.
• At mean position, y=0 • The direction of velocity is
• V = Aω (maximum) either towards the mean
• At extreme positions, y=A position or away from the
mean position.
• V = 0 (minimum)
GRAPHS IN SHM
 ACCELERATION

• In SHM, the acceleration is • a=-ω^2y

proportional to the • At mean position, y=0;
displacement but opposite in • a=0 (minimum)
sign, and the two quantities
are related by the square of • At extreme position, y=A
the angular frequency. • a=-ω^2A (maximum)
• The acceleration in SHM is • The maximum value of
the projection of the acceleration is called the
acceleration vector of the acceleration amplitude in
particle of reference on y-axis SHM.
or x-axis at that instant. • This is the hallmark of SHM.
 TIME PERIOD

• It is defined as the time taken by the

particle executing SHM to complete
one vibration.
• T =2п(displacement/acceleration)^1/2
• T=1/υ (relation between time
period and frequency)
SHM AND UNIFORM CIRCULAR MOTION

Uniform circular motion is not

simple harmonic motion, only its
projection on the diameter of the
circle of reference is simple
harmonic motion.
ADD A FOOTER 25
 THE FORCE LAW IN SHM IS THE HOOKE'S LAW.
• F OR SHM IN SPRING MASS SYSTEM K IS THE SPRING
CONSTANT BUT IN SHM FOR OTHER SYSTEM K IS THE FORCE
CONSTANT OR THE FORCE REQUIRED TO GIVE UNIT
DISPLACEMENT TO THE BODY.
• WE CAN IN FACT TAKE F=-KX AS AN ALTERNATIVE DEFINITION
OF SHM. IT SAYS:
• SIMPLE HARMONIC MOTION IS THE MOTION EXECUTED BY A
PARTICLE SUBJECT TO A FORCE THAT IS PROPORTIONAL TO THE
DISPLACEMENT OF THE PARTICLE BUT OPPOSITE IN SIGN.
FORCE LAW FOR SHM
F = ma = -(mω^2)x

SHM IS NOT A WAVE

The simple harmonic motion of a body
takes place under the condition of stable
equillibrium. Whenever a body in stable
equilibrium is displaced a little from its
equilibrium position(i.e., mean position),a
restoring force may be due to gravity , or
elasticity, or it may be electrical in nature.
This restoring force is proportional to the
displacement, provided the displacement
is small.
In linear SHM, The spring factor stands for force
per unit displacement and inertia factor for mass
of the body executing SHM.
In angular SHM, the spring factor stands for
torque constant,i.e., the moment of the couple
to produce unit angular displacement or the
restoring torque per unit angular displacement
and inertia factor stands for moment of inertia of
the body executing SHM.
 TOTAL ENERGY IN SHM
POTENTIAL ENERGY-THIS IS ON ACCOUNT OF THE DISPLACEMENT
OF PARTICLE FROM ITS MEAN POSITION .
U=1/2KX^2
KINETIC ENERGY-THIS IS ON ACCOUNT OF THE VELOCITY OF THE
PARTICLE.
K=K(A^2-X^2)
TOTAL ENERGY IS THE SUM OF POTENTIAL ENERGY AND KINETIC
ENERGY.
E=1/2KA^2
SIMPLE PENDULUM

An ideal simple pendulum consists of a heavy

point mass body suspended by a weightless
inextensible and flexible string from a rigid support
about which it is free to oscillate.
In equilibrium position, the centre of
gravity lies vertically below the point
of suspension. The distance between
point of suspension and the point of
oscillation is called the effective
length of the simple pendulum.
Expression for time period:
T=2п(l/g)^1/2
SHM of simple pendulum is an example of
angular simple harmonic motion
 DAMPED AND UNDAMPED OSCILLATION

UNDAMPED OSCILLATION-When a simple harmonic system oscillates with a c

amplitude which does not change with time, its oscillations are called undam
simple harmonic oscillations.

DAMPED OSCILLATION-When a simple harmonic system oscillates with a vari

amplitude which changes with time, its oscillations are called damped simple ha
oscillations.
DAMPED OSCILLATION UNDAMPED OSCILLATION
Sometimes dreams
. are wiser than waking.
THANK YOU