CH- 1 Undamped Free Oscillations

on

OSCILLATIONS

Rupali Kharat

Dr.D.Y.Patil.Arts, Commerce and Science Womens College

Pimpri, Pune 411 018

My guide P.S Tambde

Oscillations

P. S. Tambade

Content 1. Equilibrium

2. Stable equilibrium

3. Unstable Equilibrium

4. Oscillatory Motion

5. Spring –Mass system

6. Simple harmonic Motion

7. Displacement and velocity

8. Periodic Time

9. Frequency

10.Displacement and Acceleration

11.Energy of SHM

12.Lissajous Figures

13.Angular SHM

14.Simple Pendulum

Oscillations

P. S. Tambade

Equilibrium

• Types of equilibriums

1. Stable Equilibrium

2. Unstable equilibrium

3. Neutral equilibrium

The body is said to be in equilibrium at a point

when net force acting on the body at that point is

zero.

C

Oscillations

P. S. Tambade

Stable equilibrium

If a slight displacement of particle from its equilibrium position

results only in small bounded motion about the point of

equilibrium, then it is said to be in stable equilibrium

Equilibrium

position

C

Oscillations

P. S. Tambade

Potential energy curve for stable equilibrium

-a +a- x x0

V(x)

x

Slope =dV

dx

Tangent at A

A

Positive

F = dV

dx

Force

F

Force is negative i.e. directed towards equilibrium

position

B

Tangent at B

Slope =dV

dx

Negative

Force is positive i.e. directed towards equilibrium

position

F

SimulationC

Oscillations

P. S. Tambade

Unstable equilibrium

If a slight displacement of the particle from its equilibrium position

results unbounded motion away from the equilibrium position,

then it is said to be in unstable equilibrium

Equilibrium

position

C

Oscillations

P. S. Tambade

Potential energy curve for unstable equilibrium

-a +a- x x0

V(x)

x

Slope =dV

dx

Tangent at A

A

Negative

F = dV

dx

Force

F

Force is positive i.e. directed away from equilibrium

position

B

Tangent at B

Slope =dV

dx

Positive

Force is negative i.e. directed away from equilibrium

position

F

Click for simulationC

Oscillations

P. S. Tambade

Click for simulation 1

Click for simulation 2

C

Oscillations

P. S. Tambade

Oscillatory Motion

Any motion that repeats itself after equal intervals of time is called

periodic motion.

If an object in periodic motion moves back and forth over the

same path, the motion is called oscillatory or vibratory motion

C

Oscillations

P. S. Tambade

Spring-Mass system

m

m

x = 0

x

– x

Relaxed mode

Extended mode

Compressed mode

F

F

m

We know that for an ideal

spring, the force is related

to the displacement by

kxF C

Oscillations

P. S. Tambade

Simple Harmonic Motion

kxF

Linear simple harmonic motion : When the force acting on the particle is directly

proportional to the displacement and opposite in

direction, the motion is said to be linear simple harmonic

motion

Differential equation of motion is

md2x

dt2 + kx = 0

d2x

dt2 + ω2 x = 0m

k

mk

2where

Solution is

x = a sin (ωt + )

(ωt + ) is called phase and is called epoch of SHM

C

Oscillations

P. S. Tambade

We know that for an ideal

spring, the force is related to

the displacement bykxF

But we just showed

that harmonic motion

has

xmF 2

So, we directly find out

that the “angular

frequency of motion”

of a mass-spring

system is

m

k

mk

2

C

Oscillations

P. S. Tambade

• a and are determined uniquely by the position

and velocity of the particle at t = 0

• If at t = 0 the particle is at x = 0, then = 0

• If at t = 0 the particle is at x = a, then = π/2

• The phase of the motion is the quantity (ωt + )

• x (t) is periodic and its value is the same each

time ωt increases by 2π radians

x = a sin (ωt + )

The displacement of particle from equilibrium position is

C

Oscillations

P. S. Tambade

Simple harmonic motion (or SHM) is the

sinusoidal motion executed by a particle of

mass m subject to one-dimensional net

force that is proportional to the

displacement of the particle from

equilibrium but opposite in sign

Click for simulation1

Click for simulation2

C

Oscillations

P. S. Tambade

x = a sin (ωt + )

Equation of SHM is

The velocity is

v = dxdt

v = aω cos (ωt + )

or v = ω 2xa 2

The velocity is zero at extreme positions and maximum

at equilibrium position

C

Oscillations

P. S. Tambade

Graphs of Displacement and Velocity

x

t, time

T

Tωt

π 2π3π

2

5π

2

π

23π 7π

2

4π

π

2

π

2

v

For = π

2

x = a sin (ωt + ) v = aω cos (ωt + )

The phase difference between velocity and displacement is π

2

+a

-a

ω T

ω T

ωT = 2, The period of oscillation is T = 2/ ωT is called periodic time

C

Oscillations

P. S. Tambade

Periodic Time

2T

k

m 2T

The period of SHM is defined as the time taken by the

oscillator to perform one complete oscillation

After every time T, the particle will have the same

position, velocity and the direction

ttanconsiswhenT

ttanconsiswhenT

mk

1

km

T

m

C

Oscillations

P. S. Tambade

The frequency represents the number of

oscillations that the particle undergoes per

unit time interval

• The inverse of the period is called the

frequency

1ƒ

2T

•Units are cycles per second = hertz (Hz)

m

kf

2

1

Frequency

C

Oscillations

P. S. Tambade

• The frequency and the period depend only on the mass of

the particle and the force constant of the spring

• They do not depend on the parameters of motion like

amplitude of oscillation

• The frequency is larger for a stiffer spring (large values of k)

and decreases with increasing mass of the particle

k

m 2T

m

kf

2

1

C

Oscillations

P. S. Tambade

Displacement and acceleration

x

ωtπ 2π3π

2

5π

2

π

23π 7π

2

4π

π

Ax

For = π

2

Simulation

x = a sin (ωt + ) A = - aω2 sin (ωt + )

The phase difference between acceleration and displacement is π

π

C

Oscillations

P. S. Tambade

Energy

The potential energy is

V = k x21

2

The kinetic energy is

K = m v 21

2

or K = m ω 2 (a2 – x2)1

2

The total energy is

E = K + V

or E = m ω 2 a21

2

Thus, total energy of the oscillator is constant and proportional to

the square of amplitude of oscillations

C

Oscillations

P. S. Tambade

t x v K. E. P. E. E

0 + a 0 0 1

2 m ω2a2

1

2 m ω2a2

T/4 0 – a ω 1

2 m ω2a2 0

1

2 m ω2a2

T/2 – a 0 0 1

2 m ω2a2

1

2 m ω2a2

3T/4 0 + a ω 1

2 m ω2a2 0

1

2 m ω2a2

T +a 0 0 1

2 m ω2a2

1

2 m ω2a2

x-a 0 +a

Amax

Amax

Amax

Summary …….

C

Oscillations

P. S. Tambade

-a +ax

0

P. E.

K. E.

P. E. =K. E.

a/ 2- a/ 2

Energy

Graphical Representation of K. E. and P. E.

E = m ω 2 a21

2

The total mechanical energy is constant

The total mechanical energy is proportional to the square of the amplitude

Energy is continuously being transferred between potential energy stored in the spring and the kinetic energy of the blockC

Oscillations

P. S. Tambade

Variation of K.E. and P. E. With time

x

ωtπ 2π3π

2

5π

2

π

23π 7π

2

4π

For = π

2

x = a sin (ωt + )

ωt0

E

V = k x21

2K = m ω 2 (a2 – x2)

1

2

For one cycle of oscillation of particle there are two cycles for K. E.

and P.E.. Thus frequency of K. E. or P. E. is 2n

P. E.

K. E.

Click for simulation1

0

C

Oscillations

P. S. Tambade

Lissajous Figures

When a particle is subjected to two mutually

perpendicular simple harmonic motions, it traces a path

on a plane that depends upon the frequencies, amplitudes

and phases of the component SHMs. If the frequencies of

two component SHMs are not equal, the path of the

particle is no longer an ellipse but a curve called Lissajous

curve.

These curves were first demonstrated by Jules Antonie

Lissajous in 1857.

C

Oscillations

P. S. Tambade

The equations of motion for two mutually perpendicular

simple harmonic motions acting simultaneously on a particles

are given as

x = a sin (ω1t + 1)

y = b sin (ω2t + 2)

The path traced by the particle depends on ratio ω1 / ω2 ,

the amplitudes and the phase difference = 1 - 2

If ω1 / ω2 a rational number, so that the angular frequencies

are commensurable, then the curve is closed curve

Oscillations

P. S. Tambade

ω1 : ω2 1 :1

a = b, = π/2 a > b, = π/2

/4 3/4

O x

y

O x

y

O x

y

O x

y

x = a sin (ω1t + 1) y = b sin (ω2t + 2)

Oscillations

P. S. Tambade

ω1 : ω2 1 :2

a > b, = π/2

O x

y

O x

y

y

ω1 : ω2 2 :1

O x

y

O x

y

Oscillations

P. S. Tambade

2

2

2

1

2

1

2

1

2

1

k

k

k

k

m

k

4

1

2

1

2

1

2

1

k

k

Oscillations

P. S. Tambade

Variation of Lissajous Figure with phase difference

Click here to see Lissajous figures for different

frequency ratio

Click here to see Lissajous figures for different

frequency ratio and phase change

C

Oscillations

P. S. Tambade

Angular SHM

If path of particle of a body performing an oscillatory

motion is curved, the motion is known as angular

simple harmonic motion

Definition : Angular simple harmonic motion is defined

as the oscillatory motion of a body in which the body is

acted upon by a restoring torque (couple) which is

directly proportional to its angular displacement from

the equilibrium position and directed opposite to the

angular displacement

Oscillations

P. S. Tambade

2

2

dt

d I

2

2

dt

dI 0

2

2

Idt

d

02

2

2

dt

d)( tsin0

is the torsion constant of the support wire

The restoring torque is

Newton’s Second Law gives

Angular SHM ……

I – moment of inertia

Oscillations

P. S. Tambade

• The torque equation produces a motion equation

for simple harmonic motion

• The angular frequency is

• The period is

– No small-angle restriction is necessary

– Assumes the elastic limit of the wire is not exceeded

I

2I

T

C

Oscillations

P. S. Tambade

Simple Pendulum

•The equation of motion is

Oscillations

P. S. Tambade

• When angle is very small, we have sin

+ g

l =0

d 2

dt 2

T = 2 lg

The Period is

Oscillations

P. S. Tambade

When > 100 but < 200 ,

then period is

When > 200 , then period is

T = T1 + 1

40

2

2 sin

64

9+ 0

4

2 sin + ....

T= T1 + 02

16

But when angular arc is not

small ,then we have to

solve

Click for simulation

Oscillations

P. S. Tambade

Click for simulation

Click for simulation

For comparison between linear and non-linear equations of simple

pendulum, click following link

For Oscillating dipole, click following link

C