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Chapter (3)

Oscillations

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Oscillations

Mechanical

oscillation

Nonmechanical

oscillation Simple Harmonic

Oscillation

Damped Harmonic Oscillation

Forced Harmonic Oscillation

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Periodical Motion Amplitude A

Period T

Frequency F=1/T

Angular frequency ω = 2πF

Phase (ωt+φ)

Phase constant φ

X(t)=A sin ωt at t=0, x=0

X(t)=A sin (ωt+φ) at t=0, x≠0

A

T

φ

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X(t)=A sin (ωt+φ)

A

T

φ

Simple Harmonic Oscillator Simple Harmonic Oscillator

f and T is independent of A

A is constant

Simple Harmonic Oscillator has the following characteristics:

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X(t)=A sin(ωt+φ)

v(t)= ωA cos(ωt+φ)

a(t)= -ω2A sin(ωt+φ)

Displacement, Velocity, acceleration

a(t)= -ω2 X(t)

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d2x/dt2 + ω2x(t)= 0

Simple Harmonic Motion(SHM)

Simple Harmonic Motion (SHM)

a(t)= -ω2 X(t) orFor SHM to occur, three conditions must be satisfied

1) there must be a position of equilibrium.2) there must be no dissipation of energy.3) the acceleration is proportional to X and opposite direction.

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F=-Kx, F= ma -kx= ma a=-(k/m) x

m

kf 2ω2 =(k/m) or

Hook’ s law and Simple HarmonicMotion

Hook’s law and Simple Harmonic Motion

a= -ω2 X

k

mT

m

kf

2

2

1

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Energy conservation in SHM Energy conservation in SHM In the absence of friction, the energy of the block-spring system is constant.

Potential energy

kinetic energy

Since ω2 =(k/m) and sin2θ+cos2θ=1

total energy E=K+U=

)(sin2

1

2

1 222 tkAkxU

)(cos2

1

2

1 2222 tAmmvK

222

2

1

2

1

2

1kAkxmv

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The total energy of any SHM is constant and proportional to A2

0 x

U KE=K+U

-A A

energy

tU

K

E=K+U

energy

E/2

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Example of linear and angular SHM

Simple Pendulum

F=-mg sinθ,

for small θ, sinθ θ x/L

F=-mgx/L = -(mg/L)x =-kxm

L

x

mg

mg sinθ

θ

mg cosθ

T

g

LT

k

mT

2

mg

mL2T 2

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Torsional Pendulum

F ζ (torque)

x θ (angular disp.)

m I (moment of inertia)

k k (torsional const.)

Thus, Hooke’s law takes the form ζ=-k θ

MF

k21

2T 2

2MR

k

IT