1 chapter (3) oscillations. 2 mechanical oscillation nonmechanical oscillation simple harmonic...
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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