Free Vibrations of Physical Systems III of Physical Systems III
Damping in Oscillators
if we charted the amplitude as a function of time we would get get
the resistive force of a fluid to a moving 2 object is given by R (v) = b1v + b2 v
magnitude of the
Forced Vibrations and Vib Resonance III Resonance III
Power Absorption in a Driven Oscillator
what rate must energy be fed into an oscillator to rate must energy be fed into an oscillator to maintain oscillations at a fixed amplitude recall, the instanta
Forced Vibrations and Vib Resonance II Resonance II
Harmonic Forcing with Damping
F0cost
(
F0 A = cos m F0 + A = sin m
2 0 2
)
A( ) =
F0 2 2 0
m + ( )
2 1 2
(
)
2
tan = 2 0 2
Harmonic Forcing with Damping
F0cost
m = 2
2 0
(
)
1 2
A ( )
A( ) =
F0 2 2 0
Forced Vibrations and Vib Resonance Resonance I
Harmonic Forcing
Why should we care about this topic should we care about this topic
Tacoma Narrows Bridge 1940 - 42 mph wind drives oscillations 42
Harmonic Forcing
k F0cost
0 =
k m
d 2x Newton II: m 2 = kx
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Physics 422 - Spring 2013 - Midterm Exam, March 6th
Answer all questions in the exam booklets provided.
There are 6 questions - please answer all of them.
Explain your reasoning clearly but concisely.
Clearly indicate which work is to be graded.
Each ques
Coupled Oscillations and Normal Modes Normal Modes I
Coupled Pendulums
so far we have restricted ourselves to systems with one type of vibration characterized by a single natural frequency 0 real physical systems may have multiple modes of vibration with
Coupled Oscillations and Normal Modes II Normal Modes II
Coupled Pendulums
d 2 xA 2 m 2 + m0 x A + k ( x A xB ) = 0 dt d 2 xB 2 m 2 + m0 xB k ( x A xB ) = 0 dt
xB
we seek solutions of the form x A = C cos t xB = C cos t 2 m 2C cos t + m0 C cos t + k ( C
Free Vibrations of Physical Systems II of Physical Systems II
Pendula
O
Simple Pendulum 2D problem though expressible through the single parameter through the single parameter exchange of KE and PE during the motion consider small y < x
l-y l
x2 y 2l
con
3/24/11
Free Vibrations of Physical Systems I
3/24/11
Mass-Spring System
Vast majority of deformations of physical systems (stretching, compressing, bending, twisting) result in restoring forces displacement F Simple harmonic motion Prototype system
Inert
The Superposition of Waves III
Anharmonic Periodic Waves
Superposition of harmonic waves leads to periodic but anharmonic wave to periodic but anharmonic wave
Fourier Series
Fouriers Theorem: any function f(x) that has a spatial period can be synthesized
Th The Superposition of Waves II II
Standing Wave
In general: ( x, t ) = C1 f ( x vt ) + C2 g ( x + vt )
two waves traveling in opposite direction
Consider 2 waves, incident and reflected:
EI = E0 I sin(kx t + I )
ER = E0 R sin(kx + t + R )
sin + sin = (
The Superposition of Waves The Superposition of Waves I
Principle of Superposition
Wave equation:
1 2 2 2 2 + 2+ 2= 2 2 2 z x y v t
(r , t ) = Ci i (r , t )
i =1
n
If i are solutions of the wave equation, then their linear solutions of the wave equation
Electromagnetic Theory Electromagnetic Theory, Photons and Light III Photons, and Light III
The Poynting Vector: Polarized Harmonic Wave
E = E0 cos k r t
0
[ ] B = B cos[k r t ]
( )
Polarized EM wave:
S=
1
0
EB
Poynting vector: 1 S= E0 B0 cos 2 k r t 0
[
Electromagnetic Theory Electromagnetic Theory, Photons and Light II Photons, and Light II
Maxwells Equations
Gausss Gausss Faradays AmpreMaxwells
B dS = 0
S
E dS = q
S 0
1
In vacuum (free space) space)
d CE dl = dt
[ B dS ]
A
E CB dl = 0 A J + 0 t dS
Electromagnetic Theory, Electromagnetic Theory, Photons, and Light I
Classical EM Waves versus Photons
The energy of a single light photon is E=h energy of single light photon is
c The Plancks constant h = 6.62610-34 Js E1 = h = h 4 1019 J Visible light w
Chapter Chapter 2 Wave Motion III Motion III
3-D Waves: Plane Waves
(simplest 3-D waves) All the surfaces of constant phase of the disturbance form parallel planes that are generally perpendicular to the propagation direction An equation of a plane that i
Chapter Chapter 2 Wave Motion II Motion II
Harmonic Waves: Summary
Functional shape: Wave parameters:
- for wave moving right + for wave moving left
= A sin k ( x vt )
Alternative forms: forms:
x t = A sin 2
= A sin[2 (x t )] = A sin[kx t ]
x = A sin
Chapter Chapter 2 Wave Motion I Motion
One Dimensional Wave
Classical traveling wave: self-sustaining disturbance of a medium, which moves through space transporting energy and momentum. Example: sound waves Longitudinal waves: waves:
the medium is displa