part two: oscillations, waves, & fluids
DESCRIPTION
Examples of oscillations & waves : Earthquake – Tsunami Electric guitar – Sound wave Watch – quartz crystal Radar speed-trap Radio telescope. Part Two: Oscillations, Waves, & Fluids. Examples of fluid mechanics : Flow speed vs river width Plane flight. - PowerPoint PPT PresentationTRANSCRIPT
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Part Two: Oscillations, Waves, & Fluids
High-speed photo: spreading circular waves on water.
Examples of oscillations &
waves:
Earthquake – Tsunami
Electric guitar – Sound wave
Watch – quartz crystal
Radar speed-trap
Radio telescope
Examples of fluid
mechanics:
Flow speed vs river width
Plane flight
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13. Oscillatory Motion
1. Describing Oscillatory Motion
2. Simple Harmonic Motion
3. Applications of Simple Harmonic Motion
4. Circular & Harmonic Motion
5. Energy in Simple Harmonic Motion
6. Damped Harmonic Motion
7. Driven Oscillations & Resonance
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Dancers from the Bandaloop Project perform on vertical surfaces,
executing graceful slow-motion jumps.
What determines the duration of these jumps?
pendulum motion: rope length & g
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Disturbing a system from equilibrium results in oscillatory motion.
Absent friction, oscillation continues forever.
Examples of oscillatory motion:
Microwave oven: Heats food by oscillating H2O molecules in it.
CO2 molecules in atmosphere absorb heat by vibrating global warming.
Watch keeps time thru oscillation ( pendulum, spring-wheel, quartz crystal, …)
Earth quake induces vibrations collapse of buildings & bridges .
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13.1. Describing Oscillatory Motion
Characteristics of oscillatory motion:
• Amplitude A = max displacement from
equilibrium.
• Period T = time for the motion to repeat itself.
• Frequency f = # of oscillations per unit time.
1fT
[ f ] = hertz (Hz) = 1 cycle / ssame period T same amplitude A
A, T, f do not specify an oscillation completely.
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Example 13.1. Oscillating Ruler
An oscillating ruler completes 28 cycles in 10 s & moves a total distance of 8.0 cm.
What are the amplitude, period, & frequency of this oscillatory motion?
Amplitude = 8.0 cm / 2 = 4.0 cm.
10
28
sT
cycles
1fT
0.36 /s cycle
28
10
cycles
s 2.8 Hz
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13.2. Simple Harmonic Motion
Simple Harmonic Motion (SHM): F k x
2
2
d xm k xd t
cos sinx t A t B t Ansatz:
sin cosd x
A t B td t
22 2
2cos sin
d xA t B t
d t 2 x
k
m
angular frequency
2T x t T x t 2
mT
k
1
2fT
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cos sinx t A t B t
sin cosd x
v t A t B td t
A, B determined by initial conditions
0 1
0 0
x
v
1A
0B cosx t t
( t ) 2
x 2A
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Amplitude & Phase
cos sinx t A t B t cosC t
cos cos sin sinC t t cos
sin
A C
B C
C = amplitude
= phase
Note: is independent of amplitude only for SHM.
Curve moves to the right for < 0.
2 2C A B
1tanB
A
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Velocity & Acceleration in SHM
cosx t A t
sind x
v t A tdt
2
22
cosd x
a t A tdt
2x t
|x| = max at v = 0
|v| = max at a = 0
cos2
A t
2 cosA t
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GOT IT? 13.1.
Two identical mass-springs are displaced different amounts from equilibrium &
then released at different times.
Of the amplitudes, frequencies, periods, & phases of the subsequent motions,
which are the same for both systems & which are different?
Same: frequencies, periods
Different:amplitudes ( different displacement )
phases ( different release time )
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Application: Swaying skyscraper
Tuned mass damper :
f damper = f building ,
damper building = .Taipei 101 TMD:
41 steel plates,
730 ton, d = 550 cm,
87th-92nd floor.
Also used in:
• Tall smokestacks
• Airport control towers.
• Power-plant cooling towers.
• Bridges.
• Ski lifts.
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Example 13.2. Tuned Mass Damper
The tuned mass damper in NY’s Citicorp Tower consists of a 373-Mg (vs 101’s 3500
Mg) concrete block that completes one cycle of oscillation in 6.80 s.
The oscillation amplitude in a high wind is 110 cm.
Determine the spring constant & the maximum speed & acceleration of the block.
2
3 2 3.1416373 10
6.80kg
s
2
T
53.18 10 /N m
2 3.1416
6.80 s
10.924 s
2
2k m
T
2m
Tk
maxv A 10.924 1.10s m 1.02 /m s
2maxa A 210.924 1.10s m 20.939 /m s
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13.3. Applications of Simple Harmonic Motion
• The Vertical Mass-Spring System
• The Torsional Oscillator
• The Pendulum
• The Physical Pendulum
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The Vertical Mass-Spring System
k
m
Spring stretched by x1 when loaded.
mass m oscillates about the new equil.
pos.
with freq
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The Torsional Oscillator
= torsional constant
I
I
2
2
dIdt
Used in timepieces
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The Pendulum
sinm g L g
2
2
dIdt
Small angles oscillation: sin
2
2
dI m g Ldt
m g L
I
Simple pendulum (point mass m):
2I m Lg
L
LT
g
Tτ 0
sin
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Example 13.3. Rescuing Tarzan
Tarzan stands on a branch as a leopard threatens.
Jane is on a nearby branch of the same height, holding a 25-m-long vine attached to a
point midway between her & Tarzan.
She grasps the vine & steps off with negligible velocity.
How soon can she reach Tarzan?
LT
g
2
1 25
2 9.8 /
mT
m s
Time needed:
5.0 s
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GOT IT? 13.2.
What happens to the period of a pendulum if
(a) its mass is doubled,
(b) it’s moved to a planet whose g is ¼ that of Earth,
(c) its length is quadrupled?
no change
doubles
doubles
LT
g
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The Physical Pendulum
Physical Pendulum = any object that’s free to swing
Small angular displacement SHM
m g L
I
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Example 13.4. Walking
When walking, the leg not in contact of the ground swings forward,
acting like a physical pendulum.
Approximating the leg as a uniform rod, find the period for a leg 90 cm long.
T
2
4 0.92 3.1416
3 9.8 /
m
m s
1.6 s
m g L
I 21
23
I m L
42
3
L
g
Table 10.2
Forward stride = T/2 = 0.8 s
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13.4. Circular & Harmonic Motion
Circular motion:
cosx t r t
siny t r t2 SHO with same A &
but = 90
x = Rx = Rx = 0
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GOT IT? 13.3.
The figure shows paths traced out by two pendulums swinging with
different frequencies in the x- & y- directions.
What are the ratios x : y ?
1 : 2 3: 2
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13.5. Energy in Simple Harmonic Motion
cosx t A tSHM: sinv t A t
21
2K m v
21
2U k x 2 21
cos2k A t
2 2 21sin
2m A t 2 21
sin2k A t
21
2E K U k A
= constant
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Potential Energy Curves & SHM
F k xLinear force:
U F d x
parabolic potential energy:
21
2k x
Taylor expansion near local minimum:
min
22
min min2
1
2x x
d UU x U x x x
d x
2
min
1
2const k x x
min
0x x
dU
d x
Small disturbances near equilibrium points SHM
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GOT IT? 13.4.
Two different mass-springs oscillate with the same amplitude & frequency.
If one has twice as much energy as the other, how do
(a) their masses & (b) their spring constants compare?
(c) What about their maximum speeds?
The more energetic oscillator has
(a) twice the mass
(b) twice the spring constant
(c) Their maximum speeds are equal.
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13.6. Damped Harmonic Motion
Damping (frictional) force:
dF b vd x
bd t
Damped mass-spring:
2
2
d x d xm k x bd t d t
Ansatz:
costx t A e t
cos sintv t A e t t
2 2 cos 2 sinta t A e t t
2 2m k b
2m b
2
b
m 2k
m
2
2
k b
m m
sinusoidal oscillation
Amplitude exponential decay
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costx t A e t 2
b
m
2
2
k b
m m
At t = 2m / b, amplitude drops to 1/e of max value.
(a) For 0 is real, motion is oscillatory ( underdamped )
(b) For is imaginary, motion is exponential ( overdamped )
(c) For 0 = 0, motion is exponential ( critically damped )
220 2
b
m
0
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Example 13.6. Bad Shocks
A car’s suspension has m = 1200 kg & k = 58 kN / m.
Its worn-out shock absorbers provide a damping constant b = 230 kg / s.
After the car hit a pothole, how many oscillations will it make before the
amplitude drops to half its initial value?
T
16.95 s
1
2e
8
costx t A e t 2
b
m
Time required is 1 1
ln2
2
ln 2m
b
2 1200
ln 2230 /
kg
kg s 7.23 s
2
2
k b
m m
2
58000 / 230 /
1200 2 1200
N m kg s
kg kg
0.904 s
# of oscillations:7.23
0.904
s
T s
bad shock !
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13.7. Driven Oscillations & Resonance
External force Driven oscillator
0 cosext dF F tLet d = driving frequency
2
02cos d
d x d xm k x b F td t d t
Prob 75: cos dx A t
0
222 2
0d
d
FA
bm
m
0
k
m = natural frequency
Resonance: 0d
( long time )
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Buildings, bridges, etc have natural freq.
If Earth quake, wind, etc sets up resonance, disasters result.
Resonance in microscopic system:
• electrons in magnetron microwave oven
• Tokamak (toroidal magnetic field) fusion
• CO2 vibration: resonance at IR freq Green house effect
• Nuclear magnetic resonance (NMR) NMI for medical use.
Collapse of Tacoma bridge is due to self-excitation described by the van der Pol equation.