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
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
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
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
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 .
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.
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
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
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
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
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
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 )
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.
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
13.3. Applications of Simple Harmonic Motion
• The Vertical Mass-Spring System
• The Torsional Oscillator
• The Pendulum
• The Physical Pendulum
The Vertical Mass-Spring System
k
m
Spring stretched by x1 when loaded.
mass m oscillates about the new equil.
pos.
with freq
The Torsional Oscillator
= torsional constant
I
I
2
2
dIdt
Used in timepieces
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
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
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
The Physical Pendulum
Physical Pendulum = any object that’s free to swing
Small angular displacement SHM
m g L
I
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
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
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
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
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
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.
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
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
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 !
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 )
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.