Part Two: Oscillations, Waves, & Fluids
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
High-speed photo: spreading circular waves on water.
Plane flight
13. Oscillatory Motion
1.
2.
3.
4.
5.
6.
7.
Describing Oscillatory Motion
Simple Harmonic Motion
Applications of Simple Harmonic Motion
Circular & Harmonic Motion
Energy in Simple Harmonic Motion
Damped Harmonic Motion
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
Wilberforce
Pendulum
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 .
Oscillation
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.
same period T
same amplitude A
f 
1
T
[ f ] = hertz (Hz) = 1 cycle / s
A, T, f do not specify an oscillation completely.
Oscillation
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.
T
f 
10 s
 0.36 s / cycle
28 cycles
1
28 cycles

T
10 s
 2.8 Hz
13.2. Simple Harmonic Motion
Simple Harmonic Motion (SHM):
F  k x
d 2x
m 2  k x
dt
2nd order diff. eq  2 integration const.
Ansatz:
x  t   A cos  t  B sin  t
dx
  A sin  t  B  cos  t
dt
d2 x
  2 A cos  t  B  2 sin  t
2
dt
x t  T   x t 

 T  2
f 
1 

T 2
angular
frequency
  2 x
T  2
m
k


k
m
x  t   A cos  t  B sin  t
A, B determined by initial conditions
v t  
dx
  A sin  t  B  cos  t
dt
x 0  1
v  0  0
A 1


B 0
x  t   cos  t
( t )  2
x  2A
Amplitude & Phase
x  t   A cos  t  B sin  t  C cos  t   
 C  cos  t cos   sin  t sin  

C = amplitude
 = phase
A  C cos 
B  C sin 
C  A2  B 2
   tan 1
B
A
Note:  is independent of
amplitude only for SHM.
Curve moves to the right for  < 0.
Oscillation
Velocity & Acceleration in SHM
x  t   A cos  t   
|x| = max at v = 0
dx
  A  sin  t   
dt


 A  cos   t    
2

v t  
|v| = max at a = 0
d2 x
a  t   2   A  2 cos  t   
dt
  2 x  t 
 A  2 cos  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 :
Damper highly damped ,
Overall oscillation overdamped.
Taipei 101 TMD:
41 steel plates,
Also used in:
730 ton, d = 550 cm,
87th-92nd floor.
Movie
Tuned Mass
Damper
• 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.
T  2
m
k

 2 
k m

 T 

2
 2  3.1416 
  373 103 kg  

 6.80 s 
2
 3.18 105 N / m
2
2  3.1416

 0.924 s 1
T
6.80 s
1
vmax   A   0.924 s  1.10 m   1.02 m / s
amax   2 A   0.924 s 1  1.10 m   0.939 m / s 2
2
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
Spring stretched by x1 when loaded.
mass m oscillates about the new equil. pos.
with freq

k
m
The Torsional Oscillator
   
 = torsional constant
 I

d 2
I 2   
dt

Used in timepieces

I
The Pendulum
τT  0
d 2
 g  m g L sin   I 2
dt
Small angles oscillation:
sin   
d 2
I 2  m g L 
dt

sin   
mgL
I
Simple pendulum (point mass m):
I mL
2

g
L
T  
L
g
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?
T  
L
g
Time needed:
1
T 
2
25 m
 5.0 s
2
9.8 m / s
GOT IT? 13.2.
What happens to the period of a pendulum if
no change
(a) its mass is doubled,
doubles
(b) it’s moved to a planet whose g is ¼ that of Earth,
doubles
(c) its length is quadrupled?
T  
L
g
Conceptual Example 13.1. Nonlinear Pendulum
A pendulum becomes nonlinear if its amplitude becomes too large.
(a)As the amplitude increases, how will its period changes?
(b)If you start the pendulum by striking it when it’s hanging vertically,
will it undergo oscillatory motion no matter how hard it’s hit?
(b) If it’s hit hard enough,
motion becomes rotational.
(a)
sin increases slower than 
 smaller  
 longer period
The Physical Pendulum
Physical Pendulum = any object that’s free to swing
Small angular displacement  SHM

mgL
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.
Table 10.2
mgL
I

T


 2
 2  3.1416
1
2
I  m  2L 
3
4L
3g
4   0.9 m 
3   9.8 m / s 2 
 1.6 s
Forward stride = T/2 = 0.8 s
13.4. Circular & Harmonic Motion
Circular motion:
x=R
x=0
x  t   r cos t
2  SHO with same A & 
y  t   r sin t
but  = 90
x=R
Lissajous Curves
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
Lissajous Curves
13.5. Energy in Simple Harmonic Motion
SHM:
x  t   A cos  t
v  t    A  sin  t
U
1
1
k x 2  k A2 cos 2  t
2
2
K
1
1
1
m v 2  m  2 A2 sin 2  t  k A2 sin 2  t
2
2
2
E  K U 
1
k A2
2
= constant
Energy in SHM
Potential Energy Curves & SHM
F  k x
Linear force:
 parabolic potential energy:
U   F d x 
Taylor expansion near local minimum:
1 d 2U
U  x   U  xmin  
2 d x2
dU
dx
0
x  xmin
 x  xmin 
x  xmin
1
k x2
2
2

1
2
 const  k  x  xmin 
2
 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
sinusoidal
oscillation
Damping (frictional) force:
Fd  b v  b
dx
dt
Damped mass-spring:
Amplitude
exponential decay
d 2x
dx
m 2  k x  b
dt
dt
Ansatz:
x  t   A e t cos  t   
v  t   A e  t   cos  t      sin  t    
m  2   2    k  b 

2m    b 
a  t   A e t  2   2  cos  t     2  sin  t    

b

2m

k
 2 
m
k  b 


m  2m 
2
x t   A e
 t
cos  t   
b

2m

k  b 


m  2m 
2

 b 
 

 2m 
2
0
 At t = 2m / b, amplitude drops to 1/e of max value.
(a) For
0  
 is real, motion is oscillatory ( underdamped )
(c) For
0  
 is imaginary, motion is exponential ( overdamped )
(b) For
0  
 = 0, motion is exponential ( critically damped )
Damped & Driven
Harmonic Motion
2
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?
b

2m
x  t   A e t cos  t   
Time  required is
e   
1
2

 
1

ln
1
2


58000 N / m  230 kg / s   6.95 s 1
 

1200 kg
2
1200
kg

 
# of oscillations:

T

7.23 s
8
0.904 s
2
2m
2 1200 kg 
ln 2 
ln 2  7.23 s
b
 230 kg / s 
2

k  b 


m  2m 
bad shock !
T


 0.904 s
13.7. Driven Oscillations & Resonance
External force  Driven oscillator
Fext  F0 cos d t
Let
d = driving frequency
d 2x
dx
m 2  k x  b
 F0 cos d t
dt
dt
x  A cos d t   
Prob 75:
F0
A
m
0 
k
m
Resonance:
( long time )

2
d


2 2
0
 b d 


 m 
2
= natural frequency
d  0
Damped & Driven
Harmonic Motion
Buildings, bridges, etc have natural freq.
If Earth quake, wind, etc sets up resonance, disasters result.
Collapse of Tacoma bridge is due to self-excitation
described by the van der Pol equation.
Tacoma Bridge
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.