1. 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
Plane flight
High-speed photo: spreading circular waves on water.

2. 13. Oscillatory Motion 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

3. 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

4. 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 .

5. 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 ] = hertz (Hz) = 1 cycle / s
A, T, f do not specify an oscillation completely.

6. 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.

7. 13.2. Simple Harmonic Motion
Simple Harmonic Motion (SHM):
Ansatz:
angular frequency  

8. A, B determined by initial conditions 
( t )  2
x  2A

9. Amplitude & Phase  C = amplitude  = phase
Note:  is independent of amplitude only for SHM.
Curve moves to the right for  < 0.

10. Velocity & Acceleration in SHM
|x| = max at v = 0
|v| = max at a = 0

11. 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 )

12. 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.

13. 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. 

14. 13.3. Applications of Simple Harmonic Motion
The Vertical Mass-Spring System
The Torsional Oscillator
The Pendulum
The Physical Pendulum

15. The Vertical Mass-Spring System
Spring stretched by x1 when loaded.
mass m oscillates about the new equil. pos.
with freq

16. The Torsional Oscillator
 = torsional constant 
Used in timepieces

17. The Pendulum
Small angles oscillation:
Simple pendulum (point mass m):

18. 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?
Time needed:

19. GOT IT? 13.2. What happens to the period of a pendulum if
its mass is doubled,
it’s moved to a planet whose g is ¼ that of Earth,
its length is quadrupled?
no change
doubles
doubles

20. The Physical Pendulum
Physical Pendulum = any object that’s free to swing
Small angular displacement  SHM

21. Example 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
Forward stride = T/2 = 0.8 s

22. 13.4. Circular & Harmonic Motion
Circular motion:
2  SHO with same A &  but  = 90
x =  R
x = R
x = 0

23. 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

24. 13.5. Energy in Simple Harmonic Motion
SHM:
= constant

25. Potential Energy Curves & SHM
Linear force:
 parabolic potential energy:
Taylor expansion near local minimum:
 Small disturbances near equilibrium points  SHM

26. 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
their masses & (b) their spring constants compare?
(c) What about their maximum speeds?
The more energetic oscillator has
twice the mass
twice the spring constant
(c) Their maximum speeds are equal.

27. 13.6. Damped Harmonic Motion
sinusoidal oscillation
Damping (frictional) force:
Damped mass-spring:
Amplitude exponential decay
Ansatz:  

28.  At t = 2m / b, amplitude drops to 1/e of max value.
(a) For
 is real, motion is oscillatory ( underdamped )
(b) For
 is imaginary, motion is exponential ( overdamped )
(c) For
 = 0, motion is exponential ( critically damped )

29. Example 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?
Time  required is 
# of oscillations:
bad shock !

30. 13.7. Driven Oscillations & Resonance
External force  Driven oscillator
Let
d = driving frequency
Prob 75:
( long time )
= natural frequency
Resonance:

31. 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.
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.