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.