Vibrations and Vibrations and WavesWaves

Extreme ExampleExtreme Example

Tacoma Narrows BridgeTacoma Narrows Bridge It stood for only 3 months before….It stood for only 3 months before….

http://www.youtube.com/watch?v=P0Fi1VcbpAI

Vibrations and wavesVibrations and waves

SHM SHM OscillationOscillation

PeriodicPeriodic

f= -KXf= -KXF= Restoring forceF= Restoring force

K= spring constantK= spring constant

X= displacementX= displacement

F= Restoring forceF= Restoring force

K= spring constantK= spring constant

X= displacementX= displacement

Equilibrium PositionEquilibrium Position DisplacementDisplacement

AmplitudeAmplitude Period-T, time needed for 1 full Period-T, time needed for 1 full

cycle (sec)cycle (sec) Frequency- f, # of cycles per Frequency- f, # of cycles per

second (Hz)second (Hz)

f= 1/T T= f= 1/T T= 1/f1/f

*Car Spring Example, pg. 311*Car Spring Example, pg. 311Section 11-2

Section 11-2 draws draws heavily from

heavily from chpt. 6chpt. 6

Read on your

Read on your

own pg. 312-own pg. 312-

314314

Section 11-3, read the deriving Section 11-3, read the deriving of formula…of formula…

TTss = 2 = 2 m/k m/k Formula for Formula for Period of Period of

SpringSpringm= massm= mass

k= spring k= spring constantconstant

Not direct relationship!Not direct relationship!

T m/kT m/k Mass must QUADRUPLE to double Mass must QUADRUPLE to double

periodperiod We can Substitute…We can Substitute…

f = 1/T = f = 1/2f = 1/T = f = 1/2 k/m k/m

What are What are T & fT & f of the car of the car example on pg.311 after example on pg.311 after hitting a bump? Assume shock hitting a bump? Assume shock absorbers are poor so car absorbers are poor so car really oscillates a lot. really oscillates a lot.

T = 2T = 2 √m/k√m/k = 6.28 = 6.28 √√1400kg/ 1400kg/ 6500N/m6500N/m

=.92sec =.92sec

F = 1/.92F = 1/.92 = 1.09Hz= 1.09Hz

A small insect (.3g) is A small insect (.3g) is caught in a spider web caught in a spider web (mass-less). The web (mass-less). The web vibrates at 15Hzvibrates at 15Hz

a.a. estimate the value of K estimate the value of K for the webfor the web

b.b. find f for an insect of find f for an insect of mass .1g.mass .1g.

f = 1/2f = 1/2 √K/m√K/m

K = (2K = (2f)f)22mm= (6.26 x 15)= (6.26 x 15)22 (3 x 10 (3 x 10-4-4kg)kg)

2.7 N/m2.7 N/m

Sub in for MSub in for M

1 x 101 x 10--

44kgkgf = 26Hzf = 26Hz

Simple PendulumSimple Pendulum

What determines T for a What determines T for a Pendulum?Pendulum?

Mass of bobMass of bobAmplitude of swingAmplitude of swing

Length of stringLength of stringGravity of locationGravity of location

××××

FormulaFormula

TTpp = 2 = 2 √√LL//gg

F = F = 11//TT = ½ = ½ √√gg//LL

For pendulums to be said to be in SHM the For pendulums to be said to be in SHM the displacement angle has to be small- less than displacement angle has to be small- less than

1515°°

For pendulums to be said to be in SHM the For pendulums to be said to be in SHM the displacement angle has to be small- less than displacement angle has to be small- less than

1515°°

Damped Harmonic Damped Harmonic MotionMotion

Amplitude of a swinging spring Amplitude of a swinging spring on a pendulum slowly on a pendulum slowly

decreases in time until the decreases in time until the oscillations stop all together. oscillations stop all together.

XXTimeTime

Damping is caused by air Damping is caused by air friction and internal friction and internal

friction within the systemfriction within the system Common dampening Common dampening

systems are shock systems are shock absorbers and door absorbers and door

closing mechanisms.closing mechanisms.

A = over damped (too slow)A = over damped (too slow)

B= Critically dampedB= Critically damped

C=under damped (still C=under damped (still oscillating)oscillating)

CC

BB

AAx

time

Resonance… Forced Resonance… Forced Vibrations Vibrations

Objects have Objects have natural natural resonant frequenciesresonant frequencies. .

When vibrations are put When vibrations are put on an object that are at on an object that are at

the the natural resonant natural resonant frequencyfrequency, an , an increaseincrease in in amplitudeamplitude is observed. is observed.

If pushing is random- swing If pushing is random- swing just bounces aroundjust bounces around

But if you push with frequency But if you push with frequency equal to Natural Resonant equal to Natural Resonant Frequency(NRF) you can get a Frequency(NRF) you can get a big amplitudebig amplitude

SwingSwing

examples…examples…

Tuning Fork

GuitarsGuitars

http://www.youtube.com/watch?v=I4jdGf3RzCs

Wave MotionWave Motion mechanical waves for now…mechanical waves for now… -OVER HEAD- -OVER HEAD-

Is the velocity of a wave moving Is the velocity of a wave moving along a cord the same as the along a cord the same as the velocity of a particle of the cord?velocity of a particle of the cord?

NO – velocities are different so are directions…

Waves don’tWaves don’t carry matter, carry matter,

they they transfer transfer energyenergy

The parts of a single The parts of a single frequency wavefrequency wave

Types of wavesTypes of waves

Wave Wave MotionMotion

Wave Wave MotionMotion

Transverse Transverse

CompressionCompression

Rare Rare factionsfactions

Longitudinal Longitudinal

S- wavesS- waves Earth quake Earth quake waveswaves

P- waves; only P P- waves; only P travels through travels through

liquidliquid

Water WavesWater Waves

Wave FormulaWave Formula

V= fV= fVelocity=wavelength x Velocity=wavelength x

frequencyfrequency

Reflection of WavesReflection of Waves

•Wave fronts

•Plane waves

•Law of reflection

Wave InterferenceWave Interference

•Principle of superposition

•Destructive interference

•Constructive Interference

•In phase or Out of phase

Standing Waves: Standing Waves: ResonanceResonance

•Standing Waves

•Nodes

•Antinodes

•Natural frequencies

•Fundamental frequencies

•Overtones

http://www.youtube.com/watch?feature=player_embedded&v=Oz53w_k_j_A#!

Refraction and DiffractionRefraction and Diffraction