CSUEB Physics 1200Category: Harmonic Motion, Waves, Sound

I. Harmonic Motion& Resonance

Physics 2703Updated 2015Mar30

Dr. Bill Pezzaglia

Outline

A. Periodic Motion

B. Harmonic Systems

C. Resonance

D. Energy in Oscillations

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A. Periodic Motion

1. Equilibrium

2. Periodic Motion

3. Frequency

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Knight Section 14.1, 14.3

1. Equilibrium

Equilibrium: a system which is not changing with time (no net force).

There are 3 types:

a) Neutral equilibrium: boring case, if you move the object to another position it will just sit there.

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b. Unstable Equilibrium

• if you displace system slightly it changes drastically,

• e.g. a ball perched on top of a steep hill

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c. Stable Equilibrium

• if you displace the system, there is a “restoring force” which opposes the change

• Strength of restoring force increases with displacement

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2. Periodic Motion

a) Oscillations:

• A system displaced from stable equilibrium will oscillate about the equilibrium point

• The motion is “periodic” (repeats in time)

• The time for one cycle is called the “period”

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3. Frequency

a). Definition:

• Frequency is the rate of vibration

• Units: Hertz=“cycles per second”

• Relation to Period:

8

Tf

1

b. Frequency is “Pitch”

• 1600 Scraper across grooved board produces notes (relates frequency of vibration to pitch of sound)

• Mathematical Discourses Concerning Two New Sciences (1638) most lucid of the frequency equivalence

• Made sound waves visible by striking a wine glass floating in water and seeing the vibrations it made on the water’s surface.

• First person to accurately determine frequency of musical pitch was probably Joseph Sauveur (1653-1716)

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Galileo Galilei(1564-1642)

c. Toothed Wheels & Sirens

• 1819 Cagnaird de la Tour’s siren used to precisely measure frequency of sound (disk with holes spun, air blown across holes)

• 1830 Savart uses card against moving toothed wheel to equate frequency and vibration

• Measures the lowest pitch people can hear is about 16 to 20 Hertz

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B. Harmonic Systems

1. Pendulum

2. Mass on Spring

3. Mathematical Description

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Knight Sections 14.2, 14.5

(a) Galileo: (1581) showed the period of oscillation depends only upon gravity “g” and length “L” of the string:

• Period is INDEPENDENT of:– Mass on end of string– Size (“amplitude”) of oscillation

(“isochronism”)

1. Pendulums 12

g

LT 2

Acceleration of gravity on earth: g=9.8 meters/second2.Gravity on moon is 1/6 as strong, so pendulum will go slower!

b. Physical Pendulum

Complex body will behave like a simple pendulum

• L is distance to CM from pivot is moment of inertia about “O”

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mgL

IT 2

L

c. Advanced DetailsAngular Equation of motion for pendulum:

Only for small oscillations is the period independent of amplitude. For large amplitude 0 (in radians) the period does depend upon the amplitude.

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1612

20

g

LT

sin

sinLmgI

a) Hooke’s Law: if you squash a spring by distance x, it will give a restoring force F proportional to x:

b) Spring Constant “k” tells the stiffness of the spring.

c) Period of Oscillation for mass m on spring:

2. Springs 15

k xF

k

mT 2

d. Advanced DetailsEquation of motion for spring oscillator:

Springs are not massless. The effective mass in the equation is approximately given by:

16

kxma

springhung mmm 31

3. Mathematical Description

A is the amplitude of oscillation

is the angular frequency

f is the frequency

P is the period

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Pf

tAta

tAtv

tAtx

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)cos()(

)sin()(

cos)(

2

3b. Velocity

The maximum speed of the mass on a spring is related to the maximum displacement (amplitude “A”) by the frequency “ f ” :

AAfv 2max

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3c. Acceleration

The acceleration for a “harmonic” system is proportional to the displacement. The minus sign means it’s a restoring force, such that the system will oscillate.

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)()( 2 txta

C. Resonance

1. Driven Oscillator

2. Resonance Curve

3. Bandwidth and Quality

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Knight Section 14.7

Unwanted Resonance (1850)

Angers Bridge: a suspension bridge over the Maine River in Angers, France. Its famous for having collapsed on April 15, 1850, when 478 French soldiers marched across it in lockstep. Since the soldiers were marching together, they caused the bridge to vibrate and twist from side to side, dislodging an anchoring cable from its concrete mooring.. 226 soldiers died in the river below the bridge.

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Tacoma Narrows Bridge Collapse (1940)

“Just as I drove past the towers, the bridge began to sway violently from side to side. Before I realized it, the tilt became so violent that I lost control of the car... I jammed on the brakes and got out, only to be thrown onto my face against the curb... Around me I could hear concrete cracking... The car itself began to slide from side to side of the roadway. On hands and knees most of the time, I crawled 500 yards [450 m] or more to the towers... My breath was coming in gasps; my knees were raw and bleeding, my hands bruised and swollen from gripping the concrete curb... Toward the last, I risked rising to my feet and running a few yards at a time... Safely back at the toll plaza, I saw the bridge in its final collapse and saw my car plunge into the Narrows.” -eyewitness account

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Video on Collapse of Bridge: http://www.youtube.com/watch?v=ASd0t3n8Bnc

Millennium Bridge (London)

You’d think that engineers don’t make mistakes like this anymore? Yet again in 2000….

The Millennium pedestrian Bridge opened June 10, 2000. Almost immediately it was discovered to resonate when people walked over it, causing it to be close 2 days later.

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Video on Millennium Bridge Problems: http://www.youtube.com/watch?v=gQK21572oSU&feature=related

1. Driven Oscillator

• Externally force a system to oscillate at “driven” frequency “f”. When you approach the system’s natural frequency, the response will be BIG.

• Example: Singing in the shower, if you hit right note, the whole room sings.

• Demo: resonances of Chladni plate or wineglass• http://www.youtube.com/watch?v=BE827gwnnk4

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2. Resonance Curve

• Consider a system with resonance at 10,000 Hertz.

• As you move away (above or below) resonance, the amplitude drops off.

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220

2

aff

aAmplitude

3. Bandwidth• The bandwidth is the

range of frequencies for which the amplitude response is above 70%.

• Or, the range for which the power is above 50% (power is proportional to square of amplitude)

• Bandwidth “f”

• Resonance f0

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12 fff

2f1f 0f

3b. “Q” factor

• Quality of Resonance: A big Q has a very sharp response, you have to be very close to resonant frequency for the system to respond (musical instruments)

• Small Q: very poor resonance, for example if you make a tube out of paper (seal bottom) and blow on it you will not get as good as a sound as blowing on a glass tube/bottle. The “damping” is too big.

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f

fQ

0

D. Energy in Oscillations

1. Maximum Speed & Amplitude

2. Kinetic & Potential Energy

3. Damping (energy loss)

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Knight Sections 14.4, 14.6

1. Velocity

The maximum speed of the mass on a spring is related to the maximum displacement (amplitude “A”) by the frequency “ f ” :

AAfv 2max

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2. Kinetic and Potential Energy of Spring

a) Kinetic Energy(maximum at middle):

b) Potential Energy (of displaced spring) where k is the spring constant.

c) Total Energy is constant. This gives a relation between maximum velocity and amplitude:

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221 m vK E

221 k xU

22max k Am v

3. Damped Oscillations

• Including friction the oscillations die out exponentially with time.

• Frictional force is velocity dependent (for wind instruments its due to viscosity of air)

• Critical Damping: If friction is big enough (small Q), system does not oscillate!

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)2sin()( tfAetx t

vmkxma )2(

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Notes

1. Demo: Spring, Pendulum

2. Comb or other similar device

3. Include massive spring

4. Include pendulum corrections

5. Break wine glass

http://www.youtube.com/watch?v=BE827gwnnk4

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