Download - Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Transcript
Page 1: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Physics 1251Physics 1251The Science and The Science and

Technology of Musical Technology of Musical SoundSound

Unit 1Unit 1

Session 5 Session 5

Simple Harmonic Simple Harmonic OscillatorsOscillators

Unit 1Unit 1

Session 5 Session 5

Simple Harmonic Simple Harmonic OscillatorsOscillators

Page 2: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5Unit 1 Session 5Simple Harmonic OscillatorsSimple Harmonic Oscillators

Foolscap Quiz:Foolscap Quiz:

What is the wavelength of a CWhat is the wavelength of a C44 (262. Hz) (262. Hz) at room temperature?at room temperature?

Frequency = 262. Hz; velocity = 343. Frequency = 262. Hz; velocity = 343. m/s m/s

v = v = ff ⋅ ⋅ λλ= (262. Hz) ⋅ = (262. Hz) ⋅ λλ = 343. m/s = 343. m/s

λ = (343. m/s) / (262. Hz) = 1.3091603 m λ = (343. m/s) / (262. Hz) = 1.3091603 m

Name, Date,Name, Date, Session Session 55

Seat # Seat #

Joe College Session 5

1/26/2002 #309

λ= 1.31 m (3 significant λ= 1.31 m (3 significant figures)figures)

Page 3: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Sound is a pressure/displacement wave Sound is a pressure/displacement wave that propagates in a material medium.that propagates in a material medium.

But what But what causescauses sound to begin with? sound to begin with?

What is the sound of one hand What is the sound of one hand clapping?clapping?

Page 4: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Oscillating Hand DemonstrationOscillating Hand Demonstration

☞♫☞♫

CompressionCompression

RarefactioRarefactionn

Page 5: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

An oscillation of a body causes a sound.An oscillation of a body causes a sound.

Page 6: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5Unit 1 Session 5Simple Harmonic OscillatorsSimple Harmonic Oscillators

11′ Lecture:′ Lecture:• A A Simple Harmonic OscillatorSimple Harmonic Oscillator is a is a

device that oscillates at one device that oscillates at one frequency, determined by the spring frequency, determined by the spring constant k and the mass m of the constant k and the mass m of the system.system.

• The natural frequency of an SHO is The natural frequency of an SHO is related to these quantities by the related to these quantities by the equation:equation:

f = 1/(2f = 1/(2π) π) ‧ √ (k/m)‧ √ (k/m)

Page 7: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Existential Physics Questions:Existential Physics Questions:• What happens when you stretch a What happens when you stretch a

“spring”?“spring”?• What happens if you stretch the “spring” What happens if you stretch the “spring”

twice as much? Or three times as much?twice as much? Or three times as much?• What is a “spring”?What is a “spring”?• What has this got to with acoustics?What has this got to with acoustics?

Page 8: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

FF

xx

1 N1 N2 N2 N3 N3 N4 N4 N

-1 -1 NN-2 -2 NN-3 -3 NN-4 -4 NN

-.3 m-.3 m-.4 m-.4 m -.2 m-.2 m-.1 m-.1 m .2 m.2 m .1 m.1 m .3 m.3 m .4 m.4 m

Page 9: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

FF

xx

xx FF

00 00

.1 m.1 m -1 N-1 N

.2 m.2 m -2 N-2 N

.3 m.3 m -3 N-3 N

F/x = - 10 N/mF/x = - 10 N/m

Page 10: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Hooke’s LawHooke’s Law

When you stretch or compress a When you stretch or compress a “spring,” the force (F) produced is “spring,” the force (F) produced is proportionalproportional to the displacement (x) and to the displacement (x) and in the direction to restore the system (-) in the direction to restore the system (-) to the original position.to the original position.

Page 11: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Spring Constant kSpring Constant kThe restoring force produced by a spring The restoring force produced by a spring is proportional to the negative of its is proportional to the negative of its extension (or compression).extension (or compression).

F = - k F = - k ‧ ‧ x (Hooke’s Law)x (Hooke’s Law)F : restoring forceF : restoring forcek: spring constantk: spring constantx: extension, the “stretch or squeeze (if negative).”x: extension, the “stretch or squeeze (if negative).”

What is the restoring force of a spring with What is the restoring force of a spring with k =25. N/m when it is stretch by 10 cm?k =25. N/m when it is stretch by 10 cm?

F = - (25. N/m )(0.10 m) = -2.5 NF = - (25. N/m )(0.10 m) = -2.5 N

Page 12: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Mass on a Mass on a SpringSpring

ForceForce

Spring Spring ————————→————————→

Mass Mass ——————————→——————————→

F = - k F = - k ‧ x‧ x

xx

- k - k ‧ ‧ xx

Page 13: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

FF

xx

F/x = - 10 N/mF/x = - 10 N/mF/x = - 25. N/m = F/x = - 25. N/m = -k-kF/x = - 12.5 N/mF/x = - 12.5 N/m

Page 14: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

xx

What determines the frequency What determines the frequency of the oscillation of a simple of the oscillation of a simple harmonic oscillator?harmonic oscillator?

The The stronger stronger the force (k), the more the force (k), the more rapidrapid is the oscillation. is the oscillation.

The The greatergreater the mass (m) the the mass (m) the slowerslower is the oscillation.is the oscillation.

Page 15: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

80/2080/20Frequency of Oscillation Frequency of Oscillation

of Mass on a Springof Mass on a Spring

f = 1/(2f = 1/(2π)π)√(k/m)√(k/m)

f = 0.1592 f = 0.1592 ‧ √ (k/m)‧ √ (k/m)

Page 16: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

If mass = 0.500 kg and k = 25.0 N/m, If mass = 0.500 kg and k = 25.0 N/m, what will be the frequency of what will be the frequency of oscillation?oscillation?

What will be the period?What will be the period?

ff = 1/(2= 1/(2π) π) ‧‧ √ (k/m) √ (k/m)

==1/(21/(2π) π) √ [(25.0 N/m)/(0.500 √ [(25.0 N/m)/(0.500 kg)]kg)]

= 0.1592 √ [50.0] = 1.13 Hz= 0.1592 √ [50.0] = 1.13 HzP = 1/ P = 1/ ff = 1/ (1.13 Hz) = 0.885 sec= 1/ (1.13 Hz) = 0.885 sec

Page 17: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

If we increase the mass from 0.5 kg to 1.0 If we increase the mass from 0.5 kg to 1.0 kg what will happen to the frequency?kg what will happen to the frequency?

Will it increase, decrease or stay the Will it increase, decrease or stay the same?same?

If it changes, by how much will it change?If it changes, by how much will it change?f = 1/(2f = 1/(2π) π) ‧‧ √ (k/m)√ (k/m)

f f ∝∝ √ (1/m) → f√ (1/m) → f22 / f / f11 = √(m = √(m11 /m /m22 ) )

= √(0.5/1.0) ) = √(0.50) = √(0.5/1.0) ) = √(0.50) =0.71=0.71

Page 18: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

If we double the springs what will happen If we double the springs what will happen to the frequency?to the frequency?

Will it increase, decrease or stay the Will it increase, decrease or stay the same?same?

If it changes, by how much will it change?If it changes, by how much will it change?

f = 1/(2f = 1/(2π)π)√(k/m)√(k/m)

f f ∝∝ √k → f√k → f22 / f / f11 = √(k = √(k22 /k /k11 ) )

= √(50./25.) ) = √(2.0) = √(50./25.) ) = √(2.0) =1.41=1.41

Page 19: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Simple Harmonic OscillatorsSimple Harmonic Oscillators• Tuning forkTuning fork

Page 20: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Simple Harmonic OscillatorsSimple Harmonic Oscillators• Tuning ForkTuning Fork (Bulova Accutron)(Bulova Accutron)

Tuning ForkTuning Fork

Page 21: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Simple Harmonic OscillatorsSimple Harmonic Oscillators

““Mouth Harp”Mouth Harp”

OscillatorOscillator

Page 22: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Simple Harmonic OscillatorsSimple Harmonic Oscillators• KalimbaKalimba(Finger Piano)(Finger Piano)

Page 23: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Simple Harmonic OscillatorsSimple Harmonic Oscillators• HarmonicaHarmonica(“Mouth Organ”)(“Mouth Organ”)

Page 24: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Simple Harmonic OscillatorsSimple Harmonic Oscillators• Helmholtz ResonatorHelmholtz Resonator

Page 25: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Hermann von HelmholtzHermann von Helmholtz

© American Institute of © American Institute of Physics,Physics, used by permission used by permission

(1821-1894)(1821-1894)

Prominent 19Prominent 19thth century century physicist and physicist and mathematician. mathematician.

Author of Author of Perception of Perception of Tone,Tone, highly influential highly influential treatise in musical treatise in musical acoustics.acoustics.

Page 26: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

A Helmholtz Resonator is a simple A Helmholtz Resonator is a simple harmonic oscillator that uses air in a harmonic oscillator that uses air in a narrow neck as a mass and air trapped narrow neck as a mass and air trapped in a volume as a spring.in a volume as a spring.

AirAirMassMass

Air Air SpringSpring

Page 27: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic oscillatorsSimple Harmonic oscillators

Examples of Helmholtz Examples of Helmholtz Resonators:Resonators:

BottleBottle Acoustic TileAcoustic Tile Cinder BlockCinder Block OcarinaOcarina

Page 28: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic Oscillators

Helmholtz ResonatorHelmholtz Resonator

• OcarinaOcarina

Page 29: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic Oscillators Simple Harmonic Oscillators

Summary:Summary:• A A Simple Harmonic OscillatorSimple Harmonic Oscillator is a is a

device that oscillates at one device that oscillates at one frequency, determined by the spring frequency, determined by the spring constant k and the mass m of the constant k and the mass m of the system.system.

• The natural frequency of an SHO is The natural frequency of an SHO is related to these quantities by the related to these quantities by the equation:equation:

f = 1/(2f = 1/(2π) π) ‧ √ (k/m)‧ √ (k/m)

Page 30: Physics 1251 The Science and Technology of Musical Sound Unit 1 Session 5 Simple Harmonic Oscillators Unit 1 Session 5 Simple Harmonic Oscillators.

Physics 1251 Physics 1251 Unit 1 Session 5 Unit 1 Session 5 Simple Harmonic Oscillators Simple Harmonic Oscillators

Summary:Summary:• A Helmholtz resonator is a A Helmholtz resonator is a

Simple Harmonic Oscillator Simple Harmonic Oscillator comprising a trapped volume comprising a trapped volume of air that acts as a spring of air that acts as a spring and a narrow neck that acts and a narrow neck that acts as a mass.as a mass.