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

• Physics 1251The Science and Technology of Musical SoundUnit 1Session 5 Simple Harmonic Oscillators

• Physics 1251 Unit 1 Session 5Simple Harmonic OscillatorsFoolscap Quiz: What is the wavelength of a C4 (262. Hz) at room temperature? Frequency = 262. Hz; velocity = 343. m/s v = f = (262. Hz) = 343. m/s = (343. m/s) / (262. Hz) = 1.3091603 m = 1.31 m (3 significant figures)

• Physics 1251 Unit 1 Session 5 Simple Harmonic Oscillators

Sound is a pressure/displacement wave that propagates in a material medium.But what causes sound to begin with?What is the sound of one hand clapping?

• Physics 1251 Unit 1 Session 5 Simple Harmonic Oscillators

Oscillating Hand Demonstration

• Physics 1251 Unit 1 Session 5 Simple Harmonic Oscillators

An oscillation of a body causes a sound.

• Physics 1251 Unit 1 Session 5Simple Harmonic Oscillators1 Lecture:A Simple Harmonic Oscillator is a device that oscillates at one frequency, determined by the spring constant k and the mass m of the system.The natural frequency of an SHO is related to these quantities by the equation:

f = 1/(2) (k/m)

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsExistential Physics Questions:What happens when you stretch a spring?What happens if you stretch the spring twice as much? Or three times as much?What is a spring?What has this got to with acoustics?

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsFx

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsFxxF00.1 m-1 N.2 m-2 N.3 m-3 NF/x = - 10 N/m

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsHookes LawWhen you stretch or compress a spring, the force (F) produced is proportional to the displacement (x) and in the direction to restore the system (-) to the original position.

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsSpring Constant kThe restoring force produced by a spring is proportional to the negative of its extension (or compression).

F = - k x (Hookes Law)F : restoring forcek: spring constantx: extension, the stretch or squeeze (if negative).What is the restoring force of a spring with k =25. N/m when it is stretch by 10 cm?F = - (25. N/m )(0.10 m) = -2.5 N

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsMass on a SpringForceSpring Mass

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsFxF/x = - 10 N/mF/x = - 25. N/m = -kF/x = - 12.5 N/m

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsxWhat determines the frequency of the oscillation of a simple harmonic oscillator?The stronger the force (k), the more rapid is the oscillation.The greater the mass (m) the slower is the oscillation.

• Physics 1251 Unit 1 Session 5 Simple Harmonic Oscillators80/20Frequency of Oscillation of Mass on a Spring

f = 1/(2)(k/m)

f = 0.1592 (k/m)

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsIf mass = 0.500 kg and k = 25.0 N/m, what will be the frequency of oscillation? What will be the period? f = 1/(2) (k/m) =1/(2) [(25.0 N/m)/(0.500 kg)]= 0.1592 [50.0] = 1.13 HzP = 1/ f = 1/ (1.13 Hz) = 0.885 sec

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsIf we increase the mass from 0.5 kg to 1.0 kg what will happen to the frequency? Will it increase, decrease or stay the same?If it changes, by how much will it change?f = 1/(2) (k/m)f (1/m) f2 / f1 = (m1 /m2 ) = (0.5/1.0) ) = (0.50) =0.71

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsIf we double the springs what will happen to the frequency? Will it increase, decrease or stay the same?If it changes, by how much will it change?f = 1/(2)(k/m)f k f2 / f1 = (k2 /k1 ) = (50./25.) ) = (2.0) =1.41

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic OscillatorsTuning fork

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic OscillatorsTuning Fork (Bulova Accutron) Tuning Fork

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic OscillatorsMouth Harp Oscillator

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic OscillatorsKalimba(Finger Piano)

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic OscillatorsHarmonica(Mouth Organ)

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsSimple Harmonic OscillatorsHelmholtz Resonator

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsHermann von Helmholtz American Institute of Physics, used by permission(1821-1894)Prominent 19th century physicist and mathematician. Author of Perception of Tone, highly influential treatise in musical acoustics.

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsA Helmholtz Resonator is a simple harmonic oscillator that uses air in a narrow neck as a mass and air trapped in a volume as a spring. Air MassAir Spring

• Physics 1251 Unit 1 Session 5 Simple Harmonic oscillatorsExamples of Helmholtz Resonators:BottleAcoustic TileCinder BlockOcarina

• Physics 1251 Unit 1 Session 5 Simple Harmonic OscillatorsHelmholtz ResonatorOcarina

• Physics 1251 Unit 1 Session 5 Simple Harmonic Oscillators Summary:A Simple Harmonic Oscillator is a device that oscillates at one frequency, determined by the spring constant k and the mass m of the system.

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

f = 1/(2) (k/m)

• Physics 1251 Unit 1 Session 5 Simple Harmonic Oscillators Summary:A Helmholtz resonator is a Simple Harmonic Oscillator comprising a trapped volume of air that acts as a spring and a narrow neck that acts as a mass.