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Lecture 08Lecture 08
Adding the Adding the PhysicsPhysics to the mix to the mix
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Where are we?Where are we?
We have looked at some history of We have looked at some history of music from the “monkey” to the music from the “monkey” to the work of Helmholtz.work of Helmholtz.
We found that the Greeks noted that We found that the Greeks noted that certain fractions of the string lengths certain fractions of the string lengths led to tones that sounded “well” led to tones that sounded “well” together.together.
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What We FoundWhat We FoundThe Original String Length is LThe Original String Length is L00
Length Name Comment
L0
Fundemental Tone
L0/2 OctaveMen/Women tend to sing
one octive apart
(2/3)L0 FifthTenors and Bass will sing
one fifth apart
(4/3)L0 FourthDivide by 2 =2/3 = Reduce
to same octave
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We also defined repetitive pulses We also defined repetitive pulses as having a frequency (f) and a as having a frequency (f) and a
period (T)period (T) Frequency, f, is the number of pulses Frequency, f, is the number of pulses
that happen each second.that happen each second.– Unit is Hertz=1 cycle/secondUnit is Hertz=1 cycle/second
The Period (T) is the time BETWEEN The Period (T) is the time BETWEEN pulses of the length of a single pulse.pulses of the length of a single pulse.– T=1/fT=1/f
ConsequentlyConsequently– fT=1 (IMPORTANT RELATIONSHIP)fT=1 (IMPORTANT RELATIONSHIP)
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We introduced the siren We introduced the siren (Helmholtz)(Helmholtz)
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The sirenThe siren
Sends out a Sends out a KNOWN number of KNOWN number of pulses per second.pulses per second.
The pulses sound The pulses sound like tones.like tones.
We defined the We defined the frequency as the frequency as the number of puffs number of puffs produced per produced per second.second.
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ExampleExample
We turn the belt so that the disk We turn the belt so that the disk makes 2 rotations per second.makes 2 rotations per second.
The disk has 50 holes around the The disk has 50 holes around the circumference.circumference.
The frequency = (number of rotations The frequency = (number of rotations per second) x (number of holes in the per second) x (number of holes in the circumference) = 100 hertzcircumference) = 100 hertz
This is a sound that we can hear.This is a sound that we can hear.
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NOTENOTE
The lowest tone the human ear can The lowest tone the human ear can hear is in the 25-50 hertz region.hear is in the 25-50 hertz region.– It varies with the individual.It varies with the individual.
The highest tone the human ear can The highest tone the human ear can hear is about 20,000 hertz.hear is about 20,000 hertz.
Dogs hear higher!!!Dogs hear higher!!! But they can’t sing.But they can’t sing.
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The Helmholtz ResonatorThe Helmholtz Resonator
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The Helmholtz ResonatorThe Helmholtz Resonator
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The Helmholtz ResonatorThe Helmholtz Resonator
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For a PARTICULAR sized For a PARTICULAR sized resonatorresonator
frequency
Soundlevel
F0 = Resonant Frequency
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Let’s Look at some Let’s Look at some SpringsSprings
Demo-01Demo-01
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An Observation from N-1
M M
Mg
Tension=T
M
Tension=T
Mg
FREE BODY DIAGRAM
Because spring isWeightless, T=constant
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Remember from a few weeks back???
F=-kx
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We looked at:
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An Observation from N-1
M M
Mg
Tension=T
M
Mg
FREE BODY DIAGRAM
½ the lengthrequires twice
the force to stretchthe same length!
k is Bigger
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The Vertical Spring
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The Vertical Spring
Un-stretched
Hold andSuddenlyRelease Weight=Mg
acceleration
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Continue On
Un-stretched
Hold andSuddenlyRelease Weight=Mg
acceleration
Spring Force
v
a
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Oscillation
Repeat
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OscillationPosi
tion
T
T
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Example
0.1 seconds
HertzT
f
Tperiod
10sec
110
sec1.0
11
sec1.0
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For the SPRING it is found that
m
k
Tf
andk
mT
kxF
2
11
2
Law) s(Hooke'
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Conclusions
• The bigger the mass the lower the frequency.
• The bigger the spring constant (stiffer) the higher frequency.m
k
Tf
andk
mT
kxF
2
11
2
Law) s(Hooke'
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Let’s Look at a Let’s Look at a “Springy String”“Springy String”
DEMO-02DEMO-02
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The TonesThe Tones
m
k
Tf
2
11f0
022
12
4
2
1
2/
2
2
1f
m
k
m
k
m
kf
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The FIFTH (???)The FIFTH (???)
FIFTHf
ff
0scale
0
2
3f
2by divide
so octavenext in the
3
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Take a lookTake a look
ff00 is the first tone. is the first tone.
The octave is 2fThe octave is 2f00..
The NEXT octave is The NEXT octave is 4ff00..
The NEXT octave is The NEXT octave is 8ff00..
DOUBLE FOR EACH DOUBLE FOR EACH OCTAVEOCTAVE
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Another PerspectiveAnother Perspective
f0 2f0 4f0 8f0
octave
3f0
(3/2)f0
fifth
Nextoctave
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Another ViewAnother View
(3/4)f0=FOURTH
Why would astring looklike this???