Lecture 08Lecture 08

Adding the Adding the PhysicsPhysics to the mix to the mix

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.

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

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)

We introduced the siren We introduced the siren (Helmholtz)(Helmholtz)

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.

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.

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.

The Helmholtz ResonatorThe Helmholtz Resonator

The Helmholtz ResonatorThe Helmholtz Resonator

The Helmholtz ResonatorThe Helmholtz Resonator

For a PARTICULAR sized For a PARTICULAR sized resonatorresonator

frequency

Soundlevel

F0 = Resonant Frequency

Let’s Look at some Let’s Look at some SpringsSprings

Demo-01Demo-01

An Observation from N-1

M M

Mg

Tension=T

M

Tension=T

Mg

FREE BODY DIAGRAM

Because spring isWeightless, T=constant

Remember from a few weeks back???

F=-kx

We looked at:

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

The Vertical Spring

The Vertical Spring

Un-stretched

Hold andSuddenlyRelease Weight=Mg

acceleration

Continue On

Un-stretched

Hold andSuddenlyRelease Weight=Mg

acceleration

Spring Force

v

a

Oscillation

Repeat

OscillationPosi

tion

T

T

Example

0.1 seconds

HertzT

f

Tperiod

10sec

110

sec1.0

11

sec1.0

For the SPRING it is found that

m

k

Tf

andk

mT

kxF

2

11

2

Law) s(Hooke'

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'

Let’s Look at a Let’s Look at a “Springy String”“Springy String”

DEMO-02DEMO-02

The TonesThe Tones

m

k

Tf

2

11f0

022

12

4

2

1

2/

2

2

1f

m

k

m

k

m

kf

The FIFTH (???)The FIFTH (???)

FIFTHf

ff

0scale

0

2

3f

2by divide

so octavenext in the

3

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

Another PerspectiveAnother Perspective

f0 2f0 4f0 8f0

octave

3f0

(3/2)f0

fifth

Nextoctave

Another ViewAnother View

(3/4)f0=FOURTH

Why would astring looklike this???