Some Problems

Musical Interlude – Stringed Instruments

What do Springs Have To Do With It?

1

What are we doing???

• Today we finish up chapter 3,• Do some problems and• Move on to springs/music

• EXAM IS STILL APPROACHING!

2

Examination Date

Exam #1 Feb-11

Where are we??

3

aF

avv

vx

m

t

t

0

Let’s talk more about friction.

4

StationaryBoth forces the same

Stationary – Pushing harder. Both the same.

MovingPush bigger than frictional force.

Graph

5

Force Newtons

Fric

tion N

ew

tons

Stationary

Moving

6

7

Applied force

f

N

W

8

Applied Force

Stationary Sliding

Fric

tional Fo

rce

Friction

9

NF

Moving

NF

Static

d

s

Dynamic -

Which coefficient of friction is the largest?

A. StaticB. DynamicC. They are both the same.

10

The Violin – Friction in Motion!

11

The Bow

12

Horsehair

The Bowing Process of a Violin

13

Performer pushes downAnd to the right

Reaction ForceString on Bow

Motion of Bow

Frictional Force

Focus in on therelative motion of

the bow and the string

The Process

14

Static MotionString moves withbow

Still StaticFriction aboutto change tosliding friction

Sliding friction. Thestring slips back dueto inertia until the stringpressure goes in the otherdirection

Process repeats

APPLICATION ?? PHYSICS LATER

15

Things that go back and forth

• Pendulum• Mass on Spring

16

Guitar Strings

The Spring

17

Spring Force Equatiom

• F=-kx

• The “-” sign indicates that the force and the displacement are in opposite directions.

18

Springs Oscillate

19

Graph

20

21

Important Result for a Spring:

m

kf

kxF

2

1

So ….m

kf

2

1

22

23

Concept … Tension

24

The Musical String

25

Force = F

x

Linitial T

T TThe Bigger the angle themore T points UP!

The distance “x” is the samesort of thing as the x in F=-kx.

ANGLE

Spring /String Motion

26

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25

Time (seconds)

dis

turb

an

ce H

eig

ht

The Guitar Strings

27

Stringed Instruments

28

PLUCK

Momentum

Important Definitions

29

Tf

or

periodfrequency

1

1

The PERIOD, T is the time it takes to go from one condition to the next time that exact condition is repeated.The frequency, the number of oscillations per second, is given by:

Example:

If T=2 seconds

F=1/2 (sec-1)=0.5 per second

Question

30

What is a tone and how do you prove it??

Remember Helmholtz

• In physiology and physiological psychology,• he is known for his mathematics of the eye, • theories of vision, • ideas on the visual perception of space, • color vision research, • the sensation of tone, • perception of sound. • In physics, he is known for his theories on the

conservation of force, • work in electrodynamics, chemical thermodynamics, • A mechanical foundation of thermodynamics.

31

1821 - 1894

32

Helmholtz Today

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

Time in milli-seconds

"air

spee

d"

- re

lati

ve

The SINE curve

Two Fuzzy Sine Waves

33

34

We Know (And will know even more later)

Tone

Today’s Approach

35

Speaker

36

Into the air …

37Credit: http://www.soundonmind.com/

Helmholtz’s ResultsNote from Middle C Frequency

C 264

D 297

E 330

F 352

G 396

A 440

B 496

38

39

We can study these tones with electronics

Tone

Or:

Oscilloscope

40

http://commons.wikimedia.org/wiki/Main_Page

One More Tool

41

Tone

Signal GeneratorElectrical

In using these modern tools

1. We postpone understanding how some of these tools work until later in the semester.

2. We must develop some kind of strategy to convince us that this approach is appropriate.

42

Another piece of the string!

43

LinitialLstretchl

Lfinal

F

The Guitar Strings

44

Consider Two Situations

45

For the same “x” therestoring force is doublebecause the angle is double.

The “mass” is about halfbecause we only havehalf of the stringvibrating.

So…

m

kf

kxF

2

1

46

For the same “x” therestoring force is doublebecause the angle is double.

The “mass” is about halfbecause we only havehalf of the stringvibrating.

k doubles

m -> m/2

f doubles!

Guitar

• Pressing the fret that is in the middle of the string doubles the frequency~– Walla … the octave

• In general … the frequency is proportional to the length of the string.

• Next time we will examine the monochord and Dr. Koons will show us how we develop (a) musical scale(s).

47

Now …. lets look at the MONOCHORD

48

HAVE FUN!

49

It has been shown that …

k""constant spring a like Looks

area sectional-cross theisA

material afor constant a is E

withpullyou force theis F

initial

stretchinitial

L

EA

LL

EAF

50

More about this when we do the string thing.

Octave

51

0.001 0.002 0.003 0.004 0.005

-1

-0.5

0.5

1

0.001 0.002 0.003 0.004 0.005

-1

-0.5

0.5

1

0.001 0.002 0.003 0.004 0.005

-1.5

-1

-0.5

0.5

1

1.5