Transcript Springs and Things

1.3 The Ups and Downs
Science of Music
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Things that go back and forth
 Pendulum
 Mass on Spring
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Another Graph .. a biggy!
1.5
disturbance
Height
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0.5
0
0
5
10
15
20
25
-0.5
-1
-1.5
Time (seconds)
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Important Definitions
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:
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frequency 
period
or
1
f 
T
Example:
If T=2 seconds
F=1/2 (sec-1)=0.5 per second
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Question
What is a tone and how
do you prove it??
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Hemholtz
1821 - 1894
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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.
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Siren … a scientific instrument
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The Graph
1.2
"airspeed" - relative
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0.8
0.6
0.4
0.2
0
-0.2
0
2
4
6
8
10
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Time in milli-seconds
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100 Bottles of beer on the wall
…
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Loudness
Resonance (later)
Rotational
Speed
(Turns/second)
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Helmholtz Resonators
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Helmholtz’s Results
Note from Middle C
C
D
E
F
G
A
B
Frequency
264
297
330
352
396
440
496
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We need to try to remember a few
things from High School Physics
 Definitions of acceleration and force.
 Newton’s Laws
 Directional Thingys … vectors.
 Relax … it ain’t that hard.
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Velocity (speed)
 Velocity is the distance you have
traveled divided by the time it took
you to travel the distance.
 If you travel the 20 miles from
Orlando to Lake Buena Vista in 20
minutes, you have traveled
 speed = 20 miles/20 minutes = 1 mi/min
or 60 miles per hour.
 On I-4 you would also get a ticket if you
could drive that fast.
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Acceleration
 Something that is moving at one
speed and then starts moving at a
faster (or slower) speed is said to be
accelerating.
 Your car accelerates when you start
driving it or when you enter a highway
(except for I-4 on occasion).
 You accelerate if you jump out of the
window of a tall building. The
acceleration continues until the splat.
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Definition
 If you start at a speed of vinitial and finish at
a speed of vfinal and it takes a time “t” to
accomplish this, then
change in velocit y (speed)
acceleration 
t (time it took to do it)
a
v final  vinitial
t
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Sir Isaac Newton (1642-1727)
Didn’t really happen but let’s talk
about it.
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Interesting Facts about Sir Issac
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He lived a long and fruitful life.
He died a virgin.
He invented the calculus.
So did Leibnitz.
They fought about it.
Nobody one.
The apple story is probably untrue but if it
IS true, Newton also must have invented
applesauce.
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Newton’s Simplified Three Laws
 An object moving at a constant velocity will
continue moving at that same constant
velocity if NOT acted upon by an external
force.
 If an external force acts on an object it will
accelerate in proportion to the force.
 F=ma
 the mass is the proportionality constant
 When you push on something, that something will
push back on you with the same force that you are
pushing with. (Action = Reaction)
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Who cares about Newton??
 If Newton’s laws were not correct,
there would not be any music!!!
 We will use this stuff in many places
but we will not get to deep into it.
 See Bolemon (reference list) for more
information if you want it.
 For more … any College Physics Text
will suffice.
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Now …. lets look at the
MONOCHORD but with an important
detour.
Dr. Koons will play the monochord for us soon
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The Spring
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Spring Force Equatiom

F=-kx
The “-” sign indicates that
the force and the
displacement are in
opposite directions.
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Springs Oscillate
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Graph
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Important Result for a Spring:
F  kx
1
f 
2
k
m
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So ….
1
f 
2
k
m
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Concept … Tension
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The Musical String
Linitial
T
x
T
The Bigger the angle the
more T points UP!
The distance “x” is the same
sort of thing as the x in F=-kx.
T
Force = F
ANGLE
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The Guitar Strings
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Consider Two Situations
For the same “x” the
restoring force is double
because the angle is
double.
The “mass” is about half
because we only have
half of the string
vibrating.
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So…
F  kx
1
f 
2
k
m
For the same “x” the
restoring force is double
because the angle is
double.
k doubles
The “mass” is about half
because we only have
half of the string
vibrating.
m -> m/2
f doubles!
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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).
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1
Octave
0.5
0.001
0.002
0.003
0.004
0.005
0.001
0.002
0.003
0.004
0.005
0.001
0.002
0.003
0.004
0.005
-0.5
-1
1
0.5
-0.5
-1
1.5
1
0.5
-0.5
-1
-1.5
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Take a wire (String on Guitar ..)
Linitial
Lstretchl
F
Lfinal
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It has been shown that …
 EA 
F 
 Lstretch
 Linitial 
F is the force you pull with
E is a constant for a material
A is the cross - sectional area
 EA 

 Looks like a spring constant " k"
 Linitial 
More about this when we do the string thing.
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HAVE FUN!
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