Transcript Strings and Things

Springs and Things
Science of Music
Week 2 notes (no pun intended) …
well, maybe.
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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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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 and
we have defined it previously.
 When you push on something, that
something will push back on you with the
same force that you are pushing with. (Action
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= Reaction)/
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 get back to the
MONOCHORD via an important
detour.
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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.
 Somewhat explains the monochord results from
last session when we built the scale.
 Let’s look some more at the consonant
intervals that Dr. Koons examined with the
monochord as he built the major scale.
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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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1000 Hz.+
1273 Hz.
2
1
(a)
0.001
0.002
0.003
0.004
0.005
-1
(b)
-2
2
1
0.005
0.01
0.015
0.02
0.025
0.03
-1
-2
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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 cress - sectional area
 EA 

 Looks like a spring constant " k"
 Linitial 
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