Transcript Physics 1251 The Science and Technology of Musical Sound

Physics 1251
The Science and Technology
of Musical Sound
Unit 1
Session 5
Simple Harmonic Oscillators
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Foolscap Quiz:
What is the wavelength of a C4 (262. Hz) at
room temperature?
Joe College Session 5
1/26/2002
Name, Date, Session 5
Seat #
Frequency = 262. Hz; velocity = 343. m/s
v = f ⋅ λ= (262. Hz) ⋅ λ = 343. m/s
λ = (343. m/s) / (262. Hz) = 1.3091603 m
λ= 1.31 m (3 significant figures)
#309
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Sound is a pressure/displacement wave that
propagates in a material medium.
But what causes sound to begin with?
What is the sound of one hand clapping?
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Oscillating Hand Demonstration
☞♫
Compression
Rarefaction
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
An oscillation of a body causes a sound.
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
1′ Lecture:
•
•
A Simple Harmonic Oscillator is a device
that oscillates at one frequency, determined
by the spring constant k and the mass m of
the system.
The natural frequency of an SHO is related
to these quantities by the equation:
f = 1/(2π) ‧ √ (k/m)
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Existential Physics Questions:
• What happens when you stretch a “spring”?
• What happens if you stretch the “spring” twice
as much? Or three times as much?
• What is a “spring”?
• What has this got to with acoustics?
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
F
4N
3N
2N
1N
-.4 m -.3 m -.2 m -.1 m
.1 m .2 m .3 m .4 m
-1 N
x
-2 N
-3 N
-4 N
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
F
x
F
0
.1 m
.2 m
.3 m
0
-1 N
-2 N
-3 N
x
F/x = - 10 N/m
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Hooke’s Law
When you stretch or compress a “spring,” the
force (F) produced is proportional to the
displacement (x) and in the direction to
restore the system (-) to the original position.
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Spring Constant k
The restoring force produced by a spring is
proportional to the negative of its extension (or
compression).
F = - k ‧ x (Hooke’s Law)
F : restoring force
k: spring constant
x: extension, the “stretch or squeeze (if negative).”
What is the restoring force of a spring with k =25. N/m
when it is stretch by 10 cm?
F = - (25. N/m )(0.10 m) = -2.5 N
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Mass on a Spring
Spring ————————→
Mass ——————————→
-k‧x
x
Force
F=-k‧x
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
F
x
F/x = - 12.5
10 N/m
25.
N/m
N/m= -k
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
What determines the frequency of the
oscillation of a simple harmonic
oscillator?
x
The stronger the force (k), the more rapid is the
oscillation.
The greater the mass (m) the slower is the
oscillation.
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
80/20Frequency
of Oscillation
of Mass on a Spring
f = 1/(2π)√(k/m)
f = 0.1592 ‧ √ (k/m)
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
If mass = 0.500 kg and k = 25.0 N/m, what will
be the frequency of oscillation?
What will be the period?
f = 1/(2π) ‧ √ (k/m)
=1/(2π) √ [(25.0 N/m)/(0.500 kg)]
= 0.1592 √ [50.0] = 1.13 Hz
P = 1/ f = 1/ (1.13 Hz) = 0.885 sec
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
If we increase the mass from 0.5 kg to 1.0 kg
what will happen to the frequency?
Will it increase, decrease or stay the same?
If it changes, by how much will it change?
f = 1/(2π) ‧ √ (k/m)
f ∝ √ (1/m) → f2 / f1 = √(m1 /m2 )
= √(0.5/1.0) ) = √(0.50) =0.71
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
If we double the springs what will happen to the
frequency?
Will it increase, decrease or stay the same?
If it changes, by how much will it change?
f = 1/(2π)√(k/m)
f ∝ √k → f2 / f1 = √(k2 /k1 )
= √(50./25.) ) = √(2.0) =1.41
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Simple Harmonic Oscillators
• Tuning fork
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Simple Harmonic Oscillators
• Tuning Fork
(Bulova Accutron)
Tuning Fork
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Simple Harmonic Oscillators
“Mouth Harp”
Oscillator
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Simple Harmonic Oscillators
• Kalimba
(Finger Piano)
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Simple Harmonic Oscillators
• Harmonica
(“Mouth Organ”)
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Simple Harmonic Oscillators
• Helmholtz Resonator
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Hermann von Helmholtz
(1821-1894)
Prominent 19th century
physicist and mathematician.
Author of Perception of Tone,
highly influential treatise in
musical acoustics.
© American Institute of Physics,
used by permission
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
A Helmholtz Resonator is a simple harmonic
oscillator that uses air in a narrow neck as a
mass and air trapped in a volume as a spring.
Air
Mass
Air
Spring
Physics 1251
Unit 1 Session 5
Simple Harmonic oscillators
Examples of Helmholtz Resonators:
 Bottle
 Acoustic Tile
 Cinder Block
 Ocarina
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Helmholtz Resonator
•
Ocarina
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Summary:
•
A Simple Harmonic Oscillator is a device
that oscillates at one frequency, determined
by the spring constant k and the mass m of
the system.
•
The natural frequency of an SHO is related
to these quantities by the equation:
f = 1/(2π) ‧ √ (k/m)
Physics 1251
Unit 1 Session 5
Simple Harmonic Oscillators
Summary:
•
A Helmholtz resonator is a Simple
Harmonic Oscillator comprising a
trapped volume of air that acts as a
spring and a narrow neck that acts as
a mass.