Physics 1251 Unit 1 Session 5
Simple Harmonic Oscillators
Foolscap Quiz:
What is the wavelength of a C room temperature?
4
(262. Hz) at
Joe College Session 5
1/26/2002 #309
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)

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
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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
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•
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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
-.4 m -.3 m -.2 m -.1 m
F
4 N
3 N
2 N
1 N
.1 m
-2 N
-3 N
-4 N
.2 m

Physics 1251 Unit 1 Session 5
Simple Harmonic Oscillators
F x
0
.1 m
.2 m
.3 m
F
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

Physics 1251 Unit 1 Session 5
Simple Harmonic Oscillators
F x

Physics 1251 Unit 1 Session 5
Simple Harmonic Oscillators
What determines the frequency of the oscillation of a simple harmonic oscillator?
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
80/20 Frequency 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) → f
2
/ f
1
= √(m
1
/m
2
)
= √(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 → f
2
/ f
1
= √(k
2
/k
1
)
= √(50./25.) ) = √(2.0) =1.41

Physics 1251 Unit 1 Session 5
Simple Harmonic Oscillators
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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
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Harmonica
(“Mouth Organ”)

Physics 1251 Unit 1 Session 5
Simple Harmonic Oscillators
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Simple Harmonic Oscillators
Helmholtz Resonator

Physics 1251 Unit 1 Session 5
Simple Harmonic Oscillators
(1821-1894)
Prominent 19 th 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
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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
•
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.