Download Background Experiment 1: Open-Mic and Oscillating Air Molecules

Experiment # 1
Harmonic Motion
PHYS/MUS 102—Spring 2016
Background
Simple and damped harmonic motion characterize the behavior of a great many things in the
universe including ships bobbing up and down in a harbor and, of course, musical instruments and
sound waves! The purpose of this lab is to observe and quantify a system that undergoes simple
and/or damped harmonic motion, and understand its relation to music. Another major objective
of this lab is to familiarize you with standard lab equipment. First, you’ll examine a spring-mass
system. Then you’ll see how this type of macroscopic motion is related to a microscopic motion—air
molecules moving back and forth to transmit sound!
Experiment 1: Open-Mic and Oscillating Air Molecules
Let’s get our feet wet with oscillations and see what this simple harmonic motion business has to do
with describing sound waves! You’ll do a brief sound demo and then briefly analyze the resulting
sound waves.
Figure 1: It’s open-mic time in 102. Image credit: metromusicmakers.com.
For this part of the experiment, you will use a sound recording software package called Audacity. Download a free copy from: http://web.audacityteam.org/download/. It is a wonderful,
easy to use piece of software that is essential for sound analysis.
Connect a Radioshack microphone to your laptop (or desktop computer). Make sure to turn
on the power switch. Note that if you have a laptop, the built-in microphone fidelity is often quite
good (especially on newer devices), but typically the external mic is still better.
A microphone is simply a device which converts air pressure waves (oscillating air molecules!)
into an electrical signal. Audacity records these electrical signals, allowing you to display and
analyze them.
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If you have an instrument, play a steady note. Strings, winds, and percussion all work well. In
your report, be sure to state what instrument you used. You’ll do the same analysis below for a
range of notes you play.
Observe and record what you see on the the audacity display. Be sure to zoom in sufficiently
such that you can see the fine detail of several oscillations/sound waves (about 5–10 oscillations
usually a good number). To analyze what you just played and heard in terms of physical terms:
• What was the damping time τ ?
• What was the measured fundamental frequency of the note you played? Note that the
waveform you record will probably like look perfectly sinusoidal. We’ll see later in the term
these extra “bumps” are actually the various harmonics summing up—they are what make
music sound interesting to our ears. Be sure to measure the fundamental period T1 in order
to determine the fundamental frequency f1 = 1/T1 .
• What was the name of the note you played (for instance concert A is A4 ≈ 440 Hz )? How does
the frequency you measured compare to the “standard” in Western music? See the table here:
http://auditoryneuroscience.com/topics/fundamental-frequencies-notes-western-music.
Repeat the above analysis for at least 3 notes—one bass, one mid, one treble—comparing and
contrasting results for each. Which frequency was higher? Which note was higher on the scale?
Which note damped out the fastest? Importantly, did the results match your expectations (theory)?
These questions are meant to get the thinking machine cranking in terms of connecting oscillations
to music.
Hopefully you are now convinced that simple harmonic motion has something to do with how
sound is produced and transmitted!
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Experiment 2: Mass and spring oscillations
While doing the experiments, the critical questions you should be thinking about are:
1. How do my experimental results compare to theory?
2. Are my data seemingly sensible?
3. What’s different about my actual setup compared to our prettier, idealized physics world?
4. How do observations and results in the lab relate to actual musical principles?
Undamped Oscillation—dependence of frequency on mass and spring constant
The objective of this experiment is to analyze how the frequency of oscillation varies with mass
and spring constant. While our setup doesn’t imminently resemble a musical instrument, there
are some close analogies. Perhaps most obvious is thinking of guitar strings and/or piano strings.
The bass strings are big and thick—they have relatively lots of mass. This makes sense because
bass notes are lower frequency, and to get things to oscillate at lower frequencies, we need larger
masses (or lower spring constants). In terms of spring constants, treble strings are strung at higher
tension. They take more force to displace, thus act like stiffer springs. This makes sense because
treble notes are at high frequency, and higher spring constants make for higher frequencies.
Using the same spring, determine the frequency of oscillation using (at least) 5 different masses.
Of course, carefully record the spring constant and each mass. Make a log-log plot to investigate
the power-law relation between frequency and mass. It is highly recommended to plot as you
go along. More details on the experimental setup are below. You should gather enough data over
a large enough range that you can see a clear trend emerging.
Damped Oscillations—dependence of decay time on mass
The objective of this experiment is to analyze how the damping time constant varies with mass.
Using the same spring, determine the damping constant τ using (at least) 5 different masses. Of
course, carefully record the spring constant and each mass. Then make a log-log plot to investigate
the power-law relation between τ and m.
The Setup
Set up the spring mass system as illustrated in Fig. 2(a). All equipment (spring, mass, stands,
clamps, etc.) is available in the physics lab. Ask the instructor or TA if you need help finding
anything. Do your very best to figure out how the equipment works; that’s one of the essential lab
skills you should start to develop this term.
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(a) Basic Experiment Configuration
(b) Exciting the System
Figure 2: Spring-Mass System Experiment Setup. Figure credits: (a)http://www.batesville.
k12.in.us/physics/phynet/mechanics/newton3/Labs/SpringScale.html; (b) http://www.
saburchill.com/physics/chapters/0015.html
First you need measure and record values for the mass m and spring constant k. The former
should be printed right on the metal itself; or use a scale. The value of k can be determined from
Eqn 1. When the mass is at its equilibrium position and not moving, the spring force must be
equal to the gravitational force (because Sir Isaac newton said so): Fs = Fg = mg, where g = 9.8
m/s2 is the the gravitational constant on Earth.
Next, you will take digital video of the spring in motion. Your mobile device should work great
. There are multiple apps available (e.g., in the iTunes store).
You’ll use Kinovea motion analysis software kinovea.org to quantify the motion of the system1
Kinovea works best if its computer-automated tracking algorithm can easily “lock on” to a
high-contrast region in video. Best suggestion: Place a small, bright piece of tape on the oscillating
mass . Then you ask tell Kinovea to track this bright spot. Lighting conditions may be important
too.
As for using Kinovea, please pay close attention to how the manual is written; the steps need
to be followed in the precise order specified, else unexpected things can happen with the software.
When you are ready, start the oscillations by modestly displacing the mass by a comfortably small
amount, as illustrated in Fig. 2(b). Your goal is to generate data to measure the frequency
and damping (decay) time of oscillation. Be sure to take a long enough video that you can
clearly determine the latter.
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Kinovea is Windows only; sorry Mac users. Mac users might consider using teaming up with a lab partner who
has PC laptop, or use Windows machine in the in the IQ center.
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Lab Reports: What you need turn in
Each student will submit a single document written in a prose style (not bullet point check list).
The focus of this lab is writing up the Results and Discussion sections. Therefore, you do NOT
need to write Introduction or Methods sections. You’ll have 2 sort of independent sections to the
lab report (1 for initial experiment, and another one with 3 sort of inter-related subsections for the
mass and spring experiments) The reporting for each section should be include:
1. One or two representative figures showing raw data that you analyzed. These should be
annotated appropriately to indicate, for example, how you used them to measure the period
of oscillation. Do NOT include a representative figure for each data point/measurement.
2. One exceptionally clear and pretty figure summarizing the main result (e.g. your log-log
plots). This figure should tell a story in its own right. Thus, it should illustrate both your
experimental result and theoretical result.
3. Complementary data table that summarizes, experiment, theory, and % difference.
4. Concise and precise description of the main findings.
5. Concise and precise discussion (typically 1 paragraph to 1 page max) interpreting the results.
Were the results what you expected based on theory? If so, great. If not, why not? Try to
pinpoint specific departures between experiment and theory? Was there something about
your lab setup that wasn’t the same as the pretty, idealized physics model? Were there
shortcomings in theoretical model that couldn’t account for the actual physical behavior?
Again, build your results and discussion around a beautifully done figure—science eye-candy
are really effective for communicating results.
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Good Vibrations: Theory of Simple and Damped Oscillations
Before diving into the experiment, it might be helpful to make sure we have the necessary theory
under our belts. Here we go!
Theory: Hooke’s Law of Springs
Simple harmonic motion occurs whenever a restoring force acts linearly proportional to, and in
the direction opposite to, an object’s displacement. Let’s dissect that statement with the help of
a musical example. Imagine you stretch a guitar string upward. Which direction does it want to
move? It wants to “pull back” downward, of course! And if you push the string downward, it wants
to move back upward. Thus the string always wants to move in a direction opposite of the original
stretch.
Now what if you stretch the guitar string just a little bit vs. a lot. In which scenario does the
guitar string pull back harder with more force? The latter, of course! Equivalently, the further
your finger displaces a guitar string, the more force that is required of you.
Figure 3: Spring-mass system in equilibrium. The spring force is equal and opposite to the gravitational force. Picture credit: answers.com.
These observation can be quantified by Hooke’s Law :
Fs = −k∆y
(1)
This equation simply says that the spring—or spring-like object, such as a reed or vocal cords—
produces a force Fs that acts in the direction opposite of its displacement ∆y, and is proportional
to the spring constant k. The negative sign just makes sure we have a restoring force that pulls
the object in the direction opposite the stretch. Note that k is always a positive quantity. A “stiff”
spring has a high spring constant; it is very difficult to stretch or compress. And vice-versa for
“soft” spring.
Theory: Simple Harmonic Oscillation
The result of all of this physics-talk is a type of motion called simple harmonic oscillation
(SHO). That’s a long-winded, fancy term which really just means an object bobs up and down, or
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moves back and forth about an equilibrium position, as illustrated in Fig. 4.
This motion results because as the mass moves downward, it goes cruising right past its equilibrium position, so the spring starts to pull it back up. As the spring pulls up, the mass goes
cruising right past its equilibrium position again, which compresses the spring, so the mass starts
moving downward, beginning a new cycle all over again.
It’s not just masses on springs that oscillate as such. Air molecules act this way inside of wind
instruments, reeds on oboes and clarinets oscillate as such! And of course we already know strings
on stringed instruments bounce back and forth undergoing SHO (at least for a short amount of
time; see Damped Oscillations below).
Figure 4: Simple harmonic motion of a spring-mass system. Figure credit: http://hyperphysics.
phy-astr.gsu.edu/hbase/soushm.html.
The vertical position y(t) vs time t is mathematically described as a sin wave:
y(t) = A sin(2πf t + φ)
(2)
The musically important variables in Eqn 2 are the amplitude A and the frequency of (undamped)
oscillation f . The period is just the reciprocal of the frequency, T = 1/f . (The phase angle φ just
lets us start measuring an arbitrary point in the oscillation, and will not concern us here.)
It turns out that the undamped frequency of oscillation for a mass m bobbing on a spring with
spring constant k is given by:
r
1
k
f=
(3)
2π m
Damped Oscillations
For our final piece of theory, we’ll consider the decay time of the oscillations 2 . Damping happens
because some of the some of the kinetic energy involved with mechanical oscillations is lost with
each cycle in other forms of energy, such as heat3 .
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As Axel Rose of Guns’n’Roses sang “Nothing lasts forever, not even November Rain. For the uninitiated, see
and listen here: https://www.youtube.com/watch?v=8SbUC-UaAxE
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Guitars, pianos, oboes, etc. don’t spontaneously combust. It would take a lot more energy dissipated in a much
shorter amount of time to generate some smoke or fire
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(a) Idealized decay. The decay time τ is about 3.05 s, and occurs where y(t) ≈ 0.37ymax . In this
illustration, ymax = 1 and the dotted red line is drawn at about 0.37.
(b) Real world guitar string
Figure 5: Idealized damping (a) versus real damping of a plucked guitar string (b). Figure credits
for (b): http://hyperphysics.phy-astr.gsu.edu/hbase/sound/timbre.html.
An idealized version of damped oscillation is shown in Fig. 5(a). For instance, pluck a guitar
string, and listen for the sound (volume) decreasing over time. If you watched the very middle
of the string (of a reeeaaaaaallly low bass note) and marked its position vs. time, it might look
something sort of like what you see in Fig. 5(a). A real world plucked guitar string example is
shown in Fig. 5(b).
In math terms, we write damped oscillations as follows:
y(t) = A e|−t/τ
t + φ)
{z } sin(2πf
|
{zd
}
damping
(4)
oscillations
The musically important parameters in Eqn 4 are the amplitude A, the damped frequency of
oscillation fd and the damping time constant (or decay time) τ . The amplitude is related to
the initial displacement (or initial volume of a note sounding).
The damped frequency of oscillation fd is similar to the undamped frequency f , but damping
usually makes objects oscillate more slowly compared to no damping. It is always true that is
fd < f . However, In most musical contexts, the damping is so slight that there is only a small
effect, such the damped and undamped frequencies are approximately equal:
fd ≈ f
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.
The damping time constant τ tells us how long it takes for the oscillation to die off (aka decay).
A large value for τ means it takes a really long time for the oscillations to die off. Conversely, a
small value indicates that the oscillations damp very quickly. A quick bit of intuition based on a
piano. If you strike a bass note, the sound lingers for quite some time, maybe a few seconds (larger
τ ). By contrast, if you play a note far into the treble, you’ll hear a high-pitched plink and then
the sound is gone—the oscillations decayed very quickly (smaller value for τ ). Vibrational theory
tells us that we expect the decay time to be proportional to the mass of the object oscillating m.
The bigger the mass, the longer it takes to damp down the oscillations—just like our heavy bass
strings!. In summary 4 ,
τ ∝m
(5)
So how do we measure τ experimentally? The typical way to quantify how long this decay
process requires is too look for where the signal drops to e−1 ≈ 0.37 = 37% of the maximum
amplitude A. Often times you have to imagine drawing a decaying exponential “envelope” around
the oscillating part, then interpolate where the amplitude y(t = τ ) = Ae−1 is reached.
One last important note (pun intended, ha!). Comparing the ideal and real-world damped
oscillations in Figs. 5(a) and 5(b), you may notice that the guitar string oscillations to “turn on.”
This is called the attack time. These attack transients are very important for sound quality, but
we’ll save discussion them for another day.
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We also expect damping to depend on a material properties. For example, rubber bands are faster at damping
things than metal. The time constant τ will be inversely proportional to the material damping, such materials that
damp a lot have short time constants.
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