Experiment 4 Lab Manual Phys 115L Spring 2018

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4.

Coupled oscillations

A.

Objectives

B.



Observe the vibrations of a coupled oscillator consisting of two masses hanging on
springs. Measure the frequencies of the two normal modes of this system, and see if you
can verify the theoretical result for these frequencies.



Individually excite each of the two normal modes of two masses hanging on springs.
Observe the relative displacements of the two masses for each mode. See if you can
verify the theoretical results for these relative displacements.

Equipment required

1. 80-20 frame with eyebolts for suspending masses
2. Three ringstands, each consisting of a ½" diameter, 24" tall stainless steel post on a 5" x 8"
stainless steel base
3. White cardboard sheet to set behind frame
4. Ball and spring assembly consisting of two stainless steel balls with tapped 8-32 holes, two
extension springs, three 8-32 eyebolts, and a 1/4" diameter magnet in a ½" diameter holder
5. Stopwatch
6. Drive coil assembly and two patch cables with banana plug ends
7. Machine vision camera with varifocal lens on ½" post and USB 3.0 cable
8. Incandescent light bulb with power supply on ½" post
9. 4" square plate on ½" post with taped-on cardboard strip
10. Five ½" right-angle post clamps and one additional ½" post
11. Computer data acquisition system including Pasco 850 interface and Pasco Capstone,
XiCamTool, and Tracker software

C.

Introduction

Often in physics, we encounter multiple oscillators with motions that are coupled to each other.
Many mechanical structures exhibit them. Even seemingly isolated oscillators, such as two
identical pendulum clocks set near each other in the same room, can show effects of coupling.
Atoms in molecules sometimes vibrate as coupled oscillators. And, there are many examples of
coupled oscillators in electronic circuits.

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1.

Two masses hanging on springs

In this lab, we'll study two masses hanging on springs, as illustrated in Figure 4.1. Specifically, a
mass m1 hangs from a support on a spring of spring constant k1 , and a mass m2 hangs from the
first mass on a spring of spring constant k 2 . If we allow the masses to move undisturbed for
some time, eventually they'll settle down into their static equilibrium positions shown in Figure
4.1(a). In this configuration, each spring exerts just enough force to cancel the downward
gravitational force on the objects below.

Figure 4.1. Two masses hanging on springs. (a) Configuration of the masses when hanging in static
equilibrium. (b) Configuration of the masses when displaced from equilibrium.

Now suppose that the masses are displaced from their equilibrium positions as shown in Figure
4.1(b). Let y1 be the displacement of mass m1 and y2 the displacement of mass m2 , both
measured relative to the equilibrium positions. With this definition of displacement, the
equations of motion of the two masses are
d 2 y1
m1 2  k1 y1  k2 ( y1  y2 )
dt
m2

(4.1)

d 2 y2
 k2 ( y2  y1 )
dt 2

(4.2)

We can rewrite these as
d 2 y1
m1 2   k1  k2  y1  k2 y2
dt
m2

(4.3)

d 2 y2
 k2 y2  k2 y1
dt 2

(4.4)

If the right hand side of each of the equations (4.3) and (4.4) were zero, then each equation
would be that of a single harmonic oscillator – one with mass m1 and spring constant k1  k2 ,
and another with mass m2 and spring constant k 2 . However on the right-hand side of equation
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(4.3) for the first oscillator there is a term proportional to the displacement of the second
oscillator. Similarly on the right-hand side of equation (4.4) for the second oscillator, there is a
term proportional to the displacement of the first oscillator. These terms couple the motion of the
first oscillator to the second, and vice-versa. In other words, we have a system of two coupled
oscillators.
In order to solve for the motion of the two oscillators, we'll guess that there is a solution in which
both oscillators have a sinusoidal oscillation at the same frequency  i :

y1  A1i cos i t  1i 

(4.5)

y2  A2i cos i t  2i 

(4.6)

In these equations, we're anticipating that there might be more than one solution for the
frequency, so we label the different solutions with an index i. Substituting these expressions into
the equations of motion (4.3) and (4.4), we find a solution only if 1i  2i  i , and

 2 k1  k2 
k2
 i 
 A1i   A2i
m1 
m1


(4.7)

 2 k2 
k2
A1i
 i   A2i  
m
m

2 
2

(4.8)

Equations (4.7) and (4.8) can each be solved for the ratio A1i / A2i ; if these equations are both
satisfied the ratio must be the same, so that
 2 k2 
k2
 i 

m2 
A1i
m1



k2
A2i
 2 k1  k2 
 i 

m2
m1 


(4.9)

From the second equality in equation (4.9) we deduce that (4.5) and (4.6) are solutions of (4.3)
and (4.4) if and only if
2

 k2 k1  k2 
k2 (k1  k2 )  k2 2
1  k2 k1  k2 
i   



 
 4
m1 
m1 
m1m2
2  m2
 m2

(4.10)

or
2

 k2 k1  k2 
k2 (k1  k2 )  k2 2
1  k2 k1  k2 
i   



 
 4
m1 
m1 
m1m2
2  m2
 m2

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(4.11)

So for this coupled oscillator, there are solutions of the form (4.5) and (4.6) for two and only two
specific frequencies, which we label as i , with i   and i  . Oscillations of this form, in
which all components of the system vibrate at the same frequency, are called normal modes.
The two normal modes differ not only in frequency, but in the displacements A1 and A2 . It
turns out that the ratio A1 / A2  is always positive, so for the "minus" mode the two masses are
always moving in the same direction, as illustrated in Figure 4.2(b). On the other hand, for the
"plus" mode the ratio A1 / A2  is always negative, so the masses are always moving in the
opposite direction, as illustrated in Figure 4.2(c).

Figure 4.2. (a) Illustration of the masses when in equilibrium and at rest. (b) Displaced positions of the
masses when oscillating only in the "minus" frequency mode, at (i) the time when the displacement of
mass m1 is a maximum, and (ii) one-half cycle of oscillation later. (c) Displaced positions of the masses
when oscillating only in the "plus" frequency mode, at (i) the time when the displacement of mass m1 is a
maximum, and (ii) one-half cycle of oscillation later.

In the minus mode, the two masses vibrate up and down in the same direction. There is also a
vibration in the distance between two masses, but the vibration of the center of mass position is
more pronounced. The plus mode consists of a "stretching mode" vibration, in which the two
masses vibrate in opposite directions. There is also a vibration of the center-of-mass, but the
vibration in the distance between the masses is more pronounced.
From Figure 4.2, it's easy to see why the "plus" mode has the higher frequency. In this mode,
spring k 2 is stretched to a much greater degree. Therefore the restoring forces are larger and the
frequency is higher in the “plus mode” than in the "minus" mode.
Since our system is linear, any superposition of solutions is a solution. This allows us to write
down the general solution to the equations of motion (4.3) and (4.4) as a superposition of the two
normal mode vibrations:
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y1 (t )  A1 cos t     A1 cos t   

(4.12)

y2 (t )  A2 cos t     A2 cos t   

(4.13)

As expected, this general solution to a system of two coupled second order differential equations
contains four arbitrary constants: A1 ,  , A1 , and  . The other two parameters, A2 and A2 , are
not arbitrary constants, because once A1 and A1 are specified, A2  and A2  are given by
equation (4.9) i.e.

A2   A1

A2   A1

k2
m2

(4.14)

 2 k2 
  

m2 

k2
m2

(4.15)

 2 k2 
  

m2 


If the coupling were absent, we'd have a system with just two frequencies – a frequency 1 for
mass 1 and a frequency 2 for mass 2. With the coupling, we find that the system still oscillates
with only two frequencies, but these frequencies are   and  . Both are different from the
uncoupled frequencies 1 and 2 .
If we displace just one of the masses and let go, we'll generally find that both A1 and A2 are
different from zero. The resulting oscillation will not be a pure sinusoid of the displaced mass.
Instead, both masses will oscillate in superpositions (4.12) and (4.13) of the two normal modes.
Each mass will oscillate at both frequencies.
It is possible to excite just one of the two normal modes. We could do this by exciting the
oscillation with a drive at just one of the frequencies   . Or, we could set the initial
displacements of the two masses to match the amplitude ratio (4.14). In either case, we would
produce an oscillation with A1  0. Then the displacements would be given by
y1  A1 cos t   

(4.16)

y2  A2 cos t   

(4.17)

and the masses would oscillate at just the one frequency  . This is referred to as oscillation in a
single mode. It is still a coupled oscillation: both masses oscillate at that frequency, with a

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definite amplitude ratio given by equation (4.14). Of course, it would also be possible to excite
an oscillation with A1  0, in which both masses oscillate only at the frequency  .

D.

Experimental Procedure

The apparatus is illustrated in Figure 4.3. It contains a frame made of 80-20. This is an extruded
aluminum framing material that can be ordered with standardized brackets, nuts, and bolts, and is
commonly used to make frames and supports for scientific apparatus. Your frame includes nine
¼-20 eyebolts that can be used to support springs. You can easily loosen the eyebolts by turning
them slightly counterclockwise, move them to a desired position, and tightening them back
down. For this experiment you'll only use the top center eyebolt as shown in Figure 4.3(b). This
will support the system of two masses hanging on springs that we analyzed in the introduction.

Figure 4.3. Experimental apparatus (a) "Ringstand" with machine vision camera, varifocal lens, and light
bulb. (b) Additional components including 80-20 frame, masses and springs, coil, and cardboard flap.

In this experiment we'll measure the displacement of the masses with video analysis. We'll take
data with a machine vision camera. This type of camera is generally capable of high frame rates
of the order of a few hundred frames per second, comes with support for detailed control of the
camera parameters, and produces video that is compatible with sophisticated video analysis
software. These are most often used for industrial applications to monitor manufacturing
processes, but they're sometimes also used in scientific applications such as high-speed video
microscopy. Your camera is a Ximea model MQ013MG-ON, which is a black-and-white 1280 x
1024 pixel camera that can record up to 170 frames per second.
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Your camera is equipped with a varifocal lens. This is similar to a zoom lens, in that the focal
length (i.e. the magnification) can be set within some range of values. Both a zoom and a
varifocal lens have two focusing adjustments. The difference is that one of the adjustments on a
zoom lens adjusts the focal length only – so that the image remains in focus as the focal length is
adjusted. On a varifocal lens the two adjustments are not orthogonal, so you always have to
move both adjustments if you want to change the focal length. The camera and lens will be
supported on a tall "ring-stand" consisting of a tall ½" post on a 5"x 7" base.
1.

Set up your masses and springs

To begin the experiment, take three ring stands, one 80-20 frame, and one of the large white
cardboard sheets. Set the frame on your lab bench with the square part of the frame towards the
back of the bench. Set two of the ringstands behind the frame, with their bases tucked under the
square frame. Set the third ringstand in the position labeled (*) in Figure 4.3(b). Finally, place
the white cardboard sheet at the very back, behind the ringstands. The purpose of this sheet is to
provide a uniform background for the video recording.
Next, select one of the mass-and-spring sets. Each set has a number. Take a note of this number
– write it down in your lab book, and include it in your report. This is important because we want
to see if you got the correct results for your particular mass-and-spring set.
The masses consist of stainless steel balls of two different sizes from the set 1-1/2", 1-1/4", 11/8", or 1" diameter. Use the larger ball for mass m1 and the smaller ball for mass m2 . Use the
stronger spring for spring k1 and the weaker spring for spring k 2 . Make sure that two 8-32
eyebolts are screwed into opposite sides of your ball m1 . Make sure that one 8-32 eyebolt is
screwed into your ball m2 , and a magnet into the opposite side of your ball m2 . The balls,
springs, and magnet should be assembled as shown in Figure 4.3(b). You won't use the magnet in
the first part of the experiment, but the magnet should be placed on the ball anyway, because it
will contribute to the mass m2 , and you want that mass to remain constant throughout the
experiment.
For each set, we have selected springs and masses that should work well together. The
requirements are:
(i) When mass m1 hangs by itself from spring k1 , the extension of spring k1 should be greater
than 1 cm and less than 5 cm.
(ii) When mass m2 hangs by itself from spring k 2 , the extension of spring k 2 should be greater
than 1 cm and less than 5 cm.
(iii) When mass m1 , spring k 2 and mass m2 hangs from spring k1 , the extension of spring k1
should be greater than 2 cm and less than 8 cm. Also, when this combination hangs from the
central ¼-20 eyebolt as shown in Figure 4.3(b), you should have at least 3" of clearance between
75

the magnet and the bottom of the frame. Consult with your instructor if you notice these
conditions aren’t satisfied.
2.

Measure the masses and spring constants

Next, measure each of your two masses m1 and m2 on the electronic scale at the front of the lab.
To keep the masses from rolling around, you can place them in a pyrex dish. (If you turn the
scale on with just the dish, it will zero out the mass of the dish.) When you measure the masses,
make sure that the 8-32 eyebolts and magnet are installed onto the masses, because the mass of
the eyebolts and magnet is a part of the mass that oscillates. (The springs also contribute a little
to the oscillating mass, but this is a tricky thing to get right for coupled oscillations, so just leave
the springs off for your mass measurements, and ignore the effect of the spring masses for this
lab.)
Next, we want to determine the spring constants k1 and k 2 . You might think we could use
Hooke's law mg  kx for this, where x is the extension, but that turns out to be a poor strategy.
One problem is that x is difficult to measure accurately. Another is that it's not true that mg  kx
for these springs. The reason is that they are under a non-zero compression force F0 when
unloaded. It takes a force greater than F0 to get a non-zero extension, so there is an offset in
Hooke's law for these springs. The correct force law is mg  F0  kx.
We'll use a different strategy: to measure the frequency  of the oscillation of each mass on its
own spring. This works much better because the frequency can be measured quite accurately,
and then we can use the result k  m 2 .
To get started with this, hang spring k1 from the central eyebolt on the frame, and hang mass m1
from spring k1. Make sure that both eyebolts are installed on m1, because we want this mass to
remain constant throughout the experiment. Now, k1 and m1 constitute a simple harmonic
oscillator. (Leave k 2 and m2 set aside.)
Then, pull the mass m1 down slightly and let go, so that you set it into oscillation. Using your
stopwatch, measure the time it takes to complete 50 periods of oscillation, and from that
measurement determine the oscillation frequency. Remember that   2 / T , where T is the
period, since there are 2 radians in one cycle. You should be able to measure  to better than
1% accuracy in this way. (Say, a relative accuracy better than ½ period/50 periods.)
Repeat this for mass m2 suspended from the center support by spring k 2 . (Mass m1 and spring
k1 should be set aside while you do this.) Also make sure that the magnet is attached to mass
m2 , since you want that mass to be constant throughout the experiment.

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At this point, you should have good measurements for m1 , k1 , m2 , and k2 .
3.

Learn to record and track a video of an oscillating mass, and re-measure 1.

Next, remove mass m2 and spring k 2 from the frame, and hang mass m1 from spring k1 again.
Then, attach the post-mounted light bulb to the ringstand with a right angle clamp as shown in
Figure 4.3(a). Plug the light bulb supply into an electrical outlet, and make sure the bulb
illuminates. Next, set up your camera and lens on the ringstand as shown in the figure. You
should use two right angle clamps and posts, so that the camera is oriented as shown. Then
position the ringstand so that the camera faces your masses. Next, connect the micro-B end of
the USB 3.0 cable into the camera, and the A end of the cable into the back of the Mac computer.
Please be gentle when making the connection to the camera so that you don't stress or
damage the connector.
Next, log in to the computer, and start the program "XiCamTool", which you can do via its
orange-colored icon along the bottom row of your screen. This is the program to operate and take
video from the camera. When the program comes up, it should automatically detect the camera.
If this has happened, you should see an orange arrow in the upper left corner of the window, and
you should see the camera model "MQ013MG-ON" indicated in the right hand column. If you
do not see these, try unplugging the USB cable and plugging it back in. If that doesn't work,
consult with your instructor. Next, press the orange arrow. This will take in a continuous stream
of images from the camera and display it.
The camera lens has three adjustments. The front two adjust the lens focal parameters (i.e. focal
length (magnification) and focus). The rear one adjusts the aperture, (i.e the amount of light
collected by the camera). Note that the little levers screw in and out in order to secure or loosen
the lens adjustments. Adjust the vertical and horizontal position of the camera and the two focus
adjustments until you see a clear image of your mass. For tracking, bigger (more magnification)
is not necessarily better. The reason is that the more pixels the tracker has to deal with, the
slower it goes. For this reason, you may want to select a relatively low magnification.
Next, uncheck the "Auto exposure" box under the "Settings" tab. Then, select “Frame Rate” from
the drop down menu next to “Control FPS” under the “Performance” tab. At this point, you can
independently adjust the frame rate using the "Control FPS" slider, and the exposure time using
the "Exposure" slider. (This is subject to the constraint that the exposure time must be less than
the inverse of the frame rate, which is enforced automatically.) Another slider allows you to
adjust the electronic gain of the camera.
Below, you'll be using automatic video tracking. One of the most difficult parts of this
experiment is to get the automatic tracking working well. The most effective strategy is to track a
small feature that has a consistent shape from frame to frame, and has very high contrast.
Tracking the entire ball has the problem that the tracking tends to be very slow, due to the
relatively large number of pixels involved. We have come up with the following strategy which
77

works reasonably well. We've chosen masses that are polished, reflective spheres, and we've
placed a small light bulb in front of them. You should see the reflection of this bulb in your video
as a compact, bright spot near the center of your ball. This spot is nearly ideal for the video
tracking. And, since the ball is spherical, the spot doesn't change its appearance with rotations of
the ball.
You now have manual control over the brightness of the image, using three controls: (1) the
camera aperture, (2) the exposure time, and (3) the gain of the camera (this controls an electronic
amplifier that boosts the signal from each pixel). Keep the exposure time set somewhere in the
range from 2 ms to 5 ms. We don’t want longer exposures, because that can result in blurring of
the bright spot due to the ball’s motion. On the other hand, exposures less than 2 ms result in a
loss of brightness without any real benefit.
After setting the exposure time, adjust the camera aperture and gain in some combination that
produces a clear image of the ball that has good contrast. You especially want to see good
contrast between the bright spot and the surrounding ball surface.
When you record video, you would like the frame rate to be fast enough to record something like
10 frames per cycle of oscillation or more. On the other hand, you'd like to follow a number of
cycles of oscillation without accumulating too many frames, because the tracker may take a long
time to complete tracking of a large number of frames. For this reason, it is best to select a frame
rate that is neither large nor small compared to 10 frames per cycle of oscillation. Your
oscillation frequency is likely to be between 1 Hz and 4 Hz for the single masses, so 40 fps
would be a maximum frame rate at this point. For now, set the frame rate to about 30 fps using
the Control FPS slider. Later, you can adjust this if you want more or fewer frames per second.
At this point, you should be looking at mass m1 hanging from spring k1 , and mass m2 and
spring k 2 should be set aside. Return to this condition if you aren’t. Make sure none of the
tapped holes on the sides of the ball faces the camera. Then, record and save a video of the
oscillation of the mass. To do this, click on the "Record Loop" icon along the top row of the
window. Set the number of frames to 200 to start with. Pull down slightly on your mass and let
go, so as to set it into oscillation. Click "Start". When the circle on the left has completed, check
"Stop." This will record a video with 200 frames into memory. To save this video, select the
diskette icon. Under "Files of type", select “.MOV". Type in a descriptive filename, then click
"Save." Note where the file is saved (probably in your “Documents” folder).
Throughout this experiment, during any file save or file conversion, wait until the save or
conversion completes before moving on. If you don’t, you may end up with a truncated video
file.
Next, open the folder with your video, and keep it open throughout the experiment. This is the
easiest way to work with these files.

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Next, start the program "Tracker" by clicking on its light blue, pink, and purple icon. Then, go to
Edit → Preferences → Video. Under “Video Engine”, select “Xuggle,” and click “Save.” (You
have to make this selection if you want Tracker to be able to work well with your .MOV format
movies.)
Drag and drop the .MOV file you just produced into the open Tracker window. If you get a
warning about frame durations, ignore it. You can now view the video with the video player
controls at the bottom of the window. Check that the video plays properly in Tracker. Then
return your movie to its first frame in the player window before proceeding to the next step.
We'll next use Tracker's Autotracker feature. To do this, click on "Create" and select "Point
Mass". Then press and hold cntrl-shift. You will see a cross-hair pattern on the screen. While
holding down cntrl-shift, position the exact center of the cross hair on the bright spot on your ball
and click. This will open up the Autotracker window. (If cntrl-shift doesn’t work, you can also
open Autotracker from Tracker’s menus.) Move this window so you can see both it and your
ball. On your ball, you'll see a small red circle in the middle of a red square. The small red circle
selects the video feature that is to be tracked. The red square selects the area of the video that is
to be searched for this feature. You can see the appearance of the tracked feature in the
"Template" image in the Autotracker window. For each successive frame, the Autotracker
searches for an image that matches this template. To save time, it searches only within the square
search window, not in the entire video frame.
If Tracker finds the feature, then it records the x (horizontal) and y (vertical) coordinate of the
feature as the location of the "point mass" that you created. If it can't find the feature, it stops and
asks you what to do. For this lab, we'll try to avoid this by producing videos that reliably
autotrack.
You can enlarge or shrink the video template area and search area by clicking and dragging on
them. There is a compromise to be struck. Increasing one or both areas may (or may not) result
in more reliable autotracking, but at the cost of increased tracking time. The default for the area
tracked might seem too small, but often it works just fine. That is because Tracker moves this
area as needed to keep up with the moving object.
The main thing you want is a highly distinct bright spot, surrounded by some dark area on all
sides, in the Template window. As long as you have this, the tracking will probably work well.
Adjust the Autotracker until you do have this condition. Then press "Search". Hopefully, the
program will then track through your entire video. If it doesn't, one or more of the following
steps might help:
* Check that you have a 2 to 5 ms exposure time, so that you don't get smearing out of the spot
due to motion.

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* Make sure that the tapped holes in the sides of the ball are off to the side of the camera image.
If a hole is near the front of the ball, it may interfere with the reflected light bulb spot.
* Increase or decrease the template area or search area.
* Block out or turn off lights other than your light bulb that give stray reflections. (For instance,
try placing a cardboard sheet above your camera and oscillators.)
* Take a video with less brightness, in order to improve contrast. Or, try a video with more
brightness, so as to saturate the bright spot (and reduce variation in its appearance).
* Try the other side of the ball, or switch your ball for one of the same size with a better polish.
Again, your goal is to have a template consisting of a bright spot in the middle of a dark area,
that is reasonably consistent throughout the video. If you have difficulty in getting good tracking
after some effort, consult with your instructor.
At this point, you will be presented with a graph of x vs. time t. This is not what you want since
x is the horizontal coordinate. Click on the x, and you'll be presented with a series of choices for
that axis of the graph. Select "y". Now you have a graph of the oscillation in displacement y vs
time t. It should look like an accurate sinusoid. You also have a table of data values for x, y, and
t. At this point you can maximize the size of the graph by clicking on the up arrow, or similarly
you can return the graph to its smaller size.
Next, right-click on the graph and select "Analyze." Then, click on "Analyze" and select "Fourier
Spectrum." You should see a Fourier Power spectrum with one peak at non-zero frequency, both
in the graph and in the data. To make it clear in the graph, you can reset the vertical scale by
clicking on the top entry on the vertical scale, and similarly for the horizontal scale. You'll
probably want to reduce the maximum plotted vertical value because there is usually a large but
uninteresting zero-frequency peak associated with the offset of your sine wave. You'll probably
also want to reduce the maximum plotted frequency.
Check that the frequency of the peak in the Fourier spectrum agrees with the value you measured
earlier for 1. If it doesn’t, figure out where you’ve gone wrong. Remember that frequency in
radians per second is equal to 2 times frequency in Hz.
4.

Measure the frequencies of the coupled oscillation.

Next, set up the double mass and spring oscillator illustrated in Figure 4.3. View the double
oscillator with XiCamTool. Adjust the camera field of view and light bulb so that you can see
both balls clearly, with a bright spot in the middle of each. Then record a video of the coupled
oscillation using the method discussed in the last section. For this video, start with 1000 frames
and a frame rate of about 50 frames per second. (You can adjust either of these in a second video
if you want.) The reason for the increased frame rate is that the highest mode frequency will be
larger than the frequencies you measured earlier. The reason to have a video with more frames is
that you will get higher frequency resolution from the Fourier Transform of the ball’s track
80

position data. A good method to get the coupled oscillation going is to displace m1 upward
slightly while holding m2 stationary, and then let go of both.
Then, save the video as discussed above, and load it into Tracker. Make a track of one of the
two balls, and display y vs. t for that ball. (The tracking might take a few minutes.) You’re now
looking at coupled oscillation of the two masses, that is a superposition of both of its modes,
mathematically described by equations (4.12) and (4.13). Take note of the qualitative appearance
of this data. Does it look like a sine wave? Or completely chaotic? Or does it differ from a sine
wave, yet still have come regularities?
Next, use the Fourier Transform tool to calculate and display the Fourier power spectrum of y vs.
t. You should see a power spectrum with two peaks at non-zero frequency. Once you can see
both peaks, determine their centroids with good accuracy from the actual data values, which are
tabulated as "frequency" and "power". (Here, "power" means the value of the Fourier power
spectrum.) One way to do this is to copy a set of data points for "frequency" and "power"
extending from five to ten data points below a peak to five to ten points above the peak from
Tracker’s table of data values, and to paste them into an Excel spreadsheet. (Use command-c and
command-v to cut and paste on a Mac.) Once you have the data in Excel, you should be able to
calculate the centroid (weighted average frequency) of the peak relatively easily. Use this
centroid for your measured mode frequency.
Once you've determined the frequency of these two modes, check whether they agree with what
you'd predict based on your measured values of m1, m2 , k1, and k2 , using equations (4.10) and
(4.11).
5.
Excite one mode at a time, observe the single-mode oscillations, and measure
the relative displacements for single mode oscillation
Next, observe single mode vibrations of your coupled oscillator. To do this, connect your coil to
Output 1 of the Pasco 850 interface. Make sure that your current passes through both the 5 
resistor and the coil in series. Position the coil just below the magnet. This seems to work best
with a gap of perhaps 4 to 8 mm between the coil and the magnet. Using Pasco Capstone, set the
Pasco 850 to 1 V voltage amplitude, and frequency of about 0.5 Hz. You should be able to see a
small mechanical response of your coupled oscillator to this current. Once you’re sure it’s
working, reduce the voltage amplitude to 0.3 V. Next, use the Capstone controls to tune the
frequency of the oscillator near your measured frequency  . Don't forget that you probably got

 in units of radians per second, whereas the Capstone frequency is specified in Hz, which
differs by a factor of 1/ 2 . See if you can excite just one mode of vibration, so that both masses
vibrate only with frequency  . When you do this, you should press the cardboard strip lightly
against one mass. The friction of the ball against the cardboard will reduce the Q of the
oscillation. Without this, the decay time of the oscillation can easily be several minutes. Thus,
81

when you make a change to the system in the absence of damping, it can take several minutes for
the system to "forget" its previous condition of oscillation, and come into equilibrium for the
new condition. In other words, more damping means the system will respond more quickly to
any changes that you make. A damping time of something like 10 to 20 seconds would be
reasonable. You will probably also want to reduce the drive amplitude when you are close to
resonance.
When completing this part of the lab, make sure you're really seeing a resonance. If you are, you
should be able to tune something like 0.3 Hz away from the mode frequency, and the vibrations
of the masses should decrease substantially. If they don't, then you are not driving a single mode
oscillation on resonance. There are two reasons why you might see oscillations, but not
resonance: (i) you don't have enough damping, so the masses continue to oscillate for a long time
regardless of what you do with the frequency, or (ii) you have your voltage turned up way too
high. (If you have your amplitude set to 1 V or more, that is too high.)
When you have succeeded in getting a single mode vibration going at frequency  , record a
video of the oscillation, load it into Tracker, and autotrack the displacement of one of the two
balls. You don't need lots of frames for this part, 200 frames would be plenty. Then autotrack the
displacement of the other ball. You can do this by pressing "create" and selecting "point mass" a
second time, and then running Autotrack a second time on the second ball. This will give you
one set of data labelled "mass A" and one labelled "mass B", which you can select for any of the
plotting or analysis functions.
Next, measure the amplitudes A1 and A2 of the vibrations of the two balls. Ideally, you’ll do
this with a fit to your data as discussed below. However the fitting in Tracker is pretty finicky, so
to start, just measure the amplitudes from the screen display, or by printing out your graphs on
paper and measuring amplitudes with a ruler (relative to the graph scale) (better). It would be a
little better to obtain your amplitudes from a fit as discussed below, but for this lab just
measuring amplitudes from the graph will be acceptable. (We’ll take off only a small bit of credit
if you don’t do the full fitting.)
[Fitting your data: To carry out a fit with Tracker, right-click on the graph and select "Analyze."
Click on "Analyze" and select "Curve Fits". Under "Fit Name," select "Sinusoid". This fit
function won't work because it lacks an offset from zero. To correct this, click on "Fit Builder."
Under "Parameters", click "Add." Edit the variable field "param" to read "D". Scroll Down and
double-click the expression. Edit it to read "A*sin(B*t+C)+D". Then press close.
Next select the data that you want to fit by clicking in the data field on the right. Selected data
points are yellow. You can select all or only part of the data. To select all points, right click on
the data area and select "Select All". Then, get a good fit to the data by some combination of
adjusting fit parameters by hand, and by checking "Autofit". Unfortunately, this fitting routine is
not good at automatically finding the best fit parameters. So, in order to get the Autofit to work,
you'll have to leave the Autofit unchecked, tweak the fit parameters by hand until the fit function
82

is quite close to the data, and then check Autofit. You'll know you have a good fit when the pink
curve goes through the data points.
An alternative would be to fit your data in another numerical environment such as Mathematica,
Matlab, or Python. ]
Check that your frequency  is the same as the frequency  that you measured earlier with the
Fourier transform. Verify that the two masses vibrate with the same phase. Finally, check
whether your measured ratio A2  / A1 agrees with the theoretical ratio (4.15).
Repeat the previous procedure for the single mode with frequency  . Again check whether  
is the same as the frequency you measured earlier. Check how well your measured amplitude
ratio A2  / A1 agrees with theory. In your report, give a qualitative description of the difference
between the two modes.

83



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