P221 Student Projects/Presentations
During the Fall, 1995 semester, students worked on a variety of projects
which involved data taking, data analysis, simulation to check data analysis
and in some cases, more advanced analysis using Mathematica. The projects
took several weeks and concluded with a write-up along with a presentation
to an audience of Physics Department faculty, staff and graduate students.
The presentations were also video-taped.
The projects:
Stacking Bricks
Large angle Oscillations of a Physical Pendulum
Moments of Inertia
Rolling Cylinders
Studying Damped Harmonic Motion
Stacking Bricks
Kay Brewer - Actually this first little project is from Fall, 1994.
I assigned the classic problem of stacking bricks. Each brick extends beyond
the brick underneath. For a given number of bricks, the idea is to see how
far the topmost brick can extend beyond the bottom-most brick and still
be stable. After you consider the requirements for equilibrium, you find
that the algorithm for stacking (assuming identical bricks) is: top brick
extends half of its length beyond the brick beneath it, the next extends
one fourth of its length beyond the brick beneath it, the next, one sixth,
and so on. The total extension for N bricks is:
When I assigned this problem, we considered the case of four bricks.
Kay Brewer, a P221 student in 1994, decided to try it with dominoes - it
didn't work ! We decided it was a problem of lack of uniformity so we decided
to ask for help from our machine shop. Vic Matteson, the shop foreman, worked
with Kay and produced a stack of ten identical aluminum blocks, each 2 in
by 6 in by 1/2 in. He also scribed a fine mark on each showing where to
stack. The photo shows the stack on my office desk. Kay and I and the others
were relieved and many thanks to Vic.
Now here is a fun problem. For ten blocks, the top-most block
extends 8.5 in beyond the bottom-most block. Suppose you wanted an extension
of 60 in ? How high would the stack be ? Hint: how fast (or slow) does the
series diverge ?
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Large Angle Oscillations of a Physical
Pendulum
Andrew DuBois and Robert Deirth used a single photogate timer located
at the lowest point of the swing of a physical pendulum. The pendulum had
a nice ball-bearing mount. The photogate measured the period and the velocity
of the pendulum at its lowest point. The maximum amplitude of swing was
about 75 degrees. Using energy conservation, the amplitude could be expressed
in terms of velocity. So they measured the dependence of period on amplitude,
and of course, found that it was not constant.
The above expression for the dependence of period on amplitude can be
derived by just using energy conservation. Now make substitutions:
Using Mathematica, the integrand of the following was expanded term
by term:
and here is the result:
For the particular pendulum used the small-angle (isochronous) period
was 1.49 sec. The plot below shows the data (open) circles compared to the
small amplitude answer (horizontal line) and the above expression keeping
2, 3 or 4 terms. Data were analyzed with the program IGOR.
They also studied simulations using Interactive Physics to test their
analysis programs and to study large angle oscillations. A phase space plot
(angular velocity vs angle) is shown below and is useful in understanding
the motion. In the plots below we look at oscillations with two different
amplitudes.
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Moments of Inertia
Julie McBain, Sara Spalding and Jessica Talaga used the setup shown
above to measure the moments of inertia of various objects. The object is
placed on a turn-table. The turn-table is turned so the string holding the
mass winds on the spindle. The mass is released and it accelerates down.
After the string has completely unwound it starts winding again, because
of the momentum of the turntable, and the mass reverses direction. The plot
of distance (as measured by the ultrasonic motion detector) vs time is shown
below. In the absence of friction, the acceleration on the way up should
equal that on the way down. Because of friction, these accelerations are
not equal in magnitude and the numerical results of the fits to the data
(using IGOR) clearly show this.
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Rolling Cylinders
Michael Pacold and Stephan Ichiriu studied the motion of cylinders
rolling down an inclined plane. Their idea was to show that the acceleration
does not depend on the mass (m) or radius (R) of the cylinder, but only
how the mass is distributed as given by I, the moment of inertia about the
center of mass. In fact, the the acceleration is uniform and given by:
where g is the accleration of gravity and the angle is the angle of incline
of the plane. For a hollow cylinder, x = 1 and for a solid cylinder x =
1/2.
The setup is shown above is self-explanatory. When I first tried this at
home, I used an eight foot long wooden plank and an empty coffee can and
a full jar of mustard. When I first took the data and tried fitting the
distance vs time data to a quadratic, the fits were not very good. The
board was sagging. So I used more supports and a straight edge and took
new data. That solved the problem. However, when I compared the measured
acceleration with the predicted, the slight disagreement was outside measurement
errors. I had determined the angle by measuring the length of the board
and the height of the support. It turns out the floor was off by 1 degree
(as determined by using a digital level). When I corrected for that, the
agreement was very good. The velocity vs time for the coffee can is shown
below.
Mike and Steve did similar experiments and obtained good agreement with
their predictions.
Using their measurements, they were also able to plot the energy versus
time, as shown in the plot below. In one case, they considered only the
kinetic and potential energies (red data points). In the second case, they
included the energy of rotation (blue data points). The plot shows how
important that contribution becomes.
One Friday evening, I stopped by the P221 lab on the way home to find Mike
and Steve hard at work. They were getting nice results and I felt things
were too easy, so I threw in a monkey wrench. We used a heavy magnet and
attached it to the inside of a hollow cylinder. This destroyed the nice
symmetry. The measured plot of velocity vs time was no longer a straight
line ! The plot below shows this. I gave them the challenge of analytically
analyzing the motion to see if they could understand the data. Steve decided
to do the theoretical analysis using Mathematica to check the algebra.
Mike took careful measurements. A true collaboration was born - a theorist
and experimentalist.
The analysis resulted in the theoretical prediction for velocity vs time
as shown above. To call this a prediction, however, is slightly misleading
since Steve knew what the data looked like. In the process of his analysis,
Steve realized that if the cylinder reached a critical velocity, it would
jump off the plane. He tried it out and behold - it jumped. Now that
is a prediction. The Interactive Physics simulation shown below, also
shows the phenomenon and agrees with the predicted critical velocity.
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Studying Damped Harmonic Motion
Under construction pal ! You should some back later. OK please
?
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