Physics 15a
Lab 1: Measurement and Uncertainty
Spring 2008
: Lab 1:
Measurement and Uncertainty
Pendulum Period
Introduction
Physics is primarily an experimental science. Physics theories are tested and refined
and are only retained when they are proven to be an accurate description of nature.
The key concept that distinguishes experimental science from theoretical science is
uncertainty. In an experimental setting, it is just as important to specify not only what
you know, but how well you know it. The purpose of this first lab is to introduce some
of the basic ideas of measurement and uncertainty.
There are two parts to the lab. In the first part, you'll try to measure the period of a
pendulum using different techniques, and see how the uncertainty in the measurement
depends on both the technique and the number of times you repeat the measurement.
In the second, we will use our data to try to measure the acceleration of gravity, g in
Boston.
The information needed for this lab was covered in the experimental lecture you had
early on. You should also consult the Lab Companion which will help guide your work.
It includes hints for conducting a successful lab; suggestions for writing your report
and a background tutorial on error analysis. Please read it carefully before beginning
your first lab and bring it to lab every week for reference.
Lab Goals
I.
Understand the different types of uncertainty involved in a measurement
II.
Observe how repeated physics measurements follow a Gaussian distribution
III. Learn to use the Logger Pro software to collect and analyze data using histograms
and Gaussians distributions
IV. Measure the value of the acceleration of gravity in Boston. Discuss your
measurement taking into account its uncertainty.
Prelab Assignment
You should read through this lab handout in its entirety before your lab section. The
prelab work this time is mostly for you to become acquainted with the Logger Pro
software. This is critical for you to be successful in lab since the time will be limited.
The software is not terribly complicated but it does require some training before you
can master it.
The Logger Pro software for Mac or Windows can be downloaded from http://
www.fas.harvard.edu/computing/download . You will need to log in with your Harvard
ID and PIN. Logger Pro 3.5 requires Mac OS X version 10.3.9 or later, or Windows
2000/XP/Vista. If you have a PC or Mac but are unable to install Logger Pro, you can
contact us for help. If you don't have a PC or a Mac of your own, you can use Logger
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Physics 15a
Lab 1: Measurement and Uncertainty
Spring 2008
Pro on a computer in the Science Center computer lab.
Familiarize yourself with the software:
I.
Start Logger Pro and click the
Open button.
II.
Under the Experiments folder, find the sub-folder named "Tutorials."
III. Open the tutorial called "01 Getting Started."
IV. Follow the instructions in the tutorial. This will give a general idea of the program.
V.
When you have finished complete the following subsequent tutorials:
05 Manual Data Entry
07 Viewing Graphs
08 Stats, Tangent, Integral
09 Curve Fitting
Exercise:
Download the file prelab1.cmbl from the course website. There is a link on the
Laboratory page of the course website. Right-click on the link and save the file to your
computer.
Open the file in Logger Pro and follow the instructions. When you reach the end,
answer the following questions (on a separate sheet of paper):
I.
Briefly sketch each the shapes of the first two histograms (Die #1 and Sum 2). Why
isn't the second one flat? Can you guess the shape of the "true" underlying
distribution for the sum of two dice?
II.
Qualitatively describe how the shape of the histogram changes as you increase the
number of dice in the sum from one to five. Does it start to look like any other
shape you've seen before?
III. In the lower left of each page of the Logger Pro file is a box that says "N 1000." N
is the number of times each die is rolled (i.e. the number of rows in the data
table). You can click the up and down arrows in this box to change the value of N
in steps of 100. Does anything interesting happen to the histograms if you
increase the value of N? What if you decrease it?
In the lab, you will calculate the acceleration due to gravity, g in Boston using a
pendulum. As preparation for lab, review your knowledge of the pendulum and write
down the equation for g in terms of the pendulum length and period. Finally, using the
propagation of errors rules from your lecture notes and lab companion, determine the
equation for the relative uncertainty Δg/g. We will help you in class if you have trouble
with this last piece.
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Physics 15a
Lab 1: Measurement and Uncertainty
Spring 2008
Material
Pendulun
Composed of a stainless steel cylinder, a thin aluminum rod, a couple of setting rings
and screws and two razor blades as hinge points.
Measuring sensors
I.
Digital stopwatch.
A. Press the right button to start or stop the timer.
B. Press the left button to reset the timer to zero.
II.
LabPro Hardware
A. LabPro Interface: The LabPro interface is the green box
that allows you to connect a wide variety of different
sensors to the computer via the Logger Pro software.
There are connections for two digital sensors on one
side and four analog sensors on the other. Today we will
use two digital sensors.
B. Photogate sensor: This is a U-shaped device (gate) with a
small infrared source in one arm and a fast IR detector in
the other. The small shutter over the IR detector needs
to be open for the photogate to operate. When the gate
is blocked the red LED on the top of the gate will be
illuminated (try it using your finger between the two
arms).
C. Motion Detector (sonar): The motion detector uses
ultrasound to measure distance. Ultrasonic pulses are
emitted by the motion detector, reflect from a target,
and then detected by the device. The time it takes for
the reflected pulses to return is used to calculate the
position of the target (also velocity and acceleration).
The motion detector can measure objects as close as 15
cm and as far away as 6 m.
Photogate documentation:
- User Guide: http://www2.vernier.com/booklets/vpg-btd.pdf
- Technical information: http://www.vernier.com/til/1422.html
Motion Detector documentation:
- User Guide: http://www2.vernier.com/booklets/md-btd.pdf
- Technical information: http://www.vernier.com/til/1374.html
Other miscellaneous tools
You will also use a meter stick, a caliper and a balance.
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Physics 15a
Lab 1: Measurement and Uncertainty
Spring 2008
Procedure
We start with a very detailed description of the procedure but, as the lab progresses,
we will give fewer guidelines. This is because some are repetitive and because we want
you to have the freedom to explore the lab. Please write down in the report all the
information (including graphs and tables) that you deem important. Don’t forget to
save your files often.
Pendulum construction
Before start building your pendulum, measure the length and diameter of the
aluminum rod and the cylinder, and record their weights. Try to make these
measurements as precise as possible. The small dimensions are better measured with
a caliper. Ask for help from your TF if you have never used one. Record all
measurements and their uncertainties for future use.
Use the parts provided to build your own pendulum. The cylinder slides into the long
aluminum rod. You should set it in place using one of the small rings with a set screw.
The cylinder should be located towards one end of the rod. You are free to choose the
exact position.
The pendulum is attached to the hinge fixture using a second ring and set screw (be
careful with the razor blades). You should set the razor blades on the top of the two
thick stainless steel rods.
Drive the pendulum oscillations from the top assembly. You should aim at stable
oscillations along one single axis and remember that the amplitude of the oscillation
should be small.
Measurement of the pendulum period
We will determine the period of the pendulum in three different ways. Here, we are
actually more interested in the period uncertainty, rather then the value of the period
itself. At the end, we want to keep the most precise and most accurate measurement
of the period.
I.
Stopwatch measurements
We will start by measuring the pendulum period using a stopwatch. Open the Logger
Pro file called Lab1-pendulum.cmbl (inside the Lab1 folder on the Desktop). Save the
file giving it a new name including your initials. Make sure to start on page 1.
Each member of your group should get a stopwatch. You will each measure the period
of the pendulum several times. You can write your measurements on a piece of paper
or directly on the computer.
A. Single period measurement
1. Before you begin, discuss with your lab partners the exact procedure you will
use to time the pendulum. Decide at which point of the trajectory you will be
measuring the time. Use same feature in the background to help you. Record
this location in your notes. Make sure you are all using the same technique.
2. Time a single period of the pendulum using the stopwatch. You should start
the watch at the instant when the pendulum passes the point agreed upon in
(1), wait for it to swing across to the opposite end, and come back. Stop the
watch at the moment when the pendulum returns to its original position.
3. Record the measured period and repeat the procedure until you have
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Physics 15a
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measured the period ten separate times each.
4. Enter all of the results recorded by your group in the data table, under the
column labeled 1T. If there are three of you in the group, there should be a
total of 30 entries in the column.
5. Create a histogram of the column 1T. In Logger Pro, the histogram feature
can be found under the menu Insert→Additional Graphs. Once the histogram
is displayed, you will need to double-click on the histogram to set the
options. Choose the correct column to display, and then try to make a
reasonable choice for the bin width. You don't want most of the bins to be
empty or have only one data point, but neither do you want the bins to be so
wide that almost all of the values are in just one or two bins.
6. From the histogram, click on the
Statistics button to calculate the
statistical mean and standard deviation of your 30 data points. Record these
values.
7. Try fitting a Gaussian to the histogram by clicking the
Curve Fit button.
In the box for "General Equation," scroll to the bottom and select “Gaussian
15a”. This function has three parameters: the amplitude A (which we don't
care about), the mean M, and the standard deviation S. What are the mean
and standard deviation of the best-fit Gaussian? (Note: you might find that
the program calculates a best-fit value of S which is negative. If so, ignore
the minus sign; only S2 enters into the equation for the Gaussian, anyway.)
Record these values.
a) Check to make sure these values are similar to the statistical mean and
standard deviation that you found in the previous step. If they are not, or
if you are having other problems getting a Gaussian fit, you may need to
adjust the bin settings for your histogram. Ask a TF for help.
b) If everything has gone well, though, you should find that your histogram
resembles a Gaussian (peaked in the middle and roughly bell-shaped), in
which case the values of M and S that you obtained from the fit are
probably a more useful indicator of the underlying Gaussian distribution
than the statistical mean and standard deviation of the original data
points.
c) Save a screenshot or a printout of your histogram in your report.
d) Based on these 30 data points, what is your best estimate of the
pendulum period? What is the uncertainty of this estimate? Don't forget
to specify the correct units in your answers.
8. Save your work in Logger Pro (and logbook editor if you are using one)
before moving on to the next part.
B. Measuring 10 consecutive periods (the procedure here is similar to the part
above)
1. Repeat the measurement of the period using the same technique as before,
however, instead of measuring a single period, wait for ten periods to elapse
before stopping the stopwatch. (You should expect your answer to be ten
times as big. If it only looks nine times as big, you probably stopped the
watch one period too early.)
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Physics 15a
Lab 1: Measurement and Uncertainty
Spring 2008
2. Record the duration of ten periods, and repeat the measurement ten times
each.
3. Enter the 30 data points into the Logger Pro data table under the column
labeled 10T.
4. Based on what you already know about the measurement process, and
before performing any calculations, what do you expect the standard
deviation of these data points to be? the mean? Why?
5. Repeat the graphs and statistical analysis that you have done in the previous
part and record your results. Do they agree with your previous results? What
is your best estimate of the length of ten consecutive pendulum periods?
What is the uncertainty of this estimate? (Don't forget units.)
6. What is your best estimate of a single period based on your thirty
measurements of ten periods? What is the uncertainty on this estimate?
II.
Measuring period with a photogate
Now the fun begins. We will now measure the period using a photogate. When the
pendulum passes in front of the photogate, it will interrupt the beam of infrared light
between the two arms of the photogate, i.e. it will close the gate. The time
measurement between every three instances that the gate is closed will give you the
pendulum period.
The photogate will behave in a very similar fashion each time the pendulum passes
through, therefore this method will result in a more precise measurement of a single
period then the stopwatch technique. There are, however, other potential advantages
to this technique. Can you come up with a couple of them?
1. Make sure to position the photogate below the pendulum so that, the rod
swings passing between the arms of the photogate.
2. Move to the second page (Photogate) of your Logger Pro file. You will find that
this file is already setup to measure the period of the pendulum and it already
has a graph of the period defined.
3. Click the button
data collection.
to start collecting data. Pressing the red button will stop
4. Make sure the pendulum is swinging quite stably and take data for 5 min or so.
Keep an eye on the two graphs. Is the data stable? Is there any trend of the
period versus time? How do you expect the histogram to look like? Does it?
5. If your pendulum period is not stable, you can always the data taking and start
again. It is important to obtain a fairly stable set of data.
6. Fit the histogram to a gaussian and determine the mean and standard deviation.
What is the error on the mean? Is it similar to what you would expect?
Measurement of the acceleration of gravity, g
The period of the pendulum depends on its length and the acceleration of gravity.
Determine the length of the pendulum as precise as you can. Write that value with its
uncertainty in your report. Make sure to specify exactly how you obtained it and the
corresponding uncertainties.
Write down the equation for g and determine the equation for the relative uncertainty
Δg/g using the propagation of errors rules. Then, tabulate the values of g that you find
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Physics 15a
Lab 1: Measurement and Uncertainty
Spring 2008
corresponding to the three periods you measured above. Your table should have three
columns: technique label, period, and g. Don’t forget units and uncertainties.
Which one is the most precise? Are you satisfied with your results given the value for
the acceleration of gravity in Boston shown in the table below?
Table with values of acceleration due to gravity, g (in cm/s2) at different places of
the world. From the book “Theory of Physics” by Joseph Sweetman Ames, 1927.
If you are not satisfied with your result, what could be going wrong?
The pendulum motion
If you have time, here is another way to look into the pendulum motion. Your setup
includes a sonar motion detector that is able to determine the distance between its
sensor and an object.
The sonar should be aiming at the cylinder and at least 20 cm away from it. Move to
the third page (Motion Sensor) of your LoggerPro file. Connect the sensors cable to the
second digital input of your LabPro interface. The sensor should be automatically
detected. You will notice that a column called “position” pops-up. Make a graph of this
column. Do you know what to expect? Try to make a fit (look for an appropriate
function). What is the period you get this way?
Compare this result with the previous values for the period taking into account the
uncertainties. Comment.
Challenge exercises
I.
II.
LoggerPro Page: Length. The relation between the length of a simple pendulum and the
square of its period is linear. The incline of the graph between those two variables would
give you another measurement of g. Make measurements of the period at different lengths,
plot them and verify this relationship. What insight can you get from plotting these two
variables?
LoggerPro Page: Angle. We are constantly reminded to keep the pendulum oscillation angle
small and as you know, the common period equation does not depend on the oscillation
angle. Will that still hold true for large angles? Make a plot of the period versus the angle of
oscillation and comment on your results.
Discussion
The last 15 minutes of the session should be spent discussing with your TF and colleagues the
results of the lab. Please provide your TF with your results for the acceleration of gravity and the
length of your pendulum as soon as you are done. Everyone is encourage to participate and
comment on the results and difficulties of the lab.
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