PHY 241 Lab 1- Measurement of a Pendulum
Background on Accuracy vs Precision
Consider the following illustration where four different players are trying to hit the center of a dart
board:
1. Accurate
and Precise
2. Accurate only
3. Neither Accurate
nor Precise
4. Precise only
Our goal as scientists is very similar to these players, we want to be both accurate and precise. Instead
of throwing darts, we are making measurements. The dots show the results of our measurements, and
the center of the dart board is the “true” value; the value we need to find with our experiment
We measure these two concepts with “Random Uncertainty” and “Systematic Uncertainty/Error.”
Random Uncertainty, 𝛿𝑟, is a measure of the precision of our data. It can be visualized by
drawing a circle around our data (capturing 68% of our data points), and then measuring the radius of
that circle as shown for player 3 below. Most of the time we will measure 𝛿𝑟 by taking a “Standard
Deviation” or “=STDEV.S” in Excel. Because 𝜹𝒓 is a measurement of our inaccuracy, we always round
this value to one Significant Figure when writing our final report for an experiment.
𝛿𝑟
Random Uncertainty
Example: Consider data set 1 shown to the right. As a collection of numbers
it is very hard to tell what is going on with this data.
Trial
1
2
3
4
5
6
7
8
Time (s)
1.23
1.35
1.36
1.32
1.25
1.34
1.2
1.27
Data Set 1: Measurement
of Time for X
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Measurement of time for X
1.4
1.35
time (s)
If we represent Data Set 1 as a scatter plot we can see
that there is no obvious pattern in our data, so we will use
the “Average( )” and “StDev.S()” functions within to find
the average and the uncertainty.
1.3
1.25
1.2
1.15
0
2
4
6
8
10
trial number
=AVERAGE(B3:B10)
Average (s)
Measurement of time for X
(version 2)
1.29
=STDEV.S(B3:B10)
0.060474316
This average and uncertainty can be added to our chart
as a single new data point with error bars;
Measurement of time for X (version 2). Make sure the
Error bar on the orange point is “Custom” and “Specify
Value” = Uncertainty cell in your spreadsheet.
time (s)
Uncertainty (s)
1.4
1.35
1.3
1.25
1.2
1.15
0
2
4
6
8
10
trial number
This single orange dot (with error bars) summarizes our entire collection of 8
trials!
Systematic Uncertainty or Systematic Error, 𝛿𝑅, is a measure of the accuracy of our data and is
much more important and usually more difficult to determine. It can be visualized by measuring the
distance between the center of the target and the center of the data that has been collected. To
calculate this value we can simply find the difference between our average of our data and the “True” or
“Expected” value:
𝛿𝑅 = |𝑋𝑇𝑟𝑢𝑒 − 𝑋̅|
𝛿𝑅
Systematic Uncertainty
An experiment is successful when the random uncertainty is as small as reasonably possible AND the
systematic uncertainty is even smaller than the random uncertainty. Whenever the systematic
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uncertainty is larger than the random uncertainty we must do some additional work explaining and
verifying that the experiment did not measure what we intended.
Today, we will focus on Random uncertainty, 𝜹𝒓 only.
Introduction:
A pendulum is a mass swinging back and forth at the end of a string. A pendulum is one of the simplest
interesting physical systems, because they have one value, the Period, which is easy to measure. The
Period, represented by the symbol 𝑻, is defined as the amount of time it takes an object to repeat its
motion. There are many questions we might ask:
1)
2)
3)
4)
5)
Does the amount of mass make a difference?
Does the string make a difference?
Does the size of the swing make a difference?
Does the planet that the pendulum is on make a difference?
Does the support that is holding the pendulum at the top make a difference?
The answers to most of the questions above are surprisingly simple, however before we can answer any
of them, we must be very specific with the phrase “make a difference.” Almost certainly if you measure
the period of a pendulum two times in a row, you will get slightly different measurements. If it is exactly
the same pendulum then those differences must be part of the uncertainty of the measurement, 𝛿𝑇.
Equipment:
Mass
String
Metal stand
Stopwatch
Photogate
protractor
Computer with Logger Pro/Lite and Excel
Task 1: The lies our equipment tell us.
1) For this first lab we will be repeatedly measuring the same “Control” system. Naively, if we
measure the same system multiple times, we should get identical answers… Therefore, set up
your pendulum so that it is swinging from the metal stand. Once you have set the length of
string and amount of mass, do not change these throughout the rest of this lab period.
2) Get the pendulum swinging gently, at most the string should be 10° or less from vertical. The
exact angle does not matter for today’s lab.
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3) Using stopwatches, smart phones, and or the computer (search the web for “stopwatch”) have
all three members of the group measure the period of exactly the same swing of the pendulum.
We want three independent measurements of the same swing. We have space in the “Lab 1
Template” under Task 1, for you to record any mistakes if it requires multiple attempts for the
group to be happy with the results. Feel free to add additional columns to the excel sheet so
you can record all the data in the accompanying excel file.
Remember: Falsifying data is a major breach of the student code of conduct and carries a minimum
punishment of a zero on the assignment in question.
4) Group- Evaluate the data you acquired so far. Based solely on this data, report the period for
your pendulum using the 𝑇̅ ± 𝛿𝑇 notation introduced in Prelab 1.
Task 2: Learning how to use the Photogate
5) Connect the photogate to the “LabQuest Mini” and then to the computer using the USB cable.
Open “LoggerPro” or “Logger Lite.” If the Collect button is unavailable:
a. In Logger at the top go to “Experiment” and “Connect Interface” until you get the
LabQuest Mini recognized by the software.
b. Again, go to “Experiment” and “Set Up Sensors” and “Show all Interfaces” to make sure
a “Photogate” appears in the proper port of the Mini.
6) Spend some time playing with the photogate while it is
collecting data. Make sure the “Shutter” is open so the
photogate will collect data. Pretend that your hand is the
pendulum and move slowly through the photogate until you
can determine how the “times” reported by the photogate
can be turned into the “Period” you want to measure.
“Shutter”
7) The photogate always creates five columns; “Time” and “GateState” are important. Make sure
you figure out what these numbers mean.
The “Distance,” “Velocity,” and “Acc” columns are total nonsense for the photogate. Feel free
to ignore or delete the “Distance,” “Velocity” and “Acceleration” columns.
8) Without adjusting the string or the mass, add the Photogate to the stand so it can measure the
period of the pendulum. While some groups like for the photogate to “watch” the mass, you
might have better success if the photogate “watches” the string instead.
9) Give the pendulum a small push and collect about 15 seconds worth of data. Use this data to
check your work from Task 1.
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10) Rather than use a hand-held calculator to check the period from the photogate data, copy/paste
the entire “Time” column from Logger Pro to the yellow cells in the provided Excel sheet.
11) In one of the blue cells next to the data we want to add a formula to teach Excel how to
calculate the period for us:
a. All equations in Excel start with an “=” (no quotation marks).
b. To do a subtraction between two cells, we can click on the individual cells we want to
subtract with just a “-” keystroke between the two clicks.
c. When you are finished hit “Enter” to save your equation. If you ever need to exit an
equation without saving, hit “Esc.”
d. Your final equation should look like “=X18-Y12” where X18 and Y12 are replaced by the
correct cell references for your formula.
e. Double check that the value returned by excel is the same as the value you got in step 9.
12) To copy your formula, click on the cell with your formula so it is highlighted. In the bottom-right
corner of the cell, as shown below with the red arrow, there is a small box. Double click this box
to “fill down” your formula. (you can also “Copy” the cell that has the formula in it and the
“Paste” that into the blue cells that are below.
13) At the top of Excel go to “Insert” and select the “Scatter” chart
14) Copy your column of “Periods” and paste them into chart you just made. You should get
something like:
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Period of Pendulum
1.118
Period (s)
1.117
1.116
1.115
1.114
1.113
1.112
0
5
10
15
20
Swing number
15) Notice: I have added a title and axis labels. To add these and other features click on the chart
you want to edit and then use the “Add Chart Element” found under “Chart Tools” and “Design.”
If you need additional help refer to the internet, the “Appendix A: Making Charts…” in Canvas,
or the Excel Help function.)
16) Data Runs 2-4 are for you to work on reducing the random uncertainty in your data. We may
not change the mass or length of string in the pendulum, but you can make changes to the
location of the photogate, or the way that you release the pendulum. Presumably these will not
change the period of the pendulum, but they might change the uncertainty in the measurement.
Investigate and explore so you can find the “best” way to take measurements that involves the
least uncertainty in your measurements.
17) The goal of the rest of the lab is to fully understand the measurements you are making with the
photogate. Collect the data you need to answer the following questions. To be clear you will be
submitting the Excel sheet into Canvas. The entire group is responsible for collecting data
needed to answer all questions. Questions assigned to specific individuals will alter the grade of
only those individuals. Therefore, it is forbidden for a single individual to answer all questions
on the behalf of the group.
A) DA- What is the connection between your measurements between task 1 and task
2. What is the period of the pendulum? (Your answer should include both text and
a graph of your data-I suggest a single graph that summarizes Task 1 as one data
series and Task 2 as a separate data series.)
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PHY 241 Fall 2021
B) Researcher- What is the Random Uncertainty of your Period measurements? What
adjustments did you make to the experimental setup or procedure to reduce (or
increase) this uncertainty?
C) PI- Compare your results for two different Task 2 data runs by building a second
graph. Is there a “systematic shift” between these two data sets? Can you estimate
the size of this shift? Could this be accounted for by the Random uncertainty of
your system?
D) Group- The data in the “Period of Pendulum” graph in the manual has a weird
pattern in it. The 1st, 5th, 9th, and 13th points are low while the 4th, 8th, and 12th
points are high. If you are getting a similar pattern where is it coming from? If no
similar pattern appears then you don’t need to worry about this question.
18) Before leaving the classroom, make sure you email the data out to the entire group and clean up
your work station, returning small equipment to the appropriate storage.
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PHY 241 Fall 2021