Lab 3
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Rose-Hulman Institute Of Technology *
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Course
111
Subject
Aerospace Engineering
Date
Jan 9, 2024
Type
docx
Pages
15
Uploaded by BrigadierLorisPerson984
Jackson Summers and Jonathon Fairchild
October 20, 2023
Simple Pendulum and Conservation of Energy Lab
Introduction
Purpose:
The purpose of this lab is to create and use a pendulum to measure and investigate the effect of
the pendulum's amplitude on its period of oscillation; determine maximum angular velocity; and carry
out an analysis of conservation of energy. This will be completed by using Logger Pro and its Pendulum
and rotary motion sensor.
Equipment:
Pendulum and rotary motion sensor (Vernier CI 6625)
LabQuest Mini Interface
Laptop computer running Logger Pro software.
Meter stick, pan (or digital) balance/scale.
Equations:
T
=
2
√
2
L
g
∗
S
∅
We use this equation to find the predicted and measured periods during the trials where we vary the
amplitude
Procedure:
Image 1-1:
Image 1-2
Shows the pendulum in motion.
Shows the setup of the pendulum and the
Length of the pendulum.
Image 1-3: Drawing of a simple pendulum where it was
released at point P. Also notes the LabQuest Mini Interface, and Laptop Computer running Logger
Pro.
1)
Measure the mass of the nut and verify the confidence of the measurement by taking multiple
readings on multiple scales. The average mass of the nut was 16.56 grams.
2)
Assemble the pendulum by cutting a string of about 1.4 meters. Then tie a knot to keep the nut
on the string.
3)
Measure the length of the pendulum from the center of mass of the nut to the point of
revolution. This should be around 0.6 meters to 0.7 meters.
4)
Connect Sensor to computer running logger pro via the LabQuest mini and ensure the
connection is registered.
5)
Record data by having one person release the mass from a chosen amplitude, while the other
starts the recording process on the computer and ensure that the mass only swings on the
desired axis, preventing twisting and drifting.
6)
Measure the data first at the angular displacement of 30
0
.
7)
Repeat the data recording process 5 times at different amplitudes between 10 and 60
o
, excluding
30
0
.
Data Collection:
Trial #
Release Angle
Amplitud
e
Period from Equation
1-1
Period from Logger Pro
Percent
Error
1
10
0.22 rad
1.5315 s
1.5000 s
2.10%
2
20
0.40 rad
1.5403 s
1.5333 s
0.46%
3
25
.50 rad
1.5470 s
1.5333 s
0.89%
4
30
.58 rad
1.5543 s
1.5666 s
0.79%
5
35
.66 rad
1.5650 s
1.5666 s
0.10%
6
40
.81 rad
1.5765 s
1.6000 s
1.47%
7
50
.94 rad
1.6046 s
1.6333 s
1.76%
Table 1-1
Logger Pro
LabQuest Mini
Interface
Figure 1-1
The figure above shows the angle and time graph as well as the velocity and time graph when the
angular displacement started at 10
0
. In this data collection amplitude and the period can be calculated.
This may not be entirely accurate as data is taken every .0333 seconds.
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Figure 1-2
The figure above shows the angle and time graph as well as the velocity and time graph when the
angular displacement started at 10
0
. In this data collection amplitude and the period can be calculated.
This may not be entirely accurate as data is taken every .0333 seconds.
The following is the measured amplitude maximums and minimums when the pendulum has an angular
displacement of 25
o
:
Angular Amplitude Max 1
0.02
Angular Amplitude Min 1
-1.03
Max 2
0.02
Min 2
-1.01
Max 3
0
Min 3
-0.99
Max 4
-0.02
Min 4
-0.98
Max 5
-0.02
Min 5
-0.98
Max 6
-0.03
Min 6
-0.98
Max 7
-0.03
Min 7
-0.96
Max 8
-0.05
Min 8
-0.96
Max 9
-0.05
Min 9
-0.94
Max 10
-0.05
Min 10
-0.94
Table 2-1
Figure 1-3
The figure above shows the angle and time graph as well as the velocity and time graph when the
angular displacement started at 25
0
. In this data collection amplitude and the period can be calculated.
This may not be entirely accurate as data is taken every .0333 seconds.
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Figure 1-4
The figure above shows the angle and time graph as well as the velocity and time graph when the
angular displacement started at 30
0
. In this data collection amplitude and the period can be calculated.
This may not be entirely accurate as data is taken every .0333 seconds.
Figure 1-5
The figure above shows the angle and time graph as well as the velocity and time graph when the
angular displacement started at 35
0
. In this data collection amplitude and the period can be calculated.
This may not be entirely accurate as data is taken every .0333 seconds.
Figure 1-6
The figure above shows the angle and time graph as well as the velocity and time graph when the
angular displacement started at 40
0
. In this data collection amplitude and the period can be calculated.
This may not be entirely accurate as data is taken every .0333 seconds.
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(
Figure 1-7
The figure above shows the angle and time graph as well as the velocity and time graph when the
angular displacement started at 50
0
. In this data collection amplitude and the period can be calculated.
This may not be entirely accurate as data is taken every .0333 seconds.
Analysis:
Part 1:
All data points are gathered in Figures 1-(1-7).
Figure 1-8
Part 2:
The table below was created using values of the LoggerPro software and the given S(θ
m
)
values. Using figures 1-(1-8) this table was developed:
Θ
m
(radians)
S(θ
m
)
0.1745
2.2257
0.3491
2.2385
0.5236
2.2601
0.6981
2.2911
0.8727
2.332
1.0472
2.384
1.2217
2.4484
Part 3:
The table below was formed using Figure 1-3 and Table 2-1 to collect data:
Time (s)
θ
(radians)
ω (radians/s)
U (Joules)
K (Joules)
TE (Joules)
1.600
0.02
0.116
1.915*10^-5
3.878*10^-5
5.793*10^-5
1.900
-0.28
-1.818
0.03894
9.526*10^-3
0.04847
2.200
-0.86
-1.614
0.3476
7.508*10^-3
0.35511
2.500
-0.96
0.611
0.42648
1.076*10^-3
0.42756
2.800
-0.49
1.774
0.01127
9.071*10^-3
0.02034
3.100
-0.03
0.596
4.308*10^-5
1.024*10^-3
1.067*10^-3
3.400
-0.17
-1.513
1.380*10^-3
6.598*10^-3
7.978*10^-3
3.700
-0.73
-1.614
0.02440
7.508*10^-3
0.03191
Table 4-1
Examples of Calculations for Part 3 and Table 4-1:
θ
1
−
cos
¿
U
(
2.5
s
)=
mgL
¿
m
¿¿
=
(
.01656
kg
)
(
9.8
m
s
2
)
(
.59
m
)
(
1
−
cos
(
−
0.96
)
)
=
0.42648
J
K
(
2.5
s
)
=
1
2
m L
2
ω
2
=
1
2
(
0.01656
kg
) (
.59
m
)
2
(
.611
rad
s
)
2
=
1.076
∗
10
−
3
J
TE
(
2.5
s
)
=
U
(
2.5
s
)
+
K
(
2.5
s
)
=
0.42648
J
+
1.076
∗
10
−
3
J
=
0.42756
J
Part 4:
Total Energy at swing 1 = .0481 J
Total Energy at swing 10 = .0355
Total energy lost = .0136
Total Energy lost per swing = .00136
Energy at swing 1 – energy lost per swing = .0468
(Energy at swing 2 / energy at swing 1) * 100
= 2.624%
These rudimentary calculations show that between each swing the system loses 2.6% of its total energy
to friction.
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Calculations:
T
=
2
√
2
L
g
∗
S
∅
Test for 10
0
angular displacement:
T
=
2
√
(
2
∗
.59
)/
9.8
∗
2.2257
=
1.5315
seconds
10
0
angular displacement through LoggerPro was 1.5000 seconds
Percent Error:
(
1.5315
−
1.5000
)/
1.5000
= 2.10% error
T
=
2
√
2
L
g
∗
S
∅
Test for 20
0
angular displacement:
T
=
2
√
(
2
∗
.59
)/
9.8
∗
2.2385
=
1.5403
seconds
20
0
angular displacement through LoggerPro was 1.5333 seconds
Percent Error:
(
1.5403
−
1.5333
)/
1.5333
= 0.46% error
T
=
2
√
2
L
g
∗
S
∅
Test for 25
0
angular displacement:
T
=
2
√
(
2
∗
.59
)/
9.8
∗
2.2482
=
1.5470
seconds
25
0
angular displacement through LoggerPro was 1.5333 seconds
Percent Error:
(
1.5470
−
1.5333
)/
1.5333
= 0.89% error
T
=
2
√
2
L
g
∗
S
∅
Test for 30
0
angular displacement:
T
=
2
√
(
2
∗
.59
)/
9.8
∗
2.2601
=
1.5543
seconds
30
0
angular displacement through LoggerPro was 1.5666 seconds
Percent Error:
(
1.5543
−
1.5666
)/
1.5666
= 0.79% error
T
=
2
√
2
L
g
∗
S
∅
Test for 35
0
angular displacement:
T
=
2
√
(
2
∗
.59
)/
9.8
∗
2.2744
=
1.5650
seconds
30
0
angular displacement through LoggerPro was 1.5666 seconds
Percent Error:
(
1.5650
−
1.5666
)/
1.5666
= 0.10% error
T
=
2
√
2
L
g
∗
S
∅
Test for 40
0
angular displacement:
T
=
2
√
(
2
∗
.59
)/
9.8
∗
2.2911
=
1.5765
seconds
40
0
angular displacement through LoggerPro was 1.6000 seconds
Percent Error:
(
1.5765
−
1.6000
)/
1.6000
= 1.47% error
T
=
2
√
2
L
g
∗
S
∅
Test for 50
0
angular displacement:
T
=
2
√
(
2
∗
.59
)/
9.8
∗
2.3320
=
1.6046
seconds
40
0
angular displacement through LoggerPro was 1.6333 seconds
Percent Error:
(
1.6046
−
1.6333
)/
1.6333
= 1.76% error
Conclusion
In conclusion, we can see that our results are fairly accurate, having only an average percentage
error of 1.08% for the periods. From these calculations we can see that our experiment was successful in
showing the relationship between period of oscillation and angular displacement. Additionally, from the
graph in part one of the analysis we can see that between an angular displacement of 0 and pi/2, as
displacement is increased, period increases exponentially. Another Conclusion we found was that the
system lost a certain percentage of energy per swing due to friction which we found to be about 2.6%
per swing. In the end our experiment did not agree with the Conservation of Energy. This could have
been because of a simple mistake. Our graph was not smooth, so that may have been our problem. In
the next experiment similar to this, it would be necessary to make sure our data is carefully calculated
and accurate before we leave the lab.
Reflections:
Jon: I feel as though my work was inconsistent and spotty, and that I have let my partner down at
multiple sections of the project, and that Jackson has helped me through a good portion of work that I
should have been able to figure out on my own. I feel as though my prep for this lab was quite lacking,
and that is most likely the reason for my heavy dependance on my partner to get the lab done. I feel like
I should focus on participating more actively before and during the lab if I want to fix this problem, as
this will leave me more well prepared to do my work when writing the lab report with my partner.
Jackson: The work completed in this lab may not have been the most accurate, but we completed the
goal of the lab. Next time we complete a lab, we should focus on efficiency while in the lab setting,
which will make the lab report easier to write, especially with more depth in our post lab writing. This
lab tested our writing and compressive skills of physics, and I believe our lab report shows our
capabilities. In our next lab, which is on momentum, we will spread the work more evenly and complete
the lab in the required time slot with more than the necessary amount of information.
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