Lab 5_ Circular Motion
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University of Windsor *
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Course
03-64130
Subject
Aerospace Engineering
Date
Dec 6, 2023
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docx
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Lab 5: Circular Motion
Introduction
The purpose of this lab is to determine the centripetal acceleration of the iOLab using various
methods. The accelerometer, at point A, reads the centripetal acceleration, whereas the
gyroscope, at point G, is the point from which the iOLab gets spun in the air and determines the
angular velocity,
ω
. After doing this five times and obtaining data with little wobbling (a
consistency/stability in the z-direction), we could measure the centripetal acceleration of the
iOLab. In this lab, we also manually measure the radius of the iOLab simply from the point of A
to the point of G. Finally, we use Excel to plot
ω
versus
a
c
using both a linear fit, as well as a
parabolic/quadratic/polynomial-order-2 fit. Altogether, the exercises within this lab teach us how
to calculate and interpret centripetal acceleration using data from a real world scenario.
Exercise 2: Measuring
r
Directly - Data Collection
In exercise 2, we directly measured the distance from the point A on the iOLab to the point B
on the iOLab.
Figure 1- To determine the Ax and Ay of the iOLab a ruler was used to measure distance between the
accelerometer and gyroscope, and the accelerometer and the iOLab center
Exercise 2: Measuring
r
Directly - Data Analysis
Distance: 4.1cm
Uncertainty (half of the smallest digit): 0.05cm
r
: 4.1cm +/- 0.05cm
Exercise 2: Measuring
r
Directly - Conclusion
We measured the distance between the G and A on the iOLab using a ruler, then we calculated
the uncertainty of this value.
Exercise 3: Measuring
r
Using the Accelerometer and Gyroscope - Data Collection
In exercise 3, the iOLab was calibrated on the computer to measure accelerometer, gyroscope,
and magnetometer. The accelerometer and gyroscope graphs were selected. The iOLab was
thrown in the air five times about its z-axis, beginning with slow flips, and increasing in speed
gradually (approximately). The raw data was then collected from the graphs in an Excel sheet.
Data from each of the throws is included in the form of screenshots of acceleration and
gyroscope graphs. Finally, the centripetal acceleration and angular velocity (with their
uncertainties) were calculated for each of the throws.
Figure 3- Data collected from the first 5 flips of the iOLab
Figure 4- Data collected from second throw of 5 flips
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Figure 4- Data collected from third throw
Figure 5- Data collected from fourth throw
Figure 6- Data collected from fifth throw
Exercise 3: Measuring
r
Using the Accelerometer and Gyroscope - Data Analysis
Acceleration
of first throw:
a
c
=
A
tot
=
√
❑
¿
√
❑
¿
1.935
m
/
s
2
Uncertainty for acceleration
of first throw:
Δa
c
=(
1
/
a
c
)
√
❑
¿
(
1
/
1.935
m
/
s
2
)
√
❑
¿
0.034
m
/
s
2
Angular velocity
of first throw:
−
7.523
rad
/
s
¿
2
?
w
2
=
¿
¿
56.60
rad
2
/
s
2
Uncertainty of
w
2
for the first throw
:
(
?
w
2
/
w
2
)=
2
⎸
w
?
/
w
⎸
2
∨
w
?
/
w
∨
¿
?
w
2
=
w
2
¿
2
∨
0.0078
rad
/
s
/−
7.523
rad
/
s
∨
¿
−
7.523
rad
/
s
¿
2
¿
?
w
2
=
¿
?
w
2
=(
56.60
rad
2
/
s
2
)[
15.03
rad
/
s
]
?
w
2
=
850.698
ra d
2
/
s
2
Raw Data
*
Calculations
A
x
Uncertaint
y on A
x
(ΔA
x
)
A
y
Uncertainty
on A
y
(ΔA
y
)
ω
z
Uncertainty
on ω
z
(Δω
z
)
a
c
Uncertainty
on
a
c
(Δ
a
c
)
(
ω z
)
❑
2
Uncertainty on
(
ω z
)
❑
2
(
Δ
ω z
)^2
0.741
m/s
2
0.039m/s
2
-
1.788m
/s
2
0.033m/s
2
-7.523
rad/s
0.0078rad/s
1.935
m/s
2
0.034m/s
2
56.6
(rad/s)
2
850.698(rad/s)
2
0.832
m/s
2
0.12m/s
2
-
2.506m
/s
2
0.14m/s
2
-8.558
rad/s
0.086rad/s
2.640
m/s
2
0.951m/s
2
73.24(r
ad/s)
2
1.47(rad/s)
2
1.453
m/s
2
2.4m/s
2
-
3.514m
/s
2
4.9m/s
2
-11.58
rad/s
0.075rad/s
3.802
m/s
2
4.59m/s
2
134.1(r
ad/s)
2
1.74(rad/s)
2
1.726
0.064m/s
2
-
0.16m/s
2
-
0.02rad/s
4.60
0.15m/s
2
134.7(r
0.46(rad/s)
2
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m/s
2
4.262m
/s
2
11.606
rad/s
m/s
2
ad/s)
2
1.783
m/s
2
0.068m/s
2
-
3.985m
/s
2
0.084m/s
2
-
11.068
rad/s
0.06rad/s
4.365
m/s
2
0.08m/s
2
122.5(r
ad/s)
2
1.33(rad/s)
2
*The values from trials 2-5 were calculated on the side, but they followed the procedure for calculation 1
Exercise 3: Measuring
r
Using the Accelerometer and Gyroscope - Conclusion
In this exercise the value of r (radius of the circular path) was obtained from the raw data
collected from the iOLab. By flipping the device and collecting 5 trials the centripetal
acceleration and angular velocity were obtained (along with their corresponding uncertainties).
Using
a
c
=
A
tot
=
√
❑
, the centripetal acceleration for the first throw was calculated to be;
1.935
m
/
s
2
. The next four trials came out to the following; 2.640
m
/
s
2
, 3.802
m
/
s
2
,
4.60
m
/
s
2
, 4.365
m
/
s
2
. The uncertainty corresponding to these values were; 0.034
m
/
s
2
, 0.951
m
/
s
2
, 4.59
m
/
s
2
, 0.15
m
/
s
2
, and 0.08
m
/
s
2
, respectively. Possible
sources of error include the force of air resistance skewing the iOLab away from the exact z-axis.
As well as human error, someone cannot throw the device at the same height, acceleration, or
spin each time. The other component of the activity was finding the angular velocity. The values
collected are as follows; 56.6
(
rad
/
s
)
❑
2
, 134.1
(
rad
/
s
)
❑
2
, 134.7
(
rad
/
s
)
❑
2
, and 122.5
(
rad
/
s
)
❑
2
. The uncertainties of the angular velocity are as follows; 850.698
(
rad
/
s
)
❑
2
,
1.47
(
rad
/
s
)
❑
2
, 1.74
(
rad
/
s
)
❑
2
, 0.46
(
rad
/
s
)
❑
2
, 1.33
(
rad
/
s
)
❑
2
. Possible sources
of error for the angular velocity fall along the same lines as the centripetal acceleration where air
resistance, and human error are factors. The results for both the centripetal acceleration and
angular velocity were varying, especially the beginning trials versus later on. The data represents
the human error and how by flipping the iOLab several times the individual gets better at
throwing, resulting in more consistent data.
Exercise 4: Measuring
r
by Plotting
ω vs.
a
c
and
ω
2
vs.
a
c
- Data Collection
For this exercise, we will be plotting
w
vs.
a
c
on a graph to generate a non linear fit graph
which can be denoted as either a parabolic fit, a quadratic fit, or a polynomial of order-2 fit. After
inputting our data obtained into Excel, we will obtain 2 separate plots with one being
ω
on
the x-axis and
a
c
on the y-axis. While the second plot displays
w
2
on the x-axis and
a
c
on the
y-axis. The value for the radius, known as
r
from both plots are then obtained and compared to
one another as well as to the value obtained when measuring with a ruler.
Exercise 4: Measuring
r
by Plotting
ω vs.
a
c
and
ω
2
vs.
a
c
- Data Analysis
Figure 7: plot of w vs a
c
to generate a polynomial order-2 fit to the data
*Note: We also used the magnitude of our angular velocity values because the negatives were problematic in Excel
Figure 8: a plot of
w
2
vs a
c
to generate a linear fit to the data
R-value obtained from the polynomial order-2 graph
: 0.0878m (negative can be dismissed),
with an uncertainty of the R^2 value (0.9239m).
R-value obtained from the linear fit graph
: 0.0296m, with an uncertainty of the R^2 value
(0.9034m).
Radius
calculated in exercise 2: 4.1 cm, with an uncertainty of 0.05cm.
Exercise 4: Measuring
r
by Plotting ω vs.
a
c
and
ω
❑
2
vs.
a
c
- Conclusion
The R values obtained from each graph was the value in front of the x^2 term on the
polynomial order-2 graph and the x term on the linear graph. The R value on the polynomial
graph was 0.0878m (8.78 cm) and on the linear graph it was 0.0296m (2.96 cm). The measured
value we calculated with a ruler in exercise 2 was 4.1 cm. These values are quite different, and
definitely out of the uncertainty of the value we measured with the ruler. The uncertainty of the
polynomial order-2 graph does include the measured value from exercise 2, however. Despite
this, the values are still very different. The R^2 in both the polynomial order-2 graph and the
linear fit graph are close to 1.00, which means that they are slightly reliable, since they are still
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relatively far from 1.00. As a result, it cannot be guaranteed that the values we calculated are
comparable to the value we measured in exercise 2, which is know is correct because we
conducted it ourselves using a simple method. The larger
r
values can be explained by the iOLab
wobbling within each throw we took, altering the acceleration values, as well as the fact that we
only took data from five throws, which is inaccurate because the trendline will then be greatly
affected by the outliers.
END OF REPORT