Lab 5: Moment of Inertia
Christopher Jones
CCID: cjones6
Student ID #: 1899048
Lab Partners: Nahil Nigar, Sina Kahsay
TA: Faustina Awafu
April 7, 2026
1: Introduction
​
1.1: Objectives
In this lab, the moment of inertia is explored using the rotation of a bicycle
wheel, and by connecting it to a pulley, can be related to linear nonuniform
motion.
​
1.2: Background
The moment of inertia is crucial to the understanding of physics, and is
one of the three fundamental rules of motion that Issac Newton established,
being that an object at rest remains at rest, and an object in motion remains in
motion unless acted upon by an external force. The objective of this lab is to
investigate the moment of inertia of a rotating bicycle wheel, and how it interacts
with Newton’s second law of inertia.
​
1.3: Equations
This lab uses the following formulae:
​
Force-acceleration formula: (Newton’s second law)
2
(𝐹 is force (𝑁), 𝑔 and 𝑎 are gravity and acceleration (𝑚/𝑠 ), m is mass (𝑘𝑔), and
𝑇 is tension (𝑁). 𝐹𝑁𝐸𝑇 = 𝑚𝑔 − 𝑇 = 𝑚𝑎 (when isolating for T, this becomes
𝑇 = 𝑚(𝑔 − 𝑎)). This can be made into a torque formula by multiplying by 𝑟,
representing the radius of the wheel hub (𝑚), giving τ𝑇 = 𝑟𝑇 = 𝑟𝑚(𝑔 − 𝑎) )This
formula can be modified to obtain torque and inertia: τ𝑁𝐸𝑇 is the net torque, τ𝑇 and τ𝑓
being torque from tension, and torque from friction respectively (𝑁𝑚). 𝐼 being inertia (
2
2
𝑘𝑔𝑚 ), and α being angular acceleration (𝑚/𝑠 ). τ𝑁𝐸𝑇 = τ𝑇 − τ𝑓 = 𝐼α
𝑎
This can be combined with the formula of angular acceleration α = 𝑟 to obtain the
formula τ𝑇 = 𝑟𝑇 = 𝑟𝑚(𝑔 − 𝑎) to linearize it, giving the formula and linear relation
𝑎
𝑟𝑚(𝑔 − 𝑎) = 𝐼 𝑟 + τ𝑓. A theoretical value of 𝐼 can be obtained using the following
2
formula: 𝐼 = 𝑀𝑅 , where 𝑀 is the mass of the wheel, and 𝑅 is the radius from the centre
of the hub to the middle of the tire.
2: Methods
​
2.1: List of materials
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-​
-​
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Bicycle wheel
Set of weights
String
Smart pulley
Device with LoggerPro
2.2: Procedures
In this experiment, my partners and
I connected the to the wheel via string.
The wheel is wound, spooling the string
to raise one end, and attaching the weight
to the raised end. This weight is then
released, causing the wheel to rotate. The
linear motion of the weight is measured
using loggerPro using the smart pulley,
specifically the velocity mass versus time.
Taking the slope of this gives the
acceleration, and can also grant the error
in acceleration as well. The loggerPro
graph should resemble as such:
3: Results
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3.1: Raw data
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​
​
To view the results and raw data, refer to the tables present in the appendix.
(Figs 3 and 4)
3.2: Sample calculations:
𝑎
To calculate the moment of inertia, take the formula 𝑟𝑚(𝑔 − 𝑎) = 𝐼 𝑟 + τ𝑓, and
𝑎
by using the values 𝑟𝑚(𝑔 − 𝑎) (torque) as 𝑦, and 𝑟 (angular acceleration) as 𝑥, a linear
graph can be formed. For example take the first trial’s values.
2
The mass, 𝑚 = 200𝑔 = 0. 200𝑔, the acceleration is 𝑎 = 0. 002707𝑚/𝑠 , and the
2
radius of the cylinder, 𝑟 = 1. 885𝑐𝑚 = 0. 01885𝑚, while gravity is constant at 9. 81𝑚/𝑠
−2
This gives a torque of 3. 70 * 10
−3
2
𝑁𝑚, and an angular acceleration of 3. 82 * 10 𝑚/𝑠 .
Doing so should result in a linear graph, and the values (refer to fig. 5 in appendix)​
3.3: Graph
When both the torque and angular acceleration are determined for all masses,
𝑎
the following graph of 𝑟𝑚(𝑔 − 𝑎) = 𝐼 𝑟 + τ𝑓 can be created:
​
​
Using the LINEST function, the moment of Inertia, 𝐼 , and the torque generated
from friction, τ𝑓, can be found along with their errors. They are 𝐼 = 0. 212 ± 0. 001 and
τ𝑓 = 0. 0065 ± 0. 0004 respectively.
3.4: Analysis
2
Using the formula 𝐼 = 𝑀𝑅 ,the uncertainty formula of if 𝑓 = 𝑐𝑥, then δ𝑓 = 𝑐δ𝑥,
and the values 𝑀 = 2. 0 ± 0. 2 𝑘𝑔, and 𝑅 = 67. 5𝑐𝑚 = 0. 675𝑚, the theoretical value of
𝐼 = 0. 91 ± 0. 09.
4: Discussion
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4.1: Results/Comparison
The recorded values of 𝐼 are significantly smaller than that of the theoretical
value of 𝐼 (within more than 5 margins of error). This is to be expected, as the formula
2
𝐼 = 𝑀𝑅 makes the assumption that all of the mass is concentrated at the very edge of
the wheel, maximizing the moment of inertia, when in reality this is not the case, causing
the measured values to result in a smaller moment of inertia. Friction, possibly generated
from the smart pulley or the overall rotation of the wheel, also causes a loss of energy,
reducing the amount of inertia measured, as demonstrated in the structure of the linear
function.
4.2: Uncertainties/Improvements
It is expected for measured values of 𝐼 to be smaller, but some steps can be
taken to mitigate these effects. The first is to use a wheel with more of its mass
concentrated at the edge of the wheel to better fit the theoretical model. The second
change that would make this experiment more accurate is to use a different measuring
device such as a rotary motion sensor to reduce the effect that friction has on the
experiment.
5: Conclusion
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5.1: Description of Lab
In this experiment, the rotational movement of a bicycle wheel and its relation to
the moment of inertia was investigated, and using its motion in relation to a series of
loads, it was possible to identify the wheel’s moment of Inertia
​
5.2: Main Result
The value 𝐼 = 0. 212 ± 0. 001 was obtained. While this was not found to be in
agreement with the theoretical value, this was determined to be a result of the weight
distribution of the wheel, and energy loss through friction in the system.
6: References and Acknowledgements
6.1: References
[1] Department of Physics, EN PHYS 131 Laboratory Manual, University of
Alberta, Edmonton, AB, Canada, Winter 2026.​
[2] Department of Physics, EN PHYS 131 Lab 5: Moment of Inertia, University of
Alberta, Edmonton, AB, Canada, Winter 2026.
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6.2: Acknowledgements
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Acknowledgements to my two lab partners, Nahil Nigar, and Sina Kahsay who
helped me gather the data, and process it in loggerPro.
​
7: Appendix
Values measured from experiment: (Fig 3)
Average
Average(m Acceleration
/s²)
uncertainty
Mass
(g)
Trial 1
Slope(m/s²)
Slope 1
Uncertainty
Trial 2
Slope(m/s²)
Slope 2
Uncertainty
200
0.002702
9.66E-06
0.002712
7.74E-06
0.002707
8.15E-06
0.0000059802804
0.0035455
28
1.06E-05
0.0000109450319
0.0043545
8
1.54E-05
0.0000101361050
0.0051105
2
250
300
350
0.003538
0.00441
0.005099
8.75E-06
1.92E-05
1.32E-05
0.003553
0.004299
0.005122
0.0000061876072
92
400
0.005937
1.50E-05
0.00591
1.40E-05
0.0000102839984
0.0059235
9
450
0.006778
1.37E-05
0.006712
1.87E-05
0.006745 0.00001158751268
500
0.007613
2.03E-05
0.007633
1.72E-05
0.007623
0.0000133284329
5
550
0.008456
1.97E-05
0.008428
1.96E-05
0.008442
0.0000138841717
4
600
0.009266
2.20E-05
0.009233
2.38E-05
0.0092495
0.0000162264429
3
Angular
acceleration
Uncertainty
Torque
Uncertainty
Dimensions: (Fig 4)
Diameter of Wheel:
67.5cm
Diameter of Cylinder:
3.77cm
Radius of Cylinder
1.885cm
Mass of wheel
2.0 ± 0.2 kg
Calculated values for the graph (Fig 5)
Mass(kg
)
Acceleration
Angular
accleration
Torque
0.2
0.002707
0.1436074271
0.03697349461
0.003823332922
0.0009814536
415
0.25
0.0035455
0.1880901857
0.04621291683
0.0049992065
0.0012267123
17
0.3
0.0043545
0.2310079576
0.0554509253
0.006154980879
0.0014719335
21
0.35
0.0051105
0.2711140584
0.06468775848
0.007211430141
0.0017171235
71
0.008353220793
0.0019622644
59
0.009511263384
0.0022073628
16
0.01075015372
0.0024524060
14
0.4
0.45
0.5
0.0059235
0.006745
0.007623
0.3142440318
0.3578249337
0.404403183
0.07392273681
0.08315611054
0.09238740323
0.55
0.008442
0.4478514589
0.1016176526
0.0119021618
0.0026974215
57
0.6
0.0092495
0.4906896552
0.1108464882
0.01304407579
0.0029423996
29