Experiment 8.1
Rotational Inertia
AP Physics 1 Lab:
Purpose
In this experiment you will experimentally measure the rotational inertia of a solid disk and a ring.
Theory
The calculated rotational inertia can be calculated by the rotational equivalent of Newton’s Second Law (F=ma):
 = I  OR after rearranging:  /  = Icalculated
As long as both the applied torque and resulting angular acceleration are known. This will provide you with an
experimental measure of the rotational inertia; to find the theoretical value we can use the formula for the
rotational inertia of a disk:
Itheoretical = ½MR2
,where M is the mass of the
platter and R is the radius of the platter
Prelab Question
1. How will the small aluminum step pulley affect your
theoretical result? Will it tend to make the theoretical
value too large or too small?
Procedure
1. Set up the apparatus as shown in the diagram to the
right. Use the bubble level to ensure that the
apparatus is level.
2. Plug the Smart Pulley into digital channel one of the
Photogate Adapter and start up the Capstone
program on the computer to record the angular
velocity of the Smart Pulley.
Load the “08-01
Rotational Inertia Lab Setup.cap” file to setup the
program.
3. Attach a 1.5 meter long piece of string to the step
pulley and wind it up on the largest of the three
spindles. You may wish to tie it through the tiny hole
in the spindle. Attach a 50-gram mass holder to the
other end. You may wish to measure the radius of
the largest spindle at this time.
4. Click the “Record” button to start recording data and
simultaneously release the mass, stopping the data
recording just before the mass hits the floor.
5. Maximize the graph window then zoom in on the
graph with the
tool. If you need to zoom in more
use the
tool by dragging a selection rectangle
over the area you wish to zoom in on.
6. Determine the angular acceleration (the slope of the
graph of angular velocity) by dragging across the
section of interest and then clicking the statistics
button
tool and then the
drop-down menu.
Choose 'curve fit' and then 'linear fit' – the value of a2 then indicates the slope (the angular acceleration).
Record the angular acceleration and the hanging mass on your data table.
7. Repeat steps 5 - 7 using hanging masses of 100 and 150 grams.
8. Repeat steps 5 - 8 using the other two step pulleys.
9. Flip the platter over and place the ring on top. Using the smallest radius step pulley and 100 grams,
measure the combined rotational inertial of the two platters using procedure steps 4 - 7. Record this data
in the second data table.
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Experiment 8.1
Analysis
The torque can be calculated by the following formula:
 =Fd
where F is the force in Newtons and d is the moment arm (the
distance from the point the force is applied to the pivot point).
In this case d is the radius of the spindle (d) and the angle is
always 90 degrees, so the equation can be reduced to the
force times the radius of the spindle.
Data Table
Radius of platter: 0.125 m
Mass of platter: 0.962 kg
Platter
Trial #
Step Pulley
Radius (m)
Hanging
Mass (kg)
0.025
0.025
0.025
0.020
0.020
0.020
0.015
0.015
0.015
1
2
3
4
5
6
7
8
9
0.050
0.100
0.150
0.050
0.100
0.150
0.050
0.100
0.150
Force on
Hanging
Mass (N)
0.55
1.05
1.55
0.55
1.05
1.55
0.55
1.05
1.55
Radius of ring: 0.0625 m
Torque
(Nm)
0.0138
0.0263
0.0388
0.011
0.021
0.031
0.00825
0.0158
0.0233

Angular
Acceleration
α (rad/s2)
1.75
3.39
4.97
1.41
2.78
4.02
1.07
2.08
3.05
Experimental
Rotational
Inertia (kgm2)
I=/ α
0.00786
0.00774
0.00780
0.00780
0.00755
0.00771
0.00771
0.00757
0.00762
Theoretical
Rotational
Inertia (kgm2)
I = ½ M R2
Error (%)
0.00752
0.00752
0.00752
0.00752
0.00752
0.00752
0.00752
0.00752
0.00752
4.52
2.93
3.72
3.72
0.399
2.53
2.53
0.665
1.33
Theoretical
Rotational
Inertia (kgm2)
I = I platter +
MR2
Error (%)
0.0103
0.0103
0.0103
0.0103
0.0103
6.80
2.91
0.971
0
0
Mass of ring: 0.713 kg
Platter & Ring
Trial #
1
2
3
4
5
Step Pulley
Radius (m)
0.025
0.025
0.025
0.025
0.025
Hanging
Mass (kg)
0.020
0.040
0.060
0.080
0.100
Force on
Hanging
Mass (N)
0.25
0.45
0.65
0.85
1.05
Torque
(Nm)

0.00625
0.01125
0.01625
0.02125
0.02625
Angular
Acceleration
α (rad/s2)
0.569
1.06
1.57
2.06
2.56
Experimental
Rotational Inertia
(kgm2)
I=/ α
0.0110
0.0106
0.0104
0.0103
0.0103
Questions
1. Graph the applied torque versus angular acceleration for platter. What is the slope of the graph? What
is its physical significance?
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Experiment 8.1
The slope of the graph is 0.0078 N*m*s^2. The physical significance is that the slope correlates with rotational
inertia. In the equation  = I * α, I is the slope of between the relationship of torque and angular acceleration. This is
validated through the calculated rotational inertia values which are extremely close to the slope generated from the
graph. Additionally this relationship is evident through the units rotational inertia has which are kg*m^2, while the
slope’s units are N*m*s^2 where N is equal to (kg*m)/s^2 when substituted in ((kg*m)/s^2)*m*s^2 equals to
kg*m^2, which confirm the relationship between the slope vs rotational inertia.
2. Graph the applied torque versus angular acceleration for platter and ring. What is the slope of the
graph? What is its physical significance?
The slope of the graph is 0.01 N*m*s^2. Just like the previous graph for the platter, the inertia is the slope of the
graph. The inertia in this scenario is for both the platter and the ring while the previous graph was only for the
platter. This means that by subtracting the platter’s “slope” (inertia) from the platter + ring’s inertia, it would leave
only the ring’s rotational inertia. By going through with the calculations, 0.01 kg*m^2 – 0.0078 kg*m^2 = 0.0022
kg*m^2 for the ring’s rotational inertia.
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Experiment 8.1
3. What is the theoretical Rotation Inertia for the ring? What was your experimental value for the ring’s
Rotational Inertia? [Hint: Subtract the I for the disk from the I for the disk and ring]
The theoretical rotational inertia for the ring for the ring would be derived from subtracting the theoretical rotational
inertia of the platter from the theoretical rotational inertia of the platter and ring. Doing the calculations, 0.0103
kg*m^2 - 0.00752 kg*m^2 = 0.00278 kg*m^2. The theoretical rotational inertia of the ring would be 0.00278
kg*m^2. The same calculations can be done with the experimental values to find the experimental rotational inertia.
This is done by using the slopes of the best fit lines. As established in previous questions the slopes of the graphs
between applied torque and angular acceleration is equivalent to the experimental inertias. Thus, 0.01 kg*m^2 –
0.0078 kg*m^2 = 0.0022 kg*m^2, which means the experimental rotational inertia of the ring is 0.0022 kg*m^2.
4. A disk, a ring, and a bar have approximately the same mass and radius. Arrange them from the smallest
rotational inertial to the largest.
The smallest rotational inertia would be the bar, then the disk, and then the ring. This is because the bar has a
rotational inertia formula of I = 1/12*M*L^2 which then when put in terms of R, I = 1/12*M*(2R)^2, when
simplified is equal to I = 1/3*M*R^2. Then the disk has a rotational inertia equation of I = ½*M*R^2. Then the
ring which has the rotational inertia equation of I = M*R^2. Given that they have approximately the same mass
and radius, it negates the M*R^2 in each rotational inertia equation leading to the following, bar – 1/3, disk –
1/2, ring – 1.
5. Was the tension really constant for a given hanging mass? (You may wish to consider the hanging mass’
acceleration) What effect, if any, would this have on your results? Which hanging mass would tend to give
you the best results?
Extra Credit: Write an equation that takes into account the reduced tension in the string due to the fact
that the hanging mass is falling (accelerating downward). This reduces the tension and
therefore torque on the platter. Solve this new equation for the rotational inertia of the disk.
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