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PHYS 161204
The moment of Inertia Abstract
11/5/2019
Dr. Jason P. Lee
In this lab, we studied the relationship between torque, a moment of inertia and angular
acceleration of a rod with two masses attached. The experiment began by setting up the pulley so
that the rotary motion sensor turns counterclockwise as the mass on the pully drops. The distance
r(m) with the corresponding masses was measured and used to create a table. Distance r was the
distance from the point mass to the axis of rotation. We created three graphs of angular position
(!), angular velocity !, angular velocity, (#) and angular acceleration ($) against time. We
determined the average angular acceleration (α) to be 2.05754 %&'/(2, the average applied torque
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(τ) to be 0.012176 Nm and the average moment of inertia to be 0.005686 kgm2. We calculated the
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standard deviation of the experimental moment of inertia to be 0.005232 kgm2. Ns = 1.060509.
The value we obtained for the moment of inertia was more than one standard deviation from the
theoretical value (0.0112 kgm2) and hence demonstrate an agreement with the theory. In
conclusion, the mean value of 0.005686 kgm2 with an uncertainty of 0.005232 kgm2 the results
are in agreement with the theory on the interpretation of Newton's law. Newton's law on the angular
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motion was conserved in this experiment.
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Physics 161
Rotational Motion and Angular Momentum
Introduction
You will be exploring various aspects of rotational motion in this lab. In Part I, you will use
Video Point to examine a twodimensional circular motion of an object. You will investigate the
relationships among acceleration, velocity and position of the object. In Part II, you will observe
the conservation of angular momentum by drop an object onto a rotating disk. You will measure
the initial and final angular velocities and calculate the initial and final angular momenta and
their uncertainties to test if angular momentum is conserved during this collision.
Reference
Young and Freedman, University Physics, 13th Edition: Chapter 10, sections 15.
Theory
Part I: If an object is moving around a circle at any given instant, the distance of the object to
the origin must be constant. Therefore:
𝑟=
𝑥! + 𝑦!
(1)
where x and y are the coordinates of the center of the object and r is the radius of the circular
path. Uniform circular motion can be described as the motion of an object in a circle at a
constant speed. The speed at any given instant can also be calculated as:
𝑣=
𝑣!! + 𝑣!!
(2)
where 𝑣! is the velocity in the 𝑥 direction, 𝑣! is the velocity in the 𝑦 direction, and 𝑣 is the
tangential velocity of the object. The angular velocity is expressed as:
𝜔=
𝑣
𝑟
(3)
The acceleration of the object is called centripetal acceleration which points towards the center
of the circle. This acceleration can be calculated using the following equation:
𝑎=
𝑣!
= 𝑟𝜔!
𝑟
(4)
You will use these equations to analyze the video of an object moving around a circle.
Part II: Angular motion is similar to translational motion in many ways. For example, instead
of linear velocity, 𝑣, we have angular velocity, 𝜔, of a rotating body. Similarly, there are
equivalent angular quantities for acceleration, momentum, force and mass. In the case of a
rotating mass, we use the quantity moment of inertia, 𝐼, to describe how readily that body can
undergo angular acceleration. The moment of inertia, 𝐼 = 𝑚! 𝑟!! , depends upon the mass of the
object and how the mass is distributed about the axis of rotation.
In the absence of any external torques, such as friction, on an object, angular momentum is
conserved, i.e. initial and final angular momenta are equal. Conservation of angular momentum
means that if the moment of inertia, I, of a rotating object is changed, the angular velocity, 𝜔,
will change by some factor so that the total angular momentum, L is conserved:
𝐿! = 𝐿!
(1)
𝐿 = 𝐼𝜔
(2)
𝐼! 𝜔! = 𝐼! 𝜔!
(3)
The moment of inertia for a disk of uniform density rotating about its center is:
1
𝐼 = 𝑀𝑅!
2
(4)
Procedure
Part I: Object moving around a circle.
You should open the UCM.avi file into the Video Point program. This is a video of a turntable with
two dimes placed on top of it. As the turntable rotates at constant angular speed, the dimes will
travel in circular paths.
1. Click on the VideoPoint Physics Fundamentals icon on your screen; you may need to go to
the Start bar and select Programs>Video Point.
2. Once you have opened VideoPoint Physics Fundamentals, select File>Open to open the
video file. The file is located under Desktop > videos_shortcut. Select Quick Time in the
lower righthand box and open the file UCM.mov. You should see the turntable with two
dimes, see Figure 1.
Figure 1: Open file UCM.mov.
3. In the calibration step, the diameter of the turntable is 30.5 cm, see Figure 2. In the setup
pane, set the origin of your coordinates at the center of the turntable. The point you mark
should be the tip of the shaft that comes out of the middle of the turntable, see Figure 3.
Figure 2: Calibration.
Figure 3: Set the origin of the coordinate system.
4. Place your cursor over the center of mass of the dime closest to the origin. Mark the center of
this dime in each frame by clicking the mouse. The video will automatically advance to the
next frame. Continue in this fashion, clicking for a data point in each frame until you reach
the end of the video. A dialog box will pop up and ask if you want to take another data set,
click on “Yes.” Repeat the above steps for the second dime.
5. Select File>Export to export the data in .xls format. Save your data to the desktop. Open the
data file in Excel.
6. Your spreadsheet should have separate columns for t, 𝑥! , 𝑦! , 𝑥! , and 𝑦! . The time in seconds
is in the first column. The second and third columns include the 𝑥 𝑦 positions for the first
dime, and the 𝑥 𝑦 positions of the second dime should be listed in the fourth and fifth
columns.
7. In a new column in Excel, calculate the radius of the circular path for the first dime using
equation 1. . Repeat this for the second dime.
8. Now calculate the x and y components of the velocity for both dimes just as you did in the
Projectile Motion lab. Calculate the speeds, 𝑣! and 𝑣! , for each time step using equation 2. Is
𝑣! constant? Is 𝑣! constant? Is 𝑣! equal to 𝑣! ? For the closest dime to the center of rotation,
produce plots of xposition, yposition and 𝑟 versus time, place all these in one graph. Now,
for the same dime, plot 𝑣! , 𝑣! , and 𝑣 versus time, place all these in one graph.
9. Finally for each time step use the linear speed 𝑣 and radius 𝑟 to calculate the angular speed 𝜔
at each time point for both dimes. From the time series calculate the average and standard
deviation for the angular speed for each dime. Add the uncertainties in quadrate as you did
for the momentum conservation lab and use the total uncertainty to assess whether the two
dimes move with the same angular speed. If not, what could account for this difference?
10. Calculate the centripetal acceleration using equation 4 for both dimes.
Part II: Conservation of Angular Momentum
1. Measure and record the radius and mass of the disk and calculate I.
2. Open Pasco Capstone. Choose Create Experiment. Select Hardware Setup and set a
Rotary Motion Sensor, see Figure 4. Set the Rotary Motion Sensor to 20 Hz at the bottom
near Recording Conditions
Figure 4: Rotary motion sensor.
3. Click on Graph under the Display Tab to the right and set a plot area. Set the 𝑦 axis of the
graph to Angular velocity 𝜔 in radians/s.
4. Now with the disk attached to the rotary motion sensor, give the system a spin in the
clockwise direction. Press Record to begin collecting data.
5. Carefully drop a second disk on top of the spinning disk. Make sure that there is very little
excess motion when the disk lands. It may take a few practice trials to achieve minimal
disturbance of motion. Wait a couple of seconds and press Stop.
6. You should get a graph resembling the one shown in Figure 5.
Figure 5: Angular velocity vs time.
7. As you did in the linear momentum lab export at least five points just before and after the
sudden change to excel.
8. Calculate the initial and final angular momenta from the angular velocity measurements by
multiplying by their respective moments of inertia. Calculate the mean and uncertainty in
these values from the data. Based on these values is angular momentum conserved?
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