Date: Apr. 4th, 2024
Student Name(s): Kasey White
Rotational Dynamics
Physics 12
Experiment 10
Lab Report
Spring 2024
Purpose/Goal: The purpose of Experiment 10 was to investigate the principles of rotational
dynamics. Explicitly, the experiment included measuring the masses of a ring and a disk, as well
as their diameters and the diameter of the apparatus shaft. The angular acceleration of both
the disk and then the combination of the disk and the ring were also found, which assisted in
finding the moment of inertias for the systems. Finally, the angular momentums were
determined and compared through the application of angular speed. The manipulation of
equations involving moment of inertia, angular acceleration and torque, and conservation of
angular momentum allow experimentation to confirm the principles of rotational dynamics.
List of Materials:
1. Rotational Apparatus
2. Disc
3. Ring
4. String
5. Plastic Washer
6. Assortment of Masses
7. Hanger
8. Level
9. Triple Beam Balance
10. Vernier Caliper
11. Ruler
12. Computer
13. Capstone Program
14. Exp10.Data File & Experiment_10 Capstone File
15. Excel
16. Writing Utensil
17. Paper
Diagram of Apparatus:
Procedure A (The Disk):
1. Open Exp.10Data in the shared directory on the laboratory computer. This is will your
experiment results will be recorded.
2. Measure the mass of the ring and the disk three times using the triple beam balance.
The excel should calculate the average for you, convert it into kilograms.
3. Measure the diameter of the disk three times. The excel should calculate the average
for you, convert it into meters.
4. Due to the ring's thickness you need to measure the diameter of the inside and outside
of the ring. Measure it three times and record the results in the excel sheet. The average
will be calculated for you in the excel sheet, convert into meters.
5. Calculate the theoretical values of the moments of inertia of the disk as well as the disk
and ring together by using the formulas in Table 8.1.
6. Measure the diameter of the stand in the shaft three times using the vernier calipers.
The excel sheet will find you average, convert it into meters. Then calculate the average
radius of the shaft.
7. Open the Experiment 10 Capstone file in the shared directory. You will use this to
measure angular speed in your experiment.
8. The hanger provided should be attached to the end of the string.
9. Put the disk onto the shaft of the rotational apparatus and tighten the knob on the
device to secure it.
10. Use the Capstone file to measure the angular acceleration of the rotating disk three
times by following the steps below.
a. Put a 200g mass on the hanger and record the total mass (including the 50g
hanger) in the excel sheet.
b. A washer attached to a string is provided. Slip the washer on the small post of
the rotational apparatus shaft and the other end over the mass hanger. Wrap
the string around the shaft a few times and stretch the string over the pulley.
Wind up the string around the shaft until the hanger is a foot and a half from the
ground.
c. Hit the record button on the Capstone program and release the mass to let it fall
on the floor. Let the disk continue for a few rotations after the mass has fallen
and then hit the stop button.
d. Click on Scale Axis in the Capstone program to make the graph larger. Then,
after clicking the highlight button on Capstone, drag the shaded box over two
linear portions of the graph (one at a time). Adjust the size of graph to include all
data points if necessary.
e. Click on the Curve Fit button in the menu bar and select Linear: mt + b. The slope
and intercept values for the best fit line will appear.
f. Record the values of aplus and aminus from the lines of best fit. Note that the
two slopes of the best fit lines are equal to aplus and aminus.
11. The average angular acceleration should be found for you in the excel sheet with the
recorded data after completing all three trials.
12. Determine the average angular acceleration using the same procedure as above but
with the masses: 350g, 450g, 550g, and 650g. There should be a total of 15 angular
acceleration values and five average angular acceleration values by the end. All of the
data should be included in the excel sheet and lab report.
13. Plot the angular acceleration values as a function of the hanging mass on Excel. Find the
slope of the best fit line using the trendline function.
14. Use equation 10.13 to determine the moment of inertia for the disk.
15. Compare the experimental value to the calculated value of moment of inertia through a
percent error calculation. Are the values close? If not, why?
Procedure B (The Disk and Ring):
1. Using the same device and apparatus setup as the previous procedure, place the ring on
top of the disk.
2. Follow through the experimental procedure of Part A (*see step 10. Above) Start off
with 250g of total hanging mass for three trials. Record the average angular acceleration
in the excel sheet. The average should be calculated for you. Then, just like in Part A,
repeat for 350g, 450g, 550g, and 650g to total fifteen angular acceleration and five
average angular acceleration values.
3. Highlight the a vs. m chart in the excel sheet and make it into a graph.
4. Use the trendline function on the graph to obtain the line of best fit and its slope.
5. Determine the experimental moment of inertia of the disk and ring together using Eq.
10.13.
6. Compare the experimental value to the calculated value of moment of inertia through a
percent error calculation. Are the values close? If not, why?
Procedure C (Conservation of Angular Momentum):
1. Remove the ring that is on top of the apparatus from the previous step, keep the disk.
2. Place the level on top of the disk to make sure that the apparatus is level. If not, use the
knobs on the apparatus until it is.
3. Open a new Capstone file titled Experiment 10b that is in the shared directory. This will
be used to measure angular velocity.
4. Use your hands to spin the disk and click record on Capstone. After 10 seconds drop the
ring into the disk, making sure it goes into the groove. Wait another 10 seconds then hit
stop.
5. Scroll down in the table to find a drop in the angular velocity which will occur during
collision. Record the angular velocity before and after the collision which will act as
winitial and wfinal.
6. Use equation 10.15, the initial angular velocity, and experimental moment of inertia for
the disk to calculate the angular momentum of the system before collision.
7. Use equation 10.16, the final angular velocity, and the average experimental value of
the disk and ring to calculate the angular momentum of the system after collusion.
8. Compare the angular momentum for before and after the collisions using the percent
difference equation. Based on the result determine if angular momentum is conserved.
If not, why?
General Data:
Part A Theoretical Calculations:
Part A Disk Data & Graph:
Part A Disk Results:
Part B Disk and Ring Data & Graph:
Part B Disk and Ring Results:
Part C Data: Here you need to include all the table you created for the capstone data provided.
The table should be done in Excel, so you need to copy the completed table here in this
document.
time (s)
Angular speed (rad/s)
11.55
15.37
11.74
10.70
Angular speed initial = 15.37, Angular speed final = 10.70 rad/s
Part C Results:
Questions:
Compare the experimental value of the moment of the inertia for the disk with the theoretically
calculated value for the disk by calculating the percent error between the theoretical value and
the experimental value. Are the values close to each other? If not, discuss why they are not close
to each other in value.
The experimental value for the moment of inertia for the disk is .0089 kgm^2. The theoretical
value is .0049 kgm^2. The percent error was calculated for these vastly different numbers and
ended up being around 92%. There could be many reasons for this large percentage error, but
the leading factors are the inaccuracy of the triple beam balance and confusion caused by
labels on a particular mass used in the experiment. This idea will be elaborated on below in the
error analysis.
Compare the experimental value of the moment of the inertia for the disk and ring together
with the theoretically calculated value for the disk and ring together by calculating the percent
error between the theoretical value and the experimental value. Are the values close to each
other? If not, discuss why they are not close to each other in value.
The experimental value for the moment of inertia for the disk is .0084 kgm^2. The theoretical
value is .0139 kgm^2. The percent error was calculated and ended up being around 50%. Again,
there could be many reasons for this large percentage error, but the leading factors are the
inaccuracy of the triple beam balance and confusion caused by labels on a particular mass used
in the experiment.
Compare the angular momentum of the system before and after the collision, by calculating the
percent difference between the two values. Determine if the angular momentum is conserved
and justify your answer? If it is not conserved, discuss why this may be so.
The initial angular momentum or momentum of the system before collision is .1377 kgm^2/s
and final angular momentum or momentum after collision is .1487 kgm^2/s. These two values
are relatively close and when plugged into the percent difference equation, resulted in 7.7%.
This indicates that momentum is conserved. .1487 and .1377 are close in value and could have
been closer if not rounded. These velocities were also handpicked, so any angular velocities in
that were around 11 seconds could have been used, and could have resulted in a lower percent
difference.
Conclusion:
The goal to explore the principles of rotational dynamics was successful. After recording the
mass and diameters of two objects, a disk and a ring, and utilizing a derivation of Newton’s
second law to relate the objects to torque and angular acceleration, the moment of inertia for
the systems were found. The theoretical moment of inertia is found through the known
equation for inertia of a disk and was applied before experimentation to get .0046kgm^2. After
experimentation and the realization that inertia can be found by relating gravity, the radius of
the shaft, and the slope of an a vs. t. graph it resulted in an inertia for .00895kgm^2. Despite
these contradictory values, I learned how to use and manipulate the moment of inertia for a
disk. Similarly, the moment of inertia was found for a system that included the disk and the
radius. When the properties of the two objects are combined, they can be used to find the
inertia of the system as a whole. The theoretical inertia for the disk and ring was .0069kgm^2
while the experimental was .0139kgm^2. The values do not match but this is expected since the
numbers of the inertia of the disk were off. Again, despite the contradiction I still learned how
to calculate and manipulate the moment of inertia for a system of a disk and ring. Lastly, the
conservation of angular momentum was tested. Angular momentum can be found using the
previously calculated inertias and angular velocity. Once the initial and final angular velocities
were found (15.39 and 10.70 rad/s) the result was an initial angular momentum of .1377
kgm^2/s and final angular momentum of .1487kgm^2. These values are relatively close and
indicate that the rotational momentum is conserved. Overall, through rotational motion I was
see the relationship between moments of inertia and conservation of momentum.
Error Analysis:
This experiment resulted in very high percentage errors when finding the moments of inertia,
and as explained in the “Questions” section the route cause involved mass. Firstly, the ring and
disc were heavier than the triple beam balance’s capacity. An extra mass had to be attached to
the scale for measurements, yielding a higher margin of inaccuracy. Secondly, the added mass
had two labels on it: one that read 295 grams and another as 1000 grams. Initially I performed
calculations with the assumption that the mass was 1000 grams and it resulted in a significantly
lower percentage error. However, after lifting the mass and realizing that it could not be 1000
grams, the calculations were redone with the 295-gram value. This resulted in a higher
percentage error which is displayed in my lab report. Either way, the true mass of the mass is
still unknown and greatly affects the values for the moments of inertia. Other contributors to
error include friction which was caused by the string hanging over a gear and could affect the
angular acceleration, as well as rounding and calculation mistakes.