15-May-2017: Angular Acceleration

Lab 16: Angular Acceleration

15-May-2017

Idea:

1. Use a torque-hanging mass system to demonstrate the effect of changing the hanging mass, the radius of the hanging mass' torque, and changing the rotating mass. 
2. Use this data to derive an expression for the moment of inertia for each of the disks, and solve.

Summary:

The torque-hanging mass system

Part 1

1. The first thing that we did was take measurements of all of our equipment. We took the mass and diameter of the disks, the small and large torque pulley, and the mass of the hanging mass.
2. In LoggerPro, we set the sensor to record at 200 counts per rotation. Then, we obtained graphs of angular position and angular velocity vs time.
3. Afterwards, we look at the effect various changes in the system have on the angular acceleration recorded by our sensor. We do this by coiling the string around the disk, and releasing the gas valve so that the hanging mass will rise, descend, multiple times.
1. To do this, we first record the α_down, α_up, and calculate their average with a single hanging mass, with a small torque pulley, with a steel topmost disk.
2. Then we do the same with a twice the hanging mass, with a small torque pulley, with a steel topmost disk.
3. Next, we have thrice the hanging mass, with a small torque pulley, with a steel topmost disk.
4. Now, we again have a single hanging mass, but with a large torque pulley, with a steel topmost disk.
5. Then a single hanging mass, with a large torque pulley, but with an aluminum topmost disk.
6. Lastly, we have a single hanging mass, with a large torque pulley, but with an aluminum topmost disk AND a bottom steel disk.

Part 2

1. In order to find the moment of inertia of each of our disks, we utilize our data from Part 1 to derive an expression for the moment of inertia, and solve.

Analysis:

Part 1

The purpose of the first part of this lab was to study the effects of various changes in the system, and how they ultimately affected the acceleration of the system. To that end, I discovered various things.

As shown in my data table for the various experiments two images below, changing the mass, changing the radius of the pulley, changing the mass of the disks, all had a direct impact on the angular acceleration.

• Doubling the mass doubled the angular acceleration. Likewise, tripling the mass tripled the angular acceleration, by about 1.97 times and 2.92 times, respectively. The "initial" condition of the system was: 1 hanging mass, a small torque pulley, and a top steel disk.
• Increasing the radius of the torque pulley by approximately 50% (from .0131 m to .01812m) doubled the angular acceleration of the system by by approximately 1.93 times..
• Changing the top steel disk to an aluminum steel disk that was 34% as heavy nearly doubled the angular acceleration by 1.921 times.

Part 2

The purpose of the second part of this lab was to first examine how friction could complicate our deriving of an expression for the moment of inertia (although we ignore it in our actual calculations), and more importantly solve for the moment of inertia for each of our disks. Using the measurements we obtained in part 1 will allow us to solve for the moment of inertia for each of our disks--the aluminum, top steel, and bottom steel. While our measurements let us directly solve for the inertia of 

• Both top+bottom steel, 
• Top steel by itself
• The top aluminum disk 

We cannot directly solve for the inertia of the bottom disk. However, because we can find the inertia of both top and steel disks directly, we can subtract from that the inertia of the top steel in order to solve for the bottom steel disk.

Because the bottom steel disk and the top steel disk have similar masses and identical dimensions, I expected their moment of inertia to be relatively identical as well--and they seemed to be, with a final percent error of approximately 18.1%.

Measured Data:

Mass and radii of all masses in the system

Data recorded from Logger Pro

Graphs:

Expt #1

Expt #2

Expt #3

Expt #4

Expt #5

Expt #6

Calculated Data:

Calculated Data table

Deriving expression for moment of inertia

Calculating inertia of the aluminum disk, and the top steel disk.

Calculating the inertia of the total steel-disk system, and then solving for the inertia of the bottom steel disk by subtracting the inertia of the top steel disk.

Conclusion:

Based on my analysis of my data in part 1, I determined that angular acceleration does indeed change directly proportional to mass and radius, and furthermore, was able to see that such changes were reflected in the changing of various components of the disk-machine. Lastly, I was able to calculate the various inertia of the disks in the system using the data I found in part 1.

Apart from the usual minor sources of error - with our measuring scale, the imperfect string, and a slightly inconsistent power source, the biggest source of error was friction - its impact could be seen on our graphs, as our angular velocity continued to decrease over time, indicating a gradual loss of energy. This means that our values of angular acceleration read by LoggerPro is inaccurate. In addition, friction would complicate our calculations for the moment of inertia. All in all, ignoring friction in our calculation for inertia, and while recording our data, lead to a final percent error of approximately 18.1%