29-Mar-2017: Centripetal Acceleration vs Angular Frequency

Lab 8: Centripetal Acceleration vs Angular Frequency
29-Mar-2017
Idea: Model and plot a relationship between Force, mass, radius, and rotational velocity by taking data from a mass on a rotating disk under various conditions.

Summary
1) As a class, we observed a spinning disk, with a mass on a string attached to its center. We recorded data at various masses, lengths of the string (radius), forces, and power settings. We did this via a force sensor attached to the mass, and a photo gate at the edge of the disk, recording the time over 10 revolutions.
2) Then, we had to organize our data in order to create three graphs, whose slopes would give me mass, radius, and m*r, because as it would turn out, our value of omega was not linear enough. In order to do this, we arranged this formula to get my graphs using Excel.

Analysis:
I began to rearrange/organize my data, so that it would fit the cases where 1) mass and the power of the machine were consistent 2) radius and power were consistent and 3) mass and radius are constant. But first, I had to calculate the average time for each individual case by subtracting the first time interval from the time at the 10th, and dividing by 10. Then, I calculated rotational velocity, which was 2*pi over the average time. In addition, for convenience' sake, I converted all of the radius measurements to centimeters. Overall, this is what I observed by organizing my table:
Case 1: As the radius increased, force clearly increased. Omega was constant until the radius of 0.343m.
Case 2: As the mass increased, the force also increased, and Omega was not consistent.
Case 3: As the power increased, the force also increased, and again Omega was not consistent.

Afterwards, I began graphing my data exactly as I described in number 2 of my summary, using each of my test cases. slopes would give me mass, radius, and m*r. Afterwards, I could compare these values to my actual data for mass, radius, and m*r. 

As I could see by comparing these results from that of the table (shown in red), none of my results strayed far from the experimental values. In fact, without yet having done the percent error calculation, most of my results appeared close to within 8%.

After actually doing the percent error calculations for each of the test cases, this was confirmed. I got percent errors of 9.3%, 2.5%, and 1.9%, respectively for each case.

Conclusion/Uncertainty:
Mass and radius were demonstrated to be directly proportional to Force, and thus angular velocity increases as Force increases, and decreases as mass or radius increases. Additionally, I was able to draw from this the notion that my data could still come out somewhat accurate by running multiple cases despite the larges sources of uncertainty. Overall, I discovered that under conditions where angular velocity is too unreliable to use as a constant, I could still somewhat compensate for the uncertainty by solving for the product of its two components, m and r, instead. 
In the end, there were some glaring sources of uncertainty, the largest of which came from the rotating disk itself.
1) First, the disk had a glaring wobble to it, which only got worse as the power was cranked up. The sheer instability definitely affected the force readings on the sensor, as the non-average force readings were all over the place in logger pro. Inaccurate force readings would subsequently hurt the accuracy of my three graphs, as I noticed various outliers within them.
2) Of course, the measurements of the radius had an uncertainty to them, possibly to within 1 cm. Just like the force readings, this too would affect my graphs, but clearly not as severely.
3) In addition, although we assumed that the power would remain constant to what we chose, the power supply naturally has slight uncertainty in its measurement over time. This had consequences to everything in my data table from the time, angular velocity, etc.