9/28/2016 - Centripetal Acceleration vs Angular Frequency (Lab 8)

PURPOSE:
To determine the relationship between centripetal acceleration and angular speed.

THEORY:
We know that the net centripetal force of a rotating object is given by F = mrω². If we compare this equation to Newton's second law, F = ma, then it becomes apparent that if the net force equal mass times acceleration, then the net centripetal force must be equal to mass time centripetal acceleration. Therefore, a_cent = rω² = v²/r

APPARATUS:
The apparatus was assembled beforehand for this lab.

Wooden rotating disk with power supply and voltage regulator.

Setup:

1. Place the wireless accelerometer on the wooden disk.
2. Adjust the voltage on the power supply, turn the scooter motor on, and let the disk come to a constant speed.
3. Collect period and acceleration data for a variety of rotational speeds by varying the voltage.

EXPERIMENTAL PROCEDURE:

1. Use a photogate to measure the rotational period of the given mass.
2. Determine the mass of the object on the rotating disk.
3. Measure the radius of the object from the center of the disk.
4. Record the mean force exerted on the rotating mass.
5. Calculate the angular speed of rotating mass.
6. Repeat these steps 7-8 more times w/ varying ω and record them on a spreadsheet.

DATA/GRAPHS:

Recorded data

F vs. mω² 

F vs. rω²

F vs. ω²

Sample Calculation:
F = mrω²
F/(mr) = ω²
ω = [F/(mr)]^1/2
F = 2.35 N
m = 0.2 kg
r = 0.58 m
ω = 4.5 rad/s

ANALYSIS:
For this experiment, we made graphs of F vs. mω², F vs. rω², and F vs. ω². Ideally, each graph was supposed to have a slope of r, m, and mr respectively. If the slopes of the graphs match the variable that it is missing from the equation for centripetal force, then we have verified that the relationship between angular speed and centripetal acceleration is credible. The reason this method works is because it allows us to see how a change in a single variable can effect the others. For example, from F vs. rω², we can see that the slope of the linear fit of the graph is 0.2026, which is very close to the mass of the object on the wooden disk: 0.2 kg. The same hypothesis held true for the F vs. ω² graph, which yielded a slope value of  0.1249; a very close approximation of mr = 0.116. On the other hand, when we tested it with the graph of F vs. mω², we found that our prediction did not hold to be true. In fact, our data shows that the slope of the graph was 0.8249 when it  should have been closer to 0.58: the measured radius of the spinning mass.

CONCLUSION:
The data that we collected allowed us to analyze the relationship between centripetal acceleration and angular speed. More specifically, the graphs we created allowed us to illustrate how each of the variables we collected -- m,r, ω -- can effect this relationship. By comparing the slope of each graph to its corresponding "missing variable", we were able to successfully prove that centripetal acceleration did in fact depend on angular speed, among other factors. This mathematical relationship, however, was not entirely consistent with our data. Therefore, there must have been some uncertainty that was influencing our results. The most likely source of this discrepancy was human error. After comparing our group's data to another, we discovered that one of our data points for force varied by .05 N. Additionally, the other group used more data points in their graph for F vs. mω², which yielded a slope that was relatively more consistent with the radius of the mass around the disk. Another likely source of uncertainty in our calculations was our assumptions about the experiment. For example, when we began the experiment, we assumed that friction was negligible, when in reality there were many sources of friction because the wooden disk was in contact with several wheels while spinning.

GROUP MEMBERS: Xavier C., Billy J., Matthew I.