Lab 8: March 29, 2017: Centripetal Acceleration vs. Angular Frequency

Lab 8: Centripetal Acceleration vs. Angular Frequency

Amy, Chris, and John

March 29, 2017

Today's lab was about finding the relationship between centripetal force and angular speed, with multiple varying scenarios. Then graph our findings and find a graph with the best linear fit that shows us the best correlation. 

Theory: We know for an object moving in uniform circular motion, the direction of acceleration is towards the center. The force that corresponds with this acceleration is called the centripetal force and in equation, centripetal force is the ma. It can be interpreted as the net force. If we had different forces that contribute to the circular motion, the sum of those forces would be equal to your centripetal force.

In our lab, looking at the setup from the side of the disk, the free body diagram of the mass looks something like:  

We derived the equations in order to change our variables one at the time and figure out the best angular acceleration from the original formula given to us. The formula we used was 

omega=2*pi/period of oscillations.

Apparatus and Procedure: We have a circular wooden disc that sits on top of a scooter motor with a pole going through the middle. The force sensor is attached to this pole and a piece of string is tied to it with a mass on the other end. The motor is turned on via a power supply and the disc and the mass spins. We allow the disc to spin until it reaches a constant speed and begin to record the force. The force fluctuates during data collection so we find the mean value of it. We also have a photogate with a piece of paper on the disc so we can measure how much time it takes to make a certain amount of rotations. This experiment is run multiple times keeping one variable constant for a number of tries and repeating the process while keeping another variable constant. We are able to manipulate these variables in the experiment by altering radius, mass or the power supply's output. Radius simply requires us to increase the length of the string that holds the force sensor and the mass together. The mass can be changed by just swapping out a different mass. The change in output from the power supply alters the angular speed of the disc thus altering our omega.

With the data gathered, we can plot graphs for each situation where we keep one variable constant. As explained above, this allows us to find how high the correlation is between the data by applying a linear fit.

Data: 
he data recorded from the apparatus:

trial #	rotations	time initial (s)	time final (s)	Δtime (s)	radius (cm)	mass (kg)	Average Force (N)
1	10	1.428	14.769	13.341	22.86	0.2	1.2
2	10	1.728	14.879	13.151	29.21	0.2	1.47
3	5	13.36	20.11	6.75	34.29	0.2	1.691
4	10	1.57	15.6	14.03	46.99	0.2	1.981
5	10	2.42	17.44	15.02	59.44	0.2	2.198
6	10	2.68	17.02	14.34	46.99	0.1	1.052
7	10	1.451	14.97	13.519	46.99	0.05	0.57
8	10	2.35	16.46	14.11	46.99	0.3	2.801
9	10	1.24	11.48	10.24	46.99	0.3	5.393
10	10	1.12	10.75	9.63	46.99	0.3	5.94
11	10	1.46	10.6	9.14	46.99	0.3	6.434

Force vs. Mass graph. Force= r*omega^2. mass in changed in intervals

Force vs. radius graph. Constant mass and omega. Radius is changed in intervals

Force vs. mr*omega^3. Mass and radius are constant. Omega is changed in intervals

Measured values: 

constant	value	uncertainty	correlation
r, ω	9.615	0.2489	0.9955
m, ω	0.04246	0.002837	0.7589
m, r	0.1393	0.001616	0.9971

Conclusion: 
Explanations for the poor correlation number obtained in our constant m and ω model is most likely due to the omega value. Since our mass is constant throughout, the only variable that could produce such error is from the ω. We can also see that the uncertainty obtained from that model is a rather large ratio (0.002837/0.04246 = 0.0668 = 6.68%) and stems from the value for omega being squared. Just visually, the speed of the disc seemed unstable based on shakiness and rocking of the wooden disc during its spin. The readings for force that were obtained showed spikes and dips which suggests the angular speed of the apparatus was not quite as constant as we would have liked it to be.
Ruling out that option, we are left with constant r, ω and m, r which both produced high correlation but I opted for the constant m, r. The correlation is indeed higher but constant m, r doesn't use a constant value for ω that the other one does. Judging by the poor results obtained from constant m and ω, it seems reasonable to opt for the model where ω is not assumed to be constant.

Comparing the slope value we obtained that represents m times r with the actual value, we see that the 0.1393 from the slope is very close to the actual value of 0.14097. With an uncertainty value of 0.001616, 0.1393 +/- 0.001616 gives us a range of values from 0.137684 to 0.140916 with a percent error of 1.16%. The actual value of 0.14097 is just barely out of the range but still very close. The reason for these small discrepancies in numbers may have to do with the setup that hindered accurate collection of data for the force reading.