5 Apr 2017: Lab 8: Centripetal Acceleration vs Angular Frequency

Lab 8: Centripetal Acceleration vs Angular Frequency

Joel Cook
Nina Song

Lab performed March 27 and 29, 2017

In this lab, we determined the relationship between centripetal acceleration and angular speed in a rotating system.

Introduction: In this lab, we used a rotating apparatus to calculate the relationship between force, centripetal acceleration, mass, distance and angular frequency. By attempting to keep certain variables constant, we were able to see the affect of varying one variable on the relationship.

Apparatus/Procedure: The apparatus used in this experiment (shown below) consists of a wooden disk, driven by a motor and belt, on a shaft through the center of the disk. A force sensor is attached to the shaft and various known masses were tied to the force sensor. The motor is connected to a variable voltage unit to be able to control the speed of rotation. A photogate measures the speed of rotation. 

We collected data using logger pro and the two previously mentioned sensors. We initially varied the mass attached to the force sensor for four trials, and then completed three trials with the length of string varying,  and finally varied the speed of rotation. For all trials, the intent was to keep all other variables constant. Measurements were taken in each trial after the apparatus was at a constant speed. Distance (r) was measured using a tape measure attached to the surface of the disk (pictured below), force (F) was measured with a force sensor, known masses were used for mass (m), and time of one rotation was determined by dividing the time for 10 rotations, as recorded by the photogate, by 10. The variable voltage was used to vary the voltage sent to the motor to vary the rotational speed of the apparatus.

Data:

An example of a set of data showing the force for one trial is shown below. Since the force is inconsistent during the trial, we used the mean average force for each trial. 

The following data was recorded for the experiment.

m (kg)	r (m)	F (N)	t0	tf	Δt	# rotations
0.3	0.47	1.31	3.4	23.92	20.52	10
0.2	0.47	0.8031	3.07	23.96	20.89	10
0.1	0.47	0.4332	4.57	25.92	21.35	10
0.05	0.47	0.2792	2.3	21.71	19.41	10
0.2	0.548	1.66	3.33	20.37	17.04	10
0.2	0.47	0.8031	3.07	23.96	20.89	10
0.2	0.39	1.651	1.8	16.43	14.63	10
0.2	0.23	0.873	1.87	16.3	14.43	10
0.2	0.39	1.651	1.8	16.43	14.63	10
0.2	0.39	4.304	1.79	10.59	8.8	10
0.2	0.39	6.4	1.48	8.75	7.27	10
0.2	0.39	10.36	0.68	6.54	5.86	10

Results:

As evident in the previous table of data, the time for 10 rotations varied in cases where speed of rotation was kept constant. Since the intent of the experiment was to only change one variable in each trial and determine the relationship, we were unable to analyze the data to that end. Since two variables were changing in each portion of the experiment, we graphed the data with only one variable as a constant to analyze the collected data and generate our graphs. The data was plotted such that the slope of a line through the data points should be equal to the constant parameter. 

ω, in radians per second, was calculated by dividing 2 times pi by the time for one rotation. Formulas were inputed to get the values for m*ω^2 and r*ω^2 from values on the spreadsheet. 

m (kg)	r (m)	F (N)	ω	m*ω^2	r*ω^2	slope
0.3	0.47	1.31	3.061978558	2.812713806		0.4635
0.2	0.47	0.8031	3.007745333	1.809306397
0.1	0.47	0.4332	2.942941452	0.866090439
0.05	0.47	0.2792	3.237083977	0.523935634
0.2	0.548	1.66	3.687312207		7.450756677	0.218
0.2	0.47	0.8031	3.007745333		4.251870034
0.2	0.39	1.651	4.294723172		7.193412377
0.2	0.23	0.873	4.354248094		4.360679587
0.2	0.39	1.651	4.294723172	3.688929424	7.193412377	0.2264
0.2	0.39	4.304	7.139977273	10.19585509	19.88191743	0.4414
0.2	0.39	6.4	8.64261348	14.93895355	29.13095943
0.2	0.39	10.36	10.72215017	22.99290086	44.83615667

Graphs were created for Force vs. m*ω^2 and Force vs. r*ω^2.

The above graph shows Force vs. m*ω^2 for the portion of the experiment in which r was kept constant. Since F=r*m*ω^2, a graph of the data points for Force vs. m*ω^2 should have a slope of the constant of the equation, in this case r. A proportional fit of the data points shows a slope of 0.4635 (meters) and r for this portion was 0.47 meters. A proportional fit was chosen because when  m*ω^2 is 0, force will be zero. In addition, we expect the data to be linear.

The above graph shows Force vs. m*ω^2 for another portion in which r was constant. The slope is shown as 0.4414 (meters) and the value for r was 0.39 meters.

The above graph shows Force vs. r*ω^2. In this portion of the experiment, when force and r*ω^2 are varying, m is constant. The slope of the line through the data should have a slope of m. The picture shows that the slope was 0.2180 (kg) and the mass for this portion was 0.2 kg. As before, a proportional fit was chosen because when r*ω^2 is 0, force will be zero. In addition, we expect the data to be linear.

A second set of data for Force vs. r*ω^2 was plotted above and shows a slope of 0.2264 (kg) and the mass in this portion of the experiment was 0.2 kg.

Conclusion:

As the following table shows, the values observed from the slopes of the data graphed was off from the actual value in the experiment by between 7.1-11.7%. The values are accurate enough to confirm the relationships we were attempting to verify but various errors and uncertainties in the experiment caused our data to be less accurate than expected. First, as we discovered in experimentation, the speed of the apparatus was inconsistent from trial to trial. The design and construction of the apparatus does not allow for better consistency of speed. Second, our force sensor may not have been perfectly zeroed/calibrated causing error in our measurement. In addition, the force varies greatly throughout each trial, hence the need for using an average. Even though an average was a good approximation, it was just an approximation and introduces error. 

Lastly, at high speeds the apparatus, and wheels that guide the rotation, would vibrate severely. This vibration may have affected the consistency in speed of rotation or even force measured. 

Value from experiment	Value calculated from graph	% Difference
0.4635 meters	0.47 meters	7.1%
0.4414 meters	0.39 meters	11.6 %
0.218 kilograms	0.2 kilograms	8.3 %
0.2264 kilograms	0.2 kilograms	11.7 %

In summation, even though the values calculated were approximately 10% off, the experiment was a success and the values calculated confirm the assumption of a relationship between the values in the experiment.