25-March-2015: Centripetal Acceleration vs. Angular Frequency

The purpose of this demonstration was to determine the relationship between centripetal acceleration and angular speed.

The Idea Behind The Experiment:
Given, we already have a relationship for centripetal acceleration and angular speed. a=rω^2, where a is centripetal acceleration, r is the radius at which the object is rotating, and omega is the objects angular speed. Given this relationship, our goal now was to prove it.

Materials and Setup:
For this demonstration, we used the following materials:

• A heavy rotating disk
• An accelerometer
• A photogate
• A motorized wheel
• LoggerPro

To setup the demonstration, the heavy rotating disk was placed flat onto a lab table. There, the accelerometer was secured near the edge of the heavy rotating disk, with its axis pointing towards the center of the disk. To secure it, a vast amount of tape was used. Once secured, we placed a photogate which was attached to a ring stand to the side of the heavy rotating disk. From here, a long piece of tape was put at the edge of the accelerometer. This piece of tape hung off the edge of the heavy rotating disk and would be used to activated the photogate. After this, we placed the motorized wheel parallel and to the side of the heavy rotating disk. This would ensure the the heavy rotating disk was spinning at a constant speed. Finally all of our LoggerPro equipment was attached to the accelerometer and photogate. For our final set up, see Fig. 1-1 and Fig. 1-2.

Fig. 1-1
Since we didn't have enough equipment for everyone to use, the entire class just witnessed one single demonstration.

Fig. 1-2
The hanging piece of tape would alert the photogate every time a rotation had occurred. 

Collecting Data:

Once we had our lab set up and our accelerometer calibrated, we proceeded to collect our data. Initially, we started the motorized wheel at some low voltage which gave us a low constant velocity. We used LoggerPro to measure the centripetal acceleration. However, since this was not a perfect environment, we got data that seemed a bit "noisy." In our acceleration data chart, we used the mean of the accelerations to find our value for centripetal acceleration (See Fig. 2).

Fig. 2
Since this was not a perfect environment, we had to choose the average value of our
recorded accelerations. 

Now that we had centripetal acceleration, we needed to find our value for angular speed. Since we did not have a device that could directly measure angular speed, finding our value for omega would be a little bit more tricky. Since we had the photogate, we were able to find the exact times in which the heavy disk rotated one time. We had a starting point of time which we considered zero rotations. We then allowed the disk to rotate ten times, giving us an ending time after ten rotations. When we took the difference of these two times, we had the exact time it took for the disk to rotate ten times. To find its period, we simply divided the time it took for ten rotations by ten. Once we had the period, we knew that angular speed was simply 2π/T, where T was the objects period. Thus we were able to find the angular speed for this trial. (see Fig. 3)

Fig. 3 
Although we were able to directly measure angular speed, we
were able to derive it with the period of the rotating object.

We repeated this process five times, each time using a different voltage to drive the mechanical wheel, which in turn gave us different accelerations and velocities. With all our data, it was now time to plot our centripetal a vs. ω^2 graph.

Analyzing Our Data:

Once we had all of our data (see Fig. 4),

Fig. 4

we opened a new LoggerPro file which we would use to graph our a vs. ω^2. Our graph appeared to be linear. Thus, we added a linear fit which would give us the slope of the graph. If our given model of  a=rω^2 was correct, we should see a slope that is similar to the radius at which we were spinning the accelerometer at. The actual radius we physically measured was 0.139 meters. The value our graph gave us was 0.1386 meters, which is extremely close to our actual value. (see Fig. 5)

Fig. 5

Possibility of Error and Uncertainty:

We calculated our percent error from the actual value of our radius to be 0.288%. This indicated that the relationship we had between centripetal acceleration and angular speed was correct. Due to the experiment being relatively straight forward, there were only a few areas where we may have hit some uncertainty. For example, when we measured the length from the center of the heavy rotating disk to the accelerometer, we had to essential guess from which point of the accelerometer the device was reading. So in a sense, we had to guess the actual value of our radius. Another possible source of uncertainty may have came from the mechanical wheel. We put all our faith into the notion that the wheel would keep the heavy disk rotating at a constant speed. If there was some flaw in the wheel, it would have affected our results. Another source of uncertainty may have also came from our mean value for acceleration. However, in all the experiment was conducted effectively and yielded us the relationship we were initially looking for.