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

     The purpose of this experiment was to determine the relationship between centripetal acceleration and angular speed. The formula a=rw^2 was used as a reference point for this experiment. It was hypothesized that a graph of acceleration vs angular speed would produce a slope, which would represent the radius from the object in motion to the center of its circular path.
     The demonstration was done using a flat, heavy rotating disk. A scooter wheel, powered by a motor, was placed next to the disk. The wheel was touching the side of the disk so that when it was powered up, the wheel rotated the disk. An Accelerometer was taped to the top of the disk. A piece of tape was sticking outward from the accelerometer so that it passes through a photogate sensor at the same place once every revolution.

Setup of the demonstration. Photogate sensor to the left, disk in the middle, and scooter wheel to the right.

   
     The motor was powered up and set the wheel into motion, which set the disk into motion. The first trial was done with the motor running at 4.8 volts. Logger Pro was used to make recordings of each time the accelerometer passed through the photogate. It was also setup to the accelerometer itself, providing graphs of acceleration vs time. This procedure was repeated five more times for a total of six trials, with each trial having a larger voltage going through the motor.
     Here is an example of data collected from the photogate sensor. The table on the left shows the time when the accelerometer passes through the photogate. Under the "State" column, when a 1 shows up is the time that was being looked at. The first "1" was the starting point. Ten "1's" were counted after that to find the time after ten rotations. This data was used for calculations later.

Table and graph given by the photogate sensor.

     Here are the graphs the accelerometer produced. For each trial (shown by the voltage used), a statistical analysis of the graph was done. That produced a mean value of the graph. That mean value was used as the acceleration for that trial. This data was saved for later use. 

Graphs produced by accelerometer.

     Once all data was collected, a graph of acceleration vs angular speed needed to be produced. The collected data was input into a table in Logger Pro. Below are the first three columns. 

     Next, angular speed needed to be found for each trial and input into a new column. The formula used for angular speed, represented by the greek symbol "omega", was

     To get the time for one rotation, the time after 10 rotations was subtracted by the start time, and then divided by ten. The calculation for angular speed was done for each trial. The last calculation was to square the angular speed. This was because a graph of acceleration vs angular speed squared was needed to find the slope, which would be the radius of where the accelerometer was located on the disk. 

     The radius was measured by with a ruler to be 13 cm. The first and last column were plotted in a graph and a linear fit was done.

     From the linear fit data above, the slope shows 0.1366, which translates to 13.66 cm. The experiment outcome matched the hypothesis. The measured radius and the experimental radius did have some difference in it. This could have been caused by uncertainties or errors in the lab. It was assumed that there was no friction on the spinning disk. There were also uncertainties in how accurate the photogate sensor was in reading the tape going through it. Would the data have been more accurate if the tape going through the sensor was thinner? For now, a rough approximation is good enough to show that, yes, there is indeed a relationship between centripetal acceleration and angular speed, and it is the radius of the circular path.