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

Purpose: We will be exploring the relationship between centripetal acceleration and angular speed.

Allow me to define angular frequency, angular speed, and centripetal acceleration. 
Angular frequency (ω) is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function. Angular speed is the rate at which an object changes its angle, measured in radians, in a given time period. Angular speed has a magnitude (a value) only. 

Centripetal acceleration is the rate of change of tangential velocity.

For our experiment we will be sharing data with the class. Due to the lack of equipment available for each group our professor ran a series of trials in which we recorded the initial time (one rotation), final time (ten rotations), and acceleration in the x axis.

If we were to do it on our own, our apparatus for this experiment would look like the above picture. There is an accelerometer on the disk with tape sticking out to the photogate. The wheel on the left is powered by a battery source and will turn the disk at a desired voltage. We took our measurements at six different voltages; 4.8, 6.2, 7.8, 8.8, 10.8, and 12.6. Each one we recorded the initial time (one rotation), final time (ten rotations), and acceleration in the x axis.

Above image: We obtained our initial and final time from these recordings.

Above image: Minus the 6.2 volts, we had the rest on one chart. Using the mean gave us a value for acceleration.


We then took the data we gather and recorded them onto excel. Using excel's calculation capabilities we input the formula for ω. Refer to the top for the equation of ω, where 2*pi is in radians and time is for one rotation.

We then plotted an acceleration vs. angular speed^2 graph. The slope gives us the radius in meters.

Conclusion

The slope in our graph above tells us the distance from center and end point of the disk. We were able to use acceleration and angular speed to find a general length of our circle. Theory predicts that if either our acceleration or angular speed were to increase or decrease, so would the radius of the disk. Essential by using the relationship of these two we can determine the size of any circular disk.

Errors that may pertain to this experiment could be the equipment. The photogate is old and the tape that it is suppose to read can give an error. The equipment was taped onto the disk. Knowing that, as the disk spins faster and faster whose to tell that the equipment may or may not have slipped from its original position. We noted that the initial time was not making sense at first, but came to realize that the tape passing through the photogate started at different positions. That is why we took the time at ten rotations and divided it by ten for an average. Even ten may not have been enough, with more rotations, say 50, would have possibly provided a better value.