25 March 2015. Centripetal Acceleration vs Angular Frequency.

Purpose
to determine the relationship between centripetal acceleration and angular speed.

Apparatus:

1. Heavy rotating disk.
2. Logger pro acceleration detector.

Procedure

1. We first place the accelerometer on the disk and make sure that the accelerometer reads 0 in the x and y-directions and -9.8m/s² in the z-direction.
2. pin the disk at some speed corresponding to 4.4 Volts, 6.4 Volts, 8.6 Volts, 9.6 Volts, and 12.8 Volts
3. Using Logger Pro to record the acceleration as a function of time and also time the number of rotations with a stopwatch. 
4. To make it easy, the apparatus of the experiment includes a photo-gate and a bit of tape sticking out from the edge of the disk to determine how long it takes to make one rotation.
5. Plotting acceleration vs. angular speed and compare the slope to the theoretical value.

Data Analysis:

• For each time we spin the disk with different electrical power, we record how many rotations, total time to finish those rotations, and the acceleration. Below is an example how we collected the data.  We started to spin the disk with 4.4 Volts. We collected the rotations and the total time to finish those rotations.

this is the 4.4 data from the accel detector

• Then we used the Logger Pro to record the value of acceleration as a function of time (for the 4.4 Volts data). We take the mean value which is the average.

This is the acceleration vs time plot for 4.4

• Next, we do this repeatedly with increasing amount of voltage. This is the one with 6.4 volts and so on (8.6 volts, 9.6 V, and 12.8 V).

Same thing but with 6.4 volts.

the accel vs time plot with 6.4 volts.

• Then we choose some of data collected, which is the rotations, the total time for all the rotations, and the acceleration to calculate the angular speed. From the data, we need to figure out the value of angular speed which can be calculated by multiplying 2 phi and the number of rotations, then divide it by the total time that the disk needs to spin these rotations. 

•  After calculating the value of angular speed to the power of 2, we then enter the data into the Logger Pro and plot acceleration vs. angular speed to check the relationship. 

Here is the data calculated!

Here is the graph!

• From the graph, the centripetal acceleration has a positive relationship with angular speed which is shown as the value of the slope of the graph m = 0.1386. We know that centripetal acceleration is related to angular speed through an equation:
  a = rw²    
• Then the experimental value of the radius of the disk is 13.86 cm
• After doing this experiment, we can verify that the value showing the relationship between centripetal acceleration and angular speed is the radius of the accelerometer. We then measure the radius of the accelerometer and then compare it with the value we got from the graph. The percent error would show us how well we do in predicting the relationship between centripetal and angular speed. The smaller the percent error is, the more accurately we did in determining the relationship.
• Measuring the radius of accelerometer. r = 13.8 cm.
• Then the percent error is

Conclusions:

• Through this experiment, we determine the relationship between centripetal acceleration and angular speed. The result we got after plotting the data is that the centripetal acceleration is positively related to the angular speed through an equation: 
  a = rw²
• We also calculate the percent error to check how well we did in determining the relationship. Our percent error is % error = +0.435% which is almost perfect.
• It shows that our experimental value is so sufficiently close to the theoretical value. The percent error is not equal 0 might be caused by several factors such as errors in measuring the radius of the accelerometer or rounding number of significant figure.