The purpose of this experiment was to measure different accelerations of a rotating disk and the corresponding time it takes for one revolution (known as the period) to determine the relationship between centripetal acceleration and angular speed.
Apparatus:
Figure 1: Diagram of apparatus
The apparatus consisted of a heavy disk that is able to rotate about its center. A scooter wheel and a transformer were also used to cause the disk to rotate and give it centripetal acceleration at varying levels, depending on the amount of voltage run through the scooter. A wireless accelerometer was also placed on top of the disk to measure the centripetal acceleration of the rotating disk. In addition, tape and a photogate were used to measure and record the time it takes for one rotation to happen. Tape was also used to hold down the accelerometer on the disk. Lastly, a computer with LoggerPro software was needed to connect the photogate and accelerometer to.
Procedure:
The disk was rotated at different speeds using different voltages run through the scooter. Then, the centripetal acceleration was measured on LoggerPro every trial using the accelerometer. As expected, the higher the voltage run through the scooter wheel, the larger the centripetal acceleration would be. The number of rotations performed over a time interval was also measured. These measurements were made for five trials, with each trial increasing the voltage run through the scooter wheel and therefore increasing the centripetal acceleration of the rotating disk. The voltages used for each trial are listed below:
- Trial 1: 4.4 V
- Trial 2: 6.4 V
- Trial 3: 8.6 V
- Trial 4: 9.6 V
- Trial 5: 10.8 V
The acceleration for Trial 1 was measured to be 1.557 m/s/s. The acceleration of trial 2 was 5.074 m/s/s. The acceleration of Trial 3 was 10.70 m/s/s. The acceleration of Trial 4 was 11.89 m/s/s. The acceleration of Trial 5 was 18.15 m/s/s.
The radius of the rotating disk was then measured to be 13.8 cm. The radius is used to find the relationship between angular speed and acceleration.
Then, Excel was used to create a graph of acceleration vs angular velocity squared.
Discussion:
In order to find the angular velocity of the rotating disk, the period must be found first, since w = (2pi)/T, where w is angular velocity and T is the period, the time it takes for one revolution. The period was calculated by taking the time it took for "n" amount of revolutions to be done, then dividing that time by "n", which gives the time it takes for one revolution. For Trial 1, n = 8 revolutions. For the remaining trials, n = 10 revolutions. Once the period for each trial was found, the angular speed (w) was found by dividing 2pi by the period. Next, "w" was squared for each trial, and the graph of acceleration vs angular velocity squared was created.
|
Voltage (V)
|
To (s)
|
Tf (s)
|
Period (s)
|
w^2 (rad^2/s/s)
|
a (m/s/s)
|
|
4.4
|
1.6716
|
16.461
|
1.848675
|
11.5515047
|
1.557
|
|
6.4
|
0.1789
|
10.5305
|
1.03516
|
36.84213
|
5.074
|
|
8.6
|
0.3783
|
7.57
|
0.71917
|
76.3302297
|
10.7
|
|
9.6
|
0.0534
|
6.7843
|
0.67309
|
87.1391697
|
11.89
|
|
10.8
|
0.377
|
5.865
|
0.5488
|
131.078508
|
18.15
|
Figure 3: Graph of centripetal acceleration vs angular velocity squared
Knowing that acceleration is equal to the product of the radius of the disk and the squared angular velocity, the slope of the graph above provides the radius of the disk, as shown below:
a = r w^2
y = m
x
As found, the slope of the graph (experimental finding of radius) is 0.1383 m, or 13.83 cm. This is very close to the measured radius value of 13.8 m, which shows that this model was indeed very accurate for finding the relationship between centripetal acceleration and angular velocity.
Conclusion:
One source of uncertainty in this experiment is the uncertainty in the measurement of the acceleration by the accelerometer and the measurement of the period by the photogate. However, these uncertainties are very small since, as shown, the experimental radius is very similar to the measured radius. Another source of uncertainty is the measurement of the radius with a ruler. This would have increased or decreased the percent difference, depending on whether the value was rounded down or up.
Most sources of error, such as neglecting friction between the axle of rotation and the disk, assuming all power goes into rotating the disk, etc., were eradicated in this experiment since almost everything was measured via equipment and many assumptions that contain error, such as all power goes into rotating the disk, were not used in the data analysis and in the model since they have no effect on the relationship between centripetal acceleration and angular velocity.
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