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

Lab 8: Centripetal Acceleration vs. Angular Speed

Purpose: To determine the relationship between centripetal acceleration and angular speed.

For this lab, the class used the same data. Due to budget restraints, Mr. Wolf demonstrated how the lab worked and ran the experiment. The data was then saved and all calculations were done individually by all the groups.

Materials: Rotating Disk, Accelerometer, Photogate, Scooter wheel

Apparatus: The accelerometer was taped down to the rotating disk and measured the disk's acceleration. The photogate would then help record the wheel's period (time for 1 rev).  The scooter wheel provided the force required for this lab. The scooter wheel was in contact with the disk; so as the wheel turned, the disk also turned. The speed of the scooter wheel was regulated by reducing or increasing the voltage allotted to it.

 

Procedure:
In order to determine the relationship between centripetal acceleration and angular speed, we conducted five runs with the scooter wheel at different speeds and calculated the following for each run:

• Centripetal Acceleration
• Period of Rotating Disk
• Angular Speed

The results where then plotted on an Acceleration vs Angular Speed graph to determine their relationship.

Data:

Measured Radius (Distance from Accelerometer to Center of Rotating Disk): 13.84 cm

Run 1:  In run 1, 4.4 volts was given to the scooter wheel. The rotating disk began to rotate and an acceleration was recorded.

Trial 1 Acceleration

As you can see above, the accelerometer recorded the disk's acceleration during the run and we calculated an average of the data.

• Calculated Acceleration: 1.557 m/s^2

Through use of the photogate, we were determined the rotating disk's period (time for 1 rotation).

The table above shows the times in which the rotating disk passed through the photogate. In order to calculate the period, we subtracted the time the disk last passed through the gate minus the first time it passed through the gate. We then divided the difference by the number of rotations that occurred during that time. Every two spots on the table is one rotation.

• Run 1: (16.461 sec - 1.6716 sec)/8 rotations = 1.85 sec
• T=1.85 sec for 1 rotation

We then found the angular speed using w = (2(pi))/T

• Angular Speed (w) = 3.4 rad/sec

This process was repeated for four other runs.




Run 1: 4.4 Volts

• T=1.85 sec
• w= 3.4 rad/sec
• a= 1.557 m/s^2

Run 2: 6.4 Volts

• T=1.035 sec
• w= 6.07 rad/sec
• a= 5.074 m/s^2

Run 3: 8.6 Volts

• T=.7192 sec
• w= 8.74 rad/sec
• a= 10.70 m/s^2

Run 4: 9.6 Volts

• T=.6731 sec
• w= 9.33 rad/sec
• a= 11.89 m/s^2

Run 5: 10.8 Volts

• T=.5488 sec
• w= 11.44 rad/sec
• a= 18.15 m/s^2

Plotting Results: 
In this part of the lab, we determined the relationship of centripetal acceleration and angular speed. We determined the relationship was proportional. In order to prove this assumption, we graphed        a vs w^2. The equation for the graph should look like this a = rw^2. In this case, r is the slope of the graph. If done correctly, r should match the radius measured from the accelerometer to the center of the disk. To recap, r = 13.84 cm.
We plotted the following data:

Graph Data



a=rw^2

• X Column: w^2 (angular speed)
• Y Column: a (centripetal acceleration)

Then, we took a proportional fit of the data to determine the slope.

• Slope of Graph: r= 0.1384 +/- 0.0006436 m

As you can see, the slope of the graph (r) is identical to the measured radius.
In both cases, r = 13.84 cm. This proves that the relationship between centripetal acceleration and angular speed is proportional.

Sources of Error:  The sources of error in this lab come from the precision of the instruments we used. The experiment depends upon the precision of the accelerometer and photogate. As you can see, the slope of the graph has a very small uncertainty. This uncertainty is the result of squaring the angular speed. In all, the sources of error were limited since our calculated and measured radii matched.