5-4-2015: Angular Acceleration

Lab 16: Angular Acceleration Part 1 & 2

Purpose:
The purpose of the experiment was to apply a known torque to an object that can rotate and measure its angular acceleration. Using this information, we can find a measured value for the moment of inertia of the rotating object.

Apparatus:

Apparatus used during experiment

 The apparatus shown above consisted of a rotational stand, rotational sensor, a hanging mass and a nearly frictionless pulley. The metal disks and torque pulleys are placed on the stand and are allowed to rotate. The disks can rotate together or independent of one another through the use of running air through a hose. A hose clamp is used to divert the air. If the clamp is open the disks rotate independent of each other and if it is closed the disks rotate together. On the sides of disks you can find 200 marks, alternating between black and white. The rotational sensor will use these marks to measure the angular velocity of the disks. Lastly, we have a frictionless pulley that allows for the hanging mass to travel freely.

Furthermore, we used LoggerPro to record and analyze the data of the experiment. First, we setup a Rotary Motion sensor for the apparatus and adjusted the sensor settings to 200 counts per rotation.

Calipers were also used to measure the diameters of the disks and pulleys used during the experiment.

Part 1: Angular Acceleration

Procedure:
 During this part of the lab, we conducted six experiments. Each with the same apparatus but sometimes using different disks, torque pulleys, and hanging masses.

 

Measurements of Disks and Pulleys used in the experiment: d= diameter, m=mass

• Top Steel Disk: d=12.630 cm  m=1360g
• Bottom Steel Disk: d=12.630 cm  m=1348g
• Top Aluminum Disk: d=12.630 cm  m=465g
• Smaller Torque Pulley: d=2.81 cm  m=10g
• Larger Torque Pulley: d=5.36 cm  m=36.3g
• Hanging Mass: Varies

Next, LoggerPro was setup as stated above and we connected the rotational sensor to the computer.     For the first five experiments conducted, the hose clamp was left open so that the disks would rotate independently of each other but a pin was placed in the stand so that only the top disk would rotate. For the sixth experiment the hose clamp was closed so that both disks would rotate.

Now the actual experiment. First, we would turn the air on. Then we wrapped the string of the hanging mass around the torque pulley so that the hanging mass was at its highest point. Here, we would start the measurements and then release the mass. Next, we used the angular velocity graphs to measure the angular acceleration as the hanging mass moved up and down. The angular acceleration was found using the slope of the angular velocity graphs.

In the picture above, the positive slope is the mass going down and the negative slope is the mass going up. Using both angular accelerations, an average angular acceleration was calculated. This process was repeated for all six experiments.

Essentially, what we are doing during this lab is seeing how changing one component of the experiment effects the angular acceleration of the system.

Expts 1,2, and 3: Effect of changing the hanging mass
Expts 1 and 4: Effect of changing the radius and which the hanging mass exerts a torque
Expts 4, 5, and 6: Effect of changing the rotating mass

The data table below shows the components of each experiment: mass of the hanging mass, torque pulley used, which disk is rotating, angular acceleration when mass goes up/down and the calculated average.

Furthermore, we conducted a separate experiment during experiment 5 to ensure that the given value of the angular acceleration of the system was correct. We achieved this by measuring the linear acceleration of the hanging mass. During expt. 5, we placed a motion sensor on the floor so that it measured the velocity of the hanging mass. We then took the slope of the velocity graph and found the acceleration.

 acceleration of hanging mass = 0.165 m/s

Now, we can relate this acceleration to the angular acceleration from expt 5 using the following relationship:  a = alpha*radius of torque pulley

 

From expt 5, Alpha = 6.588 rad/s^2 and radius of large torque pulley= 0.0268 m

Using these values, a= 6.588*0.0268= 0.176 m/s while expt value a= 0.165 m/s

As you can see the values are fairly close, this shows us that the given angular acceleration from the experiment is correct. The small discrepancy can be the result of friction present during the experiment as well as the uncertainties in the measurements.

 

Conclusion:  

If we look at experiments 1 through 3, we notice that the hanging mass does effect the angular acceleration of the disks. When the hanging  mass doubles from 24.6g to 49.6g the angular acceleration also doubles from 1.13 rad/s^2 to 2.28 rad/s^2. When the mass triples from 24.6 to 74.6g, the angular acceleration also nearly triples from 1.13 to 3.56 rad/s^2. Therefore, we can assume that the hanging mass is proportional to the angular acceleration of the disks.

 

In experiments 1 and 4, we can see that the radius of the torque pulley effects the angular acceleration of the disks. Here, the radius was nearly doubled from 1.40 cm to 2.68 cm and we see that the angular acceleration also doubles from 1.13 to 2.19 rad/s^2. Therefore, the radius of the torque pulley is proportional to the angular acceleration of the disks.   

 

Lastly, in experiments 4 through 6, we can see that changing the rotating mass has an effect on the angular acceleration of the system. From experiments 4 and 5, the mass changes from 1360g to 465 g nearly three times lighter. Here, the angular acceleration nearly triples from 2.196 to 6.588 rad/s^2. Furthermore, when the mass was almost doubled from 1360g to 2708g, the angular acceleration was almost twice as slow, from 2.196 to 1.107 rad/s^2. Therefore, we can assume that the rotating mass is inversely proportional to the angular acceleration.

 

 

Part 2:

Procedure: Using the data from Part 1, we found the moment of inertia of the disk in each experiment using a derived equation. We compared these values to a theoretical value of the moment of inertia of each disk.

The derived equation for the moment of inertia can be seen below, as well as the standard moment of inertia equation for a disk. All moments of inertia are in kg*m^2

Experimental and Actual moments of inertia for experiments 1 and 2

Experimental and Actual moments of inertia for experiments 5 and 6



Experimental and Actual moments of inertia for experiments 3 and 4

 

Next, we found the percent error in each experiment by comparing the experiment value of inertia to the actual value of inertia.
Percentage Error:

• Experiment 1: 9.96 % Error
• Experiment 2: 9.93 % Error
• Experiment 3: 5.90 % Error
• Experiment 4: 7.75 % Error
• Experiment 5: 3.99 % Error
• Experiment 6: 7.96 % Error

As you can see, the percentage error varies between the experiments. However, the percentage error is less than 10% so we can assume that the values are accurate and the discrepancy between the values is the result of uncertainties. The uncertainties may come from the measurements of disks, such as the mass and radii. Furthermore, the apparatus used for the experiment was not entirely frictionless.

 

Conclusion:

As stated above, we found the moments of inertia for each experiment and compared them to the theoretical values of inertia of the disks. We found that uncertainties exist within the experiment, since our values did not entirely equal each other. The uncertainties can be found in the measurement of the diameters and mass of the disks and torque pulleys. Uncertainty can also be found in the measurement of the angular acceleration of the rotating mass and the linear acceleration of the hanging mass. Finally, friction present in the system could also slightly effect the experimental values. If we were to reduce the uncertainty, better equipment would be essential as well as a larger data pool to minimize errors.