Lab 16: Angular Acceleration
Chris Ceron, Amy Chung, John Choi
Goal of Lab:
Find out what factors affect angular acceleration using our rotating disk with Pasco air sensor setup and apply this knowledge to our calculation of torque.
Theory/Introduction:
Torque is a quantity defined by the equation:
Where delta r is distance from the axis of rotation, I is the moment of inertia of the rotating object, and alpha is the angular acceleration. Our setup for the lab allows us to apply a known force (gravitational force from hanging mass) which rotates our pulley and disk system. Friction is reduced significantly by using air between the rotating tracks.
For the first part of our experiment, we use LoggerPro to record the angular velocity of the object. We can then find find angular acceleration by taking the slope of the graph. We find both the angular acceleration going up and going down with a couple of variables changed in each trial run to see how these changes affect the angular acceleration (changes in hanging mass, change in radius of the pulley and the change of the rotating mass itself).
With the data we collected in part 1 of our experiment, we are able to numerically calculate the torque in the system of different combinations of disks and pulleys. We neglect the small frictional torque in our system. Although our setup uses air to reduce friction to a minimum, this is still a quantity we cannot ignore in our experiment as well as some small mass in the pulley we used.
We can calculate for different moments of inertia for disk/disk combinations in our experiment using the derivations outlined in our Lab handout:
Apparatus and Procedure:
We connect our Pasco rotational sensor to LoggerPro in order to read the rotation of the disk. There are 200 different markings on the disks we use, so we ensured that LoggerPro accurately records one rotation of the disk by counting the markings. We also ensured that both disks in our apparatus can move freely by checking that the hose clamp on the bottom of the apparatus was open. We wrap our string around the torque pulley and allow the hanging mass to accelerate up and down to the find the angular velocity and derive the angular acceleration from the slope of our graphs. We take the magnitude of these measurements and average them for each of the 6 experimental trial runs to see how changes in mass, pulley size and disk(s) affects the value of angular acceleration.
Once we have obtained these values, we carry out the calculations outlined above in Theory/Introduction.
Data:
Measured mass and radii of the disks and pulleys:
The data we collected for the first part of the experiment and an example of the data collection graphs:
Find out what factors affect angular acceleration using our rotating disk with Pasco air sensor setup and apply this knowledge to our calculation of torque.
Theory/Introduction:
Torque is a quantity defined by the equation:
Where delta r is distance from the axis of rotation, I is the moment of inertia of the rotating object, and alpha is the angular acceleration. Our setup for the lab allows us to apply a known force (gravitational force from hanging mass) which rotates our pulley and disk system. Friction is reduced significantly by using air between the rotating tracks.
For the first part of our experiment, we use LoggerPro to record the angular velocity of the object. We can then find find angular acceleration by taking the slope of the graph. We find both the angular acceleration going up and going down with a couple of variables changed in each trial run to see how these changes affect the angular acceleration (changes in hanging mass, change in radius of the pulley and the change of the rotating mass itself).
With the data we collected in part 1 of our experiment, we are able to numerically calculate the torque in the system of different combinations of disks and pulleys. We neglect the small frictional torque in our system. Although our setup uses air to reduce friction to a minimum, this is still a quantity we cannot ignore in our experiment as well as some small mass in the pulley we used.
We can calculate for different moments of inertia for disk/disk combinations in our experiment using the derivations outlined in our Lab handout:
Apparatus and Procedure:
We connect our Pasco rotational sensor to LoggerPro in order to read the rotation of the disk. There are 200 different markings on the disks we use, so we ensured that LoggerPro accurately records one rotation of the disk by counting the markings. We also ensured that both disks in our apparatus can move freely by checking that the hose clamp on the bottom of the apparatus was open. We wrap our string around the torque pulley and allow the hanging mass to accelerate up and down to the find the angular velocity and derive the angular acceleration from the slope of our graphs. We take the magnitude of these measurements and average them for each of the 6 experimental trial runs to see how changes in mass, pulley size and disk(s) affects the value of angular acceleration.
Once we have obtained these values, we carry out the calculations outlined above in Theory/Introduction.
Data:
Measured mass and radii of the disks and pulleys:
Radius (m)
|
Mass (kg)
| |
Steel
|
0.0632+/-.0001m
|
1.358
|
Aluminum
|
0.0683
|
0.465
|
Small Pulley
|
0.0125
|
0.0101
|
Large Pulley
|
0.02745
|
0.0362
|
 |
| Fig. 2. The digital scale reading for the small pulley. Shows the hundredths decimal place. |
The data we collected for the first part of the experiment and an example of the data collection graphs:
Experiment #
|
mass (kg)
|
torque pulley
|
Disk
|
|αdown|
|
|α up |
|
|α avg|
|
1
|
0.02461
|
small
|
Top steel
|
1.204
|
1.062
|
1.133
|
2
|
0.04961
|
small
|
Top steel
|
2.34
|
2.187
|
2.2635
|
3
|
0.07461
|
small
|
Top steel
|
3.509
|
3.279
|
3.394
|
4
|
0.02461
|
large
|
Top steel
|
2.313
|
2.12
|
2.2165
|
5
|
0.02461
|
large
|
Top aluminum
|
6.659
|
5.853
|
6.256
|
6
|
0.02461
|
large
|
top steel + bottom steel
|
2.363
|
2.051
|
2.207
|
 |
| Fig. 3. The positive slope of the graph represents our angular acceleration going up. |
 |
| Fig. 4. The negative slope of the graph represents our angular acceleration going down. |
We consider the counter clockwise direction as positive and the mass is descending when the torque pulley spins counter clockwise.
Calculated Data:
Following the equations derived in our lab handout, taking trial 1 for example:
plugging in the values we get:
We apply this same calculation for each of the trials and obtain these values:
Experiment #
|
Idisk (kg*m^2)
|
1
|
0.002656989
|
2
|
0.002677128
|
3
|
0.002681249
|
4
|
0.002968299
|
5
|
0.001039694
|
6
|
0.002981155
|
Analysis of Graph and Data:
From our initial data table, we can see what affects our angular acceleration. Increasing the hanging mass increased the angular acceleration and increasing the torque pulley both caused the angular acceleration to increase . Using a much lighter disk from steel to aluminum (aluminum being much lighter) also increased the angular acceleration. When we used two steel disks instead of one, the mass of the disk was greater than any of the trial runs and as expected the angular acceleration decreased.
Conclusion:
We can compare these calculated experimental values with the moment of inertia equations derived using calculus:
and to find percent error between experimental and theoretical:
Experiment #
|
Experimental Value
|
Inertia Equation Value (Theoretical)
|
% Error
|
1
|
0.002656989
|
0.00273226
|
2.754898875
|
2
|
0.002677128
|
0.00273226
|
2.017816752
|
3
|
0.002681249
|
0.00273226
|
1.866989232
|
4
|
0.002968299
|
0.002784385
|
6.605192888
|
5
|
0.001039694
|
0.001169021
|
11.0628466
|
6
|
0.002981155
|
0.005496474
|
45.76241059
|
We see that our derived values for Inertia are very close to the true value for most of the trials. The percent error found in the first experiments were minimal in comparison to the last one. For trial 6, one of the factors that contributed to the percentage of error was the air we had coming into the system. The introduction of the lab stated to use only enough air so the disks can rotate. Initially, our two steel disks were not rotating together as they were supposed to. We fixed this issue by turning the air down. Even after we did this though, the disks didn't completely rotate together which explains the discrepancy in data above.
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