21 Rotational Motion
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Introduction
The rotary encoder is a rotational version of a motion detector. As the encoder rotates, its angular position is measured and displayed as a graph of angular position vs. time. In constructing the angular position vs. time graph, the orientation of the encoder when the LabPro first begins collecting data always serves as the origin of the coordinate system.
I. Constant Angular Velocity
A. Angular Position vs. Time Graphs
Open the file Rotation.
Collect angular position vs. time graphs for the following three motions:
 Rotating clockwise
 Rotating clockwise at a greater angular speed
 Rotating counterclockwise
To collect data, press the Collect button, wait for the encoder to begin collecting data, and then give the encoder a quick spin. If the data is clean, select Experiment/Store Latest Run. If the data is noisy, diagnose the problem and repeat the observation until the problem is solved.
Once you’ve prepared your graph for presentation, print your graph and attach it to the end of this activity.
Question: Clearly explain how the direction of rotation of the encoder can be determined from an angular position vs. time graph.
Question: Clearly explain how the angular velocity of the encoder can be determined from an angular position vs. time graph.
B. Angular Velocity vs. Time Graphs
Display a graph of angular velocity vs. time.
Question: Clearly explain how the direction of rotation of the encoder can be determined from an angular velocity vs. time graph.
C. Angular Acceleration vs. Time Graphs
Display a graph of angular acceleration vs. time.
Question: In all three trials, the angular acceleration of the encoder should be approximately constant and close to zero after you release it. Clearly explain both why the angular acceleration is constant and why its magnitude is close to zero.
II. Angular Acceleration
Display angular position, velocity, and acceleration graphs.
Hang 10 g from the end of a string wrapped around the encoder. Wrap the string so that the encoder will rotate in the positive direction, and place a small piece of tape on the end of the string so that the string won’t slip off the encoder. Press Collect, and then release the mass.
If the data is clean, select Experiment/Store Latest Run. If the data is noisy, diagnose the problem and repeat the observation until the problem is solved. Replace the 10 g with 20 g and repeat the data collection.
Once you’ve prepared your graph for presentation, print your graph and attach it to the end of this activity.
Question: Clearly explain how to determine the angular acceleration of the encoder based only on an angular position vs. time graph. Perform this analysis and record the results in the table below.
Question: Clearly explain how to determine the angular acceleration of the encoder based only on an angular velocity vs. time graph. Perform this analysis and record the results in the table below.
Question: Clearly explain how to determine the angular acceleration of the encoder based only on an angular acceleration vs. time graph. Perform this analysis and record the results in the table below.
Trial 
Angular Acceleration (rad/s^{2}) determined from: 

vs. t 
vs. t 
vs. t 

10 g 
± 
± 
± 
20 g 
± 
± 
± 
Question: Are the three quantities measured above consistent, within measured uncertainties, for each motion analyzed?
III. Unbalanced Forces and Torques
Display only an angular acceleration vs. time graph.
In the previous experiment, you measured the angular acceleration of the encoder when an unbalanced torque acted on the encoder. In this activity, you will apply the same unbalanced torque to a slightly different system.
Question: Imagine hanging identical 100 g masses from opposite ends of a string passing over the encoder, and then placing an additional 10 g on one side of the string. If the system is released from rest, will the angular acceleration be greater than, less than, or approximately the same as it was when the two sides of the pulley were unbalanced by 10 g above? Explain.
Perform the experiment described above, and measure the resulting angular acceleration. Record your result and complete the table below.
Hanging Masses 
Angular Acceleration (rad/s^{2}) 
0 g and 10 g 
± 
100 g and 110 g 
± 
200 g and 210 g 
± 
Question: Although the difference in mass on the two sides of the pulley is equal in all three trials, clearly describe why the resulting angular acceleration is not the same in all three trials.
III. More with Unbalanced Forces and Torques
A. Relating the Outer and Inner Pulley Radii
In this activity, you will flip the encoder around to use the 3step pulley.
Hang 10 g from the end of a string wrapped around the outer radius pulley. Hang 20 g from the end of a second string wrapped the opposite direction around the inner radius pulley. Secure both strings to the pulley with small pieces of tape.
Press Collect, and then release the masses.
Question: Clearly explain how it is possible for a falling 10 g mass to lift a 20 g mass.
Record your result and complete the table below.
Inner Pulley Mass (g) 
Angular Acceleration (rad/s^{2}) 
20 g 
± 
30 g 
± 
40 g 
± 
50 g 
± 
Create a graph of angular acceleration, with error bars, vs. inner pulley mass. Select an appropriate bestfit function and display it on your graph.
Finish preparing your graph, then print and attach it to the end of this activity.
Question: Based on your bestfit function, determine the ratio of the outer radius to the inner radius (R_{outer}/R_{inner}) of the pulley. Describe how you determined this ratio, as well as how you determined the uncertainty in this ratio.
B. Relating the Outer and Middle Pulley Radii
Determine the ratio of the outer radius to the middle radius (R_{outer}/R_{middle}) of the pulley by collecting and analyzing the following data. In all cases, attach 10 g to the outer radius.
Middle Pulley Mass (g) 
Angular Acceleration (rad/s^{2}) 
10 g 
± 
20 g 
± 
30 g 
± 
40 g 
± 
Create a graph of angular acceleration, with error bars, vs. middle pulley mass. Select an appropriate bestfit function and display it on your graph.
Finish preparing your graph, then print and attach it to the end of this activity.
Question: Based on your bestfit function, determine the ratio of the outer radius to the middle radius (R_{outer}/R_{middle}) of the pulley. Describe how you determined this ratio, as well as how you determined the uncertainty in this ratio.