Laboratory Exercise

Rotary Motion

INTRODUCTIONExercise 1 Rotational Dynamics

Up to this point we have been dealing primarily with translational motion. Translational motion is the motion of an object

When you studied Newtonian dynamics you learned that when an object underwent some form of translational motion (whether in a straight line, parabolic, or circular path), the net force applied to the object is proportional to the acceleration. The constant of proportionality is the mass of the accelerating object. When a torque (the rotational analogue to force), is applied to an object that is free to rotate, the object will undergo rotational acceleration. In this experiment, you will investigate the relationship between torque and angular acceleration.

Objectives Collect angular acceleration data for objects subjected to a torque. Determine an expression for the torque applied to a rotating system. Determine the relationship between torque and angular acceleration. Relate the slope of a linearized graph to system parameters.

Make and test predictions of the effect of changes in system parameters on the constant of proportionality.

MaterialsVernier data-collection interface Vernier Rotary Motion SensorLogger Pro Software Vernier Rotary Motion Accessory kitring stand balance

hanging massesstring

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Physics

Making predictions - your hypothesis

1. Tie a length of string to the edge of the largest pulley on the Rotary Motion Sensor. Pull the string taut and mark the string one meter from the point of attachment. Wind the string on the pulley until the one-meter mark is at the edge of the pulley.

2. Holding the sensor securely pull the string so that it unwinds from the pulley. 3. Draw what you predict what the Radians vs. time graph will look like if you were to pull the

string out at a constant velocity.

4. Connect the sensor to the interface and launch the data-collection program. The default settings work fine for this investigation.

5. Rewind the string on the pully.6. Holding the sensor securely, start data collection, then pull the string so that it unwinds

from the pulley. 7. Read the maximum angle on the graph of angle vs. time and record this value. 8. Repeat Steps 2–4, except that this time, wind the string in the opposite direction.9. Now tie the string to the edge of the middle pulley on the sensor. Repeat Steps 2–4.10. Determine the relationship between the angle through which the pulley turned, the linear

distance the edge of the pulley traveled as the string was unwound, and the radius of the pulley. Compare your findings with others in class.

Procedure

AFTER COMPLETING THE LAB COPY AND PASTE THE GRAPHS FOR ALL PREDICTIONS AND ANALYSIS INTO YOUR WEB SITE. INCLUDE DATA TABLES. INCLUDE EXPLANATIONS AND ANSWERS TO QUESTIONS IN YOUR SITE AS WELL.

1. Connect the Ultra Pulley to the Swivel Mount and then to the Rotary Motion Sensor, as shown in Figure 1. Attach the sensor to a ring stand and position it so that a weight tied to the edge of the large pulley on the sensor and hanging over the Ultra Pulley can hang freely without touching the floor.

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Part 1

2. Record the radius of the 3 different steps on the pulley. Find the mass of the solid aluminum disk with the small hole, then attach it to the 3-step pulley on the sensor.

3. Open a new file in the data-collection program. The default data-collection rate is fine, but reduce the length of the experiment to five seconds.

4. Place the 10 gram mass on the string, then wind the string onto the largest pulley on the Rotary Motion Sensor.

5. Start data collection, then release the hanging weight. Catch the weight just before the string has completely unwound.

6. To determine the angular acceleration of the disk, perform a linear fit on the appropriate portion of the angular velocity vs. time graph. Record this value in your lab notebook, along with the mass of the weight for each value you use.

7. Repeat Steps 4–6, increasing the mass, use 10g, 20g, 30g, 50g, and 70g masses.

Part 28. Find the mass of the second solid aluminum disk. Using the longer machine screw attach

both disks to the 3-step pulley on the sensor. Repeat Steps 3–7.

Part 39. Remove the disks from the sensor. Find the mass of each of the weights and the rod in the

accessory kit. Attach each of the weights to opposite sides of the rod at a distance recommended by your instructor. Record this distance.

10. Attach the rod and weights to the sensor as shown in Figure 2. Repeat Steps 3–7 as you did in Parts 1 and 2.

.

DataIn this exercise you will create your own data tables. Include those tables on your web site.

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Figure 1

Figure 2

AnalysisUse of Logger Pro will help you to calculate the value of the net torque in Parts 1–3 for your analysis.

Part 11. To evaluate the relationship between acceleration and force, disconnect the sensors from the

interface and choose New from the File menu in Logger Pro.

2. Enter your values for the hanging mass and angular acceleration. Create a new calculated column finding the values of net torque. Your instructor may guide you in the design of this file.

3. Even though you investigated how angular acceleration responded to changes in the torque, in order to facilitate your analysis, plot a graph of net torque, , vs. angular acceleration, .

4. If the relationship between net torque and angular acceleration appears to be linear, fit a straight line to your data.

5. Write a statement that describes the relationship between the net torque acting on the disk and its angular acceleration.

6. Write the equation that represents the relationship between the net torque, , acting on the disk and its angular acceleration, . Be sure to label this data set with the value of the mass of the disk.

Part 27. Choose New Data Set from the Data menu. Enter your values for the hanging mass and

angular acceleration for the stacked disks. As you did in Step 2, create a new calculated column finding the values of the net torque the hanging mass applied to the disks.

8. Plot a graph of net torque, , vs. angular acceleration, . If the relationship between net torque and angular acceleration appears to be linear, fit a straight line to your data. If possible, print a copy of the graph showing both data sets.

9. Write the equation that represents the relationship between the net torque, , acting on the pair of disks and their angular acceleration, . Be sure to label this data set with the value of the mass of the disk.

10. How does the slope of this equation compare to the one you obtained in Part 1?

As you are likely to have found before, the slope of a graph is usually some function of physical parameters of the system. For example, in the Newton’s Second Law experiment, the slope of the graph of net force vs. acceleration is the mass of the object accelerated by the force. The greater the mass of the object, the larger was the force required to produce a given acceleration. In effect, the mass is a measure of the resistance to the change in the motion of the object. Physicists call this resistance to change in motion inertia. In this experiment, the slope is also a measure of the resistance of the object to undergo an acceleration; in the case of rotational motion it is known as the moment of inertia.

11. From the previous paragraph, you might suspect that the slope is a function of the mass. What evidence do you have that supports this hypothesis?

Part 3

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12. Choose New Data Set from the Data menu. Enter your values for the hanging mass and angular acceleration for the rod and weights. As you did in Step 2, determine the values of the net torque the hanging mass applied to the rod and weights.

13. Plot a graph of net torque, , vs. angular acceleration, . If the relationship between net torque and angular acceleration appears to be linear, fit a straight line to your data.

14. Write the equation that represents the relationship between the net torque, , acting on the rod and weights and their angular acceleration, .

Part 4 Further examination of moment of inertia15. You might also expect that the slope of your graph (moment of inertia) is somehow related to

the distance, r, of the object(s) from the axis of rotation. To test this hypothesis, you will need the data from several groups that positioned the weights at different distances along the rod. If that is not feasible, you will have to repeat Part 3 at least five more times, changing the distance the weights are from the center of rotation for each set of net torque vs. angular acceleration.

16. Open a new Logger Pro file and plot moment of inertia, I, vs. radius for the data collected in Part 3. If the relationship appears to be linear, fit a straight line to the data. If not, then take the necessary steps to modify a column so as to produce a linear relationship.

17. How does the slope of this new graph compare to the mass of the weights used? Suggest a possible expression for the moment of inertia, I, for the two point masses.

18. Try to account for the fact that your graph has a non-negligible intercept.

19. Now, revisit your equations for Parts 1 and 2. How do the slopes of the lines compare to the expression you obtained for I in Step 17? Suggest a reason for any difference.

Exercise 2 Conservation of Angular MomentumINTRODUCTION

In your study of linear momentum, you learned that, in the absence of an unbalanced external force, the momentum of a system remains constant. In this experiment, you will examine how the angular momentum of a rotating system responds to changes in the moment of inertia, I.

OBJECTIVESIn this experiment, you will

Collect angle vs. time and angular velocity vs. time data for rotating systems. Analyze the -t and -t graphs both before and after changes in the moment of inertia. Determine the effect of changes in the moment of inertia on the angular momentum of

the system.

MATERIALSVernier data-collection interface ring stand or vertical support rodLogger Pro balanceVernier Rotary Motion Sensor metric rulerVernier Rotary Motion Accessory Kit

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Procedure

AFTER COMPLETING THE LAB COPY AND PASTE THE GRAPHS FOR ALL PREDICTIONS AND ANALYSIS INTO YOUR WEB SITE. INCLUDE DATA TABLES. INCLUDE EXPLANATIONS AND ANSWERS TO QUESTIONS IN YOUR SITE AS WELL.

1. Mount the Rotary Motion Sensor to the vertical support rod. Place the 3-step Pulley on the rotating shaft of the sensor so that the largest pulley is on top. Measure the mass and diameter of the aluminum disk with the smaller hole. Mount this disk to the pulley using the longer machine screw sleeve (see Figure 1).

Figure 12. Connect the sensor to the data-collection interface and begin the

data-collection program. The default data-collections settings are appropriate for this experiment.

3. Spin the aluminum disk so that it is rotating reasonably rapidly, then begin data collection. Note that the angular velocity gradually decreases during the interval in which you collected data. Consider why this occurs. Store this run (Run 1).

4. Obtain the second aluminum disk from the accessory kit; determine its mass and diameter. Position this disk (cork pads down) over the sleeve of the screw holding the first disk to the pulley. Practice dropping the second disk onto the first so as to minimize any torque you might apply to the system (see Figure 2).

5. Begin the first disk rotating rapidly as before and begin collecting data. After a few seconds, drop the second disk onto the rotating disk and observe the change in both the -t and -t graphs. Store this run (Run 2).

6. Repeat Step 5, but begin with a lower angular velocity than before. Store this run (Run 3).

7. Find the mass of the steel disk. Measure the diameter of both the central hole and the entire disk. Replace the first aluminum disk with the steel disk and hub and tighten the screw as before (see Figure 3).

8. Try to spin the steel disk about as rapidly as you did the aluminum disk in Step 3 and then begin collecting data. Store this run (Run 4).

9. Repeat Step 5, dropping the aluminum disk onto the steel disk after

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Figure 2

Figure 3

a few seconds. Store this run (Run 5) and save the experiment file in case you need to return to it.

EVALUATION OF DATA 1. Use a text or web resource to find an expression for the moment of inertia for a disk;

determine the values of I for your aluminum disks. With its large central hole, the steel disk should be treated as a cylindrical tube. Using the appropriate expression, determine the value of I for the steel disk.

2. Examine the -t graph for your runs with the single aluminum disk (Run1) and the steel disk (Run 4). Determine the rate of change of the angular velocity, , for each disk as it slowed. Account for this change in terms of any unbalanced forces that may be acting on the system. Explain the difference in the rates of change of (aluminum vs. steel) in terms of the values you calculated in Step 1.

3. Examine the -t graph for Run 2. Determine the rate of change of before you dropped the second disk onto the first. Record the angular velocity just before and just after you increased the mass of the system. Determine the time interval (t) between these two velocity readings.

In Logger Pro, drag-select the interval between these two readings. The x in the lower left corner gives the value of t.

In LabQuest App, drag and select the interval between these two readings and use the Delta function under Statistics to perform this task.

4. The angular momentum, L, of a system undergoing rotation is the product of its moment of inertia, I, and the angular velocity, .

L=Iw

Determine the angular momentum of the system before and after you dropped the second aluminum disk onto the first. Calculate the percent difference between these values.

5. Use the initial rate of change in and the time interval between your two readings to determine due to friction alone. What portion of the difference in the angular momentum before and after you increased the mass can be accounted for by frictional losses?

6. Repeat the calculations in Steps 3–5 for your third and fifth runs.

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